Principles of Sonar Performance Modelling (Springer Praxis Books Geophysical Sciences)

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Principles of Sonar Performance Modeling

Michael A. Ainslie

Principles of Sonar Performance Modeling

Published in association with

Praxis Publishing Chichester, UK

Dr Michael A. Ainslie TNO, Sonar Department The Hague The Netherlands

SPRINGER–PRAXIS BOOKS IN GEOPHYSICAL SCIENCES SUBJECT ADVISORY EDITOR: Philippe Blondel, C.Geol., F.G.S., Ph.D., M.Sc., F.I.O.A., Senior Scientist, Department of Physics, University of Bath, Bath, UK

ISBN 978-3-540-87661-8 e-ISBN 978-3-540-87662-5 DOI 10.1007/978-3-540-87662-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010921914 # Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Marı´a Pilar Ainslie and Jim Wilkie Project management: OPS Ltd, Gt Yarmouth, Norfolk, UK Printed on acid-free paper Springer is part of Springer Science þ Business Media (www.springer.com)

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv xvii

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix xxv

PART I FOUNDATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 What is sonar? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Purpose, scope, and intended readership . . . . . . . . . . 1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Part I: Foundations (Chapters 1–3) . . . . . . . . 1.3.2 Part II: The four pillars (Chapters 4–7) . . . . . 1.3.3 Part III: Towards applications (Chapters 8–11) 1.3.4 Appendices . . . . . . . . . . . . . . . . . . . . . . . . 1.4 A brief history of sonar . . . . . . . . . . . . . . . . . . . . . 1.4.1 Conception and birth of sonar (–1918) . . . . . . 1.4.2 Sonar in its infancy (1918–1939) . . . . . . . . . . 1.4.3 Sonar comes of age (1939–) . . . . . . . . . . . . . 1.4.4 Swords to ploughshares . . . . . . . . . . . . . . . . 1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 3 4 6 6 6 7 7 7 8 15 17 22 23

2

Essential background . . . . . . . . . . . . . . . . . . 2.1 Essentials of sonar oceanography . . . . . . 2.1.1 Acoustical properties of seawater 2.1.2 Acoustical properties of air . . . .

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The sonar equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Objectives of sonar performance modeling . . . . . . 3.1.2 Concepts of ‘‘signal’’ and ‘‘noise’’ . . . . . . . . . . . 3.1.3 Generic deep-water scenario . . . . . . . . . . . . . . . 3.1.4 Chapter organization . . . . . . . . . . . . . . . . . . . 3.2 Passive sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Definition of standard terms (passive sonar) . . . . . 3.2.3 Coherent processing: narrowband passive sonar . . 3.2.4 Incoherent processing: broadband passive sonar . . 3.3 Active sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Definition of standard terms (active sonar) . . . . . 3.3.3 Coherent processing: CW pulse þ Doppler filter. . . 3.3.4 Incoherent processing: CW pulse þ energy detector 3.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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53 53 53 54 55 55 56 56 58 64 80 94 94 95 99 112 122

THE FOUR PILLARS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

Sonar oceanography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Properties of the ocean volume . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Terrestrial and universal constants . . . . . . . . . . . . . . . 4.1.2 Bathymetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Factors affecting sound speed and attenuation in pure seawater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Speed of sound in pure seawater . . . . . . . . . . . . . . . . 4.1.5 Attenuation of sound in pure seawater . . . . . . . . . . . . 4.2 Properties of bubbles and marine life . . . . . . . . . . . . . . . . . . 4.2.1 Properties of air bubbles in water . . . . . . . . . . . . . . . 4.2.2 Properties of marine life . . . . . . . . . . . . . . . . . . . . .

125 126 126 126

2.3

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2.5

3

PART II 4

Essentials of underwater acoustics. . 2.2.1 What is sound? . . . . . . . . 2.2.2 Radiation of sound . . . . . 2.2.3 Scattering of sound . . . . . . Essentials of sonar signal processing 2.3.1 Temporal filter . . . . . . . . 2.3.2 Spatial filter (beamformer) . Essentials of detection theory . . . . 2.4.1 Gaussian distribution . . . . 2.4.2 Other distributions . . . . . . References . . . . . . . . . . . . . . . .

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4.3

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159 159 166 169 171 172 180 183 184

Underwater acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The wave equations for fluid and solid media . . . . . . . . . . . . . 5.2.1 Compressional waves in a fluid medium . . . . . . . . . . . 5.2.2 Compressional waves and shear waves in a solid medium 5.3 Reflection of plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Reflection from and transmission through a simple fluid– fluid or fluid–solid boundary . . . . . . . . . . . . . . . . . . 5.3.2 Reflection from a layered fluid boundary . . . . . . . . . . 5.3.3 Reflection from a layered solid boundary . . . . . . . . . . 5.3.4 Reflection from a perfectly reflecting rough surface . . . . 5.3.5 Reflection from a partially reflecting rough surface . . . . 5.4 Scattering of plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Scattering cross-sections and the far field . . . . . . . . . . 5.4.2 Backscattering from solid objects . . . . . . . . . . . . . . . 5.4.3 Backscattering from fluid objects . . . . . . . . . . . . . . . . 5.4.4 Scattering from rough boundaries . . . . . . . . . . . . . . . 5.5 Dispersion in the presence of impurities . . . . . . . . . . . . . . . . . 5.5.1 Wood’s model for sediments in dilute suspension . . . . . 5.5.2 Buckingham’s model for saturated sediments with intergranular contact . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Effect of bubbles or bladdered fish . . . . . . . . . . . . . . 5.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 192 192 194 197

Sonar signal processing . . . . . . . . . . . . . . . . . . 6.1 Processing gain for passive sonar . . . . . . . 6.1.1 Beam patterns . . . . . . . . . . . . . . 6.1.2 Directivity index . . . . . . . . . . . . 6.1.3 Array gain . . . . . . . . . . . . . . . . 6.1.4 BB application . . . . . . . . . . . . . . 6.1.5 Time domain processing . . . . . . . 6.2 Processing gain for active sonar . . . . . . . . 6.2.1 Signal carrier and envelope . . . . . 6.2.2 Simple envelopes and their spectra

251 252 252 266 271 278 279 279 280 282

4.4

4.5 5

6

Properties of the sea surface . . . . 4.3.1 Effect of wind . . . . . . . . 4.3.2 Surface roughness . . . . . 4.3.3 Wind-generated bubbles . Properties of the seabed . . . . . . 4.4.1 Unconsolidated sediments 4.4.2 Rocks . . . . . . . . . . . . . 4.4.3 Geoacoustic models . . . . References . . . . . . . . . . . . . . . .

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198 201 204 205 208 209 209 210 214 223 225 225 226 227 247

viii Contents

6.2.3

6.3 7

Statistical detection theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Single known pulse in Gaussian noise, coherent processing . . . . 7.1.1 False alarm probability for Gaussian-distributed noise . 7.1.2 Detection probability for signal with random phase . . . 7.1.3 Detection threshold . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Application to other waveforms . . . . . . . . . . . . . . . . 7.2 Multiple known pulses in Gaussian noise, incoherent processing 7.2.1 False alarm probability for Rayleigh-distributed noise amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Detection probability for incoherently processed pulse train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Application to sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Active sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Passive sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Decision strategies and the detection threshold . . . . . . 7.4 Multiple looks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 AND and OR operations . . . . . . . . . . . . . . . . . . . . 7.4.3 Multiple OR operations . . . . . . . . . . . . . . . . . . . . . 7.4.4 ‘‘M out of N ’’ operations . . . . . . . . . . . . . . . . . . . . 7.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PART III 8

Autocorrelation and cross-correlation functions and the matched filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Ambiguity function . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Matched filter gain for perfect replica . . . . . . . . . . . . 6.2.6 Matched filter gain for imperfect replica (coherence loss) 6.2.7 Array gain and total processing gain (active sonar) . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

296 300 306 307 308 309 311 312 312 313 326 327 327 328 329 344 344 344 346 348 348 350 354 356 357

TOWARDS APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . .

359

Sources and scatterers of sound . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Reflection and scattering from ocean boundaries . . . . . . . . . . . 8.1.1 Reflection from the sea surface . . . . . . . . . . . . . . . . . 8.1.2 Scattering from the sea surface . . . . . . . . . . . . . . . . . 8.1.3 Reflection from the seabed . . . . . . . . . . . . . . . . . . . 8.1.4 Scattering from the seabed . . . . . . . . . . . . . . . . . . . 8.2 Target strength, volume backscattering strength, and volume attenuation coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Target strength of point-like scatterers . . . . . . . . . . . . 8.2.2 Volume backscattering strength and attenuation coefficient of distributed scatterers . . . . . . . . . . . . . . . . . . 8.2.3 Column strength and wake strength of extended volume scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 361 362 369 375 391 399 400 409 412

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8.3

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439 440 440 459 483 484 489 490 491 492 493 494 494 495 497 500 508 508 509 510

10 Transmitter and receiver characteristics. . . . . . . . . . . . . . . . . . . . . . 10.1 Transmitter characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Of man-made systems . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Of marine mammals . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Receiver characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Of man-made sonar . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Of marine mammals, amphibians, human divers, and fish 10.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

513 514 515 542 545 545 549 565

11 The sonar equations revisited . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Passive sonar with coherent processing: tonal detector 11.2.1 Sonar equation . . . . . . . . . . . . . . . . . . . . 11.2.2 Source level (SL) . . . . . . . . . . . . . . . . . . . 11.2.3 Narrowband propagation loss (PL) . . . . . . . 11.2.4 Noise spectrum level (NLf ) . . . . . . . . . . . . 11.2.5 Bandwidth (BW) . . . . . . . . . . . . . . . . . . . 11.2.6 Array gain (AG) and directivity index (DI) . .

573 573 574 574 575 576 578 579 580

8.4 9

Sources of underwater sound . . . . . . . . . . . . . . . . 8.3.1 Shipping source spectrum level measurements 8.3.2 Distributed sources on the sea surface . . . . . 8.3.3 Distributed sources on the seabed (crustacea) References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Propagation of underwater sound. . . . . . . . . . . . . . . 9.1 Propagation loss . . . . . . . . . . . . . . . . . . . . 9.1.1 Effect of the seabed in isovelocity water 9.1.2 Effect of a sound speed profile . . . . . . 9.2 Noise level . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Deep water . . . . . . . . . . . . . . . . . . . 9.2.2 Shallow water . . . . . . . . . . . . . . . . . 9.2.3 Noise maps . . . . . . . . . . . . . . . . . . 9.3 Signal level (active sonar) . . . . . . . . . . . . . . 9.3.1 The reciprocity principle . . . . . . . . . . 9.3.2 Calculation of echo level . . . . . . . . . . 9.3.3 V-duct propagation (isovelocity case) . . 9.3.4 U-duct propagation (linear profile) . . . 9.4 Reverberation level . . . . . . . . . . . . . . . . . . . 9.4.1 Isovelocity water . . . . . . . . . . . . . . . 9.4.2 Effect of refraction . . . . . . . . . . . . . . 9.5 Signal-to-reverberation ratio (active sonar) . . . 9.5.1 V-duct (isovelocity case) . . . . . . . . . . 9.5.2 U-duct (linear profile) . . . . . . . . . . . . 9.6 References . . . . . . . . . . . . . . . . . . . . . . . . .

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11.2.7 Detection threshold (DT) . . . . . . . . . . . . . . . . . . . . . 11.2.8 Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . Passive sonar with incoherent processing: energy detector . . . . . 11.3.1 Sonar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Source level (SL) . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Broadband propagation loss (PL) . . . . . . . . . . . . . . . 11.3.4 Broadband noise level (NL) . . . . . . . . . . . . . . . . . . . 11.3.5 Processing gain (PG) . . . . . . . . . . . . . . . . . . . . . . . 11.3.6 Broadband detection threshold (DT) . . . . . . . . . . . . . 11.3.7 Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . Active sonar with coherent processing: matched filter . . . . . . . 11.4.1 Sonar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Echo level (EL), target strength (TS), and equivalent target strength (TSeq ) . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Background level (BL) . . . . . . . . . . . . . . . . . . . . . . . 11.4.4 Processing gain (PG) . . . . . . . . . . . . . . . . . . . . . . . 11.4.5 Detection threshold (DT) . . . . . . . . . . . . . . . . . . . . . 11.4.6 Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . The future of sonar performance modeling . . . . . . . . . . . . . . 11.5.1 Advances in signal processing and oceanographic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Autonomous platforms . . . . . . . . . . . . . . . . . . . . . . 11.5.3 Environmental impact of anthropogenic sound . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

581 583 591 591 592 592 593 593 597 599 606 606

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

635

Special functions and mathematical operations . . . . . . . . . . . . . . . . . A.1 Definitions and basic properties of special functions . . . . . . . . A.1.1 Heaviside step function, sign function, and rectangle function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Sine cardinal and sinh cardinal functions . . . . . . . . . . A.1.3 Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . A.1.4 Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.5 Error function, complementary error function, and righttail probability function . . . . . . . . . . . . . . . . . . . . . A.1.6 Exponential integrals and related functions . . . . . . . . . A.1.7 Gamma function and incomplete gamma functions . . . . A.1.8 Marcum Q functions . . . . . . . . . . . . . . . . . . . . . . . . A.1.9 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.10 Bessel and related functions . . . . . . . . . . . . . . . . . . . A.1.11 Hypergeometric functions . . . . . . . . . . . . . . . . . . . . A.2 Fourier transforms and related integrals . . . . . . . . . . . . . . . . A.2.1 Forward and inverse Fourier transforms . . . . . . . . . . A.2.2 Cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . . .

635 635

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11.4

11.5

11.6

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635 636 636 636 637 639 640 644 644 645 648 649 649 650

Contents xi

A.2.3 Convolution. . . . . . . . . . . . . . . . . . . . . . A.2.4 Discrete Fourier transform . . . . . . . . . . . A.2.5 Plancherel’s theorem . . . . . . . . . . . . . . . . A.3 Stationary phase method for evaluation of integrals A.3.1 Stationary phase approximation . . . . . . . . A.3.2 Derivation . . . . . . . . . . . . . . . . . . . . . . A.4 Solution to quadratic, cubic, and quartic equations . A.4.1 Quadratic equation . . . . . . . . . . . . . . . . A.4.2 Cubic equation . . . . . . . . . . . . . . . . . . . A.4.3 Quartic and higher order equations . . . . . . A.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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651 651 652 652 652 653 655 655 655 656 656

B

Units and nomenclature . . . . . . . . . . . . . . . . . . . B.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 SI units . . . . . . . . . . . . . . . . . . . B.1.2 Non-SI units . . . . . . . . . . . . . . . . B.1.3 Logarithmic units . . . . . . . . . . . . B.2 Nomenclature . . . . . . . . . . . . . . . . . . . . B.2.1 Notation . . . . . . . . . . . . . . . . . . B.2.3 Names of fish and marine mammals B.3 References . . . . . . . . . . . . . . . . . . . . . . .

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659 659 659 659 659 665 665 666 671

C

Fish and their swimbladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Tables of fish and bladder types . . . . . . . . . . . . . . . . . . . . . C.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

673 673 694

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

695

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To Anna

Preface

The science of sonar performance modeling is traditionally separated into a ‘‘wet end’’ comprising the disciplines of acoustics and oceanography and a ‘‘dry end’’ of signal processing and detection theory. This book is my attempt to bring both aspects together to serve as a modern reference for today’s sonar performance modeler, whether for research, design, or analysis, as Urick’s Principles of Underwater Sound did for sonar engineers of his day. The similarity in the title is no accident. During the process I made some valuable discoveries that I now share with the reader. The radar literature provides a deep mine of resources, with applicable results from the theories of wave propagation, signal processing, and (an especially rich vein, largely unexploited in the sonar literature) statistical detection. From oceanography we learn that each of the world’s oceans has its own unique physical, chemical, and biological signature, with sometimes profound consequences for sonar. Marine mammals have evolved a sonar of their own, the remarkable properties of which we are only beginning to unravel, as reported in the increasingly sophisticated bioacoustics literature. Governments and industry around the world have begun to take seriously the environmental consequences of man’s use, whether deliberate or incidental, of sound in the sea. I have done my best to provide a representative snapshot of this rapidly developing field. Some readers will treat this book as a repository of facts, figures, and formulas, while others will seek in it explanations and clarity. It has been my intention to satisfy the needs of both types of reader by including mathematical derivations and worked examples, supplemented with measurements or estimates of relevant input parameters. Of all readers I request the patience to overlook the flaws that undoubtedly remain, despite my best attempts to weed them out. Michael A. Ainslie TNO, The Hague, The Netherlands, March 2010

Foreword

Underwater acoustics is largely a branch of physics, perhaps merging with geophysics and oceanography, but as soon as one attempts to assess a sonar’s performance under realistic conditions, a host of other engineering factors come into play. Is the desired target signal louder than all the other natural noise from wind, waves, ship engines, strumming cables? Is it louder than sound scattered from other distant objects? How do the standard signal-processing techniques such as beamforming, spectral analysis, and statistical analysis influence the probability of achieving a target detection and the probability of a false alarm? The author, Dr. Mike Ainslie, is a physicist with a considerable academic publication record and many years’ hands-on experience in sonar assessment for the U.K.’s MOD and for TNO in The Netherlands. Through a firm foundation in physics, always taking great care over the physical units, Principles of Sonar Performance Modeling introduces rigor and clarity into the traditional sonar equation while still answering the fundamental engineering questions. As well as dealing with the more pure disciplines of sound generation, propagation, and reverberation, it tackles sound sources, targets, signal processing, and detection theory for man-made and biological sonar. Underlying all this is a desire ‘‘to see the wood for the trees’’. For instance, it is often the case with propagation that, despite all the complexities of refraction, reflection, diffraction, scattering, and so on, some simple mechanism dominates, and sometimes one can express the entire transmission loss, ambient noise level, or reverberation level by a simple formula. This insight, or even revelation, is an important bonus and check if one is to have faith in numerical assessment of complicated search scenarios. It can also become a useful shortcut when a particular scenario is to be investigated under many different acoustic, or processing, conditions. Examples of such insights will be found throughout. The cornerstone is the derivation of the sonar equations—too often presented as indisputable fact—from simple physical principles. The derivation is presented

xvi Foreword

initially in terms of ratios of simple physical quantities, and converted to decibels only at the end. Such an approach provides both clarity and a systematic rationale for determining how to evaluate each sonar equation term, and occasionally throws up unexpected new corrections. The book will provide a useful reference for acousticians, engineers, physicists, mathematicians, sonar designers, and naval sonar operators whether working in research labs, the defense industry, or universities. Chris Harrison NATO Undersea Research Centre (NURC), Italy, March 2010

Acknowledgments

The eight years it has taken me to write this book were spent working at TNO in The Hague. It has been a pleasure and a privilege to do so. The Sonar Department, despite two changes of name and two changes of leadership in that time, has provided constant support and understanding for the necessary extra-curricular activities. I wish to thank all at TNO—too many to mention all by name—who helped to make it possible. I thank D. A. Abraham, P. Blondel, D. M. F. Chapman, P. H. Dahl, C. A. F. de Jong, P. A. M. de Theije, D. D. Ellis, R. M. Hamson, C. H. Harrison, J. A. Harrison, R. A. Hazelwood, D. V. Holliday, T. G. Leighton, A. J. Robins, S. P. Robinson, C. A. M. van Moll, K. L. Williams, M. Zampolli, and two anonymous referees, all of whom reviewed at least one complete chapter and helped to improve the quality of the final product. Any remaining errors that find their way into print are entirely mine and not of the reviewers. Through his written publications, David Weston is an eternal inspiration—I have lost count of the number of times his name is cited. I also benefited from discussions with Chris Harrison, Chris Morfey, Christ de Jong, Dale Ellis, Frans-Peter Lam, Mario Zampolli, Peter Dahl, and Tim Leighton. Data or artwork were made available to me by Pascal de Theije (Figure 7.6), Peter Dahl (Figure 8.3), Alvin Robins (Figure 8.5), Vincent van Leijen (Figure 8.13), Peter van Holstein (Figure 8.14), Henry Dol (Figures 9.24 and 9.25), Mathieu Colin (all figures in Chapter 9 making use of either BELLHOP or SCOOTER), Robbert van Vossen (Figures 9.28 and 9.29), Wim Verboom (miscellaneous seal and porpoise audiograms), Garth Mix (thumbnail images of marine mammals), and Paul Wensveen (Figure 11.20). The computer model INSIGHT (version 1.4.2) was used, with permission of CORDA Ltd., to illustrate many of the sonar performance calculations. Also used were the acoustic propagation models SCOOTER and BELLHOP from the Ocean Acoustics Library (http://oalib.hlsresearch.com). Other valuable Internet resources

xviii

Acknowledgments

include FishBase (www.fishbase.org), the Ocean Biogeographic Information System (www.iobis.org), Mathworld (http://mathworld.wolfram.com) and Wikipedia (www. wikipedia.org). Phillipe Blondel and Clive Horwood were always available when needed for advice. Neil Shuttlewood is responsible for a professional end-product. Last but not least, none of this would have been possible without the unquestioning love and support from my wife Pilar and patience of my daughter Anna, whose teenage years are forever tinted with shades of sonar performance. Michael A. Ainslie TNO, The Hague, The Netherlands, March 2010

Figures

1.1 1.2 1.3 1.4 1.5 1.6 1.7 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13

Sketch of Beudant’s experiment of ca. 1816 . . . . . . . . . . . . . . . . . . . . . . . . Sketch of the Colladon–Sturm experiment of 1826 . . . . . . . . . . . . . . . . . . . Inventor Reginald Fessenden and physicist Jean Daniel Colladon . . . . . . . . Physicists Paul Langevin and Robert William Boyle . . . . . . . . . . . . . . . . . . French statesman and mathematician Paul Painleve´ . . . . . . . . . . . . . . . . . . Installation of early U.S. passive-ranging sonar with two towed eels . . . . . . Sound absorption vs. frequency in seawater . . . . . . . . . . . . . . . . . . . . . . . . Attenuation coefficient and audibility vs. frequency in seawater . . . . . . . . . . Radiation from a point source of power W in free space . . . . . . . . . . . . . . Radiation from a point source in the presence of a reflecting boundary . . . . Radiation from a sheet source element of width
r . . . . . . . . . . . . . . . . . . . Beam patterns for L= ¼ 5 and steering angles 0, 45 deg. . . . . . . . . . . . . . . Probability density functions of noise and signal-plus-noise observables . . . Principles of passive detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral density level of the radiated power at the source and intensity at the receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral density level of the transmitter source factor and mean square pressure at the receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coherent propagation loss vs. range and target depth . . . . . . . . . . . . . . . . . Spectral density level of background noise . . . . . . . . . . . . . . . . . . . . . . . . . Spectral density level of signal and noise . . . . . . . . . . . . . . . . . . . . . . . . . . ROC curves for a Rayleigh-distributed signal in Rayleigh noise. . . . . . . . . . Propagation loss and figure of merit vs. target range . . . . . . . . . . . . . . . . . Signal level vs. target range, and in-beam noise level . . . . . . . . . . . . . . . . . Linear signal excess and twice detection probability vs. range for NB passive sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal excess vs. target range and depth . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral density level of the transmitter source factor and mean square pressure at the receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation loss vs. frequency and target range . . . . . . . . . . . . . . . . . . . . .

8 9 9 11 13 15 19 30 33 35 38 46 50 56 57 65 66 67 68 72 76 77 78 79 81 83

xx Figures 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19

Spectral density level of signal and noise . . . . . . . . . . . . . . . . . . . . . . . . . . ROC curves for a BB signal in Rayleigh noise . . . . . . . . . . . . . . . . . . . . . . Propagation loss and figure of merit vs. range . . . . . . . . . . . . . . . . . . . . . . Signal spectrum level vs. range, and in-beam noise spectrum level . . . . . . . . Linear signal excess and twice detection probability vs. range for BB passive sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation loss vs. range and depth for the BB passive worked example . . Principles of active detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation loss and figure of merit vs. target range at fixed array depth and vs. array depth for fixed range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal level and in-beam noise level vs. target range at fixed array depth and vs. array depth for fixed range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear signal excess and twice detection probability for coherent CW active sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal excess vs. target range and array depth . . . . . . . . . . . . . . . . . . . . . . Signal and (in-beam) background levels vs. target range at fixed array depth and vs. array depth for fixed range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total background, background components, and in-beam background level vs. target range at fixed array depth and vs. array depth for fixed range . . . . . . Propagation loss and figure of merit vs. target range at fixed array depth and vs. array depth for fixed range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear signal excess and twice detection probability for incoherent CW active sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global bathymetry map derived from satellite measurements of the gravity field Annual average temperature map at depth 3 km. . . . . . . . . . . . . . . . . . . . . Geographical variations in surface temperature for northern winter and northern summer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature profiles for locations in the northwest Pacific Ocean and northeast Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bathymetry map for the northwest Pacific Ocean . . . . . . . . . . . . . . . . . . . . Bathymetry map for the north Atlantic Ocean . . . . . . . . . . . . . . . . . . . . . . Annual average salinity map at depth 3 km . . . . . . . . . . . . . . . . . . . . . . . . Temperature salinity diagram for the World Ocean . . . . . . . . . . . . . . . . . . Seasonal variations in surface salinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . Salinity profiles for locations in the northwest Pacific Ocean and northeast Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density profiles for locations in the northwest Pacific Ocean and northeast Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global acidity (K) contours at sea surface and at depth 1 km . . . . . . . . . . . Arctic acidity (K) contours at the sea surface and at depth 1 km . . . . . . . . . Acidity (K) profiles for major oceans . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound speed profiles for locations in the northwest Pacific Ocean and northeast Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seawater attenuation coefficient vs. frequency . . . . . . . . . . . . . . . . . . . . . . Fractional sensitivity of seawater attenuation to temperature, salinity, acidity, and depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geographical distribution of herring and Norway pout in the North Sea . . . Wind speed scaling factors to convert from a 20 m reference height to the standard reference height of 10 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 86 91 92 93 94 95 108 109 110 111 118 119 120 121 127 129 130 131 132 132 133 134 135 136 137 140 142 144 145 149 150 160 164

Figures 4.20 4.21 4.22 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 8.1

Near-surface bubble population density spectra . . . . . . . . . . . . . . . . . . . . . Compressional and shear speed vs. density of rocks . . . . . . . . . . . . . . . . . . Compressional and shear speeds vs. density for all rocks and for basalts . . . Illustration of compressional and shear wave propagation. . . . . . . . . . . . . . Fluid sediment layer between two uniform half-spaces . . . . . . . . . . . . . . . . Form function j f ðkaÞj vs. ka for a rigid sphere, a tungsten carbide sphere, and spheres made of various metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance frequency vs. bubble radius for air bubbles in water. . . . . . . . . . Resonant bubble radius vs. frequency for air bubbles in water. . . . . . . . . . . Sinc beam patterns for steering angles 0, 30, 60, and 90 deg . . . . . . . . . . . . Beam patterns for continuous line array: cosine and Hann shading . . . . . . . Beam patterns for continuous line array: raised cosine shading . . . . . . . . . . Hamming family shading patterns and beam patterns. . . . . . . . . . . . . . . . . Beam pattern of unshaded circular array . . . . . . . . . . . . . . . . . . . . . . . . . . Directivity index for an unsteered continuous line array vs. normalized array length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directivity index vs. steering angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shading factor vs. steering angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power spectrum for a Gaussian LFM pulse . . . . . . . . . . . . . . . . . . . . . . . . Power spectrum for a rectangular LFM pulse . . . . . . . . . . . . . . . . . . . . . . Generic ambiguity surface for Gaussian CW pulse . . . . . . . . . . . . . . . . . . . Ambiguity surfaces for Gaussian CW pulses of duration 0.5 s and 2.0 s . . . . Generic ambiguity surfaces for Gaussian LFM pulse . . . . . . . . . . . . . . . . . ROC curves for non-fluctuating amplitude signal in Rayleigh noise . . . . . . . Rayleigh, one-dominant-plus-Rayleigh, Dirac, and Rice probability distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROC curves for Rayleigh-fading signal in Rayleigh noise . . . . . . . . . . . . . . ROC curves for Rician fading signal in Rayleigh noise . . . . . . . . . . . . . . . . Rice probability density functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROC curves for 1D þ R signal in Rayleigh noise . . . . . . . . . . . . . . . . . . . . Graph of xðMÞ vs. M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROC curves (Albersheim approximation) for a non-fluctuating amplitude signal: variation of detection threshold with M for fixed pfa . . . . . . . . . . . . ROC curves (Albersheim approximation) for a non-fluctuating amplitude signal: variation of detection threshold with pfa for fixed M . . . . . . . . . . . . ROC curves for a non-fluctuating amplitude signal: incoherent addition with M ¼ 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROC curves for a non-fluctuating amplitude signal: incoherent addition with M ¼ 1 to M ¼ 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROC curves for a broadband signal: limit of large M . . . . . . . . . . . . . . . . . Supplementary ROC curves for a broadband non-fluctuating signal. . . . . . . ROC curves for a Rayleigh fading signal: incoherent addition with M ¼ 30 . ROC curves for a 1D þ R signal: incoherent addition with M ¼ 30 . . . . . . . Fusion gain vs. pfa for OR operation (fixed pd ) . . . . . . . . . . . . . . . . . . . . . Fusion gain vs. F for OR operation (fixed D) . . . . . . . . . . . . . . . . . . . . . . ROC curves for a non-fluctuating signal: effect of AND and OR fusion. . . . ROC curves for a 1D þ R signal: effect of AND and OR fusion . . . . . . . . . ROC curves for a Rayleigh-fading signal: effect of AND and OR fusion . . . Variation of surface reflection loss with wind speed (1–4 kHz) . . . . . . . . . . .

xxi 170 181 183 195 202 211 238 241 254 258 260 262 265 268 269 270 288 289 302 303 305 315 318 319 321 323 324 331 332 333 335 336 338 339 341 343 352 353 354 355 356 366

xxii 8.2 8.3 8.4 8.5 8.6 8.7 8.8

8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15

Figures Surface reflection loss in nepers calculated vs. angle and frequency . . . . . . . Surface reflection loss vs. wind speed (30 kHz) . . . . . . . . . . . . . . . . . . . . . . Seabed reflection loss vs. grazing angle for uniform unconsolidated sediments Seabed reflection loss vs. angle and frequency–sediment thickness product for a layered unconsolidated sediment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seabed reflection loss vs. angle for rocks . . . . . . . . . . . . . . . . . . . . . . . . . . Seabed reflection loss vs. angle and frequency–sediment thickness product for a sand sediment overlying a granite basement and clay over basalt . . . . . . . . . Seabed reflection loss vs. angle and frequency–sediment thickness product for a sand sediment of thickness 10 m overlying a granite basement and a clay sediment of thickness 300 m over basalt. . . . . . . . . . . . . . . . . . . . . . . . . . . Seabed backscattering strength for a medium sand sediment and frequency 1–30 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between predicted and measured seabed backscattering strength for a fine sand sediment and frequency 35 kHz. . . . . . . . . . . . . . . . . . . . . . Seabed backscattering strength for a coarse clay sediment. . . . . . . . . . . . . . Comparison between predicted and measured seabed backscattering strength for a medium silt sediment and frequency 20 kHz. . . . . . . . . . . . . . . . . . . . Typical ambient noise spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical values of sound pressure level and peak pressure level. . . . . . . . . . . Measured equivalent source spectral density levels: commercial and industrial shipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimated third-octave monopole source level: cargo ship Overseas Harriette Areic dipole source spectrum: wind noise . . . . . . . . . . . . . . . . . . . . . . . . . Areic dipole source spectrum: rain noise . . . . . . . . . . . . . . . . . . . . . . . . . . Measured waveform and frequency spectrum of a single shrimp snap . . . . . Geometry for bottom reflections in deep water . . . . . . . . . . . . . . . . . . . . . Propagation loss vs. range for reflecting seabed at f ¼ 250 Hz . . . . . . . . . . . Bottom-refracted ray paths travel through the sediment and form a caustic in the reflected field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation loss vs. range for a reflecting and refracting seabed at f ¼ 250 Hz Propagation loss vs. range for a reflecting and refracting seabed: sensitivity to sediment properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reflection loss vs. angle for sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation loss vs. range, and reflection loss vs. angle for sand and mud in shallow water at frequency 250 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound speed profile in the northwest Pacific . . . . . . . . . . . . . . . . . . . . . . . Propagation loss vs. range for northwest Pacific summer and winter at f ¼ 1500 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation loss vs. range and depth for northwest Pacific winter profile: effect of upward refraction in surface duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depth factor vs. receiver depth in surface duct . . . . . . . . . . . . . . . . . . . . . . Ray trace illustrating formation of caustics and cusps in surface duct up to a range of 40 km, for a source depth of 30 m, and for the same case as Figure 9.10 Propagation loss vs. frequency and range for a surface duct . . . . . . . . . . . . Ray trace illustrating the formation of convergence zones at the sea surface Propagation loss vs. range and depth: effect of downward refraction on Lloyd mirror interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

368 369 376 382 385 387

390 393 394 395 396 416 418 422 423 426 428 430 441 442 445 446 450 455 456 460 463 465 469 470 473 475 477

Figures 9.16 9.17 9.18 9.19 9.20 9.21 9.22 9.23 9.24 9.25 9.26 9.27 9.28 9.29 9.30 9.31 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14

Propagation loss vs. range for shallow water with a mud bottom for two different sound speed profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximation to D= for fixed min . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predicted deep ocean noise spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity of deep-water ambient noise spectra to rain rate. . . . . . . . . . . . . Sensitivity of deep-water noise spectra to wind speed . . . . . . . . . . . . . . . . . Predicted ambient noise spectral density level vs. frequency and depth . . . . . Effect of the seabed on the ambient noise spectrum in isovelocity water . . . . Effect of the sound speed profile on the ambient noise spectrum for a clay seabed Dredger noise map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bathymetry used for Figure 9.24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reverberation for problem RMW11 and frequency 3.5 kHz . . . . . . . . . . . . Reverberation depth factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reverberation for problem RMW12 and frequency 3.5 kHz . . . . . . . . . . . . Reverberation for problem RMW12 and frequency 3.5 kHz (close-up) . . . . . Ray trace illustrating formation of caustics and cusps in a bottom duct, and propagation loss vs. range and depth at f ¼ 3.5 kHz. . . . . . . . . . . . . . . . . . SRR depth factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum multibeam echo sounder and sidescan sonar source levels vs. transmitter frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unweighted and Gaussian-weighted cosine pulses from Table 10.16 . . . . . . Exponentially damped sine and decaying exponential pulses from Table 10.17 Mean square pressure vs. energy fraction. . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of echolocation pulses made by the harbor porpoise and killer whale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Underwater audiograms for harbor porpoise . . . . . . . . . . . . . . . . . . . . . . . Underwater audiograms for killer whale . . . . . . . . . . . . . . . . . . . . . . . . . . Underwater audiograms for harbor seal . . . . . . . . . . . . . . . . . . . . . . . . . . Underwater audiograms for human divers . . . . . . . . . . . . . . . . . . . . . . . . . Underwater sound level weighting curves for three groups of cetaceans plus pinnipeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directivity index DI ¼ 10 log10 GD for an unsteered continuous line array vs. normalized array length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROC curves for 1D þ R amplitude signal in Rayleigh noise . . . . . . . . . . . . Propagation loss vs. range for NWP winter case . . . . . . . . . . . . . . . . . . . . In-beam signal and noise levels vs. range for NWP winter and Chapter 3 NBp worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input parameters for northwest Pacific (NWP) problem . . . . . . . . . . . . . . . Signal excess vs. range and depth for NWP winter . . . . . . . . . . . . . . . . . . . Signal excess vs. range and depth for NWP winter: close-up of first convergence zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Albersheim’s approximation for the detection threshold . . . . . . . . . . . . . . . Propagation loss vs. range and depth for SWS and for the Chapter 3 BBp worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal and noise spectra for SWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In-beam signal and noise levels vs. range for SWS and BBp . . . . . . . . . . . . Signal excess vs. range and depth for SWS . . . . . . . . . . . . . . . . . . . . . . . . Input parameters for shallow-water sand (SWS). . . . . . . . . . . . . . . . . . . . . In-beam signal and noise spectra for SWS . . . . . . . . . . . . . . . . . . . . . . . . .

xxiii

479 482 484 486 487 488 489 490 491 492 499 503 504 505 506 509 519 530 532 535 548 551 552 554 556 561 581 582 584 586 588 589 590 598 600 601 602 603 604 605

xxiv 11.15 11.16 11.17 11.18 11.19 11.20 11.21 11.22 11.23 11.24 11.25 A.1 A.2 A.3

Figures Signal excess vs. range and rainfall rate for SWS . . . . . . . . . . . . . . . . . . . . Geometry for worked example involving killer whale hunting salmon . . . . . Example measurements of orca pulse shapes and power spectra . . . . . . . . . Variation in orca source level with distance from target . . . . . . . . . . . . . . . Propagation loss vs. distance and broadband correction . . . . . . . . . . . . . . . Orca audiogram and individual hearing threshold measurements . . . . . . . . . Echo level and noise level vs. distance between orca and salmon: wind speed 2 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Echo level and noise level vs. distance between orca and salmon: wind speed 2 to 10 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background level vs. distance between orca and salmon: wind speed 10 m/s . Array gain vs. distance between orca and salmon: wind speed 10 m/s . . . . . . Signal and background levels vs. distance between orca and salmon: wind speed 10 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The complementary error function erfcðxÞ and three approximations . . . . . . The gamma function and four approximations. . . . . . . . . . . . . . . . . . . . . . The modified Bessel function and Levanon’s approximation . . . . . . . . . . . .

606 614 615 617 618 620 621 624 628 629 630 638 643 647

Tables

2.1 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20

Detection truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sonar equation calculation for NB passive example . . . . . . . . . . . . . . . . . . Sonar equation calculation for BB passive example . . . . . . . . . . . . . . . . . . Sonar equation calculation for CW active sonar example with Doppler filter Sonar equation calculation for CW active sonar example with incoherent energy detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average salinity and potential temperature by major ocean basin . . . . . . . . Seawater parameters used for evaluation of attenuation curves plotted in Figure 4.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass, length, and aspect ratio of selected sea mammals . . . . . . . . . . . . . . . Volume and surface area of ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustical properties of fish flesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustical properties of whale tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustical properties of euphausiids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of zooplankton density and sound speed ratios . . . . . . . . . . . . . . . . North Sea fish population estimates by species. . . . . . . . . . . . . . . . . . . . . . WMO Beaufort wind force scale and estimated wind speed. . . . . . . . . . . . . Comparison of wind speed estimates for Beaufort force 1–11 based on WMO code 1100 and CMM-IV with those of da Silva . . . . . . . . . . . . . . . . . . . . . Definition of sea state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beaufort wind force: relationship between wind speed and wave height . . . . Sea state: relationship between wave height and wind speed . . . . . . . . . . . . Sediment type vs. grain diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of sediment grain sizes and qualitative descriptions . . . . . . . . . . . Default HF geoacoustic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Default MF geoacoustic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Names of sedimentary rocks resulting from the lithification of different sediment types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geoacoustic parameters for sedimentary and igneous rocks. . . . . . . . . . . . .

48 76 90 107 122 133 150 154 155 155 156 157 157 158 162 165 166 168 168 172 174 176 178 180 183

xxvi 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 9.1 9.2 9.3 9.4

Tables Compressional speed, shear speed, and density used to calculate the form factors for the four metals shown in Figure 5.3 . . . . . . . . . . . . . . . . . . . . . Backscattering cross-sections of large rigid objects . . . . . . . . . . . . . . . . . . . Backscattering cross-sections of large fluid objects . . . . . . . . . . . . . . . . . . . Water and solid grain sediment parameter values needed for Buckingham’s grain-shearing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of physical constants used for the evaluation of the bubble resonance characteristics in Figures 5.4 and 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of properties for various taper functions . . . . . . . . . . . . . . . . . . . Summary of beam properties for selected shading . . . . . . . . . . . . . . . . . . . Summary of frequency domain properties of simple pulse envelopes . . . . . . Summary of time domain properties of simple pulse shapes (envelope). . . . . Summary of time domain properties of simple pulse shapes (phase) . . . . . . . Summary of amplitude envelopes required to synthesize simple power spectra Autocorrelation functions for CW and LFM pulses . . . . . . . . . . . . . . . . . . Derivation of matched filter gain for pulse duration and sample interval . . . Effect of multipath on matched filter gain . . . . . . . . . . . . . . . . . . . . . . . . . Comparison table: moments of probability distribution functions . . . . . . . . DT þ 5 log10 M vs. M and pfa for three different pd values . . . . . . . . . . . . . Application of the detection theory results of Section 7.1 to active sonar CW and FM pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations for the detection probability for different signal amplitude distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of detection theory results to NB and BB passive sonar . . . . . . Detection threshold for various statistics . . . . . . . . . . . . . . . . . . . . . . . . . . Detection threshold for a 1D þ R amplitude distribution . . . . . . . . . . . . . . ROC relationships and fusion gain for AND and OR operations for fixed SNR Sediment properties at top and bottom of the transition layer . . . . . . . . . . . p and s critical angles for representative rock parameters . . . . . . . . . . . . . . Parameters for uniform fluid sediment and rock half-space . . . . . . . . . . . . . Defining parameters for a layered solid medium. . . . . . . . . . . . . . . . . . . . . Measurements of the Lambert parameter . . . . . . . . . . . . . . . . . . . . . . . . . Target strength measurements for bladdered fish . . . . . . . . . . . . . . . . . . . . Target strength measurements for whales . . . . . . . . . . . . . . . . . . . . . . . . . Target strength measurements for euphausiids and bladder-less fish . . . . . . . Target strength measurements for jellyfish . . . . . . . . . . . . . . . . . . . . . . . . . Target strength measurements for siphonophores . . . . . . . . . . . . . . . . . . . . Second World War measurements of the target strength of man-made objects Predicted average night-time contribution to VBS, CS, and attenuation due to pelagic fish in the North Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Default advice for VBS for sparse, intermediate, and dense marine life . . . . Wake strength measurements for various WW2 surface ships . . . . . . . . . . . Wake strength for various WW2 submarines . . . . . . . . . . . . . . . . . . . . . . . Third-octave source levels of various commercial and industrial vessels . . . . Characteristic properties from Chapter 4 of medium sand and mud . . . . . . . Sound speed profiles for the northwest Pacific location . . . . . . . . . . . . . . . . Nomenclature used for shipping densities . . . . . . . . . . . . . . . . . . . . . . . . . Seabed parameters for problems RMW11 and RMW12 . . . . . . . . . . . . . . .

212 213 215 228 239 264 265 284 285 285 291 296 306 308 320 334 345 345 346 347 348 353 381 385 388 389 397 401 403 404 407 407 408 410 411 414 414 421 454 461 484 500

Tables 9.5 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 10.21 10.22 10.23 10.24 10.25 10.26 10.27 10.28 10.29 10.30 10.31 10.32 10.33 10.34 10.35 10.36

Caustic ranges and corresponding two-way travel arrival times for a source at depth 30 m and receiver at depth 50 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . Source level of single-beam echo sounders . . . . . . . . . . . . . . . . . . . . . . . . . Source level of sidescan sonar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Source level of multibeam echo sounders. . . . . . . . . . . . . . . . . . . . . . . . . . Source level of depth profilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Source level of fisheries search sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Source level of hull-mounted search sonar . . . . . . . . . . . . . . . . . . . . . . . . . Source level of helicopter dipping sonar . . . . . . . . . . . . . . . . . . . . . . . . . . Source level of active towed array sonar . . . . . . . . . . . . . . . . . . . . . . . . . . Source level of miscellaneous search sonar (including coastguard and risk mitigation sonar). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Source level of low-amplitude acoustic deterrents . . . . . . . . . . . . . . . . . . . . Source level of high-amplitude acoustic deterrents . . . . . . . . . . . . . . . . . . . Source level of acoustic communications systems . . . . . . . . . . . . . . . . . . . . Source level of selected acoustic transponders and alerts . . . . . . . . . . . . . . . Source level of acoustic cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Source level of miscellaneous oceanographic sonar . . . . . . . . . . . . . . . . . . . Relationships between different source level definitions for two symmetrical wave forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationships between different source level definitions for two asymmetrical wave forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative MSP, averaged over time window during which local average exceeds specified threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative MSP, averaged over time window during which pulse energy accumulates to specified proportion of total. . . . . . . . . . . . . . . . . . . . . . . . Dipole source level of air guns and air gun arrays . . . . . . . . . . . . . . . . . . . Zero-to-peak source level of generator–injector air guns . . . . . . . . . . . . . . . Zero-to-peak source level of seismic survey sources other than air guns . . . . Summary of peak pressure and pulse energy for three types of explosive . . . Specific pulse energy and apparent specific SLE for pentolite . . . . . . . . . . . . Echolocation pulse parameters for selected animals . . . . . . . . . . . . . . . . . . Maximum peak-to-peak source levels of high-frequency marine mammal clicks Peak equivalent RMS and peak-to-peak source levels of low-frequency marine mammal pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hearing thresholds and sensitive frequency bands of selected cetaceans . . . . MSP and EPWI hearing thresholds in air and water for four pinnipeds plus human subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hearing thresholds in water for 10 species of fish . . . . . . . . . . . . . . . . . . . . Parameters of bandpass filter used in M-weighting . . . . . . . . . . . . . . . . . . . Genera represented by the functional hearing groups . . . . . . . . . . . . . . . . . Proposed thresholds of M-weighted sound exposure level for permanent and temporary auditory threshold shift in cetaceans and pinnipeds . . . . . . . . . . Proposed thresholds of peak pressure for permanent and temporary auditory threshold shift in cetaceans and pinnipeds . . . . . . . . . . . . . . . . . . . . . . . . . Outline of the severity scale from Southall et al. (2007). . . . . . . . . . . . . . . . Spread of sound pressure level values resulting in the specified behavioral responses in cetaceans and pinnipeds for nonpulses . . . . . . . . . . . . . . . . . . .

xxvii

507 516 517 518 520 520 521 521 522 522 524 525 526 527 527 528 529 531 533 534 536 537 538 540 541 543 546 548 553 555 557 560 561 562 563 564 564

xxviii

Tables

10.37

Spread of sound pressure level values resulting in the specified behavioral responses in cetaceans and pinnipeds for multiple pulses . . . . . . . . . . . . . . . List of applications of man-made active and passive underwater acoustic sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error in DT incurred by assuming 1D þ R statistics . . . . . . . . . . . . . . . . . . Sonar equation calculation for NWP winter . . . . . . . . . . . . . . . . . . . . . . . Filter gain vs. bandwidth in octaves for a white signal and colored noise . . . Sonar equation calculation for shallow-water sand . . . . . . . . . . . . . . . . . . . Active sonar example, limited by hearing threshold . . . . . . . . . . . . . . . . . . Active sonar example, limited by wind noise . . . . . . . . . . . . . . . . . . . . . . . Integrals of integer powers of the sine cardinal function . . . . . . . . . . . . . . . Selected values of the gamma function GðxÞ for 0 < x  1 . . . . . . . . . . . . . Examples of Fourier transform pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . SI prefixes for indices equal to an integer multiple of 3. . . . . . . . . . . . . . . . SI prefixes for indices equal to an integer between þ3 and 3. . . . . . . . . . . Frequently encountered non-SI units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of abbreviations and acronyms, and their meanings . . . . . . . . . . . . . . . Bladder presence and type key used in Tables C.3, C.4, and C.7 . . . . . . . . . Reference key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bladder type by order for ray-finned fishes (Actinopterygii) . . . . . . . . . . . . Bladder type by family. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ‘‘Catchability’’ key (Yang groups) used in Table C.7 . . . . . . . . . . . . . . . . . Length key used in Table C.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fish and their bladders, sorted by scientific name. . . . . . . . . . . . . . . . . . . .

11.1 11.2 11.3 11.4 11.5 11.6 11.7 A.1 A.2 A.3 B.1 B.2 B.3 B.4 C.1 C.2 C.3 C.4 C.5 C.6 C.7

565 575 582 587 596 604 620 624 636 640 650 660 661 662 667 674 674 675 676 677 677 678

Part I Foundations

1 Introduction

Wee represent Small Sounds as Great and Deepe; Likewise Great Sounds, Extenuate and Sharpe; Wee make diverse Tremblings and Warblings of Sounds, which in their Originall are Entire. Wee represent and imitate all Articulate Sounds and Letters, and the Voices and Notes of Beasts and Birds. Wee have certaine Helps, which sett to the Eare doe further the Hearing greatly. Wee have also diverse Strange and Artificiall Eccho’s, Reflecting the Voice many times, and as it were Tossing it; And some that give back the Voice Lowder then it came, some Shriller, and Some Deeper; Yea some rendring the Voice, Differing in the Letters or Articulate Sound, from that they receyve. Wee have also meanes to convey Sounds in Trunks and Pipes in strange Lines, and Distances. Francis Bacon (1624)

1.1

WHAT IS SONAR?

Sonar can be thought of as a kind of underwater radar, using sound instead of radio waves to interrogate its surroundings. But what is special about sound in the sea? Radio waves travel unhindered in air, whereas sound energy is absorbed relatively quickly. In water, the opposite is the case: low absorption and the presence of natural oceanic waveguides combine to permit propagation of sound over thousands of kilometers, whereas the sea is opaque to most of the electromagnetic spectrum. The word sonar is an acronym for sound navigation and ranging. The primary purpose of sonar is the detection or characterization (estimation of position, velocity, and identity) of submerged, floating, or buried objects. Electronic systems capable of

4

Introduction

[Ch. 1

underwater detection and localization were developed in the 20th century, motivated initially by the sinking of RMS Titanic in 1912 and the First World War (WW1), and spurred on later by the Second World War (WW2) and the Cold War. Nevertheless, by comparison with marine fauna, man remains a novice user of underwater sound. Deprived of light in their natural habitat, dolphins have evolved a sophisticated form of sonar over millions of years, without which they would be almost blind. They transmit bursts of ultrasound, and sense the world around them by interpreting the echoes. Many fish and other aquatic animals are also capable of both producing and hearing sounds.

1.2

PURPOSE, SCOPE, AND INTENDED READERSHIP

This book is aimed at anyone, novice and experienced practitioner alike, with an interest in estimating the performance of sonar, or understanding the conditions for which a particular existing or hypothetical system is likely to make a successful detection. This includes sonar analysts and designers, whether for oceanographic research, navigation, or search sonar. It also includes those studying the use of sound by marine mammals and the impact of exposure of these animals to sound. Regardless of application, the objective of sonar performance modeling is usually to support a decision-making process. In the case of man-made sonar, the decision is likely to involve the optimization of some aspect of the design, procurement, or use of sonar. (What frequency or bandwidth is appropriate? How many sonars are needed to complete the task in the time available?) For bio-sonar there is increasing interest in the assessment (and mitigation) of the risk of damage to marine life due to anthropogenic sources of underwater sound. (What level of sound might disrupt a dolphin’s ability to locate and capture its prey? How can the risk of hearing damage be prevented or minimized?) The nature of the sought object, known as the sonar target, depends on the application. Examples include man-made objects of military interest (a mine or submarine), shipwrecks (as a navigation hazard or archeological artifact), and fish (the target of interest to a whale or fisherman). In general, sonar can be grouped into two main categories. These are active sonar and passive sonar, which are distinguished by the presence and absence, respectively, of a sound transmitter as a component of the sonar system. .

.

An active sonar system comprises a transmitter and a receiver and works on the principle of echolocation. If a signal (in this case an echo from the target) is detected, the position of the target can be estimated from the time delay and direction of the echo. The echolocation principle is also used by radar, and by the biological sonar of bats and dolphins. A passive sonar includes a receiver but no transmitter. The signal to be detected is then the sound emitted by the target.

Examples of man-made sonar include

Sec. 1.2]

.

.

.

.

.

1.2 Purpose, scope, and intended readership

5

Echo sounder: perhaps the most common of all man-made sonars, an echo sounder is a device for measuring water depth by timing the delay of an echo from the seabed. The strength and character of the echo can also provide an indication of bottom type. Fisheries sonar: sonar equipment used by the fisheries industry exploits the same principle as the echo sounder, except that the purpose is to detect fish instead of the sea floor. Military sonar: modern navies deploy a wide variety of sonar systems, designed to detect and track potential military threats such as surface ships, submarines, mines, or torpedoes. The diverse nature of these threats and of the platforms on which the sonar systems are mounted means that military sonars are themselves diverse, with each specialized system dedicated to a particular task. Oceanographic sensor: scientific work aimed at understanding and surveying the sea (acoustical oceanography) makes extensive use of a variety of different kinds of sonar, many of which are variants of the echo sounder. Shadow sensor: in exceptional cases, the sonar ‘‘signal’’, instead of being the sound emitted or scattered by the target, might actually be some perturbation to the expected background. For example, the shadow of an object lying on the seabed might be detectable when the object itself is not.

Many readers will be familiar with Urick’s classic Principles of Underwater Sound for Engineers,1 which provided its readers with the tools they needed to carry out sonar design and assessment studies. These tools come in the form of a set of equations relating the predicted signal-to-noise ratio to known parameters such as the radiated power of the sonar transmitter, or the size and shape of the target. This set of equations is known as the ‘‘sonar equations’’. The same basic requirement remains today, but the modeling methods have increased in sophistication during the 25 years that have elapsed since Urick’s third and final edition, with a bewildering array of computer models to choose from (Etter, 2003). The present objective is to meet the needs of the modern user or developer of such models by documenting established methods and relevant research results, using internally consistent definitions and notation throughout. The discipline of sonar performance modeling is perceived sometimes as a black art. The purpose of this book is, above all, to demystify this art by explaining the jargon and deriving the sonar equations from physical principles. The book’s scope includes underwater sound, the properties of the sea relevant to the generation and propagation of sound, and the processing that occurs after an acoustic signal has been converted to an electrical one2 and then digitized. The estimation of sonar performance is taken as far as the detection (and false alarm) probability, but no further than that. While the scope excludes localization, 1

See Urick (1967) and two later editions (Urick, 1975, 1983). Conversion between electrical and acoustical energy (known as transduction), whether on transmission or reception, is excluded from the scope. The interested reader is referred to Hunt (1954) and Stansfield (1991). 2

6

Introduction

[Ch. 1

classification, and tracking tasks, such as the estimation of position and velocity of a sonar target, a satisfactory detection capability is a prerequisite for any of these.

1.3

STRUCTURE

Sonar performance modeling is a multidisciplinary science, requiring knowledge of subjects as diverse as mathematics, physics, electrical engineering, chemistry, geology, and biology.3 It is convenient to group the material into four foundation categories (or ‘‘pillars’’), on which the science of sonar performance modeling is built: sonar oceanography, underwater acoustics, sonar signal processing and statistical detection theory. The book has three main parts, described below. 1.3.1

Part I: Foundations (Chapters 1–3)

Part I comprises this Introduction and two further chapters, also of an introductory nature. The purpose of Chapter 2 is to describe the essential concepts required for a basic understanding of the sonar equations, which are derived in Chapter 3. Four generic types of sonar are introduced, with a simple worked example provided for each. The material in Chapters 2 and 3 is intended as a primer, to illustrate the principles, and generally preferring simplicity to realism. Advanced readers might prefer to skip the introductory part and start reading from Chapter 4, consulting Chapter 3 only for definitions. 1.3.2

Part II: The four pillars (Chapters 4–7)

Each of the four chapters in Part II is devoted to one of the four pillars. The one on oceanography (Chapter 4) describes the sea as a medium for sound propagation. Relevant properties of the oceans’ contents and boundaries are considered, such as the geoacoustical properties of sediments and rocks, sea surface waveheight spectra, near-surface bubble density, and the acoustical properties of marine life. The chapter on acoustics (Chapter 5) provides a theoretical foundation for understanding the behavior of sound in the sea, including reflection and scattering from its contents and boundaries. Cumulative propagation effects associated with multiple boundary reflections are the subject of Chapter 9. An acoustic signal arriving at a sonar receiver is converted to an electrical signal by a device known as a ‘‘transducer’’. This electrical signal is subjected to a series of operations designed to determine the presence or otherwise of a sonar target. These operations are known collectively as signal processing, which is the subject of Chapter 6. The purpose of signal processing can be thought of as either to enhance the signal from the target or to reduce the background noise. These two points of view are 3 The reader is assumed to have completed a degree-level course in a numerate discipline such as physics, applied mathematics, or engineering.

Sec. 1.4]

1.4 A brief history of sonar

7

entirely equivalent, as in the end what matters is the ratio of signal power to noise power. Finally, to be of practical use, the output of the signal processing must be interpreted by a decision-maker. The chance that a sonar operator correctly (or incorrectly) deduces that a target is present is known as the probability of detection (or false alarm). The quantitative study of detection and false alarm probabilities is known as statistical detection theory, and this is the subject of Chapter 7. 1.3.3

Part III: Towards applications (Chapters 8–11)

The purpose of the final chapters is to show how to apply the principles from Parts I and II to more realistic situations. Chapter 8 provides quantitative information about the sources, reflectors, and scatterers of underwater sounds, while Chapter 9 describes sound propagation in the sea and its impact on both the signal and background. Chapter 10 describes the characteristics of both man-made and biological sonar, including the sensitivity of marine animals to underwater sound. Chapter 11 brings together information from all the preceding chapters and applies it to a set of problems partly based on the worked examples of Chapter 3, introducing more advanced concepts and definitions where necessary. It closes with a speculative account of possible future development of sonar performance modeling in the 21st century. 1.3.4

Appendices

In addition to the 11 chapters, there are three appendices. Two of these provide information needed for the correct interpretation of the main text, describing special functions and mathematical operations (Appendix A), and units and nomenclature (Appendix B). Finally, Appendix C can be thought of as an extension to Chapter 4. It contains information about fish and their swim bladders that will be of use to a reader interested in the interaction of sound with fish or fish shoals.

1.4

A BRIEF HISTORY OF SONAR

The remainder of this Introduction is devoted to a historical account of the development of sonar. It is the author’s tribute to the work of Constantin Chilowski,4 Daniel Colladon, Pierre and Jacques Curie, Maurice Ewing, Reginald Fessenden, Harvey Hayes, Paul Langevin, H. Lichte, Leonard Liebermann, J. Marcum, Stephen Rice, and Albert Beaumont Wood. It owes its existence in no small part to the detailed accounts of Hunt (1954), Wood (1965), and Hackmann (1984). The history focuses on developments in France, Britain, and the U.S.A., as these are the places where the main early advances took place, especially during WW1. Developments in Germany and the U.S.S.R. are mentioned only briefly, partly due to 4

Zhurkovich (2008) transcribes this name as ‘‘K.V. Shilovsky’’.

8

Introduction

[Ch. 1

Figure 1.1. Sketch of Beudant’s experiment of ca. 1816 (reprinted fom Girard, 1877).

the difficulty in finding reliable sources for them (in the case of Russian and Soviet acoustics, corrected recently by the publication of the History of Russian Underwater Acoustics, edited by Godin and Palmer, 2008).

1.4.1 1.4.1.1

Conception and birth of sonar (–1918) Discovery and ingenuity

The concept of echo ranging, by which the distance to an object is determined by measuring the time delay to an echo from that object, originates from at least as far back as the 17th century. More recent origins of sonar can be traced to two seemingly unrelated scientific developments in the 19th century, the first being the measurement of the speed of sound in seawater, ca. 1816, by Franc¸ois Beudant, in the French Mediterranean. Beudant used a crude but effective method (illustrated in Figure 1.1), involving an underwater bell and a swimmer waving a flag. A more precise determination, with improved light–sound synchronization (Figure 1.2), was made in 1826 by Colladon (Figure 1.3) and Sturm, in Lake Geneva.5 Both measurements are described by Colladon and Sturm (1827), and in both cases the values obtained (1,500 m/s and 5 Their purpose was not to measure the speed of sound for its own sake, but to determine the bulk modulus of water, which can be calculated from the sound speed if its density is known.

Sec. 1.4]

1.4 A brief history of sonar

9

Figure 1.2. Sketch of the Colladon–Sturm experiment of 1826 (reprinted fom Girard, 1877).

Figure 1.3. Inventor Reginald Fessenden (left) and physicist Jean Daniel Colladon (right). The image of Fessenden is reprinted from http://www.ieee.ca/millennium/radio/radio_unsung.html, last accessed October 22, 2009, # RadioScientist.

10 Introduction

[Ch. 1

1,435 m/s) are consistent with modern expectation for the respective measurement conditions. The second important development is the discovery of piezoelectricity by Pierre and Jacques Curie in 1880. Experiments with certain special dielectric crystals (especially quartz and Rochelle salt) revealed that these materials respond to an applied pressure by developing a small potential difference. The converse effect, whereby an applied electric field distorts the shape of the crystal, was predicted shortly afterwards by Gabriel Lippmann and confirmed by the Curie brothers in 1881. In the late 1890s and early 1900s, some lightships were fitted with underwater bells, which were rung to alert approaching vessels of danger in conditions of poor visibility. In good visibility these sounds provided an indication of distance as well, by estimating the time delay between light and sound signals, as when estimating the distance from an electrical storm by counting seconds to the thunder following a bolt of lightning. These early underwater signaling systems would eventually mature into what we now call sonar. 1.4.1.2

The Titanic and the Fessenden oscillator

The tragic collision and subsequent sinking of RMS Titanic on the night of April 14/ 15, 1912 resulted in a flurry of activity and ideas directed at providing advance warning of nearby icebergs. Lewis Richardson filed patents first for an airborne echolocation system in April 1912 and a month later for an underwater one. Reginald Fessenden (Figure 1.3) patented an electromagnetic transducer in 1913 and demonstrated its use by detecting the presence of an iceberg on April 27, 1914 at a distance of ‘‘nearly two miles’’ (i.e., approximately 3–4 km). This device became known as the Fessenden oscillator (Waller, 1989). 1.4.1.3

WW1: a sense of urgency

It took an even greater tragedy, the loss of life inflicted by U-boats during WW1, to provide the focus of intellect and resources that would lead to the development of a working underwater detection system. French and British efforts began in 1915, with Paul Langevin (Figure 1.4) working in Paris with Russian engineer Constantin Chilowski, while A. B. Wood worked with Harold Gerrard in Manchester. The focus of the French research was on echolocation (‘‘active sonar’’ in modern terminology), while the British team concentrated initially on listening devices known as hydrophones (‘‘passive sonar’’). At the outset of WW1, Lord Rutherford had assembled an extraordinary group of physicists at his laboratory at the University of Manchester, including the household names Bohr, Geiger, and Chadwick. In his autobiographical account, A. B. Wood recalls (Wood, 1965): ‘‘It would be difficult to find anywhere such a galaxy of scientific talent, either before or since, working together in the same physics laboratory at the same time.’’ Of particular relevance here are the arrivals of Wood himself in 1915 and of the Canadian physicist Robert Boyle (Figure 1.4) the following year. The Board of Invention and Research (BIR) was established in 1915, with

Sec. 1.4]

1.4 A brief history of sonar

11

Figure 1.4. Physicists Paul Langevin (left) and Robert William Boyle (right). The image of Langevin is reprinted from Anon. (wp, a) and that of Boyle from http://www.100years.ualberta. ca, last accessed October 26, 2009.

facilities at Hawkcraig (in Fifeshire, Scotland), and expanded in 1917 to a team of more than 80 scientists and technicians working at Parkeston Quay (Harwich, England) under the leadership of Professor W. H. Bragg. Amongst them were Boyle and Wood from Rutherford’s group, responsible, respectively, for research investigating echolocation and passive listening. Boyle made promising initial progress with the Fessenden oscillator, such that by late 1917 a submarine detection had been reported at a distance of 1,000 yd (910 m) (Hackmann, 1984, p. 75).6 Nevertheless, this line of work was abandoned because the frequency of Fessenden’s transmitter (1 kHz) was too low to obtain the necessary resolution in bearing for its intended purpose of locating submarines. A highfrequency transducer was needed to achieve this. In France, Langevin had begun to experiment with quartz early in 1917 after obtaining a small supply from a Paris optician. Quartz is a piezoelectric material suitable for the radiation of high-frequency sound,7 but the unamplified received 6

The yard (symbol yd) is a unit of length defined as 0.9144 meters (see Appendix B). Use here of the term ‘‘sound’’ is not restricted to the audible frequency range, but refers also to ‘‘ultrasound’’, which means that the frequency is above the upper limit of normal human hearing (i.e., 20 kHz). In general, it can also refer to sounds below 20 Hz, known as ‘‘infrasound’’. Langevin’s early experiments with quartz (April 1917) were at a frequency of 150 kHz. The frequency was later lowered to 40 kHz in order to reduce absorption. 7

12 Introduction

[Ch. 1

signals were found to be very weak. Fortunately, a suitable valve amplifier, designed by Le´on Brillouin and G. A. Beauvais,8 was made available to Langevin soon after, enabling him to build a system by November 1917 that ‘‘gave a signalling distance of up to six kilometres’’ (Hackmann, 1984, p. 81). The real breakthrough came when the French and British teams started sharing their findings after a series of high-level meetings held in Washington, D.C. between May and July 1917. Boyle visited Langevin shortly afterwards, when he would have learnt of the French advances. On his return to England, Boyle started working on quartz transducers, and the French amplifier was made available to the British team at Parkeston Quay. The reliance on quartz was such that, until a suitable supply was identified from Bordeaux, Boyle threatened to ‘‘raid the crystal exhibits in several geological museums’’. Meanwhile, Langevin continued with his own work in Toulon, and by February 1918 had obtained echoes from a submarine using the high-frequency (40 kHz) quartz transducers. Boyle followed suit a month later with a submarine echo from a distance of 500 yd (about 460 m). The Armistice of November 1918 led to the cancellation of plans to fit both British and French navy ships in early 1919, but asdics (as the technology of high-frequency echolocation was then called) was born.9 The term sonar was coined during WW2. The origin of the term asdics as an acronym for Anti-Submarine Division -ics, where the ‘‘ics’’ meant ‘‘activities pertaining to’’ in the same way as in ‘‘physics’’, is recounted by Wood (1965). The alternative explanation (for the term asdic, without the second ‘‘s’’) as an acronym for ‘‘Allied Submarine Detection Investigation Committee’’ appears to be a myth created by the British Admiralty in 1939 in response to a question by Oxford University Press (Hackmann, 1984, p. xxv). During the initial development of the sensor at Parkeston Quay, secrecy was such that even the material quartz was referred to by its codename ‘‘asdivite’’. On the subject of semantics, it is worth mentioning the change in meaning of the word ‘‘supersonic’’ after the end of WW2. Between the two world wars, this term was used in the U.S.A. to mean ‘‘pertaining to sound whose frequency is too high to be heard by the human ear’’, synonymous with the European term ‘‘ultrasonic’’ (Klein, 1968). Today the European term has been adopted worldwide, presumably as a consequence of the modern use of ‘‘supersonic’’ to describe ‘‘faster than sound’’ flight. The first working active sonar was built in November 1918 by Boyle, a Canadian scientist working in England. Reading an account of the early history of echo ranging, however, one cannot help being struck by a series of key contributions made by 8

This work was assisted by a wireless expert, Paul Pichon. Having deserted from the French army he found himself importing some American valve amplifiers to his adoptive Germany early in WW1. Realizing the military value of these, he took them instead to France where he— though immediately arrested—handed over his equipment to the French authorities. These early valves provided the basis for the Beauvais–Brillouin design (Hackmann, 1984, pp. 80–81). 9 Boyle’s quartz system was fitted to a trawler on November 16, 1918, five days after the end of WW1.

Sec. 1.4]

1.4 A brief history of sonar

13

French scientists, including: — the earliest known description of the echo-ranging concept, by Mersenne (1636); — the measurement of the speed of sound in seawater, by Beudant (ca. 1816); — the discovery of piezoelectricity, by the Curie brothers and Lippmann (1880– 1881); — the development of the valve amplifier, by Beauvais and Brillouin (ca. 1916); — pioneering research on the use of quartz transducers, including the first ever detection of an echo from a submarine, by Langevin10 (1917–1918). To this impressive list one can add the work of a remarkable statesman named Paul Painleve´ (Figure 1.5). In January 1915, Chilowski had written a letter urging the French government to develop an underwater echolocation device as a defense against U-boats. Recognizing its importance and urgency, Painleve´ forwarded this letter to Langevin without delay, thus facilitating the early Langevin–Chilowski collaboration. Painleve´ also saw the value in Anglo-French co-operation, requesting a scientific exchange agreement between France and Britain in December 1915. Despite delays caused by opposition from the Admiralty, the agreement, without which the co-operation between Langevin and Boyle might not have flourished, was eventually approved by the British Government in October 1916 (Hackmann, 1984, p. 39). 1.4.1.4

Origins of passive sonar

By comparison with active sonar, invented in a race against time between Chilowski’s 1915 letter and the first successful French and British tests in 1917, the arrival of passive sonar was a gradual affair that lasted centuries. Its 15th-century conception in Leonardo da Vinci’s device able to detect ships ‘‘at a great distance’’ was followed by a 400-year gestation, including the 18th-century observations of Benjamin Franklin (see Section 1.4.3.3), and culminating in the listening equipment fitted to shipping vessels at the end of the 10

Figure 1.5. French statesman and mathematician Paul Painleve´—reprinted from Anon. (wp, b). Painleve´ was Minister for Public Instruction and Inventions during the period 1915–1917, and later served two brief periods as Prime Minister in 1917 and 1925.

Langevin is one of five sonar scientists after whom the Pioneers of Underwater Acoustics Medal, awarded to this day by the Acoustical Society of America, is named. The others are H. J. W. Fay, R. A. Fessenden, H. C. Hayes, and G. W. Pierce. In 1959, Hayes became the first ever recipient of this medal, which was also awarded to Wood (in 1961) and to Urick (1988).

14 Introduction

[Ch. 1

19th century to notify them of the presence of nearby lightships: in 1889, the U.S. Lighthouse Board described an invention of L. I. Blake comprising an underwater bell and microphone receiver, and a similar system—patented in 1899 (Hersey, 1977)—was developed a few years later by Elisha Gray and A. J. Mundy (Lasky, 1977).11 In common with the echolocation devices of Langevin and Boyle, it was WW1 that provided the final impetus for the birth of passive sonar. An important difference, though, is that underwater listening equipment was put to practical use well before the end of the war. Portable omnidirectional hydrophones were available as early as 1915, and directional ones followed in 1917. Towed hydrophones were operational before the end of WW1, and in 1918 a prototype passive-ranging system was fitted to an American destroyer. British listening devices used during WW1, based on early American work, were developed at BIR by Wood and Gerrard (occasionally assisted by Rutherford) at Parkeston Quay and by Captain C. P. Ryan at Hawkcraig. To reduce noise, directional hydrophones could be towed behind the ship in a streamlined capsule known as a ‘‘fish’’, developed by G. H. Nash. Ryan constructed a network of up to 18 underwater listening stations positioned strategically in British coastal waters. These listening stations, each comprising a field of hydrophones, were manned with shore-based operators, who listened for distinctive U-boat sounds and reported their position to the nearest anti-submarine flotilla. Some minefields were also equipped with special listening devices (magnetophones), with which it was possible to determine the precise moment at which a U-boat was passing overhead. The mines could then be detonated remotely from a shore-based monitoring facility. According to Hackmann (2000), such minefields were responsible for the destruction of four U-boats towards the end of WW1, the first taking place on August 29, 1918. Early in WW1, Rutherford had proposed the use of an array of multiple hydrophones, in theory able to both amplify the signal and provide bearing information. The Royal Navy considered the proposed device too unwieldy and the idea was dropped in Britain, but American scientists pursued it and by the end of the war had developed the most sophisticated listening devices of that time (Hayes, 1920). This American research took place at the Naval Experimental Station in New London, under the direction of Harvey Hayes. The property of sound waves that Rutherford wished to exploit is that they retain their phase coherence over distances of at least several wavelengths. The first American device to use this property was the ‘‘M-B tube’’, comprising two groups of eight hydrophones each. The (acoustic) signals from each group were combined coherently by a sequence of equal-length delay lines before being presented (binaurally, one coherently summed group in each ear) to a human listener. The construction was such that coherent reinforcement took place from only one direction at a time, so in order to scan over different bearings it was necessary to rotate this device in the water. The inconvenience of the M-B tube—it needed to be lowered into the sea each time it was 11 Gray coined the term ‘‘hydrophone’’ to describe their underwater microphone, while Mundy went on to co-found the Submarine Signal Company (now part of Raytheon) in 1901.

Sec. 1.4]

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15

Figure 1.6. Installation of early U.S. passive-ranging sonar with two towed eels of length 40 ft (12 m), and 12 ft (4 m) apart, and two hull-mounted M-V tubes of the same length. The eel was towed about 300–500 ft (100–150 m) behind the ship (reprinted with permission from Lasky, 1977, copyright 1977 American Institute of Physics).

used—was overcome by the introduction of variable-length delay lines, which permitted the operator to select the direction of listening without any form of mechanical rotation. This meant that the entire device, known as the ‘‘M-V tube’’, could be fixed to a ship’s hull, and used with the ship in motion. The M-V tube had two groups of six hydrophones (later, two groups of ten), the signals from which were presented binaurally in the same way as for the M-B tube. The capability to use the M-V tube in motion was a huge advantage, but it came at a price—the din from a ship underway. To counter the noise problem the ‘‘U-3 tube’’ (nicknamed the ‘‘eel’’), was invented. The eel comprised two groups of six hydrophones towed behind the ship, thus benefiting from lower noise levels. The U-3’s streamlined housing gave it the appearance of a snake or eel—hence its nickname. The key advance that made this possible was the use of electrical instead of acoustical delay lines, making the equipment less bulky. An experimental device comprising two towed eels and two ship-mounted M-V tubes was fitted to an American destroyer in April 1918 (Figure 1.6). The combined system was capable of passive ranging by triangulation of the two different bearings (Hayes, 1920). The first working sonar capable of localization in range and bearing was neither a French nor a British invention, but an American one.

1.4.2 1.4.2.1

Sonar in its infancy (1918–1939) Fathometers and fish finders

In peacetime, the thoughts of sonar engineers turned away from U-boats and back initially to maritime safety, and later to fishing. The principle of acoustic echo ranging was applied to measuring water depth, and Fessenden’s oscillator turned out to be

16 Introduction

[Ch. 1

ideally suited to this purpose. For this new application,12 its low operating frequency became an advantage because of reduced absorption, and there was no need for directivity because the direction to the seabed is known in advance. The first patent is attributed by Hersey (1977) to A. F. Wells as early as 1907, while Hackmann (1984) credits the first workable system to Alexander Behm in 1912.13 The first commercial echo sounders (called ‘‘fathometers’’) were designed by Fessenden at the Submarine Signal Company, using his electromagnetic oscillator as a transmitter in combination with a conventional carbon microphone receiver. A recording echo sounder, enabling a permanent paper record to be kept of the echo sounder output, was invented by Marti and Langevin in 1922. The next challenge for echo ranging was to be the detection of fish shoals. Although these produce weaker signals than the seabed, echoes from fish were recorded by the trawler Glen Kidston in the North Sea in 1933 (Cushing, 1973). Even before then, echo sounders were used by Belloc to identify the location of fish shoals in the Bay of Biscay (Belloc, 1929a, b), and by the innovative fisherman Captain R. Balls to find shoals of herring (Hersey and Backus, 1962, p. 499). 1.4.2.2

National research laboratories

The 1920s marked the beginning of nationally co-ordinated peacetime research efforts in both Britain and the U.S.A., with both Wood and Hayes continuing at their respective national research laboratories. In Britain, the Applied Research Laboratory (ARL) was founded in 1921, led first by B. S. Smith (1921–1927) and later by Wood. The achievements of this group include the development of the magnetostrictive transducer in 1928 and of the recording echo sounder used on the Glen Kidston.14 The U.S. Naval Research Laboratory (NRL) followed in 1923. The NRL Sound Division, led by Hayes, was responsible for an oddly named listening device called the ‘‘JK projector’’, installed on U.S. Navy ships in 1931 (Klein, 1968). This listening system made use of Rochelle salt—a more efficient piezoelectric material than quartz—housed in a special material known as ‘‘rho-c rubber’’, providing an impedance match with water while keeping the transducer dry (Rochelle salt dissolves in water). The device was later adapted to enable its use for echo ranging also, leading to an early American active sonar known as the ‘‘QB’’, produced commercially by the Submarine Signal Company. 1.4.2.3

Temperature and the ‘‘afternoon effect’’

Soon after the end of WW1, both British and American scientists working on the recently developed asdics sets noticed that their performance was inconsistent. A 12

The idea itself was not new, but 19th-century attempts had been unsuccessful (Drubba and Rust, 1954; Maury, 1861; Newman and Rozycki, 1998). 13 An entire chapter of Hackmann’s book is devoted to the development of echo sounders. 14 Wood’s work is honored by the A. B. Wood Medal, awarded annually by the U.K. Institute of Acoustics.

Sec. 1.4]

1.4 A brief history of sonar

17

system that had achieved reliable detections to several kilometers on one day would perform erratically the next. More specifically, a diurnal cycle was noticed, whereby the detection performance would deteriorate noticeably during a warm afternoon and recover again the next morning. This so-called ‘‘afternoon effect’’ was explained in the summer of 1937 by Columbus Iselin of the Woods Hole Oceanographic Institution and L. Batchelder of the Submarine Signal Company in terms of refraction due to unexpectedly high temperature gradients they discovered in the Cayman Sea, south of Guantanamo Bay.15 Their measurements were confirmed a year later using Spilhaus’s recently invented bathythermograph.

1.4.3 1.4.3.1

Sonar comes of age (1939–) WW2: a giant awakes

A detailed account of the role that sonar played in the outfolding of WW2 is given by Hackmann (1984). Although alternatives to quartz had been investigated in the interwar years, especially in the U.S.A., the asdics sets in use by the Royal Navy still relied on the quartz technology developed during WW1. Quartz supplies were available locally, while essential cutting equipment and expertise were rescued from a factory in Antwerp (Belgium) shortly after the German invasion in May 1940. The entry of the U.S.A. in WW2 shortly after the Japanese attack on Pearl Harbor resulted in an investment of scientific resources in military objectives on an unprecedented scale, and sonar was no exception to this. The recently formed National Defense Research Committee (NDRC) was restructured, while the NRL was expanded to a strength of several thousand. American research at the Universities of California and Columbia led to major developments in the theoretical understanding of the propagation and scattering of underwater sound. Much of this research was published in the compilation volume Physics of Sound in the Sea, edited by Lyman Spitzer Jr. (Spitzer, 1946), a document that remains of value today. According to Hackmann (1984), the term ‘‘sonar’’ was coined by F. V. Hunt in 1942 not as an acronym but as a phonetic analogue to ‘‘radar’’, while the modern explanation as an acronym for sound navigation and ranging was invented later. Whatever its origin, Hunt’s term would eventually replace the name ‘‘asdics’’ preferred by the Royal Navy. 1.4.3.2

Passive sonar in Germany

While Allied efforts focused mainly on the detection and prosecution of U-boats using active sonar, German scientists developed sophisticated passive sonar equipment known as Gruppenhorchgera¨t (GHG). This GHG equipment was fitted to the cruiser Prinz Eugen, providing it with effective early warning of torpedo attacks, permitting timely evasive action. 15

Sound speed was known at this time to vary with temperature.

18 Introduction

1.4.3.3

[Ch. 1

The anomalous absorption of seawater

It has long been known that sound is able to travel remarkable distances in water. The statement ‘‘if you cause your ship to stop, and place the head of a long tube in the water, and place the other extremity to your ear, you will hear ships at a great distance from you’’ is attributed to the 15th-century scientist Leonardo da Vinci (Urick, 1967),16 and underwater sound was used even before then by ancient Asian fishermen (Hackmann, 1984). In July 1762, Benjamin Franklin wrote ‘‘. . . the sound [of two stones struck under water nearly a mile away] did not seem faint, . . . but smart and strong, and as if present just at the ear . . .’’ The main explanation for these observations is the rapid decrease in absorption with decreasing frequency, permitting long-range propagation of low-frequency sound. In freshwater, the absorption of sound increases quadratically with frequency. This functional form (though not the magnitude) was consistent with 19th-century predictions of viscous absorption by G. Stokes. After the end of WW2, oceanographers directed their efforts to understanding the discrepancy in magnitude and the (more complicated) frequency dependence in seawater. In a landmark publication, Liebermann (1948) explained the discrepancy with Stokes’s theory by introducing the effects of dilatational viscosity. He also suggested an ionic relaxation as a possible mechanism for the anomalous frequency dependence below 1 MHz in seawater. In the 1950s it was realized that the salt responsible for Liebermann’s relaxation effect was magnesium sulfate, which has a relaxation time close to 1 ms at room temperature. The realization appears to have been a gradual one, with contributions from Leonard et al. (1949), Wilson and Leonard (1954), Fisher (1958), and Schulkin and Marsh (1962). A lower frequency relaxation with a relaxation time of order 100 ms, due to boric acid, was suggested by Yeager et al. (1973) and confirmed later by Fisher and Simmons (1975) and Simmons (1975). The combined effect of the two relaxations, plus viscosity at high frequency (above 300 kHz), is illustrated by Figure 1.7.17 An equation for the absorption coefficient of seawater that came into widespread use after it appeared in the second edition of Urick’s book (Urick, 1975), in units of decibels per kilometer, is18 ! ! 0:11F 2 44F 2 a¼ þ þð3  10 4 F 2 ÞH2 O ; ð1:1Þ 1 þ F2 4100 þ F 2 BðOHÞ3

MgSO4

where F is the acoustic frequency expressed in kilohertz. Although Equation (1.1) is known as ‘‘Thorp’s equation’’, in fact only the first of the three terms originates from 16

Urick’s source is an unpublished report by T. G. Bell (Bell, 1962). An alternative source, from Burdic (1984), is due to E. MacCurdy (MacCurdy, 1942). Neither Bell (1962) nor MacCurdy (1942) have been seen by the present author. 17 The possibility of a third chemical relaxation, associated with magnesium carbonate ions, is suggested by Mellen et al. (1987). 18 Urick’s original equation was expressed in units of decibels per kiloyard. The metric version quoted here is due to Fisher and Simmons (1977).

Sec. 1.4]

1.4 A brief history of sonar

19

Figure 1.7. Sound absorption vs. frequency in seawater (reprinted with permission from Fisher and Simmons, 1977, copyright 1977 American Institute of Physics).

Thorp (1967). Urick combined the viscosity and magnesium relaxation terms from Horton (1959) with the low-frequency relaxation term introduced by Thorp.

1.4.3.4

SOFAR, SOSUS, and the Roswell Incident

1.4.3.4.1 The speed of sound in seawater In perhaps the first-ever application of Snell’s law to underwater sound, Lichte (1919) showed that the sensitivity of compressibility to temperature, pressure, and salinity means that the sound speed of seawater varies with depth, and that this depth

20 Introduction

[Ch. 1

variation influences sound propagation in a predictable manner. Such predictions became a reality once accurate tables of sound speed as a function of these three parameters became available in the 1920s (Matthews, 1927) and were later improved by Kuwahara (1939). The essence of these tables is captured in a simple formula due to Medwin (1975) giving the sound speed c, in meters per second, as cðS; T; zÞ ¼ 1,449:2 þ 4:6T þ 0:016z  0:055T 2 þ ½ð1:34  0:010TÞðS  35Þ þ 2:9  10 4 T 3 

ð1:2Þ

where T is the temperature expressed in units of degrees Celsius; S is the salinity in parts per thousand (g/kg); and z is the depth from the sea surface in meters. Under typical ocean conditions, the first four terms are the most important ones. While Equation (1.2) is not accurate by modern standards, its simplicity makes it particularly suitable for investigating the sensitivity to temperature and pressure (parameterized as depth), which are of interest to this historical account. Neglecting the terms in square brackets, the partial derivatives with respect to temperature and depth are  @c  4:6  0:110T m/s per degree Celsius ð1:3Þ @T S;z and

 @c  0:016 @z T;S

m/s per meter:

ð1:4Þ

Putting T ¼ 10 C in Equation (1.3) gives @c=@T 3.5 m/s per C. Close to the sea surface, the most important effect usually arises from the temperature gradient, especially in summer when the temperature at the surface can be 20 C higher than at depth. For example, using a temperature gradient of 5 C per 100 m gives a sound speed gradient of 0.18 m/s per meter due to temperature alone, much larger than the pressure effect of Equation (1.4). 1.4.3.4.2 Sound fixing and ranging: the SOFAR channel Because of the temperature gradient referred to above, the sound speed tends to decrease with increasing depth close to the sea surface, and this process continues until the temperature stabilizes at its deep-water value (ca. 2 C). At this point, the pressure effect—which works in the opposite direction—takes over, resulting in a sound speed minimum at a depth that depends on the surface temperature and hence on latitude, below which sound would be refracted upwards, back towards the sea surface. A typical depth for the minimum is ca. 1 km. Lichte realized this in 1919, but the implications of his remarkable insight were not properly investigated until the 1940s. Maurice Ewing understood that Lichte’s upward refraction would create a huge waveguide, possibly spanning entire oceans. Coupled with low absorption at low frequency, such a waveguide creates a means for communicating over vast distances, and the existence of the predicted channel was

Sec. 1.4]

1.4 A brief history of sonar

21

confirmed in 1943 (Ewing and Worzel, 1948).19 Amongst other applications, Ewing proposed a system to aid ditched air force pilots. The scheme involved a device that, when dropped into the sea, would sink and explode at a depth close to the channel axis. The idea was to record the arrival time of the resulting acoustic pulse at three or more listening stations and calculate the pilot’s location by triangulation. This remarkable recovery system, the principle of which was demonstrated during WW2 (Stifler and Saars, 1948), was called sofar (for sound fixing and ranging) and would later give its name to the channel itself. 1.4.3.4.3 The Sound Surveillance System (SOSUS) During the Cold War, a series of so-called ‘‘Summer Studies’’ was held at various American universities between 1948 and 1957 to work on the major defense problems of the time (Holbrow, 2006). The second of these, held at MIT in 1950 and directed by Professor J. Zacharias, posed the problem of ‘‘undersea defense’’ (i.e., defense against Soviet submarines). The proposed solution was a global network of hydrophone arrays in the sofar channel to exploit long-range waveguide propagation. Such a network, a natural successor to sofar and known as SOSUS, for sound surveillance system, was indeed designed and built. This network has since the end of the Cold War been used for non-military purposes (Stafford et al., 1998). 1.4.3.4.4 Project MOGUL and the Roswell Incident A U.S. Air Force report published in 1995 (Weaver and McAndrew, 1995) makes a fascinating connection between SOSUS and a famous UFO story dating to 1947 known as the Roswell Incident. Even before SOSUS, Ewing had proposed an ambitious project to exploit an atmospheric analogue of the SOFAR channel. He suggested that high-altitude balloons equipped with microphones might be able to detect Soviet weapons tests. The project, known as MOGUL, went ahead, and the claim made in the USAF report, after 50 years shrouded in secrecy, is that one of the MOGUL balloons crashed at Roswell in 1947, thus spawning the UFO story. An illustrated account, including a general history of the SOFAR channel, is given by Richard Muller of the University of California at Berkeley (Muller, www). 1.4.3.5 1.4.3.5.1

Advances in detection theory and processing technology Statistical detection theory

Important advances in the statistical theory of detection of signals in a noisy background were made by radar and telecommunications scientists in the 1940s and 1950s (Marcum, 1947; Rice, 1948; Swerling, 1954). This seminal work, epitomized by the Marcum function, Rician distribution, and Swerling distributions, provides the foundation for modern detection theory. 19 According to Goncharov (2008), the effect was discovered independently by L. D. Rozenberg in 1946 and explained by L. M. Brekhovskikh in 1948.

22 Introduction

[Ch. 1

1.4.3.5.2 FM processing and electronic scanning In the 1950s and 1960s, new types of processing technology became feasible. The polar circumnavigation by USS Nautilus in 1958 was made possible by a newly developed FM ( frequency modulation) sonar pulse that provided accurate navigation data under the ice cap. The high resolution of this new technique also made it suitable for detecting small objects such as sea mines. The sonar used by the Nautilus was rotated mechanically. In the 1960s such mechanical devices were replaced by sophisticated electronic systems that could scan horizontally without the need for moving parts. 1.4.3.5.3 The computer era The widespread availability of digital computers in the late 20th century has opened up new possibilities that would have been inconceivable in the early days of sonar. In modern sonar systems, beamforming, FM processing, Doppler filtering, localization, and tracking algorithms are implemented in software rather than analogue hardware. Other processing methods, relying on sophisticated computer models of sound propagation, include time reversal acoustics, synthetic aperture sonar, matched field processing, and noise correlation (Kuperman and Lynch, 2004). Most computer models of sound propagation available today (Jensen et al., 1994) have their roots in one (or more) of normal mode theory (Pekeris, 1948), flux propagation theory (Weston, 1959), the fast-field method (DiNapoli, 1971), and the parabolic equation method (Tappert, 1977). 1.4.4 1.4.4.1

Swords to ploughshares Oceanographic instruments

The first non-military spin-off of sonar was Fessenden’s echo sounder, and fisheries sonar followed soon after, as did early uses for seismic prospecting (Hersey, 1977). Today, an increasingly sophisticated understanding of underwater sound propagation has brought new applications in acoustical oceanography, marine archeology, and deep-sea exploration. Examples are seabed classification and mapping, classification of marine flora and fauna (Fernandes et al., 2002), weather and climate observation, and high-resolution acoustic imaging. Perhaps the most striking achievement of sonar was the discovery in September 1985, by Robert Ballard and Jean Louis Michel, of the wreck of the Titanic. The wreck was found at a depth of 3,800 m with the aid of a deep-diving submersible equipped with side-scan sonar. In the early 1990s an ambitious global experiment took place, known as the Heard Island Feasibility Test (HIFT), the purpose of which was to determine whether it was possible to build a global acoustic thermometer by exploiting the dependence of sound speed, and hence travel time, on temperature. A location in the southern Indian Ocean was identified, close to Heard Island, from which sound following great circle paths could be heard on both the east and west coasts of the U.S.A., involving propagation distances of up to 18,000 km (Munk et al., 1994; Collins et al., 1995). HIFT was a precursor to measurements carried out both on a large scale in the

Sec. 1.5]

1.5 References

23

Atlantic and Pacific Oceans and in smaller basins such as the Arabian, Barents, and Mediterranean Seas (Munk et al., 1995). 1.4.4.2

Discovery of dolphin sonar and concern over the effects of anthropogenic sound

With man’s increasingly sophisticated use of sound in the sea came also a gradual awareness that we are not alone in this use. The use of echolocation by bottlenose dolphins was first suspected in the 1940s by Arthur McBride, curator of Marine Studios (Florida). This speculation was confirmed in the 1950s by W. N. Kellogg, William Schevill, and others (Au, 1993). It is now known that many odontecetes use echolocation for both hunting and navigation (Au, 1993). A growing concern is that whales might rely on features such as the SOFAR channel for long-distance communication, and that increasing levels of anthropogenic sound due to shipping, underwater explosions, high-power military sonar, and even low-power systems such as used in basin-scale acoustic thermometry, might be disrupting their ability to do so. This concern has led to an increasing interest and awareness in the sonar of dolphins and other marine mammals (Richardson et al., 1995).

1.5

REFERENCES

Anon. (wp, a) Paul Langevin, available at http://en.wikipedia.org/wiki/Paul_Langevin (last accessed October 21, 2009). Anon. (wp, b) Paul Painleve´, available at http://en.wikipedia.org/wiki/Paul_painleve (last accessed October 21, 2009). Au, W. W. L. (1993) The Sonar of Dolphins, Springer, New York. Bell, T. G. (1962) Sonar and Submarine Detection, U.S. Navy Underwater Sound Lab. Rep. 545. Belloc, M. G. (1929a) La Croisie`re de la Tanche en Aouˆt 1927, Rapports et Proce`s-verbaux des Re´unions, Conseil P, 55, 139–150 [in French]. Belloc, M. G. (1929b) La Croisie`re de la Tanche en Juillet–Aouˆt 1928, Rapports et Proce`sverbaux des Re´unions, Conseil P, 62, 66–78 [in French]. Bjørnø, L. (2003) Features of underwater acoustics from Aristotle to our time, Acoustical Physics, 49(1), 24–30. Burdic, W. S. (1984) Underwater Acoustic Systems Analysis, Prentice-Hall, Englewood Cliffs NJ. Colladon, J.-D. and Sturm, J. C. F. (1827). Me´moire sur la compression des Liquides, Annales de Chimie et de Physique [in French]. Collins, M. D., McDonald, B. E., Heaney K. D., and Kuperman, W. A. (1995) Threedimensional effects in global acoustics, J. Acoust. Soc. Am., 97, 1567–1575. Cushing, D. (1973) The Detection of Fish (International Series of Monographs in Pure and Applied Biology Vol. 52), Pergamon. DiNapoli, F. R. (1971) Fast Field Program for Multilayered Media (NUSC Technical Report 4103), Naval Underwater Systems Center.

24 Introduction

[Ch. 1

Drubba, H. and Rust, H. H. (1954) On the first echo-sounding experiment, Annals of Science, 10(1), March, 28–32. Etter, P. C. (2003) Underwater Acoustics Modeling and Simulation: Principles, Techniques and Applications, Spon Press, New York. Ewing, M. and Worzel, J. L. (1948) Long-range sound transmission, Geological Society of America Memoir, 27. Fernandes, P. G., Gerlotto, F., Holliday, D. V., Nakken O., and Simmonds, E. J. (2002) Acoustic applications in fisheries science: The ICES contribution, ICES Marine Science Symposia, 215, 483–492. Fisher, F. H. (1958) Effect of high pressure on sound absorption and chemical equilibrium, J. Acoust. Soc. Am., 30, 442–448. Fisher, F. H. and Simmons, V. P. (1975) Discovery of boric acid as cause of low frequency sound absorption in the ocean, IEEE Oceans ’75, 21–24. Fisher, F. H. and Simmons, V. P. (1977) Sound absorption in sea water, J. Acoust. Soc. Am., 62, 558–564 Girard, M. (1877) Le son dans l’air et dans l’eau, La Nature, Revue des Sciences et de leurs applications aux arts et a` l’industrie (pp 247–250), G. Masson, Paris [in French]. Godin, O. A. and Palmer, D. R. (Eds.) (2008) History of Russian Underwater Acoustics, World Scientific, Hackensack, NJ. Goncharov, V. V. (2008) The development of sound propagation theory in the USSR and in Russia, in O. A. Godin and D. R. Palmer (Eds.), History of Russian Underwater Acoustics (pp. 71–120), World Scientific, Hackensack, NJ. Hackmann, W. (1984) Seek & Strike: Sonar, Anti-submarine Warfare and the Royal Navy 1914– 54, HM Stationery Office, London. Hackmann, W. (2000) Asdics at war, IEE Review, 15–19. Hayes, H. C. (1920) Detection of submarines, Proc. Amer. Phil. Soc., LXIX, March 19, 1–47. Hersey, J. B. (1977) A chronicle of man’s use of ocean acoustics, Oceanus, 20(2), 8–21. Hersey, J. B. and Backus, R. H. (1962) Sound scattering by marine organisms, in M. N. Hill (Ed.), The Sea, Ideas and Observations on Progress in the Study of the Seas, Vol. 1: Physical Oceanography (pp. 498–539), Interscience, New York. Holbrow, C. H. (2006) Scientists, security, and lessons from the cold war, Physics Today, 59(7), 39–44. Horton, J. W. (1959) Fundamentals of SONAR (Second Edition). United States Naval Institute, Annapolis. Hunt, F. V. (1954, reprinted 1982) Electroacoustics: The Analysis of Transduction, and Its Historical Background, American Institute of Physics, New York. Hunt, F. V. (1992) Origins of Acoustics, Acoustical Society of America, New York. Jensen, F. B., Kuperman, W. A., Porter, M. B., and Schmidt, H. (1994) Computational Ocean Acoustics, American Institute of Physics, New York. Klein, E. (1968) Underwater sound and naval acoustical research and application before 1939, J. Acoust. Soc. Am., 43, 931–947. Kuperman, W. A. and Lynch, J. F. (2004). Shallow-water acoustics, Physics Today, October, 55–61. Kuwahara, S. (1939) Velocity of sound in sea-water and calculation of the velocity for use in sonic sounding, Hydrographic Review, 16, 123–140. Lasky, M. (1977) Review of undersea acoustics to 1950, J. Acoust. Soc. Am., 61, 283–297. Leonard, R. W., Combs, P. C., and Skidmore Jr., L. R. (1949) Attenuation of sound in ‘‘synthetic sea water’’, J. Acoust. Soc. Am., 21, 63.

Sec. 1.5]

1.5 References 25

Lichte, H. (1919) U¨ber den Einfluß horizontaler Temperaturschichtung des Seewassers auf die Reichweite von Unterwasserschallsignalen, Physikalische Zeitschrift, 17, 385–389 [in German].20 Liebermann, L. N. (1948) The origin of sound absorption in water and sea water, J. Acoust. Soc. Am., 20, 868–873. Liebermann, L. N. (1949) Sound propagation in chemically active media, Phys. Rev., 76, 1520. MacCurdy, E. (1942) The Notebooks of Leonardo da Vinci (Chap. X), Garden City Publishing, New York. Marcum, J. I. (1947) A Statistical Theory of Target Detection by Pulsed Radar (Research memorandum RM-754), The RAND Corporation. Marcum, J. I. (1948) A Statistical Theory of Target Detection by Pulsed Radar: Mathematical Appendix (Research memorandum RM-753), The RAND Corporation. Matthews, D. J. (1927, reprinted 1939) Tables of the Velocity of Sound in Pure Water and Sea Water for Use in Echo-sounding and Sound-ranging (H. D. 282). Hydrographic Department, The Admiralty, London. Maury, M. F. (1861, reprinted 1963) The Physical Geography of the Sea, and Its Meteorology (Eighth Edition). Harvard University Press. Medwin, H. (1975) Speed of sound in water: A simple equation for realistic parameters, J. Acoust. Soc. Am., 58, 1318–1319. Mellen, R. H., Scheifele, P. M., and Browning, D. G. (1987) Global Model for Sound Absorption in Sea Water, Naval Underwater Systems Center, Newport. Muller, R. A. (www) Government Secrets of the Oceans, Atmosphere, and UFOs, University of California at Berkeley, available at http://muller.lbl.gov/teaching/Physics10/old%20 physics%2010/chapters%20(old)/9-SecretsofUFOs.html (last accessed September 29, 2008). Munk, W. H., Spindel, R. C., Baggeroer, A., and Birdsall, T. G. (1994) The Heard Island Feasibility Test, J. Acoust. Soc. Am., 96, 2330–2342. Munk, W., Worcester, P., and Wunsch, C. (1995) Ocean Acoustic Tomography, Cambridge University Press, Cambridge. Newman P. G. and Rozycki, G. S. (1998) The history of ultrasound, Surgical Clinics of America, 78(2), 179–195. Pekeris, C. L. (1948) Theory of propagation of explosive sound in shallow water, Geol. Soc. Amer. Mem., 27. Rice, S. O. (1948) Statistical properties of a sine wave plus random noise, Bell Syst. Tech. J., 109–157,January. Richardson, W. J. Greene, C. R., Malme, C. I., and Thomson, D. H. (1995) Marine Mammals and Noise, Academic Press, San Diego. Schulkin, M. and Marsh, H. W. (1962) Sound absorption in sea water, J. Acoust. Soc. Am., 34, 864–865. Simmons, V. P. (1975) Investigation of the 1 kHz sound absorption in sea water, PhD thesis, University of California, San Diego. Spitzer, L. (1946) Physics of Sound in the Sea (NAVMAT P-9675, Summary Technical Report of Division 6, Volume 8). National Defense Research Committee, Washington, D.C. [Reprinted by Department of the Navy Headquarters Naval Material Command, Washington, D.C., 1969.] Stafford, K. M., Fox, C. G., and Clark, D. S. (1998) Long-range detection and localization of blue whale calls in the northeast Pacific Ocean, J. Acoust. Soc. Am., 104, 3616–3625. 20

Urick (1983) refers to an English translation of this work by A. F. Wittenborn.

26 Introduction

[Ch. 1

Stansfield, D. (1991) Underwater Electroacoustic Transducers, Bath University Press, Bath. Stifler Jr., W. W. and Saars, W. F. (1948) SOFAR, Electronics, 21, 98–101. Swerling, P. (1954) Probability of Detection for Fluctuating Targets (Research Memorandum RM-1217), The RAND Corporation. Tappert, F. D. (1977) The parabolic approximation method, in J. Keller and J. S. Papadakis (Eds.), Wave Propagation and Underwater Acoustics (pp. 224–287), Springer-Verlag, Berlin. Thorp, W. H. (1967) Analytic description of the low-frequency attenuation, J. Acoust. Soc. Am., 42, 270. Urick, R. J. (1967) Principles of Underwater Sound for Engineers, McGraw-Hill, New York. Urick, R. J. (1975). Principles of Underwater Sound for Engineers (Second Edition), McGrawHill, New York. Urick, R. J. (1983) Principles of Underwater Sound (Third Edition), Peninsula, Los Altos, CA. Waller, A. (1989) Unsung Genius: Canadian Reginald Fessenden pioneered radio, invented sonar and laid the groundwork for television. Why has his own country ignored him? Equinox, 44, March/April. Weaver, R. L. and McAndrew, J. (1995) The Roswell Report: Fact versus Fiction in the New Mexico Desert (Accession No. ADA326148), Defense Technical Information Center. Weston, D. E. (1959) Guided propagation in a slowly varying medium, Proc. Royal Society, LXXIII, 365–384. Wilson, O. B. and Leonard, R. W. (1954) Measurements of sound absorption in aqueous salt solutions by a resonator method, J. Acoust. Soc. Am., 26, 223–226. Wood, A. B. (1965) From Board of Invention and Research to Royal Naval Scientific Service, Journal of the Royal Naval Scientific Service, 20, Jul, No. 4 (A. B. Wood, O.B.E., D.Sc., Memorial Number), 16–97 (200–281). Yeager, E., Fisher, F. H., Miceli, J., and Bressel, R. (1973) Origin of the low-frequency sound absorption in sea water, J. Acoust. Soc. Am., 53, 1705–1707. Zhurkovich, M. V. (2008) Hydroacoustics: What is it?, in O. A. Godin and D. R. Palmer (Eds.), History of Russian Underwater Acoustics (pp. 3–6), World Scientific, Hackensack, NJ.

2 Essential background

Plurality should not be posited without necessity William of Ockham (ca. 1285–1349).

The purpose of this chapter is to introduce the basic knowledge required by the reader to understand the description of the sonar equations introduced in Chapter 3, and no more than this. The knowledge is sub-divided into four general subject areas: oceanography, acoustics, signal processing, and detection theory. Further details of these four areas, omitted here for simplicity, are described in Chapters 4 and following.

2.1

ESSENTIALS OF SONAR OCEANOGRAPHY

This section describes those basic physical properties of the sea and the air–sea boundary of relevance to the generation, propagation, and scattering of sound at sonar frequencies. The speed of sound and the density of water influence the generation and propagation of sound. These and other related parameters are described in Section 2.1.1, followed by the relevant properties of air in Section 2.1.2. A more comprehensive description of these oceanographic properties is provided in Chapter 4. Many parameters of relevance to underwater acoustics vary with temperature T, salinity S, and hydrostatic pressure (often parameterized through the depth from the sea surface z). Where ‘‘representative’’ numerical values are quoted, they are

28 Essential background

[Ch. 2

evaluated for the following conditions: T ¼ 10  C; S ¼ 35; depth z ¼ 0: By convention the depth co-ordinate z is zero at the sea surface and increases downwards to the seabed. For simplicity, in Chapters 2 and 3 the ocean is assumed to extend to infinite depth, with uniform properties occupying the entire half-space satisfying z > 0. For example, the speed of sound and density are assumed independent of depth and range. The purpose of the quantitative numerical calculations based on this idealization (see Worked Examples in Chapter 3) is to illustrate the main principles of sonar performance modeling, rather than to provide realistic estimates of detection performance. More realistic examples are presented at the end of the book, in Chapter 11. 2.1.1

Acoustical properties of seawater

The two most important acoustical properties of seawater are the speed at which sound waves travel (abbreviated as sound speed ) and the rate at which they decay with distance traveled (the decay rate, or absorption coefficient). A third parameter that can influence sound propagation, through its effect on interaction with boundaries, is density. The density of seawater, and the speed and absorption of sound in seawater, all depend on salinity, temperature, and pressure. At low frequency the absorption also depends on acidity or pH (see Chapter 4). 2.1.1.1

Speed of sound

For the representative conditions described above, the sound speed in seawater, denoted cwater , is 1490 m/s. More generally, the parameters S, T, and P all vary with depth and therefore so too does the sound speed, resulting in significant refraction. A discussion of these gradients and their important acoustic effects is deferred to Chapters 4 and 9. Here and in Chapter 3 they are neglected for simplicity. The wavelength  at frequency f is  ¼ cwater =f : 2.1.1.2

ð2:1Þ

Density

The density of seawater (water ) under the representative conditions introduced above is 1027 kg/m 3 . Departures of seawater density from this value are small and for most sonar performance applications may be neglected. 2.1.1.3

Attenuation of sound

Attenuation is the name given to the process of decay in amplitude due to a combination of absorption and scattering of sound. The term ‘‘absorption’’ implies conversion

Sec. 2.1]

2.1 Essentials of sonar oceanography

29

to some other form of energy, usually heat, whereas ‘‘scattering’’ implies a redistribution in angle away from the original propagation direction, with no overall loss of acoustic energy. The rate of attenuation of sound in water is less than in air and much less than that of electromagnetic waves in water. Low-frequency sound, of order 1 Hz to 10 Hz, can travel for thousands of kilometers, but high-frequency sound is attenuated more rapidly. The attenuation coefficient water increases monotonically with frequency by about four orders of magnitude in the frequency range from 30 Hz to 300 kHz, and quadratically with frequency thereafter. For frequencies f exceeding 200 Hz, it can be written (see Chapter 4) water ¼ 1

f2 f2 þ  þ 3 f 2 : 2 2 f 2 þ f 21 f þ f 22

ð2:2Þ

For the specified representative conditions, the three coefficients i are1 1 ¼ 1:40  10 2

Np km 1 ;

2 ¼ 5:58

Np km 1

3 ¼ 3:90  10 5

Np km 1 kHz 2 :

and ð2:3Þ

The frequencies f1 and f2 , explained further in Chapter 4, are known as relaxation frequencies and for the representative conditions are equal, respectively, to 1.15 kHz and 75.6 kHz. Numerical evaluation of Equation (2.2) gives (to the nearest order of magnitude) water  10 3 , 10 1 , and 10 þ1 Np/km at 300 Hz, 10 kHz, and 300 kHz, respectively. A graph of water vs. frequency, computed using Equation (2.2), is plotted in Figure 2.1. The reciprocal of the attenuation coefficient (i.e.,  1 ), plotted on the same graph, provides a rough measure of the distance that sound can travel in water if unimpeded by physical obstacles. This quantity is referred to here as ‘‘audibility’’, the acoustical analogue of optical ‘‘visibility’’, and varies between 10 2 m Np1 at 300 kHz and 10 6 m Np 1 at 300 Hz. By comparison, the attenuation coefficient of green light2 is at least 10 2 Np m1 for clear seawater, so that the optical visibility in water never exceeds 10 2 m Np1 and is usually less than 10 1 m Np1 . Thus, for acoustic frequencies up to 300 kHz, the audibility of sound exceeds the maximum visibility of light by up to six orders of magnitude. This is the reason why sound waves have 1

Two sound waves are said to differ in level by 1 Np if their amplitudes are in the ratio 1 : e. The neper (Np) and the related unit the decibel (dB) are defined in Appendix B. 2 The sea is opaque to electromagnetic radiation with the exception of visible light, very low frequency radio waves, and gamma rays. The extinction coefficient is a measure of the decay of light intensity with distance, and in the present notation is equal to 2opt ,where opt is the optical attenuation coefficient in units of nepers per unit distance. For example, the value quoted by (Clarke and James, 1939) of 4 % per meter for the extinction coefficient in the Sargasso Sea means that expð2opt xÞ ¼ 0:96 when the distance x ¼ 1 m. Taking logarithms gives 2opt ¼ 0:04/m.

30 Essential background

[Ch. 2

Figure 2.1. Numerical value of attenuation coefficient vs. frequency of sound in seawater  (expressed in units of nepers per megameter) and of its reciprocal,  1 (in kilometers per neper), calculated using Equation (2.2) for the specified representative conditions: S ¼ 35; T ¼ 10  C; z ¼ 0.

become the most successful means of probing the underwater environment. It is the raison d’eˆtre of sonar. 2.1.2

Acoustical properties of air

Together with those of seawater, the properties of air determine the reflection coefficient at the air–sea boundary. The sound speed and density of air depend on temperature (T) and pressure (P). For the representative conditions, these are cair ¼ 337 m/s and air ¼ 1.25 kg/m 3 . Thus, both  and c in air are considerably lower than their counterparts in water, which has important implications for the behavior of underwater sound.

2.2 2.2.1

ESSENTIALS OF UNDERWATER ACOUSTICS What is sound?

Steady-state pressure increases with increasing depth z and is equal to the total weight per unit area of water plus atmosphere supported above that depth. This quantity is called the static pressure (or hydrostatic pressure) and can be expressed quantitatively

Sec. 2.2]

2.2 Essentials of underwater acoustics

31

in the form Pstat ðzÞ ¼ Patm þ Pgauge ðzÞ;

ð2:4Þ

where Patm is the atmospheric pressure (approximately 101 kPa); and Pgauge is the additional pressure due to the weight of the water above depth z (the gauge pressure) ðz Pgauge ðzÞ ¼ water ðÞgðÞ d: ð2:5Þ 0

Underwater disturbances result in departures P from this value, for an arbitrary position vector x, Ptot ðx; tÞ ¼ Pstat ðzÞ þ Pðx; tÞ: ð2:6Þ Once created, provided that certain basic conditions are met (Pierce, 1989), a pressure disturbance propagates with the speed of sound, and P is known as the acoustic pressure, henceforth denoted qðx; tÞ and assumed small by comparison with static pressure.3 Such an acoustic disturbance is known as underwater sound. The study of this sound is called underwater acoustics. The assumption of small q simplifies the mathematics and is generally justified because atmospheric pressure is large compared with typical acoustic pressure fluctuations. A brief account is given here of radiation and scattering of underwater sound from simple sources and for a simple geometry. First, radiation is considered from a point source in an infinite uniform medium, with and without a perfectly reflecting plane boundary (Section 2.2.2). Then the scattering of plane waves is considered, first from a point object and then from a rough surface (Section 2.2.3). The sea surface is considered as an example of a reflecting surface, a radiating surface, and a scattering boundary. A more complete treatment of these phenomena is presented in Chapters 5 and 8. 2.2.2

Radiation of sound

2.2.2.1 2.2.2.1.1

Radiation from a point monopole source Spherical spreading

Consider a point monopole4 source of power W. To generate sound at a given frequency, the source must expand and contract at that frequency. During expansion the source motion causes an increase in density of the surrounding fluid, with a corresponding increase in its pressure. The resulting high-pressure disturbance propagates outwards in the form of a spherical wave, traveling at the speed of sound cwater . The same sequence follows a contraction, except with a low-pressure disturbance replacing the high-pressure one. At any fixed moment in time the radiated field comprises a series of concentric ‘‘rings’’ (actually spherical shells in three dimensions) of alternating high and low 3 The symbol p, introduced in Section 2.2.2, is reserved for a complex variable representing the acoustic pressure. See footnote 5. 4 A monopole source is one with a fluctuating volume, such as a pulsating bubble.

32 Essential background

[Ch. 2

pressure. The potential of these rings to do work on the surrounding medium can be expressed in terms of their potential energy density (Pierce, 1989) ðPÞ

EV ¼

q2 ; 2Bwater

ð2:7Þ

where EV denotes energy per unit volume; and Bwater is the bulk modulus of water, a measure of its opposition to compression or rarefaction, analagous to the stiffness of a spring, and equal to Bwater ¼ water c 2water : ð2:8Þ The rings also contain kinetic energy, due to the particle velocity u, given by ðKÞ

EV ¼

water juj 2 : 2

ð2:9Þ

The superscripts ðPÞ and ðKÞ in Equations (2.7) and (2.9) denote potential and kinetic energy, respectively. If the pressure and particle velocity are in phase, it can be shown that the kinetic and potential densities are equal (Pierce, 1989), so that the average rate of energy flux (i.e., intensity) is   ðKÞ ðPÞ ðPÞ I ¼ cwater E V þ E V ¼ 2cwater E V ¼

q2 ; water cwater

ð2:10Þ

where the overbar indicates an average in time. Conservation of energy demands that the total radiated power at distanceqsffiffiffiffiffifrom the source, 4 s 2 I, be constant, which means that the RMS pressure (i.e.,

q 2 ) must vary as 1=s with distance.

A point monopole source radiates omni-directionally (i.e., with equal power in all directions). At a distance s from the source, in the assumed uniform medium the energy is distributed uniformly on a sphere of surface area 4 s 2 (Figure 2.2), so the component of acoustic intensity normal to the surface of the sphere at a distance s is IðsÞ ¼

W : 4 s 2

ð2:11Þ

Also of interest is the acoustic pressure resulting from the point source. Using standard complex variable notation for a diverging harmonic spherical wave of angular frequency !, the complex pressure field p varies with time t and distance s according to Pierce (1989)5 pffiffiffi e iðks!tÞ pðs; tÞ ¼ 2p0 s0 ; ð2:12Þ s where p0 is the RMS pressure at a distance s0 from the source; and k is the acoustic wave number, so that k ¼ !=cwater : ð2:13Þ 5 The real part of the complex variable pðs; tÞ is the acoustic pressure qðs; tÞ. Unless otherwise stated, an expði!tÞ time convention is used for traveling waves throughout.

Sec. 2.2]

2.2 Essentials of underwater acoustics

33

Figure 2.2. Radiation from a point source of power W in free space. The intensity at a distance s is I0 ¼ W=ð4 s 2 Þ. At distance 2s the same power has spread into four times the area, reducing the intensity by a factor of 4.

The true acoustic pressure is obtained by taking the real part of Equation (2.12), so that pffiffiffi cosðks  !tÞ qðs; tÞ ¼ 2p0 s0 : ð2:14Þ s From Equation (2.10) it follows that I¼

j pj 2 ; 2w cw

ð2:15Þ

where the ‘‘water’’ subscript is abbreviated henceforth as ‘‘w’’. From Equations (2.11), (2.12), and (2.15) it then follows that   W 1=2 p0 s0 ¼ w cw ; ð2:16Þ 4

or more generally (for a directional source) p0 s0 ¼ ðw cw WO Þ 1=2 ;

ð2:17Þ

where WO indicates the radiated power per unit solid angle (the radiant intensity). It is convenient to define a steady-state propagation factor FðsÞ in terms of the ratio of the mean square pressure at the receiver to that at a small distance ðs0 Þ from the source, such that FðsÞ ¼

q2 j pj 2 ¼ 2 2: 2 2 p 0 s 0 2p 0 s 0

ð2:18Þ

Defined in this way, the propagation factor has dimensions [distance] 2 . For a spherical wave in a medium of uniform impedance it is equal to the ratio of received intensity I to the radiant intensity WO .

34 Essential background

[Ch. 2

It follows by substituting for pðs; tÞ from Equation (2.12), scaled by expðsÞ to account for absorption, that the propagation factor for a point source in a uniform medium is FðsÞ ¼

e 2s ; s2

ð2:19Þ

where  is the sound attenuation coefficient introduced in Section 2.1.1.3. The above arguments apply to a steady-state field due to a source of constant radiant intensity. If the power is transmitted for a short time only, it is useful to think in terms of the transient field resulting from the total radiated energy per unit solid angle EO . The appropriate propagation factor under these conditions is obtained by integrating the numerator and denominator of Equation (2.18) over time instead of averaging them: ð q 2 dt FðsÞ  : ð2:20Þ w cw EO To summarize, the steady-state mean square pressure for a source of radiant intensity WO , from Equation (2.18), is q 2 ¼ w cw WO FðsÞ

ð2:21Þ

and for a transient field, the time-integrated pressure squared, from Equation (2.20), is ð q 2 dt ¼ w wEO FðsÞ: ð2:22Þ Either way, FðsÞ is given by Equation (2.19) for an omni-directional point source in an infinite uniform medium. The same equation applies also for a directional source, provided that WO (or EO ) is measured in the direction of the receiver. The behavior described by Equation (2.19), characterized by its s 2 dependence due to the spherical nature of the expanding wave front, is known as spherical spreading. 2.2.2.1.2

Reflection from the sea surface

Now consider the effect of placing a reflecting boundary, such as the sea surface, close to the point source of Section 2.2.2.1.1. There are two straight-line ray paths connecting the source to any given receiver position: the direct path and a surface reflected one. If the source is at depth z0 below the surface (see Figure 2.3), the contribution to the pressure field at the receiver due to the direct path is given by Equation (2.12) with a source–receiver separation equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ r 2 þ ðz  z0 Þ 2 : ð2:23Þ

Sec. 2.2]

2.2 Essentials of underwater acoustics

35

Figure 2.3. Radiation from a point source in the presence of a reflecting boundary.

The reflected path can be thought of as originating from an image source at height z0 above the boundary, with image–receiver separation of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sþ ¼ r 2 þ ðz þ z0 Þ 2 : ð2:24Þ Using the method of images, the two contributions from source and image are added coherently to obtain the total pressure at the receiver, scaling the reflected path by the surface reflection coefficient R " # pffiffiffi e ðikÞs e ðikÞsþ i!t p ¼ 2s 0 p0 þR e : ð2:25Þ s sþ If R is real (implying a phase change on reflection of 0 or ) it follows that Fðsþ ; s Þ ¼

e 2s R 2 e 2sþ 2Re ðs þsþ Þ þ þ cosð2kxÞ; s sþ s 2 s 2þ

ð2:26Þ

where x

sþ  s 2z0 z ¼ : 2 s þ sþ

ð2:27Þ

Equation (2.26) can be interpreted as follows. The first and second terms on the right-hand side are associated with the energy from the direct and surface-reflected ray paths, respectively. The third term is due to interference between these two paths, resulting from the coherent addition of complex pressures. The expression is useful for broadband applications because the third term vanishes after averaging over frequency. An (equivalent) alternative version, convenient for narrowband applications, is

 x   e e x 2 4R ðs þsþ Þ 2 Fðsþ ; s Þ ¼ e þR  sin ðkxÞ : ð2:28Þ s sþ s sþ

36 Essential background

[Ch. 2

It is often the case that the distances sþ and s are approximately equal, such that the product x is sufficiently small to neglect the x terms, and sþ  s sþ þ s . Furthermore, for many applications the sea surface can be treated as a perfect reflector with a -phase change (i.e., R ¼ 1; see Box on p. 37). It then follows that   4e 2s 2 kz0 z Fcoh ðsÞ sin ; ð2:29Þ s s2 where s ¼ ðs þ sþ Þ=2: ð2:30Þ The sequence of sinusoidal peaks and troughs predicted by Equation (2.29) is known as a Lloyd mirror interference pattern. The ‘‘coh’’ subscript stands for ‘‘coherent addition’’, indicating that the two contributions to the total pressure, from the direct and reflected paths, respectively, are added with regard to their phase, before squaring. This means that the phase difference information is used for the purpose of combining the contributions from the two paths. Specifically, if Equation (2.25) is written in the form pffiffiffi p ¼ 2s0 p0 ðFþ þ F Þ; ð2:31Þ where F ¼

e ðikÞs i!t e s

ð2:32Þ

and Fþ ¼ R

e ðikÞsþ i!t e ; sþ

ð2:33Þ

it follows that Fcoh ¼ jFþ þ F j 2 :

ð2:34Þ

The ‘‘incoherent’’ propagation factor is obtained by discarding the phase terms. In other words Finc ¼ jFþ j 2 þ jF j 2 :

ð2:35Þ

Alternatively, averaging Fcoh over (say) frequency hFcoh i ¼ hjFþ j 2 i þ hjF j 2 i þ h2jFþ jjF j cosð2kxÞi:

ð2:36Þ

The first two terms are hardly affected by the average and together approximate to Finc . If the average is over several cycles of the cosine function, the third term can be expected to average out to zero, and hence hFcoh i Finc :

ð2:37Þ

For this reason, there are many situations in which a coherent sum (add and square), followed by an average over frequency, gives the same result as an incoherent sum (square and add).

Sec. 2.2]

2.2 Essentials of underwater acoustics

37

The reflection coefficient at the air–sea boundary The reflection coefficient at the sea surface is determined by the impedance of air relative to that of water. The characteristic impedance of air for the assumed representative conditions is given by Zair ¼ air cair ¼ 420 kg m 2 s 1 ; more than three orders of magnitude smaller than that of water, which is equal to Zwater ¼ water cwater ¼ 1:53  10 6 kg m 2 s 1 : The low impedance of air compared with that of water means that the acoustic pressure required to achieve a given acoustic intensity is much smaller in air than in water. From the continuity of pressure across the boundary it follows that the pressure on the boundary itself must also be small, and to first order this can be approximated by the boundary condition p ¼ 0 at z ¼ 0. The only way this can be achieved for an incident plane wave of finite amplitude in water is for a reflected wave to be generated at the surface of the same amplitude and opposite phase. Let the horizontal and vertical wave numbers be and , respectively, so that the incident wave can be represented by pincident ¼ e ið x zÞ e i!t and the reflected wave by preflected ¼ Re ið xþ zÞ e i!t : By adding these two terms it can be seen that the only way the total pressure pincident þ preflected can be zero everywhere on the z ¼ 0 boundary is if R ¼ 1. This has two important consequences. First, the unit magnitude corresponds to 100 % reflection of energy, so that sound becomes trapped in the sea, potentially traveling very long distances. Second, the negative sign means a phase change of the reflected wave relative to the incident one, which results in near-perfect cancellation of acoustic pressure close to the sea surface. In general, the reflection coefficient is a function of frequency, and of the physical properties of the reflecting boundary. For example, the sea surface reflection coefficient depends on the wave height and on the population of near-surface bubbles created by breaking waves (see Chapters 5 and 8.)

2.2.2.2

Radiation from an infinite sheet of uniformly distributed dipoles

One of the factors that limit sonar performance is the presence of background noise in the sea. Much of this background noise originates at the sea surface (e.g., due to breaking waves). Consider an infinitesimal patch of sea surface with surface area A and radiating power per unit surface area and solid angle WAO . The contribution from this patch to the mean square pressure at a receiver situated at a distance s (see

38 Essential background

[Ch. 2

Figure 2.4. Radiation from a sheet source element of width r.

Figure 2.4) is q 2 ¼ w cw WAO A

e 2s : s2

ð2:38Þ

The sea surface behaves like a sheet of dipoles,6 with a radiation pattern proportional to sin 2 , so that7 3 WAO ¼ W sin 2 ; ð2:39Þ 2 A where  is the ray grazing angle (the angle between the ray path and the horizontal), so that 3 e 2s q 2 ¼ WA A w cw 2 sin 2 : ð2:40Þ 2

s The solid angle O subtended by the surface element r at the receiver, for an azimuthal increment , is O ¼ cos    ð2:41Þ 6 A dipole source is one made out of two out-of-phase monopole sources, placed an infinitesimal distance apart (in practice, they must be separated by at most a small fraction of a wavelength). An important distinction between a monopole and a dipole is that in the case of the dipole source, there is no net change in volume. See Crocker (1997) for details. 7 The constant 3=ð2 Þ ensures thatÐ the power radiated per unit area, integrated over all solid angles (into the lower half-space) 2 WAO dO is WA . This follows from the use of dO ¼ cos  d d and the result ð 2

ð =2 d d sin 2  cos  ¼ 2 =3; 0

where  is the azimuth angle.

0

Sec. 2.2]

2.2 Essentials of underwater acoustics

39

where (Figure 2.4)  ¼

r sin  : s

ð2:42Þ

The corresponding element of area at the sea surface is A ¼ r r 

ð2:43Þ

s 2 O ; sin 

ð2:44Þ

so that A ¼

and hence (taking the limit of infinitesimal O)   dq 2 3 2z ¼  c W sin  exp  : dO 2 w w A sin 

ð2:45Þ

This is the mean square pressure per unit solid angle, at the receiver position. Now assume that the contribution to the pressure field from each patch of the surface, covering an infinitesimal solid angle dO, is uncorrelated with all other contributions. This means that the energy contributions (mean square pressures) may be added incoherently. The contribution from a concentric ring centered on the origin O is therefore obtained by replacing d with 2 , so that dO becomes 2 cos  d. The total mean square pressure is then found by integrating over  ð =2 2 dq q 2 ¼ 2

cos  d ¼ 3w cw WA E3 ð2zÞ; ð2:46Þ dO 0 where E3 ðxÞ is a third-order exponential integral (see Appendix A). For small arguments, the limiting form may be used lim E3 ðxÞ ¼ 12

x!0

ð2:47Þ

so that, in the case of negligible attenuation (z 1), Equation (2.46) becomes q 2 32 w cw WA :

ð2:48Þ

As an example, consider the radiation of sound from the sea surface, already identified as an important source of background noise. The acoustic power radiated by the sea surface due to wind (per unit area and bandwidth) can be written in the form8 (see Chapter 8 for details) WAf ¼ 8

2

K; 3w cw

ð2:49Þ

The subscripts A and f denote derivatives with respect to area and frequency such that dW dW d2W Wf  WA  WAf  : df dA dA df This is a generalization of the notation introduced previously for power and energy per unit solid angle dW dE WO  EO  . dO dO

40 Essential background

[Ch. 2

where the parameter K varies with frequency f and wind speed v. An empirical expression for K, based on measurements over a wide range of frequencies is K ¼ 1:32  10 4

^v 2:24 1:5 þ F 1:59

mPa 2 Hz 1

ð2:50Þ

where F is the frequency in units of kilohertz F

f 1 kHz

ð2:51Þ

and ^v is the wind speed in meters per second ^v 

v : 1 m/s

ð2:52Þ

Throughout this book, standard SI units and prefixes are used so that 1 mPa (one micropascal) is equal to 10 6 Pa. See Appendix B for a complete list of SI prefixes.

2.2.3 2.2.3.1

Scattering of sound Scattering from a small object

The likelihood that an echo from a distant object is detected depends on how much sound is reflected (i.e., scattered) in the direction of the receiving system. The ability of a small underwater object to scatter sound is quantified by its scattering crosssection, defined as the ratio of total scattered power W to incoming intensity I (from a specified grazing angle in ), of an incident plane wave ðin Þ ¼

W : Iðin Þ

ð2:53Þ

Thus,  is the scattered power per unit incident intensity and has dimensions of area. A related quantity is the differential scattering cross-section, proportional to the power per unit solid angle scattered in a specified direction9 (elevation out , and bearing  relative to that of the incident plane wave): O ðin ; out ; Þ ¼ 9

WO ðout ; Þ Iðin Þ

That is, the radiant intensity of the scattered sound.

ð2:54Þ

Sec. 2.2]

so that

2.2 Essentials of underwater acoustics

ð ðin Þ ¼ O ðin ; Oout Þ dOout ;

41

ð2:55Þ

where the shorthand Oout denotes the direction (out ; ) and dOout is an element of solid angle such that dOout ¼ cos out dout d: ð2:56Þ The backscattering cross-section is defined as the differential cross-section evaluated in the backscattering direction, multiplied by 4 . In equation form10 (Pierce, 1989; Morfey, 2001):11  back ðÞ  4 O ð; ; Þ:

ð2:57Þ

The backscattering cross-section of a rigid sphere of radius a at high frequency (ka  1), is (see Chapter 5)  back ¼ a 2 :

2.2.3.2

ð2:58Þ

Scattering from a rough surface

The echo from an underwater object can be masked by sound that happens to arrive at the same time, after it has been scattered from a rough boundary. For a rough surface, the scattered power is proportional to the area ensonified by the incident wave. In this situation it makes sense to define a (dimensionless) scattering coefficient as the differential scattering cross-section per unit scattering area; that is, OA ðin ; Oout Þ 

WOA ðOout Þ : Iðin Þ

ð2:59Þ

The parameter OA is also known as the scattering coefficient of a rough surface. Provided that the wind speed is low enough to neglect the influence of nearsurface bubbles, the scattering coefficient for the sea surface is approximately (except in directions close to that of specular reflection) (see Chapter 8) OA ðin ; out ; Þ

CPM tan 2 in tan 2 out ; 16

ð2:60Þ

where CPM is a constant equal to 0.0081. Often the scattering coefficient is written as a function of a single angle , in which case the backscattering direction is implied; 10 By ‘‘in the backscattering direction’’ is meant that the propagation direction of the scattered wave is taken to be the same as that of the incident wave, except that its sense is reversed. 11 An alternative definition of backscattering cross-section as the differential scattering crosssection in the backscattering direction (i.e., omitting the factor 4 from Equation 2.57) is sometimes used. In this book the definition of Equation (2.57) is used throughout. This point is discussed further in Chapter 5.

42 Essential background

[Ch. 2

that is, OA ðÞ  OA ð; ; Þ

CPM tan 4 : 16

ð2:61Þ

Substituting for the numerical value of CPM gives OA ðÞ 1:61  10 4 tan 4 :

2.3

ð2:62Þ

ESSENTIALS OF SONAR SIGNAL PROCESSING

When a sound wave reaches a sonar receiver, the acoustic pressure is first converted to an electrical voltage by an underwater microphone, or hydrophone. This voltage could be displayed on an oscilloscope and monitored for evidence of something other than background noise, such as the voltage exceeding some pre-established threshold. Such a simple system might work in practice for a strong signal, while a weak one would first need to be enhanced by signal processing. For example, the signal-to-noise ratio can be increased by filtering out sound at unwanted frequencies, or from unwanted directions or ranges, or a combination of these. Before any such enhancement begins, the electrical signal is passed through an anti-alias filter and then digitized.12 The details of subsequent processing depend on the characteristics of the expected signal, but almost all sonar systems use either a temporal filter to remove noise at unwanted frequencies or a spatial filter to remove noise from unwanted directions or both. No distinction is made in the following between acoustic and electrical signals. The justification for this is that the waveform, once digitized, can be rescaled by an arbitrary constant factor to represent either the voltage or the original pressure. A time domain filter operation involves sampling a waveform in time and combining successive samples in such a way as to remove any unwanted sound, whereas a spatial filter (or beamformer) samples in space instead of time. Both are described below, with the purpose of introducing some basic relationships between time duration and frequency bandwidth (Section 2.3.1) and between spatial aperture and beamwidth (Section 2.3.2). A more complete treatment of signal processing is presented in Chapter 6. 2.3.1

Temporal filter

Imagine a receiving system that passes frequencies between fmin and fmax and blocks frequencies outside this range. Such a system is called a passband filter of bandwidth Df  fmax  fmin . If fmin is zero it is a low-pass filter; if fmax is infinite it is a high-pass filter. General filter theory is beyond the present scope, but it is useful to introduce 12

An anti-alias filter is one that removes high-frequency signals above a threshold that depends on the sampling rate of the subsequent digital sampler. According to the Nyquist–Shannon sampling theorem, the maximum acoustic frequency that can be correctly sampled is half of the sampling rate. This maximum permissible frequency is known as the Nyquist frequency.

Sec. 2.3]

2.3 Essentials of sonar signal processing

43

some basic concepts. A special kind of filter of particular interest is a discrete Fourier transform (DFT),13 the basic properties of which are outlined below. Let FðtÞ denote the time domain waveform of interest, sampled at discrete times tn at fixed intervals t. The DFT of FðtÞ is the spectrum Gð!Þ (see Appendix A for details) N 1 X Gð!Þ  Fðtn Þ expði!tn Þ; tn ¼ t0 þ n t: ð2:63Þ n¼0

The inverse transform is the operation that recovers the original function FðtÞ from the spectrum at discrete frequencies !m : FðtÞ ¼

X 1 N1 Gð!m Þ expðþi!m tÞ; N m¼0

!m ¼

2

m: N t

ð2:64Þ

For the special case of simple harmonic time dependence of angular frequency ! it follows that

FðtÞ ¼ e i!t ;

ð2:65Þ

 Dt sin ð!  ! Þ N1 m X 2

; Gð!m Þ ¼ exp½ið!  !m Þtn  ¼ t n¼0 sin ð!  !m Þ 2

ð2:66Þ

where the time origin is chosen for convenience to be at the center of the sequence of time samples (such that t0 þ tN1 ¼ 0) and Dt is given by14 Dt ¼ N t;

ð2:67Þ

approximately equal to the signal duration. If the signal is well sampled in time (such that jð!  !m Þ tj 1), the denominator of Equation (2.66) may be approximated by the argument of the sine function. The spectrum is then given by Gð!m Þ N sincðyÞ;

ð2:68Þ

where y ¼ ð!  !m Þ

Dt 2

ð2:69Þ

and sincðyÞ is the sine cardinal function, defined (see Appendix A) as sincðyÞ 

sin y : y

ð2:70Þ

Written in this form it can be seen that the DFT operation, with output Gð!m Þ, is a passband filter, centered on !. The parameter Dt (the total time duration) determines 13

When the number of points in a DFT is a power of 2, a particularly efficient implementation is possible. This efficient implementation is also known as a fast Fourier transform (FFT). 14 The time between first and last samples is equal to Dt  t, which is approximately equal to Dt if t is assumed small.

44 Essential background

[Ch. 2

the frequency resolution of the filter through the argument of the sinc function. Specifically, the full width at half-maximum (fwhm), i.e., the spectral width between half-power points, is given by   4 1 1 pffiffiffi : !fwhm ¼ sinc ð2:71Þ Dt 2 Evaluation of the inverse sinc function gives pffiffiffi sinc 1 ð1= 2Þ 1:3916;

ð2:72Þ

so the frequency resolution, as defined by !fwhm ; 2

is approximately equal to the reciprocal of the total time duration Dt pffiffiffi 2 sinc 1 ð1= 2Þ 1 0:886 ffwhm ¼

:

Dt Dt ffwhm 

2.3.2

ð2:73Þ

ð2:74Þ

Spatial filter (beamformer)

Mathematically, there is no difference between spatial and temporal filtering, except that spatial filtering can be carried out in more than one dimension. For simplicity, the scope is limited here to a single dimension, so the expression for a spatial DFT can be obtained from Equations (2.63) and (2.64) by replacing the time variable ðtÞ with the spatial one ðxÞ. It is also customary to represent the spatial ‘‘frequency’’ (the wave number) variable by the symbol k. Thus, the forward and inverse transforms are, respectively, GðkÞ 

N1 X

Fðxn Þ expðikxn Þ;

xn ¼ x0 þ n x;

ð2:75Þ

n¼0

and FðxÞ ¼

X 1 N1 Gðkm Þ expðþikm xÞ; N m¼0

km ¼

2

m: N x

ð2:76Þ

A collection of hydrophones whose output is combined to carry out spatial filtering is known as a hydrophone array or, if it extends in only one dimension, a line array. Consider a pressure field whose spatial distribution along such an array is of the form FðxÞ ¼ e ikx :

ð2:77Þ

By an exact analogy with the time domain filter, if the origin is at the geometrical center of the array it follows that

 Dx sin ðk  km Þ 2

; Gðkm Þ ¼ ð2:78Þ x sin ðk  km Þ 2

Sec. 2.3]

2.3 Essentials of sonar signal processing

45

or (for sufficiently small hydrophone spacing x) Gðkm Þ N sincðyÞ;

ð2:79Þ

where y ¼ ðk  km Þ

Dx ; 2

ð2:80Þ

and the parameter Dx is Dx ¼ N x:

ð2:81Þ

Here, the distance Dx plays the role of Dt in the time domain filter, and determines the resolution of the spatial filter in the wavenumber domain. It is approximately equal to the length of the array. The approximation Equation (2.79) requires the signal to be well sampled in space such that jk  km j x 1: ð2:82Þ The fwhm in wave number, by analogy with Equation (2.71), is   4 1 kfwhm ¼ sinc 1 pffiffiffi : Dx 2

ð2:83Þ

The output wavenumber spectrum of the hydrophone array is referred to as the array response. The squared magnitude of the normalized array response for an incident plane wave, known as the beam pattern of the array, is   2  Gðkm Þ 2  ¼ sin y ðsinc yÞ 2 : B   ð2:84Þ  y Gmax N 2 sin 2 N Because of its ability to amplify selectively acoustic waves arriving from a narrow range of angles (a ‘‘beam’’), the spatial filter is called a beamformer. To see how the angle selection process works, consider an array aligned along the x-axis (y ¼ z ¼ 0) and an acoustic plane wave traveling in a direction parallel to the x–y plane. The field of this plane wave as a function of space and time can be written pðx; tÞ ¼ e ikEx e i!t ¼ expðikx xÞe iðky y!tÞ :

ð2:85Þ

Defining  as the angle between the wavenumber vector and the plane normal to the array axis, the along-axis wavenumber component is15 kx ¼

2

sin ; 

ð2:86Þ

where  is the acoustic wavelength. Specializing to the field along the array (y ¼ 0), Equation (2.85) becomes pðx; tÞ ¼ expðikx xÞe i!t :

ð2:87Þ

Now consider the field at an arbitrary instant in time (say t ¼ 0) and use this field as 15

Similarly, ky ¼ ð2 =Þ cos .

46 Essential background

[Ch. 2

Figure 2.5. Beam patterns for L= ¼ 5 and steering angles 0, 45 deg as indicated.

input to the beamformer, so that FðxÞ  pðx; 0Þ ¼ expðikx xÞ: The response is Equation (2.79), with   2

Dx y¼ sin   km :  2

ð2:88Þ

ð2:89Þ

If the magnitude of km does not exceed 2 =, there exists an arrival angle  at which the beamformer output is maximized, corresponding to y ¼ 0. This value of  is given by k m ¼ arcsin m ð2:90Þ 2 = and is known as the beam steering angle. It is measured from the direction perpendicular to the array axis, known as the broadside direction. Beam patterns for two different steering angles are shown in Figure 2.5. The angular width of the beam varies with steering angle as follows. Taking a finite difference of Equation (2.86) kx

2

cos  ; 

ð2:91Þ

the fwhm beamwidth is obtained by equating the right-hand sides of Equations (2.83)

Sec. 2.4]

2.4 Essentials of detection theory 47

and (2.91): fwhm

pffiffiffi 2 sinc 1 ð1= 2Þ  :

Dx cos m

ð2:92Þ

 Dx cos m

rad

ð2:93Þ

deg:

ð2:94Þ

In radians this is fwhm 0:886 and in degrees fwhm 50:8

 Dx cos m

This approximation for the beamwidth works best at angles close to the broadside direction. For the case of Figure 2.5, the predicted and observed half-power widths are about 10 deg at broadside ( ¼ 0) and 14 deg at ¼ 45 deg. The approximation breaks down at angles close to 90 deg from broadside (i.e., parallel to the array axis, known as the endfire direction), due to the singularity in the derivative d=dkx in that direction. The equation for wavenumber width (Equation 2.83) is valid at any angle (see Chapter 6).

2.4

ESSENTIALS OF DETECTION THEORY

The calculation of detection probability is the whole point of sonar performance modeling and the ultimate goal of this book. Hence, considerable attention is paid to its calculation. The end result of the processing, after all filtering, is presented to a sonar operator, whose job it is to report the detection or not of a (potential) sonar target, based on the information provided by the sonar. Depending on the signal-tonoise ratio, the probability of making a detection might be high or low, but it is never certain. The objective of statistical detection theory is to quantify this probability. 2.4.1

Gaussian distribution

In this section, expressions are derived for the probability of detection (denoted pd ) for a simple case involving a constant signal in Gaussian noise.16 The signal is represented by the constant xS and noise by the variable xN ðtÞ. The signal, if present, is always accompanied by a noise background, and the combination of both is represented by xSþN ðtÞ. The parameter x can be the amplitude or energy of an acoustic wave, depending on the processing, and is referred to below as the ‘‘observable’’. The two possibilities ‘‘signal present’’ and ‘‘signal absent’’ are represented by the total observable xtot given by either xtot ðtÞ ¼ xSþN ðtÞ

ðsignal presentÞ

ð2:95Þ

16 The choice of constant signal and Gaussian noise is for mathematical convenience and does not necessarily represent a realistic situation for a sonar system. More realistic distributions are considered in Section 2.4.2.

48 Essential background

[Ch. 2

Table 2.1. Detection truth table; pd is the probability of deciding correctly that a signal is present (‘‘detection probability’’) and pfa is the probability of declaring a detection when there is no signal. Threshold exceeded xtot > xT

Threshold not exceeded xtot < xT

Signal present xtot ¼ xSþN

Correct decision (probability pd )

Incorrect decision (probability 1  pd )

Signal absent xtot ¼ xN

Incorrect decision (probability pfa Þ

Correct decision (probability 1  pfa )

or xtot ðtÞ ¼ xN ðtÞ

ðsignal absentÞ:

ð2:96Þ

A decision-maker (the sonar operator) presented with the data sequence xtot ðtÞ deems a signal to be present whenever the value of xtot exceeds some threshold xT . To avoid too many false alarms it is desirable for the threshold xT to exceed the noise, in some average sense, but how should the noise be averaged, and by how much must this average be exceeded? The answer depends, among other things, on the rate of false alarms considered acceptable. The larger the threshold, the fewer false alarms will result, at the expense of a reduced probability of detection. Assuming that the operator must always choose between the two decisions ‘‘signal present’’ and ‘‘signal absent’’, irrespective of the chosen threshold there are always four possible outcomes, according to Table 2.1. Because of the statistical fluctuations in the noise there is always a chance that the threshold is exceeded when there is no signal, and conversely there is also a chance that the threshold is not exceeded even when the target is present. Both situations lead to an incorrect decision, indicated in the table by gray shading. The probability of making a correct ‘‘signal present’’ decision is known as the detection probability and denoted pd . The false alarm probability pfa is the probability of making an incorrect ‘‘signal present’’ decision. One’s objective is to make the correct decision as often as possible. In other words, to maximize pd , implying a low threshold, while at the same time minimizing pfa , which requires a high threshold. These conflicting requirements are resolved in practice by deciding in advance on a highest acceptable false alarm rate, and then determining the threshold consistent with this rate. Thus, the values of pd and pfa depend on the choice of xT as well as on the statistical fluctuations of noise and of signal þ noise. The following calculations assume a randomly fluctuating noise observable xN ðtÞ with Gaussian statistics, and a non-fluctuating signal so that the signal-plus-noise (xSþN ) has the same statistics (the same Gaussian distribution with the same standard deviation) as noise alone (xN ). Expressions for pfa and pd are derived below for these assumptions.

Sec. 2.4]

2.4.1.1

2.4 Essentials of detection theory 49

Noise only

Let the probability density function (pdf ) of the noise observable be fN ðxÞ, so that the mean and variance of the distribution are ð þ1 xN ¼ x fN ðxÞ dx ð2:97Þ 1

and 2 ¼

ð þ1

ðx  xN Þ 2 fN ðxÞ dx;

ð2:98Þ

1

respectively. The Gaussian distribution with these properties (see Figure 2.6, upper graph) is " # 1 ðx  xN Þ 2 fN ðxÞ ¼ pffiffiffiffiffiffi exp  : ð2:99Þ 2 2 2  2.4.1.2

Signal plus noise

Similarly, if the non-fluctuating signal is added " # 1 ðx  xSþN Þ 2 fSþN ðxÞ ¼ pffiffiffiffiffiffi exp  ; 2 2 2 

ð2:100Þ

illustrated by the lower graph of Figure 2.6. Assuming that the observable terms add linearly, if the signal is constant, the signal-plus-noise can be written xSþN ðtÞ ¼ xS þ xN ðtÞ and therefore xSþN 

ð þ1

x fSþN ðxÞ dx ¼ xS þ xN :

ð2:101Þ ð2:102Þ

1

Thus, the signal-plus-noise distribution has the same pdf as the noise alone but with a higher mean value. Suppose that a detection is declared by the operator whenever the threshold xT is exceeded. The probability of this occurring as the result of a single observation is equal to the area under the pdf curve to the right of the threshold. This area, depending on whether in reality a signal is absent or present, is either pfa or pd . In other words, respectively,   ð1 1 xT  xN pffiffiffi pfa ¼ fN ðxÞ dx ¼ erfc ð2:103Þ 2 2 xT where erfcðxÞ is the complementary error function (Appendix A), or   ð1 1 xT  xSþN pffiffiffi pd ¼ fSþN ðxÞ dx ¼ erfc : 2 2 xT

ð2:104Þ

In this treatment, a large negative result is arbitrarily not considered a threshold crossing. This choice might be justified if the observable is a positive definite quantity, such that the negative tail of the Gaussian has no physical meaning. If large negative

50 Essential background

[Ch. 2

Figure 2.6. Probability density functions of noise (upper graph) and signal-plus-noise (lower) observables. The threshold for declaring a detection, xT , is shown as a vertical dashed line. The shaded areas are the probabilities of false alarm (upper graph) and detection (lower). The example shown is for the case xN ¼ 2, xSþN ¼ 5, and xT ¼ 4.

Sec. 2.4]

2.4 Essentials of detection theory 51

values were considered to be threshold crossings, the expressions for both pfa and pd would then need to include contributions from values of x between 1 and xT . Both pfa and pd vary between 0 and 1. It is convenient to replace xT in Equation (2.104) by expressing it as a function of pfa (from Equation 2.103). The result is

 1 xS 1 pd ¼ erfc erfc ð2pfa Þ  pffiffiffi : ð2:105Þ 2 2

2.4.2

Other distributions

The analysis of sonar detection problems requires the consideration of more complicated distributions than that of a constant signal in Gaussian background noise. A preview of some important results from Chapter 7 is presented below. In each case, expressions are quoted for the false alarm probability pfa and detection probability pd as a function of the signal-to-noise ratio (SNR). 2.4.2.1

Coherent processing (Rayleigh statistics)

Coherent processing for Gaussian noise results in a Rayleigh distribution for the noise amplitude A. For an amplitude threshold AT , and assuming a Rayleigh distribution for the signal as well as for the noise, the false alarm and detection probabilities are ! A 2T pfa ¼ exp  2 ð2:106Þ 2 and 1=ð1þRÞ

pd ¼ pfa

;

ð2:107Þ

where R is the SNR R¼

2.4.2.2

A2 : 2 2

ð2:108Þ

Incoherent processing (chi-squared statistics with many samples)

Incoherent addition of a number of Rayleigh-distributed samples results in a chisquared (or ‘‘ 2 ’’) distribution for the total energy. The detection and false alarm probabilities can be expressed for this distribution in terms of special functions as described in Chapter 7. If the number of samples M is sufficiently large (M > 100), these expressions simplify to those presented below. For an energy threshold ET , the false alarm probability becomes "rffiffiffiffiffi # 1 M ET pfa erfc 1 ; ð2:109Þ 2 2 2M 2 where  is the standard deviation of the noise samples before any averaging. The

52 Essential background

[Ch. 2

detection probability simplifies to

" pffiffiffiffiffiffiffiffiffiffi # erfc 1 ð2pfa Þ  M=2R 1 pd erfc ; 2 1þR

ð2:110Þ

where R is the power signal-to-noise ratio. Equation (2.110) can be rearranged for R: R¼

erfc 1 ð2pfa Þ  erfc 1 ð2pd Þ pffiffiffiffiffiffiffiffiffiffi : M=2 þ erfc 1 ð2pd Þ

ð2:111Þ

If M 1=2 is large compared with erfc 1 ð2pd Þ, this simplifies further to R

erfc 1 ð2pfa Þ  erfc 1 ð2pd Þ pffiffiffiffiffiffiffiffiffiffi : M=2

ð2:112Þ

The condition on M makes Equation (2.112) mainly relevant to situations involving a low SNR. If R and M are both large, there is usually no need for a detailed analysis, because in this situation the detection probability is always close to unity.

2.5

REFERENCES

Clarke, G. L. and James, H. R. (1939) Laboratory analysis of the selective absorption of light by sea water, J. Opt. Soc. Am., 29, 43–55. Crocker, M. J. (1997) Introduction, in M. J. Crocker (Ed.), Encyclopedia of Acoustics, Wiley, New York. Morfey, C. L. (2001) Dictionary of Acoustics, Academic Press, San Diego. Pierce, A. D. (1989) Acoustics: An Introduction to Its Physical Principles and Applications, American Institute of Physics, New York.

3 The sonar equations

If you cause your ship to stop, and place the head of a long tube in the water, and place the other extremity to your ear, you will hear ships at a great distance from you Leonardo da Vinci (15th century).

3.1

INTRODUCTION

The objective of this chapter is to illustrate the basic principles of sonar performance modeling. This is achieved by deriving the most important passive and active sonar equations, each accompanied by a worked example. These worked examples are intended to be didactic rather than realistic: enough realism is included in them to illustrate the underlying principles, but no more—where there is a conflict between simplicity and realism then simplicity is preferred, except at the expense of the principle itself.1 3.1.1

Objectives of sonar performance modeling

The objective of sonar performance modeling is to quantify sonar performance, enabling a decision-maker to: — predict the likelihood that a given sonar task, such as the detection of a submerged object, will be carried out successfully; 1

More realistic examples are provided in Chapter 11.

54 The sonar equations

[Ch. 3

— compare the effectiveness of different sonar designs in carrying out a given task; — compare the effectiveness of different strategies for carrying out a given task. Examples of possible sonar tasks, in addition to detection, are localization, classification, evasion,2 surveillance, and communication. Irrespective of the application, sonar effectiveness must depend on the probability of making a successful detection each time the sonar is used. Less obvious, but equally important, is the observation that sonar effectiveness also depends on the number of false alarms,3 because of the time and other resources wasted on investigating these. Much of sonar performance modeling, and the main thrust of this book, is concerned with the calculation of the probabilities of detection and false alarm for a given scenario or scenarios. 3.1.2

Concepts of ‘‘signal’’ and ‘‘noise’’

A sonar receiver is a complicated piece of equipment, typically comprising — a hydrophone, or an array of hydrophones, to convert an underwater pressure disturbance into an electronic one; — a suite of signal-processing algorithms,4 to enhance the signal-to-noise ratio; — a display unit, to help the sonar operator determine whether an object of interest (a target) is present. Pressure fluctuations5 at the receiver can be thought of as a linear sum of two different kinds: — those caused by the presence of the target (the signal); — all other pressure fluctuations (the noise). The ‘‘target’’ is any object that we wish to detect. The above definitions of signal and noise are necessarily vague, as the distinction between them depends on details of the signal processing that have not yet been specified. The noise definition as ‘‘all sound that is not part of the signal’’ means that 2

Although the task of evasion is not performed directly by the sonar, the modeling of evasion and the development of evasion tactics are nevertheless an important application of sonar performance modeling. 3 Fluctuations in noise levels alone can result in a sonar detection system erroneously reporting the presence of a target. Such an event is known as a ‘‘false alarm’’. 4 The algorithms can be implemented either in hardware or software. 5 Strictly speaking, what matters is the voltage in an electrical or electronic circuit, after filtering. For simplicity we assume for now that, to within a multiplying constant, the pressure and voltage fluctuations are identical. The validity of this assumption requires the hydrophone sensitivity and filter response (or at least their product) to be independent of frequency, within the bandwidth of interest.

Sec. 3.1]

3.1 Introduction

55

many different potential sound sources need to be taken into account.6 Each case is different, and knowledge of which noise sources to include in a model is acquired by experience. Common sources of ambient noise are wind and shipping. Also important is self-noise, especially from the sonar platform. A special kind of noise that is unique to active sonar, known as reverberation, is the sound originating from the sonar transmitter and subsequently scattered by underwater boundaries and obstacles other than the target, before arriving back at the receiver. The combined effect of ambient noise, self-noise, and reverberation is known as the background. The sonar equation is an expression for the signal-to-noise ratio (or more generally signal-to-background ratio) written as a product of energy ratios, and usually expressed in decibels. The conversion to decibels turns the product of ratios into a sum of the logarithms of these ratios.

3.1.3

Generic deep-water scenario

For the purpose of the present chapter, attention is restricted to a specific deep-water scenario, in which the following simplifying assumptions and approximations are made: — reflections from the seabed are neglected, equivalent to assuming an infinite water depth; — density and sound speed are assumed to be uniform everywhere in the sea; — all background noise (including reverberation) is assumed to originate at the sea surface. The scenario resulting from these assumptions is not a realistic one, but it contains enough realistic features (the reflecting sea surface, a source of reverberation, and some basic signal processing) to illustrate the main principles involved. More realistic applications, without these simplifying assumptions, are described in Chapter 11. A sonar equation is derived for each main category of sonar, followed by a worked example. These examples make some additional assumptions that serve to simplify the calculations. For example, for passive sonar the background noise is assumed to arise only from wind; and the target is assumed to be located in the broadside beam of a horizontal line array (a sequence of hydrophones placed along a straight horizontal line).

3.1.4

Chapter organization

The remainder of this chapter is divided into two main sections, one on passive sonar (Section 3.2) and one on active sonar (Section 3.3). These sections are further divided into sub-sections concerned with coherent and incoherent processing. In each of the 6

In some situations non-acoustic sources of noise can also be important.

56 The sonar equations

[Ch. 3

four sub-sections the relevant sonar equation is derived, and illustrated by means of a worked example. 3.2 3.2.1

PASSIVE SONAR Overview

This section is concerned with analysis of the performance of a passive sonar system, which relies on detecting sounds emitted by an underwater object (the ‘‘target’’). In the situation illustrated by Figure 3.1, the whale on the left (whale ‘‘A’’, representing the target) emits a communication signal that is detected by the one on the right (whale ‘‘B’’, representing the sonar). A stylized radiated power spectrum, comprising a series of lines (tonals) superimposed on a smoothly varying background, is illustrated by Figure 3.2 (upper panel ). Collectively, the tonals are referred to as the narrowband spectrum of the target, whereas the smooth background is its broadband spectrum. The source of sound (i.e., the target) is assumed here to radiate sound continuously and uniformly in all directions (omni-directionally). The signal is further assumed to be infinite in duration and statistically stationary. The acoustic signal is transmitted through the propagating medium (seawater) and can be detected by means of suitable receiving equipment, such as the human ear or an underwater hydrophone. Some frequencies propagate better than others through the same medium, so the received spectrum, while retaining the same qualitative features as the transmitted one, has a different shape, illustrated by Figure 3.2 (lower panel ). By the time it arrives at the receiver, the sound power radiated by the target has spread from a point to a finite area, so the physical parameter of interest is power per unit area (i.e., intensity, instead of power). Thus, the lower panel shows the spectral density of sound intensity at the receiver. In principle both narrowband and broadband spectra can be exploited by a suitable detector, requiring different processing techniques to do so. Both are considered below, after the definition of some standard nomenclature.

Figure 3.1. Principles of passive detection: the sonar target (whale A) emits a sound wave that travels through the sea and is detected by the sonar receiver (whale B).

Sec. 3.2]

3.2 Passive sonar 57

Figure 3.2. Spectral density level of the radiated power at the source (upper) and intensity at the receiver (lower).

58 The sonar equations

3.2.2 3.2.2.1

[Ch. 3

Definition of standard terms (passive sonar) Mean square pressure, sound pressure level, and the decibel

The concept of mean square (acoustic) pressure (abbreviated MSP) is one of key importance to sonar performance modeling and plays a central role in this book. For acoustic pressure qðtÞ and averaging time T, the mean square pressure Q is defined as ð 1 T 2 Q q ðtÞ dt: ð3:1Þ T 0 The result of Equation (3.1) (MSP) is independent of T for large T if qðtÞ is statistically stationary in time. Another important parameter is the acoustic intensity I, a vector quantity whose magnitude I, for a plane-propagating wave, is equal to the MSP (Q) divided by the characteristic impedance of the medium I  jIj ¼

Q c

(plane wave).

ð3:2Þ

For types of wave other than a plane wave, the relationship between intensity and MSP is a more complicated one, but one can define the equivalent plane wave intensity (EPWI) of any statistically stationary pressure field as the intensity of a plane wave of the same MSP. In other words, denoting this quantity IEPWI , IEPWI 

Q c

(any pressure field).

ð3:3Þ

In some publications the term ‘‘intensity’’ is used as a synonym for EPWI, and the sonar equation is expressed as a product of ‘‘intensity’’ ratios. In the following, a deliberate choice is made to characterize sound waves by their MSP and not EPWI, as this avoids ambiguities associated with the choice of impedance, and is consistent with the definition of propagation loss in common use (see Appendix B). The distinction becomes an important one if the impedance at the source is different from that at the receiver. Except in some special situations, neither MSP nor EPWI are proportional to the true acoustic intensity. The sonar equations derived in this chapter and in Chapter 11 are all expressed in terms of MSP ratios and do not rely on any particular relationship between sound pressure and intensity. Sound pressure level is the mean square pressure expressed in decibels. The decibel is a logarithmic unit of power or energy (see Appendix B for details). The conversion to decibels involves the following three operations: divide by a standard reference value of the parameter, take the base-10 logarithm, and multiply by 10. Thus, the sound pressure level (SPL) is: SPL  10 log10

Q : p 2ref

ð3:4Þ

The reference value of MSP is p 2ref and the internationally accepted value of pref for use in underwater acoustics is one micropascal (1 mPa).

Sec. 3.2]

3.2 Passive sonar 59

Consider a point source at the origin. Regardless of the acoustic environment and source–receiver geometry, the mean square pressure Q at an arbitrary location x can always be written in the form QðxÞ ¼ p 20 s 20 FðxÞ;

ð3:5Þ

where FðxÞ is the propagation factor, which is defined by Equation (3.5); and p0 is the RMS pressure at a small distance s0 from the source.7 It is conventional to write Equation (3.5) in logarithmic form ! Q p 20 s 20 F 10 log10 2 ¼ 10 log 2 2 þ 10 log10 2 ; ð3:6Þ p ref p ref r ref r ref where rref is a constant reference distance, with an internationally accepted value of 1 meter (1 m). The left-hand side of Equation (3.6) is the sound pressure level. It is conventional to present all sonar equation terms in decibels (dB). Using the above recipe, a power-like quantity x is converted to its dB equivalent Lx by dividing by its reference unit xref and applying the relationship Lx  10 log10

x : xref

ð3:7Þ

For this reason the combination 10 log10 appears repeatedly. Often, for notational convenience, the denominator of the argument is omitted, in which case the reference value is quoted instead as a qualifier to the dB unit. For example, the meaning of X  10 log10 x

dB re xref

ð3:8Þ

is identical to that of Equation (3.7). Thus, the power level of a 1-watt source is 10 log10 (1 W/1 pW) ¼ 120 dB re pW. Similarly, a sinusoidal pressure wave of amplitude 1 Pa has a mean square pressure of 0.5 Pa 2 , and consequently a sound pressure level of 10 log10 [0.5 Pa 2 /(1 mPa) 2 ] ¼ 117.0 dB re mPa 2 . The choice of mPa 2 as a reference unit for SPL, adopted here in preference to the more conventional mPa, follows naturally from Equation (3.8) (or Equation 3.7) with x equal to the mean square pressure. In general, a level quoted in decibels needs to be accompanied by a statement of the corresponding reference unit, even if an international standard exists for its value. This is partly because standards can, and do, change with time and circumstances and partly because not all users of decibels adhere to these standards.8 The only safe 7

Equation (3.5) holds for a point source. The distance s0 from the source at which p0 is measured must be small enough for distortions due to absorption, refraction, reflection, or diffraction to be negligible. 8 In water, reference pressures of 20 mPa and 1 mbar were both used before the modern value of 1 mPa became widespread. At the time of writing, a reference pressure of 1 mbar (and a reference distance of 1 yd) is still in use by at least one sonar manufacturer. Further, the reference pressure used for sound in air is 20 mPa, not 1 mPa, making it unclear which value to use in situations involving a mixture of air and water, such as in foam caused by breaking waves.

60 The sonar equations

[Ch. 3

exception to this general rule occurs with dimensionless quantities, for which it seems reasonable to assume a reference unit of 1. 3.2.2.2

Source level

On the right-hand side of Equation (3.6) there are two terms. The first, a measure of source power, is known as the source level SL  10 log10 S0

dB re mPa 2 m 2 ;

ð3:9Þ

where S0 is the product S0 ¼ p 20 s 20 :

ð3:10Þ

It is remarkable that a parameter of such fundamental importance to sonar as S0 does not have a widely accepted name. The term source factor is adopted here.9 It is more conventional to define source level as the sound pressure level at a standard reference distance (rref ) from the source (ASA, 1994; IEC, www). For a point source in free space, the numerical value is the same provided that rref is the unit distance in whatever units system is used (i.e., 1 meter in the SI system), but the conventional definition leads to difficulties for an extended source such as a ship or an array of sonar projectors (see Chapter 11). Because p0 varies with distance in such a way that p0 s0 is constant, the source factor is also a constant, independent of measurement position close to the source, making it a natural physical quantity with which to characterize the source. Although source level is defined in terms of pressure p0 , due to the s0 scaling (through Equation 3.10), in practice it is actually a measure of radiated power. Thus, an alternative expression for the source factor is S0 ¼ cWO ;

ð3:11Þ

where WO is the radiant intensity (power per unit solid angle). For an omni-directional source of power W, the source factor is equal to cW=4. For example, if the source power is 1 watt (W ¼ 1 W), then from Equation (3.9), S0 ¼ 0.122 kPa 2 m 2 , corresponding to a source level of 170.9 dB re mPa 2 m 2 .10 3.2.2.3

Propagation loss

The second term on the right-hand side of Equation (3.6) is (minus) the propagation loss11 PL  SL  SPL ¼ 10 log10 FðxÞ dB re m 2 ; ð3:12Þ 9

The combination p0 s0 , which is the square root of the source factor, is referred to by ASA (1989) as the ‘‘source product’’. 10 For the assumed conditions (T ¼ 10  C, S ¼ 35, at atmospheric pressure). 11 In underwater acoustics a synonymous term to ‘‘propagation loss’’ is ‘‘transmission loss’’. The term ‘‘propagation loss’’ is used here to avoid possible confusion with alternative definitions of ‘‘transmission loss’’ from other branches of acoustics such as sound transmission through a wall (Morfey, 2001).

Sec. 3.2]

3.2 Passive sonar 61

or equivalently, PL ¼ 10 log10

S0 Q

dB re m 2 :

ð3:13Þ

This term plays the role of transfer function between source and receiver. The ratio S0 =Q has dimensions of area, so the unit of propagation loss is dB re m 2 . 3.2.2.4

Noise spectrum level and array response

The mean square pressure of background noise within a specified bandwidth (usually the processing bandwidth of the sonar receiver) is denoted Q N . Thus, the SPL of background noise in the same bandwidth, known as the noise level, is NL  10 log10 Q N

dB re mPa 2 :

ð3:14Þ

This background noise is often broadband in nature, so it is useful to consider noise spectral density (denoted Q N f ) and the corresponding noise spectral density level (or noise spectrum level ) NLf , defined as12 NLf  10 log10 Q N f

dB re mPa 2 Hz 1 :

ð3:15Þ

The background against which the signal is to be detected is that at the output of the beamformer (i.e., the array response, denoted Y N ), not at the hydrophone. The spectral density of this quantity (denoted Y N f ) can be written as the integral of the noise spectral density over all solid angles O, weighted by the beam pattern BðOÞ: ð N Y f ¼ QN ð3:16Þ f O ðOÞBðOÞ dO: For the special case of isotropic noise, meaning that the magnitude of noise spectral density is independent of direction such that QN fO ¼

QN f ; 4

ð3:17Þ

it follows that N YN f ¼ Qf

O ; 4

ð3:18Þ

where O is the solid angle footprint of the beam pattern (in steradians), defined by ð O ¼ BðOÞ dO: ð3:19Þ 12 The use of the subscript f for logarithmic quantities (expressed in decibels) indicates a spectrum level (i.e., a spectral density expressed in decibels).

62 The sonar equations

3.2.2.5

[Ch. 3

Signal-to-noise ratio, array gain, and directivity index

Now consider a monochromatic signal13 whose MSP per unit solid angle is Q SO . By analogy with Equation (3.16), the array response to this signal (denoted Y S ) is ð Y S ¼ Q SO BðOÞ dO: ð3:20Þ Assuming further that this signal is in the form of an incoming plane wave from the direction O S ¼ ð S ;  S Þ, such that Q SO ¼ Q S ðO  O S Þ;

ð3:21Þ

where ðxÞ is the Dirac delta function (see Appendix A), the array response is then ð Y S ¼ Q S ðO  O S ÞBðOÞ dO ¼ Q S BðO S Þ: ð3:22Þ The value of the signal-to-noise ratio (SNR) depends on where it is measured in the processing chain. In particular, its value after beamforming (denoted Rarr ) is different from its value at the hydrophone, before any processing (Rhp ). The ratio of these two SNR values, expressed in decibels, is the array gain; that is, AG  10 log10 GA

dB;

ð3:23Þ

where GA ¼

Rarr : Rhp

ð3:24Þ

Here, the hydrophone SNR (Rhp ) is defined as the ratio of signal mean square pressure (MSP) to noise MSP at the receiving hydrophone Rhp 

QS ; QN

ð3:25Þ

where Q N is the noise MSP integrated over the sonar processing bandwidth ð QN ¼ QN ð3:26Þ f df : Similarly Rarr is the SNR at the output of the beamformer, for a ‘‘flat response filter’’ (i.e., one whose response is independent of frequency within a specified passband14 and zero everywhere else) YS Rarr  N : ð3:27Þ Y A related parameter is the directivity index (DI) of an array, which is the array gain for the special case of a plane wave signal and isotropic noise. Unlike AG, DI is a property of the array and the acoustic frequency only. Thus, it is independent of 13

The single frequency assumption is relaxed in Section 3.2.4.4. Unless otherwise stated, this passband is understood to be the processing bandwidth (denoted f or Df depending on context). 14

Sec. 3.2]

3.2 Passive sonar 63

medium and target properties and is usually easier to calculate. For this reason, DI is often used as an approximation to AG in the sonar equation. 3.2.2.6

Signal gain and noise gain

The array gain can be expressed in terms of the signal gain SG: SG  10 log10

YS QS

dB

ð3:28Þ

NG  10 log10

YN QN

dB;

ð3:29Þ

and noise gain NG:

such that AG ¼ SG  NG:

ð3:30Þ

According to these definitions, SG and NG are both negative. For a well-designed beamformer, the magnitude of NG is usually greater than that of SG, in which case AG is positive. 3.2.2.7

Detection threshold and signal excess

The SNR threshold R50 is the value of Rarr (more generally, that of the SNR after all processing) required to achieve a detection probability of precisely 50 %.15 (The value of R50 depends on statistical fluctuations present in both signal and noise, and on the false alarm probability). When expressed in decibels, this quantity is known as the detection threshold DT ¼ 10 log10 R50 dB: ð3:31Þ Unlike the amplitude threshold AT (in volts, or pascals) introduced in Chapter 2, which is the value of the signal þ noise amplitude in volts, or pascals (or energy threshold ET in V 2 s or Pa 2 s) above which a ‘‘signal present’’ decision is triggered, the detection threshold (R50 ) is a dimensionless parameter. The signal excess is defined as the amount by which the SNR exceeds the detection threshold, in decibels: SE  10 log10 Rarr  DT:

ð3:32Þ

It follows from this and from the definition of array gain that SE ¼ 10 log10 Rhp þ AG  DT:

ð3:33Þ

Equation (3.33) is the sonar equation. Written like this it looks simple, and in some special circumstances it is. To a large extent, the purpose of this book is to explain how to calculate each of the terms on the right-hand side and corresponding ones for active sonar (see Section 3.3). The examples and special cases considered in the remainder of Chapter 3 are deliberately simplified in order to illustrate the main principles involved. 15 The symbol X50 is used to denote the value of any variable X required to achieve a detection probability of 50 %.

64 The sonar equations

3.2.3

[Ch. 3

Coherent processing: narrowband passive sonar

This section is concerned with the calculation of the probability of detection ( pd ) for a narrowband passive sonar. The term ‘‘narrowband’’ (abbreviated ‘‘NB’’) implies that the signal may be described, to a first approximation, by sound of a single frequency (see Figure 3.3). The processing considers a very narrow range of frequencies at a time, thus minimizing the noise in each processing band. The bandwidth is then assumed to be large enough to contain the entire signal, but sufficiently small for the noise power to be directly proportional to the bandwidth. Under these circumstances the signal-to-noise ratio, and hence also the detection probability, increase with decreasing bandwidth. A narrowband signal is called a tonal because of its resemblance to a single frequency tone in music. The following sub-sections look first at the sonar signal, then the background noise, signal-to-noise ratio, and finally the probabilities of detection and false alarm. A special case involving a horizontal line array is introduced in Section 3.2.3.7, followed by a worked example for this special case (Section 3.2.3.8). 3.2.3.1

Signal (single hydrophone)

For a NB system the sonar signal is the received mean square pressure Q S associated with one of the transmitted tonals (red line in Figure 3.3). With the assumption of an infinite water depth, there are two contributions to the received signal, one from the direct path and one from a surface reflection. We assume that the sea surface is smooth, so that the direct and surface-reflected paths add coherently. If the tonal power is W S it follows from Chapter 2 that Q S ¼ c where

zarr is ztgt is r is FNB is

WS F ðr; zarr ; ztgt Þ; 4 NB

ð3:34Þ

array depth; target depth; horizontal separation between array and target; and the coherent propagation factor.

Assuming that the horizontal separation r is large compared with the product zarr ztgt , where is the attenuation coefficient, this can be written:   4 2 r 2 kzarr ztgt FNB ¼ Fcoh ðr; zarr ; ztgt Þ 2 e sin : ð3:35Þ r r The sine-squared behavior in Equation (3.35) is a result of alternate constructive and destructive interference between the two paths, illustrated in Figure 3.416 for a frequency of 300 Hz. The separation between successive peaks (or troughs) is r=kzarr in depth and r 2 =kzarr ztgt in range. This fringe pattern is known as a Lloyd mirror interference pattern after an analogous effect from optics. For the example 16 This graph and many subsequent ones, as acknowledged in the individual captions, are calculated using the sonar performance model INSIGHT (Ainslie et al., 1996).

Sec. 3.2]

3.2 Passive sonar 65

Figure 3.3. Spectral density level of the transmitter source factor (upper) and mean square pressure at the receiver (lower).

66 The sonar equations

[Ch. 3

Figure 3.4. Coherent propagation loss PL ¼ 10 log10 ðFcoh Þ [dB re m 2 ] vs. range r and target depth ztgt for array depth zarr ¼ 30 m and frequency f ¼ 300 Hz (INSIGHT).

shown (the array depth is 30 m), at a range of 300 m the fringe spacing in depth is approximately 25 m. 3.2.3.2

Noise (single hydrophone)

We consider a broadband noise spectrum illustrated by Figure 3.5. Conceptually, the total noise entering the processing bandwidth f is the area under the curve between the dashed lines; that is, Q N ðzÞ ¼ fQ N f ðzÞ;

ð3:36Þ

where Q N f is the spectral density of the ambient noise MSP. For the isovelocity case, with infinite water depth, this can be written (see Chapter 2) N QN f ðzÞ 3cE3 ð2 zÞW Af ;

ð3:37Þ

where W N Af is the power spectral density of the noise source per unit of sea surface area, and E3 ðxÞ is a third-order exponential integral (see Appendix B). Here, the processing bandwidth is the analysis bandwidth of the Fourier transform, equal to the reciprocal of the coherent processing time.

Sec. 3.2]

3.2 Passive sonar 67

Figure 3.5. Spectral density level of background noise. The noise term Q N is the contribution in the processing bandwidth, marked f .

3.2.3.3

Signal-to-noise ratio, signal excess, and narrowband passive sonar equation

To derive the NB sonar equation we start by calculating the signal-to-noise ratio for the array (rearranging Equation 3.24 for Y S =Y N and substituting the result in Equation 3.27) as

Rarr ðr; zarr ; ztgt Þ 

YS QS ¼

GA : Y N QN

ð3:38Þ

Recall that the ratio Q S =Q N is the SNR measured at a hydrophone, before any processing apart from initial filtering into the frequency band f (see Figure 3.6). It follows from Equations (3.34), (3.36), and (3.11) that

Rarr ðr; zarr ; ztgt Þ ¼

S0 FNB ðr; zarr ; ztgt Þ GA

: f QN f ðzarr Þ

ð3:39Þ

Recall from Section 3.2.2.7 that the threshold R50 is the SNR required, after spatial and temporal filtering, to achieve a detection probability of 50 %. The sonar equation is obtained by dividing Equation (3.39) through by R50 , and converting to decibels in the usual way. Specifically, the left-hand side becomes the signal excess (SE), defined

68 The sonar equations

[Ch. 3

Figure 3.6. Spectral density level of signal 10 log10 Q Sf (red) and noise 10 log10 Q N f (cyan).

as the ratio by which the SNR exceeds R50 , expressed in decibels SENB  10 log10

Rarr : R50

ð3:40Þ

Thus, SENB ðr; zarr ; ztgt Þ ¼ ½SL  PLðr; zarr ; ztgt Þ  ½NLf ðzarr Þ ðAG  BWÞ  DT;

ð3:41Þ

where ðsource level: dB re mPa 2 m 2 Þ;

ð3:42Þ

ðpropagation loss: dB re m 2 Þ;

ð3:43Þ

2 NLf ðzarr Þ  10 log10 Q N f ðzarr Þ ðnoise spectrum level: dB re mPa =HzÞ;

ð3:44Þ

SL ¼ 10 log10 S0 PLðr; zarr ; ztgt Þ ¼ 10 log10 FNB ðr; zarr ; ztgt Þ

AG  10 log10 GA

ðarray gain: dBÞ;

ð3:45Þ

DT ¼ 10 log10 R50

ðdetection threshold: dBÞ;

ð3:46Þ

ðbandwidth: dB re HzÞ:

ð3:47Þ

and BW ¼ 10 log10 f

Equation (3.41) is the NB passive sonar equation in logarithmic form. The processing

Sec. 3.2]

3.2 Passive sonar 69

bandwidth term BW, originating from the denominator of Equation (3.39), is grouped for convenience with the array gain.17 The figure of merit (FOM) is defined as the propagation loss at which the detection probability is 50 % (i.e., FOM  PL50 ). From this definition it follows that FOMNB ðzarr Þ ¼ SL þ ðAG  BWÞ  NLf ðzarr Þ  DT

ð3:48Þ

and the sonar equation is then SENB ðr; zarr ; ztgt Þ ¼ FOMNB ðzarr Þ  PLðr; zarr ; ztgt Þ: 3.2.3.4

ð3:49Þ

Array gain and directivity index for a horizontal line array

The array gain for a horizontal line array (neglecting the signal gain) is given by GA ¼ ð

QN QN O BðOÞ

;

ð3:50Þ

dO

where (if the hydrophone spacing is small compared with the acoustic wavelength) BðOÞ ¼

sin 2 u u2

ð3:51Þ

and, expressing the direction in terms of the spherical co-ordinates  (elevation) and  (bearing) k Dx cos  sin  u ¼ uð; Þ ¼ ; ð3:52Þ 2 where Dx is the array length. Given that for a sheet dipole source, Q N O is proportional to sin  (see Chapter 2), and using dO ¼ cos  d d, it follows that ð =2 ð 2 d sin  cos  d 0 0 GA ¼ ð =2 ð3:53Þ  : ð 2  sin uð; Þ 2 d sin  cos  d uð; Þ 0 0 The numerator of Equation (3.53) is , and its denominator can be simplified by defining the horizontal beamwidth DðÞ as  ð 2  sin uð; Þ 2 DðÞ  d ; ð3:54Þ uð; Þ 0 17

An alternative convention is to group f instead with the narrowband detection threshold so that DTNB would be given instead as 10 log10 ðfR50 Þ, with implied units dB re Hz. The convention of Equation (3.41) is preferred because it makes it possible to standardize on a single definition of DT (as 10 log10 R50 ) for both coherent and incoherent processing, as well as for both passive and active sonar.

70 The sonar equations

[Ch. 3

so that 2

GA ¼ ð =2

:

ð3:55Þ

sin 2 DðÞ d 0

The integrand of Equation (3.54) is periodic in , with period , which means that the beamwidth can be written   ð =2 sin u 2 D ¼ 2 d : ð3:56Þ u =2 The beamwidth calculation can be further simplified by replacing the beam pattern with a top-hat approximation of the same area18   u sin u 2 P ; ð3:57Þ u  where PðxÞ is the rectangle function (see Appendix A), so that ð þ D 2 d ¼ 2ðþ   Þ;

ð3:58Þ



where

  ¼ arcsin min 1;

 : k Dx cos    D ¼ 4 arcsin min 1; : k Dx cos 

It follows that

ð3:59Þ ð3:60Þ

For a sufficiently long array (many wavelengths), Equation (3.55) may be written  GA ð =2 : ð3:61Þ  2 sin 2 arcsin d k Dx cos  0 Approximating the arcsine function by its argument, this becomes k Dx GA ð =2 ; 4 sin  d

ð3:62Þ

0

and hence k Dx  Dx ¼ : 4 2

ð3:63Þ

AG ¼ 10 log10 GA :

ð3:64Þ

GA The array gain is

Now consider the directivity index of the array. This can be calculated as DI ¼ 10 log10 GD ;

ð3:65Þ

where GD is the array gain for isotropic noise, which can be calculated by replacing 18

ð þ1 

See Appendix A: 1

 sin u 2 du ¼ . u

Sec. 3.2]

3.2 Passive sonar 71

the sin  terms in Equation (3.53) with unity. Following the steps as outlined above for GA then gives 2Dx GD  10 DI=10 ¼ : ð3:66Þ

Thus, Equation (3.63) can be written GA

 G ; 4 D

ð3:67Þ

which means that noise directionality (for the dipole sheet noise source considered) has eroded about 22 % of the array’s directivity (a degradation of 0.9 dB). The broadside beam of a horizontal line array is less effective at discriminating against noise from the vertical direction than against isotropic noise. 3.2.3.5

Probability of detection, detection threshold, and ROC curves

The probability that a sonar detects a given signal in a noisy background depends on its ability to discriminate between signal plus noise and noise alone. In turn this depends on the statistical fluctuations in both the signal and noise separately, after signal processing. As described in Chapter 2, if it is assumed that individual noise pressure samples follow a Gaussian distribution, the noise amplitude after NB processing will follow a Rayleigh distribution. If, further, the signal amplitude also follows a Rayleigh distribution, the appropriate relationship between the probabilities of detection ( pd ) and false alarm ( pfa ) is 1=ð1þRarr Þ

pd ¼ p fa

:

ð3:68Þ

Using RT to denote the SNR threshold required to achieve a specified detection probability equal to pT , it is convenient to write Equation (3.68) in the form log pd ¼

1 þ RT log pT ; 1 þ Rarr

ð3:69Þ

log pfa  1: log pT

ð3:70Þ

where RT ¼

The detection threshold was introduced above as 10 log10 R50 , indicating that this threshold is based on a detection probability of 50 %. Detection thresholds based on other pT values are sometimes used. Throughout this book, if a value is not stated explicitly, a detection threshold based on pT ¼ 0.5 is implied. Figure 3.7 shows the receiver operating characteristic (ROC) curves calculated using Equation (3.70) in the form DT ¼ 10 log10 RT vs. pfa , for values of pT between 0.1 and 0.9. For example, the detection threshold for a 30% detection probability and false alarm probability of 10 6 is DT30 10 dB. Of special interest is the case pT ¼ 0.5, for which it is convenient to use base-2 logarithms in Equation (3.69), which then simplifies to log2 pd ¼ 

1 þ R50 ; 1 þ Rarr

ð3:71Þ

72 The sonar equations

[Ch. 3

Figure 3.7. ROC curves in the form 10 log10 RT vs. pfa for specified pT values and for a Rayleigh-distributed signal in Rayleigh noise. These ROC curves are suitable for NB sonar with a strongly fluctuating signal amplitude.

where R50 ¼ log2

3.2.3.6

1 : 2pfa

ð3:72Þ

Probability of false alarm

The SNR threshold R50 is related to the probability of false alarm ( pfa ) through Equation (3.72). The value of pfa can be estimated by relating it to the false alarm rate nfa , the total number of detection opportunities per ‘‘look’’ Ntot , and the duration of eack look Dt (the coherent integration time) ðnfa Þtrue ¼

Ntot pfa : Dt

ð3:73Þ

The number of opportunities Ntot depends on the number of sonar beams (Nbeams ) and frequencies (NFFT ): Ntot ¼ Nbeams NFFT : ð3:74Þ The true pfa value (and therefore also nfa ) is determined by the threshold R50 . However, this threshold should be chosen to achieve an acceptable false alarm rate. Thus, one can estimate the pfa by rearranging Equation (3.73) and removing the

Sec. 3.2]

3.2 Passive sonar 73

‘‘true’’ qualifier (i.e., assuming ‘‘true’’ and ‘‘acceptable’’ rates to be approximately equal), such that n nfa pfa fa Dt ¼ ; ð3:75Þ Ntot Nbeams Df where Df is the total system bandwidth Df ¼ NFFT f :

3.2.3.7

ð3:76Þ

Special case: low-frequency tonal in the broadside beam of a horizontal line array

As a prelude to the worked example in the following section we consider here the case of a horizontal line array receiver operating with a target in its broadside beam. This special case is analyzed with a view to simplifying the sonar equation, such that the signal-to-noise ratio can be expressed in terms of readily understood parameters such as acoustic wavelength, source power, and array length. The frequency is assumed to be low enough to justify neglecting attenuation, so (putting ¼ 0 in Equation (3.37)19) the ambient noise spectral density is N N 3 QN f 3cE3 ð0ÞW Af ¼ 2 cW Af :

ð3:77Þ

It is convenient to define the signal excess in linear units as 

Rarr : R50

ð3:78Þ

For the idealized situation considered (small z, unsteered horizontal line array, large k Dx, and infinitely deep isovelocity water) the following approximation for the sonar equation then follows from Equation (3.39) for the signal-to-noise ratio—using Equation (3.63) for the array gain and replacing FNB with Fcoh !   Fcoh ðr; zarr ; ztgt Þ W S Dx ðr; zarr ; ztgt Þ ¼ : ð3:79Þ fR50 12

WN Af The first factor on the right-hand side, which is independent of frequency and has dimensions of [distance] multiplied by [time], is a measure of the spatial (Dx) and temporal ðf Þ 1 aperture of the sonar, and is a useful indicator of a sonar’s effectiveness. The sonar performance, as measured by the detection probability alone, improves with increasing Dx or decreasing f or decreasing R50 . However, for fixed R50 this is at the expense of a greater false alarm rate, because a larger number of beams (or frequency bins) is needed for the increased spatial (or temporal) resolution. For fixed SNR, any reduction in R50 (the SNR threshold at which pd ¼ 12), will increase pd at the expense of an increased pfa . Thus, the false alarm rate increases in this situation also. 19 That is, neglecting the product zarr . The quantity zarr ztgt =r has already been neglected in the derivation of Equation (3.35) (see Chapter 2 for details).

74 The sonar equations

[Ch. 3

The second factor in Equation (3.79) is a complicated term that depends on the properties of the radiated tonal, the background noise, and the propagation conditions. It has dimensions [distance] 1 multiplied by [time]1 and is strongly dependent on analysis frequency and sonar–target geometry. Use of Equation (3.79) is illustrated in Part (vii) of the following worked example. 3.2.3.8

Worked example

An underwater target in deep water, at a depth of 10 m beneath the sea surface, radiates a NB signal (a tonal) at a frequency of 300 Hz. The tonal power is 0.2 mW, radiated uniformly in all directions. A horizontal receiving array of length 45 m is placed at a depth of 30 m. The received signal is passed through a NB filter of resolution 0.25 Hz and then beamformed. The target is in the broadside beam and the wind speed is 5 m/s. For the purpose of calculating pfa , assume that 32 beams are formed for each of 1,024 frequency bins between 256 Hz and 512 Hz, and that one per hour is an acceptable rate of false alarms. It is convenient to introduce the variables L S (signal level) and L N (in-beam noise level) as ð3:80Þ L S  SL  PL and L N  NLf  ðAG  BWÞ; ð3:81Þ so that the sonar equation can be written SENB ¼ L S  L N  DT:

ð3:82Þ

(i) Calculate the source level (SL) of the target using Equation (3.42). (ii) Calculate the background noise spectrum level (NLf ) using Equation (3.44). (iii) Using Equations (3.45) to (3.47), calculate the array gain (AG), detection threshold (DT), and processing bandwidth (BW) of the sonar. (iv) What is the sonar figure of merit (FOM)? (v) Using the method of Chapter 2 or a propagation model of your choice, plot propagation loss as a function of range PLðrÞ. What is the significance of the range at which PL ¼ 78 dB re m 2 (denote this range r50 )? (vi) Calculate the signal level defined by Equation (3.80). What is its value at the range r50 ? What is the value of the in-beam noise level (Equation 3.81)? What is the significance of the difference between L S and L N at this range? (vii) Using Equation (3.79), or otherwise, suggest ways in which the sonar or sonar geometry might be altered in order to improve its performance. 3.2.3.8.1

Part (i): source level SL

First use Equation (3.11) to calculate the source factor: S0 ¼ 24.4 Pa 2 m 2 ¼ 2.44 10 13 mPa 2 m 2 . The conversion to decibels is effected by dividing by p 2ref r 2ref , taking the base-10 logarithm, and multiplying the result by 10, giving SL ¼ 133.9 dB re mPa 2 m 2 .

Sec. 3.2]

3.2 Passive sonar 75

3.2.3.8.2 Part (ii): noise spectrum level NLf The noise level is calculated from the sea surface–radiated power spectral density, which follows from Chapter 2 WN Af ¼

2 K; 3c

ð3:83Þ

where the parameter K, which is a measure of the source factor per unit area of the sea surface (see Chapter 8 for details) and has dimensions of spectral density, is K ¼ 1:32 10 4

^v 2:24 1:5 þ F 1:59 kHz

mPa 2 Hz 1 ;

ð3:84Þ

where FkHz is the numerical value of the frequency in kilohertz; and ^v is the wind speed in meters per second. Substituting v ¼ 5 m/s (i.e., ^v ¼ 5) into Equation (3.84) 2 we obtain W N Hz 1 The noise spectrum level is then calculated, Af ¼ 0:403 pW m using Equations (3.44) and (3.77), as 0.926 mPa 2 Hz1 , or in decibels: 59.7 dB re mPa 2 Hz1 . This value is independent of depth due to the assumed (isovelocity) environment and the negligible attenuation at low frequency. 3.2.3.8.3 Part (iii): AG, BW, and DT Calculation of the bandwidth term BW is straightforward using Equation (3.47), giving 6.0 dB re Hz. For AG, Equation (3.63) gives GA ¼ 14.2, which in decibels is 11.5 dB. For the detection threshold we need the SNR threshold R50 , given by Equation (3.72), which depends on the desired pfa . Using Equation (3.75) for pfa in the form nfa pfa ¼ ; ð3:85Þ DfNbeams with Df ¼ 256 Hz, we obtain pfa ¼ 3:4 10 8 and hence (from Equation 3.72) R50 ¼ 23.8 (i.e., DT ¼ 13.8 dB). Answers to Parts (i), (ii), and (iii) are summarized in Table 3.1.20 3.2.3.8.4 Part (iv): figure of merit FOM For the figure of merit, Equation (3.48) gives FOM ¼ 78.0 dB re m 2 . 3.2.3.8.5

Part (v): propagation loss PL(r)

A graph of PLðrÞ is plotted in Figure 3.8. The 78 dB re m 2 mentioned in the question is a reference to the figure of merit, which by definition is the propagation loss 20

Intermediate calculations (see Table 3.1) are rounded to one decimal place in decibels, and to three significant figures in the linear form. While this level of accuracy might not be justified for all of the terms, it is advisable to prevent accumulation of errors by delaying any further rounding (say, to the nearest decibel) until the end of the calculation.

76 The sonar equations

[Ch. 3

Table 3.1. Sonar equation calculation for NB passive example. For each term in the sonar equation a linear form and a dB form are quoted. In each case the dB form is equal to 10 log10 (linear form) so that, for example, SL ¼ 10 log10 ðS0 =mPa 2 m 2 Þ and S0 ¼ 10 SL=10 mPa 2 m 2 . Description

dB form Symbol

Source level (Equation 3.42)

SL

Value 133.9 dB re mPa 2 m 2

Linear form Expression

Numerical value in reference units

S0

2.44 10 þ13

3 N 2 cW Af

9.26 10 þ5

Noise spectrum level (Equation 3.77)

NLf

59.7 dB re mPa 2 Hz1

Array gain (Equation 3.63)

AG

11.5 dB re 1

Dx 2

14.2

Detection threshold (Equation 3.46)

DT

13.8 dB re 1

R50

23.8

Analysis bandwidth (Equation 3.47)

BW

6.0 dB re Hz

Df =NFFT

0.250

Figure 3.8. Propagation loss (blue) and figure of merit (cyan) vs. target range. The probability of detection is 50 % when the propagation loss is equal to the figure of merit, corresponding to the intersection between the two lines.

Sec. 3.2]

3.2 Passive sonar 77

Figure 3.9. Signal level L S vs. target range in blue (solid), and in-beam noise level L N in red (dashed). The probability of detection is 50 % when L S exceeds L N by the detection threshold (DT ¼ 13.8 dB), corresponding to the intersection between solid blue and cyan lines.

resulting in a detection probability of 50 %. The range at which this happens is known as the detection range, which for this example is about 2.5 km.21

3.2.3.8.6

Part (vi): signal level L S ðrÞ and in-beam noise level L N

A graph of L S ðrÞ (i.e., SL  PL) is shown in Figure 3.9 as a solid blue line. At the detection range (2.5 km) the signal level is ca. 56 dB re mPa 2 , 14 dB higher than the inbeam background (L N ) of 42 dB re mPa 2 (dashed). The difference is the detection threshold in decibels (Equation 3.82). The third line (solid, cyan) is L N þ DT. This line intersects the signal at exactly the same range as the FOM intersects the PLðrÞ curve in Figure 3.8, thus illustrating an alternative way to calculate detection range. The two methods are equivalent, making the choice between them a matter of personal preference. Either way, the intersection between blue and cyan lines separates the region to its left, where detections are likely ( pd > 50%), from that to its right, where they are unlikely ( pd < 50%). The point of intersection itself is the detection 21

Use of different propagation models, making different inherent assumptions and approximations, could lead to small differences in this value. For this problem it is important to use a model that takes into account the phase difference between direct and surface-reflected paths (such a model is sometimes referred to as a ‘‘coherent’’ model, or it might have a ‘‘coherent’’ option that can be switched on for this purpose).

78 The sonar equations

[Ch. 3

Figure 3.10. Linear signal excess (Equation 3.86) and twice detection probability (Equation 3.68) vs. range for NB passive sonar (Rayleigh statistics); the two curves cross when  ¼ 2pd ¼ 1. The range at which this happens (2.45 km) is the detection range. (The second crossing, at 3 km, has no special significance.)

range and denoted r50 . At this location the signal excess (Equation 3.41) is 0 dB, which means that the detection probability is exactly 50 %. Detection probability vs. range is plotted in Figure 3.10, calculated using Equation (3.71). The slight dip in detection probability at a range of 120 m is caused by the interference null at that range (see Figure 3.8).

3.2.3.8.7 Part (vii): sensitivity to sonar parameters Part (vii) requires an understanding of the sensitivity of the signal excess to the sonar parameters. To gain this understanding, a graph of signal excess (equal to FOM minus PL) is plotted vs. target range and depth in Figure 3.11. By definition, contours of zero signal excess correspond to a detection probability of 50 %. For r > 500 m, the signal excess is seen to increase monotonically with increasing target depth in the depth range 0 m to 40 m.22 This is because the argument of the sin 2 function in Equation (3.35) for Fcoh is small (about 17 at 2.5 km).23 Thus, sin x may be replaced by 22

The  phase change at the sea surface leads to cancellation between the direct and surface reflected paths if their path lengths are equal. 23 Absorption is also small (0.0086 dB/km) and is considered negligible ( R50 for detection gives for the sphere’s radius sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 2 Q f R50 a> : ð3:225Þ F GA S0 t Using Equation (3.212) for pfa in Equation (3.72) results in R50 ¼ log2

Nbeams Df ; 2nfa

ð3:226Þ

which, with Df ¼ 500 Hz, Nbeams ¼ 128, and nfa ¼ 1=h, gives R50 ¼ 26.8. Equations (3.205), (3.218), and (3.179) give for the gain GA ¼ 99.1, one-way propagation factor 2 F ¼ 5:34 10 9 m 2 , and noise spectral density Q N f ¼ 0.0197 mPa /Hz. Further, the SLE =10 2 2 energy source factor (defined as 10 mPa m s) is S0 t ¼ 20 kPa 2 m 2 s. With these values, the smallest sphere that satisfies Equation (3.225) has radius a ¼ 6.10 m. The relevant parameter values are listed in Table 3.3, expressed in standard reference units, as well as converted to decibels. For readers who prefer to work in decibels, the equivalent of Equation (3.225) is obtained by requiring SE > 0 dB in Equation (3.188): TSE > NLf þ DT  ðAG  2PL þ SLE Þ;

ð3:227Þ

from which it follows that TSE must exceed 9.7 dB re m 2 , consistent with the previous calculation for the radius a. 39

See footnote 38.

Sec. 3.3]

3.3 Active sonar

107

3.3.3.8.2 Part (ii): figure of merit There are two ways of calculating the FOM. The first and simplest uses the definition of FOM from Equation (3.196), that is, FOM ¼ PL50 ¼ 10 log10 Fðr50 Þ;

ð3:228Þ

with r50 ¼ 1.3 km. The value of the propagation factor from Part (i) gives FOM ¼ 82.7 dB re m 2 . The alternative is to use the second half of Equation (3.196) with the help of Table 3.3. While the result is the same either way, the second method is more susceptible to rounding errors, illustrating the motivation for retaining precision to the first decimal place for quantities expressed in decibels. Figure 3.21 shows propagation loss plotted vs. range (upper graph) and vs. depth (lower graph), both calculated using Equation (3.218). As expected, the FOM of approximately 83 dB re m 2 intersects the PLðrÞ curve at 1.3 km and PLðzarr Þ at 100 m. When the array depth exceeds about 300 m, the PL starts to increase (the signal decreases) with increasing sonar depth due to the increased path length. The same conclusion is reached from Figure 3.22, which compares signal and in-beam noise levels. This behavior results in a second threshold crossing in the depth plot, close to 900 m. 3.3.3.8.3

Part (iii): detection probability

Probability of detection is plotted in Figure 3.23. A value of 50 % occurs at the detection range of 1.3 km (upper) and 100 m array depth (lower). Also shown is the signal excess in linear form ðÞ, defined in the same way as for passive sonar 

Rarr : R50

ð3:229Þ

Table 3.3. Sonar equation calculation for CW active sonar example with Doppler filter. Description

dB form Symbol

Value

Linear form Expression

Numerical value in reference units

Energy source level (Equation 3.189)

SLE

193.0

dB re mPa 2 m 2 s

S0 t

2.00 10 þ19

Noise spectrum level (Equation 3.193)

NLf

42.9

dB re mPa 2 Hz1

3cW N Af E3 ð zÞ

1.97 10 þ4

Array gain (Equation 3.194)

AG

20.0

dB re 1

Dx cos " 2

99.1

Detection threshold (Equation 3.195)

DT

14.3

dB re 1

log2

Nbeams Df 2nfa

26.8

108 The sonar equations

[Ch. 3

Figure 3.21. Propagation loss (blue) and figure of merit (cyan) vs. target range for array at depth 100 m (upper) and vs. array depth for target at 1.3 km (lower). The probability of detection is 50 % when the propagation loss is equal to the figure of merit, corresponding to the intersection between solid blue and cyan lines. The target depth is 150 m.

Sec. 3.3]

3.3 Active sonar

109

Figure 3.22. Signal level L SE (blue, solid) and in-beam noise level L N f (red, dashed) vs. target range for array at depth 100 m (upper) and vs. array depth for target at 1.3 km (lower). The probability of detection is 50 % when the signal exceeds L N f by the detection threshold, corresponding to the intersection between solid blue and cyan lines. The target depth is 150 m.

110 The sonar equations

[Ch. 3

Figure 3.23. Linear signal excess (Equation 3.229) and twice detection probability (Equation 3.68) for coherent CW active sonar. The target depth is 150 m. Upper: vs. range for array depth 100 m—the two curves cross at the detection range, where  ¼ 2pd ¼ 1; lower: vs. array depth for target range 1.3 km.

Sec. 3.3]

3.3 Active sonar

111

A more precise value of the depth at which the second threshold crossing occurs, namely 870 m, can be read from Figure 3.23 (lower graph).

3.3.3.8.4 Part (iv): best depth At depths exceeding a few hundred meters, Figure 3.21 shows that propagation loss increases with increasing depth. The same graph shows that the figure of merit (FOM) also increases—a consequence of the decreasing in-beam noise level (L N f ). The result is that the variation in signal excess is not monotonic, with a maximum occurring at a depth of about 430 m (see Figure 3.23). A contour plot of signal excess vs. depth and range is shown in Figure 3.24. Although the predicted best depth of 430 m is unchanged, the detection range inferred from this graph is about 20 % less than the value expected from the above calculations. This difference, which is attributed to the use of different noise and propagation models, is typical of comparisons between different sonar performance calculations with slightly different assumptions, even for nominally the same conditions.

Figure 3.24. Signal excess [dB] vs. target range and array depth for target depth 150 m and sonar frequency 50 kHz, illustrating a maximum detection range at a depth of about 430 m (INSIGHT).

112 The sonar equations

3.3.4

[Ch. 3

Incoherent processing: CW pulse þ energy detector

Consider now a sonar that transmits identical pulses to those of Section 3.3.3: a CW pulse of duration t but with a simple energy detector as receiver instead of a Doppler filter. By ‘‘energy detector’’ is meant a receiver that squares and adds successive (equally spaced) time samples, without regard to their frequency content. This change does not affect the signal energy, but it has important consequences for the background. The pulses are spaced at regular intervals Dt. 3.3.4.1

Signal (single hydrophone)

As for Section 3.3.3.1, the signal energy for a single hydrophone is Q SE ¼ Q S t;

ð3:230Þ

Q S ¼ S0 F2 ;

ð3:231Þ

where (using Equation 3.150) and S0 is the source factor. 3.3.4.2

Background (single hydrophone)

The relevant hydrophone noise is that in the total receiver bandwidth Df , typically several orders of magnitude greater than the processing bandwidth of a Doppler filter. Further, the ability to discriminate between moving and stationary targets is lost, meaning that reverberation also forms part of the background. (To be sure of containing the whole signal the receiver bandwidth must also be large enough to contain all reverberation.) The total background energy arriving in a time interval t is Q BE ¼ Q B t; where B

ð

R Q ¼ QN f df þ Q ðÞ:

3.3.4.3

ð3:232Þ ð3:233Þ

Signal-to-background ratio, signal excess, and incoherent active sonar equation

Although the transmitted pulse is CW, the sonar equation derived below is termed ‘‘BB’’ because the receiver adds up energy incoherently across the whole sonar bandwidth. From the definition of AG, the signal-to-background ratio (SBR) is Rarr ¼ Rhp GA ;

ð3:234Þ

where Rhp ¼

Q SE : Q BE

ð3:235Þ

Sec. 3.3]

3.3 Active sonar

113

Substituting Equations (3.230) and (3.232) in Equation (3.235), it follows from Equation (3.234) (using Equation 3.231 for Q S ) that Rarr ¼

S0 t F2 ; t Q B ðÞ=GA

ð3:236Þ

where F2 is the two-way propagation factor. The sonar equation is obtained in the usual way by dividing by R50 and converting to decibels: SE  10 log10

Rarr ¼ SLE  TPL  ðBLE  AGÞ  DT: R50

ð3:237Þ

This equation has the same form as its NB counterpart, Equation (3.182), except that NLf is replaced by BLE . The expressions for some of the individual terms are also different. As in the case of NB active sonar, TPL is given by TPL ¼ PLTx þ PLRx  TSE :

ð3:238Þ

Thus the BB active sonar equation for a point target and with incoherent processing, including reverberation is SE ¼ ½SLE  ðPLTx þ PLRx  TSE Þ  ðBLE  AGÞ  DT:

ð3:239Þ

The individual sonar equation terms are given by SLE  10 log10 ðS0 tÞ PLTx ¼ 10 log10 FTx PLRx ¼ 10 log10 FRx

ðenergy source level: dB re mPa 2 m 2 sÞ;

ð3:240Þ

ðpropagation loss Tx to target: dB re m 2 Þ;

ð3:241Þ

2

ðpropagation loss target to Rx: dB re m Þ;

ð3:242Þ

ðtarget strength: dB re m 2 Þ;

ð3:243Þ

ðbackground energy level: dB re mPa 2 sÞ;

ð3:244Þ

ðarray gain: dBÞ;

ð3:245Þ

back

TSE ¼ 10 log10

 4

BLE ¼ 10 log10 ½t Q B ðÞ AG ¼ 10 log10 GA and DT ¼ 10 log10 R50

ðdetection threshold: dBÞ: ð3:246Þ

A figure of merit can be defined as previously for coherent processing, although the concept is a less useful one in the presence of reverberation because the background is a function of distance to the target. The result is FOMðrÞ  12 ðPLRx þ PLTx Þ50 ¼ 12 ½SLE þ TSE  BLE ðrÞ þ AGðrÞ  DT ;

ð3:247Þ

and hence SEðrÞ ¼ 2 FOMðrÞ  ½PLRx ðrÞ þ PLTx ðrÞ :

ð3:248Þ

114 The sonar equations

[Ch. 3

Finally, in analogy with Equation (3.244), it is useful to introduce energy levels of noise and reverberation separately:  ð  NLE  10 log10 t Q N df ð3:249Þ f and RLE  10 log10 ½t Q R ðÞ :

3.3.4.4

ð3:250Þ

Array gain

The array gain depends on the relative importance of noise and reverberation and therefore depends on the delay time  in the following manner GA ðÞ ¼ ð

Q N þ Q R ðÞ

;

ð3:251Þ

R ½Q N O þ Q O ðÞ BðOÞ dO

where (for a continuous line array) BðOÞ ¼

sin 2 u u2

ð3:252Þ

and u¼

k Dx cos  sin  : 2

ð3:253Þ

The reverberation arriving after time  does so at a well-defined grazing angle  , determined by 2z sin  ¼ arr ; ð3:254Þ c such that QR QR ð   Þ: ð3:255Þ O ðÞ ¼ 2 cos  The array response to reverberation, calculated using the method of Section 3.2.3.4, is ð 1 R ðQ R Q Dð Þ; ð3:256Þ O ÞBðOÞ dO ¼ 2 where D is given by Equation (3.60). For a long array, away from the endfire direction:  D ¼ 4 arcsin ; ð3:257Þ k Dx cos  so that ð 2Q R  ðQ R arcsin ; ð3:258Þ O ÞBðOÞ dO ¼  k Dx cos 

Sec. 3.3]

3.3 Active sonar

115

which may (assuming the argument of the arcsine function to be small) be approximated further by ð

2Q R ðQ R ÞBðOÞ dO : ð3:259Þ O 2 Dx  cos  It follows from Equation (3.205) for the noise gain that ð

4Q N ðQ N : O ÞBðOÞ dO ¼ 2 Dx  cos "

ð3:260Þ

Substitution of this result and Equation (3.259) in Equation (3.251) yields Q N þ Q R ðÞ #: GA ðÞ ¼ G0 " 4 QN Q R ðÞ þ  cos " 2 cos 

ð3:261Þ

The gain against noise is the directivity factor multiplied by ð=4Þ cos ", as previously (Section 3.3.3.4). The gain against reverberation includes an additional factor 2 cos  . Thus, if the reverberation is from a distant scatterer, such that  is close to zero, the gain against reverberation (in the broadside beam, and neglecting absorption of wind noise) is approximately twice the gain against surface-generated wind noise. This is because the broadside beam is narrower in azimuth than in the elevation direction. 3.3.4.5

Probability of detection, detection threshold, and ROC curves

The calculation of detection probability follows that for active sonar with coherent processing, with Rayleigh statistics assumed for both background and signal amplitudes. The corresponding ROC relationships are identical to those of Section 3.2.3.5 and are not repeated here. 3.3.4.6

Probability of false alarm

Following Section 3.2.4.6, the false alarm probability can be estimated from the desired false alarm rate nfa N p nfa tot fa ; ð3:262Þ Dt where Ntot ¼ Nbeams Nranges : ð3:263Þ Range resolution is determined by the pulse duration t, so that the total number of range cells is Dt Nranges ¼ ; ð3:264Þ t and hence n t pfa ¼ fa : ð3:265Þ Nbeams

116 The sonar equations

3.3.4.7

[Ch. 3

Special case: point target in the broadside beam of a horizontal line array, CW pulse with incoherent processing

Consider now the same special case as previously for NB, except with incoherent receiver processing. In this situation it is no longer legitimate to neglect reverberation, so the signal-to-background ratio (SBR) after beamforming is Rarr ¼

Q SE Q S ð Þ G ¼ GA ðÞ; A Q BE Q N þ Q R ðÞ

where QN ¼

ð fm þDf =2

QN f df ;

ð3:266Þ

ð3:267Þ

fm Df =2

fm is the center frequency; and Df is the total receiver bandwidth. If the spectrum Q N f varies linearly with frequency within the sonar bandwidth, the integral simplifies to QN ¼ QN f ð fm Þ Df :

ð3:268Þ

The signal and reverberation MSP terms are given by Q S ð Þ ¼ S0 F 2

 back 4

ð3:269Þ

and Q R ðÞ ¼ S0 FðÞ 2 OA ðÞ AðÞ;

ð3:270Þ

where AðÞ ¼ 2

c c t 2 2

ð3:271Þ

is the scattering area and OA ðÞ ¼ ðCPM =16Þ tan 4 

ð3:272Þ

is the surface scattering coefficient, evaluated in the backscattering direction. The time variable  is proportional to the distance s  ¼ ð2=cÞs

ð3:273Þ

and the grazing angle  is given by tan 2  ¼

s2

z 2arr :  z 2arr

ð3:274Þ

Substituting back into Equation (3.266), with Equation (3.218) for the propagation factor, and assuming small angles (such that zarr  s), the SBR is Rarr ¼

4GA ð2s=cÞ  back : 4 3  16s 4 e 4 s Q N f ð fm Þ Df =S0 þ CPM cz arr t=s

ð3:275Þ

Sec. 3.3]

3.3.4.8

3.3 Active sonar

117

Worked example

(i) For otherwise the same problem as Section 3.3.3.8—Parts (i)–(iii)—what is the effect on (a) signal level, (b) in-beam background level, and (c) detection threshold of replacing the Doppler filter with an energy detector? (ii) What is the detection range for the energy detector if the sonar is kept at its original depth of 100 m? (iii) Calculate pd ðrÞ for a sonar depth of 100 m and pd ðzarr Þ at a range of 0.9 km. It is convenient to introduce the in-beam background energy level L BE  BLE  AG:

ð3:276Þ

The signal level takes the same form as for coherent processing (Equation 3.221) L SE ¼ SLE  ðPLTx þ PLRx  TSE Þ

ð3:277Þ

so that the sonar equation becomes SE ¼ L SE  L BE  DT:

ð3:278Þ

3.3.4.8.1 Part (i): signal, background, and detection threshold (a) signal: the signal level, as defined by Equation (3.277) and plotted in Figure 3.25, is independent of receiver processing and therefore identical to that of Figure 3.22. Small differences arise in the depth plots only because these are evaluated at a range of 900 m instead of 1300 m previously. (b) background: the in-beam noise level is 20 dB higher than for the same problem with Doppler processing (compare Figure 3.26 with Figure 3.22); at short range (up to 400 m) the contribution to the background from reverberation is higher still; the reverberation peaks at a depth of about 900 m, causing an inflexion in the total background. (c) detection threshold: the incoherent processing results in a 100-fold reduction in the number of detection opportunities per ping, which (if pfa is kept fixed) results in a decrease in DT50 of 1.2 dB compared with Doppler processing. 3.3.4.8.2 Part (ii): detection range The net result of the large increase in background level and small decrease in detection threshold is a decrease of about 19 dB in long-range signal excess, with an even larger decrease at short range due to reverberation. Consequently, the detection range is reduced to 0.9 km (the intersection of L SE and L BE þ DT in Figure 3.25, or of PL and FOM in Figure 3.27). 3.3.4.8.3 Part (iii): detection probability Figure 3.28 shows detection probability vs. range at 100 m depth and vs. depth at 900 m range. The increased reverberation at short range (Figure 3.27) manifests itself here as a dip in the detection probability for ranges less than 200 m. While the presence of reverberation must affect sonar performance (by reducing the detection

118 The sonar equations

[Ch. 3

Figure 3.25. Signal L SE and (in-beam) background L BE levels vs. target range at an array depth of 100 m (upper) and vs. array depth at range 900 m (lower). The probability of detection is 50 % when L SE exceeds L BE by the detection threshold (DT ¼ 13.0 dB), corresponding to the intersection between solid blue and cyan lines.

Sec. 3.3]

3.3 Active sonar

119

Figure 3.26. Total background BLE , background components NLE , RLE , and in-beam background level L BE vs. target range at an array depth of 100 m (upper) and vs. array depth at range 900 m (lower).

120 The sonar equations

[Ch. 3

Figure 3.27. Propagation loss vs. target range for array at depth 100 m (upper) and vs array depth for target at 0.9 km (lower). The probability of detection is 50 % when propagation loss is equal to the figure of merit, corresponding to the intersection between solid blue and cyan lines.

Sec. 3.3]

3.3 Active sonar

121

Figure 3.28. Linear signal excess and twice detection probability (Equation 3.68) for incoherent CW active sonar. Upper: vs. range for array depth 100 m—the two curves cross at the detection range, where  ¼ 2pd ¼ 1; lower: vs. array depth for target range 900 m.

122 The sonar equations

[Ch. 3

Table 3.4. Sonar equation calculation for CW active sonar example with incoherent energy detector. Description

dB form Symbol

a

Value

Linear form Expression

Numerical value in reference units

Energy source level (Equation 3.240)

SLE

193.0

dB re mPa 2 m 2 s

S0 t

2.00 10 þ19

Noise spectrum level (Equation 3.193)

NLf

42.9

dB re mPa 2 Hz1

3cW N Af E3 ð2 zÞ

1.97 10 þ4

Array gain a (Equation 3.245)

AG

20.0

dB re 1

G0 ½Q N þ Q R ðÞ   4 QN Q R ðÞ þ  cos " 2 cos 

99.1

Detection threshold (Equation 3.246)

DT

13.0

dB re 1

R50 ¼ log2

Nbeams 2nfa t

20.1

The numerical values quoted correspond to evaluation of the stated formula in the limit of large .

probability), it has no effect here on the predicted detection range. This is because reverberation has decreased to a negligible contribution at that range. In this situation, the detection is said to be noise-limited.

3.4

REFERENCES

Ainslie, M. A., Harrison, C. H., and Burns, P. W. (1996) Signal and reverberation prediction for active sonar by adding acoustic components, IEE Proc. Radar, Sonar, Navig., 143(3), 190–195. [Special issue on recent advances in sonar.] ASA (1989) American National Standard: Reference Quantities for Acoustical Levels, ANSI S1.8-1989 [ASA 84-1989, Revision of ANSI S1.8-1969(R1974)], Acoustical Society of America, New York. ASA (1994) American National Standard: Acoustical Terminology, ANSI S1.1-1994 [ASA 1111994, Revision of ANSI S1.1-1960(R1976)], Acoustical Society of America, New York. Horton, J. W. (1959) Fundamentals of SONAR (Second Edition), United States Naval Institute, Annapolis, MD. IEC (www) Electropedia (IEV online), Acoustics and electroacoustics/IEV 801 (International ELectrotechnical Commission), available at http://www.electropedia.org/iev/iev.nsf (last accessed June 23, 2009). Morfey, C. L. (2007) Dictionary of Acoustics, Academic Press, San Diego. Southall, B. L., Bowles, A. E., Ellison, W. T., Finneran, J. J., Gentry, R. L., Greene Jr., C. R., Kastak, D., Ketten, D. R., Miller, J. H., Nachtigall, P. E., Richardson, W. J., Thomas J. A., and Tyack, P. L. (2007) Marine mammal noise exposure criteria: Initial scientific recommendations, Aquatic Mammals, 33(4), 411–521.

Part II The Four Pillars

4 Sonar oceanography

All science is either physics or stamp collecting Ernest Rutherford (ca. 1910)

This is the first of four chapters dealing further with each of the four main subjects introduced in Chapter 2. The purpose is to describe them in sufficient detail to equip the reader with the necessary knowledge to carry out predictions of sonar performance in realistic situations. The first subject of the four, and that of the present chapter, is oceanography. The remaining three are underwater acoustics (see Chapter 5), sonar signal processing (Chapter 6), and detection theory (Chapter 7). By ‘‘sonar oceanography’’ is meant a description of those properties of the sea, its contents, and its boundaries of relevance to sonar. As implied by the introductory quotation from Lord Rutherford, the material is presented as a collection of empirical facts—an organized ‘‘stamp collection’’ of the acoustical properties of the sea. Readers are encouraged to treat this chapter as they might an encyclopedia, skimming through it on first reading, and returning to consult it as often as necessary to look up details for later chapters. Wherever possible, simple equations are given for key parameters such as the speed of sound in seawater. More accurate (and more complicated) expressions are available if required (Fisher and Worcester, 1997; Fofonoff and Millard, 1983; Leroy, 2001). For application to sonar performance modeling, very high accuracy is rarely needed, justifying some simplifications made in the interests of clarity, without sacrificing realism. The following are described: — the bulk physical and chemical properties of seawater that affect the propagation of underwater sound (Section 4.1);

126 Sonar oceanography

[Ch. 4

— the properties of the acoustically significant contents of the sea, especially those with an air-filled enclosure (Section 4.2); — the properties of wind-generated waves and associated bubble clouds (Section 4.3); — the acoustically significant parameters of the seabed (Section 4.4).

4.1 4.1.1

PROPERTIES OF THE OCEAN VOLUME Terrestrial and universal constants

A parameter that appears repeatedly in this chapter is the acceleration due to gravity g. Except where stated otherwise, the value used for g is 9.806 65 m/s 2 , corresponding to one standard gravity. Other important parameters are atmospheric pressure, for which a standard atmosphere is used, defined as (IAPSO, 1985)1 PSTP
101:325

kPa;

ð4:1Þ

and the freezing point of water at one standard atmosphere, given by YSTP ¼ 273:15

K:

ð4:2Þ

Also needed is a value for the Boltzmann constant K: K ¼ 1:38065  10 23 4.1.2

J/K:

ð4:3Þ

Bathymetry

An important characteristic of the sea is its depth. In the oceans, water depth values between 2 km and 5 km are common, increasing to about 10 km in the deepest ocean trenches. The continental shelves have a water depth of typically 20 m to 200 m. Regions of intermediate water depth (200–2000 m) are relatively rare and usually limited to the continental slopes. Water depths traditionally of interest to sonar performance modeling are mostly between 20 m and 5 km, with increasing emphasis on the shallow end of this range. The lateral variations in water depth within a geographical region are referred to as the bathymetry of that region. A global bathymetry map, available from the website of the University of California, San Diego,2 is shown in Figure 4.1. The data are derived from satellite measurements of geographical variations in Earth’s gravity field (see Smith and Sandwell, 1997). 4.1.3

Factors affecting sound speed and attenuation in pure seawater

Seawater covers more than two-thirds of Earth’s surface. Each of the three main oceans, the Pacific, Atlantic and Indian Oceans, has a distinct oceanographic signature determined, for example, by its temperature and salinity. All three are 1 2

The subscript STP denotes ‘‘standard temperature and pressure’’. http://topex.ucsd.edu/marine_topo (last accessed February 16, 2008).

Sec. 4.1]

4.1 Properties of the ocean volume 127

Figure 4.1. Global bathymetry map from NOAA, derived from satellite measurements of the gravity field. Scale in meters above sea level (reprinted from Sandwell et al., www).

interconnected via the Southern (or Antarctic) Ocean. The properties of these oceans that affect the propagation speed (c) or rate of attenuation () of underwater sound are described below. Advice for calculating c and , respectively, is provided in Sections 4.1.4 and 4.1.5. 4.1.3.1

Density and static pressure

The static pressure Pw (denoted Pstat in Chapter 2) increases monotonically with depth, starting from atmospheric pressure at the surface, and is given in terms of the density  and acceleration due to gravity g (a function of depth here) by ðz Pw ðzÞ ¼ Patm þ ðÞgðÞ d: ð4:4Þ 0

Leroy (1968) provides the following convenient expression for gðzÞ as a function of latitude  gðzÞ ¼ ð9:7805 m s 2 Þð1 þ 5:28  10 3 sin 2 Þ þ ð2:4  10 6 s 2 Þz:

ð4:5Þ

Similarly, the density profile can be written in terms of the pressure, salinity, and temperature profiles (Pierce, 1989, p. 34) ^ðzÞ ¼ 1027 þ 4:3 10 7 P^w ðzÞ þ 0:75½SðzÞ35  0:16½T^ðzÞ10  0:004½T^ðzÞ10 2 : ð4:6Þ Here, ^, P^w , and T^ are dimensionless variables, equal to the numerical values of density, hydrostatic pressure, and temperature, when expressed in units of kilograms

128 Sonar oceanography

[Ch. 4

per cubic meter, pascals, and degrees Celsius, respectively. In other words  ^
; 1 kg m 3

ð4:7Þ

P P^w
w ; 1 Pa

ð4:8Þ

Y  YSTP ; T^
1K

ð4:9Þ

and

where Y is the absolute temperature YðTÞ ¼ T þ YSTP :

ð4:10Þ

More generally, the notation x^ is used throughout for the numerical value of the variable x when expressed in SI units. For example, the variable ^v introduced in Chapter 2 is the wind speed expressed in meters per second. The integral of Equation (4.4) cannot be calculated directly, because it is an implicit equation, with the pressure that we are seeking to evaluate appearing itself inside the integrand. However, for realistic conditions in seawater, Leroy derives the following quadratic approximation for the variation of pressure with depth and latitude (Leroy, 1968):3 P^w ðzÞ ¼ 98066:5½1:04 þ 0:102506ð1 þ 5:28 10 3 sin 2 Þ^ z þ 2:524 10 7 z^2 ; ð4:11Þ where z^


z : 1m

ð4:12Þ

The pressure at any given depth z increases with increasing latitude  due to the increasing gravitational force, which is at its greatest close to the poles. If the latitude is not known, the sin 2  term can be approximated by its average value of 0.5. 4.1.3.2

Temperature

Apart from the polar regions, the temperature of the deep ocean has an almost constant value of about 2 C (see Figure 4.2). There is little seasonal variation, so only the annual average is shown. The Atlantic Ocean is about one degree warmer than the Indian and Pacific Oceans. This is a consequence of the oceanic thermohaline circulation, a global conveyor belt that begins its cycle as surface water in the North Atlantic.4 Surface temperature is higher and less uniform in space and time (Figure 4.3). 3

Leroy gives separate equations for the Baltic and Black Seas, but for most applications the general equation is sufficient. 4 Strong winds in the North Atlantic evaporate the surface water, simultaneously cooling it and increasing its salinity. The combined effect is a sudden increase in density that causes the surface water to sink and begin a long journey south as fresh North Atlantic deep water (NADW). Because of its recent contact with the atmosphere, the NADW is warmer than deep water in other oceans (Brown et al., 1989).

Sec. 4.1]

4.1 Properties of the ocean volume 129

Figure 4.2. Annual average temperature map at depth 3 km, from the World Ocean Atlas (WOA, 1999). The deep-water temperature of major non-polar oceans is between 1 C and 3 C. Green indicates land (or water depth less than 3000 m).

Seasonal changes are particularly noticeable in temperate seas. For example, the Mediterranean region, including the Black Sea, exhibits changes in surface temperature of up to about 15 C between summer and winter. Similar variations can be seen in the northwest corner of the Pacific (the Yellow Sea and Sea of Japan). Between the warm near-surface water and the cold deep water there is a region of rapidly decreasing temperature with depth, known as a thermocline. Variations of sound speed with depth resulting from the temperature profile have a strong influence on underwater sound propagation and hence on sonar performance. A temperature profile can be measured using a conductivity–temperature–depth (CTD) probe, although the use of this type of probe might require a stationary measurement platform. An alternative is an expendable bathythermograph (XBT), which can be deployed while the measurement ship is in motion. The main benefit of the CTD probe is that, by combining the conductivity and temperature data, the salinity can also be calculated. Two typical deep-water temperature profiles are shown in Figure 4.4, representative of deep-water profiles in the northwest Pacific (see Figure 4.5) and northeast Atlantic (Figure 4.6), respectively. 4.1.3.3

Salinity

The traditional definition of salinity is a ratio by mass of dissolved salts in water. Today this traditional definition, referred to as absolute salinity, is superseded by practical salinity. Practical salinity is defined as a dimensionless (and unitless) ratio in terms of the conductivity of the salt-water solution (IAPSO, 1985), in such a way that its value is almost identical to that of absolute salinity expressed in parts per thousand by mass (i.e., grams per kilogram).

130 Sonar oceanography

[Ch. 4

Figure 4.3. Geographical variations in surface temperature for northern winter (upper graph) and northern summer (lower) (from WOA, 1999). Green indicates land.

Compared with that of temperature, the effect of salinity on sound speed is usually small; a simple estimate of its value is often sufficient. According to Fisher and Worcester (1997), three-quarters of the ocean has a salinity within 1.0 of its median value of 34.69. This point is illustrated by Figure 4.7, showing that the salinity of the major oceans falls mostly between 34.5 and 35.0. Hence, if no better information is available for salinity, a default value of 34.7 is suggested. However, there are systematic differences between the average salinity of major oceans (see Table 4.1 and Figure 4.8) and this information can be used to obtain improved estimates. Notice especially the high salinity prevalent in the deep Atlantic Ocean (Figure 4.7), neces-

Sec. 4.1]

4.1 Properties of the ocean volume 131

Figure 4.4. Temperature profiles from WOA (1999) for locations in the northwest Pacific Ocean (thin curves: 19 N, 150 E) and northeast Atlantic (thick curves: 42 N, 13 W). Upper: full profile; lower: zoom of upper 300 m.

132 Sonar oceanography

[Ch. 4

Figure 4.5. Bathymetry map for the northwest Pacific Ocean from NOAA, derived from satellite measurements of the gravity field (reprinted from Sandwell et al., www). For scale see Figure 4.1.

Figure 4.6. Bathymetry map for the north Atlantic Ocean from NOAA, derived from satellite measurements of the gravity field (reprinted from Sandwell et al., www). For scale see Figure 4.1.

Sec. 4.1]

4.1 Properties of the ocean volume 133

Figure 4.7. Annual average salinity map at depth 3 km, from WOA (1999). White indicates land (or water depth less than 3000 m).

sary to maintain a uniform density at a depth of 3000 m, compensating for the higher temperature there relative to the other oceans (Figure 4.2). Seasonal variations in salinity are difficult to discern even at the sea surface (Figure 4.9). Nevertheless, where significant departures of salinity from its median Table 4.1. Average salinity and potential temperature by major ocean basin (Worthington, 1981). a Mean salinity

Mean potential temperature b / C

North Pacific

34.57

3.13

South Pacific c

34.63

3.50

Indian c

34.79

4.36

North Atlantic

35.09

5.08

South Atlantic c

34.84

3.81

Southern

34.65

0.71

World Ocean

34.72

3.51

Ocean

a The difference in salinity between the Pacific and Atlantic (and a similar difference in temperature already described), is well documented (Worthington, 1981). b Potential temperature is the temperature the water would have if transported adiabatically to a standard depth or reference pressure (usually atmospheric) (Brown et al., 1989). c Excluding Southern Ocean.

134 Sonar oceanography

[Ch. 4

Figure 4.8. Temperature salinity (T–S) diagram for the World Ocean (adapted from Worthington, 1981, # MIT Press, reprinted with permission).

value do occur, these need to be considered for an accurate prediction of attenuation (see, for example, Equation 4.33). For most locations a reasonable estimate can be obtained from climatology. In special situations,5 the effect of salinity on sound speed can also be important. Salinity profiles for the same situations as Figure 4.4 are shown in Figure 4.10. Salinity can be measured with a CTD probe. Alternatively, if only temperature is measured in situ (e.g., using an XBT), the salinity profile can be calculated from an estimate of the density profile using the procedure outlined below. It is assumed that representative T and S profiles are available from a nearby CTD cast (or, if no CTD data are available, from climatological data). The method is based on the presumption that the density profile does not change significantly between the CTD and XBT measurements. The force of gravity can usually be relied on to stabilize the density profile, even if the sea is in a state of flux. An illustration of this point is found in the upper graph of Figure 4.11. Even though salinity and temperature profiles in the deep Atlantic Ocean are different from those in the deep Pacific, the density profiles between 1 km and 5 km are almost identical. The first step in the procedure is to estimate the density profile using Equation (4.6) from representative temperature and salinity profiles. We call this density profile

5 Regions of exceptionally low salinity (e.g., the Baltic Sea) can occur due to the influx of freshwater from rivers, precipitation or melting ice, and of high salinity (e.g., Persian Gulf ) due to freezing or evaporation.

Sec. 4.1]

4.1 Properties of the ocean volume 135

Figure 4.9. Seasonal variations in surface salinity (from WOA, 1999). White indicates land.

est ðzÞ and use it as an estimate of in situ density. An estimate of the in situ salinity profile can then be obtained by rearranging Equation (4.6): Sest ðzÞ ¼ 35 þ 43 ^est ðzÞ  43 f1027 þ 4:3  10 5 P^w ðzÞ  0:16½T^meas ðzÞ  10  0:004½T^meas ðzÞ  10 2 g

ð4:13Þ

where Tmeas ðzÞ is the in situ temperature measurement; and Pw ðzÞ is the pressure profile from Equation (4.11). Other variants of this procedure are also possible. For example, if in situ sound speed and temperature profiles are both known (the sound speed can be measured directly using a velocimeter), salinity can be estimated by rearranging Equation 4.21.

136 Sonar oceanography

[Ch. 4

Figure 4.10. Salinity profiles from WOA (1999) for locations in the northwest Pacific Ocean (thin curves: 19 N, 150 E) and northeast Atlantic (thick curves: 42 N, 13 W). Upper: full profile; lower: zoom of uppermost 300 m.

Sec. 4.1]

4.1 Properties of the ocean volume 137

Figure 4.11. Density profiles from WOA (1999) for locations in the northwest Pacific Ocean (thin curves: 19 N, 150 E) and northeast Atlantic (thick curves: 42 N, 13 W). Upper: full profile; lower: zoom of uppermost 300 m.

138 Sonar oceanography

4.1.3.4

[Ch. 4

Acidity (pH)

The absorption of underwater sound is sensitive to the pH of seawater. Several different pH scales are in use, each resulting in a different numerical value for pH under identical conditions (see Appendix B for their definitions). This problem can be mitigated by presenting acidity data in terms of a parameter ‘‘K’’, whose precise definition is adjusted according to the scale used in such a way that its numerical value is approximately independent of that scale (Brewer et al., 1995). Pioneering work by Mellen et al. (1987) with the U.S. National Bureau of Standards scale ( pHNBS ) used the definition KNBS ¼ 10 ðpHNBS 8Þ : ð4:14Þ Other scales in use include the total proton scale ( pHT ), with KT ¼ 10 ðpHT 7:858Þ

ð4:15Þ

KSWS ¼ 10 ðpHSWS 7:85Þ :

ð4:16Þ

and the seawater scale ( pHSWS )

These three pH scales are related approximately as follows pHNBS pHSWS þ 0:15

ð4:17Þ

pHT pHSWS þ 0:01;

ð4:18Þ

and from which it can be seen that the differences between the three K definitions, of up to a factor of order 10 0:002 (about 1.005), are small and for the present purpose may be neglected. For this reason, no further distinction is made between them and the subscript is dropped. (The subscript is retained for pH scales, as differences between these are significant). To avoid the risk of confusion between these scales, the convention is adopted here to report acidity values in terms of K rather than pH. Modern use of the NBS scale for seawater is discouraged by Brewer et al. (1995) and Millero (2006). Instead, Brewer et al. (1995) use the seawater scale ( pHSWS ), which is recommended by a UNESCO report (Dickson and Millero, 1987). More recent literature (Wedborg et al., 1999; Ternon et al., 2001) adopts the total proton scale ( pHT ). In the following, where a pH value is mentioned, the SWS scale is used, accompanied by a conversion to NBS in order to facilitate comparison with early literature. Figure 4.12 shows the global contours of K from Mellen et al. (1987). The North Pacific is renowned for its low attenuation at low frequency, caused by low pH values (i.e., low K) close to the sound channel axis (see Figure 4.12).6 Figure 4.13 shows similar maps for the Arctic Ocean and Figure 4.14 shows K profiles for major oceans, calculated using GEOSECS data from Mellen et al. (1987). Modern (higher resolu6

The reason for the low pH is a 10 % higher concentration of carbon (in the form of carbonic acid) than in the Atlantic. The carbon originates from CO2 formed by the respiration and decomposition of living matter (Brown et al., 1989, p. 113). The CO2 concentration reaches a peak at a depth of about 1 km, resulting in a minimum of pH at that depth (Millero, 2006).

Sec. 4.1]

4.1 Properties of the ocean volume 139

tion) measurements for the equatorial Atlantic Ocean are reported by Ternon et al. (2001). The lowest reported value of K (based on Figure 4.12) is 0.5, meaning that pHSWS is greater than 7.55 (i.e., pHNBS > 7.70). More precisely, the pH of seawater is mostly in the range 7.55 < pHSWS < 8.15 (i.e., 7.7 < pHNBS < 8.3), which is slightly alkaline. A good global default value for pHSWS is 7.85, increasing to 8.10 close to the sea surface.7 4.1.3.5

Viscosity

Shear viscosity (commonly abbreviated as ‘‘viscosity’’) can be described in simple terms as resistance to shear flow, of the kind encountered when stirring honey. Formally it is a constant of proportionality relating viscous stress to the rate of strain of a fluid (Morfey, 2001). Water has a much lower shear viscosity than honey, but it is nevertheless sufficient to have a noticeable effect on the attenuation of high-frequency sound. The shear viscosity of seawater, denoted S , is equal to 1.4 mPa s at a temperature of 10 C, salinity 35, and atmospheric pressure. It varies with temperature from about 1.9 mPa s at T ¼ 0 C to 0.9 mPa s at 30 C (Horne, 1969, p. 96). The effect of salinity is small by comparison, viscosity decreasing by no more than 10 % with a reduction in salinity from 40 to 5. The effect of pressure is also relatively small, with a reduction of viscosity of at most 5 % from an increase in pressure to 50 MPa (500 atmospheres). Dependence on salinity S and absolute temperature Y is given by the empirical formula (Francois and Garrison, 1982a)8 ^ expð2431=Y ^Þ S ðS; YÞ ¼ 0:924½1 þ 0:0018ðS  35Þ Y

nPa s:

ð4:19Þ

A related parameter is bulk viscosity (also known as volume viscosity or compression viscosity). The measured value of bulk viscosity is given by Liebermann (1948) as B ¼ 2:2: S

4.1.4

ð4:20Þ

Speed of sound in pure seawater

The speed of sound in pure seawater is a complicated function of salinity S, temperature T, and pressure P (Fofonoff and Millard, 1983; Fisher and Worcester, 1997; Leroy, 2001). A useful summary of seawater properties, including on-line calculators for the speed of sound in seawater and freshwater, is provided by NPL (2007a). A simplified formula for the speed of sound in seawater, suitable for sonar 7 The pH of the sea surface is predicted to drop at a rate of up to 0.05 per decade during the 21st century (Caldeira and Wickett, 2005; Raven et al., 2005). 8 For a more complicated expression, including the effect of pressure, see Mattha¨us (1972).

140 Sonar oceanography

[Ch. 4

Figure 4.12. GEOSECS global K contours at sea surface (left) and at depth 1 km (right) (reprinted from Mellen et al., 1987).

performance modeling, is given by Mackenzie (1981) c^ðS; T; zÞ ¼ 1448:96 þ 4:591T^  0:05304T^ 2 þ 2:374  10 4 T^ 3 þ ð1:340  0:01025T^ÞðS  35Þ þ 0:01630^ z þ 1:675  10 7 z^2  7:139  10 13 T^z^3 ;

ð4:21Þ

Sec. 4.1]

4.1 Properties of the ocean volume 141

where, as previously, the circumflex indicates the numerical value of each variable expressed in SI units. Thus, evaluation of the right-hand side of Equation (4.21) gives the sound speed in units of meters per second.9 A graph of sound speed vs. depth is known as a sound speed profile (see Figure 4.15 for an example) The expression is

9

The temperature T^ is in degrees Celsius, not kelvin.

142 Sonar oceanography

[Ch. 4

Figure 4.13. GEOSECS Arctic K contours at the sea surface (upper: left ¼ winter; right ¼ summer) and at depth 1 km (lower left); the 1 km bathymetry contour is also shown (lower right) (reprinted from Mellen et al. (1987).

Sec. 4.1]

4.1 Properties of the ocean volume 143

144 Sonar oceanography

[Ch. 4

Figure 4.14. GEOSECS K profiles for major oceans derived using Nutall’s algorithm (Mellen et al., 1987).

valid (to within a precision of 0.07 m/s) in the following parameter ranges: 9 2 < T^ < þ30 > > > = ð4:22Þ 25 < S < 40 > > > ; 0 < z^ < 8000: The dependence on pressure is parameterized in Equation (4.21) by means of the depth z. This is made possible by the near-universal relationship between pressure and depth described previously (see Equation 4.11). A slightly more complicated equation (with 14 terms instead of 8), given by Leroy et al. (2008), retains its precision even in extreme situations such as the Black Sea, Baltic Sea, Arctic Ocean, and Mariana Trench. Leroy’s 2008 equation is c^ðS; T; z; Þ ¼ 1402:5 þ 5T^  0:0544T^ 2 þ 2:1  10 4 T^ 3 þ ð1:33  0:0123T^ þ 8:7  10 5 T^ 2 ÞS þ ½0:0156 þ 1:2  10 6 ðdeg  45Þ þ 3  10 7 T^ 2 þ 1:43  10 5 S ^ z þ 2:55  10 7 z^2  ð7:3  10 12 þ 9:5  10 13 T^Þ^ z3;

ð4:23Þ

where deg is the latitude in degrees; and the pressure-dependent terms are written as a power series in depth to facilitate comparison with Equation (4.21). Leroy et al.

Sec. 4.1]

4.1 Properties of the ocean volume 145

Figure 4.15. Sound speed profiles calculated using Equation (4.21) with TðzÞ and SðzÞ from WOA (1999), for locations in the northwest Pacific Ocean (thin curves: 19 N, 150 E) and northeast Atlantic (thick curves: 42 N, 13 W). Upper: full profile; lower: zoom of uppermost 300 m.

146 Sonar oceanography

[Ch. 4

(2008) demonstrate the accuracy of their equation (to within 0.2 m/s) for so-called ‘‘Neptunian’’10 waters in the range 9 1 < T^ < þ21 > > = ð4:24Þ 12 < S < 40:5 > > ; 0 < z^ < 12000: Equations (4.21) and (4.23) do not include the influence of bubbles on the speed of sound. A discussion of this important effect is deferred to Chapter 5. Sound speed calculated using Equation (4.21) is plotted vs. depth for the northwest Pacific and northeast Atlantic locations considered previously.

4.1.5

Attenuation of sound in pure seawater

The limiting factor for long-range sonar performance can sometimes be the absorption of sound in seawater. At very high frequency, of order 1 MHz and higher, sound absorption is caused mainly by water viscosity, which depends mainly on temperature and salinity. At lower frequencies (up to about 300 kHz) chemical relaxations are important, resulting in an additional dependence on pressure and acidity (parameterized through the parameter K as described in Section 4.1.3.4). The presence of bubbles can have a significant effect on the attenuation of sound.11 Also important, especially in coastal regions, is the possible presence of large numbers of fish. Discussion of the effects of both bubbles and fish on attenuation is deferred to Chapter 5. The present focus is on the effects of S, T, z, and K. The amplitude attenuation coefficient  introduced in Chapter 2 was expressed in units of nepers per unit distance (e.g., Np/km). It is conventional to express this quantity in units of decibels per unit distance (e.g., dB/km), for which the symbol a is used here. The two quantities are related by a=(dB km 1 ) ¼ ð20 log10 eÞ=(Np km 1 ):

ð4:25Þ

Denoting the individual contributions due to viscous and chemical relaxation effects as avisc and achem , respectively, the total (decibel) absorption coefficient in pure seawater due to viscosity and chemical relaxations can be written awater ¼ avisc þ achem :

4:26Þ

It is convenient to write the viscous and chemical terms in the following forms avisc ¼ Avisc f 2 10

ð4:27Þ

The colorful adjective ‘‘Neptunian’’ refers to those water masses interconnected with the World’s oceans. The term originates from the notion that King Neptune would have been unable to reach landlocked seas and lakes, which are thus excluded from his ‘‘kingdom’’. 11 The word ‘‘attenuation’’ is used here as a synonym of ‘‘extinction’’; in general, including the effects of both absorption and scattering.

Sec. 4.1]

4.1 Properties of the ocean volume 147

and achem ¼ AB

f2 f2 þ AMg 2 ; 2 f þfB f þ f 2Mg 2

ð4:28Þ

where fB and fMg are the respective relaxation frequencies of boric acid (B(OH)3 ) and magnesium sulfate (MgSO4 ), which vary with temperature and salinity according to  1=2 S ^ fB ¼ ð0:78 kHzÞ  e T =26 ð4:29Þ 35 and ^ fMg ¼ ð42 kHzÞ  e T =17 : ð4:30Þ The coefficients of Equations (4.27) and (4.28) are given by " !# T^ z^ 4 Avisc ¼ 4:9  10 exp  þ dB km 1 kHz 2 ; 27 17000 AB ¼ 1:06  10 4 f^B K 0:776 and AMg ¼ 5:2  10

4

!  T^ S ^ 1þ f e ^z=6;000 43 35 Mg

ð4:31Þ

dB km 1 ;

4:32Þ

dB km 1 :

ð4:33Þ

The above empirical equations, from Ainslie and McColm (1998), are based on the more complicated expressions of Francois and Garrison (1982b).12 They agree with Francois–Garrison in the following parameter ranges 9 > 200 < f^ < 10 6 > > > > > 0:5 < K < 2 ð7:55 < pHSWS < 8:15Þ ði.e., 7:7 < pHNBS < 8:3Þ > > > = 8 < S < 40 ð4:34Þ > > > > > 2 < T^ < þ30 > > > > ; 0 < z^ < 3000 and were used to calculate the coefficients of the formula quoted in Chapter 2 for the representative conditions T ¼ 10 C, S ¼ 35, z ¼ 0, and K ¼ 1. A more accurate simplification of the Francois–Garrison equation is given by van Moll et al. (2009). An alternative equation, with particular emphasis on accuracy in the frequency range 10 kHz to 120 kHz, is given by Doonan et al. (2003). Attenuation models are reviewed in NPL (2007b). 12 The term K 0:776 in Equation (4.32) is written by Ainslie and McColm (1998) as e ðpH8Þ=0:56 . The applicable pH scale is not stated (the author was unaware at that time of the ambiguity), but is the same scale as used by Francois and Garrison (1982b). In the absence of a statement to the contrary it is assumed here that Francois and Garrison used the same pH scale as Mellen et al. (1987), which, according to Brewer et al. (1995), is the NBS scale. The conversion from pH to K to obtain Equation (4.32) was therefore made using Equation (4.14).

148 Sonar oceanography

[Ch. 4

At very low frequency, of order 100 Hz and below, measurements show a residual attenuation, of unknown origin, and between 0.0002 dB/km and 0.004 dB/km in magnitude (Francois and Garrison, 1982b). This residual is not included in the above equations. At slightly higher frequency (say, 300 Hz to 3 kHz), attenuation increases rapidly with increasing pH (Figure 4.16, upper graph, cyan curves). The low value of K in the north Pacific (see Section 4.1.3) results in exceptionally low absorption of low-frequency sound in the deep-sound channel there (Figure 4.16, lower graph, solid blue curve). The blue curve (upper graph) is a reference curve calculated using the representative parameters of Chapter 2. At high frequency, the attenuation coefficient is sensitive to temperature, salinity, and pressure. The effect of varying temperature between 0 C and 20 C is illustrated by the dashed red lines (upper graph). At 100 kHz the attenuation coefficient at 20 C is about 60 % higher than at 0 C. Figure 4.16 shows the predicted attenuation coefficient for various realistic ocean conditions as listed in Table 4.2. Notice the low values of attenuation in the Baltic, caused by a combination of low temperature and very low salinity, and high values in the Red Sea (where temperature and salinity are both high). Figure 4.17 shows fractional sensitivity sð f Þ to a parameter x, calculated using the expression x0 @að f ; xÞ ; ð4:35Þ sð f Þ ¼ að f ; x0 Þ @x x¼x0 where x is any one of T, S, K, and z, and x0 is the representative value of the appropriate parameter except for the case of depth, for which z0 ¼ 1 km is chosen.

4.2

PROPERTIES OF BUBBLES AND MARINE LIFE

Gas bubbles form an acoustically important part of the sea’s constituents, as they are involved in the creation, scattering, and destruction of sound. Animals and plants, many of which contain a gas enclosure of some kind, are also important. The acoustical properties of both are reviewed here.13 4.2.1 4.2.1.1

Properties of air bubbles in water Properties of air under pressure

The (adiabatic) sound speed in air can be calculated as a function of temperature T as: cair ðTÞ ¼ ½ air Rair YðTÞ 1=2 ð4:36Þ 13 For a discussion of the influence of solid suspensions, see Chapter 5. Suspensions can usually be ignored in the open sea, but are sometimes important in coastal waters, especially near river outlets.

Sec. 4.2]

4.2 Properties of bubbles and marine life

149

Figure 4.16. Seawater attenuation coefficient vs. frequency calculated using Equations (4.26) to (4.33). Upper graph: sensitivity to temperature and acidity; lower graph: curves for different oceans with parameters from Table 4.2.

150 Sonar oceanography

[Ch. 4

Table 4.2. Seawater parameters used for evaluation of attenuation curves plotted in Figure 4.16 (adapted from Ainslie and McColm, 1998).

a

Ka

S

T/ C

z/km

Arctic Ocean

1.58

30

1.5

0.0

Atlantic Ocean

1.00

35

4.0

1.0

Baltic Sea

0.79

8

4.0

0.0

Pacific Ocean

0.50

34

4.0

1.0

Red Sea

1.58

40

22.0

0.2

The parameter K is defined in Section 4.1.3.4.

where the specific heat ratio of air is air


ðCP Þair ¼ 1:4011; ðCV Þair

ð4:37Þ

Rair is the gas constant of air, equal to 287 J kg1 K1 ; and Y is the absolute

Figure 4.17. Fractional sensitivity sð f Þ of seawater attenuation (Equation 4.35) to temperature (T), salinity (S), acidity (parameterized through K), and depth ðzÞ. The attenuation coefficient að f Þ is calculated using Equations (4.26) to (4.33).

Sec. 4.2]

4.2 Properties of bubbles and marine life

151

temperature. The ideal gas law gives the equilibrium air density under pressure air ðzÞ ¼

STP YSTP Pair ðzÞ ; PSTP YðzÞ

ð4:38Þ

where STP is the density of air at standard temperature and pressure (STP), equal to 1.29 kg m 3 . The rate at which heat can be transported in a gas bubble is controlled by the thermal diffusivity of the gas. The higher the diffusivity, the more quickly heat can be dissipated and the more acoustical energy is lost to heat when the bubble pulsates. It is defined by Morfey (2001)14 as Dair


Kair ; air ðCP Þair

ð4:39Þ

where ðCP Þair is the specific heat capacity of air, equal to 1.005 J g1 K1 (Leacock, 2003); and Kair is the thermal conductivity of air, equal to 0.0249 W m1 K1 at a temperature of 10 C. For other temperatures it can be calculated using (Pierce, 1989, p. 513)

 YðTÞ 3=2 Y0 þ YA expðYB =Y0 Þ Kair ðTÞ ¼ K0 ; ð4:40Þ Y0 YðTÞ þ YA exp½YB =YðTÞ where Y0 , YA , and YB are constant temperatures, given by Y0 ¼ 300:0 K;

ð4:41Þ

YA ¼ 245:4 K;

ð4:42Þ

YB ¼ 27:6 K:

ð4:43Þ

and The remaining constant, K0 , is the value of Kair at temperature Y0 , equal to 2.624  10 2 W m1 K1 . The value of Dair at 10 C and atmospheric pressure is about 20 mm 2 /s. 4.2.1.2

Properties of water that affect the behavior of a pulsating bubble

The pressure inside a submerged gas bubble exceeds hydrostatic pressure by an amount that depends on the surface tension of water. The attractive force between water molecules results in a tension at the surface of the bubble. The surface tension can be defined as the force acting tangentially to the (air–water) interface, per unit length of that interface. For clean water it is equal to 0.072 N/m. The surface tension of slightly dirty bubbles is lower (a value of 0.036 N/m is quoted by Thorpe, 1982), whereas for bubbles in salt water it is slightly higher. 14 Alternative definitions are sometimes encountered. Weston (1967) uses specific heat at constant volume instead of at constant pressure, and Stephens and Bate (1966, p. 765) omits the specific heat factor altogether.

152 Sonar oceanography

4.2.1.3

[Ch. 4

Properties of bubbly water

A presentation of the acoustical properties of bubbly water (such as sound speed and attenuation of the air–water mixture) is deferred to Chapter 5. 4.2.2 4.2.2.1

Properties of marine life Basic physiological properties

4.2.2.1.1 Zooplankton Greenlaw and Johnson (1982) give expressions for the volume V of individual euphausiids, decapods, and copepods, as a function of their length L. For example, the average for all euphausiids is   L 3:10 V ¼ ð5:75  10 3 mm 3 Þ ; ð4:44Þ 1 mm and for a species of decapod (Sergestes similis) V ¼ ð3:74  10

3



L mm Þ 1 mm 3

3:00 :

ð4:45Þ

For arthropods, Stanton et al. (1987) provide expressions relating animal length L and volume V to their weight. Combining them gives the following relationship between length and volume   L 2:295 3 4 3 V ¼ 7:7 mm þ ð4:06  10 mm Þ : ð4:46Þ 1 mm 4.2.2.1.2 Fish The single most important physiological property of fish, from an acoustical point of view, is the presence or absence of a gas-filled bladder. It is known that entire families of fish, such as gadoids and clupeoids, possess such a bladder. The effect of the gasfilled enclosure is to enhance the scattering properties of the fish, with a particularly dramatic effect close to the resonance frequency of the enclosure. Two types of bladdered fish can be distinguished. Some, known as physostomes, are equipped with a connecting tube between the bladder and the gut, enabling the exchange of air between these two organs. Others, called physoclists, have a completely closed bladder. According to MacLennan and Simmonds (1992), all gadoids (e.g., cod or haddock) are physoclists, and all clupeoids (e.g., herring) are physostomes. Other classes of fish, such as mackerel, have no bladder at all. Appendix C contains a list of species, compiled from various sources, with information for each species concerning the presence or absence of a bladder, and the type of bladder if present. For a valuable and comprehensive Internet resource describing the taxonomy and physiology of fish generally, see fishbase (Froese and Pauly, 2007).15 15

The bladder is absent in the gadoids Melanonus and Squalogadus (Froese and Pauly, 2007).

Sec. 4.2]

4.2 Properties of bubbles and marine life

153

Assuming a fish has a bladder, the volume and surface area of the bladder can be estimated from the length L of the fish by means of the equations (Haslett, 1962) Vbladder 3:40  10 4 L 3

ð4:47Þ

Sbladder 0:0291L 2 :

ð4:48Þ

and Weston (1995) The volume of the whole fish is approximately (Haslett, 1962) Vfish 0:0083L 3 :

ð4:49Þ

An estimate of the surface area of the fish can be made by a simple geometrical scaling of the form16   Vfish 2=3 Sfish Sbladder ¼ 0:238L 2 : ð4:50Þ Vbladder 4.2.2.1.3

Marine mammals

Typical values of mass m and length L are given for selected species of marine mammal in Table 4.3. The ratio m=L 3 is also included, providing information relating to the aspect ratio of the animal. A prolate spheroid of volume V and length L has a breadth-to-length aspect ratio X given by (see Table 4.4) rffiffiffiffiffiffiffiffiffi 6V X¼ : ð4:51Þ L 3 The final column of Table 4.3 shows this aspect ratio X, estimated for each species by replacing the volume V with that of the animal m=. The value of X varies between 0.11 (using m=L 3 7 kg/m 3 , for the franciscana dolphin and sperm whale) and 0.26 (m=L 3 37 kg/m 3 , for the northern sea lion, walrus, and elephant seal). 4.2.2.2

Acoustical properties

4.2.2.2.1 Fish flesh The response of a fish bladder to sound is similar to that of an air bubble, but influenced in a non-trivial way by the surrounding fish flesh. The sound speed and density of fish flesh exceed those of seawater by a few percent, as summarized in Table 4.5. The elasticity of fish flesh is determined by the (complex) shear modulus . The real part determines the pressure Pe exerted by the bladder wall on the gas contents (Andreeva, 1964) 4 Pe ¼ ReðÞ: ð4:52Þ 3 a The imaginary part of  determines losses due to vibration of the flesh. The value of  is subject to considerable uncertainty but a typical value, attributed by Love (1978) to 16

Use of Equation (4.50) implies an assumption that the bladder and fish are of similar shape.

154 Sonar oceanography

[Ch. 4

Table 4.3. Mass, length, and aspect ratio of selected sea mammals (based on Pabst et al., 1999). Species a

a

Mass m/kg

Length m L 3 / Aspect L/m kg m 3 ratio b X

Northern fur seal (female) (Callorhinus ursinus)

30

1.4

10.9

0.14

Franciscana dolphin (Pontoporia blainvillei)

32

1.7

6.5

0.11

Sea otter (Enhydra lutris)

45

1.5

13.3

0.16

Harbor seal (Phoca vitulina)

140

1.9

20.4

0.19

Amazonian manatee (Trichechus inunguis)

450

3.0

16.7

0.18

Bottlenose dolphin (Tursiops truncatus)

650

4.0

10.2

0.14

Northern sea lion (Eumetopias jubatus)

1,100

3.2

33.6

0.25

Walrus (Odobenus rosmarus)

1200

3.2

36.6

0.26

Elephant seal (Mirounga leonina, Mirounga angustirostris)

5000

5.0

40.0

0.27

Steller’s sea cow (Hydrodamalis gigas) c

10000

7.0

29.2

0.23

Sperm whale (male) (Physeter macrocephalus)

45000

18.5

7.1

0.11

Right whale (Lissodelphis borealis, Eubalaena glacialis)

90000

17.7

16.2

0.17

Thumbnail images # Garth Mix, GMIX Designs. Reprinted with permission. Of an equivalent prolate spheroid with the same length and mass (see Equation 4.51). c Extinct species. b

Sec. 4.2]

4.2 Properties of bubbles and marine life

155

Table 4.4. Volume and surface area of ellipsoids with semi-axes a  b  c. Shape General ellipsoid, semi-axes a, b, and c Prolate spheroid, a semi-major axis a, semi-minor axes b, ellipticity e Oblate spheroid, b semi-major axes a, semi-minor axis c, ellipticity e Sphere, radius a a b

Volume

Surface area

Ellipticity

4 3 abc

Expressable in terms of an elliptic integral (Weisstein, www)

N/A

4 2 3 ab

4 2 3 a c

  arcsin e 2 b 2 þ ab e



  c2 1þe 2 a 2 þ loge 2e 1e



4 3 3 a

4a 2

1

b2 a2

1

c2 a2

1=2

1=2

0

A prolate spheroid has one major axis and two minor ones, like an airship. An oblate spheroid has two major axes and one minor one, like a flying saucer.

Table 4.5. Acoustical properties of fish flesh. Species

Sound speed ratio c=cw

Density ratio Reference =w

Cod (Gadus morhua)

1.050

1.040

Clay and Horne (1994)

Unspecified fish with swimbladder

1.033

1.023

Love (1978)

Andreeva (1964), is  ¼ ð300 þ 90iÞ

kPa;

ð4:53Þ

consistent with a value of Pe close to 300 kPa. Weston (1995) suggests a smaller value for Pe in the range 50 kPa to 100 kPa, based on the measurements of Love (1978) and Løvik and Hovem (1979). An alternative model introduced by Love (1978) treats fish flesh as a viscous fluid medium described by a viscosity parameter , defined as  ¼ 43 S þ B :

ð4:54Þ

Nero et al. (2004) suggest for fish flesh a value of  ¼ 50

Pa s:

ð4:55Þ

156 Sonar oceanography

[Ch. 4

Table 4.6. Acoustical properties of whale tissue. Attenuation measurements are standardized for ease of comparison by dividing them by frequency and presenting in units of dB/(m kHz). The actual measurement frequencies are 100 kHz (Miller and Potter, 2001) and 10 MHz (Jaffe et al., 2007).  Species Tissue type c=m s1 =kg m 3  dB m1 Reference f kHz1 Atlantic northern Skin right whale Blubber (Eubalaena glacialis) Florida manatee Connective (Trichechus manatus tissue latirostris) Blubber Muscle

1700 1600

1200 900

0.09

Miller and Potter (2001) Miller and Potter (2001)

1680–1710

1030–1150

0.3–0.6

Jaffe et al. (2007)

1520–1530 1600–1630

960–1060 1020–1070

0.5–0.8 0.3–0.5

Jaffe et al. (2007) Jaffe et al. (2007)

4.2.2.2.2 Whale tissue The acoustical properties of whale tissue, as reported by Miller and Potter (2001) and Jaffe et al. (2007), are summarized in Table 4.6. 4.2.2.2.3 Zooplankton The density and sound speed of krill (euphausiid) flesh are correlated with animal length. Chu and Wiebe (2005) give the following correlation equations, valid for krill length L exceeding 25 mm (i.e., L^ > 0:025): ckrill ¼ 1:009 þ 0:50L^ cw krill ¼ 1:002 þ 0:54L^: w

ð4:56Þ ð4:57Þ

A summary of the acoustical properties of euphausiids is given in Table 4.7 and for other zooplankton in Table 4.8 (see also Lavery et al., 2007). 4.2.2.3

Population estimates

4.2.2.3.1 Fish in the North Sea: population density and case study An order of magnitude estimate of the average areic17 mass of all marine fauna is about 10 g/m 2 in both deep and shallow water. It is difficult to improve on this estimate except for well-studied locations. Despite this uncertainty, the effects can be large and should therefore be considered. Population estimates for the North Sea are provided as an example although, at the time of writing, the data on which the estimates are based are already 20 years out of date. The total North Sea biomass is estimated (Sparholt, 1990; Yang, 1982) to be 17 Following Taylor (1995), the adjectives ‘‘areic’’ and ‘‘volumic’’ are used, respectively, to mean ‘‘per unit area’’ and ‘‘per unit volume’’.

Sec. 4.2]

4.2 Properties of bubbles and marine life

157

Table 4.7. Acoustical properties of euphausiids (from Simmonds and MacLennan, 2005). Species

Sound speed ratio c=cw

Density ratio =w

Euphausia pacifica

1.005–1.015

1.035–1.040

Euphausia superba

1.028  0.002

1.021–1.040

Thysanoessa raschii

1.010

1.027

Thysanoessa spp.

1.025

1.026–1.044

Meganyctiphanes norvegica

1.035

1.029–1.048

Table 4.8. Values of zooplankton density and sound speed ratios (from Clay and Medwin, 1998, Table 9.5 and Simmonds and MacLennan, 2005, p. 277). Class Amphipods

Sound speed ratio c=cw

Density ratio =w

1.000–1.009

1.055–1.088

Cladocerans

1.011–1.017

Copepods

1.006–1.012

Decapods

0.997–1.006

Mysids Cephalopods Cod eggs

1.023–1.049

1.075 1.007

1.003

1.017–1.026

0.979–0.992

about 10 Tg (1 teragram is equal to 10 12 grams, or 1 million metric tons). Assuming for the North Sea a total surface area of 575 000 km 2 and volume 42 300 km 3 , the average areic and volumic biomass densities for the North Sea are 17 g/m 2 and 0.24 g/ m 3 , respectively. Data by individual species appear in Table 4.9. Additional information about North Sea fish can be found in Knijn et al. (1993). Also worth mentioning are the argentines (Argentina spp.), pelagic physostomes of length about 13 cm, common in the Norwegian Deep. The estimated biomass of argentines in the Norwegian Deep is 0.4 Tg (Yang, 1982). The geographical distribution of two important North Sea species is shown in Figure 4.18, including an indication of the variation with season (summer vs. winter) and fish size (adults vs. juveniles). The data show that, during the 1980s, herring were common in both summer and winter throughout the North Sea except in the

Table 4.9. North Sea fish population estimates by species, in order of decreasing biomass B. Estimates apply to various periods between 1977 and 1992. Species

B/Tg

Reference

L50 /m a (Knijn et al., 1993)

Sandeel (Ammodytes spp.)

1.82

Sparholt (1990) b

(0.20)

26 700

Demersal (no bladder)

Herring (Clupea harengus)

1.33

Sparholt (1990)

0.24

11 300

Pelagic (physostome)

Norway pout (Trisopterus esmarkii)

1.20

Sparholt (1990)

0.13

64 100

Pelagic (physoclist)

Dab (Limanda limanda)

0.74

Yang (1982) c

0.12

50 300

Demersal (bladder absent in adults)

Grey gurnard (Eutrigla gurnardus)

0.64

Yang (1982)

0.19

11 000

Demersal

Plaice (Pleuronectes platessa)

0.55

ICES (1994) d

0.33

1800

Demersal flatfish (bladder absent in adults)

Haddock (Melanogrammus aeglefinus)

0.50

ICES (1994)

0.30

2200

Pelagic (physoclist)

Starry ray (Raja radiata)

0.45

Yang (1982)

0.47

500

Demersal

Mackerel (Scomber scombrus)

0.44

Sparholt (1990)

(0.30)

1900

Pelagic (no bladder)

Whiting (Merlangius merlangus)

0.40

ICES (1994)

0.20

5900

Pelagic (physoclist)

Cod (Gadus morhua)

0.35

ICES (1994)

0.70

100

Pelagic (physoclist)

Saithe (Pollachius vireus)

0.30

ICES (1994)

(0.45)

400

Pelagic (physoclist)

Silvery pout (Gadiculus argenteus)

0.25

Yang (1982)

(0.06)

135 800

Long rough dab (Hippoglossoides platessoides)

0.23

Yang (1982)

0.17

5500

Demersal flatfish (bladder absent in adults)

Horse mackerel (Trachurus trachurus)

0.22

Yang (1982)

0.24

1900

Pelagic (no bladder)

Sprat (Sprattus sprattus)

0.20

Sparholt (1990)

0.10

23 500

Pelagic (physostome)

Estimated Description North Sea population/ millions

Pelagic (physostome)

a The length L50 is the fish length ‘‘at which 50 % of the individuals sampled in that length class are sexually maturing/ mature’’ (Knijn et al., 1993) (values in brackets are estimates). b Three-year average (1983–1985). c Two-year average (1977–1978).

4.3 Properties of the sea surface

159

northernmost region, and that most were juveniles (with L < L50 ). By comparison, Norway pout were present mainly in the north and northwest, with a population that was more evenly balanced between adults and juveniles. 4.2.2.3.2

Marine mammals

A biogeographical database, including information from marine mammal sightings, is available from the Ocean Biogeographic Information System (obis, www). The abundance of different species varies enormously. Global population estimates from Bowen and Siniff (1999) for various periods in the 1980s and 1990s include — for pinnipeds, between 12 000 000 crabeater seals18 or 6 000 000 harp seals to fewer than 500 Mediterranean monk seals; — for cetacea, between 2 000 000 sperm whales or 2 000 000 spinner dolphins19 to 500 Indus river dolphins and fewer than 1000 northern right whales; — for sirenians, between 100 000 dugongs or 100 000 sea otters to just 2500 Florida manatees.

4.3

PROPERTIES OF THE SEA SURFACE

If the sea surface is perfectly calm, its only effect on underwater sound in water is to reflect it like a mirror. The consequences of a disturbance (e.g., due to local wind, currents, precipitation, and distant storms), described in more detail in Chapters 5 and 8, include — scattering of sound at a roughened air–sea boundary; — generation, scattering, refraction, and absorption of sound by near-surface bubbles created by breaking waves and other mechanisms. The most important single parameter required to model these effects is local wind speed (denoted v). Knowledge of wind speed permits estimating surface roughness and near-surface bubble population. 4.3.1

Effect of wind

Visible effects of wind can be described qualitatively by means of the Beaufort wind force scale, ranging from force 0 (calm) to force 12 (a hurricane). Many versions of the Beaufort scale exist, the most widely used being WMO code 1100.20 However, WMO 1100 is known to be biased in the sense that wind speeds estimated visually using WMO 1100 are systematically lower than the true wind speed for v less than 18

The crabeater seal is described by Bowen and Siniff (1999) as ‘‘probably the most abundant marine mammal in the world’’. 19 Eastern tropical Pacific population. 20 WMO: World Meteorological Organization.

160 Sonar oceanography

[Ch. 4

Figure 4.18. Geographical distribution in the North Sea by season, 1985–1987: herring . . . (reprinted from Knijn et al., 1993).

12 m/s (for higher wind speeds, the opposite is true). Kent and Taylor (1997) compare various alternatives to WMO 1100 and show that, of the scales considered, those due to da Silva et al. (1994) and to Lindau (1995) agree best with observation. Table 4.10 shows the revised Beaufort scale of da Silva et al. (1994), preferred here over Lindau

Sec. 4.3]

4.3 Properties of the sea surface

161

Figure 4.18 (cont.). . . . and Norway pout (reprinted from Knijn et al., 1993).

(1995) because, in addition to an average value of v for each wind force, an indication is given of the likely spread in v. The first column shows the Beaufort force number and corresponding WMO description. An alternative description used for the same scale, referred to by Bowditch (1966) as the ‘‘seaman’s term’’, is included for compar-

162 Sonar oceanography

[Ch. 4

Table 4.10. WMO Beaufort wind force scale and estimated wind speed v20 due to da Silva et al. (1994) at a measurement height of 20 m. The final column shows the wind speed v10 converted to a standard height of 10 m, assuming air temperature is equal to water temperature (see Figure 4.19). Beaufort force WMO description Seaman’s term (Bowditch 1966) 0 Calm Calm

1 Light air Light air

2 Light breeze Light breeze

3 Gentle breeze Gentle breeze

4 Moderate breeze Moderate breeze

5 Fresh breeze Fresh breeze

6 Strong breeze Strong breeze

Photograph reproduced from NOAA (http)

Appearance at sea if fetch and duration of the blow have been sufficient to develop the sea fully (WMO, 1970)

Estimated Wind speed wind speed at standard height v20 /m s1 v10 /m s1

Sea like a mirror

0.0–1.0

0.0–1.0

Ripples with the appearance of scales are formed, but without foam crests

1.0–3.0

1.0–2.9

Small wavelets, still short but more pronounced; crests have a glassy appearance and do not break

3.0–4.6

2.9–4.3

Large wavelets; crests begin to break; foam of glassy appearance; perhaps scattered white horses

4.6–6.8

4.3–6.4

Small waves, becoming longer; fairly frequent white horses

6.8–9.8

6.4–9.2

Moderate waves, taking a more pronounced long form; many white horses are formed (chance of some spray)

9.8–12.0

9.2–11.3

Large waves begin to form; the white foam crests are more extensive everywhere (probably some spray)

12.0–15.0

11.3–14.1

Sec. 4.3]

4.3 Properties of the sea surface

Appearance at sea if fetch and duration of the blow have been sufficient to develop the sea fully (WMO, 1970)

Estimated Wind speed wind speed at standard height v20 /m s1 v10 /m s1

Sea heaps up and white foam from breaking waves begins to be blown in streaks along the direction of the wind

15.0–17.8

14.1–16.5

8 Gale Fresh gale

Moderately high waves of greater length; edges of crests begin to break into the spindrift; the foam is blown in wellmarked streaks along the direction of the wind

17.8–21.0

16.5–19.5

9 Strong gale Strong gale

High waves; dense streaks of foam along the direction of the wind; crests of waves begin to topple, tumble and roll over; spray may affect visibility

21.0–24.2

19.5–22.5

10 Storm Whole gale

Very high waves with long overhanging crests; the resulting foam, in great patches, is blown in dense white streaks along the direction of the wind; on the whole, the surface of the sea takes a white appearance; the tumbling of the sea becomes heavy and shock-like; visibility affected

24.2–27.8

22.5–25.9

11 Violent storm Storm

Exceptionally high waves (small and medium sized ships might be for a time lost to view behind the waves); the sea is completely covered with long white patches of foam lying along the direction of the wind; everywhere the edges of the wave crests are blown into froth; visibility affected

27.8–31.4

25.9–29.0

The air is filled with foam and spray; sea completely white with driving spray; visibility very seriously affected

>31.4

>29.0

Beaufort force WMO description Seaman’s term (Bowditch 1966) 7 Near gale Moderate gale

12 Hurricane Hurricane

Photograph reproduced from NOAA (http)

163

164 Sonar oceanography

[Ch. 4

Figure 4.19. Wind speed scaling factors to convert from a 20 m reference height to the standard reference height of 10 m (Dobson, 1981). The legend shows the temperature difference jTair  Twater j in C. Red curves indicate stable conditions (Tair > Twater ); blue curves indicate unstable conditions (Tair < Twater ).

ison. Some of these descriptions are ambiguous unless the Beaufort force is also quoted. For example, the same word ‘‘storm’’ can mean either force 10 or force 11 depending on whether the WMO or seaman’s term is intended. A more complete physical description is provided in columns 2 and 3 in the form of an image (NOAA, http) and text (WMO, 1970). Finally, columns 4 and 5 contain the likely spread of wind speeds (at two different measurement heights of 20 and 10 m, denoted v20 and v10 respectively, and averaged over 10 min in time) associated with these conditions. Figure 4.19 shows the conversion factors between these two measurement heights. The defining characteristics of the Beaufort scale are the physical descriptions under the heading ‘‘appearance at sea’’. All other parameters, including quoted wind speed values, have the status of derived or likely parameters for the stated appearance. Wind speed is given here in units of meters per second, in keeping with the adoption of SI units throughout this book. For wind speed values reported in knots (kn), the conversion is (see Appendix B) 1 kn ¼ ð1852=3600Þ m=s 0:5144 m=s: In an attempt to address the shortcomings of code 1100, which is based on measurements made in the late 19th and early 20th centuries, in 1970 the WMO published an updated Beaufort Scale (dubbed ‘‘proposed new Code 1100’’ and referred to by Kent

Sec. 4.3]

4.3 Properties of the sea surface

165

Table 4.11. Comparison of wind speed estimates for Beaufort force 1–11 based on WMO code 1100 and CMM-IV with those of da Silva. All are average values in meters per second except the shaded column, which shows wind speed ranges in knots for the original WMO code 1100. Beaufort force

WMO 1100 WMO 1100 (NOAA, http) spread/kn av./m s1

da Silva da Silva WMO CMM-IV (Table 4.10) (Table 4.10) (WMO, 1970) av./m s1 av./m s1 av./m s1

meas. height:

10 m

10 m

10 m

20 m

20 m

1

1–3

1.0

2.0

2.0

2.0

2

4–6

2.6

3.6

3.8

3.6

3

7–10

4.4

5.4

5.7

5.6

4

11–16

6.9

7.8

8.3

7.9

5

17–21

9.8

10.3

10.9

10.2

6

22–27

12.6

12.7

13.5

12.6

7

28–33

15.7

15.3

16.4

15.1

8

34–40

19.0

18.0

19.4

17.8

9

41–47

22.6

21.0

22.6

20.8

10

48–55

26.5

24.2

26.0

24.2

11

56–63

30.6

27.5

29.6

28.0

and Taylor, 1997 as CMM IV21), intended for scientific use (WMO, 1970). Wind speed estimates based on the WMO 1100 and WMO CMM-IV scales are compared in Table 4.11 with those of the modern values of da Silva from Table 4.10. Treating da Silva’s scale as a reference, WMO 1100 underestimates wind speed by up to 1 m/s for Beaufort force 1–5, and overestimates it by up to 3 m/s for force 7-11, while the CMM-IV scale generally tends to underestimate wind speed. This bias (the difference between columns 5 and 6 of Table 4.11) increases with increasing wind force up to a maximum of about 1.8 m/s for force 9 and above. Kent and Taylor conclude that ‘‘the operationally used WMO1100 seemed to be better than the CMM-IV scale recommended for scientific use.’’ Another widely used measure of sea surface conditions is sea state, which is a measure of the height of surface waves rather than wind speed, although the two are related. As with the Beaufort scale, there is no single, universally accepted definition, but the one known as WMO code 3700 is in widespread use (see Table 4.12). 21

CMM IV stands for ‘‘Commission for Maritime Meteorology IV’’.

166 Sonar oceanography

[Ch. 4

Table 4.12. Definition of sea state (WMO code 3700). WMO code 3700 (NODC, www)

Sea state code

Description

a

4.3.2

Significant wave height a in meters (exact)

0

Calm (glassy)

0

1

Calm (rippled)

0–0.1

2

Smooth (wavelets)

0.1–0.5

3

Slight

0.5–1.25

4

Moderate

1.25–2.5

5

Rough

6

Very rough

4–6

7

High

6–9

8

Very high

9–14

9

Phenomenal

>14

2.5–4

See Equation (4.65).

Surface roughness

Sea surface roughness is determined by the spectrum of surface waves propagating along it. An overview of surface waves and associated spectra, including the effect of wind fetch, is provided by Robinson (2004). Two surface roughness spectra are described in Sections 4.3.2.1 and 4.3.2.2, both applicable to open ocean conditions. Of these, the Pierson–Moskowitz spectrum is in modern use, while the Neumann– Pierson spectrum is needed for comparison with results from older literature. The RMS slope  of the sea surface has been measured optically by Cox and Munk (1954) and related empirically to the wind speed. Their equation relating these two parameters is22  2 ¼ ð3 þ 5:12^v10 Þ  10 3 :

4.3.2.1

ð4:58Þ

Pierson–Moskowitz spectrum

The statistics of sea surface waves can be represented by a spectrum due to Pierson and Moskowitz (1964). The Pierson–Moskowitz (PM) wave height spectral density 22 The actual measurement height for wind speed was 12.5 m (41 ft). For simplicity, it is assumed here that the difference between the wind speeds at 10 m and 12.5 m may be neglected.

Sec. 4.3]

4.3 Properties of the sea surface

167

(of squared displacement from the mean surface) vs. surface wave frequency O, for wind speed between 0 m/s and 20 m/s, is of the form

  g2 g 4 SðOÞ ¼ CPM 5 exp BPM ; ð4:59Þ Ov20 O where23 v20 is the wind speed at an anemometer height of 20 m; g is acceleration due to gravity; and BPM and CPM are dimensionless constants given by BPM ¼ 0:74

ð4:60Þ

CPM ¼ 0:0081:

ð4:61Þ

and The RMS roughness is the square root of the variance of the sea surface elevation about its mean value. Thus (Chapman, 1983)  2PM ¼

CPM v 420 : 4BPM g 2

ð4:62Þ

Substituting for numerical values, this becomes  2PM ¼ DPM v 420 ;

ð4:63Þ

where DPM ¼ 2:85  10 5

m 2 s 4 :

ð4:64Þ

A common descriptor of the sea surface is significant wave height Hsig , often referred to simply as ‘‘wave height’’. This parameter is historically defined as the average peak-to-trough height of the highest third of all waves. With this definition Hsig is given to a good approximation by four times the RMS roughness, whereas the American Meteorological Society defines Hsig as precisely this value, that is, Hsig
4:

ð4:65Þ

 , given Another descriptor sometimes used is the mean peak-to-trough wave height H approximately by  2:5: H ð4:66Þ Using Equation (4.62) one can estimate the RMS roughness and hence the wave height for each Beaufort force, shown in Table 4.13 for Beaufort force 0 to 7. The inverse operation converts wave height to wind speed, as shown in Table 4.14 for sea states 0 to 6. 4.3.2.2

Neumann–Pierson spectrum

An alternative spectrum that is of historical importance, as it is used in much early theoretical work on surface wave scattering, is the Neumann–Pierson (NP) spectrum, 23

The actual wind speed measurement height of 19.5 m is rounded here to 20 m.

168 Sonar oceanography

[Ch. 4

Table 4.13. Beaufort wind force: relationship between wind speed and wave height. Beaufort WMO description force

Wind speed (Table 4.10)

Wave height (Pierson–Moskowitz spectrum)

v10 =m s1 v20 =m s1 0

Calm

0–1.0

0–1.0

1

Light air

1.0–2.9

2

Light breeze

3

=m

Approximate sea state equivalent

Hsig =m

0.000–0.006 0.000–0.024

0–1

1.0–3.0

0.006–0.050

0.024–0.20

1–2

2.9–4.3

3.0–4.6

0.050–0.11

0.20–0.44

2

Gentle breeze

4.3–6.4

4.6–6.8

0.11–0.24

0.44–0.97

2–3

4

Moderate breeze

6.4–9.2

6.8–9.8

0.24–0.51

0.97–2.03

3–4

5

Fresh breeze

9.2–11.3

9.8–12.0

0.51–0.77

2.03–3.10

4–5

6

Strong breeze

11.3–14.1 12.0–15.0

0.77–1.19

3.10–4.77

5–6

7

Near gale

14.1–16.5 15.0–17.8

1.19–1.68

4.77–6.72

6–7

Table 4.14. Sea state: relationship between wave height and wind speed. Sea state Description (WMO code 3700)

Significant wave height Hsig /m

RMS Wind speed of Approximate roughness corresponding Pierson– Beaufort Moskowitz spectrum force equivalent /m v10 =m s 1 v20 =m s 1

0

Calm (glassy)

0.00

0

0.0

0.0

0

1

Calm (rippled)

0.00–0.10

0–0.025

0.0–2.1

0.0–2.2

0–1

2

Smooth (wavelets) 0.10–0.50

0.025–0.12

2.1–4.6

2.2–4.8

1–2

3

Slight

0.50–1.25

0.12–0.31

4.6–7.2

4.8–7.7

3–4

4

Moderate

1.25–2.50

0.31–0.62

7.2–10.2

7.7–10.8

4–5

5

Rough

2.50–4.00

0.62–1.00

10.2–12.9

10.8–13.7

5–6

6

Very rough

4.00–6.00

1.00–1.50

12.9–15.7

13.7–16.8

6–7

Sec. 4.3]

4.3 Properties of the sea surface

given by (Neumann and Pierson, 1957, 1966)

  1 g 2 SðOÞ ¼ ANP 6 exp 2 ; Ov5 O

169

ð4:67Þ

where ANP is a constant equal to 2.4 m 2 s 5 ; and v5 is the wind speed at an anemometer height24 of 5 m. The RMS wave height roughness corresponding to the NP spectrum is given by (Ainslie, 2005)   3 1=2 v5 5 2  NP ¼ ANP 11=2 : ð4:68Þ g 2 Substituting for numerical values yields  2NP ¼ DNP v 55 ;

ð4:69Þ

where DNP ¼ 3:11  10 6

4.3.3

m 3 s 5 :

ð4:70Þ

Wind-generated bubbles

Wind-generated bubbles close to the sea surface are caused primarily by breaking waves. The number of whitecaps, and hence the bubble population density, is highly correlated with wind speed. Bubble density is highest close to the sea surface and decreases with increasing depth away from the sea surface. Some measurements of bubble population density, from Trevorrow (2003), are shown in Figure 4.20. The calculation of sound speed and attenuation (see Chapter 5) requires as input the bubble population density nðaÞ as a function of wind speed, depth, and bubble radius, in principle vs. position in three-dimensional (3D) space. A detailed 3D model of near-surface bubble distribution, including the bubble plumes, is given by Novarini et al. (1998). A range-averaged model, retaining only depth dependence, is described below. The following recipe, based on measurements by Johnson and Cooke (1979), is due originally to Hall (1989) and modified by Novarini as described by Keiffer et al. (1995). The resulting bubble population model, referred to henceforth as the ‘‘Hall–Novarini’’ model, is nða; zÞ ¼ n0 uðv10 ÞDðz; v10 ÞGða; zÞ;

ð4:71Þ

where n0 is a constant equal to n0 ¼ 1:6  10 10

m 4 :

ð4:72Þ

The other factors are uðv10 Þ, Dðz; v10 Þ, and Gða; zÞ which describe the dependence, respectively, on wind speed at 10 meters v10 , depth from the surface z, and bubble 24 The wind speed measurement height is not specified explicitly by Neumann and Pierson (1957), but the implied value is approximately 5.5 m, rounded here to 5 m.

170 Sonar oceanography

[Ch. 4

Figure 4.20. Measurements of the near-surface population density of wind-generated bubbles vs. bubble radius; wind speed 12 m/s (reprinted from Trevorrow, 2003, # American Institute of Physics).

radius a. The first of these three factors depends only on the wind speed  uðvÞ ¼

3 v : 13 m/s

ð4:73Þ

The depth factor D also includes some wind speed dependence through the e-folding depth LðvÞ

 z Dðz; vÞ ¼ exp  ; ð4:74Þ LðvÞ where  LðvÞ ¼

0:4 m

v  7:5 m/s

0:4 m þ 0:115 sðv  7:5 m/sÞ

v > 7:5 m/s.

ð4:75Þ

Finally, the bubble radius distribution G varies with depth according to 8 0 > > > < ½aref ðzÞ=a 4 Gða; zÞ ¼ > ½aref ðzÞ=a x > > : 0

a < amin amin  a 4 aref ðzÞ aref ðzÞ < a  amax a > amax ,

ð4:76Þ

Sec. 4.4]

4.4 Properties of the seabed 171

where aref ðzÞ ¼ 54:4 mm þ 1:984  10 6 z;

ð4:77Þ

amin ¼ 10 mm;

ð4:78Þ

amax ¼ 1000 mm;

ð4:79Þ

and x ¼ xðzÞ ¼ 4:37 þ



z 2 : 2:55 m

ð4:80Þ

The bubble radius appears only in Gða; zÞ, so the void fraction can be written

 ð1 z UðzÞ ¼ VðaÞnða; zÞ da ¼ n0 uðv10 Þ exp  IðzÞ; ð4:81Þ Lðv10 Þ 0 where VðaÞ is the volume of a single bubble; and IðzÞ is the integral Ð1 VðaÞGða; zÞ da. It follows that 0 ( ) aref ðzÞ 1  ½aref ðzÞ=amax xðzÞ4 4 4 IðzÞ ¼ 3 a ref ðzÞ loge þ : ð4:82Þ amin xðzÞ  4 An especially important parameter is the surface void fraction, as this controls the sound speed and, through Snell’s law, the angle of acoustic interaction with the sea surface. For the Hall–Novarini bubble population model it is given by Uð0Þ ¼ n0 uðv10 ÞIð0Þ; where

" Ið0Þ ¼

4 4 3 a 0

a 1  ða0 =amax Þ 0:37 loge 0 þ amin 0:37

ð4:83Þ # ð4:84Þ

and a0 ¼ aref ð0Þ:

ð4:85Þ

Uð0Þ ¼ 2:04  10 6 uðv10 Þ:

ð4:86Þ

Thus,

4.4

PROPERTIES OF THE SEABED

The seabed is a complicated layered medium whose acoustical properties vary with depth on length scales from a few millimeters to hundreds of meters. For simplicity it can be convenient to represent this layered medium by means of a uniform seabed with representative depth-averaged properties. However, to be useful the average must be over a depth scale relevant to the frequency of interest—typically a few wavelengths. In the following, some representative depth-averaged properties are provided, with guidance on the frequency range for which they are applicable. A distinction is made between consolidated and unconsolidated sediments as follows.

172 Sonar oceanography

4.4.1

[Ch. 4

Unconsolidated sediments

The individual grains of surface sediments are usually interconnected only loosely and are able to slide relative to one another. A common assumption is that such unconsolidated sediments are unable to support a large shear stress and can be characterized by their compressional properties alone. This contrasts with harder consolidated sediments and rocks, whose ability to propagate shear waves is not negligible (see Section 4.4.2). The most common unconsolidated marine sediments are clastic sediments, which are made up from tiny fragments of weathered rock. Other sediment types are chemical sediments (made from salts previously dissolved in seawater) and biogenic sediments (made of decomposing plants and animals). Here the main emphasis is placed on clastic sediments as these are the most abundant. According to Buckingham (2005), many of the properties of unconsolidated sediments can be obtained from the knowledge of a single parameter: sediment porosity. The basic relationships derived by Buckingham are described in Chapter 5. The emphasis here in Chapter 4 is on measured values of geoacoustic parameters, and the empirically determined relationships between them. 4.4.1.1

Pure samples and porosity

Pure sediment samples (i.e., those Table 4.15. Sediment type vs. grain diameter. whose grains all have the same size) Sediment type Grain diameter can be classified according to grain diameter as in Table 4.15. SubClay 2 mm approximated as a fluid medium. Acoustical behavior can be characterized to a large extent by the plane wave reflection coefficient RðÞ. For a fluid, RðÞ depends on density sed , sound speed csed , and attenuation coefficient sed . The depth variation of these parameters is such that the characteristic impedance sed csed tends to increase with increasing depth in the sediment. Grain size is a useful descriptor of the acoustical properties of the seabed because it is strongly correlated with sound speed and density and, for a given sediment type, is independent of depth. The term ‘‘grain size’’, denoted M, is a logarithmic measure of grain diameter d defined as M
log2

d ; dref

ð4:87Þ

where dref
1

mm:

ð4:88Þ

Sec. 4.4]

4.4 Properties of the seabed 173

The porosity of a fluid–solid mixture is the ratio of the volume occupied by the fluid to the total volume of fluid plus solid. Porosity is related to density via sed ¼ w þ ð1  Þgrain ;

ð4:89Þ

where the grain density grain is approximately 2680 kg/m 3 (Hamilton and Bachman, 1982). Therefore, using 1027 kg m 3 for w , ¼ 1:62  0:622

sed : w

ð4:90Þ

Like grain size, sediment porosity is correlated with its density and sound speed. The grains become more tightly packed with increasing pressure, and consequently porosity decreases with increasing depth into the sediment. 4.4.1.2

Mixed samples and the ‘‘phi’’ scale

Naturally occurring sediments are mixtures of different sediment types, with different grain diameters. They can be characterized by an average value of the logarithmic grain size, denoted Mz , and defined as (Folk, 1966)   1 d16 d50 d84 Mz
 log2 þ log2 þ log2 ; ð4:91Þ 3 dref dref dref where the subscript denotes the percentile by weight (e.g., d50 is the median grain diameter). The parameter Mz is referred to in the literature as ‘‘mean grain size’’. By convention the averaging is done in log space, so that Mz is a measure of the geometric mean diameter and not the arithmetic mean. This can be seen by rewriting the definition as d Mz ¼ log2 GM ; ð4:92Þ dref where dGM ¼ ðd16 d50 d84 Þ 1=3 : ð4:93Þ Grain size is often expressed in so-called ‘‘phi units’’ (Appendix B). If the right-hand side of Equation (4.92) is equal to x, this is written as Mz ¼ x, meaning that the mean grain size in phi units is x. Table 4.16 lists sub-divisions of sediment type using the Udden–Wentworth25 classification scheme as presented by Krumbein and Sloss (1963). Also included in the table are the names of typical mixed samples using the Shepard (1954) and Folk (1954) classification schemes. The Shepard scheme groups sediment types according to the relative proportions of sand, silt and clay in a sample. For example, ‘‘sandy silt’’ is mostly silt but with a significant proportion of sand, whereas ‘‘sand–silt–clay’’ indicates roughly equal proportions of all three. Folk’s classification scheme is similar, but more suited for classifying coarser sediments 25 The sediment classification scheme by grain size is attributed by Krumbein and Sloss (1963) to Wentworth (1922), but the naming of silts and clays owes at least as much to Udden (1914). The Udden–Wentworth scheme is in widespread use for the classification of marine sediments.

174 Sonar oceanography

[Ch. 4

Table 4.16. Definition of sediment grain sizes and qualitative descriptions. Sediment description (Udden–Wentworth)

Grain size parameter MðÞ

Grain diameter d/mm

Typical mixed sample of the same mean grain size a Shepard

Folk

< 8

>256

Gravel

Large cobble gravel

8 to 7

128–256

Gravel

Small cobble gravel

7 to 6

64–128

Gravel

Very large pebble gravel

6 to 5

32–64

Gravel

Large pebble gravel

5 to 4

16–32

Gravel

Medium pebble gravel

4 to 3

8–16

Gravel

Small pebble gravel

3 to 2

4–8

Sandy gravel

Granule gravel

2 to 1

2–4

Muddy sandy gravel

1 to 0

1–2

Sand

Gravelly sand

Coarse sand

0 to 1

1 2 –1

Sand

Gravelly sand

Medium sand

1 to 2

1 1 4–2

Sand

Muddy gravelly sand b

Fine sand

2 to 3

1 1 8–4

Silty sand

Gravelly muddy sand

Very fine sand

3 to 4

1 1 16 – 8

Silty sand

Muddy sand

Coarse silt

4 to 5

1 1 32 –16

Sandy silt

Sandy gravelly mud b

Medium silt

5 to 6

1 1 64 –32

Silt

Gravelly sandy mud

Fine silt

6 to 7

1 1 128 – 64

Sand–silt–clay

Sandy mud

Very fine silt

7 to 8

1 1 256 – 128

Clayey silt

Mud

Coarse clay

8 to 9

1 1 512 – 256

Silty clay

Mud

Medium clay

9 to 10

1 1 1024 – 512

Clay

Mud

Fine clay

10 to 11

1 1 2048 – 1024

Clay

Mud

Boulder gravel

Very coarse sand

a Guide of typical sediment mixtures only. Mean grain size is not enough on its own to determine either the Folk or Shepard class unambiguously. b The terms ‘‘muddy gravelly sand’’ and ‘‘sandy gravelly mud’’ are not included in Folk’s classification but there seems no obvious reason for excluding them. See Krumbein and Sloss (1963, p. 158) for a more general scheme if needed.

Sec. 4.4]

4.4 Properties of the seabed 175

as it takes into account the proportion of gravel, and makes no distinction between silt and clay, adopting the term ‘‘mud’’ for both. The Folk and Shepard classification schemes are both based on triangles and hence limited to a maximum of three primary sediment types (gravel, sand, and mud for one, and sand, silt, and clay for the other). Thus the user of either one must choose between ignoring the (acoustically important) distinction between silt and clay or ignoring the gravel content altogether. An alternative classification scheme, based on a tetrahedron and thus allowing all four primary sediment types, is suggested by Krumbein and Sloss (1963). 4.4.1.3

Near-surface (high-frequency) properties

The boundary between water and sediment is characterized by a transition between the bulk properties of water and those of the sediment, across a distance of a few millimeters or centimeters (Lyons and Orsi, 1998; Pouliquen and Lyons, 2002; Tang et al., 2002; Tang, 2004). At high frequency (above 10 kHz) the properties of the transition layer become significant. APL-UW (1994) presents empirical parameters representative of the average properties of the top few centimeters, partly based on the work of Hamilton (Hamilton, 1972; Hamilton and Bachman, 1982) and applicable to frequencies between 10 kHz and 100 kHz (APL-UW, 1994; Sternlicht and de Moustier, 2003). For this reason the subscript ‘‘HF’’ is used to denote these nearsurface properties. The equations and values are given in Table 4.17 for grain sizes 1  Mz  þ9 (correcting a typographical error in the equation for sound speed from Sternlicht and de Moustier, 2003).26 The dimensionless ratios are approximately independent of salinity, temperature, and pressure. The attenuation coefficients (in Np/m) are, to a first approximation, proportional to frequency. This results in a constant value in nepers per wavelength (conventionally abbreviated Np/) or decibels per wavelength (dB/).27 The table presents values of attenuation  in units of dB/. These values can be related to  in Np/m according to sed ¼ 20 log10 e

4.4.1.4

csed sed : f

ð4:94Þ

Bulk (medium frequency) properties

Bulk geoacoustic parameters representative of the uppermost few meters of sediment are described by Hamilton (1972) and Bachman (1985). These are applicable to intermediate acoustic frequencies approximately in the range 1 kHz to 10 kHz, and 26

More recent work by Richardson is described by Jackson and Richardson (2007). Despite appearances, the symbols ‘‘Np/’’ and ‘‘dB/’’ both represent dimensionless units. This can be confirmed by inspection of Equation (4.94), the right-hand side of which is dimensionless. Therefore, so too must be the left-hand side. The paradox is resolved by realizing that the notation ‘‘dB/’’ is used as shorthand for a decibel per wavelength which is the same as a decibel per meter multiplied by the wavelength in meters. In other words 1 ‘‘dB/’’ ¼   1 dB/m, which is dimensionless. 27

176 Sonar oceanography

[Ch. 4

Table 4.17. Default HF geo-acoustic parameters (10–100 kHz). Near-surface sediment properties vs. grain size. Sediment description (see Table 4.16) Very coarse sand Coarse sand Medium sand Fine sand Very fine sand Coarse silt Medium silt Fine silt Very fine silt Coarse clay

Representative Sound speed

ratio cHF =cw

Density ratio HF =w

Attenuation coefficient HF ðdB=Þ

Porosity fraction HF

1

1.3370

2.492

0.91

0.07

0.5

1.3067

2.401

0.89

0.13

grain size MðÞ

0

1.2778

2.314

0.87

0.18

0.5

1.2503

2.231

0.87

0.23

1

1.2241

2.151

0.88

0.28

1.5

1.1782

1.845

0.86

0.47

2

1.1396

1.615

0.86

0.62

2.5

1.1073

1.451

0.85

0.72

3

1.0800

1.339

0.92

0.79

3.5

1.0568

1.268

1.00

0.83

4

1.0364

1.224

1.07

0.86

4.5

1.0179

1.195

1.15

0.88

5

0.9999

1.169

0.67

0.89

5.5

0.9885

1.149

0.36

0.91

6

0.9873

1.149

0.20

0.91

6.5

0.9861

1.148

0.16

0.91

7

0.9849

1.147

0.13

0.91

7.5

0.9837

1.147

0.10

0.91

8

0.9824

1.146

0.09

0.91

8.5

0.9812

1.145

0.08

0.91

9

0.9800

1.145

0.08

0.91

8 > 1:2778  0:056452Mz þ 0:002709M 2z cHF < ¼ 1:3425  0:1382798Mz þ 0:0213937M 2z  0:0014881M 3z > cw : 1:0019  0:0024324Mz 8 2:3139  0:17057Mz þ 0:007797M 2z > HF < ¼ 3:0455  1:1069031Mz þ 0:2290201M 2z  0:0165406M 3z > w : 1:1565  0:0012973Mz 8 0:4556 > > > > > > 0:4556 þ 0:0245Mz > > > cHF < 0:1978 þ 0:1245Mz HF ðdB=Þ ¼ 1:490 2 cw > > > 8:0399  2:5228Mz þ 0:20098M z > > 2 > > > 0:9431  0:2041Mz þ 0:0117M z > : 0:0601

1  Mz < 1 1  Mz < 5:3 5:3  Mz  9 1  Mz < 1 1  Mz < 5:3 5:3  Mz  9 1  Mz  0 0  Mz < 2:6 2:6  Mz < 4:5 4:5  Mz < 6:0 6:0  Mz < 9:5 9:5  Mz

Sec. 4.4]

4.4 Properties of the seabed 177

are referred to here as ‘‘medium frequency’’ (MF) parameters. Sound speed and density can be estimated using the following correlations from (Bachman, 1985) ) c^MF ¼ 1952  86:3Mz þ 4:14M 2z  1:5% ð0:81 < Mz < 9:70Þ ð4:95Þ ^MF ¼ 2380  172:5Mz þ 6:89M 2z  7:5% ð0:81 < Mz < 10:69Þ: These equations give sound speed in m/s and density in kg/m 3 , for standard conditions involving atmospheric pressure, a temperature of 23 C, and salinity 35. Thus, to obtain the ratio of these parameters to the corresponding ones in water (needed for calculating the plane wave reflection coefficient) it is necessary to divide by the value of the same parameter in water and for the same conditions (i.e., cw ¼ 1529.4 m/s and w ¼ 1024.2 kg/m 3 ). The attenuation coefficient, in units of dB/, is approximately independent of frequency in the range 1 kHz to 100 kHz (Kibblewhite, 1989). This means that a value of HF can be converted to MF by multiplying it by the ratio of sound speeds cMF =cHF . Results and equations for 1  Mz  þ10 are given in Table 4.18. (The equations leave cMF and MF , and by extension also MF , undefined for 0:50  Mz  0:81. It is reasonable to interpolate  and c linearly in Mz for intermediate values.) Equation (4.95) is intended for computing c or , given the grain size Mz . Bachman (1985) advises against its use for any other purpose, including the reverse conversion from c or  to Mz . Instead Bachman provides separate correlation equations for this and other similar conversions. A useful measure of uncertainty is provided by the percentage error from Equation (4.95). This is the standard error computed by Bachman for his data set. Similar uncertainties apply to high-frequency equations (for cHF and HF ). The uncertainty in the values of , for both HF and MF cases, is an order of magnitude larger. 4.4.1.5

Low-frequency properties

A vertical sound speed gradient normally exists in the seabed on a depth scale of 1 m to 100 m, affecting sound propagation at frequencies of 1 kHz or below. Although medium-frequency (MF) parameter values can sometimes be used as default lowfrequency parameters, there are significant complications. Some of these complications, discussed below, result in an increase in the reflection coefficient, and others in a decrease. Typical effects in deep water are quite different from those in shallow water, so they are treated separately. 4.4.1.5.1

Deep water

Deep sea sediments comprise very fine particles, typically of sizes corresponding to clayey silt or silty clay (see Table 4.16), the sound speed of which is close to and often less than that of seawater. Sound speed then increases with increasing depth in the sediment, with gradient dc=dz of order 1/s (Hamilton, 1979, 1980, 1987). The lowfrequency reflection coefficient is enhanced by this sound speed gradient because of the additional contribution to the reflected field from refracted sound. In deep water

178 Sonar oceanography

[Ch. 4

Table 4.18. Default MF geo-acoustic parameters (1–10 kHz). Bulk sediment properties vs. grain size. Sediment description (see Table 4.16)

Very coarse sand

Representative Sound speed

ratio cMF =cw

Density ratio MF =w

Attenuation coefficient MF ðdB=Þ

Porosity fraction MF

1

1.3370

2.492

0.91

0.07

0.5

1.3067

2.401

0.89

0.13

0

1.2778

2.314

0.87

0.18

0.5

1.2503

2.231

0.87

0.23

1

1.2226

2.162

0.87

0.28

1.5

1.1978

2.086

0.88

0.32

2

1.1743

2.014

0.88

0.37

2.5

1.1522

1.945

0.89

0.41

3

1.1314

1.879

0.96

0.45

3.5

1.1120

1.817

1.05

0.49

4

1.0939

1.758

1.13

0.53

4.5

1.0772

1.702

1.22

0.56

5

1.0619

1.650

0.71

0.60

5.5

1.0479

1.601

0.38

0.63

6

1.0352

1.555

0.21

0.65

6.5

1.0239

1.513

0.17

0.68

7

1.0140

1.474

0.13

0.70

7.5

1.0054

1.439

0.11

0.73

8

0.9982

1.407

0.09

0.75

8.5

0.9923

1.378

0.08

0.76

9

0.9877

1.353

0.08

0.78

9.5

0.9846

1.331

0.09

0.79

0.9827

1.312

0.09

0.81

grain size MðÞ

Coarse sand

Medium sand

Fine sand

Very fine sand

Coarse silt

Medium silt

Fine silt

Very fine silt

Coarse clay

Medium clay

10 cMF ¼ cw MF ¼ w

( (

cHF =cw

1  Mz  0:5

1:2763  0:05643Mz þ 0:002707M 2z

0:81 < Mz < 9:70

cHF =cw

1  Mz  0:5

2:3237  0:16842Mz þ

c =c MF ðdB=Þ ¼ MF w HF ðdB=Þ cHF =cw

0:006727M 2z

0:81 < Mz < 10:69

Sec. 4.4]

4.4 Properties of the seabed 179

the sediment layer is usually several hundred meters thick, which means that for most sonar frequencies the interaction with the solid earth crust (a layer of igneous rock beneath sediment and sedimentary rock) may be ignored. In addition to the sound speed gradient, a second complication is that the attenuation coefficient, in units of dB/, is known to be lower at frequencies below 1 kHz than above this frequency (Kibblewhite, 1989; Potty et al., 2003). Finally, both density and attenuation coefficient vary with depth, although the effect of these gradients on low-frequency sound is minor compared with that of the sound speed gradient.

4.4.1.5.2 Shallow water In shallow water, the effect of a sound speed gradient is generally less important than for deep water, and this is for two main reasons. The first is that in shallow water the predominant sediment type is sand, whereas in deeper water it is more likely to be clay. The acoustical properties of sand (primarily its high sound speed) are such that little sound penetrates into the sediment and this means that the sound speed gradient in the sediment is relatively unimportant. The second reason is that the sediment thickness in shallow water is typically less than in deep water, so for the same gradient there is a smaller contrast between the sound speed at the top of the sediment and that at the bottom. The main complication in shallow water arises from the interaction of sound with harder layers of igneous or sedimentary rock beneath the sediment. These rocks must be treated as solids that support shear waves.28 The speed of propagation of shear waves is called the shear speed, and denoted cs . In such conditions the rigidity of the sediment layer, though relatively low, also becomes important. According to Hamilton (1987) the shear speed of sediments can be parameterized in the form c^s ¼ A^ z x;

ð4:96Þ

where A depends on the grain size; and x is a constant, approximately equal to 0.31. An empirical fit to expressions from Hamilton (1987) for different sediment types is A ¼ 79 þ 41 expð0:4Mz Þ

ðMz  1Þ:

ð4:97Þ

Because of their low rigidity, unconsolidated sediments support shear waves only weakly, resulting in very low shear wave speeds. Typical values of the ratio cs =cp are less than 0.1 near the sediment surface for unconsolidated sediments, increasing to 0.4 at a depth of 1000 m (Hamilton, 1979). The appropriate correlation equations are 28 Shear waves can arise when the medium has non-zero rigidity, such that there exists a restoring force for particle displacements that do not involve a local change in volume (see Chapter 5). In a shear wave the displacement of individual particles away from their mean position is normal to the wave propagation direction. This contrasts with an ordinary sound wave (compressional wave), for which the displacement is always parallel to the propagation direction.

180 Sonar oceanography

[Ch. 4

(Hamilton, 1979, 1980) 8 3:884^ cp  5757 1512 < c^p  1555:15 > > > < 1:137^ cp  1485 1555:15 < c^p  1657:13 c^s ¼ 2 > 0:47^ c p  1:136^ cp þ 991 1657:13 < c^p  2150:59 > > : 0:78^ cp  962 2150:59 < c^p ,

ð4:98Þ

where the precise transition values have been adjusted slightly to ensure that cs approximates to a continuous function of cp . The shear waves are also heavily attenuated, with a typical value for the attenuation coefficient of between 3 dB m1 kHz1 and 30 dB m1 kHz1 , and no obvious correlation with grain size (Kibblewhite, 1989, Fig. 11). A discussion of possible frequency dependence in these parameters is deferred to Chapter 5. 4.4.2

Rocks

Once a layer of sediment has settled on the seabed, it is gradually covered with more layers. On a geological time scale, physical and chemical changes take place that convert the sediment material into a rigid structure that can no longer be separated into its individual grains. This process is known as lithification. The solid material resulting from lithification is called sedimentary rock. Other rock types are igneous rocks (made from re-solidified molten rock) and metamorphic rocks (very hard material, such as quartzite or marble, produced under extreme conditions of temperature and pressure from sedimentary or igneous rock). Metamorphic rocks are rarely encountered near the sea floor and we therefore concentrate here on sedimentary and igneous rocks. Sedimentary rocks are further classified according to the size of the constituent sediment grains as listed in Table 4.19 (Amateur Geologist, www). 4.4.2.1

Wave speed—density correlation equations

Rocks are characterized by their high rigidity, and they cannot reasonably be represented by means of a fluid model. Instead they must be treated as solids that Table 4.19. Names of sedimentary rocks resulting from the lithification of different sediment types. Grain size

Original sediment type

Resulting sedimentary rock

M < 1

Gravel

Conglomerate

1 < M < 4

Sand

Sandstone

4 > ¼ 16:23  16:61 þ 4:798 > 1000 m s 1 1000 kg m 3 1000 kg m 3 =  2 > ð4:99Þ > cs rock rock ; ¼ 2:17  3:67 þ 1:515 : > 1000 m s 1 1000 kg m 3 1000 kg m 3 For rocks whose density exceeds 3000 kg/m 3 (mostly metamorphic rocks) the dependence is nearly linear:   9 cp rock >  0:63 > ¼ 2:63 > = 1000 m s 1 1000 kg m 3 : ð4:100Þ   > cs rock > > ; ¼ 1:47  0:27 1000 m s 1 1000 kg m 3 Combining the linear form of Equation (4.100) (denoted clin below) with the quadratic one of Equation (4.99) (denoted cquad ), an overall fit can be obtained that is valid for rock  1900 kg/m 3 8 1=8 call ¼ ðc 8 quad þ c lin Þ

500 m s 1 :

ð4:101Þ

Equation (4.101) is applicable to both compressional (p) and shear (s) waves. Graphs of call are plotted for both p-wave and s-wave speeds in Figure 4.22. For the special case of basalt (an igneous rock common in the oceanic crust), a large number of measurements is presented by Christensen and Salisbury (1975), who fit their data (for a pressure of 50 MPa) to the following empirical equations, valid for density values in the range rock ¼ 2100–3000 kg/m 3  3:63 9 cp rock > ¼ 2:33 þ 0:081  0:03 > > = 1000 m s 1 1000 kg m 3 : ð4:102Þ  4:85 > > cs rock > ; ¼ 1:33 þ 0:011  0:03 1000 m s 1 1000 kg m 3 The Christensen–Salisbury curves for basalt are also shown in Figure 4.22. 4.4.2.2

Typical parameter values

In this section, typical values of geoacoustic parameters are given for a number of important types of sedimentary and igneous rocks. Values are rounded to the nearest 50 kg/m 3 , 50 m/s, and 0.05 dB/. The variation around these representative values can be large as exemplified by Figure 4.21. The main sources used to construct Table 4.20 are Carmichael (1982) for wave speeds and Jensen et al. (1994) for attenuation. Other source references are Christensen and Salisbury (1975) for selected properties of basalt, plus Assefa and Sothcott (1997) and Hamilton (1979).

Sec. 4.4]

4.4 Properties of the seabed 183

Figure 4.22. Fit to Ludwig’s data for all rocks (including basalts, Equation 4.101) and Christensen–Salisbury equations for basalts (Equation 4.102).

Table 4.20. Representative geoacoustic parameters for typical sedimentary and igneous rocks, in order of increasing density. /(kg/m 3 )

cp /(m/s)

p /(dB/)

cs /(m/s)

s /(dB/)

Mudstone

1500

2050

0.15

600

0.40

Chalk

2200

2400

0.20

1000

0.50

Sandstone

2400

4350

0.10

2550

0.25

Basalt

2550

4750

0.10

2350

0.20

Granite

2650

5750

0.10

3000

0.20

Limestone

2700

5350

0.10

2400

0.20

Type of rock

4.4.3

Geoacoustic models

The lower the acoustic frequency, the deeper the sound penetrates into the seabed. Thus for accurate modeling of low-frequency sound propagation, a description of seabed acoustic properties (e.g., sound speed and attenuation, shear speed and

184 Sonar oceanography

[Ch. 4

attenuation, and density) vs. depth is needed. The collection of such profiles for a given location is known as a geoacoustic model. For example, a geoacoustic model representative of the continental shelf might comprise a thin sand sediment over a layer of sandstone, and a granite crust beneath. A typical deep-water model could be a thick layer of clay over mudstone, with a basalt crust below these. Either or both of the sediment and sedimentary rock layers can be absent, especially in shallow water (as in the case of exposed granite). For examples, see Hamilton (1980).

4.5

REFERENCES

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Leroy, C. C. (2001) The speed of sound in pure and neptunian water, in M. Levy, H. E. Bass, and R. R. Stern (Eds.), Handbook of Elastic Properties of Solids, Liquids and Gases (Volume IV), Academic Press, London. Leroy, C. C., Robinson S. P., and Goldsmith, M. J. (2008) A new equation for the accurate calculation of sound speed in all oceans, J. Acoust. Soc. Am., 124, 2774–2782. Liebermann, L. N. (1948) The origin of sound absorption in water and sea water, J. Acoust. Soc. Am., 20, 868–873. Lindau, R. (1995) A new Beaufort equivalent scale, Proceedings International COADS Workshop, Kiel, Germany: Berichte aus dem Institu¨t fur Meereskunde (pp. 232–252). Love, R. H. (1977) Target strength of an individual fish at any aspect, J. Acoust. Soc. Am., 62, 1397–1403. Love, R. H. (1978) Resonant acoustic scattering by swimbladder-bearing fish, J. Acoust. Soc. Am., 64, 571–580. Løvik, A. and Hovem, J. M. (1979) An experimental investigation of swimbladder resonance in fishes, J. Acoust. Soc. Am., 66, 850–854. Ludwig, W. J., Nafe, J. E., and Drake, C. L. (1970) Seismic refraction, in A. E. Maxwell (Ed.), The Sea (Vol. 4, Part I, pp. 53–84), Wiley, New York. Lyons, A. P. and Orsi, T. H. (1998) The effect of a layer of varying density on high-frequency reflection, forward loss, and backscatter, IEEE J. Oceanic Eng., 23, 411–422. Mackenzie, K. V. (1981) Nine-term equation for sound speed in the oceans, J. Acoust. Soc. Am., 70, 807–812. MacLennan, D. N. and Simmonds, E. J. (1992) Fisheries Acoustics, Chapman & Hall, London. Mattha¨us, W. (1972) Die Viskosita¨t die Meerwassers, Beitra¨ge zur Meereskunde, 29, 93–107 [in German]. Medwin, H. and Clay, C. S. (1998) Fundamentals of Acoustical Oceanography, Academic Press, Boston. Mellen, R. H., Scheifele, P. M., and Browning, D. G. (1987) Global Model for Sound Absorption in Sea Water, Part II: Geosecs PH Data Analysis (NUSC Technical Report 7925, May). Naval Underwater Systems Center, Newport, RI. Miller, J. H. and Potter, D. C. (2001) Active high frequency phased-array sonar for whale shipstrike avoidance: Target strength measurements, paper presented at Oceans, pp. 2104– 2107. Millero, F. J. (2006) Chemical Oceanography (Third Edition), CRC Taylor & Francis, Boca Raton, FL. Morfey, C. L. (2001) Dictionary of Acoustics, Academic Press, San Diego. Nero, R. W., Thompson C. H., and Jech, J. M. (2004) In situ acoustic estimates of the swimbladder volume of Atlantic herring (Clupea harengus), ICES J. Marine Sciences, 61, 323–337. Neumann, G. and Pierson, W. J. (1957) A detailed comparison of theoretical wave spectra and wave forecasting methods, Deutsch. Hydrogr. Z., 10(3), 73–92. Neumann, G. and Pierson, W. J. (1966) Principles of Physical Oceanography, Prentice-Hall, Englewood Cliffs, NJ. NIST (www) CODATA internationally recommended values of the fundamental physical constants, http://physics.nist.gov/cuu/constants, last accessed March 8, 2008. NOAA (www) Guide to: sea state, wind and clouds, http://pajk.arh.noaa.gov/forecasts/ beaufort/beaufort.html, last accessed July 7, 2009.

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5 Underwater acoustics

We know very little about the murmur of the brook, the roar of the cataract, or the humming of the sea Marcel Minnaert (1933)

5.1

INTRODUCTION

While the scope of this chapter is not limited to bubble acoustics, this statement by Minnaert, published at the height of the quantum revolution, as part of his classic article ‘‘On musical air-bubbles and the sounds of running water’’ (Minnaert, 1933), is a stark reminder that we do not understand fully even the mundane. Nevertheless, important advances have been made in the intervening years, and the present chapter documents current theoretical knowledge of reflection, scattering, attenuation, and dispersion of underwater sound,1 starting with a derivation of the wave equations for fluid and solid media in Section 5.2. The plane wave reflection coefficients for fluid– fluid and fluid–solid boundaries are described in Section 5.3, including the effects of layering. Section 5.4 deals with the scattering of sound from rigid and non-rigid bodies and from rough boundaries. Finally, the dispersive effect of impurities, in the form of bubbly water or suspended sediment, is described in Section 5.5. 1

The propagation of sound in the sea is strongly influenced by small variations of sound speed with depth. This behavior has important consequences for the propagation of ocean acoustic signals of all kinds, including ambient noise and reverberation. A separate chapter (Chapter 9) is devoted to ocean acoustic propagation.

192 Underwater acoustics

5.2

[Ch. 5

THE WAVE EQUATIONS FOR FLUID AND SOLID MEDIA

Underwater sound is a manifestation of acoustic pressure waves traveling through the sea. These waves comprise successive regions of compression and rarefaction in which the local density is, respectively, slightly higher and slightly lower than the equilibrium density. The particle motion giving rise to these density changes is subject to a restoring force, which is determined by a parameter known as the bulk modulus of water. Because water is a fluid, to a large extent the theoretical study of underwater sound concerns itself with solutions to the wave equation for a fluid medium. There are also times when sound reflects from solid objects such as the seabed. In order to understand such interactions it is sometimes necessary to consider a second type of particle motion, known as shear or transverse motion, that does not result in a local change in density. The restoring force associated with shear motion is determined by the shear modulus. For a fluid medium, the shear modulus is zero. The purpose of this section is to provide a mathematical description of both kinds of motion in terms of bulk and shear moduli, and to explain how these are related to other acoustical parameters such as the speed of sound. 5.2.1 5.2.1.1

Compressional waves in a fluid medium Equations of motion

The behavior of a compressible fluid is described by two fundamental equations of motion. The first, a statement of conservation of mass, can be written @ þ rEðuÞ ¼ 0; @t

ð5:1Þ

where  is the fluid density; and u is the particle velocity. The second is a statement of Newton’s second law (force equals mass times acceleration), which simplifies, if the forces of gravity and viscosity are neglected, to the following equation due to Euler relating particle acceleration to the gradient of the pressure P (Pierce, 1989). 1 @u rP þ þ ðuErÞu ¼ 0:  @t

ð5:2Þ

Expanding in powers of particle velocity (whose magnitude is assumed small compared with the speed of sound), and retaining only the lowest order terms, these equations become @ þ 0 rEu  0 @t

ð5:3Þ

1 @u rP þ  0; 0 @t

ð5:4Þ

and

where 0 is the equilibrium density. Taking the time derivative of Equation (5.3) and the divergence of Equation (5.4), and eliminating the velocity terms gives the following linear second-order

Sec. 5.2]

5.2 The wave equations for fluid and solid media 193

differential equation

5.2.1.2




 1 1 @ 2 rE rP  ¼ 0: 0 0 @t 2

ð5:5Þ

Bulk modulus and the acoustic wave equation

Bulk modulus is a parameter that quantifies the restoring force of a fluid to a local change in volume or density. If a material is subject to a dilatation D, defined as the fractional increase in volume V D¼ ; ð5:6Þ V0 there is an associated change in pressure P. Assuming a linear relationship between them, the bulk modulus is the constant of proportionality B in the equation P ¼ BD:

ð5:7Þ

Thus, B is like the stiffness of a spring in Hooke’s law, a measure of the material’s elastic strength, and equal to the reciprocal of the fluid’s compressibility K K  1=B:

ð5:8Þ

From the definition of B it follows that B¼

dP : d

Substituting this expression into Equation (5.5) gives
 1 @ 2P BrE rP  2 ¼ 0: 0 @t

ð5:9Þ

ð5:10Þ

Equation (5.10) is the linear wave equation satisfied by a pressure wave in a compressible fluid medium of density 0 and bulk modulus B. The corresponding wave equation satisfied by the dilatation D is
 1 @ 2D rE rðBDÞ  2 ¼ 0: ð5:11Þ 0 @t Having neglected second-order and higher order terms, the scope is limited hereafter to the regime of linear acoustics. Leighton (2007a) describes the effects of discarded non-linear terms and the conditions under which they might become significant. 5.2.1.3

Compressional wave speed

Equation (5.10) is the linear wave equation for a compressible fluid medium whose density varies with position. The generic equation for a field variable F describing a wave traveling at speed c is r 2F 

1 @ 2F ¼ 0: c 2 @t 2

ð5:12Þ

194 Underwater acoustics

[Ch. 5

Comparing Equation (5.10) with Equation (5.12) (and assuming the equilibrium density 0 to be spatially uniform) it can be seen that the speed of sound c is related to density and bulk modulus according to sffiffiffiffiffi B c¼ : ð5:13Þ 0

5.2.2

Compressional waves and shear waves in a solid medium

Whether in a fluid or solid, motion that results in local changes in density (through compression or rarefaction) is opposed by a force proportional to bulk modulus. In a solid, a second type of motion is possible, known as shear or transverse motion, that does not result in density changes. Waves associated with compressional and transverse motion in a solid are the subject of this sub-section. 5.2.2.1

Shear modulus and the wave equations for a solid

Transverse motion in a solid is opposed by a restoring force proportional to the displacement. For this kind of force, the constant of proportionality (i.e., the ratio of shear stress to shear strain) is known as the shear modulus (or rigidity modulus) and denoted . Kolsky (1963) derives the following differential equation relating the displacement vector x to the dilatation D 0

@ 2x ¼ ðB þ 13 ÞrD þ r 2 x; @t 2

ð5:14Þ

valid if B, , and 0 are all independent of position. The wave equation for the dilatation follows from Equation (5.14) by taking its divergence and noting that rEx ¼ D:

ð5:15Þ

The result is 0

@ 2D ¼ ðB þ 43 Þr 2 D: @t 2

ð5:16Þ

Traveling-wave solutions to Equation (5.16) are known as compressional waves (also ‘‘longitudinal waves’’ or ‘‘p waves’’), and are illustrated by the upper panel of Figure 5.1. Taking the curl of Equation (5.14) instead of its divergence results in the following equation for curl x 0

@2 ðr ^ xÞ ¼ r 2 ðr ^ xÞ; @t 2

ð5:17Þ

which is the wave equation describing the propagation of shear motion (i.e., motion due to lateral displacements that are not accompanied by a change in volume). Traveling wave solutions to Equation (5.17) are known as shear waves (also ‘‘transverse waves’’ or ‘‘s waves’’), as illustrated by the lower panel of Figure 5.1.

Sec. 5.2]

5.2 The wave equations for fluid and solid media 195

Figure 5.1. Illustration of compressional (p) and shear (s) wave propagation (from Leighton, 2007a, # Elsevier, reprinted with permission).

5.2.2.2

Lame´ parameters, Young’s modulus, and Poisson’s ratio

It is often the case that the elastic properties of materials are described in terms of the so-called Lame´ parameters, denoted  and . Of these,  is the shear modulus introduced in Section 5.2.2.1, while  is related to bulk and shear moduli according to  ¼ B  23 : ð5:18Þ

196 Underwater acoustics

[Ch. 5

Other parameters sometimes encountered are Young’s modulus E (the ratio of longitudinal stress to longitudinal strain) and Poisson’s ratio (the ratio of lateral contraction to the longitudinal extension of the material). These are related to the Lame´ parameters via (Kolsky, 1963) E¼

3 þ 2  þ

ð5:19Þ

¼

 : 2ð þ Þ

ð5:20Þ

and

5.2.2.3

Compressional and shear wave speeds

Comparing Equations (5.17) and (5.16) with Equation (5.12), it can be seen that the wave speeds of p and s waves are, respectively, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B þ 43  cp ¼ ð5:21Þ 0 and rffiffiffiffiffi  : ð5:22Þ cs ¼ 0 In a common approximation known as the ‘‘fluid sediment’’ model, the ocean sediment is modeled as if it were a fluid by assuming that the effects of shear waves are negligible. The name of this model can be misleading because it seems to imply that sediment rigidity is neglected. However, even when cs is small, the rigidity term in Equation (5.21) provides an important correction to the sediment compressional wave speed. Thus, what is neglected is usually not  but the energy associated with shear motion. The p-wave and s-wave speeds of ocean sediments can be estimated using the model described in Section 5.5.2. For most materials, an extension along one axis results in a compression in a direction perpendicular to that axis, implying that Poisson’s ratio must be positive, and requiring in turn that Lame´’s  parameter also be positive. Writing the compressional speed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi cp ¼

 þ 2 ; 0

ð5:23Þ

and requiring a positive shear modulus, it follows from Equation (5.22) that cs and cp are related according to c 2s 1 < : ð5:24Þ c 2p 2 Thus, a p wave in a solid medium always travels faster than an s wave in the same solid. If a given event (say, an earthquake in Earth’s crust) generates both types of wave simultaneously, a distant receiving station detects the arrival of the p wave before that of the s wave. The terms ‘‘p wave’’ and ‘‘s wave’’ were originally coined as

Sec. 5.3]

5.3 Reflection of plane waves 197

abbreviations of ‘‘primary wave’’ (meaning the wave that arrives first) and ‘‘secondary wave’’. In modern use they are associated with ‘‘compressional wave’’ and ‘‘shear wave’’. A further condition (Equation 5.31) follows from the requirement for the complex bulk modulus to have a negative imaginary part (otherwise a pure compression would violate the principle of conservation of energy). A derivation of this condition follows. The complex bulk modulus can be defined as B~  ~ c 2p  43 ~ c 2s ; where c~p and c~s are the complex p-wave and s-wave speeds defined as cp c~p  1 þ ip cp =! and cs c~s  ; 1 þ is cs =!

ð5:25Þ

ð5:26Þ ð5:27Þ

where cs are cp are the wave speeds of s and p waves; and s and p are the corresponding attenuation coefficients in nepers per unit distance. They are equal to the imaginary part of the respective complex wavenumber of the shear wave ! ks ¼ þ is ð5:28Þ cs and compressional wave ! kp ¼ þ ip : ð5:29Þ cp The complex bulk modulus is then B~ ¼

c 2p 4 c 2s  : ð1 þ ip cp =!Þ 2 3 ð1 þ is cs =!Þ 2

ð5:30Þ

Assuming the imaginary parts to be small, the requirement for B~ to have a negative imaginary part leads to the inequality s c 3s 3 < : p c 3p 4

5.3

ð5:31Þ

REFLECTION OF PLANE WAVES

Many sources of underwater sound can be approximated to first order as points that generate spherical waves. After traveling some distance these spherical waves expand and the wavefront curvature is reduced. Eventually the curvature becomes negligible and for some applications the wave may then be approximated by a plane wave, simplifying the analysis. The reflection of plane waves from plane boundaries is the present subject.

198 Underwater acoustics

5.3.1 5.3.1.1

[Ch. 5

Reflection from and transmission through a simple fluid–fluid or fluid–solid boundary Amplitude reflection coefficient

Consider a plane wave of amplitude Ainc incident on a plane boundary separating two uniform fluid half-spaces2 pinc ¼ Ainc expði 1 z  ixÞ e i!t :

ð5:32Þ

3

If the wavenumber and ray grazing angle in the medium of the incident and reflected waves are denoted, respectively, k1 and 1 , the horizontal and vertical wavenumbers can be written, respectively,  ¼ k1 cos 1 ð5:33Þ and

1 ¼ k1 sin 1 : ð5:34Þ If the reflected wave has amplitude Aref such that pref ¼ Aref expði 1 z  ixÞ e i!t ;

ð5:35Þ

the plane wave amplitude reflection coefficient is defined as the ratio of the two amplitudes A R  ref : ð5:36Þ Ainc The simplest non-trivial case is that of a plane boundary between two uniform fluid media. Denoting the sound speed and density of each layer by ci and i , where i ¼ 1 or 2, the reflection coefficient is (Brekhovskikh and Lysanov, 2003) R¼

1 ; þ1

ð5:37Þ

¼

2 1 ; 1 2

ð5:38Þ

where  is the impedance ratio

and the horizontal and vertical wavenumbers are related by

2i þ  2 ¼ k 2i ;

i ¼ 1 or 2

ð5:39Þ

where (for an angular frequency !), ! ; c1

ð5:40Þ

! þ i2 : c2

ð5:41Þ

k1 ¼ and k2 ¼

2 For the remainder of this chapter the complex variable p is used to represent the acoustic pressure as described in Chapter 2. 3 The angle between the wave vector and the horizontal plane.

Sec. 5.3]

5.3 Reflection of plane waves 199

The expression for R given by Equation (5.37) is known as the Rayleigh reflection coefficient. The true grazing angle of the refracted wave is (using Snell’s law)
 c2 ð2 Þtrue ¼ arccos cos 1 : ð5:42Þ c1 It is convenient to represent this angle by means of the closely related complex angle 2 ¼ arccosð=k2 Þ;

ð5:43Þ

the real part of which is approximately equal to the true grazing angle given by Equation (5.42). The purpose of the imaginary part of k2 is to simulate the effect of a decaying wave in layer 2. If layer 2 is a solid, Equation (5.37) still applies if  is generalized to (Brekhovskikh and Godin, 1990)  ¼ p cos 2 2s þ s sin 2 2s ;

ð5:44Þ

where the ‘‘p’’ and ‘‘s’’ subscripts indicate properties of the compressional and shear waves, respectively, in the solid. Specifically, the impedance ratios are then given by  ð5:45Þ X ¼ 2 1 ðX ¼ p or sÞ 1 X and X denotes the (complex) grazing angle of the p wave or s wave in layer 2 such that
  X ¼ arccos ; ð5:46Þ !=cX þ iX where cX and X are the corresponding wave speed and attenuation coefficient, respectively. Finally, each of the two vertical components of wavenumber X is related to the magnitude of the corresponding wavenumber kX ¼ !=cX according to

2X ¼ k 2X   2 ;

ð5:47Þ

X ¼ kX sin X :

ð5:48Þ

or, equivalently,

5.3.1.2

Amplitude transmission coefficients

In addition to the reflected compressional wave there is also a transmitted one. In the case of a fluid–solid boundary there is further a transmitted shear wave. Amplitude transmission coefficients for both types of wave are given in this section. It is convenient to describe the incident displacement field xinc in terms of a scalar potential inc , such that the corresponding displacement is equal to the gradient of this potential xinc ¼ rinc ; ð5:49Þ and similarly for the reflected field xref ¼ rref :

ð5:50Þ

A potential defined in this way represents a compressional wave. In general, the transmitted field in a solid comprises a shear wave as well as the usual compressional

200 Underwater acoustics

[Ch. 5

one. This situation can be represented by including in the analysis a vector potential t, such that the total (compressional plus shear) displacement is (Achenbach, 1975; Miklowitz, 1978)4 xtrans ¼ rtrans þ r ^ t: ð5:51Þ If the incident wave is in the x–z plane, any transverse displacement associated with the shear wave is confined to this plane, which means that the transmitted shear wave is associated with the y component of t, denoted y . Two transmission coefficients are needed to characterize the transmitted field, one for each of the two types of transmitted waves (p and s). The compressional wave transmission coefficient, denoted Tp , is defined as the ratio of the amplitude of trans to that of inc , including any phase change in the same way as for the pressure waves considered in Section 5.3.1.1. The result is (Miklowitz, 1978; Brekhovskikh and Godin, 1990) 1 p cos 2s Tp ¼ 2 ; ð5:52Þ 2 ð1 þ Þ where the impedance ratio  is given by Equation (5.44); p and s by Equation (5.45); and the shear wave grazing angle s by Equation (5.46) (with X ¼ s). In the same way, the shear wave transmission coefficient Ts is the ratio of the amplitude of y to that of inc , and is equal to   sin 2s Ts ¼ 2 1 s : ð5:53Þ 2 ð1 þ Þ 5.3.1.3

Energy reflection and transmission coefficients

In the following, a new set of reflection and transmission coefficients is introduced, defined as ratios of energy instead of amplitude. This is useful because energy is a conserved quantity, making it possible to derive simple relationships between the coefficients. Consider, as previously, a plane wave traveling in a fluid medium and incident on a plane fluid–solid boundary. The incident energy flux is partly reflected and partly transmitted. Making the simplifying assumption that the shear modulus  of the solid is real, it is possible unambiguously to associate certain proportions of the reflected and transmitted energy with each of the various reflected and transmitted paths: namely, the reflected p wave, the transmitted p wave, and the transmitted s wave.5 The energy reflection coefficient (i.e., the proportion of incident normal energy flux carried by the reflected wave) is then given by V ¼ jRj 2 :

ð5:54Þ

The p-wave energy transmission coefficient (the proportion of incident normal energy 4

The potentials  and t satisfy the same wave equations, respectively, as div x and curl x. If  has a non-zero imaginary part (implying the presence of an attenuation mechanism), such unambiguous association is no longer possible because the total energy sum includes a term proportional to ImðÞ and the product of p and s amplitudes (Ainslie and Burns, 1995). 5

Sec. 5.3]

5.3 Reflection of plane waves 201

flux carried by the transmitted p wave) is 2 Re p 1 Re 1

ð5:55Þ

2 Re s : 1 Re 1

ð5:56Þ

Wp ¼ jTp j 2 and, similarly, for the s wave Ws ¼ jTs j 2

Conservation of energy demands that incoming and outgoing energy flux be equal, and therefore V þ Wp þ Ws ¼ 1:

ð5:57Þ

The V and W terms, as defined above, are ratios of energies, and consequently are real-valued parameters. In general, the various amplitude coefficients (R, Tp , and Ts ) can take complex values, and there exists a similar relationship to Equation (5.57) that relates the closely related complex parameters (Deschamps and Changlin, 1989; Ainslie and Burns, 1995):
 1 2 2 ~ VR ¼ ; ð5:58Þ þ1 2 ~ p  T 2p 2 p ¼ 4 cos 2s 2 1 W 1 1 ð þ 1Þ 2 1 p

ð5:59Þ

2 ~ s  T 2s 2 s ¼ 4 sin 2s 2 1 : W 1 1 ð þ 1Þ 2 1 s

ð5:60Þ

and

These complex coefficients are not energy ratios, but they have the remarkable property that ~p þ W ~ s ¼ 1; V~ þ W ð5:61Þ thus providing a useful check on the calculation of individual reflection and transmission coefficients. In contrast with Equation (5.57), Equation (5.61) holds even if Im  6¼ 0.

5.3.2

Reflection from a layered fluid boundary

Consider now a more complicated boundary involving a uniform fluid transition layer sandwiched between uniform fluid half-spaces above and below. For example, the transition layer might be a uniform sediment between water and substrate. This is a special case of the situation illustrated by Figure 5.2 with the substrate shear speed cs equal to zero. If the incident wave has unit amplitude, the reflection coefficient R is equal to the amplitude of the reflected wave. From the figure this is found by adding the following

202 Underwater acoustics

[Ch. 5

Figure 5.2. Example of layered boundary comprising a uniform fluid transition sediment layer between two uniform halfspaces above (water) and below (solid substrate).

infinite series R ¼ R12 þT12 ½R23 expð2i 2 hÞ f1þR21 ½R23 expð2i 2 hÞ þR 221 ½R23 expð2i 2 hÞ 2 þ  gT21 ; ð5:62Þ where Ri j and Ti j denote partial reflection and transmission coefficients (ratios of displacement potentials) for a wave incident on layer j from layer i. Thus, i j  1 i j þ 1

ð5:63Þ

Ti j ¼

i ð1 þ Ri j Þ; j

ð5:64Þ

j i i j

ði; j ¼ 1, 2, or 3Þ:

ð5:65Þ

Ri j ¼ and

where i j ¼

In Equation (5.62), the factor R23 always appears multiplied by the phase term expð2i 2 hÞ. This is because a reflection from the lower boundary is always accompanied by a two-way transit of the sediment layer. Thus, it is convenient to define the factor S23 as S23 ¼ R23 expð2i 2 hÞ: ð5:66Þ This factor is the reflection coefficient of the lower (sediment–substrate) boundary, modified by the two-way phase shift relative to the origin at the upper one. In fact, it is the sediment–substrate amplitude reflection coefficient for a depth origin at the water–sediment boundary. Substituting Equation (5.66) in Equation (5.62) it follows

Sec. 5.3]

5.3 Reflection of plane waves 203

that T12 S23 T21 : 1  R21 S23

ð5:67Þ

S23 ð1 þ R12 Þð1 þ R21 Þ 1  R21 S23

ð5:68Þ

R ¼ R12 þ Two equivalent forms are R ¼ R12 þ and R¼

R12 þ S23 : 1  R21 S23

ð5:69Þ

Equation (5.68) continues to hold even if the sediment layer is not uniform, provided that the following expressions are used for the individual reflection coefficients (Ainslie, 1996):   f þ ð0Þ R12 ¼ 12 ; ð5:70Þ 12 þ f þ ð0Þ R21 ¼

21 f  ð0Þ  1 ; 21 f þ ð0Þ þ 1

ð5:71Þ

S23 ¼

þ pþ 2 ðhÞ23 f ðhÞ  1 ;  p 2 ðhÞ23 f ðhÞ þ 1

ð5:72Þ

and

 where p þ 2 ( p 2 ) is the downward (upward) traveling pressure field in the sediment; and

f  ðzÞ ¼

dp  2 ðzÞ=dz : i 2 ðzÞp  2 ðzÞ

ð5:73Þ

An example of particular interest, because an analytical solution is known for p2 ðzÞ, involves the density profile due to (Robins, 1991) 2 ðzÞ ¼ 

2 ð0Þ z z 2 ;  cosh  sinh 2 2 ð0Þ 2

ð5:74Þ

combined with a linear k 2 profile in the sediment: k2 ðzÞ 2 ¼ k2 ð0Þ 2 ð1  2qzÞ:

ð5:75Þ

The parameters q, , and  are constants controlling the derivatives of density and sound speed at depth z ¼ 0. In particular,  is given by
 3 d 2 2 d 2  2 ¼ 2  : ð5:76Þ  dz 2  dz The sound speed and attenuation profiles are related to the wavenumber by ! c2 ðzÞ ¼ ð5:77Þ Re k2 ðzÞ

204 Underwater acoustics

[Ch. 5

and 2 ðzÞ ¼

40 Im k2 ðzÞ ; loge 10 Re k2 ðzÞ

ð5:78Þ

where the units of  are decibels per wavelength. For this special case, the functions p2 ðzÞ and f ðzÞ are given in terms of the Airy functions AiðxÞ and BiðxÞ (see Appendix A) by  2 ðzÞ 1=2 Ai½ ðzÞ  i Bi½ ðzÞ  p 2 ðzÞ ¼ ð5:79Þ 0 Ai½ ð0Þ  i Bi½ ð0Þ and 1 d2 ðzÞ Ai 0 ½ ðzÞ  i Bi 0 ½ ðzÞ i 2 ðzÞ f  ðzÞ ¼  0 ðzÞ ; ð5:80Þ 22 ðzÞ dz Ai½ ðzÞ  i Bi½ ðzÞ where GðzÞ 2 ðzÞ ¼ ð5:81Þ ð2qk 20 Þ 2=3 and 2 GðzÞ 2 ¼ 2 ðzÞ 2  : ð5:82Þ 4 An alternative recursive approach for evaluating the reflection coefficient of an arbitrarily layered fluid sediment, by means of multiple uniform sub-layers, is described by Jensen et al. (1994). 5.3.3

Reflection from a layered solid boundary

If one or more of the layers is a solid (in the sense of having a non-negligible shear speed), calculation of the reflection coefficient can become considerably more complicated. The simplest case arises if only the substrate is solid, with the sediment layer still a fluid, which is the situation depicted in Figure 5.2. For this case, Equation (5.68) still holds provided that Equation (5.44) is used for 23 in Equation (5.72). If the sediment layer is also a solid, Equation (5.67) generalizes to the matrix equation R ¼ R12 þ T12 ð1  S23 R21 Þ 1 S23 T21 ; where Ri j ¼

Ti j ¼

S23 ¼

R pp ij R sp ij T pp ij T sp ij

! R ps ij ; R ss ij ! T ps ij ; T ss ij

R pp 23 exp ipp R sp 23 exp isp

ð5:83Þ ð5:84Þ

ð5:85Þ ! R ps exp i ps 23 : R ss 23 exp iss

ð5:86Þ

and the two-way phase terms are XY ¼ ð X þ Y Þh:

ð5:87Þ

Sec. 5.3]

5.3 Reflection of plane waves 205

XY In these equations, the partial p–p and p–s coefficients R XY i j ; T i j are the ratios of velocity (or displacement) potentials as described by Miklowitz (1978). For an example of their application, see Ainslie (1995). For the case of multiple solid layers the above method can be generalized with a recursive solution due to Kennett (1974) (see also Chapman, 2004, pp. 263–268). An alternative calculation method, described by Schmidt (1988), is implemented in the widely used OASES-OASR model (Schmidt, ca. 2000).

5.3.4

Reflection from a perfectly reflecting rough surface

The reflective properties of a randomly rough—but otherwise perfectly reflecting— surface are determined by the statistics of a rough surface, characterized by the RMS height displacement6 of the surface , and its correlation length L. Together with the grazing angle  and acoustic wavenumber k, roughness determines the Rayleigh parameter Q for a plane wave Q ¼ 2k sin :

ð5:88Þ

If the correlation length is small, on the scale of a wavelength (kL  1), then the effects of rough surface scattering are usually small, scaling with the fourth power of frequency and the second power of correlation length. For this reason only largescale roughness, such that kL  1, is considered below. If a plane wave is reflected from a rough but otherwise plane surface, the reflected wave is a distorted version of the incident one. Small phase differences are introduced due to the interaction of the incident plane wave with different parts of the distorted boundary, and these phase differences manifest themselves as slight imperfections in reflected wavefronts.7 If the roughness height is small compared with the acoustic wavelength, the wavefronts are still recognizable as approximately planar, but with small imperfections mirroring those of the rough surface. If, further, the roughness has a random nature, one can define the coherent reflection coefficient Rc as an ensemble average of the ratio of the reflected wave amplitude to that of the incident wave. This ensemble average is taken over an infinite number of realizations of the randomly rough surface. 5.3.4.1

Perturbation theory (small Q)

This section follows Brekhovskikh and Lysanov (2003), who, for the case of small Q and large correlation length, use the method of ‘‘small perturbations’’ to obtain the result jRc j 2 ¼ 1  4k 2  2 Y sin ; ð5:89Þ 6

That is, the RMS departure from mean surface height, known sometimes as surface ‘‘roughness’’. 7 The Rayleigh parameter is a measure of these phase differences (Brekhovskikh and Lysanov, 2003).

206 Underwater acoustics

[Ch. 5

where Y is a dimensionless parameter related to the 2D wavenumber spectrum G2 ðsÞ of sea surface (or seabed) displacement. Specifically, if the 2D wavenumber vector s of the roughness spectrum has magnitude  and bearing , then Y is related to the 2D spatial roughness spectrum G2 ðsÞ according to ! ð 2  2 1=2 2 2 Y  G2 ðsÞ sin   cos  cos   2 ds; ð5:90Þ k k where the notation ds indicates a 2D element of ‘‘area’’ in wavenumber space such that ds ¼  d d, and the limits of integration are over all real values of the integrand. The roughness spectrum is related to the spatial correlation function BðÞ via the 2D Fourier transform ð 1 G2 ðsÞ ¼ 2 BðÞ e isEo do; ð5:91Þ 4 and is normalized such that ð

G2 ðsÞ ds ¼  2 :

ð5:92Þ

If 0 and  are the incident and scattered grazing angles, and  is the bearing of the scattered path, relative to that of the incident one, then the Bragg scattering vector is
 cos  cos ’  cos 0 s¼k : ð5:93Þ cos  sin ’ For example, the specular direction ( ¼ 0 ;  ¼ 0) corresponds to s ¼ 0. 5.3.4.1.1 Near-grazing (kL sin 2   1) The limit of large kL and small kL sin 2  is an important one in underwater acoustics, because the propagation geometry can result in grazing angles close to zero. The consequence of assuming a large correlation length is that the spectrum GðÞ vanishes at large distances from the wavenumber origin, and consequently the contribution to the integral from the third of the three bracketed terms in Equation (5.90) becomes negligible. If, further, the grazing angle  is sufficiently small compared with ðkLÞ 1=2 , the first term is also negligible, leaving
1=2 ð1 ð 3=2 2 Y   2 d d G2 ðsÞ  cos  : ð5:94Þ k 0 =2 Assuming G2 ðsÞ to be separable, such that G2 ðsÞ ¼ 2G1 ðÞKðÞ;

ð5:95Þ

where G1 ðÞ is the 1D spectrum; and KðÞ is a dimensionless function describing the

Sec. 5.3]

5.3 Reflection of plane waves 207

azimuthal dependence of the 2D spectrum, normalized according to ð þ KðÞ d ¼ 1;

ð5:96Þ



it follows that Y


 ð ð1 2 2 1=2 3=2 1=2 ðcos Þ KðÞ d  3=2 G1 ðÞ d: 2 k =2 0

ð5:97Þ

For an isotropic spectrum, that is, with this becomes

KðÞ ¼ 1=2;

ð5:98Þ


 ð 2EPT 2 1=2 1 Y¼ G1 ðÞ 3=2 d; k 2 0

ð5:99Þ

where EPT is a constant defined as EPT and equal to EPT

ð þ=2

pffiffiffiffiffiffiffiffiffiffiffi cos  d

ð5:100Þ

rffiffiffi 3 4 Gð4Þ ¼  0:3814:  Gð14Þ

ð5:101Þ

1  2

=2

As an example, consider the special case of an isotropic Gaussian roughness spectrum with correlation radius L ! 2 2 BðÞ ¼  exp  2 ; ð5:102Þ L so that the roughness spectrum takes the form (using Equation 5.91) G1 ðÞ ¼

2L2 expð 14  2 L 2 Þ: 4

It then follows from Equation (5.99) that ð EPT L 2 Y ¼ pffiffiffiffiffi  3=2 expð 14  2 L 2 Þ d: 2k Substituting for the integral
 ð Gð5=4Þ 2 5=2  3=2 expð 14  2 L 2 Þ d ¼ ; 2 L this becomes
 1 1=2 Y¼ Gð3=4Þ: kL

ð5:103Þ

ð5:104Þ

ð5:105Þ

ð5:106Þ

The coherent reflection coefficient is obtained by substituting this expression back into Equation (5.89) for Rc .

208 Underwater acoustics

[Ch. 5

5.3.4.1.2 Non-grazing (kL sin 2   1) For larger angles—that is, large compared with ðkLÞ 1=2 —the second bracketed term of Equation (5.90) may be neglected. Retaining only the first of the three terms, it follows that ð Y ¼  2 sin  G2 ðsÞ ds;

ð5:107Þ

Y ¼ sin ;

ð5:108Þ

and therefore irrespective of the roughness spectrum. 5.3.4.2

Heuristic extension for large Q

According to Brekhovskikh and Lysanov (2003), the above perturbation theory holds for small values of the Rayleigh parameter Q. For large Q, the Kirchhoff approximation may be used to obtain: ð þ1 Rc ¼ expð2ik sin ÞwðÞ d: ð5:109Þ 1

If a Gaussian distribution is assumed for the sea surface height displacement  ! 1 2 wðÞ ¼ pffiffiffiffiffiffi exp  2 ; ð5:110Þ 2 2 and putting jR0 j ¼ 1 for a perfect reflector, it follows that (Eckart, 1953) jRj 2 ¼ expðQ 2 Þ:

ð5:111Þ

This equation can be compared with the equivalent expression for small Q, obtained by substitution of Equation (5.108) in Equation (5.89) jRc j 2 ¼ 1  Q 2  expðQ 2 Þ ðQ 2  1Þ:

ð5:112Þ

Thus, for this case (i.e., for a Gaussian-distributed displacement, and in the limit of 2 large kL sin 2 ) the expression jRc j 2 ¼ e Q may be used for any Q, large or small. A heuristic extension of this is to assume that Equation (5.89) may be replaced with jRc j 2  expð4k 2  2 Y sin Þ;

ð5:113Þ

2

for any value of the argument (i.e., for any kL sin  and for any surface statistics), with Y from Equation (5.90) or (depending on the magnitude of the kL sin 2  parameter) one of the two limits given by Equations (5.99) and (5.108). 5.3.5

Reflection from a partially reflecting rough surface

So far, scattering has been considered from a perfect reflector (i.e., one that reflects 100 % of the incident energy). Assume now that some energy is transmitted across the boundary, such that the reflection coefficient of the surface, if perfectly smooth, would be R0 .

Sec. 5.4]

5.4 Scattering of plane waves

209

Further, let Rc denote the coherent reflection coefficient corresponding to the true roughness. Brekhovskikh and Lysanov (2003, p. 205) show that for a Gaussian surface elevation, in the Eckart (i.e., large Q) limit: Rc ¼ R0 e Q

2

=2

:

ð5:114Þ

A heuristic generalization of this expression, permitting both small and large values of Q, is Rc ¼ R0 expð2k 2  2 Y sin Þ: ð5:115Þ

5.4 5.4.1

SCATTERING OF PLANE WAVES Scattering cross-sections and the far field

The ability of an object to scatter sound can be characterized by its total scattering cross-section  tot , defined for an incident plane wave as the ratio of total scattered power W to the magnitude of the incident intensity, I0  tot  W=I0 :

ð5:116Þ

If the power is not re-directed uniformly in all directions it is useful to quantify the scattered power per unit solid angle in a given direction. This can be done in terms of the differential scattering cross-section O , defined as the ratio of scattered radiant intensity8 (WO ) to the incident intensity O  WO =I0 ;

ð5:117Þ

where the radiant intensity is understood to be measured in the far field of the scattering object.9 A related parameter is the backscattering cross-section (abbreviated BSX)10  back ðÞ  4O ð; ; Þ;

ð5:118Þ

where the meaning of the arguments is explained in Chapter 2. If the power is scattered uniformly in all directions, then  back is a constant (independent of angle ), and equal to  tot . The scattering cross-sections O and  back are therefore relevant to the far field. 8

Radiant intensity, denoted WO , is defined as power per unit solid angle. The field point is said to be in the far field of the object if the scattered pressure and particle velocity fields are in phase, such that the scattered intensity is purely radiative (Morse and Ingard, 1968). 10 This definition of backscattering cross-section, from Pierce (1989) and ASA (1994), is adopted consistently throughout this book. The reader’s attention is drawn to the potential for confusion with an alternative convention that is common in work related to fisheries acoustics (Clay and Medwin, 1977; MacLennan et al., 2002) that omits the factor 4 in Equation (5.118). With the alternative convention (indicated by the subscript ‘‘alt’’), the backscattering crosssection is defined as  back alt ðÞ  O ð; ; Þ, while the definition of total scattering cross-section (Equation 5.116) is unchanged. 9

210 Underwater acoustics

[Ch. 5

Theoretical expressions for BSX are given below for various solid objects (Section 5.4.2) and fluid ones (Section 5.4.3), including spherical gas bubbles and fish. Scattering from rough boundaries is considered in Section 5.4.4. Measurements of target strength (a logarithmic measure of BSX) of real submerged objects are summarised in Chapter 8, as well as the scattering strength of the sea surface and seabed, and the volume backscattering strength of scatterers that are extended in three dimensions. 5.4.2

Backscattering from solid objects

Exact solutions for the scattering cross-section are known for a limited number of simple shapes such as a sphere or cylinder (Dragonette and Gaumond, 1997). Figure 5.3 shows the magnitude of the form function, given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  back ðkaÞ ; ð5:119Þ j f ðkaÞj ¼ a 2 plotted vs. dimensionless radius (or frequency) for two hard spheres (one perfectly rigid and one made of tungsten carbide) in the uppermost two graphs. The remaining graphs show the same function calculated for spheres made of various metals. The wave speeds and density of tungsten carbide and the metals are listed in Table 5.1. For the tungsten carbide sphere, the departure of the form factor from that of a rigid sphere is small for ka < 5. The remainder of this section concentrates on approximate solutions for scattering from rigid bodies of various simple shapes, including the sphere. 5.4.2.1

Small rigid object of approximately spherical shape

For objects that are small compared with the acoustic wavelength in water, the scattering cross-section is approximately determined by the volume of the object V and the acoustic wavelength —independent of the shape of the object. For the case of a bistatic scattering angle  (defined as the angle between incident and scattered wave vectors), the differential scattering cross-section is (Urick, 1983) O ðÞ ¼

2V 2 ð1  32 cos Þ 2 : 4

ð5:120Þ

Substituting  ¼  to give the backscatter direction and using Equation (5.118) gives the following equation for the BSX11 3  back LF ¼ 25

V2 : 4

ð5:121Þ

Equation (5.121), derived by Rayleigh for a perfect sphere, is assumed here to apply more generally to slightly deformed spheres and in particular to ellipsoids with a 11 The subscript is used here to indicate special cases, such as applicability to a low-frequency (LF) or high-frequency (HF) limit.

Sec. 5.4]

5.4 Scattering of plane waves

Rigid

Tungsten carbide

Brass

Steel

Figure 5.3. Form function j f ðkaÞj vs. ka for a rigid sphere (uppermost graph), a tungsten carbide sphere (second graph), and spheres made of various metals in water (lower graphs). For the form function of cylinders made of the same materials, see Dragonette and Gaumond (1997) (reprinted from Dragonette and Gaumond, 1997, # Wiley, with permission).

Aluminum

211

212 Underwater acoustics

[Ch. 5

Table 5.1. Compressional speed cp , shear speed cs , and density  used to calculate the form factors for the four metals shown in Figure 5.3 (Dragonette and Gaumond, 1997). cp /m s1

cs /m s1

/kg m 3

Tungsten carbide

6860

4185

13800

Brass

4700

2110

8600

Steel

5950

3240

7700

Aluminum

6376

3120

2710

Material

moderate aspect ratio. Expressions for the volume of various ellipsoids are given in Chapter 4. 5.4.2.2

Large rigid object

For objects that are large in comparison with the acoustic wavelength in water, the BSX is determined by the size and shape of the object. Making use of the Kirchhoff approximation, Neubauer (ca. 1982, p. 18) provides a simple facet summation formula that can be used to compute the scattering cross-section of any shape at high frequency. For some simple shapes it is possible to express the result in closed form, and Table 5.2 summarizes the most important results from Urick (1983). For the first entry in the table (the arbitrary convex shape), there are some important special cases that are worthy of further attention. First, consider an ellipsoid whose semi-major axes are a1 , a2 , and a3 . For a plane wave incident parallel to the axis of length a3 , the radii of curvature in question are A1 ¼

a 21 a3

ð5:122Þ

A2 ¼

a 22 : a3

ð5:123Þ

and

Thus, the BSX of a large ellipsoid ensonified along the a3 axis is
 a1 a2 2  back ¼  ; HF a3

ð5:124Þ

which reduces trivially to the result for a sphere of radius a 2  back HF ¼ a ;

ka  1;

ð5:125Þ

For the case of a convex object ensonified at random aspect, the BSX depends on the total surface area of the object. The surface area of an ellipsoid is considered in Chapter 4.

Sec. 5.4]

5.4 Scattering of plane waves

213

Table 5.2. Backscattering cross-sections of large rigid objects (Urick, 1983). Shape

 back HF 4

Conditions

Arbitrary convex object with principal radii of curvature A1 and A2 . (The curvature is measured at the leading edge, where the plane wave first intersects the convex surface, making a tangent with it). Important special cases, such as a perfect sphere, are described in the text.

A1 A2 4

kA1  1; kA2  1

Cylinder of radius a and length L, ensonified at an angle  to the normal to the cylinder axis

 aL 2 sinðkL sin Þ 2 2 cos  2 kL sin 

ka  1

Arbitrary-shaped flat plate of area S, ensonified at normal incidence


2 S 

kl  1, where l is the smallest dimension


2  ab sinðka sin Þ 2 2 cos   ka sin 

ka > kb  1

Circular plate of radius a, ensonified at an angle  to the normal


2 2  a 2J1 ðka sin Þ 2 2 cos   ka sin 

ka  1

Infinite cone of half-angle , ensonified at an angle  to the cone’s axis

ð=Þ 2 tan 4 ð1  sin 2 =cos 2 Þ 3

Arbitrary convex object with surface area S, ensonified at random aspect (ensemble average cross-section)

S 16

kl  1, where l is the smallest radius of curvature

Circular plate of radius a, ensonified at random aspect (ensemble average cross-section)

a2 8

ka  1

Rectangular plate of sides a and b, ensonified at an angle  to the normal. The direction of the incident and scattered ray paths is in the plane containing the side a

5.4.2.3


ð1  "ÞL > > > > > > :0

jxj  " "

L 2

L L < jxj  : 2 2

jxj >

ð6:31Þ

L 2

The corresponding beam pattern is

cos½ð1  "Þu=2 u 2 bTukey ðuÞ ¼ sinc : 2 1  ð1  "Þ 2 ðu=Þ 2

ð6:32Þ

The Tukey window is used for shading sonar pulses in the time domain. For communications signals, it is used in the frequency domain, where it is known as the raised cosine spectrum (Proakis, 1995); the resulting (predictable and periodic) zero crossings in the time domain (corresponding to u ¼ n, for integer n 6¼ 0) are exploited to minimize intersymbol interference (van Walree, pers. commun., 2009). 6.1.1.1.5

Summary

The main purpose of shading is to reduce the sidelobes. The numerical values of the reduced sidelobe levels for various shading functions are listed in Table 6.1.

Sec. 6.1]

6.1 Processing gain for passive sonar 261

Additional properties for some of these windows are listed by Harris (1978). Harris (1978) and Nuttall (1981) describe further windows not included in Table 6.1, some of which, such as the Barcilon–Temes, Blackman–Harris, Dolph–Chebyshev, and Kaiser–Bessel windows, feature particularly low sidelobe levels. An unwanted side-effect of shading is a broader main beam than for the unshaded case, as illustrated by Figure 6.2. By tapering the contributions from the edges, the apparent length of the array is reduced, so its angular resolution is also reduced. The fwhm of the broadside beam is also included in Table 6.1, relative to its value for an unshaded array. To convert these relative values to absolute beamwidths, they need to be multiplied by the fwhm of an unshaded array, which for the broadside beam of a long array is pffiffiffi 2 sinc 1 ð1= 2Þ  fwhm  : ð6:33Þ  L Substituting numerical values gives  L

rad;

ð6:34Þ

 L

deg:

ð6:35Þ

fwhm  0:8859 or fwhm  50:76

6.1.1.2 6.1.1.2.1

Unsteered planar arrays Piston arrays

The beam pattern of an unshaded circular array of diameter D is (Tucker and Gazey, 1966, p. 180) bðuÞ ¼ ½2J1 ðuÞ=u 2 ;

ð6:36Þ

where J1 ðuÞ is a first-order Bessel function of the first kind (Appendix A), the argument u is u ¼ ðD=Þ sin ð6:37Þ and is the angle from the circle’s axis of symmetry. The function bðuÞ is plotted in Figure 6.5. The basic properties of a circular array with Taylor shading are listed in Table 6.2. The half-power beamwidth (fwhm) of an unshaded circular array is    fwhm ¼ 2 arcsin u0 ; ð6:38Þ D where u0 is the value of u for which bðuÞ is equal to u0  1:614:

1 2

ð6:39Þ

262 Sonar signal processing

[Ch. 6

Figure 6.4. Hamming family shading patterns (left) and beam patterns (right) for continuous line array with various " as labeled: " ¼ 0:2; " ¼ 0:08; . . .

Sec. 6.1]

6.1 Processing gain for passive sonar 263

Figure 6.4 (cont.) . . . " ¼ 0.06 and " ¼ 0.

Table 6.1. Summary of properties for various taper functions. Window

Highest sidelobe level (dB)

Sidelobe fall-off (dB per octave a )

Half-power beamwidth (relative to unshaded)

Shading factor FS at broadside

Main source

Rectangle (Dirichlet)

13.3

6.0

1.00

1.00

Harris (1978)

Triangle

26.5

12.0

2.05

0.75

Harris (1978)

Cosine family (cos n ) n¼0 n¼1 n ¼ 2 (Hann) n¼3 n¼4

13.3 23.0 31.5 39 47

6.0 12.0 18.1 24.1 30.1

1.00 1.35 1.63 1.87 2.10

1.00 0.81 0.67 0.58 0.52

Dirichlet window Harris (1978) Harris (1978) Harris (1978) Harris (1978)

Hamming family " ¼ 1:0 " ¼ 0:3 " ¼ 0:2 " ¼ 0:1 (20 dB pedestal) " ¼ 0:08 (Hamming) " ¼ 0.076711 " ¼ 0:04 " ¼ 0:0

13.3 26.8 31.6 40.1 42.7 43.19 36.2 31.5

6.0 6.0 6.0 6.0 6.0 6.0 6.0 b 18.1

1.00

1.00

1.45 1.47

0.75 0.74

1.63

0.67

Dirichlet window Figure 6.4 Figure 6.4 Cheston and Frank (1990) Harris (1978) Figure 6.3 Figure 6.4 Hann window

Tukey windows " ¼ 1.00 " ¼ 0.75 " ¼ 0.50 " ¼ 0.25 " ¼ 0.0

13.3 14 15 19 31.5

6.0 18.1 18.1 18.1 18.1

1.00 1.14 1.30 1.48 1.63

1.00 0.91 0.82 0.74 0.67

Dirichlet window Harris (1978) Harris (1978) Harris (1978) Hann window

Riesz

21

12.0

1.31

0.83

Harris (1978)

Riemann

26

12.0

1.42

0.77

Harris (1978)

de la Valle´e-Poussin

53

24.1

2.05

0.52

Harris (1978)

Bohman

46

24.1

1.93

0.56

Harris (1978)

58.1

18.1

1.90

0.58

Harris (1978), Nuttall (1981)

1.18 1.33 1.47

0.9 0.8 0.73

Cheston and Frank (1990) Cheston and Frank (1990) Cheston and Frank (1990)

Blackman Taylor n¼3 n¼5 n¼8

26 36 46

a The terminology ‘‘per octave’’ arises from the alternative use of these windows in the time domain to construct passband filters (Harris, 1978). In the present context it means ‘‘per doubling of the argument u’’ as defined in Equation (6.5) or Equation (6.8). b 6 dB/octave is the theoretical fall-off rate of the Hamming family in the limit of large u if " 6¼ 0. However, if " is non-zero but still small, the large u limit might not be reached, in which case the fall-off rate of practical interest is close to that of the Hann window, about 18 dB/octave.

6.1 Processing gain for passive sonar 265

Figure 6.5. Beam pattern 10 log10 bðuÞ of unshaded circular array.

Table 6.2. Summary of beam properties for selected shading (circular arrays) (based on Cheston and Frank, 1990). Window

Highest sidelobe level (dB)

Half-power beamwidth (relative to unshaded array)

Shading factor FS

Uniform

17.57

1.00

1

Taylor n ¼ 3

26.2

1.10

0.91

Taylor n ¼ 5

36.6

1.21

0.77

Taylor n ¼ 8

45

1.31

0.65

At high frequency (D  ) this can be approximated by    fwhm   58:9 deg; D or    fwhm   1:03 rad: D The above equations apply to an unsteered 2D circular plate.

ð6:40Þ

ð6:41Þ

266 Sonar signal processing

[Ch. 6

6.1.1.2.2 Rectangular arrays The beam pattern of a rectangular array whose sides have length L1 and L2 is (Tucker and Gazey, 1966, p. 177, from Eq. 6.22) B ¼ sinc 2 ða1 cos Þ sinc 2 ða2 sin Þ;

ð6:42Þ

where L1 sin 

ð6:43Þ

L2 sin ; 

ð6:44Þ

a1 ¼  and a2 ¼ 

where and are the polar azimuth and elevation angles, respectively. Specifically, the azimuth is the angle between the axis parallel to the side of length L1 , measured in the plane of the array. The projector axis is normal to the plane of the array. The angle is measured from this axis, reaching a value of =2 in the plane of the array. Special cases of Equation (6.42) include the square array of sides L (i.e., L1 ¼ L2 ¼ L) B ¼ sinc 2 ða cos Þ sinc 2 ða sin Þ; ð6:45Þ where L sin a¼ ð6:46Þ  and the line array of length L (L1 ¼ L; L2 ¼ 0) B ¼ sinc 2 ða cos Þ;

ð6:47Þ

where a¼

6.1.2

L sin : 

ð6:48Þ

Directivity index

The directivity index (DI) is a measure of the angular resolution of an array. It can be defined as DI  10 log10 GD ; ð6:49Þ where GD is the directivity factor GD ¼

4 ; O

and O is the solid angle ‘‘footprint’’ of the beam pattern: ð ð 2 ð þ=2 O  BðOÞ dO ¼ d

d cos Bð ; Þ: 4

0

ð6:50Þ

ð6:51Þ

=2

The cases of a steered line array (Section 6.1.2.1) and an unsteered planar array (Section 6.1.2.2) are considered next.

Sec. 6.1]

6.1.2.1

6.1 Processing gain for passive sonar 267

Steered line array

To find the footprint of a steered line array, consider first the solid angle subtended by ˘ a circular ring between angles  and  þ d from the array axis, which is 2 cos  d. The contribution to the footprint is then obtained by multiplying this solid angle by the beam pattern, so that the total footprint for an unshaded line array is ð þ=2 sin 2 u O ¼ 2 d cos  2 ; ð6:52Þ u =2 where u is given by Equation (6.5). Changing the integration variable from  to u yields ð  ðL=Þð1sin Þ sin 2 u O ¼ 2 du 2 : ð6:53Þ L ðL=Þð1sin Þ u It is convenient to introduce the function ðx sin 2 u sin 2 x ðxÞ  du 2 ¼ Sið2xÞ  ; x u 0 where SiðxÞ is the sine integral function (Appendix A) ðx sin u SiðxÞ  du : u 0 It follows that    

4 G0 G0 O ¼  ð1  sin Þ þ  ð1 þ sin Þ ; G0 2 2

ð6:54Þ

ð6:55Þ

ð6:56Þ

where G0 is the high-frequency limit of GD for the broadside beam8 G0 ¼

2L : 

ð6:57Þ

At high frequency, the integration limits of Equation (6.53) tend to 1, in which case the integral is equal to . An exception occurs with the endfire case, for which one of the integration limits is zero and the integral drops to =2. From this it follows that in the high-frequency limit the directivity index tends to 10 log10 ð4L=Þ near endfire, and to 10 log10 ð2L=Þ for all other steer directions. The low-frequency limit is always 10 log10 ð1Þ ¼ 0 dB. Generally speaking, the smaller the angular footprint (i.e., the larger DI), the less noise will enter the beam, and the better the array is likely to perform. Near endfire the footprint halves in size, thus eliminating half of the noise (and doubling the signalto-noise ratio) if the noise is isotropic. However, the optimum steer direction of a horizontal line array is often not close to endfire, partly because the noise field is rarely isotropic, but rather has strong peaks in predictable directions. For example, the low-frequency noise field tends to be dominated by contributions from distant 8 Further, G0 is also equal to the high-frequency limit of GD for all steering angles except those close to endfire.

268 Sonar signal processing

[Ch. 6

sources, propagating at angles close to horizontal (  0). An HLA has its best horizontal resolution at broadside and consequently at low frequency the signalto-noise ratio (SNR) tends to be largest for contacts close to the broadside beam. At high frequency the strongest noise source is likely to be the sea surface immediately above the sonar, resulting in a peak in the noise field from that direction. For an HLA, this would lead to a higher SNR for endfire contacts at high frequency. The directivity factor increases from 1 for a very short array (L  ) to G0 for a long one, and can be written (for the broadside case) GD ¼

 G; 2ðG0 =2Þ 0

ð6:58Þ

where ðxÞ is defined by Equation (6.54). This function increases monotonically from zero at x ¼ 0 to =2 for x ! 1. For small x, ðxÞ is approximately equal to x. The directivity index (DI ¼ 10 log10 GD ) is plotted in Figure 6.6 as a function of G0 using Equation (6.58) for G, with Equation (6.54) for ðxÞ (solid blue line). The dashed line is an alternative approximation calculated using ðxÞ 

x  : 2x 1 þ tanh x  18

ð6:59Þ

Figure 6.6. Directivity index DI ¼ 10 log10 GD for an unsteered continuous line array vs. normalized array length 2L= (¼ G0 ), evaluated using Equation (6.58) with Equation (6.54) (solid blue line) or Equation (6.59) (dashed red); the third curve is the high-frequency approximation GD ¼ G0 (solid cyan).

Sec. 6.1]

6.1 Processing gain for passive sonar 269

Figure 6.7. Normalized directivity index vs. steering angle for Hann-shaded and unshaded arrays of length L= ¼ 0.5 (——), 5 (– –), 50 (    ), and 500 (–  –). The broadside direction is at ¼ 0.

Also plotted is the high-frequency approximation calculated using G  G0 . It is apparent that the high-frequency approximation is in error for 2L= < 1.5, whereas Equation (6.59) retains reasonable accuracy at all frequencies. Figure 6.7 shows a graph of the normalized directivity factor GD =G0 , plotted (in decibels) vs. steering angle. This normalized gain is close to unity (0 dB) for a long and unshaded array, except when steered close to endfire. Close to the endfire direction (90 deg), the ratio increases to 2 (i.e., 3 dB) for long arrays, as can be expected from the discussion following Equation (6.53). An unwanted side-effect of shading, caused by broadening of the main beam, is a small reduction in DI compared with an unshaded array of the same length. For isotropic noise, this would usually also result in a reduction in the SNR. The benefit of shading is the cancellation of noise peaks that might otherwise enter through a sidelobe. This degradation can be quantified in terms of a shading degradation factor FS , defined as the ratio of the directivity factors with and without shading, that is, ð Brect dO ð FS  4

; B dO

4

ð6:60Þ

270 Sonar signal processing

[Ch. 6

Figure 6.8. Shading factor (in decibels) vs. steering angle for Hann-shaded and unshaded arrays of length L= ¼ 0.5 (——), 5 (– – –), 50 (     ), and 500 (–  –). The broadside direction is at ¼ 0.

so that DI ¼ DIrect þ 10 log10 FS ;

ð6:61Þ

where DIrect is the directivity index of an unshaded array steered at the same angle. Table 6.1 lists values of FS for unsteered shaded arrays. The theoretical highfrequency value for Hann shading at broadside is 2/3 (i.e., 1.8 dB). Figure 6.8 shows that, at least for arrays longer than 5 wavelengths, departures from this value are small except in the immediate vicinity of endfire. There is no new information in Figure 6.8, as each curve is just the ratio between pairs of curves from Figure 6.7. Its purpose is to illustrate the behavior of the shading degradation on its own, without the complication of the variation with steering angle of the directivity index.

6.1.2.2

Unsteered planar array

The solid angle footprint of a baffled planar array of beam pattern Bð ; Þ is O ¼

ð =2

ð 2 d

0

d sin Bð ; Þ;

ð6:62Þ

0

where is the angle measured from the normal9; and is the azimuth angle about the 9

That is, the normal to the plane of the array.

Sec. 6.1]

6.1 Processing gain for passive sonar 271

normal. Apart from this change to the definition of , Equation (6.62) is the same as Equation (6.51). For a circular array of diameter D, it becomes ð 2 D= O ¼ du tan ðuÞbðuÞ; ð6:63Þ D 0 where  sin ðuÞ ¼ u ð6:64Þ D and bðuÞ is given by Equation (6.36). For a large array (D  ), Equation (6.63) simplifies to ð 8 2 1 ½J1 ðuÞ 2 O  du: ð6:65Þ u D 2 0 Using the standard integral (see Appendix A),  ð1 J1 ðxÞ 2 1 x dx ¼ ; x 2 0 it follows that GD ¼

4  2 D 2  : O 2

ð6:66Þ

ð6:67Þ

More generally, the directivity index of a large baffled planar array of area S (and extending many wavelengths in both dimensions) is given by (Barger, 1997)   S DI  10 log10 4 2 : ð6:68Þ  The above expressions are for a baffled array. The significance of the baffling is that it reduces the footprint by a factor of 2 and hence doubles the directivity factor. The equivalent expression for an unbaffled pulsating plate would therefore be   S DIunbaffled  10 log10 2 2 : ð6:69Þ 

6.1.3 6.1.3.1

Array gain Definition

The gain in signal-to-noise ratio achieved by spatial filtering (also known as beamforming) is called array gain (AG). This term is defined (see Chapter 3) as AG  10 log10

Rarr ; Rhp

ð6:70Þ

272 Sonar signal processing

[Ch. 6

where Rhp and Rarr are the signal-to-noise ratios before and after beamforming, respectively. Specifically, Rhp is the SNR at the hydrophone Rhp ¼

QS ; QN

ð6:71Þ

where Q S and Q N denote the mean square pressure of the signal and noise. Similarly, Rarr is the SNR at the output from the beamformer10 Rarr ¼

YS ; YN

ð6:72Þ

where Y S and Y N are the array response to signal and noise. Of all terms in the passive sonar equation, array gain is the hardest to calculate precisely. The reason for this is apparent from Equation (6.70), which shows that the signal-to-noise ratio must be calculated not just once, but twice, with and without the effects of the beamformer.11 For each calculation of SNR, all remaining terms of the sonar equation except DT are needed. If the array beam pattern is BðOÞ, the signal and noise terms after beamforming (assuming a narrowband signal) are12 ð Y S ¼ Q SO BðOÞ dO ð6:73Þ and YN ¼

ðð

QN f O ðOÞBðOÞ dO df :

ð6:74Þ

Similarly, the signal and noise terms before beamforming, the ratio of which gives Rhp in Equation (6.71), are ð Q S ¼ Q SO dO

and QN ¼

ðð

QN f O ðOÞ dO df :

ð6:75Þ

ð6:76Þ

Because of the complications associated with using AG, it is common to approximate this parameter by making simplifying assumptions about the directionality of signal and noise. Specifically, the signal tends to come from a single predominant direction, whereas the noise tends to come from all around. In the limit of a plane wave signal and isotropic noise, the AG becomes equal to the directivity index (DI), which is the subject of Section 6.1.2. The shape of the beam pattern, described in Section 6.1.1, is needed for calculations of both AG and DI. However, whereas detailed knowledge of 10

Strictly speaking, there is not one output from a beamformer, but several, one for each beam (i.e., one for each steering angle) and each with a different array gain. It is presumed that one of the beams is steered in the direction of the target and hence has a higher SNR than the others. By convention it is the beam with the highest SNR that determines the array gain. 11 Thus, AG is not only a property of the sonar, but also a complicated function of the propagation conditions, noise sources, and the target. 12 Broadband signals are considered in Section 6.1.4.

Sec. 6.1]

6.1 Processing gain for passive sonar 273

the sidelobe levels is important for AG, for DI it is often sufficient to parameterize the beam pattern in terms of the footprint of the main beam alone. It is convenient to define GA as 10 AG=10 , with GA written as a ratio of signal gain GS to noise gain GN : GA ¼ GS =GN ; ð6:77Þ where ð Q SO BðOÞ dO

GS ¼ and

ð6:78Þ

QS ð

QN O BðOÞ dO

GN ¼

QN

:

ð6:79Þ

Both GS and GN are less than 1. The hope (and expectation) is that the value of GS exceeds that of GN , such that a net gain results overall. Four special cases are considered below. 6.1.3.2

Special cases (noise gain for horizontal line array)

The noise gain of an array (and hence also the total array gain) depends on the anisotropy of the noise field. This dependence is examined here by considering the noise gain for four different noise fields (including the isotropic case). For each case, GN is calculated. If the signal gain is known, the array gain follows using Equation (6.77). For example, in the simplest case the signal can be approximated as a plane wave, meaning that GS  1, and then GA  1=GN : 6.1.3.2.1

ð6:80Þ

Noise gain for isotropic noise

The first case considered is the trivial one involving no anisotropy. For the broadside beam of a long line array it can be shown (see Chapter 3) that GN ¼ =2L:

ð6:81Þ

This result is almost independent of steering angle except close to endfire, for which the noise gain halves (causing the array gain to double) to (see Section 6.1.2.1) GN ¼ =4L:

ð6:82Þ

In general, GN ¼

1 O ¼ ; GD 4

ð6:83Þ

where O is given by Equation (6.56). These results are not limited to a horizontal line array, but apply to a line array of any orientation. Further, if the signal is a plane wave, the array gain and directivity index are identical for the case of isotropic noise.

274 Sonar signal processing

[Ch. 6

6.1.3.2.2 Noise gain for horizontal isotropic noise If all of the incoming noise is restricted to the horizontal plane, but is independent of the azimuth angle, the angular distribution of the noise field can be written QN O ð ; Þ ¼

QN ð Þ: 4

ð6:84Þ

Therefore, the noise gain is ð

QN ð ÞBðOÞ cos d d

4 GN ¼ : QN

ð6:85Þ

Carrying out the integral first, Equation (6.85) becomes GN ¼ where D 

D

; 2

ð 2 Bð0; Þ d :

ð6:86Þ

ð6:87Þ

0

This integral is a measure of the width of the beam pattern in the horizontal plane. It can be written ð 2 2 sin u D ¼ d ; ð6:88Þ u2 0 where kL u¼ ðsin  sin Þ: ð6:89Þ 2 Following Chapter 3, Equation (6.88) can be approximated by13 ð þ=2 D  2 Pðu=Þ d ;

ð6:90Þ

=2

and hence D  2

ð þ

d ¼ 2ð þ   Þ;

ð6:91Þ



where

  

 ¼ arcsin min 1;  sin : kL

ð6:92Þ

It follows that       D

 arcsin min 1; þ sin þ arcsin min 1;  sin : 2 kL kL

ð6:93Þ

For a short array (kL < ), this approximation gives the correct limiting value of D  2 13

See Appendix A:

 ð þ1  sin u 2 du ¼ : u 1

ð6:94Þ

Sec. 6.1]

6.1 Processing gain for passive sonar 275

and hence (using Equation 6.86) GN  1:

ð6:95Þ

For a long array, the behavior depends on the proximity of to the endfire direction (=2). If j j is small (not close to endfire)     D ¼ 2 arcsin sin þ  2 arcsin sin  : ð6:96Þ kL kL At high frequency (=kL  1), and using the result (valid for small ") 2 arcsinðx þ "Þ  arcsinðx  "Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi "; 1  x2

ð6:97Þ

Equation (6.96) becomes D

2  2 kL cos

;

ð6:98Þ

2 kL cos

:

ð6:99Þ

and hence (away from endfire) GN 

Thus, in horizontal isotropic noise, AG exceeds DI (Equation 6.57) by 2 dB (¼ 10 log10 ð=2Þ) in the broadside beam. The width of the endfire beam can be found using Equation (6.96) with ¼ =2:  D   ¼  arcsin 1  ; ð6:100Þ 2 2 kL which simplifies for large kL to D

 2

rffiffiffiffiffiffi 2 : kL

ð6:101Þ

The noise gain at endfire is therefore rffiffiffiffiffiffiffiffiffi 2 ðGN Þef ¼ : kL

ð6:102Þ

6.1.3.2.3 Noise gain for a uniform sheet of dipole noise sources The case of a uniform sheet of dipoles is considered next. This situation was considered in Chapter 3 for the broadside beam. The analysis is now extended for arbitrary steering angle. The noise gain for a horizontal line array is given by Equation (6.79) with (if the hydrophone spacing is small compared with the acoustic wavelength) BðOÞ ¼

sin 2 u u2

ð6:103Þ

276 Sonar signal processing

[Ch. 6

and uð ; Þ ¼

L ðcos sin  sin Þ: 

ð6:104Þ

Given that, for a dipole (see Chapter 2 for details), Q N O is proportional to sin , and the solid angle is proportional to cos according to dO ¼ cos d d ; it follows that

ð =2 GN ¼

0

ð6:105Þ

  sin u 2 d d

sin cos u 0 : ð =2 ð 2 d d sin cos ð 2

0

ð6:106Þ

0

The denominator of Equation (6.106) is . If the numerator is denoted N, the noise gain can be written N GN ¼ ; ð6:107Þ  where ð =2 N¼ D ð Þ sin cos d ð6:108Þ 0

and D ð Þ 

ð 2  0

 sin u 2 d : u

ð6:109Þ

If D is independent of , Equation (6.107) simplifies to GN ¼

D

: 2

ð6:110Þ

More generally, following Section 6.1.3.2.2, Equation (6.109) can be approximated by ð þ=2   u D  2 P d ; ð6:111Þ  =2 and hence ð þ D  2 d ¼ 2ð þ   Þ; ð6:112Þ



where

8 1 1 <  < 1;

ð6:113Þ

 < 1

and  ¼

=kL þ sin cos

:

ð6:114Þ

For sufficiently large kL, and provided also that the elevation angle and steering

Sec. 6.1]

angle

6.1 Processing gain for passive sonar 277

are not too large,14 such that  is between 1 and þ1, D

=kL þ sin  arcsin 2 cos

þ arcsin

=kL  sin cos

:

ð6:115Þ

Using Equation (6.97), Equation (6.115) becomes D 2  ðcos 2  sin 2 Þ 1=2 : 2 kL

ð6:116Þ

Substituting this approximation into Equation (6.108) and integrating over real values of the resulting integrand gives for the numerator of Equation (6.107) ð 4 arccosðsin Þ N ðcos 2  sin 2 Þ 1=2 sin cos d : ð6:117Þ kL 0 The solution to the indefinite integral is ð ðcos 2  sin 2 Þ 1=2 sin cos d ¼ ðcos 2  sin 2 Þ 1=2

ð6:118Þ

and hence GN 

4 cos : kL

ð6:119Þ

At broadside this expression predicts twice the gain of the corresponding case for horizontal noise (Equation 6.99). Further, the noise gain decreases towards endfire according to Equation (6.119), whereas for horizontal noise it increases. Both differences can be understood qualitatively by realizing that the noise from a dipole sheet of noise sources originates mainly from overhead, a direction to which the array is sensitive at broadside and not at endfire. For the endfire case ( ¼ =2), Equation (6.113) gives

þ ¼ and

 2

  1  =kL

 ¼ arcsin min 1; : cos

It follows that cos

  D

1  =kL  min 1; ; 2 cos

or (expanding for small D and small =kL) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   D

2=kL  sin 2  max 0; : 2 cos 2 14

Neither may approach =2.

ð6:120Þ

ð6:121Þ

ð6:122Þ

ð6:123Þ

278 Sonar signal processing

[Ch. 6

It follows from Equation (6.107) that pffiffiffiffiffiffiffiffiffiffi ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 arcsin 2=kL GN ¼ 2=kL  sin 2 sin d :  0 If kL is large, is small in the range of integration, and hence   2  3=2 GN  : 3 L 6.1.3.2.4

ð6:124Þ

ð6:125Þ

Noise gain for multiple point sources of noise

The noise is sometimes best characterized as a sum of one or more incoming plane waves, each contributing Ni to the differential spectral density from a specific direction represented by Oi , such that QN O ¼ N1 ðO1 Þ þ N2 ðO2 Þ þ    X ¼ Ni ðOi Þ

ð6:126Þ

i

and hence

ð

QN O BðOÞ dO ¼

X

Ni BðOi Þ:

ð6:127Þ

i

It then follows from Equation (6.79) that X GN ¼

i

Ni BðOi Þ X : Ni

ð6:128Þ

i

6.1.4

BB application

The definition of beam pattern results in a function of angle for a specified frequency, so the concept is a narrowband one. Nevertheless, the definition of AG (and hence of DI) in terms of the ratio of two SNR values applies more generally, as follows. Substituting for Rhp and Rarr in Equation (6.70), using ð Q Sf ð f Þ df Rhp ¼ ð ð6:129Þ N Q f ð f Þ df and

ð Rarr ¼ ð

Y Sf ð f Þ df ; YN f ð f Þ df

ð6:130Þ

Sec. 6.2]

gives

6.2 Processing gain for active sonar 279

0ð

ð

1 N Q ð f Þ df f B C C: ð AG  10 log10 B @ð N A S Y f ð f Þ df Q f ð f Þ df Y Sf ð f Þ

df

ð6:131Þ

At each frequency f , it is understood that Yð f Þ is calculated using the appropriate beam pattern at that frequency. 6.1.5

Time domain processing

In Chapter 3, two types of time domain processing were considered for passive sonar. These were coherent averaging, resulting in the narrowband passive sonar equation, and incoherent averaging, resulting in the broadband equivalent. These two types of passive sonar processing are discussed briefly below. An alternative type of coherent processing, beyond the present scope, involves (for example) the correlation of the received waveform with itself to exploit shape information contained in one part of the signal to enhance the SNR in another part. 6.1.5.1

Coherent averaging

Coherent averaging (or coherent integration) over time has the effect of reinforcing a stable signal relative to random noise. In the frequency domain the effect of coherent averaging over a time duration T is that of a filter whose bandwidth is 1=T. This results in a gain in SNR represented by the bandwidth term (BW) in the sonar equation (see Chapter 3). The gain in SNR is contingent on both the tonal bandwidth and any fluctuations in tonal frequency being small compared with the processing bandwidth 1=T. An additional benefit of NB processing arises because a high resolution in frequency makes it possible to characterize a signal in terms of a sequence of tonals. This provides an acoustic signature that can be used to aid the classification process. 6.1.5.2

Incoherent averaging

Incoherent averaging (or incoherent integration) does not alter the SNR, but instead decreases the fluctuations in both signal and noise. This makes a sonar operator less likely to mistake a noise fluctuation as signal and hence permits detection at a lower SNR. Thus, the gain is achieved not by increasing SNR but by decreasing the SNR threshold required for detection (known as the detection threshold ). Details are deferred to Chapter 7.

6.2

PROCESSING GAIN FOR ACTIVE SONAR

Consider a generic active sonar system whose processing chain consists of a beamformer followed by a time domain filter. For such a sonar, the processing gain

280 Sonar signal processing

[Ch. 6

is the sum in decibels of the array gain (AG) and the gain of this filter. Of particular interest is a special kind of filter known as a matched filter, described further in Sections 6.2.1 to 6.2.6. Compared with passive sonar, calculation of AG, considered together with the total processing gain in Section 6.2.7, is complicated here by the presence of reverberation.

6.2.1

Signal carrier and envelope

Active sonar signals are usually designed as relatively small perturbations, in amplitude or phase, to a well-defined sinusoidal wave form known as the carrier wave. Denoting the carrier frequency !0 , the total signal (carrier plus modulation) of such a signal can be written xðtÞ ¼ AðtÞ cos½!0 t þ FðtÞ :

ð6:132Þ

The functions AðtÞ and FðtÞ vary in complexity depending on the task to be carried out by the sonar. For simplicity they are considered initially to vary slowly by comparison with cos !0 t and !0 t, respectively. For a simple CW sonar, the phase term would be constant and the amplitude would vary in a simple manner like (say) a Gaussian or rectangle function. A more complex variation (modulation) is needed in order to carry out the transmission of underwater messages. In general, the two functions can be chosen by the designer to optimize either the processing gain (i.e., to maximize the detection probability) or the information content of an echo from an underwater target (e.g., to maximize resolution in range or frequency). For applications concerning the acoustic transmission of a message, the modulation is a coded representation of that message. The significance of the amplitude AðtÞ and phase FðtÞ is described below, first intuitively (Section 6.2.1.1) and then formally (Section 6.2.1.2). The formal derivation follows that of Burdic (1984, pp. 197–198). 6.2.1.1

Intuitive concept

Consider some real signal xðtÞ and write it as the real part of a complex one zðtÞ xðtÞ ¼ Re zðtÞ;

ð6:133Þ

with the imaginary part to be determined. For example, if xðtÞ is a sinusoidal function of the form xðtÞ ¼ A cosð!0 t þ FÞ; ð6:134Þ with A and F both real constants, for zðtÞ one intuitively thinks of a complex exponential zðtÞ ¼ A expð2if0 t þ iFÞ; ð6:135Þ where ! f0 ¼ 0 : ð6:136Þ 2 Now consider the more general case of Equation (6.132) (i.e., with either or both of A

Sec. 6.2]

6.2 Processing gain for active sonar 281

and F varying with time). A heuristic generalization of Equation (6.135), if the time variation is slow, is zðtÞ ¼ AðtÞ exp½2if0 t þ iFðtÞ : 6.2.1.2

ð6:137Þ

Formal methodology: analytic signals and the Hilbert transform

Now consider the spectrum Xð f Þ of an arbitrary real signal xðtÞ Xð f Þ ¼ I½xðtÞ ;

ð6:138Þ

where the operator I½xðtÞ indicates the Fourier transform of the function xðtÞ (see Appendix A). The result is a complex spectrum containing both positive and negative frequencies. By comparison, the Fourier transform of zðtÞ (Equation 6.137) is concentrated around the positive frequency f0 , with negligible contributions from negative frequencies. In order to construct a spectrum similar to the one obtained intuitively, it is necessary to remove the negative frequencies. This is achieved by zeroing the negative part of the spectrum and doubling the positive part,15 that is, Yð f Þ ¼ 2Hð f ÞXð f Þ;

ð6:139Þ

where Hð f Þ is the Heaviside step function (Appendix A). In the time domain this becomes yðtÞ ¼ I 1 ½2Hð f ÞXð f Þ ; ð6:140Þ or, equivalently, the Fourier transform product rule (see Appendix A) gives yðtÞ ¼ 2I 1 ½Hð f Þ  I 1 ½Xð f Þ ; where the symbol  denotes a convolution operation. Using the result i 2I 1 ½Hð f Þ ¼ ðtÞ þ ; t it follows that   ð i i þ1 xðÞ 1 yðtÞ ¼ I ½2Hð f ÞXð f Þ ¼ ðtÞ þ  xðtÞ ¼ xðtÞ þ d: t  1 t  

ð6:141Þ

ð6:142Þ

ð6:143Þ

In this equation yðtÞ is known as the analytic signal, the imaginary part of which is the Hilbert transform of xðtÞ, denoted x^ðtÞ. Thus, yðtÞ ¼ xðtÞ þ i^ xðtÞ;

ð6:144Þ

where x^ðtÞ is the integral16 1 x^ðtÞ ¼ 

ð þ1

xðÞ d: t 1  

ð6:145Þ

15 This operation removes the term expði!tÞ and transforms expðþi!tÞ into 2 expðþi!tÞ. In this way, cosð!tÞ is converted into expðþi!tÞ. 16 Cauchy principal value.

282 Sonar signal processing

[Ch. 6

With this prescription, the cosine function transforms into a sine: ð 1 þ1 cosð!0 Þ d ¼ sinð!0 tÞ  1 t  

ð6:146Þ

and hence, for a real signal cosð!0 tÞ, the analytic signal is yðtÞ ¼ cosð!0 tÞ þ i sinð!0 tÞ ¼ expði!0 tÞ;

ð6:147Þ

precisely as required. The factor 2 introduced on the right-hand side of Equation (6.139) ensures that the sine wave in the imaginary part (of Equation 6.147) has the correct amplitude. It is convenient to write the analytic signal in the following exponential form: yðtÞ ¼ ðtÞ expði!0 tÞ;

ð6:148Þ

where ðtÞ is known as the envelope function ðtÞ ¼ aðtÞ exp½i ðtÞ :

ð6:149Þ

The amplitude and phase terms, both real, are then recovered as aðtÞ ¼ ½x 2 ðtÞ þ x^2 ðtÞ 1=2

ð6:150Þ

x^ðtÞ  !0 t: xðtÞ

ð6:151Þ

and

ðtÞ ¼ arctan

This procedure provides a formal recipe for generating aðtÞ and ðtÞ unambiguously. Recall that xðtÞ is an arbitrary function of time, so there is no longer a restriction on amplitude and phase to vary slowly. In Burdic’s words ‘‘If ðtÞ is a narrow-band function, relative to f0 , it will have properties we intuitively associate with an envelope. Otherwise, it is simply a convenient mathematical representation.’’ The analytic signal contains twice the energy of the original real signal. This can be seen from Equation (6.139) ð þ1 ð þ1 2 jYð f Þj df ¼ 4 jXð f Þj 2 df ð6:152Þ 1

and hence

ð þ1

0

jYð f Þj 2 df ¼ 2

1

6.2.2

ð þ1

jXð f Þj 2 df :

ð6:153Þ

1

Simple envelopes and their spectra

Some examples of simple signal envelopes ðtÞ are given in Tables 6.4 (envelope amplitude) and 6.5 (phase). The effective pulse duration (Burdic, 1984) is ð þ1 2 jðtÞj 2 dt eff  ð1 : ð6:154Þ þ1 jðtÞj 4 dt 1

Sec. 6.2]

6.2 Processing gain for active sonar 283

The spectra of these pulses, calculated using Mð f Þ ¼ I½ðtÞ ;

ð6:155Þ

are listed in Table 6.3. The effective bandwidth is defined in a similar way as effective pulse duration (Burdic, 1984, p. 229) ð þ1 2 2 jMð f Þj df eff  ð1 : ð6:156Þ þ1 jMð f Þj 4 df 1

The normalization convention adopted for envelopes and spectra is ð þ1 ð þ1 2 jðtÞj dt ¼ jMð f Þj 2 df ¼ 1: 1

ð6:157Þ

1

Using this normalization, Equations (6.154) and (6.156) simplify to eff ¼ ð þ1

1

ð6:158Þ

jðtÞj 4 dt

1

and eff ¼ ð þ1

1

:

ð6:159Þ

jMð f Þj 4 df

1

The product eff eff is included in Table 6.3. This product is closely related to the gain due to time domain processing (see Section 6.2.5). The instantaneous angular frequency (in radians per unit time) can be defined as the rate of change of phase . Dividing this quantity by 2 gives the instantaneous frequency finst in cycles per unit time. Thus, finst 

1 d

ðtÞ: 2 dt

ð6:160Þ

The parameter ðtÞ is related in a similar way to the phase acceleration ðtÞ 

1 d2

ðtÞ; 2 dt 2

ð6:161Þ

and is referred to henceforth as the ‘‘frequency rate’’. The properties of two kinds of simple amplitude modulation, Gaussian and rectangular, are listed in Table 6.4 and those of three kinds of phase modulation in Table 6.5. Together, they provide the six combinations of Table 6.3. The first of the three phase modulations (CW or continuous wave) is a trivial one, with no modulation. The other two are linear frequency modulation (LFM) and hyperbolic frequency modulation (HFM);17 these are so called 17

An alternative name is linear period modulation (LPM), so called because the instantaneous 1 1  ð0 =f0 Þt period Tinst ðtÞ  varies linearly with time t: Tinst ðtÞ ¼ . f0 þ finst ðtÞ f0

a

HFM a

LFM

CW

HFM a

LFM

CW

Phase

This expression is valid in the limit of a large time-bandwidth product ðj0 jT 2  1Þ. The center frequency fc and spread B are defined in Section 6.2.2.3. An improved approximation is given by Equation (6.216). For an exact solution see Kroszczyn´ski (1969).

This approximation is valid in the limit of a large time-bandwidth product ðj0 jT 2  1Þ. The exact spectrum is given by Equation (6.178). sffiffiffiffiffiffiffiffiffiffiffi        f0 1 f0 f þ f0 i f  fc  exp 2i f0 loge f exp sgn 0 P f þ f0 j0 jT 0 f0 4 B

T 1=2 sincðfTÞ sffiffiffiffiffiffiffiffiffiffiffi       1 f 2 i f  exp i exp sgn 0 P j0 jT 0 4 0 T

2 1=4  eff expðf 2  2eff Þ   1=2 2 1=4  eff f 2  2eff qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  1  i0  2eff 1  i0  2eff   f f0      a f0 f0 f þ f0 i f þ f0  0 pffiffiffiffiffiffiffi  exp 2i f0 loge f exp sgn 0 f þ f0 0 f0 4 j0 j

1=2

Spectrum Mð f Þ

Stationary phase approximation.

Rectangular

Gaussian

Amplitude

Table 6.3. Summary of frequency domain properties of simple pulse envelopes.



j0 jT 2  2T 2 1þ 0 2 12f 0

 j0 jT 2

3/2

 j0 j 2eff ðj0 jeff  f0 Þ

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ  20  4eff

eff eff

284 Sonar signal processing [Ch. 6

Sec. 6.2]

6.2 Processing gain for active sonar 285 Table 6.4. Summary of time domain properties of simple pulse shapes (envelope). Description Gaussian a

Envelope amplitude aðtÞ   2 1=4 t2 exp  1=2  2eff  eff

Rectangular

T 1=2 Pðt=T Þ

a The parameter eff is related to  used pffiffiffiffiffiffi by Burdic (1984) through the equation  ¼ eff = 2.

Table 6.5. Summary of time domain properties of simple pulse shapes (phase). Description

CW LFM a HFM

a

Phase ðtÞ

Instantaneous frequency finst ðtÞ (Equation 6.160)

Frequency rate ðtÞ (Equation 6.161)

0

0

0

0 t 2     f0 0 2f0 loge 1  t þ t 0 f0

0 t

0

f0  0  f0 t f0

1



0  0 2 1 t f0

The parameter 0 is related to k used by Burdic (1984) through the equation k ¼ 0 .

because the instantaneous frequency varies, respectively, linearly and hyperbolically with time. Both LFM and HFM modulations include CW as a special case with 0 ¼ 0, where 0 is the frequency rate evaluated at the time origin, t ¼ 0. The symbol T is used to denote the total duration of the pulse. Thus, T is the elapsed time during which the envelope function is non-zero. For a rectangular envelope, T and eff are identical. The instantaneous frequency for an HFM pulse is singular at time f0 =0 , imposing a maximum pulse duration of 2f0 =j0 j. In combination with a Gaussian envelope, the more stringent requirement eff  f0 =j0 j must be satisfied. It is useful to introduce the concept of ‘‘frequency spread’’ B, defined as the difference between the largest and smallest value of the instantaneous frequency during effective pulse duration. Assuming a monotonic variation in finst this gives          B ¼  finst þ eff  finst  eff : 2 2

ð6:162Þ

The rectangular envelope (see Table 6.4) is proportional to the factor Pðt=TÞ, where PðxÞ is the rectangle function, equal to unity if jxj < 12, and zero otherwise (Appendix A). This factor is therefore equal to 1 for t between T=2 and þT=2 and 0 at all other

286 Sonar signal processing

[Ch. 6

times. For example, the envelope function for the LFM pulse can be written (see Equation 6.149) ( T 1=2 expði0 t 2 Þ jtj < T=2 ðtÞ ¼ : ð6:163Þ 0 jtj > T=2 The FM pulses are further parameterized by 0 , which is the frequency rate at time t ¼ 0. The LFM pulse has a constant frequency rate so that ðtÞ is equal to 0 at all times. For fixed 0 , the phase of the HFM pulse depends weakly on the carrier frequency. The bandwidth and duration of a pulse are related by a form of uncertainty principle that states that their product must be of order unity or greater. A more precise version of this statement can be made in terms of the variance in time and frequency ð ð tÞ 2 ¼ t 2 jðtÞj 2 dt

ð6:164Þ

ð ð f Þ 2 ¼ f 2 jMð f Þj 2 df :

ð6:165Þ

In terms of these parameters, the uncertainty principle is (Woodward, 1964) f t 

1 ; 4

ð6:166Þ

where equality is achieved with a Gaussian CW pulse, for which  2eff 4

ð6:167Þ

1 : 4 2eff

ð6:168Þ

ð tÞ 2 ¼ and ð f Þ 2 ¼

Equations (6.167) and (6.168) follow from the definite integral (see Appendix A) pffiffiffi ð1  x 2 expðAx 2 Þ dx ¼ : ð6:169Þ 3=2 2A 1 6.2.2.1

CW spectra

Spectra for the CW pulse take particularly simple forms, as shown in Table 6.3. They follow immediately from the Fourier transforms of the Gaussian and rectangle functions (Appendix A). Making use of the standard integrals (see Appendix A) ð þ1 pffiffiffi expðu 2 Þ du ¼  ð6:170Þ 1

and

ð þ1  1

 sin u 4 2 du ¼ ; u 3

ð6:171Þ

Sec. 6.2]

6.2 Processing gain for active sonar 287

the effective bandwidths (Equation 6.159) for the Gaussian and rectangular envelopes are 1=eff Gaussian eff ¼ : ð6:172Þ 3=ð2TÞ rectangular For the rectangular envelope, Equation (6.172) can be compared with the 3 dB width for this window (from Chapter 2) ffwhm 

0:886 ; T

ð6:173Þ

so for this case the effective bandwidth exceeds the 3 dB width by about 70 %. For a CW pulse, instantaneous frequency is not a useful concept. Its value is zero throughout the duration of the pulse and hence the frequency spread as defined by Equation (6.162) is also zero. 6.2.2.2

LFM spectra

The frequency spread of an LFM pulse is B ¼ j0 jeff :

ð6:174Þ

6.2.2.2.1 Gaussian envelope The LFM spectrum for a Gaussian envelope can be derived in the same way as for the CW spectrum (see Table 6.3). It is convenient to define a dimensionless spectral amplitude as   Mð f Þ : Sð f Þ   ð6:175Þ Mð0Þ  The squared modulus of Mð f Þ is known as the power spectrum.18 The dimensionless power spectrum, Sð f Þ 2 , of a Gaussian LFM pulse is plotted in Figure 6.9. For a broadband pulse, the frequency spread B is closely related to the effective bandwidth eff . Specifically, for a Gaussian LFM pulse the relationship can be quantified by writing eff from Table 6.3 in the form  1=2 1 eff ¼ B 1 þ 2 2 : ð6:176Þ B  eff Thus, if the product Beff is sufficiently large, B and eff are approximately equal. Equation (6.176) can also be written in a way that is more appropriate for small Beff , eff ¼

1 ð1 þ B 2  2eff Þ 1=2 ; eff

ð6:177Þ

demonstrating that in this limit the bandwidth is equal to the reciprocal of the pulse duration, as for a CW pulse. 18

Alternative terms are autospectral density and power-spectral density.

288 Sonar signal processing

[Ch. 6

Figure 6.9. Power spectrum 10 log10 ½Sð f Þ 2 for a Gaussian LFM pulse.

6.2.2.2.2

Rectangular envelope

For a rectangular envelope the spectrum can be written sffiffiffiffiffiffiffiffiffiffiffi ! 1 f2 Mð f Þ ¼ exp i Pð f Þ; 0 j0 jT where Pð f Þ is defined as 1 Pð f Þ  pffiffiffi 2 with

ð uþ u

   exp si x 2 dx ¼ Es ðu ; uþ Þ; 2

ð6:178Þ

ð6:179Þ

rffiffiffiffiffiffiffi  j0 j 2f u ¼  T 2 0

ð6:180Þ

s ¼ sgn 0 :

ð6:181Þ

and The function Es ð; Þ is given by the following linear combination of Fresnel integrals CðxÞ and SðxÞ (see Appendix A) Es ð; Þ ¼

CðÞ  CðÞ þ si½SðÞ  SðÞ pffiffiffi : 2

The resulting power spectrum is plotted in Figure 6.10.

ð6:182Þ

Sec. 6.2]

6.2 Processing gain for active sonar 289

Figure 6.10. Power spectrum 10 log10 ½Sð f Þ 2 for a rectangular LFM pulse.

The behavior of the function Pð f Þ is that of a low-pass filter, removing those frequencies whose magnitude exceeds j0 jT=2. For large BT (j0 jT 2  1), it can be approximated using    f  Pð f Þ  exp si P ; ð6:183Þ 4 0 T giving the spectrum quoted in Table 6.3. Using Equation (6.178) the effective bandwidth is  2T 2 eff ¼ ð þ1 0 ; jPj 2 df

ð6:184Þ

1

which can be approximated by replacing jPð f Þj with the rectangle function to give eff  j0 jT: 6.2.2.2.3

ð6:185Þ

Method of stationary phase

Sometimes the Fourier transform operations between time and frequency representations require the calculation of integrals that cannot be evaluated analytically without approximation. A powerful approximate method is described below.

290 Sonar signal processing

[Ch. 6

Consider a pulse whose analytic function is ðtÞ and that has well-defined start and end times T=2 and þT=2. It is convenient to introduce a related function ^ðtÞ ¼ a^ðtÞ e i ðtÞ that is identical to ðtÞ for the duration of the pulse but continuous through T=2 and beyond, such that ðtÞ ¼ ^ðtÞPðt=TÞ and Mð f Þ ¼

ð þT =2

a^ðtÞ e i ðtÞ dt;

ð6:186Þ ð6:187Þ

T=2

where ðtÞ ¼ ðtÞ  !t:

ð6:188Þ

Assuming that the amplitude aðtÞ (and hence also a^ðtÞ) is a slowly varying function of time in the sense that its value is essentially unchanged during a complete cycle of the exponential term, this integral may be evaluated approximately using the method of stationary phase (see Appendix A). The result is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Mð f Þ  a^ðt Þ e i ðto Þ Pð f Þ; ð6:189Þ 00 j ðto Þj o where to is the instant when the phase is stationary with respect to time, such that

0 ðto Þ  ! ¼ 0: The function Pð f Þ is given by Equation (6.179), with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  j 00 ðto Þj T u ¼   to  2 and s ¼ sgn 00 ðto Þ:

ð6:190Þ

ð6:191Þ ð6:192Þ

The instant of stationary phase to varies with frequency, and hence so also do u and s. Equation (6.189) is completely general and may be used for any slowly varying pulse. For an LFM pulse, the phase term in Equation (6.187) is LFM ¼ 0 t 2  !t:

ð6:193Þ

Differentiating Equation (6.193) and setting the result to zero yields to ð f Þ ¼

f : 0

ð6:194Þ

Using Equation (6.189) for the spectrum it follows from Equation (6.175) that   a^½t ð f Þ Pð f Þ SLFM ð f Þ ¼ o ; ð6:195Þ a0  Pð0Þ  where a0  að0Þ ¼ a½to ð0Þ : ð6:196Þ If the bandwidth is sufficiently large, the function jPð f Þj may be approximated by a

Sec. 6.2]

6.2 Processing gain for active sonar 291

Table 6.6. Summary of amplitude envelopes required to synthesize simple power spectra. Desired spectrum

Required amplitude envelope aðtÞ=a0

Sð f Þ LFM     f 0 t P P Df Df   2    2  f 0 t exp  exp  Df Df         f f 0 t 0 t cos 2  P cos 2  P Df Df Df Df

HFM   1 0 t=Df P 1  0 t=f0 1  0 t=f0    1 0 t=f0 2 exp  1  0 t=f0 1  0 t=f0     1 0 t=Df 0 t=Df cos 2  P 1  0 t=f0 1  0 t=f0 1  0 t=f0

rectangle of width j0 jT and centered on f ¼ 0. Substituting for to from Equation (6.194) then gives að f =0 Þ SLFM ð f Þ  : ð6:197Þ a0 A desired spectrum Sð f Þ can be synthesized by rearranging Equation (6.197) in the form aLFM ðtÞ ¼ a0 Sð0 tÞ ðjtj < T=2Þ: ð6:198Þ The result for various spectra is listed in Table 6.6 in the column headed ‘‘LFM’’. (The column headed ‘‘HFM’’ is discussed in Section 6.2.2.3.3). 6.2.2.3

HFM spectra

The spectrum associated with HFM modulation provides a more challenging integral than the LFM case, with a formal solution in terms of the incomplete gamma function (Kroszczyn´ski, 1969). Although exact, the complexity of this analytical solution complicates its interpretation. An approximation is applied below that describes the essential behavior of the HFM spectrum, without losing the relative simplicity of the LFM case. Specifically, Equation (6.189) is used to derive the stationary phase result for an HFM pulse. The first and second derivatives of the HFM phase (from Table 6.5) are 0 t ð6:199Þ

0 ðtÞ ¼ 2 1  0 t=f0 and 20

00 ðtÞ ¼ : ð6:200Þ ð1  0 t=f0 Þ 2 Notice the singularity in the instantaneous frequency ( 0 =2) at time f0 =0 . The need to avoid this singularity places an upper limit on the pulse duration of T
0Þ;

ð7:4Þ

where  is the standard deviation of the noise samples; and fRayleigh ðvÞ is the normalized Rayleigh pdf fRayleigh ðvÞ  v expðv 2 =2Þ

ðv > 0Þ:

ð7:5Þ

Threshold crossings caused by noise are (by definition) false alarms. If the chosen amplitude threshold is AT , the false alarm probability pfa is therefore4 ð1 pfa ¼ fN ðAÞ dA: ð7:6Þ AT

From Equation (7.4) it follows that ð1

A2 pfa ¼ fRayleigh ðvÞ dv ¼ exp  T2 2 AT =

! ð7:7Þ

or, equivalently, AT = ¼ ð2 loge pfa Þ 1=2 :

7.1.2

ð7:8Þ

Detection probability for signal with random phase

In this section, four possible amplitude distributions are considered for the signal, always with Gaussian noise, so that Equation (7.8) gives the corresponding pfa for all cases. The first of the four signal distributions represents an artificial situation with a non-fluctuating signal amplitude (referred to here as a Dirac distribution). The remaining three, all for a fluctuating signal amplitude, are the Rayleigh and Rice (or Rician)5 distributions and the so-called one-dominant-plus-Rayleigh distribution. As explained in Section 7.1.2.3, the Dirac and Rayleigh distributions are special cases of the Rice distribution. 3

For example, the Fourier amplitude in a frequency band of interest. An important property of pdfs is that the integral of the pdf f ðvÞ (with respect to v) over a given range of some physical observable A, must equal the integral of the pdf f ðuÞ with respect to a related quantity u over the same range of A. This is because the integrals represent the probability of a certain event occurring, which is independent of the variable of integration. Allowing the range of integration to vanish it follows also that f ðvÞ dv ¼ f ðuÞ du, and hence that f ðuÞ is not the same function as f ðvÞ; in other words, the functional form of a pdf depends on its argument. 5 The terms ‘‘Rice’’ and ‘‘Rician’’ are used interchangeably. 4

314 Statistical detection theory

7.1.2.1

[Ch. 7

Signal with non-fluctuating amplitude (Dirac distribution)

7.1.2.1.1

Marcum Q-function

If the signal amplitude AS does not fluctuate (i.e., takes a constant value for all pulses, say, equal to a), the amplitude distribution fS ðAÞ is non-zero only when A is identical to a. Because fS ðAÞ is a probability distribution, its area (integrated over all amplitudes A) must be equal to unity. The function that satisfies these two properties is the Dirac delta function fS ðAÞ ¼ ðA  aÞ:

ð7:9Þ

The significance of this distribution is that, if a measurement is made of the signal amplitude A, the probability of finding the value a is unity, and the probability of finding any other value is zero. When this non-fluctuating sine wave signal is added to Gaussian noise, the resulting S þ N has a Rician pdf (Rice, 1948; McDonough and Whalen, 1995): " !#   pffiffiffiffiffiffi A A A2 fSþN ðAÞ ¼ 2 exp  R þ 2 I0 2R ; ð7:10Þ   2 where I0 ðxÞ is a zeroth-order modified Bessel function of the first kind (Appendix A). The probability of detection is given by ð1 pd ¼ fSþN ðAÞ dA; ð7:11Þ AT

and hence pd ¼ Q1

pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2R; 2 loge pfa ;

ð7:12Þ

where the signal-to-noise ratio R is related to the signal amplitude a and noise standard deviation , according to R¼

a2 2 2

and Q1 is the Marcum Q-function defined as (Appendix A) ð1 Q1 ð; Þ  fRice ðv; Þ dv;

ð7:13Þ

ð7:14Þ



where fRice is the normalized Rice pdf

! v2 þ 2 fRice ðv; Þ  v exp  I0 ðvÞ: 2

ð7:15Þ

The function Q1 is a special case of the generalized Marcum function introduced later in this chapter and denoted QM .6 6

Specifically, the case M ¼ 1.

Sec. 7.1]

7.1 Single known pulse in Gaussian noise, coherent processing 315

Figure 7.1. ROC curves in the form 10 log10 ðRÞ vs. pfa for non-fluctuating amplitude signal in Rayleigh noise.

The mean square amplitude of the distribution described by Equation (7.10) is hA 2 i ¼ 2 2 ð1 þ RÞ:

ð7:16Þ

Using Equation (7.12), pd can be calculated as a function of R and pfa . A graph of this function is known as a receiver operating characteristic (ROC) curve. The information is plotted in Figure 7.1 in the form of R vs. pfa for fixed pd as shown in Figure 7.1 (solid lines). This form of the ROC curve is selected because it leads directly to an important term in the sonar equation, namely the detection threshold. This is the value of the SNR, expressed in decibels, that results in a 50% detection probability (see Chapter 3 and Section 7.3.3). Robertson’s Fig. 2 shows ROC curves for the same situation as Figure 7.1 here (solid lines), except in the form pd vs. pfa , for fixed SNR.

7.1.2.1.2

Albersheim approximation

It is desirable to obtain an explicit expression for R as a function of pd and pfa . The appearance of R in the argument of the Marcum function in Equation (7.12) makes it difficult to manipulate, but an alternative, approximate solution in the desired form due to (Albersheim, 1981) is R A þ 0:12AB þ 1:7B;

ð7:17Þ

316 Statistical detection theory

[Ch. 7

where 0:62 pfa

ð7:18Þ

pd : 1  pd

ð7:19Þ

A ¼ loge and B ¼ loge

Albersheim’s result is shown in Figure 7.1 (dashed lines). It can be seen that Equation (7.17) is accurate to within ca. 0.3 dB for 10 12 < pfa < 10 3

ð7:20Þ

0:3 < pd < 0:9:

ð7:21Þ

and Notice the very small values of pfa considered in Figure 7.1. This is necessary to compensate for the large number of false alarm opportunities arising in some applications.7 In order to keep the total number of false alarms manageable, a very low false alarm probability is needed for each opportunity. For typical orders of magnitude, see the worked examples of Chapter 3. 7.1.2.1.3

Limit of large SNR

In the limit of large SNR, the Bessel function of Equation (7.15) may be approximated by the asymptotic expression (valid for large z) (Appendix A) ez I0 ðzÞ  pffiffiffiffiffiffiffiffi ; ð7:22Þ 2z so that " # rffiffiffiffiffiffiffiffiffi v ðv  Þ 2 fRice ðv; Þ  exp  : ð7:23Þ 2 2 pffiffiffiffiffiffi If the parameter , which is equal to 2R, is sufficiently large, Equation (7.23) approximates to a Gaussian distribution centered on . In this approximation it follows that

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pd F 2 loge pfa  2R ; ð7:24Þ where FðxÞ is the right-tail probability associated with a Gaussian pdf (see Appendix A), that is, ! ð 1 1 u2 FðxÞ  pffiffiffiffiffiffi exp  du: ð7:25Þ 2 2 x The form of Equation (7.24) is reminiscent of similar results presented in Chapter 2 7

A passive narrowband system that processes (say) 100 beams and 1000 frequencies, using a coherent integration time of 1 s has 10 5 detection opportunities (and therefore also 10 5 false alarm opportunities) every second. The same result holds for an active FM sonar with 100 beams and 1000 range cells and a pulse repetition rate of 1/s. In either case, to achieve a false alarm rate of one per hour would require a false alarm probability of less than 10 8 .

Sec. 7.1]

7.1 Single known pulse in Gaussian noise, coherent processing 317

for a Gaussian signal in a Gaussian background. The differences stem from the fact that here the background (the noise amplitude) has a Rayleigh distribution and not a Gaussian one. Although the limit of large SNR is rarely an important one in its own right, the simple form of the resulting equations provides a simple test of the more complicated Marcum function in this limit. 7.1.2.2

Signal with Rayleigh fading

The term ‘‘fading’’ is used to indicate fluctuations in signal amplitude, and Rayleigh fading means that these fluctuations are random in nature, with individual amplitude values taken from a Rayleigh distribution. Only slow Rayleigh fading is considered here, meaning that fluctuations occur between, but not during individual pulses. The corresponding pdf of the signal amplitude is (McDonough and Whalen, 1995) ! A A2 fS ðAÞ ¼ 2 exp  2 ðA  0Þ; ð7:26Þ a 2a so that the expectation values of A and A 2 are rffiffiffi  hAi ¼ a 2 and hA 2 i ¼ 2a 2 :

ð7:27Þ ð7:28Þ

More generally, the expectation value of the nth power of A (i.e., its nth moment) can be written ð hA n i  a n

1

x nþ1 e x

2

=2

dx;

ð7:29Þ

0

and hence (see Appendix A) hA n i ¼ 2 n=2 Gðn=2 þ 1Þa n :

ð7:30Þ

The parameter a is the modal (i.e., most probable) amplitude in the sense that it is the value of A that maximizes fS ðAÞ. The Rayleigh distribution is plotted as a solid blue line in Figure 7.2. Also shown are the non-fluctuating distribution used in Section 7.1.2.1 (i.e., a Dirac delta function at A ¼ 1) and the one-dominant-plus-Rayleigh distribution of Section 7.1.2.4 (dotted line). The parameter a is chosen in each case to ensure that the mean square amplitude is equal to unity. The fourth curve (dashed line) is a Rician distribution whose parameters are chosen to match the first two moments of the one-dominantplus-Rayleigh (1D þ R) distribution (see Section 7.1.2.5). The distribution of signal plus noise is obtained from Equations (7.4) and (7.26) by the addition of variance ! A= 2 A 2 =2 2 exp  ; ð7:31Þ fSþN ðAÞ ¼ 1þR 1þR

318 Statistical detection theory

[Ch. 7

Figure 7.2. Rayleigh, one-dominant-plus-Rayleigh (1D þ R), Dirac, and Rice probability distribution functions, given by Equations (7.26), (7.50), (7.9), and (7.36), respectively. The parameter values are chosen in each case to satisfy hA 2 i ¼ 1.

where  2 is the noise variance; and R is now the expected SNR R

hA 2 i a 2 ¼ 2: 2 2 

The detection probability is then found by applying Equation (7.11) ! A 2T =2 2 pd ¼ exp  ; 1þR

ð7:32Þ

ð7:33Þ

or, equivalently, 1=ð1þRÞ

pd ¼ p fa

;

ð7:34Þ

with pfa from Equation (7.8). Figure 7.3 shows a graph of 10 log10 ðRÞ vs. pfa , calculated using Equation (7.34). 7.1.2.3

Signal with Rician fading

Consider a fluctuating signal comprising Gaussian-distributed fluctuations superimposed on an otherwise stable sinusoidal component of amplitude aS . Mathematically this situation is no different from adding a non-fluctuating signal to Gaussian noise, which means that the results of Section 7.1.2.1, leading to a Rician distribution for signal plus noise, apply here to the signal alone. In other words, the

Sec. 7.1]

7.1 Single known pulse in Gaussian noise, coherent processing 319

Figure 7.3. ROC curves in the form 10 log10 ðRÞ vs. pfa for Rayleigh-fading signal in Rayleigh noise, calculated using Equation (7.34).

amplitude of a sinusoid with superimposed Gaussian fluctuations follows a Rician distribution. This property is known as Rician fading. The total signal power is the sum of the coherent and incoherent contributions. Thus, the ratio of signal power to noise power (i.e., the usual SNR) for a signal with Rician fading is R¼

a 2S þ 2 2S ; 2 2N

ð7:35Þ

where  2S and  2N are variances of the signal fluctuations and noise background, respectively. By an exact mathematical analogy with Equation (7.10), the signal amplitude has the distribution " !#   pffiffiffiffiffiffiffiffi A A A2 fS ðAÞ ¼ 2 exp  RS þ 2 I0 2RS S S 2 S

ðA  0Þ;

ð7:36Þ

where RS is the ratio of coherent-to-incoherent signal power RS 

a 2S : 2 2S

ð7:37Þ

320 Statistical detection theory

[Ch. 7

Table 7.1. Comparison table: moments of probability distribution functions. The notation in ðxÞ denotes a scaled version of the modified Bessel function In ðxÞ as defined in Equation (7.65). The notation 1 F1 ð; ; xÞ denotes a hypergeometric function (Appendix A).

hAi

Dirac Rayleigh Rice 1D þ R (Section 7.1.2.1) (Section 7.1.2.2) (Section 7.1.2.3) (Section 7.1.2.4) rffiffiffiffiffiffi rffiffiffi rffiffiffi       RS RS 3 a a S ð1 þ RS Þi0 þ RS i1 a 2 2 2 2 8

hA 2 i

a2

2a 2

2 2S ð1 þ RS Þ

4 2 3a

hA n i 2 hA 2 i n

1

½Gðn=2 þ 1Þ 2

½Gðn=2 þ 1Þ 1 F1 ðn=2; 1; RS Þ 2 ð1 þ RS Þ n

2 n ½Gðn=2 þ 2Þ 2

The mean square amplitude of this distribution is (see Table 7.1 for other moments) hA 2 i ¼ 2 2S ð1 þ RS Þ: Addition of Gaussian noise to this distribution results in (Jelalian, 1992) " !#   pffiffiffiffiffiffi A A A2 fSþN ðAÞ ¼ 2 exp  S þ 2 I0 2S ; SþN  SþN 2 SþN

ð7:38Þ

ð7:39Þ

where S is the ratio of coherent signal power to incoherent signal plus noise power S¼

a 2S 2 2SþN

ð7:40Þ

and  2SþN ¼  2S þ  2N :

ð7:41Þ

Continuing further the analogy with Section 7.1.2.1 and changing the variable of integration to A

¼ ; ð7:42Þ SþN Equation (7.11) becomes !   ð1

2 þ a 2S = 2SþN a pd ¼ SþN exp  I0 S d : ð7:43Þ 2 SþN AT =SþN SþN It is convenient at this point to introduce the concept of equivalent false alarm probability, denoted qfa and given by log qfa 

log pfa ; 1 þ  2S = 2N

ð7:44Þ

where pfa is the true false alarm probability (Equation 7.7). By analogy with Equation

Sec. 7.1]

7.1 Single known pulse in Gaussian noise, coherent processing 321

Figure 7.4. ROC curves in the form 10 log10 ðSÞ vs. qfa for fluctuating amplitude (Rician fading) signal in Rayleigh noise. The equivalent signal-to-noise ratio S and equivalent false alarm probability qfa are related to R and pfa through Equations (7.46) and (7.44) (compare Figure 7.1).

(7.12), it then follows that the detection probability is

pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pd ¼ Q1 2S; 2 loge qfa :

ð7:45Þ

A graph showing S vs. qfa is shown in Figure 7.4. Apart from the change to the axis labels, this graph is identical to the curves of Figure 7.1 labeled ‘‘Marcum’’. The Albersheim approximation could also be used, but is omitted for clarity. The parameter S (see Equation 7.40) can be thought of as an equivalent SNR: S¼

R   2S = 2N ; 1 þ  2S = 2N

ð7:46Þ

where R is the true SNR (Equation 7.35). The intended use of these curves is to calculate a detection threshold (DT), for which it is necessary to obtain a value of R given pfa and pd . The procedure for doing so is as follows: — calculate qfa using Equation (7.44); — read off 10 log10 S from Figure 7.4 for this value of qfa , at the desired detection probability;

322 Statistical detection theory

[Ch. 7

— calculate R by rearranging Equation (7.46) as R ¼ ð1 þ  2S = 2N ÞS þ  2S = 2N :

ð7:47Þ

The detection threshold is then 10 log10 R. As an example, consider the case pd ¼ 0:9, pfa ¼ 10 8 , and  2S = 2N ¼ 3. Following the above procedure gives qfa ¼ 10 2 and 10 log10 S ¼ 9.4 dB from Figure 7.4. Using Equation (7.47) it then follows that DT90  10 log10 R90 ¼ 15.8 dB. This compares with DT90 ¼ 14.2 dB in the absence of fluctuations from Figure 7.1, implying a performance degradation of about 1.6 dB (meaning that for a given SNR these fluctuations reduce the signal excess by 1.6 dB). By contrast, for pd ¼ 0.1, the same fluctuations result in an enhancement (a decrease in the detection threshold DT10 ) of 10.4  8.8 ¼ 1.6 dB. For intermediate values of pd (close to 0.5), the effect of the fluctuations is small. The Rician distribution contains both the Dirac (i.e., non-fluctuating) and Rayleigh amplitude distributions as special cases in the limit of large and small RS , respectively. When the fluctuations are small, it is useful to explore the behavior of the distribution before it collapses to a constant value. In this situation the signal amplitude is described pffiffiffiffiffiffiffiffi approximately by a distribution of the form (using Equation 7.23 with  ¼ 2RS and ¼ A=S ) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # A ðA  aS Þ 2 fS ðAÞ exp  ðA > 0Þ; ð7:48Þ 2aS  2S 2 2S which for large aS =S approximates to a Gaussian distribution. Figure 7.5 shows Rice distributions for RS between 0.3 and 100 as marked. For values of RS exceeding 10 or so, the distribution approximates to a Gaussian of width S . Also shown are Rayleigh and Dirac distributions corresponding to the limits of small and large RS , respectively. As previously, the parameters are chosen to ensure a mean square amplitude hA 2 i ¼ 1, implying a variance of  2S ¼ 7.1.2.4

1 : 2ð1 þ RS Þ

ð7:49Þ

Signal with one-dominant-plus-Rayleigh distribution

An alternative signal amplitude distribution, known as the one-dominant-plusRayleigh (1D þ R) distribution (DiFranco and Rubin, 1968, p. 313), is described by the pdf ! fS ðAÞ ¼

9A 3 3A 2 exp  2a 4 2a 2

ðA  0Þ;

ð7:50Þ

where a is the modal (most probable) amplitude, related to the mean and mean square amplitudes by rffiffiffiffiffiffi 3 hAi ¼ a ð7:51Þ 8

Sec. 7.1]

7.1 Single known pulse in Gaussian noise, coherent processing 323

Figure 7.5. Rice distributions with various values of the coherent-to-incoherent-power ratio RS between 0.3 and 100 as marked; Rayleigh and Dirac probability distribution functions from Figure 7.2 are also shown for comparison.

and hA 2 i ¼ 43 a 2 :

ð7:52Þ

The 1D þ R distribution is plotted as a dotted line in Figure 7.2. In common with Rice, its form is intermediate between that of a non-fluctuating amplitude (Dirac distribution) and the Rayleigh distribution. The main benefit of 1D þ R is that its use simplifies the analysis of fluctuating signals compared with the more cumbersome Rician statistics (de Theije et al., 2008). If higher order moments of A are required, these can be calculated using ð 9a n 1 nþ3 3x 2 =2 n x e dx; ð7:53Þ hA i  2 0 so that (see Appendix A) hA n i ¼ ð2=3Þ n=2 Gðn=2 þ 2Þa n :

ð7:54Þ

The disadvantage is that there is no shape parameter equivalent to  and hence no possibility of tuning 1D þ R to match a desired ratio of coherent-to-incoherent-signal power.

324 Statistical detection theory

[Ch. 7

For 1D þ R, the probability of detection, from Equation (7.11), is (DiFranco and Rubin, 1968, Eq. (9.5-8)) " # " # 2 R A 2T 1 A T pd ¼ 1 þ exp ð1 þ R=2Þ ; ð7:55Þ ð1 þ R=2Þ 2 4 2 2 2 where the signal-to-noise ratio is R¼

2 a2 : 3 2

ð7:56Þ

Using Equation (7.8) relating the threshold AT to the probability of false alarm, the detection probability can be written

 R=2 1=ð1þR=2Þ pd ¼ 1  loge pfa p fa : ð7:57Þ ð1 þ R=2Þ 2 Given a value of the signal-to-noise ratio R (and false alarm probability pfa ), it is straightforward to calculate detection probability pd using Equation (7.57). The reverse operation, from pd to R (a necessary one to evaluate the detection threshold), can be achieved graphically (Figure 7.6), or using one of the approximate methods derived below. The first step towards an (approximate) explicit solution is to rearrange Equation (7.57) in the form R¼

log p 2fa  pffiffiffiffiffiffi  2: log pd  log 1  Rð1 þ R=2Þ 2 loge pfa

ð7:58Þ

This is an implicit equation, because R appears on both left-hand and right-hand

Figure 7.6. ROC curves in the form 10 log10 ðRÞ vs. log10 ð pfa Þ for 1D þ R signal in Rayleigh noise; coherent processing.

Sec. 7.1]

7.1 Single known pulse in Gaussian noise, coherent processing 325

sides. However, the right-hand side is relatively insensitive to the value of R, which suggests an iterative solution in the form Riþ1 ¼

log p 2fa  pffiffiffiffiffiffi  2; log pd  log 1  Ri ð1 þ Ri =2Þ 2 loge pfa

ð7:59Þ

so that the result of the first iteration is R1 ¼

2 log pfa  pffiffiffiffiffiffi  2: log pd  log 1  R0 ð1 þ R0 =2Þ 2 loge pfa

ð7:60Þ

The iteration can be initialized using as a seed the detection threshold for Rayleigh statistics, that is, log pfa R0 ¼  1: ð7:61Þ log pd With this seed, Equation (7.59) converges (to 0.1 dB) after about three iterations. At the expense of some accuracy, a simpler solution is possible by choosing to stop after only the first iteration. Applying a small (empirical) correction term then gives ( ) 2 log pfa 1  10pd  DT 10 log10 : pffiffiffiffiffiffi  2 þ 0:03 1  pd log pd  log 1  R0 ð1 þ R0 =2Þ 2 loge pfa ð7:62Þ An even simpler approach that works well for pd ¼ 0.5 is based on the observation that DTRayleigh is consistently higher than DT1DþR , and that the difference is approximately independent of pfa . For the special case pd ¼ 12, the difference is 0.8 dB, which means that a useful approximation to DT50 (accurate to within 0.1 dB for pfa in the range 10 12 to 10 4 ) for 1D þ R statistics is DT50 10 log10 ½log2 ð2pfa Þ  0:8 dB:

ð7:63Þ

Alternative approximations for the detection threshold, valid over a wide range of pd and pfa values are described by Shnidman (2002) and Barton (2005). 7.1.2.5

Summary table

Table 7.1 shows the mean and mean square amplitude for each of the four distributions considered, as well as a general (normalized) expression for the nth moment. The Rice and 1D þ R distributions are both intermediate between the non-fluctuating case and the completely random Rayleigh case. If desired, the free parameter in Rice can be adjusted to match some feature of 1D þ R. For example, matching their mean amplitudes results in the condition     pffiffiffi RS R 3 a ð1 þ RS Þi0 þ RS i 1 S ¼ ; ð7:64Þ 2 2 2 S

326 Statistical detection theory

[Ch. 7

where the function in ðxÞ is defined in terms of the nth-order modified Bessel function of the first kind In ðxÞ as in ðxÞ  e x In ðxÞ: ð7:65Þ Selecting hA 2 i ¼ 1 as before implies that a 2 ¼ 3=4

ð7:66Þ

and  2S ¼

1 ; 2ð1 þ RS Þ

so the condition on RS becomes

   2  RS R 9 2 hAi ¼ ð1 þ RS Þi0 þ RS i 1 S ¼ : 4ð1 þ RS Þ 2 2 32

ð7:67Þ

ð7:68Þ

The value of RS that satisfies this condition is (de Theije et al., 2008) RS 2:805:

7.1.3

ð7:69Þ

Detection threshold

There is an important difference between the amplitude threshold AT that appears in some of the above equations (e.g., Equation 7.33) and the detection threshold DT introduced in Chapter 3. The former is the S þ N amplitude above which an operator decision changes from ‘‘no target present’’ to ‘‘target present’’, while the latter is the SNR threshold above which the detection probability exceeds 50 %. To reduce the risk of confusion, the precise relationship between them is described below.8 Let the detection probability be written in the form pd ¼ f ðR; AT =Þ:

ð7:70Þ

At the SNR threshold corresponding to a 50 % detection probability ( pd ¼ 12, R ¼ R50 ) this becomes 1 ð7:71Þ 2 ¼ f ðR50 ; AT =Þ: For any given choice of amplitude threshold AT , this equation can be solved for R50 . Converting to decibels gives the detection threshold corresponding to a 50 % detection probability: DT50 ¼ 10 log10 R50 ðAT Þ:

ð7:72Þ

This general method can be applied to any one of the distributions considered above. As an example, consider the case of Rayleigh fluctuations, for which Equation (7.33) provides a simple equation relating the detection probability pd to the signal-to-noise ratio R and amplitude threshold AT . Substituting pd ¼ 12 and rearranging for the 8

Abraham (2010) refers to the amplitude threshold as the ‘‘detector threshold’’.

Sec. 7.2]

7.2 Multiple known pulses in Gaussian noise, incoherent processing

327

associated SNR gives (for Rayleigh statistics) DT  10 log10 R50 ðAT Þ ¼ 10 log10

7.1.4

! A 2T 1 : ð2 loge 2Þ 2

ð7:73Þ

Application to other waveforms

So far in this section, a narrowband signal has been assumed for simplicity. For other (known) waveforms, a replica correlator (cross-correlation of the received pulse with the transmitted one) can replace the Fourier transform, and the ‘‘amplitude’’ parameter A is then re-interpreted as the correlator output. In this way, all results of this section apply unaltered and the same ROC curves can be used, provided the pdf of A is known or can be estimated. The analysis requires that the shape of the waveform be fully known, but not the start time. See, for example, DiFranco and Rubin (1968, Ch. 9) or McDonough and Whalen (1995, Secs. 7.1 to 7.3).

7.2

MULTIPLE KNOWN PULSES IN GAUSSIAN NOISE, INCOHERENT PROCESSING

Consider a sequence of pulses of the kind described in Section 7.1, all of equal duration. The phase term  is assumed to take an unknown constant value within each pulse, and to vary randomly from one pulse to another. As the relative phases are unknown, the pulses cannot be combined coherently, but instead one can sum the total energy over all M pulses E

M X

A 2i ;

ð7:74Þ

i¼1

declaring a detection if E exceeds some threshold ET . This type of detector is known as an energy detector or square law detector. (Each individual pulse is processed coherently; ‘‘incoherent’’ processing refers to the way the pulses are combined.) Both fluctuating and non-fluctuating signal amplitudes are considered, as previously for coherent processing, and with the same amplitude distributions. In combination with incoherent processing over multiple pulses, Rayleigh and 1D þ R signal fluctuations are known as the Swerling II and Swerling IV fluctuation models, respectively (DiFranco and Rubin, 1968; Levanon, 1988). The corresponding case with a non-fluctuating signal is sometimes known as Swerling 0. For radar applications, a distinction is made between ‘‘pulse-to-pulse’’ fluctuations and slower ‘‘scan-to-scan’’ fluctuations, involving changes from one pulse train to the next, but not between successive pulses within a single pulse train. Such scan-to-scan fluctuations are described by the Swerling I and Swerling III

328 Statistical detection theory

[Ch. 7

models.9 The scan-to-scan cases are less relevant to sonar and not considered here.

7.2.1

False alarm probability for Rayleigh-distributed noise amplitude

The sum of squares of M Rayleigh-distributed amplitudes results in a chi-squared (or ‘‘ 2 ’’) distribution. If there are M independent noise samples, the resulting  2 distribution has 2M degrees of freedom. This can be written (McDonough and Whalen, 1995, p. 295)10   1 E M1 expðE=2 2 Þ fN ðEÞ ¼ 2 ; ð7:75Þ ðM  1Þ! 2 2 2 where  2 is the variance of the original Gaussian noise distribution. It follows that, if the energy threshold is ET , the false alarm probability is ð1 pfa ¼ fN ðEÞ dE: ð7:76Þ ET

Making the substitution v¼ results in pfa ¼

ð1 ET = 2

E 2

ð7:77Þ

f 2 ðvÞ dv; 0

ð7:78Þ

where f 2 ðvÞ is the dimensionless ‘‘ 2 ’’ distribution with 2M degrees of freedom 0

f 2 ðvÞ ¼ 0

1 ðv=2Þ M1 e v=2 : 2 ðM  1Þ!

ð7:79Þ

Equation (7.78) can be written (DiFranco and Rubin, 1968, p. 347) pfa ¼

GðM; ET =2 2 Þ ; ðM  1Þ!

where Gða; xÞ is the upper incomplete gamma function (Appendix A) ð1 Gða; xÞ  t a1 e t dt:

ð7:80Þ

ð7:81Þ

x

For fixed M, the energy threshold ET controls pfa in the same way as does AT in Section 7.1. If M increases, ET must also be increased to avoid a corresponding increase in pfa . 9

See DiFranco and Rubin (1968, p. 390) and McDonough and Whalen (1995, p. 306) for Swerling I; and DiFranco and Rubin (1968, p. 410) for Swerling III. 10 For the special case M ¼ 2, this simplifies to the so-called ‘‘one dominant plus Rayleigh’’ (1D þ R) distribution for the variable A ¼ E 1=2 .

Sec. 7.2]

7.2 Multiple known pulses in Gaussian noise, incoherent processing

329

For the special case M ¼ 1 

E pfa ¼ G 1; T2 2



  ET ¼ exp  2 ; 2

ð7:82Þ

or, equivalently, 2 loge pfa ¼

ET : 2

ð7:83Þ

For sufficiently large M, the following limit is reached (DiFranco and Rubin, 1968, p. 367) ! ET = 2  2M pffiffiffiffiffi pfa F ; ð7:84Þ 2 M where FðxÞ is the right-tailed probability function (see Appendix A). An equivalent expression for pfa is obtained by defining an (RMS) amplitude threshold AT as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi AT  ET =M ; ð7:85Þ so that " !# pffiffiffiffiffi A 2T pfa F M 1 : ð7:86Þ 2 2

7.2.2

Detection probability for incoherently processed pulse train

7.2.2.1 7.2.2.1.1

Signal with non-fluctuating amplitude General case

If all M individual pulses in the pulse train have the same amplitude a, the S þ N energy has a pdf of the form (McDonough and Whalen, 1995)11 " !# !   1 E ðM1Þ=2 2 E E 1=2 fSþN ðEÞ ¼ 2 exp  þ 2 IM1 ; ð7:87Þ 2  2  2  2 2 where  2 ¼ 2MR

ð7:88Þ

and R is the power SNR R¼

a2 : 2 2

ð7:89Þ

For sufficiently high SNR, E is just the sum over signal energies, namely Ma 2 , but the presence of noise usually complicates this simple picture. This pdf is a non-central  2 density with 2M degrees of freedom, such that ð1 ð1 pd ¼ fSþN ðEÞ dE ¼ f 2 ðv; Þ dv; ð7:90Þ ET 11

ET = 2

1

Where In ðxÞ is an nth-order modified Bessel function of the first kind (Appendix A).

330 Statistical detection theory

where

!   1 v ðM1Þ=2 v þ 2 f 2 ðv; Þ ¼ exp  IM1 ðv 1=2 Þ: 1 2 2 2

It follows that (McDonough and Whalen, 1995, Eq. 8.25) ! 1=2 ET pd ¼ QM ; ;  where QM is the generalized Marcum function ! ð 1 M x x2 þ 2 QM ð; Þ   exp  IM1 ðxÞ dx: 2 

[Ch. 7

ð7:91Þ

ð7:92Þ

ð7:93Þ

Equation (7.92) can also be written in terms of the RMS amplitude threshold   pffiffiffiffiffi AT pd ¼ QM ; M : ð7:94Þ  The previously defined Marcum Q-function (Q1 ) is a special case of the generalized Marcum function. The Marcum Q-function has been evaluated and plotted for a selection of values for the integer M by different authors (Robertson, 1967; DiFranco and Rubin, 1968). Albersheim’s approximation. It is useful to be able to express R explicitly in terms of M, pd , and pfa , but Equation (7.92) does not lend itself easily to this end. A cumbersome solution is to plot pd ðR; pfa ; MÞ for all combinations of interest (see, e.g., Robertson, 1967) and read the value of R for the desired combination of pd ; pfa ; M from the graph. A simple solution is provided by the following approximation due to (Albersheim, 1981)12 pffiffiffiffiffi 10 log10 ðR M Þ 10xðMÞ log10 ðA þ 0:12AB þ 1:7BÞ; ð7:95Þ where A and B are given by Equation (7.18) and Equation (7.19), respectively, and the function xðMÞ, plotted in Figure 7.7, is defined as 0:456 xðMÞ  0:62 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : M þ 0:44

ð7:96Þ

For large M, the accuracy of Equation (7.95) is approximately within 0.5 dB for pd 12

The precise equation proposed by Albersheim is   pffiffiffiffiffi 4:54 10 log10 ðR M Þ 6:2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi log10 ðA þ 0:12AB þ 1:7BÞ: M þ 0:44

Equation (7.95) (with Equation 7.96) is identical to this except that the constant 4.54 is replaced by 4.56 in order to match Equation (7.17) exactly for M ¼ 1.

Sec. 7.2]

7.2 Multiple known pulses in Gaussian noise, incoherent processing

331

Figure 7.7. Graph of xðMÞ vs. M. This function is used in Equation (7.95) or Equation (7.99) to calculate the detection threshold.

and pfa values satisfying the inequalities 10 12 < pfa < 10 2

ð7:97Þ

0:3 < pd < 0:7:

ð7:98Þ

and The error reduces to less than 0.3 dB in the reduced range 10 7 < pfa < 10 2 , still for large M. The ratio R given by Equation (7.95) can be written in the form R

ðA þ 0:12AB þ 1:7BÞ xðMÞ pffiffiffiffiffi : M

ð7:99Þ

The form of Equation (7.99) might give the impression that R (the SNR) varies with M, but this is not the case, because the SNR is unaffected by incoherent integration. Rather, the left-hand side of this equation should be interpreted as the detection threshold (i.e., the SNR required to achieve a certain performance). This quantity is inversely proportional to M 1=2 for large M. It is also a function of the detection and false alarm probabilities, through A and B. It is apparent from Figure 7.8 (by comparing the magnitude of 10 log10 ðM 1=2 RÞ with its large-M asymptote) that, for the range of parameters considered, this M 1=2 behavior is reached to within ca. 0.6 dB when M exceeds 100 and to within ca. 0.1 dB when M exceeds 10 4 . Figure 7.9 shows a set of ROC curves evaluated using Equation (7.99) with pd ¼ 0.5, for M between 1 and 1024.

332 Statistical detection theory

[Ch. 7

Figure 7.8. ROC curves (Albersheim approximation) in the form 10 log10 ðRÞ þ 10 log10 M 1=2 vs. M for pd ¼ 0.5 and various pfa between 10 12 and 10 4 for a non-fluctuating amplitude signal in Rayleigh noise.

In order to assess its accuracy, Albersheim’s approximation is evaluated for various pd and pfa and the results presented in Table 7.2. Where possible a comparison with Robertson’s original curves (which are computed, without approximation, for a non-fluctuating signal in a Rayleigh background) is made, in order to obtain an estimate of the error involved. This error estimate is given in brackets. For the example highlighted by gray shading (M ¼ 32, pd ¼ 0.1, pfa ¼ 10 7 ), the value of DT þ 5 log10 M is 6.2 dB, which means that the Albersheim approximation gives ðDT10 ÞAlbersheim 6:2  5 log10 M ¼ 1:3

dB:

ð7:100Þ

The correction in brackets provides an improved estimate of Robertson’s original value of: DT10 ðDT10 ÞAlbersheim  ð0:4Þ ¼ 0:9 dB:

ROC curves. The intended use of Equation (7.92) is for calculation of detection probability pd , given the signal-to-noise ratio R. Figure 7.10 shows ROC curves evaluated in this way for the case M ¼ 30, reproduced from DiFranco and Rubin (1968). This graph, and all subsequent ones from Radar Detection, show 10 log10 ð2RÞ vs. pd , for fixed values of pfa between 0:693  10 10 and 0:693  10 1 . For further

Sec. 7.2]

7.2 Multiple known pulses in Gaussian noise, incoherent processing

333

Figure 7.9. ROC curves (Albersheim approximation) in the form 10 log10 ðRÞ þ 10 log10 M 1=2 vs. pfa for pd ¼ 0.5 and various M between 1 and 1024 for a non-fluctuating amplitude signal in Rayleigh noise; the dashed line is the limit for M ! 1.

examples (2  M  3000) see DiFranco and Rubin (1968, pp. 350–358). Graphs of the form pd vs. pfa for various R, designed for ease of interpolation for arbitrary combinations of pd , pfa , and SNR, are given by Robertson (1967) for M equal to integer powers of 2 between 1 and 8,192. It is desirable to be able to calculate R directly from pd and pfa . One convenient approximation for doing so is that due to Albersheim, described above. Other approximate methods, valid for a non-fluctuating signal over a wide range of values of M, pd , and pfa , are described by Shnidman (2002) and Hmam (2005). Further ROC curves in the form 10 log10 R vs. pfa , evaluated using Albersheim’s approximation, are shown in Figure 7.11 for values of M between 1 and 300 as marked. The benefit of incoherent integration is to average out the fluctuations, making it feasible to detect a signal with a low SNR. Figure 7.11 shows that for M exceeding 100, DT can be negative even for a false alarm probability as low as 10 12 . 7.2.2.1.2

Special case M ¼ 1

For the special case M ¼ 1, Equation (7.92) reduces to 1=2

pd ¼ Q1 ð; E T =Þ; which is equivalent to Equation (7.12).

ð7:101Þ

Table 7.2. DT þ 5 log10 M vs. M and pfa for three different pd values, evaluated using Albersheim’s approximation (Equation 7.95). See text for interpretation of error values in brackets. Empty columns indicate regions outside the validity range of Equation (7.95); missing error values indicate that the point is outside the range of coverage of Robertson’s curves. The meaning of the shaded box in the table for pd ¼ 0.1 is explained in the text surrounding Equation (7.100). Detection probability pd ¼ 0:1 M

DT þ 5 log10 M DT þ 5 log10 M DT þ 5 log10 M DT þ 5 log10 M DT þ 5 log10 M (dB) (dB) (dB) (dB) (dB) ð pfa ¼ 10 1 Þ ð pfa ¼ 10 3 Þ ð pfa ¼ 10 5 Þ ð pfa ¼ 10 7 Þ ð pfa ¼ 10 9 Þ

1

6.4 (1.2)

8.9

10.5

2

5.9 (0.8)

8.1

9.6

4

5.4 (0.7)

7.5

8.8

8

5.0 (0.7)

6.9 (0.4)

8.1

32

4.5 (0.8)

6.2 (0.4)

7.3

256

4.2 (0.9)

5.8 (0.6)

6.8

8192

4.0 (1.0)

5.6 (0.6)

6.6 (0.4)

Detection probability pd ¼ 0:5 M

DT þ 5 log10 M DT þ 5 log10 M DT þ 5 log10 M DT þ 5 log10 M DT þ 5 log10 M (dB) (dB) (dB) (dB) (dB) ð pfa ¼ 10 1 Þ ð pfa ¼ 10 3 Þ ð pfa ¼ 10 5 Þ ð pfa ¼ 10 7 Þ ð pfa ¼ 10 9 Þ

1

2.6

8.1 (0.0)

10.4 (0.0)

11.9

13.1

2

2.4

7.4 (0.1)

9.5 (0.1)

10.9

11.9

4

2.2

6.8 (0.1)

8.7 (0.1)

10.0

10.9

8

2.0

6.3 (0.1)

8.1 (0.2)

9.3 (0.1)

10.1

32

1.8

5.7 (0.0)

7.3 (0.0)

8.4 (0.0)

9.1

256

1.7

5.2 (0.0)

6.8 (0.0)

7.7 (0.0)

8.5

8192

1.6

5.1 (0.1)

6.5 (0.1)

7.5 (0.2)

8.2 (0.2)

Detection probability pd ¼ 0:9 M

DT þ 5 log10 M DT þ 5 log10 M DT þ 5 log10 M DT þ 5 log10 M DT þ 5 log10 M (dB) (dB) (dB) (dB) (dB) ð pfa ¼ 10 1 Þ ð pfa ¼ 10 3 Þ ð pfa ¼ 10 5 Þ ð pfa ¼ 10 7 Þ ð pfa ¼ 10 9 Þ

1

7.8 (0.5)

10.7 (0.1)

12.5 (0.0)

13.7

14.7

2

7.1 (0.4)

9.8 (0.0)

11.4 (0.0)

12.5

13.4

4

6.5 (0.4)

9.0 (0.0)

10.4 (0.1)

11.5

12.3

8

6.1 (0.4)

8.3 (0.1)

9.7 (0.1)

10.7 (0.1)

11.4

32

5.5 (0.2)

7.5 (0.0)

8.7 (0.1)

9.6 (0.0)

10.3

256

5.1 (0.2)

7.0 (0.0)

8.1 (0.0)

8.9 (0.0)

9.5

8192

4.9 (0.3)

6.7 (0.0)

7.8 (0.1)

8.6 (0.2)

9.2 (0.3)

7.2 Multiple known pulses in Gaussian noise, incoherent processing

335

Figure 7.10. ROC curves in the form 10 log10 ð2RÞ vs. pd for various pfa for a non-fluctuating amplitude signal in Rayleigh noise; example of incoherent addition of M samples, with M ¼ 30. The pfa values are given by 0.693/n 0 , where n 0 is between 10 and 10 10 as labeled (reprinted from DiFranco and Rubin, 1968, # Scitech, Raleigh, NC).

7.2.2.1.3 Limit of large M In the large-M limit, Equation (7.92) may be approximated by pffiffiffiffiffi   ’fa  M R pd F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ 2R where ’fa  F 1 ð pfa Þ:

ð7:102Þ ð7:103Þ

Adopting a similar shorthand for the analogous function of the detection probability ’d  F 1 ð pd Þ;

ð7:104Þ

Equations (7.84) and (7.102) can be written, respectively, pfa Fð’fa Þ

ð7:105Þ

pd Fð’d Þ;

ð7:106Þ

and

336 Statistical detection theory

[Ch. 7

Figure 7.11. ROC curves in the form 10 log10 ðRÞ vs. pfa for pd ¼ 0.3, 0.5, and 0.7 as marked, for a non-fluctuating signal amplitude.

where ET = 2  2M pffiffiffiffiffi 2 M

ð7:107Þ

pffiffiffiffiffi ’fa  M R ’d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ 2R

ð7:108Þ

’fa ¼ and

It is useful to obtain an expression for R as an explicit function of ’d and ’fa , as this simplifies the calculation of ROC curves. To this end, Equation (7.108) can be recast as a quadratic equation in R pffiffiffiffiffi MR 2  2ð’ 2d þ M ’fa ÞR þ ’ 2fa  ’ 2d ¼ 0; ð7:109Þ whose solution is R¼

’ 2d þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi M ’fa  ’d M þ 2 M ’fa þ ’ 2d M

:

ð7:110Þ

Specifically, for the default situation with pd ¼ 0:5 ð’d ¼ 0Þ, this gives, without further approximation ’fa : ð7:111Þ R50 ¼ pffiffiffiffiffi M

Sec. 7.2]

7.2 Multiple known pulses in Gaussian noise, incoherent processing

337

Alternatively, for arbitrary pd , a useful approximation is obtained for large M in the form (disregarding the unphysical root)   pffiffiffiffiffi ’fa  ’d 1 pffiffiffiffiffi þ O MR : ð7:112Þ M 1 þ ’d = M Defining R0 as the asymptotic form of R for large M, that is, ’fa  ’d pffiffiffiffiffi ; M

ð7:113Þ

R0 pffiffiffiffiffi : 1 þ ’d = M

ð7:114Þ

R R0

ð7:115Þ

R0  it follows that R The approximation

is sometimes used for large M, as this simplifies calculation of the detection threshold. This approximation is compared in Figure 7.12 with the large-M limit of Albersheim’s approximation (from Equation 7.99 with B ¼ 0 and A given by Equation 7.18):13   1 0:62 0:62 Rð pfa Þ pffiffiffiffiffi loge : ð7:116Þ pfa M An important question is: how large must M be for R0 to be used as an approximation to R? The answer depends on the desired accuracy. According to DiFranco and Rubin (1968), for values of M exceeding 100, Equation (7.115) is accurate to within 1 dB (and hence so too are the solid curves of Figure 7.12), and this is supported by Figure 7.8. Greater accuracy can be achieved (still for M > 100) using Equation (7.114).14 Returning to the case of large M (say M > 100), the use of Equation (7.114) is illustrated below using a graphical method. The ratio R=R0 (calculated using Equation 7.114) is plotted vs. M in Figure 7.13. This graph may be used to obtain R for an arbitrary combination of M, pd , and pfa , assuming that M is large, in the following manner. Given pd , the first p step is to read off a value of R=R0 from Figure 7.13 for the ffiffiffiffiffiffiffiffiffiffi desired value of M, and R0 M=2 from Figure 7.12 (solid curve) for the desired value pffiffiffiffiffiffiffiffiffiffi of pfa These two factors are multiplied first together and then further by 2=M to obtain the result for R rffiffiffiffiffi pffiffiffiffiffi 2 R MR R¼   pffiffiffi 0 : ð7:117Þ M R0 2 The second factor is a function of R and pd only, independent of pfa . As an example, consider the case of 200 pulses combined incoherently and a desired detection probability of 90 %, with a false alarm probability of 10 6 . Reading 13 14

The use of B ¼ 0 implies that pd ¼ 12. For smaller values of M, Equation (7.92) or Equation (7.95) may be used.

338 Statistical detection theory

[Ch. 7

pffiffiffiffiffiffiffiffiffiffi Figure 7.12. ROC curves in the form 10 log10 ð M=2RÞ vs. pfa for fixed pd values for a broadband signal in Rayleigh noise: calculated with Equation (7.115) (i.e., the large-M approximation) and Equation (7.116) (Albersheim’s approximation, also for large M).

appropriate values from Figures 7.13 and 7.12 gives pffiffiffiffiffiffiffiffiffiffi DT90 ¼ 9:6 þ 10 log10 ðR0 M=2Þ and 10 log10

rffiffiffiffiffi! M R0 ¼ 6:3: 2

The detection threshold is therefore DT90 3:3 dB. Figure 7.12 permits assessment of the accuracy of Albersheim’s approximation in the limit of large M. It shows that the error made in this limit is less than 0.5 dB for 0:3  pd  0:7 and 10 12  pfa  10 2 ,15 and Figure 7.1 confirms that this is also the case for M ¼ 1. Further, there is no suggestion from the bracketed errors in Table 7.2 that accuracy deteriorates for intermediate values of M, so the error seems likely to be less than 0.5 dB across the entire range of M, at least for pd ¼ 0.5.16 In the limit of very large M, such that the right-hand side of Equation (7.114) is independent of M (and Equation 7.115 holds), Equation (7.102) becomes (switching 15 16

The Albersheim approximation is only plotted in places where this error is less than 0.5 dB. Greater accuracy is achieved for pd close to 0.5 and pfa in the range to 10 5  pfa  10 2 .

Sec. 7.2]

7.2 Multiple known pulses in Gaussian noise, incoherent processing

339

Figure 7.13. Supplementary ROC curves in the form 10 log10 ðR=R0 Þ vs. M for fixed pd values for a broadband non-fluctuating signal in Rayleigh noise: large-M approximation (Equation 7.114).

here to erfc notation to facilitate comparison with Chapter 2) rffiffiffiffiffi ! 1 M 1 pd erfc erfc ð2pfa Þ  R ; 2 2

ð7:118Þ

where (from Equation 7.86) 1 pfa erfc 2

"rffiffiffiffiffi !# M A 2T 1 ; 2 2 21

ð7:119Þ

and 1 is the standard deviation of the original Gaussian noise before any averaging (i.e., for a single pulse, the special case M ¼ 1), hitherto denoted . Equation (7.118) is identical in form to the corresponding expression for Gaussian statistics from Chapter 2, namely:

 1 xS 1 pd ¼ erfc erfc ð2pfa Þ  pffiffiffi ð7:120Þ 2 2 M and 1 xT  xN ffiffiffi pfa ¼ erfc p : 2 2 M

ð7:121Þ

340 Statistical detection theory

[Ch. 7

The equivalence indicates that Gaussian statistics apply for large M (the central limit theorem at work). The mean and standard deviation of the noise and signal plus noise distributions can be determined by inspection. Specifically, equating the arguments of the erfc functions for pd and pfa gives (equating the right-hand sides of Equations 7.118 and 7.120) rffiffiffiffiffi xS M pffiffiffi ¼ R ð7:122Þ 2 2 M and (from Equations 7.119 and 7.121) xT  xN pffiffiffi ¼ 2 M

! rffiffiffiffiffi M A 2T 1 ; 2 2 21

ð7:123Þ

respectively. Rearranging the first of these for M =xS gives M 1 ¼ pffiffiffiffiffi : xS MR

ð7:124Þ

The second gives (dividing through by xS ) xT A2 ¼ 2T ; xS 2 1 R

ð7:125Þ

R ¼ xS =xN :

ð7:126Þ

where Thus, if M is sufficiently large, the noise standard deviation after incoherent processing of M pulses is inversely proportional to M 1=2 (Equation 7.124). This makes the point that the gain from incoherent processing arises from a reduction in the detection threshold (for a fixed value of pd ) and not from an increase in signalto-noise ratio. 7.2.2.2 7.2.2.2.1

Signal with Rayleigh amplitude distribution (Swerling II) General case

Now consider random fluctuations in signal amplitude between successive pulses, with individual amplitude values taken from a Rayleigh distribution (Equation 7.26). For radar applications this is known as the ‘‘Swerling II’’ model. The resulting detection probability is (DiFranco and Rubin, 1968, p. 404) ! ð1 1 ET =2 2 pd ¼ fSþN ðEÞ dE ¼ G M; ; ð7:127Þ GðMÞ 1þR ET where R is the mean signal-to-noise ratio, equal to a 2 = 2 . The intended use of Equation (7.127) is for calculation of detection probability pd , given the signal-to-noise ratio R. Figure 7.14 shows ROC curves evaluated in this way for the case M ¼ 30. For additional cases (2  M  3000) see DiFranco and Rubin (1968, pp. 395–403). To calculate R from pd a different approach is needed.

Sec. 7.2]

7.2 Multiple known pulses in Gaussian noise, incoherent processing

341

Figure 7.14. ROC curves in the form 10 log10 ð2RÞ vs. pd for various pfa for a Rayleigh signal in Rayleigh noise; example of incoherent addition of M samples, with M ¼ 30. The pfa values are given by 0.693/n 0 , where n 0 is between 10 and 10 10 as labeled (reprinted from DiFranco and Rubin, 1968, # Scitech, Raleigh, NC).

Convenient approximate methods for doing so, valid for Swerling II statistics over a wide range of values of M, pd , and pfa , are given by Shnidman (2002), Hmam (2005), and Barton (2005). 7.2.2.2.2 Special case M ¼ 1 For the special case M ¼ 1, it follows from Equation (7.127) that  

 E E pd ¼ 1   1; 2 T ¼ exp  2 T ; 2 ð1 þ RÞ 2 ð1 þ RÞ

ð7:128Þ

resulting in Equation (7.34), as for a single coherently processed pulse. 7.2.2.2.3 Limit of large M If M  1, Equation (7.127) becomes pffiffiffiffiffi   ’fa  M R pd F : 1þR

ð7:129Þ

342 Statistical detection theory

[Ch. 7

Without further approximation this can be rearranged as R¼

R0 pffiffiffiffiffi ; 1 þ ’d = M

ð7:130Þ

where R0 is given by Equation (7.113). Equation (7.130) has the same form as Equation (7.114), implying that Figures 7.12 and 7.13 are applicable to this case. The 50 % threshold (R50 ) is given by Equation (7.111). 7.2.2.3

Signal with one-dominant-plus-Rayleigh amplitude distribution (Swerling IV)

7.2.2.3.1 General case The 1D þ R distribution (see Section 7.1.2.4) is given by Equation (7.50). For radar applications this case is known as the ‘‘Swerling IV’’ model. The corresponding detection probability is given by ð1 pd ¼ fSþN ðEÞ dE: ð7:131Þ ET

The result is (DiFranco and Rubin, 1968, p. 427) ! M X M! ðR=2Þ k ET =2 2 pd ¼ 1   M þ k; ð7:132Þ 1 þ R=2 ð1 þ R=2Þ M k¼0 k! ðM  kÞ! ðM þ k  1Þ! where R¼

2a 2 3 2

and  is the lower incomplete gamma function (Appendix A) ðx ða; xÞ  t a1 e t dt:

ð7:133Þ

ð7:134Þ

0

The intended use of Equation (7.132) is for calculation of detection probability pd , given the signal-to-noise ratio R. Figure 7.15 shows ROC curves evaluated in this way (with Equation 7.80 for pfa ) for the case M ¼ 30. For additional cases (2  M  3000) see DiFranco and Rubin (1968, pp. 428–436). To calculate R from pd a different approach is needed. Convenient approximate methods for doing so, valid for Swerling IV statistics over a wide range of values of M, pd , and pfa , are given by Shnidman (2002) and Barton (2005). 7.2.2.3.2

Special case M ¼ 1

For the special case M ¼ 1, Equation (7.132) simplifies to Equation (7.57), and the corresponding ROC curves are as for a single coherently processed pulse (Figure 7.6).

Sec. 7.2]

7.2 Multiple known pulses in Gaussian noise, incoherent processing

343

Figure 7.15. ROC curves in the form 10 log10 ð2RÞ vs. pd for various pfa for a 1D þ R signal in Rayleigh noise; example of incoherent addition of M samples, with M ¼ 30. The pfa values are given by 0.693/n 0 , where n 0 is between 10 and 10 10 as labeled (reprinted from DiFranco and Rubin, 1968, # Scitech, Raleigh, NC).

7.2.2.3.3 Limit of large M If M  1, Equation (7.132) simplifies to (DiFranco and Rubin, 1968, p. 439) pffiffiffiffiffi   ’fa  M R pd F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ 2R þ 2R 2

ð7:135Þ

This can be rearranged as a quadratic equation in R pffiffiffiffiffi ðM  2’ 2d ÞR 2  2ð M ’fa þ ’ 2d ÞR þ ’ 2fa  ’ 2d ¼ 0;

ð7:136Þ

whose solution, disregarding the unphysical root, is pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi ’fa  ’d ð1 þ ’fa = M Þ 1 þ ð’ 2fa  ’ 2d Þ=ð M þ ’fa Þ 2 þ ’ 2d = M pffiffiffiffiffi MR ¼ : ð7:137Þ 1  2’ 2d =M

344 Statistical detection theory

[Ch. 7

For pd ¼ 0.5, Equation (7.137) simplifies—without further approximation—to Equation (7.111). For arbitrary pd it can be written17   pffiffiffiffiffi ’fa  ’d 1 p ffiffiffiffiffi MR þO ; ð7:138Þ M 1 þ ’d = M which has the same form as Equation (7.114), implying that Figures 7.12 and 7.13 are applicable to this case. Thus, Figure 7.13 (in combination with Figure 7.12) may be used for 1D þ R and large M. ROC curves from DiFranco and Rubin (1968) can be used for smaller values of M between 2 and 100. If greater accuracy is required than obtained in this way, Equation (7.132) can be used.

7.3

APPLICATION TO SONAR

Previously in this chapter, ROC relationships were derived for two different types of processing. In order to apply the results of Chapter 7 to the four sonar types considered in Chapter 3, it is first necessary to map processing types onto sonar types. In each case the SNR must be calculated in some appropriate bandwidth that depends on signal processing. For incoherent processing there is an additional parameter M, equal to the number of pulses added incoherently. The four cases of Chapter 3 are considered separately below. A fifth type of processing, applicable to active sonar, involving the transmission of a broadband frequency-modulated (FM) pulse and replica correlation (i.e., convolution of the received echo with a replica of the transmitted waveform) of the received signal, is also considered. 7.3.1

Active sonar

For active sonar the interpretation is straightforward. A ‘‘pulse’’ is just that, the waveform transmitted by the sonar, and received at some later time, usually by the same sonar. The main possibilities are summarized in Table 7.3. This chapter contains many different equations for detection probability and many different corresponding ROC curves. There is no simple prescription for determining which of these to use for any given problem. However, once the statistics are known, it is relatively straightforward to evaluate the corresponding detection probability (see Table 7.4) to an appropriate degree of accuracy. 7.3.2

Passive sonar

For passive sonar, the concept of a pulse requires some explanation. In this context, by ‘‘pulse’’ is meant a sinusoidal or otherwise known time series, of a certain duration, received by the sonar. Specifically for a narrowband (NB) CW sonar, R0 pffiffiffiffiffi , with R0 from Equation (7.113), holds for all three 1 þ ’d = M distributions considered (Swerling II, Swerling IV, and the non-fluctuating case). 17

The approximation R ¼

Sec. 7.3]

7.3 Application to sonar 345

Table 7.3. Application of the detection theory results of Section 7.1 to active sonar cosine wave (CW) and frequency-modulated (FM) pulses. Processing

Signal-to-noise ratio ðRÞ

Amplitude ðAÞ

CW pulse with Doppler processing

SNR in Doppler processing band

Spectral amplitude (after FFT)

CW pulse with energy detector

SNR in total Rx bandwidth

Square root of total signal energy

FM pulse with matched filter

SNR after matched filter

Matched filter output

Table 7.4. Equations for the detection probability for different signal amplitude distributions. In all cases the noise amplitude is assumed to have a Rayleigh distribution, leading to Equation (7.7) for the false alarm probability of a single pulse or Equation (7.80) for multiple pulses. Distribution

Single pulse

Multiple pulses

Dirac

Equation (7.12)

Equation (7.94)

Rayleigh

Equation (7.34)

Equation (7.127)

Rice

Equation (7.45)

1D þ R

Equation (7.57)

Equation (7.132)

the input to the detector is a continuous sine wave. The processing is coherent, so the theory of Section 7.1 applies. The signal is integrated over a time DtNB . Thus, the pulse is the sinusoidal wave, the pulse duration is the coherent integration time DtNB , and the bandwidth is the reciprocal of this time B ¼ 1=DtNB . For broadband passive sonar, the appropriate assumption is that nothing is known about the received signal except that it is within the frequency band of the receiver. The processing is incoherent and the theory of Section 7.2 is applicable. Because the form of the signal is unknown, the ‘‘pulse’’ is a single time sample and the ‘‘pulse duration’’ is the sampling interval (or Nyquist interval N if larger). Assuming that the signal is sampled at the Nyquist rate, the sampling interval is

t ¼ N ¼

1 : 2B

ð7:139Þ

The number of pulses to be added incoherently is then M¼

DtBB ¼ 2B DtBB ; N

ð7:140Þ

346 Statistical detection theory

[Ch. 7

Table 7.5. Application of detection theory results to narrowband (NB) and broadband (BB) passive sonar. Processing

Section

Signal-to-noise ratio ðRÞ

Pulse duration

NB (coherent processing)

7.1

SNR in NB processing band, B ¼ 1=DtNB

Coherent integration DtNB

BB (incoherent processing)

7.2

SNR in BB bandwidth

Sampling interval, assumed equal to the Nyquist interval N (Equation 7.139)

where DtBB is the incoherent integration time. The situation for passive sonar is summarized in Table 7.5. The same choice of distribution applies here as for active sonar (see Table 7.4). 7.3.3

Decision strategies and the detection threshold

To make use of the available evidence (i.e., the received S þ N waveform) the sonar operator must interpret the information and make a decision, usually in the form of either ‘‘target present’’ or ‘‘target absent’’. Decision strategies are discussed by several authors (Selin, 1965; Helstrom, 1968; Kay, 1998; Lehmann and Romano, 2005). Ideally, one would like to consider the impact of this decision on subsequent events. For example, an archeologist detecting a sunken wreck might deploy a submersible to investigate further, whereas a ship detecting an attacking torpedo would initiate immediate countermeasures. In both cases there is a cost associated with action and a (possibly greater) cost associated with inaction. A powerful method to analyze such situations using Bayesian probabilities is described by Selin (1965, p. 11). Applications of this approach are rare, due to the difficulty in quantifying the necessary costs and a priori probabilities in a non-trivial manner (Helstrom, 1968). A pragmatic and widely adopted solution to this problem is to assign a maximum acceptable false alarm rate, and hence false alarm probability pfa . Once made, this choice determines a maximum permissible value for the amplitude threshold, from which an SNR threshold corresponding to a particular detection probability (i.e., the detection threshold, DT) can be derived using the procedure outlined in Section 7.1.3. In principle, the equations of the present chapter can be used to calculate pd (given SNR and pfa ), without the need to first calculate the detection threshold at which a particular value of pd is achieved. However, it is common practice to quantify sonar performance in terms of signal excess (the amount by which SNR exceeds DT), which is possible only if DT is known. Doing so makes it easy to make later adjustments in uncertain parameters such as target strength or source level, or to carry out sensitivity studies. Estimation of DT is straightforward from the graphs presented in this chapter of 10 log10 R vs. pfa .

Sec. 7.3]

7.3 Application to sonar 347 Table 7.6. Detection threshold 10 log10 R vs. pd for various statistics, with pfa and pd values as stated. Values are obtained from Figures 7.1, 7.3, and 7.6. Detection threshold (dB) pfa

pd ¼ 10 %

pd ¼ 50 %

pd ¼ 90 %

10 4

Constant 1D þ R Rayleigh

6.1 5.2 4.8

9.4 10.1 10.9

11.7 15.6 19.4

10 8

Constant 1D þ R Rayleigh

10.4 9.0 8.5

12.5 13.3 14.1

14.2 18.5 22.4

10 12

Constant 1D þ R Rayleigh

12.8 11.1 10.4

14.3 15.1 15.9

15.7 20.2 24.2

The presentation of ROC curves for three different signal distributions begs an important question; namely, which of them to use for any given sonar problem. The non-fluctuating signal corresponds to a very stable target and stable propagation conditions, a combination that is unlikely to be encountered in practice except in a tightly controlled experiment. At the other extreme is the Rayleigh distribution, which describes a signal with noiselike fluctuations. The third distribution considered (1D þ R) represents a signal with intermediate statistics. Table 7.6 quantifies the effect of varying pd and pfa for each of the three distributions. The differences in DT between them are relatively small (ca. 2 dB) for pd < 50 %, but significantly larger for pd > 50 %. The largest differences shown in the table (about 8 dB) are between the Rayleigh and Dirac (non-fluctuating) amplitude distributions, and arise for pd ¼ 90 %. All values in the table are for a single pulse. Averaging has the effect of reducing the differences. In general, the effect of any fluctuations is to increase DT (relative to that for a non-fluctuating signal) when pd is high, and to decrease it when pd is low. The 1D þ R distribution gives a result that is intermediate between the detection thresholds calculated using Dirac and Rayleigh distributions, as can be expected from the intermediate nature of the distribution itself (see Figure 7.2). For some applications it is convenient to use a simple approximation to the detection threshold. For this purpose Equation (7.62) is suitable. An alternative, for pd ¼ 0.5 only, is Equation (7.63). The accuracy of both approximations is examined in Table 7.7, showing that the error incurred by their use is 0.2 dB or less in the range 0:1 < pd < 0:9 and 10 12 < pfa < 10 4 . Alternative approximations for 1D þ R (Swerling IV) statistics are given by Shnidman (2002) and Barton (2005).

348 Statistical detection theory

[Ch. 7

Table 7.7. Detection threshold 10 log10 R vs. pd for a 1D þ R amplitude distribution for the same pfa and pd values as Table 7.6. Up to three values are given for each pd –pfa combination: the first is obtained from Figure 7.6; the second [in square brackets] uses the approximate Equation (7.62); the third (in round brackets, for DT50 only) uses the alternative approximation Equation (7.63). Detection threshold for 1D þ R statistics (dB) pfa

pd ¼ 10 %

pd ¼ 50 %

pd ¼ 90 %

10 4

5.2 [5.2]

10.1 [10.1] (10.1)

15.6 [15.5]

10 8

9.0 [9.0]

13.3 [13.3] (13.3)

18.5 [18.5]

10 12

11.1 [10.9]

15.1 [15.1] (15.1)

20.2 [20.3]

All of the results of this chapter make the assumption of Gaussian statistics for the noise, leading to Rayleigh-distributed amplitudes, which once averaged follow a  2 distribution. If the background is due to a large number of independent contributions, this assumption is a reasonable one. However, in some situations the passive sonar background originates from a relatively small number of discrete sources, such as individual ships, thus distorting the statistics. For active sonar it is often the case that the background is dominated by reverberation rather than ambient noise, and the reverberation can sometimes be resolved into contributions from discrete scatterers such as individual rocks or shipwrecks, with the same effect. In either case, Gaussian statistics do not provide a good description of the background (Abraham, 2003; Nielsen et al., 2008).18 Accurate approximations for the detection threshold in noise described by Weibull and K distributions,19 based on those of Hmam (2005), are derived by Abraham (2010) for both fluctuating and non-fluctuating signal models.

7.4 7.4.1

MULTIPLE LOOKS Introduction

So far, the focus of this chapter has been on detection probability associated with a single ‘‘look’’20 of a sonar—typically a single pulse for an active sonar, or for passive sonar the result of coherent or incoherent integration over a number of successive 18

Detection in noise with non-Gaussian statistics is considered by Kassam (1987) and Kay (1998). 19 A convenient summary of these and other related distributions is given by Jackson and Richardson (2007). 20 That is, a single threshold comparison with a binary outcome.

Sec. 7.4]

7.4 Multiple looks 349

time samples. More generally one can think of a number of successive independent looks, some of which might result in threshold crossings for a given target and others not. An important question is how best to combine the information from multiple detection opportunities (and multiple threshold comparisons) in such a way as to maximize the overall detection probability (for a fixed false alarm rate). This question is the subject of this section. The process of combining information from multiple detection opportunities in this way is referred to below as ‘‘fusion’’. As a simple example, consider a situation involving two nearly simultaneous (but independent—see Weston, 1989) looks on a sonar screen, or a single simultaneous look on each of two identical sonars, in nearly identical positions and orientations. For any given target the expectation value of the SNR is the same for each of the two looks. Similarly, the detection and false alarm probabilities, denoted D and F respectively, are unchanged from look to look. (The symbols pd and pfa are reserved for the corresponding probabilities after fusion). Because there are two looks, there is twice as much information as for a single one, so intuitively one might expect an improvement in the performance. Pertinent questions are: — How can this anticipated improvement be realized and quantified? — How does the improvement depend on the manner in which the information is combined? To answer these questions it is assumed, for simplicity, that within each sonar display there exists a signal due to one and only one target. If Rayleigh21 statistics are assumed for both signal and noise amplitudes, from Equation (7.34), F and D are related according to log F 1þR¼ ; ð7:141Þ log D where R is the signal-to-noise ratio, constrained to the interval ½0; 1 so that, according to Equation (7.141), F cannot exceed D. This is consistent with their respective definitions (see Chapter 2) as the probabilities of a threshold crossing, respectively, for the cases of noise only and signal plus noise. The values of pd and pfa , the new detection and false alarm probabilities after merging the two displays, depend on how the available information is combined. An equivalent SNR, denoted Req , can be defined in terms of these probabilities such that 1 þ Req 

log pfa : log pd

ð7:142Þ

Thus defined, Req is the SNR that would be required to achieve the same performance from a single look as is actually achieved by combining two independent looks. The ratio of the equivalent SNR, Req , to the true SNR, R, is a measure of the 21 The choice of Rayleigh statistics is made at this point for mathematical convenience, as this choice results in simple forms for ROC relationships. Other distributions are considered in Section 7.4.2.4.

350 Statistical detection theory

[Ch. 7

improvement in performance and is referred to henceforth as the fusion gain. This parameter, denoted G, can then be written G

Req 1  ðlog pd Þ 1 log pfa ¼ : R 1  ðlog DÞ 1 log F

ð7:143Þ

The fusion gain is a useful quantitative measure of the performance of the combined display. Its value is considered below for two different situations involving the combination (fusion) of data from two pulses for the case of a single target. The problem for multiple targets is considered by de Theije et al. (2008), including the effect of positioning errors.

7.4.2

AND and OR operations

Consider two different logical operations for combining the data from the two sonars, an AND operation, requiring a threshold crossing at the target position on both sonar screens before a detection decision is made, and an OR operation, for which a single threshold crossing is sufficient. 7.4.2.1

AND operation for Rayleigh statistics

By definition, the probability of the target causing a threshold crossing on each separate image is D. If the two observations are independent, the probability of detection after an AND operation is pd ¼ D 2

ð7:144Þ

pfa ¼ F 2 :

ð7:145Þ

and the false alarm probability is

From Equation (7.141) it follows that 1þR¼

log pfa : log pd

ð7:146Þ

Thus, for this situation the false alarm probability is reduced, but so is the detection probability, in such a way that the true and equivalent signal-to-noise ratios are identical. This can be written as GAND ¼ 1;

ð7:147Þ

where GAND is the fusion gain for an AND operation. This is a curious result. It means that, for the assumed Rayleigh statistics, the performance of the combined (AND) system can be achieved by either one of the individual systems simply by switching off the other one and adjusting the threshold to maintain the same false alarm rate.

Sec. 7.4]

7.4.2.2

7.4 Multiple looks 351

OR operation for Rayleigh statistics

For the OR operation, the detection and false alarm probabilities pd and pfa are given by pd ¼ 2D  D 2 ð7:148Þ and pfa ¼ 2F  F 2 : ð7:149Þ The detection probability increases relative to that for a single display because the number of opportunities has doubled, apparently improving the performance of the sonar. However, the false alarm probability also increases and it is not immediately obvious whether the net gain is positive or negative. The corresponding ROC curves can be obtained from the relationship pffiffiffiffiffiffiffiffiffiffiffiffiffiffi logð1  1  pfa Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi : 1þR¼ ð7:150Þ logð1  1  pd Þ The gain in performance (i.e., the reduction in detection threshold for fixed pd and pfa ) can be calculated as

GOR

log pfa 1 log pd pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ : logð1  1  pfa Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi  1 logð1  1  pd Þ

ð7:151Þ

Figure 7.16 shows the quantity 10 log10 GOR vs. pfa for fixed values of pd . This is the gain in decibels, which is positive for the entire range of values considered, indicating an improvement in detection performance (GOR > 1). All of the curves are remarkably flat for small values of the false alarm probability. The reason for this behavior can be seen by writing Equation (7.151) in the form log D log pfa  log pd pffiffiffiffiffiffiffiffiffiffiffiffiffiffi GOR ¼ ; ð7:152Þ log pd logð1  1  pfa Þ  log D which, if pfa  1, simplifies to GOR

log D 1  ðlog pfa Þ 1 log pd : log pd 1  ðlog pfa Þ 1 logð2DÞ

ð7:153Þ

It is often the case that a sonar design requires a false alarm probability that is many orders of magnitude less than pd , corresponding to the left half of Figure 7.16. In this situation, Equation (7.153) may be further approximated by neglecting terms of order ðlog pfa Þ 1 . In this limit GOR reaches an asymptotic value of pffiffiffiffiffiffiffiffiffiffiffiffiffi logð1  1  pd Þ ; ð7:154Þ GOR log pd independent of pfa and equal to 1.77 for pd ¼ 0.5. Converting to decibels this is a gain of 2.5 dB, consistent with Figure 7.16.

352 Statistical detection theory

[Ch. 7

Figure 7.16. Fusion gain 10 log10 GOR vs. log10 pfa for OR operation, with fixed values of pd as marked (Rayleigh statistics); pd values are 0.1 (smallest gain) to 0.9 (highest gain) in steps of 0.2; the asymptotic gain for pd ¼ 0.5 is 2.5 dB.

The information in Figure 7.16 can be presented alternatively in the form of gain vs. log F for fixed D values. The resulting graph is shown in Figure 7.17. 7.4.2.3

Summary table for Rayleigh statistics

The results for AND and OR operations are summarized in Table 7.8. 7.4.2.4

Simulations with Rayleigh and non-Rayleigh signal statistics

The Rayleigh distribution was chosen for the above analysis because it is particularly amenable to algebraic manipulation. The sensitivity of the fusion gain to the choice of pdf is considered by computing theoretical ROC curves for Rayleigh, Dirac, and 1D þ R distributions by means of numerical simulations. It is convenient to start with a non-fluctuating signal. ROC curves in the form pd vs. pfa are plotted in Figure 7.18 for SNR values between 9 dB and 13 dB as marked. The solid cyan lines show theoretical ROC curves for a single pulse with SNR values in decibels, as labeled. For each SNR value, in addition to the solid curve, there is also a dashed line and a dotted one. These are the theoretical ROC curves for combining a pair of pulses of the same SNR, with OR and AND fusion, respectively. For SNR ¼ 10 dB and pd ¼ 12, a fusion gain of 2 dB and 0.8 dB can be inferred from the two horizontal bars labeled AND and OR, respectively.

Sec. 7.4]

7.4 Multiple looks 353

Figure 7.17. Fusion gain 10 log10 GOR vs. log10 F for OR operation, with fixed values of D as marked (Rayleigh statistics); D values are 0.1 (smallest gain) to 0.9 (highest gain) in steps of 0.2; the asymptotic gain for D ¼ 0.5 is 3.8 dB.

These calculations are repeated for 1D þ R and Rayleigh signal statistics in Figures 7.19 and 7.20, respectively, and for the same SNR values. It can be seen that the gain for OR fusion increases (from 0.8 to 2.5 dB), while the gain for AND fusion decreases (from 2 to 0 dB), with increasing signal fluctuations. In

Table 7.8. ROC relationships and fusion gain for AND and OR operations for fixed SNR, with Rayleigh statistics. Single sonar

Combined (AND)

Combined (OR)

Detection probability pd

D

D2

2D  D 2

False alarm probability pfa

F

F2

1þR

log pfa log pd

log pfa log pd

2F  F 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi logð1  1  pfa Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi logð1  1  pd Þ

G

N/A

1

log pfa 1 log pd pffiffiffiffiffiffiffiffiffiffiffiffiffiffi logð1  1  pfa Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi  1 logð1  1  pd Þ

354 Statistical detection theory

[Ch. 7

Figure 7.18. Detection probability pd (after fusion) vs. false alarm probability pfa for nonfluctuating signal in Rayleigh background. The gain (with SNR ¼ 10 dB and pd ¼ 12 ) is 0.8 dB for OR fusion and 2 dB for AND fusion.

particular, the fusion gain for the AND operation vanishes in the Rayleigh case, so the dotted curves are hidden in Figure 7.20 (they coincide with those for a single sonar). This sensitivity of fusion gain to signal statistics means that the optimum fusion rule depends in general on the signal amplitude distribution. However, de Theije et al. (2008) show that the AND gain is severely degraded in the presence of measurement (positioning) errors, while the OR gain is less affected, making the latter potentially better suited to a multistatic sonar geometry.

7.4.3

Multiple OR operations

Consider N independent looks, each with identical detection probability per look equal to D. In principle, the N looks could correspond to simultaneous measurements using N sonars, but a more likely application would be to N consecutive looks and a single sonar. If the N looks are combined with multiple OR operations, the overall probability of detection, denoted pd (the cumulative detection probability) is the chance of at least one threshold crossing out of N opportunities. In

Sec. 7.4]

7.4 Multiple looks 355

Figure 7.19. Detection probability pd (after fusion) vs. false alarm probability pfa for 1D þ R signal in Rayleigh background. The gain is 1.8 dB for OR fusion (with SNR ¼ 10 dB and pd ¼ 12 ) and 0.9 dB for AND fusion (approximately independent of pd ).

other words22 pd ¼ 1  ð1  DÞ N

ð7:155Þ

pfa ¼ 1  ð1  FÞ N :

ð7:156Þ

and similarly

Assuming a Rayleigh distribution for the signal amplitude, the resulting ROC relationship is therefore 1þR¼

log½1  ð1  pfa Þ 1=N  : log½1  ð1  pd Þ 1=N 

ð7:157Þ

The gain in performance (i.e., the reduction in detection threshold for fixed pd 22

More generally, if the Di values are not identical, where i is the look number: N Y pd ¼ 1  ð1  Di Þ; i¼1

where pd is the probability that one or more threshold crossing occurs in N opportunities. The false alarm probability pfa is similarly increased: N Y pfa ¼ 1  ð1  Fi Þ: i¼1

356 Statistical detection theory

[Ch. 7

Figure 7.20. Detection probability pd (after fusion) vs. false alarm probability pfa for Rayleigh signal in Rayleigh background. The gain is 2.5 dB for OR fusion (with SNR ¼ 10 dB and pd ¼ 12 ) and 0 dB for AND fusion (independent of SNR and pd ).

and pfa ) is log pfa 1 log pd G¼ : log½1  ð1  pfa Þ 1=N  1 log½1  ð1  pd Þ 1=N 

7.4.4

ð7:158Þ

‘‘M out of N ’’ operations

A multiple OR operation amounts to a requirement that at least one threshold crossing is made out of N detection opportunities. Similarly, a multiple AND operation (not considered explicitly) corresponds to the much more stringent requirement of N threshold crossings out of N opportunities. The former leads to high probability of detection, and a correspondingly high false alarm rate. The latter virtually eliminates false alarms (for large N) at the expense of low detection probability. This line of thinking suggests a middle road to be explored between these two extremes, whereby M threshold crossings are required from N opportunities, with 1  M  N. For a given value of N, the parameter M can be adjusted to optimize detection performance. This ‘‘M out of N ’’ approach23 is analyzed by Reibman and Nolte (1987), Weiner (1991), and Shnidman (1998). The general result for the probability of 23

Also known as ‘‘binary integration’’.

Sec. 7.5]

7.5 References 357

obtaining M or more threshold crossings out of N independent looks in the presence of signal plus noise (i.e., the detection probability) is (Reibman and Nolte, 1987) pd ¼ 1  ð1  DÞ NM

M X

N! D q ð1  DÞ Mq : q!ðN  qÞ! q¼0

ð7:159Þ

Similarly, the probability of at least M threshold crossings if only noise is present gives the false alarm probability: pfa ¼ 1  ð1  FÞ NM

M X

N! F q ð1  FÞ Mq: q!ðN  qÞ! q¼0

ð7:160Þ

It follows from Equation (7.141) that the fusion gain for the ‘‘M out of N’’ case is, assuming Rayleigh statistics again, log pfa 1 log pd G¼ : log F 1 log D

ð7:161Þ

Weiner (1991) calculates the detection threshold vs. M for fixed N for non-fluctuating and Swerling II targets, showing that an optimum value exists for M that minimizes the detection threshold. Shnidman (1998) calculates the optimum value of M for nonfluctuating, Swerling II, and Swerling IV targets, and provides approximate expressions for this optimum. These approximations, which are all valid to within 10 % in the range 10  N  500, are M0 10 0:8 N 0:02 ;

ð7:162Þ

MII 10 0:91 N 0:38

ð7:163Þ

MIV 10 0:873 N 0:27 ;

ð7:164Þ

and where the subscript indicates the Swerling type (0 for the non-fluctuating case). Weston (1992) considers the role of prior knowledge in determining the optimum value of M, arguing, for example, that fewer (consecutive) threshold crossings are needed to confirm the presence of a contact on a sonar screen if it is known in advance that a target is in the area.

7.5

REFERENCES

Abraham, D. A. (2003) Signal excess in K-distributed reverberation, IEEE J. Oceanic Eng., 28, 526–536. Abraham, D. A. (2010) Detection-threshold approximation for non-Gaussian backgrounds, IEEE J. Oceanic Eng., to appear in 35(2), April. Albersheim, W. J. (1981) A closed-form approximation to Robertson’s detection characteristics, Proc. IEEE, 69, 839.

358 Statistical detection theory

[Ch. 7

Barton, D. K. (2005) Universal equations for radar target detection, IEEE Transactions of Aerospace and Electronic Systems, 41, 1049–1052. Crocker, M. J. (Ed.) (1997) Encyclopedia of Acoustics, Wiley, New York. de Theije, P. A. M., van Moll, C. A. M., and Ainslie, M. A. (2008) The dependence of fusion gain on signal-amplitude distributions and position errors, IEEE J. Oceanic Eng., 3(3). DiFranco, J. V. and Rubin, W. L. (1968) Radar Detection, Prentice-Hall, Englewood Cliffs, NJ. Helstrom, C. W. (1968) Statistical Theory of Signal Detection, Pergamon Press, Oxford. Hmam, H. (2005) SNR calculation procedure for target types 0, 1, 2, 3, IEEE Transactions of Aerospace and Electronic Systems, 41, 1091–1096. Jackson, D. R. and Richardson, M. D. (2007) High-Frequency Seafloor Acoustics, Springer Verlag, New York. Jelalian, A. V. (1992) Laser Radar Systems, Artech House, Boston. Kassam, S. A. (1987) Signal Detection in Non-Gaussian Noise, Springer Verlag, New York. Kay, S. M. (1998) Fundamentals of Statistical Signal Processing: Volume II, Detection Theory, Prentice-Hall, Englewood Cliffs, NJ. Lehmann, E. L. and Romano, J. P. (2005) Testing Statistical Hypotheses, Springer Verlag, New York. Levanon, N. (1995) Radar Principles, Wiley-Interscience, New York. McDonough, R. N. and Whalen, A. D. (1995) Detection of Signals in Noise (Second Edition), Academic Press, San Diego. Nielsen, P. L., Harrison, C. H., and Le Gac, J.-.C. (2008) Proc. International Symposium on Underwater Reverberation and Clutter, September 9–12, NATO Undersea Research Center, La Spezia, Italy. Reibman, A. and Nolte, L. W. (1987) Optimal detection and performance of distributed sensor systems, IEEE Transactions on Aerospace and Electronic Systems, AES-23(1), 24–30. Rice, S. O. (1948) Statistical properties of a sine wave plus random noise, Bell Syst. Tech. J., 109–157, January. Robertson, G. H. (1967) Operating characteristic for a linear detector, Bell Syst. Tech. J., 755– 774. Selin, I. (1965) Detection Theory, Princeton University Press, Princeton, NJ. Shnidman, D. A. (1998) Binary integration for Swerling target fluctuations, IEEE Transactions on Aerospace and Electronic Systems, 34, 1043–1053. Shnidman, D. A. (2002) Determination of required SNR values, IEEE Transactions on Aerospace and Electronic Systems, 38, 1059–1064. Weiner, M. A. (1991) Binary integration of fluctuating targets, IEEE Transactions on Aerospace and Electronic Systems, 27, 11–17. Weston, D. E. (1989) Independence in sonar observations, J. Acoust. Soc. Am., 85, 1612–1616. Weston, D. E. (1992) How to balance the sonar equation, Admiralty Research Establishment (ARE) Seminar, Winfrith, U.K., March.

Part III Towards Applications

8 Sources and scatterers of sound

An experiment is a question which science poses to Nature, and a measurement is the recording of Nature’s answer. Max Planck, Scientific Autobiography and Other Papers (1949) The behavior of underwater sound is central to sonar performance. A theoretical treatment is presented in Chapter 5, but on its own that is not enough. To inspire confidence, the theory must be supported by measurement. The purpose of the present chapter is to summarize relevant acoustic measurements and, where feasible, to place these in a theoretical framework. Considered first, in Section 8.1, is the interaction of sound with ocean boundaries in the form of reflection loss and scattering strength. This is followed, in Section 8.2, by measurements pertaining to the scattering and absorption of sound due to submerged objects (of interest are their target strength, volume backscattering strength, and volume attenuation coefficient). In Section 8.3 a mainly empirical description of underwater noise sources is provided.

8.1

REFLECTION AND SCATTERING FROM OCEAN BOUNDARIES

The reflective properties of the sea surface and seabed have an important influence on long-distance propagation, which often involves multiple interactions with either or both boundaries, especially in shallow water (see Chapter 9). Furthermore, scattering from these boundaries, resulting in reverberation at a sonar receiver, is often the limiting factor in determining the performance of an active sonar. Interaction of sound with the ocean’s boundaries is the subject of ongoing research (Pace and Blondel, 2005; Jackson and Richardson, 2007). In those situations for which the physics is well understood it is possible to present theoretical equations

362 Sources and scatterers of sound

[Ch. 8

with good predictive ability. More often, a limited theoretical understanding must be supplemented with an empirical element based on measurements, resulting in a semiempirical approach. It is convenient to define the parameter F as the numerical value of the frequency when expressed in units of kilohertz such that, if f^ is the frequency in hertz, F


8.1.1

f^ : 1,000

ð8:1Þ

Reflection from the sea surface

8.1.1.1

Theoretical prediction for an isotropic surface wave spectrum

8.1.1.1.1 Coherent reflection coefficient The loss due to scattering from a rough boundary can be modeled using the coherent reflection coefficient, the squared magnitude of which (see Chapter 5) is jRj 2 ¼ 1  4k 2  2 Y sin ;

ð8:2Þ

where  is the grazing angle of the incident wave; k is the acoustic wavenumber;  is the RMS boundary roughness; and the dimensionless parameter Y is given by the following integral over the roughness wavenumber 
 ð 2EPT 2 1=2 G1 ðÞ 3=2 d; ð8:3Þ Y¼ k 2 where EPT is a constant equal to 0.3814; and G1 ðÞ is the one-dimensional roughness wavenumber spectrum. The angle  is assumed to be small compared with ðkLÞ 1=2 , where L is the correlation length of the rough surface. For water of depth H, gravity waves satisfy the dispersion relation (MilneThomson, 1962) O 2 ¼ g tanhðHÞ; ð8:4Þ which in deep water (H  1) simplifies to O 2  g:

ð8:5Þ

For this situation, and assuming an isotropic wavenumber spectrum, the wavenumber and frequency spectra are related via (Brekhovskikh and Lysanov, 2003) g 1=2 pffiffiffiffiffiffi G1 ðÞ ¼ Sð gÞ: ð8:6Þ 3=2 4 8.1.1.1.2

Pierson–Moskowitz surface wave spectrum

The Pierson–Moskowitz (PM) gravity wave frequency spectrum depends in the following manner on wind speed v20 (measured at a height of 20 m): SðOÞ ¼ CPM g 2 O 5 exp½BPM ðg=Ov20 Þ 4 ;

ð8:7Þ

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 363

where BPM ; CPM are empirical constants (see Chapter 4); and g is the acceleration due to gravity. Substituting Equation (8.7) in Equation (8.6) gives ! CPM  2PM G1 ðÞ ¼ exp  2 ; ð8:8Þ 4 4  where PM ¼

pffiffiffiffiffiffiffiffiffi 2 BPM g=v 20 :

ð8:9Þ

Evaluation of the integral for Y in Equation (8.3) gives 1=4

Y¼

EPT Gð34ÞC PM ; ðkPM Þ 1=2

ð8:10Þ

where  2PM ¼

CPM v 420 : 4BPM g 2

ð8:11Þ

It is convenient to convert wind speed to a standard reference height of 10 m. Using the approximate ratio (Dobson, 1981) v10 =v20  0:94;

ð8:12Þ

the coherent reflection coefficient (see Equation 8.2) can be written in the form (Ainslie, 2005b)
3 v10 3=2 ^ loge jRPM j  1:14F sin : ð8:13Þ 10 This simple theoretical expression is known to underestimate measured sea surface scattering loss, by a factor of order 3 (Weston and Ching, 1989), illustrating the need for measurements to support theory. As explained in Section 8.1.1.2.1, the discrepancy between measurements and rough surface-scattering theory can be attributed to the formation of wind-generated bubbles close to the sea surface, which influence the interaction of sound with the air–sea boundary (Norton and Novarini, 2001; Ainslie, 2005b). 8.1.1.1.3 Neumann–Pierson surface wave spectrum An alternative to the PM spectrum sometimes used in older literature is the Neumann–Pierson (NP) spectrum, given by (see Chapter 4) SðOÞ ¼ ANP O 6 exp½2ðg=Ov5 Þ 2 ;

ð8:14Þ

where ANP is an empirical constant. Following the same procedure as previously for the PM spectrum results in
1=10 4EPT 9 2 Y¼ A NP NP ð8:15Þ 3ðgkNP Þ 1=2 2

364 Sources and scatterers of sound

and  2NP ¼ 3ð=2Þ 1=2 ANP

[Ch. 8




 v5 5 : 2g

ð8:16Þ

Converting the wind speed to a reference height of 10 m using the ratio (Dobson, 1981) v10 =v5  1:07; ð8:17Þ Equation (8.2) results in the following expression for reflection loss (Ainslie, 2005b)
4 v10 3=2 ^ loge jRNP j ¼ 0:74F sin : ð8:18Þ 10 As previously with Equation (8.13), this expression underestimates reflection loss. This discrepancy, which is attributed to the effects of wind-generated bubbles, is addressed in Section 8.1.1.2.1. 8.1.1.1.4

Effect of anisotropy

Cross-wind and downwind correlation lengths for an anisotropic Pierson–Moskowitz spectrum are calculated by Fortuin (1973) (see also Fortuin and de Boer, 1971). The resulting effect on the sea surface reflection coefficient is analyzed by Kuperman (1975). 8.1.1.2

Semi-empirical surface reflection loss models

Reflection and scattering of sound from the sea surface is the subject of ongoing research. At low frequency (< 1 kHz) and low wind speed (< 5 m/s), the surface can be regarded as perfectly flat and acoustically compliant, resulting in perfect reflection with a  phase change. At higher frequency, or if the sea surface is roughened by wind action, roughness scattering starts to become an issue. The presence of windgenerated bubbles near the sea surface plays an important part in this process. In principle these bubbles are capable of refracting, scattering, and absorbing sound, and their precise role in surface scattering is the subject of ongoing research (Norton and Novarini, 2001; Ainslie, 2005b; Dahl et al., 2008). In Section 8.1.1.2.1 a low-frequency sea surface reflection loss model, based on the measurements of Weston and Ching (1989) (henceforth abbreviated ‘‘WC89’’) and valid up to an acoustic frequency of 4 kHz, is described. The surface reflection loss (abbreviated SL) is defined as SL
10 log10 jRj 2 ;

ð8:19Þ

where R is the plane wave amplitude reflection coefficient of the sea surface. A highfrequency model developed at the University of Washington (APL-UW, 1994), intended for use above 10 kHz, is presented in Section 8.1.1.2.2.

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 365

8.1.1.2.1 Low-frequency surface loss model Regardless of the precise physical mechanism that gives rise to them, measured losses are of the form (WC89)
4 ^v loge jRWC89 j ¼ WC89 F 3=2 10 ; ð8:20Þ 10 where the parameter WC89 , equal to the reflection loss in nepers at a frequency of 1 kHz and wind speed 10 m/s, is a constant whose value appears to depend on season. On theoretical grounds, surface loss is expected to vary with grazing angle. The WC89 measurements involve propagation over a distance of 23 km. Variations in propagation loss were monitored over time and linked to changes in wind speed. The water was well mixed, resulting in an isothermal profile. For the geometry of this experiment (corresponding to a surface grazing angle of 1.7 deg for the limiting ray), measured values of WC89 are (WC89; Ainslie, 2005b)  0:0677 autumn WC89 ¼ ð8:21Þ 0:132 winterspring: Equation (8.20) is applicable to frequencies up to 4 kHz and wind speeds up to 13 m/s. Its functional form implies that the quantity f 3=2 logjRj is a straight line if plotted against the fourth power of wind speed, and this behavior is illustrated by Figure 8.1. The WC89 measurements (indicated by the symbols) do indeed follow approximate straight lines, although the gradient of the best fit straight line is different for each of the two data sets; hence the two different values of WC89 given in Equation (8.21). The assumption that the loss is proportional to angle would imply a loss in nepers per radian of 8
4 < 2:3 autumn ^ v

WC89 ¼ F 3=2 10 4:5 winterspring ð8:22Þ : 10 3:4 average. The average value quoted is the arithmetic mean of the other two. It exceeds the theoretical value due to surface scattering for the NP spectrum (Equation 8.18) by a factor of 4.6. Notice the similarity in functional form between Equations (8.13) and (8.20). Both are proportional to the product of F 3=2 and a power of v10 . Ainslie (2005b) demonstrates the need for a correction to Equation (8.13) from increased compressibility of water close to the sea surface due to the presence of wind-generated bubbles. This increase in compressibility causes a reduction in the near-surface sound speed and consequently an increase in the surface grazing angle. This refraction effect can be made explicit by writing Equation (8.13) in the form
3 v10 3=2 ^ loge jRU j  1:14F sin m ðv10 ; Þ; ð8:23Þ 10 where  is the grazing angle in bubble-free water; and the angle m is the grazing angle in bubbly water at the sea surface. If the refractive index at the sea surface is nðvÞ, it

366 Sources and scatterers of sound

[Ch. 8

Figure 8.1. Measured and predicted values of the quantity F 3=2 loge ð1=jRjÞ plotted vs. ð^v10 =10Þ 4 . Symbols are WC89 measurements; curves are theoretical predictions. The black and gray lines indicate calculations with and without bubbles, respectively. The difference between the solid and dashed lines is explained in the text (reprinted with permission from Ainslie, 2005b, # American Institute of Physics).

follows from Snell’s law that sin 2 m ðv; Þ ¼ 1 

cos 2  : n 2 ðvÞ

ð8:24Þ

A simple relationship between the refractive index n and the gas fraction U, valid for small bubbles that are well below resonance, is obtained using Wood’s equation from Chapter 5. Neglecting the density of air gives n 2  ð1  UÞ½1 þ ðBw =Ba  1ÞU:

ð8:25Þ

Assuming further that the void fraction is small (U  1) gives n 2 ðvÞ  1 þ

Bw UðvÞ; Ba

ð8:26Þ

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 367

where Bw and Ba are, respectively, the bulk moduli of water Bw ¼ w c 2w ;

ð8:27Þ

Ba ¼ P;

ð8:28Þ

and air where P is static pressure. The isothermal bulk modulus of air is used here for the bubble in preference to the adiabatic modulus (Ainslie, 2005b). For the Hall–Novarini (HN) bubble model (see Chapter 4), the gas fraction is given by
3 v 7 ^ : ð8:29Þ UðvÞ ¼ 9:29 10 10 The black lines in Figure 8.1 are obtained by evaluating Equation (8.23) using the HN gas fraction. The gray lines are calculated from Equation (8.13), amounting to an assumption that the surface void fraction is zero. The difference between the solid and dashed lines is in the assumed wave height spectrum. In each case the solid line is for the PM spectrum, and the dashed one is for the alternative NP spectrum. It is clear from the graphs that the difference between these two spectra is considerably less than the effect of the bubbles. The proposed low-frequency (LF) surface loss algorithm is the one resulting in the solid black line ð8:30Þ SLLF ¼ 10 log10 jRU j 2 ; that is, Equation (8.23) evaluated with the HN bubble model. An example is shown in Figure 8.2. The reader’s attention is drawn to the uncertainty in the near-surface bubble population, and hence also in the void fraction, for any given wind speed v (see Chapter 4). A seasonal dependence between U and v might be responsible for the variation observed in WC89 (Ainslie, 2005b). The fourth-power dependence on wind speed means that apparently minor uncertainties either in the wind speed iself, or in the height at which it is measured, are amplified into potentially significant uncertainties in surface loss. 8.1.1.2.2

High-frequency surface loss model

At high frequency (HF) the effects of refraction become less important relative to those of absorption. An empirical model of sea surface reflection loss due to absorption by near-surface wind-generated bubbles, incorporating effects due to bubble absorption, and applicable in the frequency range 10 kHz to 100 kHz is given by APL-UW (1994). A refined version of this model (Dahl, 2004) can be written 10 log10 jRHF j 2 ¼

20 log10 e SL ð^v10 ; FÞ: sin 

ð8:31Þ

where SL ð^v10 ; FÞ ¼ 10 6:45þ0:47^v10 F 0:85 :

ð8:32Þ

Figure 8.3 shows the result of evaluating Equation (8.31) at a frequency of 20 kHz, compared with measurements from Dahl et al. (2004). The rapid rise around 8 m/s to

368 Sources and scatterers of sound

[Ch. 8

Figure 8.2. Theoretical surface reflection loss in nepers calculated vs. angle and frequency using Equation (8.23) for a wind speed v10 ¼ 10 m/s (reprinted with permission from Ainslie, 2005b, # American Institute of Physics).

10 m/s at this frequency is explained by the onset of absorption effects due to windgenerated bubbles. While the predicted absorption continues to increase beyond 10 m/s, the measured losses appear to level off, possibly because some sound is reflected by the bubble cloud before being absorbed by it, as described by Dahl et al. (2004). The model of APL-UW (1994) imposes an upper limit of 15 dB to describe this effect. Dahl et al. (2008) describe an improved method for estimating bubble absorption loss for a given wind speed and frequency. Unlike the low-frequency formula, Equation (8.31) does not include a contribution from the loss of coherence due to rough surface scattering. The justification for this is a subtle but important qualitative difference between low-frequency and high-frequency propagation. The former typically takes place over long distances, allowing multiple interactions with the sea surface. In this situation only the coherent part of the field is expected to make a significant contribution at the receiver, because of the leakage that occurs out of the waveguide after multiple scattering.

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 369

Figure 8.3. Predicted surface loss (SL) vs. wind speed for a frequency of 30 kHz and grazing angle 10 deg, using Equation (8.31) (solid line) and measured SL at the same frequency (symbols).

By contrast, at high frequency the propagation distances are small, and the implicit assumption here is that only the first reflection is of interest. After a single reflection, there is no energy loss mechanism (other than absorption) because all of the incident energy is reflected by the rough boundary; hence the absence of a contribution to SL from coherence loss. However, the coherence of the signal is degraded due to rough surface scattering. Depending on the signal processing used for its detection this might affect the performance of the sonar.

8.1.2 8.1.2.1

Scattering from the sea surface Theoretical prediction for Pierson–Moskowitz surface wave spectrum

8.1.2.1.1 Non-specular scattering (perturbation theory) The general perturbation theory (PT) result for the scattering coefficient from Chapter 5, assuming an isotropic roughness spectrum, is 4 2 2  PT AO ð0 ; ; Þ ¼ 4k sin 0 sin  G1 ðÞ;

ð8:33Þ

where G1 ðÞ is the one-dimensional roughness spectrum. Substituting for the

370 Sources and scatterers of sound

[Ch. 8

Pierson–Moskowitz (PM) spectrum using Equation (8.8) gives !
 CPM k 4 2  2PM PT 2  AO ð0 ; ; Þ ¼ sin  sin 0 exp  2 ;    where the constant PM is given by PM ¼

pffiffiffiffiffiffiffiffiffi BPM g ; v 220

ð8:34Þ

ð8:35Þ

a parameter that is closely related to the reciprocal of the correlation length. The backscattering coefficient is obtained by equating  and 0 in Equation (8.34), together with  ¼ 2k cos 0 : ð8:36Þ The result is ! CPM  2PM PT 4  AO ðÞ ¼ tan  exp  2 : ð8:37Þ 16 4k cos 2  At sufficiently high frequency, the exponent in Equation (8.37) may be neglected, except for angles close to normal incidence. 8.1.2.1.2

Near-specular scattering (facet-scattering theory)

The facet-scattering formula for the near-specular scattering coefficient from Chapter 5 is
 Rð0 Þ 2 DO 2 AO ð0 ; ; Þ ¼ ð1 þ DOÞ exp  ; ð8:38Þ 8 2 2 2 where cos 2 0 þ cos 2   2 cos 0 cos  cos DO ¼ : ð8:39Þ ðsin 0 þ sin Þ 2 For in-plane scattering, Equation (8.38) simplifies to
 Rð0 Þ 2 1 1 AO ð0 ; ; Þ ¼ exp  ; 8 2 sin 4 2 2 tan 2 where is the bisector angle

(

¼

1 2 ð0 1 2 ð0

þ Þ þ   Þ

¼ :

¼0

ð8:40Þ

ð8:41Þ

The backscattering coefficient is
 RðÞ 2 1 1 AO ðÞ ¼ exp  2 : 8 2 sin 4  2 tan 2 

ð8:42Þ

The parameter  2 is the mean square slope of surface roughness, which can be estimated from wind speed using the Cox–Munk relationship from Chapter 4:  2 ¼ ð3 þ 5:12^v10 Þ 10 3 :

ð8:43Þ

Sec. 8.1]

8.1.2.2

8.1 Reflection and scattering from ocean boundaries 371

Semi-empirical surface-scattering strength models

The surface-scattering strength (SSS) is the scattering coefficient, expressed in decibels. In other words, it is defined as SSSð0 ; ; Þ
10 log10 AO ð0 ; ; Þ

dB:

ð8:44Þ

Similarly, the surface backscattering strength (SBS) is the backscattering coefficient, also in decibels SBSðÞ
10 log10 AO ðÞ

dB:

ð8:45Þ

As in Chapter 2, the use here of a single argument () implies evaluation in the backscattering direction. The scattering coefficient AO is dimensionless, so surface-scattering strength does not require a reference unit. Some sea surface scattering is caused by rough interface scattering at the air–sea boundary as described above. There is in general a further contribution due to volume scattering from wind-generated bubbles close to the boundary, the importance of which increases with increasing wind speed (Ogden and Erskine, 1994). Despite the fact that the bubbles are distributed in three dimensions, their association with the sea surface is so strong that it is difficult to separate the effects of the bubbles from those of the rough surface. Consequently, these two effects are usually lumped together in a single surface-scattering term. The semi-empirical models described below implicitly or explicitly include contributions from both effects. The lowfrequency model is valid up to 1 kHz and the high-frequency one from 12 kHz upwards. A recently developed model that bridges the gap between 1 kHz and 12 kHz is described by Gauss et al. (2006).

8.1.2.2.1

Low-frequency model

The semi-empirical model proposed by Ogden and Erskine (1994), valid in the frequency range 50 Hz to 1000 Hz and wind speed 0 m/s to 20 m/s, is described in this section. The model is constructed around two frequency-dependent wind speed thresholds UPT and UCH , given by ( UPT ¼

240 < f^ < 1,000 21:5  0:0595f^ 50 < f^ < 240 7:22

ð8:46Þ

and UCH ¼ 20:14  0:0340f^ þ 3:64 10 5 f^2  1:330 10 8 f^3 :

ð8:47Þ

For a low wind speed (less than the threshold UPT ), Equation (8.34) may be used, so that SBS can be written SBSPT
10 log10  PT AO ;

ð8:48Þ

372 Sources and scatterers of sound

[Ch. 8

1 with  PT AO from Equation (8.37). For high wind speed (exceeding UCH ) the empirical formula of Chapman and Harris (1962) is applied:
 deg SBSCH
3:3 CH log10  42:4 log10 CH þ 2:6; ð8:49Þ 30

where deg is the grazing angle  expressed in degrees deg
2

and

180  


0:58 3600 ^v10 f^1=3 CH ¼ 158 : 1852

ð8:50Þ ð8:51Þ

At intermediate wind speeds, higher than UPT and lower than UCH , the Ogden– Erskine model uses linear interpolation between SBSPT and SBSCH . Thus, the final formula for low-frequency SBS is 8 U  UPT < SBSPT SBSLF ¼ xSBSCH þ ð1  xÞSBSPT UPT < U < UCH ð8:52Þ : SBSCH U  UCH , where U  UPT x¼ : ð8:53Þ UCH  UPT As a practical matter, except in the expression for PM (Equation 8.35) the intended wind speed measurement height is 10 m, with a lower limit imposed of 2.5 m/s. Thus, U ¼ maxð^v10 ; 2:5Þ:

ð8:54Þ

The conversion to a measurement height of 20 m required for PM can be made using Equation (8.12). Because the Ogden–Erskine model is based on scattering measurements at grazing angles in the range 5 deg to 40 deg, this is the angle range in which the model may be used with confidence. Any extrapolation outside this range should take into account: — the need for a facet-scattering term close to normal incidence; — the uncertainty in the angle dependence of the scattering coefficient close to grazing incidence, especially for wind speeds exceeding UPT ; — the possible contribution from fish close to the sea surface. 8.1.2.2.2 High-frequency model (APL) For frequencies between 12 kHz and 70 kHz, the semi-empirical model proposed by APL-UW (1994) may be used, for grazing angles 0.5 deg to 90 deg, with linear 1

A lower limit of 1 deg is placed on the grazing angle. The factor 3600/1852 arises from the conversion between knots and meters per second (see Appendix B). 2

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 373

extrapolation recommended for angles less than 0.5 deg. The reported accuracy of the APL model is 4 dB for wind speeds greater than 8 m/s, and 5 dB for lower wind speeds. The APL model explicitly considers separate contributions to the scattering coefficient due to rough surface scattering (rough ) and volume scattering due to bubbles (bubble ) SBSHF ¼ 10 log10 AO ;

ð8:55Þ

AO ¼ rough þ bubble :

ð8:56Þ

where

Expressions for these two contributions are given separately below. Contribution from bubbles. The contribution to the scattering coefficient due to bubbles is (McDaniel, 1993; APL-UW, 1994, p. II-6; Dahl et al., 1997)3 bubble ¼

D0 sin  ½1  8jRj 2 loge jRj  jRj 4 ; 8res

ð8:57Þ

where the grazing angle  is evaluated in bubble-free water; D0 is the radiation damping coefficient at resonance introduced in Chapter 5 D0  0:0137

ð8:58Þ

and res is the total damping coefficient at resonance, which varies with frequency F in kilohertz according to res  2:55 10 2 F 1=3 :

ð8:59Þ

The parameter jRj is the surface reflection coefficient associated with absorption by wind-generated bubbles, such that reflection loss is proportional to pathlength, which in turn is proportional to the reciprocal of sin  loge jRj ¼

APL ð^v10 ; FÞ : sin 

ð8:60Þ

The constant of proportionality APL depends on wind speed in the following manner (APL-UW, 1994; Dahl et al., 1997; Dahl, 2003) 8
0:85 > ð5:2577þ0:4701^v10 F > > ^v10  11 < 10 25 APL ð^v10 ; FÞ ¼ ð8:61Þ
3:5 > ^ v > 10 > APL ð11; FÞ ^v10 > 11. : 11 3

Equation (8.57) is valid for large values of the Rayleigh parameter (Dahl, 2003).

374 Sources and scatterers of sound

[Ch. 8

The resulting contribution to the scattering coefficient is independent of angle if absorption is low and independent of wind speed if absorption is high 8 3D0 > > < 4 APL loge jRj  1 res bubble ¼ ð8:62Þ > D0 > : sin  loge jRj  1. 8res

Contribution from rough surface. The APL roughness-scattering model includes contributions from roughness on two different lengthscales. One contribution, denoted facet for facet scattering, is due to scattering from large-scale features at angles close to the specular direction, which in the backscattering case corresponds to normal incidence. The other contribution (ripple ) is from small-scale surface ripples. Total roughness scattering is calculated using the following non-linear interpolation algorithm rough ¼

facet þ e x ripple SLHF =10 10 ; 1 þ ex

ð8:63Þ

where the interpolation parameter is x ¼ 0:524ðfacet  deg Þ;

ð8:64Þ

and the angle facet , explained in more detail after Equation (8.66), defines the transition between ripple and facet scattering. The surface loss term SLHF is the high-frequency reflection loss associated with wind-generated bubbles (see Equation 8.31). The facet-scattering term is
 1 1 exp  2 ; ð8:65Þ facet ¼ 8 2 sin 4  2 tan 2  where  is the RMS roughness slope, given by ( 2:3 10 3 loge ð2:1^v 210 Þ ^v10  1 2  ¼ ^v10 < 1. 0:0017

ð8:66Þ

The angle facet , measured in degrees, is the angle at which 10 log10 facet falls to a level that is 15 dB below its maximum value. More precisely, it is the smallest angle that satisfies the inequality 10 log10 ½facet ð90 Þ=facet ðÞ  15 dB (APL-UW, 1994, p. II-9). Finally, the near-grazing (ripple) term is given by  1:3 10 5 ^v 210 tan 4    85 ripple ¼ ð8:67Þ 0  > 85 .

Sec. 8.1]

8.1.3

8.1 Reflection and scattering from ocean boundaries 375

Reflection from the seabed

It is conventional to quantify the loss of acoustic energy associated with reflection from the seabed using the logarithmic term bottom reflection loss, denoted BL and defined in the same way as for surface loss: BL
10 log10 jRj 2 ;

ð8:68Þ

where R is the plane wave amplitude reflection coefficient of the seabed. The main loss mechanism for sound incident on the seabed is through the transmission of sound into the sediment. If there is a second reflecting layer close to the water–seabed boundary, or if the sediment properties vary continuously with depth, then some of the transmitted energy will subsequently be reflected or scattered. Otherwise, it continues on its downward path and no longer contributes to propagation in the ocean waveguide. If there is no second reflection, the reflection loss depends only on the properties of the water–sediment boundary, and this situation is considered in Section 8.1.3.1. The effects of fine-scale layering are described for the case of an unconsolidated sediment in Section 8.1.3.2 and for a hard solid seabed in Section 8.1.3.3, including layering on a depthscale of order 10 m to 100 m. In the former the emphasis is on high frequency and in the latter on low frequency, although there is no clear-cut distinction between the two. 8.1.3.1 8.1.3.1.1

Theoretical prediction for uniform unconsolidated sediment Fluid sediment

Although in reality the seabed is never perfectly uniform, it may be approximated as such when there is a single dominant reflecting boundary at the water–sediment interface. One is dealing with sediment properties that are averaged in depth, and this averaging must be over a depthscale that is appropriate to the acoustic frequency of interest—usually a few wavelengths. In Chapter 4, two distinct sets of sediment properties are described: near-surface properties, representative of the top few centimeters of sediment and intended for use at high frequency (ca. 10–100 kHz), and bulk properties for use at lower frequency (ca. 1–10 kHz). Bottom reflection loss calculated using bulk properties is shown in Figure 8.4 (blue lines, upper graph), and using near-surface properties (lower graph). These calculations assume a semi-infinite uniform fluid model for the sediment with a perfectly smooth boundary, for which the Rayleigh reflection coefficient is applicable (see Chapter 5) RðÞ ¼

ðÞ  1 ; ðÞ þ 1

ð8:69Þ

tan  ; tan sed

ð8:70Þ

where ðÞ ¼ w

376 Sources and scatterers of sound

[Ch. 8

Figure 8.4. Predicted seabed reflection loss vs. grazing angle for uniform unconsolidated sediments. Upper: MF parameters (for grain size 0.5 to þ10 ); lower: HF parameters (0.5 to þ8:5 ). Blue lines: fluid sediment; dotted red lines: solid sediment with the same compressional speed as the fluid case, and shear speed evaluated at a depth of 10 m (see Chapter 4).

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 377

w is the density ratio w


sed ; w

ð8:71Þ

and  and sed are the ray grazing angles in water and sediment, respectively. If the fractional imaginary part of the sediment wavenumber is denoted ", such that ! ksed ¼ ð1 þ i"Þ; ð8:72Þ csed and v is the sound speed ratio v


csed ; cw

ð8:73Þ

it follows using Snell’s law in the form ksed cos sed ¼

! cos ; cw

ð8:74Þ

that sed is a complex angle given by


 cos  sed
arccos v : 1 þ i"

ð8:75Þ

The parameter " is related to the sediment attenuation coefficient sed (in decibels per wavelength) according to loge 10 "¼ : ð8:76Þ 40 sed The character of the Figure 8.4 reflection loss curves depends on grain size. The coarser (sandy) sediments exhibit a critical angle, denoted c , below which total internal reflection occurs, meaning that for  < c the magnitude of the reflection coefficient is close to unity. The critical angle is given by c

¼ arccosð1=vÞ

ðv > 1Þ:

ð8:77Þ

For example, the critical angle of coarse silt (Mz ¼ 4.5), using the MF parameters from Chapter 4, is about 22 deg. For grazing angles  < c , there is no transmitted wave and, in the absence of sediment attenuation, the angle sed is then imaginary. In this situation the reflection loss is zero and the parameter tan sed in Equation (8.70) is interpreted as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan sed ¼ i 1  k 2sed =ðk 2w cos 2 Þ: ð8:78Þ If some attenuation is present (i.e., if sed > 0), the reflection loss at sub-critical grazing angles, though small, increases slightly with increasing angle  in proportion with sin  such that jRj 2  expð2 sin Þ ð < c Þ; ð8:79Þ or equivalently BL  ð10 log10 eÞ2 sin : ð8:80Þ

378 Sources and scatterers of sound

[Ch. 8

The parameter is a function of seabed density, sound speed, and attenuation coefficient. Even if there is no critical angle, it is still possible to write reflection loss in the form of Equation (8.80) for small angles, although the value of is much higher for clay than it is for sand. A general result that holds regardless of seabed impedance is Re Zsed

¼2 ; ð8:81Þ w c w where  c 1þR  c Zsed
w w ¼ w w ; ð8:82Þ sin w 1  R sin w and  is the impedance ratio (defined as Zsed sin w =ðw cw Þ and given for a fluid sediment by Equation 8.70). The right-hand side of Equation (8.81) is understood to be evaluated at grazing incidence. Applying Equation (8.81) to the present example gives ~v

0 ¼ 2w Re ; ð8:83Þ ð1  ~v 2 Þ 1=2 where ~v is the complex equivalent sound speed, defined as v ~v
; ð8:84Þ 1 þ i" and the zero subscript in Equation (8.83) indicates that this expression for applies to the case of a fluid sediment (i.e., with a zero or negligible shear speed, irrespective of the sound speed ratio v). Coarse-grained sediments. If v > 1, a characteristic of coarse sediments, 0 is proportional to the imaginary part of the complex wavenumber (Weston, 1971) v

0 ¼ 2w" 2 ðv > 1Þ; ð8:85Þ ðv  1Þ 3=2 or equivalently, in terms of the critical angle 2

0 ¼ 2w"

cos sin 3

c

c

(see Equation 8.77)

ðcos

c

< 1Þ:

ð8:86Þ

c

Fine-grained sediments. For fine sediments (satisfying v < 1) there is no critical angle and Equation (8.83) simplifies to v

0  2w : ð8:87Þ ð1  v 2 Þ 1=2 If w > 1=v is also satisfied, there exists an angle of intromission at which the reflection coefficient passes through zero. This angle, denoted in , is given by sin 2 ðin Þ0 ¼

1  v2 : v 2 ðw 2  1Þ

ð8:88Þ

As before, the zero subscript indicates that this equation is for a fluid sediment.

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 379

8.1.3.1.2 Effect of a small non-zero shear speed If a plane wave is incident on a fluid seabed at a grazing angle steeper than the critical angle c , a compressional wave is excited in the sediment. In this situation, most of the incident energy is converted into this transmitted p-wave, carrying the energy away from the water column—hence the high reflection loss. For the case of a solid seabed, there exists additionally the possibility of exciting a shear wave, which propagates at a grazing angle s . If the shear speed in the sediment is less than the sound speed in water, a shear wave is always generated. It is convenient to define the complex sound speed ratios ~vp and ~vs using vX ~vX
; ð8:89Þ 1 þ i"X where the X subscript denotes a property of either a compressional wave (if X ¼ p) or a shear wave (X ¼ s), c vX
X ; ð8:90Þ cw and loge 10 "X ¼ : ð8:91Þ 40 X The corresponding impedance ratios can be defined as X ðÞ ¼ w

tan  ; tan X

ð8:92Þ

where X
arccosð~vX cos Þ:

ð8:93Þ

With these definitions, Equation (8.69) for the reflection coefficient still holds, provided that the impedance ratio  is generalized to  ¼ p ðÞ cos 2 2s þ s ðÞ sin 2 2s :

ð8:94Þ

The main effects on and in of a small but non-zero (relative) shear speed vs , illustrated by the red lines in Figure 8.4, are described below. Effect on reflection loss. An unconsolidated sediment is weakly capable of propagating shear waves because it has a slightly non-zero rigidity. For a nonzero sediment shear speed, Equation (8.83) becomes (Ainslie, 1992) " # ~vp 2 2 3 2 1=2

¼ 2w Re ð1  2~v s Þ þ 4~v s ð1  ~v s Þ : ð8:95Þ ð1  ~v 2p Þ 1=2 If ~vs has a negligible imaginary part, it follows that (Eller and Gershfeld, 1985)4

¼ 0 ð1  2v 2s Þ 2 þ 8wv 3s ð1  v 2s Þ 1=2

ðvp > 1 > vs Þ:

ð8:96Þ

4 A similar expression is derived by Chapman (1999), including an additional term associated with the imaginary part of ~vs .

380 Sources and scatterers of sound

[Ch. 8

For constant vs , the first term is proportional to 0 , the value for a fluid, which is given by Equation (8.83). The lowest order effect of vs is to reduce the loss for the fluid case by a factor of approximately (1  4v 2s ). This reduction is clearly visible in Figure 8.4, especially for high-frequency (APL) parameters. The second term represents a loss mechanism that does not exist for the fluid case, namely the conversion of energy into sediment shear waves. This term is usually negligible for unconsolidated sediments, but it becomes important for harder consolidated sediments like chalk or mudstone (see Section 8.1.3.3). Effect on angle of intromission. The effect of shear speed on the intromission angle can be calculated as follows. The condition for intromission is Re  ¼ 1 in Equation (8.69), with  given by Equation (8.94). This gives sin 2 w ¼

1  v 2p ½1  Dðw Þ 2 ; v 2p w 2  ½1  Dðw Þ 2

ð8:97Þ

where DðÞ
Re½ðÞ  p ðÞ;

ð8:98Þ

which can be written in the form (using the notation OðxÞ to indicate a term of order x) " ! # 2 1=2 w  1 D ¼ 4v 2s sinðin Þ0 cos 2 ðin Þ0 vp þ Oðvs Þ : ð8:99Þ 1  v 2p The intromission angle in is then the value of w that satisfies Equation (8.97). Given the assumption that vs is small, it follows that D must also be small. Equation (8.97) can then be written as a Maclaurin series in D " # 2w 2 2 2 2 sin w ¼ sin ðin Þ0 1  2 Dðw Þ þ OðD Þ ; ð8:100Þ w 1 where ðin Þ0 is the intromission angle for a fluid sediment given by Equation (8.88). Expanding Equation (8.98) in powers of vs , it can be shown that   w 2 2 2 2 3 sin in ¼ sin ðin Þ0 1 þ 8v s 2 cos ðin Þ0 þ Oðv s Þ : ð8:101Þ w 1 Thus, given that w > 1 must be satisfied for the existence of ðin Þ0 , the consequence of a small non-zero sediment shear speed is a small increase in the angle of intromission, as illustrated by Figure 8.4. 8.1.3.2

Theoretical prediction for layered unconsolidated sediment (1–100 kHz)

The somewhat arbitrary distinction made above between ‘‘high frequency’’ or ‘‘HF’’ (for which near-surface properties are used) and ‘‘medium frequency’’ or ‘‘MF’’ (bulk properties used) leads to an undesirable artifact in the predicted reflection coefficient, namely a step change in the predicted reflection loss at a frequency of 10 kHz. For some applications, the correct (continuous) dependence on frequency might be

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 381 Table 8.1. Sediment properties at top and bottom of the transition layer.

Sediment description

Mz

Sound speed ratio (c=cw ) at z ¼ 0 (at z ¼ h)

Density ratio ( =w ) at z ¼ 0 (at z ¼ h)

Attenuation coefficient (dB/) at z ¼ 0 (at z ¼ h)

Fine sand

2.5

1.1073 (1.1534)

1.451 (1.945)

0.85 (0.89)

Medium silt

5.5

0.9885 (1.049)

1.149 (1.601)

0.36 (0.38)

Coarse clay

8.5

0.9812 (0.9911)

1.145 (1.378)

0.08 (0.08)

required, in which case an improved model is necessary. This problem can be addressed by constructing a layered medium whose properties vary continuously with depth from their near-surface values at the top of the sediment to bulk properties at a depth equal to the thickness of the transition region between near-surface properties and bulk properties. The thickness of this transition layer is typically of order 1 cm to 10 cm. The layered model described below covers the approximate frequency range 1 kHz to 100 kHz. Above 100 kHz the same method is applicable except that layering on an even finer scale becomes relevant, with the near surface properties eventually becoming indistinguishable from those of water (Ainslie, 2005a). For lower frequencies (below 1 kHz), it is necessary to include large-scale features on a depthscale of tens or even hundreds of meters (Section 8.1.3.3). Three sediment types are considered, representing fine sand, medium silt, and coarse clay. The properties of these three sediments, taken from Chapter 4, are summarized in Table 8.1. Figure 8.5 shows reflection loss plotted vs. angle and frequency for each of the three cases. The precise sound speed profile used in the transition layer is the ‘‘linear k 2 ’’ case, for which the wavenumber profile kðzÞ is given by kðzÞ 2 ¼ kð0Þ 2 ð1  2qzÞ: The gradient parameter q is a complex constant given by   cð0Þ 2 1 þ i"ðhÞ 2 2qh ¼ 1  ; cðhÞ 2 1 þ i"ð0Þ where loge 10 "ðzÞ ¼ ðzÞ: 40

ð8:102Þ

ð8:103Þ

ð8:104Þ

This choice of q ensures that the correct values of sound speed and attenuation are obtained at the top and bottom of the transition layer, from Table 8.1 (at z ¼ 0 and h, respectively). The complete sound speed and attenuation profiles are ! cðzÞ ¼ ð8:105Þ Re kðzÞ

382 Sources and scatterers of sound

[Ch. 8

Figure 8.5. Predicted seabed reflection loss vs. angle and frequency–sediment thickness product for a layered unconsolidated sediment. Upper: fine sand Mz ¼ 2:5 (h ¼ 1.1092); lower: medium silt Mz ¼ 5:5 (h ¼ 1.1840); right: coarse clay Mz ¼ 8:5 (h ¼ 0.8741).

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 383

and ðzÞ ¼

40 Im kðzÞ : loge 10 Re kðzÞ

ð8:106Þ

The linear k 2 profile is adopted here because it provides a simple method for predicting the complicated frequency dependence illustrated by Figure 8.5. If the parameters were kept fixed at the values specified by Table 8.1, use of a more realistic sediment sound speed profile would alter the detailed behavior in the transition region, but would not influence either low-frequency or high-frequency limits. Robins’s density profile is taken from Chapter 5, with  chosen to ensure a zero density gradient at z ¼ h, that is,
 h  ¼ ð0Þ tanh ; ð8:107Þ 2 so that ðzÞ ¼ 

ð0Þ
 z z2 h cosh  tanh sinh 2 2 2

ð8:108Þ

and  
 z z dðzÞ ðzÞ 3=2 h ¼ ð0Þ cosh tanh  tanh : dz ð0Þ 2 2 2

ð8:109Þ

384 Sources and scatterers of sound

Continuity of the density at z ¼ h is ensured by choosing h as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0Þ h ¼ 2 artanh 1  : ðhÞ

[Ch. 8

ð8:110Þ

At depth z > h the bulk properties are used for all parameters. In the graphs of Figure 8.5, seabed reflection loss is calculated vs. angle and frequency using the method of Robins (1991). The reflection loss for this sediment model is a function of the product fh, where f is the acoustic frequency, and not of f and h separately. This symmetry is exploited here by choosing the y-axis parameter as y ¼ log10 ð f^h^Þ. The advantage of displaying the data in this way is that the graph can be applied to combinations of frequency and layer thickness over a wide range of parameter values. Below fh ¼ 100 m/s (i.e., for y < 2, corresponding to f < 2 kHz with a layer thickness h ¼ 5 cm), the plots tend to stabilize towards their low-frequency limits, as can be expected. Similarly, above 10 km/s ( y > 4, or f > 200 kHz with h ¼ 5 cm), they tend towards the high-frequency (APL) limit. The transition between these cases for the sand sediment is straightforward, showing a shift in the critical angle from 30 deg to 25 deg. For clay there is a similar shift, this time in the angle of intromission. For the silt case the behavior is more complex, especially close to grazing incidence, because the nature of the reflection process changes from one of total internal reflection at low frequency to almost total transmission at high frequency. 8.1.3.3

Theoretical prediction for layered solid seabed ( 1, there is a second critical angle, equal to arccosð1=vs Þ and denoted s . This is the critical angle for the generation of 5

See Fig. 2 from Hughes et al. (1990) for an example with a chalk sediment.

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 385

Figure 8.6. Seabed reflection loss vs. grazing angle for rocks.

sediment s-waves. At smaller angles neither p-waves nor s-waves can be generated and the result is total internal reflection. For the harder rocks the p and s critical angles are clearly identifiable in Figure 8.6 due to the abrupt change in reflection loss through these angles. For example, the basalt s and p critical angles are 51 deg and 72 deg, respectively. The critical angles for different rock types are summarized in Table 8.2. Now consider a thin layer of unconsolidated sediment sandwiched between the rock half-space below and the sea above. Two specific cases of interest are a layer of

Table 8.2. p and s critical angles for representative rock parameters (see Chapter 4). cp =m s1

cs =m s1

Mudstone

2050

600

43

—

Chalk

2400

1000

52

—

Sandstone

4350

2550

70

54

Basalt

4750

2350

72

51

Granite

5750

3000

75

60

Limestone

5350

2400

74

52

Type of rock

p =deg

s =deg

386 Sources and scatterers of sound

[Ch. 8

sand over granite and a clay layer over basalt. These two combinations are representative of typical shallow-water and deep-water seabeds, respectively. Figure 8.7 shows BL vs. angle and frequency for both cases, with the sediment layer treated as a uniform fluid. The properties of sediment and rock layers are summarized in Table 8.3. For this situation there can be up to three different critical angles: the usual critical angle for the fluid sediment ( 2 ), plus p and s , the two critical angles for rock described above. The sediment has no effect in the low-frequency limit (kh  1) because it becomes transparent to sound. In this situation, reflection loss is expected to be that for a rock half-space alone, and this is confirmed by Figure 8.7 for frequencies satisfying fh < 100 m/s. The p and s critical angles are clearly visible for both granite (upper graph) and basalt (lower graph), at the edges of the two vertical stripes. The thickness h is that of the whole sediment layer, which in this case is the medium sandwiched between seawater and rock. At high frequency the opposite situation can arise. If the frequency is high enough, none of the sound reaches the rock, and reflection loss is that of an infinite sediment half-space. For the sand case the critical angle of 30 deg is clearly visible for fh exceeding 10 km/s. For clay the asymptotic limit is not quite reached, even at 100 km/s, but the angle of intromission is nevertheless apparent. At intermediate frequencies, the reflected field includes contributions from both boundaries. The fringes visible on both graphs result from interference between the resulting multipaths. Of particular importance to the performance of sonar is the behavior at small angles, because it is this that determines the viability of long-range propagation. The clay sediment exhibits a sequence of interference nulls close to grazing incidence starting at fh  3 km/s. This behavior, described by Hastrup (1980), is characteristic of a low sound speed sediment (satisfying c2 < cw ). Hastrup’s condition for determining the frequencies at which destructive interference occurs can be generalized by first writing the reflection coefficient in the form (from Chapter 5) R¼

R12 þ R23 expð2i2 hÞ ; 1  R21 R23 expð2i2 hÞ

ð8:111Þ

where 2 is the vertical wavenumber in the sediment layer. Requiring that the numerator of Equation (8.111) be zero, and approximating the two-layer coefficients using i j  1 Ri j ¼  expð2i j Þ; ð8:112Þ i j þ 1 the condition becomes expð212 Þ þ expð2i2 h  223 Þ ¼ 0:

ð8:113Þ

Close to grazing incidence, given the assumption of small c2 , if sediment attenuation is negligible, then 12 is a real number. Assuming further that cp ; cs are both larger than cw , and neglecting rock attenuations as well, it follows that 23 is imaginary.

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 387

Figure 8.7. Predicted seabed reflection loss vs. grazing angle and frequency–sediment thickness product for a sand sediment overlying a granite basement (upper graph) and clay over basalt (lower). The sediment layer is treated as a uniform fluid medium.

388 Sources and scatterers of sound

[Ch. 8

Table 8.3. Defining parameters for the two cases involving a uniform fluid sediment and a hard rock half-space. The sediment parameters are evaluated using the Bachman correlations of Chapter 4. Rock wave speeds are from Table 8.2. Medium

fh=(m s1 ) =(kg m 3 ) cp =(m s1 ) p =(dB 1 ) cs =(m s1 ) 1

s =(dB 1 )

1027

1490

0

0

0

Sediment Fine sand (Mz ¼ 2:5) 30 to 10 5 Coarse clay (Mz ¼ 8.5) 30 to 10 5

1997 1415

1717 1478

0.89 0.08

0 0

0 0

Rock Granite Basalt

2650 2550

5750 4750

0.10 0.10

3,000 2,350

0.20 0.20

Water

1 1

Thus, the only way that the left-hand side of Equation (8.113) can vanish is for the phase of the second term to be an odd integer. In other words 22 h  2 Im 0 ¼ 2n þ 1;

ð8:114Þ

where n is a non-negative integer; and 0 is the impedance ratio at the sediment–rock boundary (23 ) evaluated at grazing incidence in water (w ¼ 0) ! "
# 1=2 c 2w 1 sin 2 s 1 1 0 ¼ iw 2  1 4  : ð8:115Þ sin p c2 cos 4 s sin s sin p It is understood that 2 is also evaluated at grazing incidence. Rearranging for the frequency, Equation (8.114) can be written ! 1=2 fh 1 c 2w ¼ 1 ½ð2n þ 1Þ þ 2 Imð0 Þ ðcw > c2 Þ: 2 cw 4 c 2

ð8:116Þ

For the sand case with a granite half-space, there is a single interference null close to grazing incidence, at a frequency–thickness product close to 200 m/s. This feature is characteristic of a seabed comprising a thin layer of sand over a hard rock basement. It is caused by a resonant evanescent wave in the sediment, which occurs at a frequency determined by (Ainslie, 2003) fh 1 ¼ cw 4 sin

2

loge

0  1 : 0 þ 1

ð8:117Þ

The impedance ratio 0 is defined in the same way as above. For the sand case, with c2 > cw , it is more convenient to write Equation (8.115) in the form "
# 1 sin 2 s 1 1 0 ¼ w sin 2 4  : ð8:118Þ sin p cos 4 s sin s sin p

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 389

Table 8.4. Defining parameters for a layered solid medium. The shear speed profile is given by c^s ðzÞ ¼ ½79 þ 41 expð0:4Mz Þ^ z 0:31 , with Mz equal to 2.5 or 8.5 (see Chapter 4). Medium

h=m

=(kg m 3 ) cp =(m s1 ) p =(dB 1 ) cs =(m s1 )

s =(dB 1 )

Water

1

1027

1490

0

0

0

Sediment Fine sand (Mz ¼ 2:5) Coarse clay (Mz ¼ 8.5)

10 300

1997 þ z^ 1415 þ z^

1717 þ z^ 1478 þ z^

0.89 0.08

c^s ðzÞ c^s ðzÞ

0.01^ cs ðzÞ 0.01^ cs ðzÞ

Rock Granite Basalt

1 1

2650 2550

5750 4750

0.10 0.10

3000 2350

0.20 0.20

For this evanescent resonance to exist at all, the critical angle (in water) for the excitation of shear waves in the rock layer must exceed about 17 deg (Ainslie, 2003). Figure 8.7 shows features that are characteristic of the interaction of sound with a fluid sediment layer overlying a solid half-space. Real sediments, even if unconsolidated, have a small but non-zero shear speed, which becomes important at low frequency. Furthermore, the sediment layer is never perfectly uniform but has a sound speed that tends to increase with increasing depth. The consequences of this gradient are particularly important in deep water, where sediment thickness can be measured in hundreds or even thousands of meters. The effects of non-zero compressional speed gradient and non-zero shear speed are considered next, by modifying the parameters of Table 8.3 as shown in Table 8.4. Uniform sound speed and density gradients are applied in the sediment layer of 1 s1 and 1 kg m 4 , respectively. Shear speed also varies with depth. The value for shear attenuation corresponds to 10 dB/ (m kHz), representative of typical values from Chapter 4. The case with a sand sediment and granite substrate represents a typical shallowwater seabed and is assigned a sediment thickness of 10 m. In deep water, represented by the clay–basalt combination, sediment thickness is usually much greater than this, and a value of 300 m is used. Reflection loss vs. angle and frequency for the sand–granite substrate, evaluated using the method of Chapman (2004) (see Chapter 5), is shown in the upper graph of Figure 8.8. The horizontal stripes between 5 Hz and 50 Hz are associated with a sequence of shear wave resonances in the sediment layer (Hughes et al., 1990; Ainslie, 1995; Tollefsen, 1998). Similar features are also visible for clay (lower graph), although they appear at a lower frequency (0.3–3 Hz) due to the greater sediment thickness. Compared with Figure 8.7, the Hastrup resonances are shifted up in frequency, to the extent that in Figure 8.8 only one remains visible, close to 200 Hz. The reason for this shift is a slightly different mechanism caused by the sediment sound speed gradient. The near-grazing sound is not reflected by rock, but refracted upwards by this gradient. The resulting resonance frequencies can be

390 Sources and scatterers of sound

[Ch. 8

Figure 8.8. Predicted seabed reflection loss vs. grazing angle and frequency for a sand sediment of thickness 10 m overlying a granite basement (upper graph) and a clay sediment of thickness 300 m over basalt (lower). The sediment is treated as a layered solid medium.

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 391

determined using the expression (Ainslie and Harrison, 1989)  3=2 vp f ¼ 32 ðn þ 14Þc 0 ; 2ð1  vp Þ

ð8:119Þ

giving a value of 194 Hz (for vp ¼ 0.9923 and n ¼ 0), independent of sediment thickness. The frequency ranges used for Figure 8.8 are chosen to correspond to the same range of fh values used in Figure 8.7 (30 m/s to 100 km/s), thus making it straightforward to compare these two figures visually. For both sand–granite and clay–basalt seabeds the main effects of non-zero sediment shear speed are the horizontal lines for grazing angles up to about 45 deg, apparent for fh < 1 km/s. The sound speed gradient is responsible for the low loss near grazing incidence for the clay–basalt case, around 10 Hz to 30 Hz in Figure 8.8. The effect of the density gradient is small.

8.1.4

Scattering from the seabed

The bottom scattering strength (BSS) and the bottom backscattering strength (BBS) are defined in an analogous way to their counterparts for surface scattering, that is, BSSð0 ; ; Þ
10 log10 AO ð0 ; ; Þ

dB

ð8:120Þ

and BBSðÞ
10 log10 AO ðÞ

dB:

ð8:121Þ

A theoretical calculation for the scattering coefficient from a rough seabed, including both boundary roughness and volume scattering, is presented in Section 8.1.4.1. However, there are many additional complications due to the possible presence of: — roughness on grossly different lengthscales, from millimeters to megameters (APL-UW, 1994); — buried shell fragments and gravel (Goff et al., 2004; Simons et al., 2007); — hard rough boundaries exposed or beneath a layer of sediment (Essen, 1994; Jackson and Ivakin, 1998; Gragg et al., 2001); — pockets of trapped gas (Boyle and Chotiros, 1995); — fine sediment in suspension close to the sea floor (e.g., after a storm); — demersal fish and other animals or plants resident in, on, or near the sea floor; — a sound speed gradient within the sediment layer (Mourad and Jackson, 1989); — a tidal or current shear flow. Any of these special effects can be modeled in principle (Jackson and Richardson, 2007), but knowing in advance which of them are actually needed in practice is a problem. See, for example, the discussion in Chapman et al. (1997). A semi-empirical approach based partly on measurements mitigates the risk of omitting an important effect. Examples of empirical or semi-empirical models are described in Section 8.1.4.2.

392 Sources and scatterers of sound

8.1.4.1

[Ch. 8

Theoretical prediction for a fluid seabed with a rough boundary and a uniform distribution of embedded scatterers

In Section 8.1.4.1.1 the theoretical scattering coefficient due to seabed boundary roughness is described. In practice, there can also be important contributions to the scattering from volume inhomogeneities in the sediment, as described in Section 8.1.4.1.2. Near the specular direction the contribution from facet scattering is important (Section 8.1.4.1.3). 8.1.4.1.1 Non-specular scattering from rough boundary (perturbation theory) The scattering coefficient for the rough boundary, with roughness spectrum WðÞ between water and a (fluid) seabed, can be written (Kuo, 1964) AO ðÞ ¼ 4k 4 sin 4 jYðÞj 2 Wð2k cos Þ;

ð8:122Þ

where YðÞ is related to the reflection coefficient as follows YðÞ ¼ RðÞ þ

2wðw  1Þ ; ðw tan  þ tan sed Þ 2

ð8:123Þ

and sed is the sediment grazing angle defined by Equation (8.75). Close to normal incidence, the correction is negligible and YðÞ may be approximated by RðÞ. Near grazing incidence, YðÞ tends to YðÞ ! RðÞ 

2wðw  1Þ ; sin 2 c

ð8:124Þ

where c is the sediment critical angle. Seabed roughness can be parameterized by means of a power law spectrum of the form (Sternlicht and de Moustier, 2003)
 100  WðÞ ¼ W0 ; ð8:125Þ ^  where the constant W0 , the roughness spectral density corresponding to a wavenumber of 1 cm1 , is correlated with grain size according to 8
2 > < 20:7 mm 4 2:03846  0:26923Mz 1  Mz < 5:0 1:0 þ 0:076923Mz W0 ¼ ð8:126Þ > : 4 5:175 mm 5:0  Mz < 9:0 and the exponent  is typically between 2 and 4 (APL-UW, 1994). A nominal value of 3.25 is suggested by Sternlicht and de Moustier (2003). Figure 8.9 shows the theoretical BBS associated with rough boundary scattering as a function of grazing angle for a sand sediment, and for various frequencies between 1 kHz and 30 kHz, as marked. The frequency dependence arises from the k 4 factor in Equation (8.122) and the   term in Equation (8.125). Combined, these give a frequency dependence of k 4 . Comparison with measurements from Jackson and Briggs (1992) at a frequency of 35 kHz is shown in Figure 8.10. At low frequency it is expected that

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 393

Figure 8.9. Predicted seabed backscattering strength for a medium sand sediment (Mz ¼ 1.5) and frequency 1 to 30 kHz, evaluated using Equation (8.122) (with Equation 8.125). The calculation is for roughness scattering only. Lambert’s rule (Equation 8.136) with a Lambert parameter of 25 dB is included for comparison.

scattering from sediment volume inhomogeneities, addressed in Section 8.1.4.1.2, would make a significant contribution to the total.

8.1.4.1.2

Scattering from sediment volume

Spatial fluctuations in both density and sound speed can occur within the sediment volume. Even if the water–sediment boundary were perfectly smooth, such fluctuations, called volume inhomogeneities, would scatter some of the incident sound. Such scattering is often treated as part of the bottom scattering coefficient, as if it had originated from a rough surface at the water–sediment boundary. This is because the two mechanisms are difficult to differentiate, similar to the situation with bubbles close to the air–sea boundary. If the differential scattering cross-section per unit volume of the sediment is denoted VO , the contribution from sediment volume scattering to the scattering coefficient can be written (Stockhausen, 1963; Jackson and Briggs, 1992) AO ðÞ ¼

1 " VO sin 2 j1  RðÞ 2 j 2 ; 4v Imðtan sed Þ sed cos 3 jtan sed j 2

ð8:127Þ

394 Sources and scatterers of sound

[Ch. 8

Figure 8.10. Comparison between predicted and measured seabed backscattering strength for a fine sand sediment (Mz ¼ 3.0) and frequency 35 kHz. Dashed line: roughness scattering only; solid line: roughness scattering þ volume scattering (reprinted with permission from Jackson and Briggs, 1992, # American Institute of Physics).

where " is the fractional imaginary part of the sediment wavenumber. Equation (8.127) is valid for all angles  between 0 deg and 90 deg, regardless of the value of the critical angle c . If  < c is satisfied—the condition for total internal reflection—it simplifies approximately to AO ðÞ 

and for  >

c

" VO cos c tan 2  4 sed cos 

j1  RðÞ 2 j 2 ! ; cos 2 c 3=2 1 cos 2 

ð8:128Þ

to AO ðÞ 

1 VO 2 sin 2  v j1  RðÞ 2 j 2 : 4 sed sin sed

ð8:129Þ

Kuo’s boundary roughness model can be combined with a volume scattering term to give (Mourad and Jackson, 1989; Jackson and Briggs, 1992) AO ðÞ ¼ rough ðÞ þ vol ðÞ;

ð8:130Þ

where rough and vol are the contributions from roughness and volume scattering as given by Equations (8.122) and (8.127), respectively. Further refinements to this model are described by APL-UW (1994). For the bistatic case, see also Williams and Jackson (1998). The ratio VO = sed in Equation (8.127) is a small dimensionless number of order 10 2 . APL-UW (1994) recommends the following values, depending on the grain

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 395

Figure 8.11. Predicted seabed backscattering strength for a coarse clay sediment (Mz ¼ 8.5), evaluated using Equation (8.127) (with Equation 8.131). The calculation is for volume scattering only; Lambert’s rule (Equation 8.136) with a Lambert parameter of 25 dB is included for comparison (reprinted with permission from Ainslie, 2007, # American Institute of Physics).

size VO 20 ¼ sed loge 10



0:002 1:0  Mz < 5:5 0:001

5:5  Mz  9:0

:

ð8:131Þ

Figure 8.11 shows the theoretical BBS due to volume scattering as a function of grazing angle for a clay sediment. The prediction is independent of frequency. Comparison with measurements from Jackson and Briggs (1992) at a frequency of 20 kHz is shown in Figure 8.12.

8.1.4.1.3 Near-specular scattering (facet-scattering theory) The near-specular scattering coefficient can be calculated in the same way as for the sea surface using Equation (8.38) (for the bistatic case) or Equation (8.37) (for backscatter). For the seabed the parameter  2 is the mean square slope of bottom roughness. Pouliquen and Lurton (1994) suggest values of roughness angle between 3 deg (for mud) and 11 deg (rock), where  ¼ tan .

8.1.4.2

Empirical and semi-empirical seabed scattering strength models

Because of the difficulties associated with theoretical predictions of bottom scattering, empirical and semi-empirical models play an important role in the

396 Sources and scatterers of sound

[Ch. 8

Figure 8.12. Comparison between predicted and measured seabed backscattering strength for a medium silt sediment (Mz ¼ 5.6) and frequency 20 kHz. Dashed line: roughness scattering only; solid line: roughness scattering þ volume scattering (reprinted with permission from Jackson and Briggs, 1992, # American Institute of Physics).

calculation of BBS for sonar performance prediction. A commonly used empirical scattering model, known as Lambert’s rule and described in Section 8.1.4.2.1, assumes arbitrarily that incoming sound is scattered equally in all possible directions. Ellis and Crowe (1991) combine Lambert’s rule with a facet-scattering term as described in Section 8.1.4.2.2. The empirical model presented in Section 8.1.4.2.3 is based on the measurements of McKinney and Anderson (1964). 8.1.4.2.1 Diffuse scattering model (empirical) Lambert’s rule If it is assumed that all incident energy is scattered diffusely (i.e., with equal radiant intensity in all directions), a simple relationship emerges of the form (Chapman et al., 1997) AO ð0 ; ; Þ ¼  sin 0 sin ;

ð8:132Þ

AO ðÞ ¼  sin 2 :

ð8:133Þ

and hence The corresponding scattering strengths are BSSð0 ; ; Þ ¼ 10 log10  þ 10 log10 ðsin 0 sin Þ

ð8:134Þ

BBSðÞ ¼ 10 log10  þ 10 log10 ðsin 2 Þ:

ð8:135Þ

and The sin  angle dependence originates from the increasing scattering area as 

Sec. 8.1]

8.1 Reflection and scattering from ocean boundaries 397 Table 8.5. Measurements of the Lambert parameter .

Seabed

10 log10  (dB)

Frequency (kHz)

Reference

Unconsolidated sediments (sand, silt, or clay)

20 30 28 35

15 8–40 22 95 16 100 15 Unspecified

Gensane (1989) Goff et al. (2004) Simons et al. (2007) Boyle and Chotiros (1995)

Gravel

19 8–40 18 to 16 95 9 to 7 100 10 to 3 Unspecified

Gensane (1989) Goff et al. (2004) Simons et al. (2007) Boyle and Chotiros (1995)

Rock

4 to 2 11 to 8

Urick (1954) McKinney and Anderson (1964)

to to to to

10–60 30–300

approaches zero. (The scattering coefficient is proportional to the ratio of scattered radiant intensity to scattering area). This relationship is often referred to as ‘‘Lambert’s law’’, but it is more appropriate to call it ‘‘Lambert’s rule’’ because, while it has a simple physical interpretation, it has no firm foundation in scattering physics. Measurements of seabed backscattering strength are often reported in terms of the coefficient , known as ‘‘Lambert’s parameter’’ or ‘‘Lambert’s constant’’. Little guidance can be given regarding a suitable choice for Lambert’s parameter. It is known to be higher for rock and gravel bottoms than for unconsolidated sediments (see Table 8.5), but the uncertainty is large. For example, Boyle and Chotiros (1995) find no clear relationship between grain size and scattering strength for unconsolidated sediments, except for the case of specially prepared laboratory sand. Lambert’s parameter for rock or gravel. Reported values of 10 log10  for rock or gravel vary between about 19 dB (Gensane, 1989) and 2 dB (Urick, 1954), both for frequencies in the range 10 kHz to 40 kHz. Thus, a typical value is 11 dB, with a large uncertainty of 8 dB. If Lambert’s rule is deemed to apply for all angles, the law of energy conservation requires that the value of  should not exceed 1= (Urick, 1983, p. 278). This means that if values larger than this (10 log10  > 5.0 dB) occur in a limited angle range, they must be offset by lower values at other angles, implying a departure from Lambert’s rule. Lambert’s parameter for unconsolidated sediments. An important factor in determining  for unconsolidated sediments, at least at high frequency, is the percentage of gravel or shell. For gravel-free (less than about 5 % gravel)

398 Sources and scatterers of sound

[Ch. 8

unconsolidated sediment, a nominal value for 10 log10  of 25 dB is suggested, with an uncertainty of about 10 dB. For medium sand (i.e., for sediment grain sizes in the range 250–500 mm), Greenlaw et al. (2004) show that the parameter 10 log10 sand increases approximately linearly with log(frequency) between 10 kHz and 400 kHz. A linear fit to their Fig. 6 in this frequency range gives
1:47 4 F sand ¼ 10 ; ð8:136Þ 5 or equivalently, in decibels 10 log10 sand ¼ 40 þ 14:7 log10 ðF=5Þ:

ð8:137Þ

A similar dependence of the scattering strength of sand on frequency (increase of 15 dB per decade between 20 kHz and 300 kHz) is observed by Williams et al. (2002). Equation (8.137) is consistent with a power law roughness spectrum (Equation 8.125) with  ¼ 2.53. Above 400 kHz, Greenlaw’s measurements of the Lambert parameter 10 log10 sand level off, reaching a maximum of about 9 dB at 700 kHz, before starting to decrease with increasing frequency at higher frequencies. At the other end of the spectrum, little variation is observed with frequency below about 10 kHz (Urick, 1983, p. 274). Effect of gravel (100 kHz). The presence of gravel or shell exceeding a proportion of about 0.05 is known to increase the Lambert parameter by several decibels at high frequency (Goff et al., 2004; Simons et al., 2007). The effect of gravel at a frequency of 100 kHz can be estimated using the empirical regression equation due to Simons et al. (2007), derived from data for d50 between 30 mm and 500 mm, gravel fraction g up to 0.7, and shell fraction s up to 0.05, 10 log10  ¼ ð9:5  3:5Þð1  g  sÞd50 =ð1 mmÞ þ ð19:2  2:1Þg þ ð173  45Þs  22:1  0:9:

ð8:138Þ

For medium sand the value predicted by Equation (8.138) is 18.5 dB, with an uncertainty of about 3 dB, which is consistent with 20.9 dB for a frequency of 100 kHz from Equation (8.137). 8.1.4.2.2

Ellis–Crowe (semi-empirical) scattering strength model

Ellis and Crowe (1991) suggest a combination of Lambert’s rule with the facetscattering term (see Sections 8.1.2.1.2 and 8.1.4.1.3), that is,
 DO 2 AO ð0 ; ; Þ ¼  sin 0 sin  þ ð1 þ DOÞ exp  2 ; ð8:139Þ 2 where cos 2 0 þ cos 2   2 cos 0 cos  cos : ð8:140Þ DO ¼ ðsin 0 þ sin Þ 2

Sec. 8.2]

8.2 Target strength, volume backscattering strength

399

The parameter  is referred to by Ellis and Crowe (1991) as the facet strength. It can be related to the seabed reflection coefficient R using ¼

Rð0 Þ 2 : 8 2

ð8:141Þ

For in-plane scattering, Equation (8.139) simplifies to

where


 1 Rð0 Þ 2 1 AO ð0 ; ; Þ ¼  sin 0 sin  þ exp  2 ; 8 2 sin 4 2 tan 2 ( 1 ð0 þ Þ

¼ ¼ 21 ; ð þ   Þ

¼0 2 0

ð8:142Þ

ð8:143Þ

and for the backscattering case
 1 R 2 ðÞ 1 AO ðÞ ¼  sin  þ exp  2 : 8 2 sin 4  2 tan 2  2

8.1.4.2.3

ð8:144Þ

McKinney–Anderson (empirical)

McKinney and Anderson (1964) presents measurements of BBS vs. angle for different bottom types and frequencies between 12.5 kHz and 290 kHz. A widely used empirical relationship that is understood to fit McKinney–Anderson6 measurements is (Jenserud et al., 2001) 16 AO ¼ 10 4:5 þ ðsin  þ 0:19Þ B cos  2:53F 3:20:8B 10 2:8B12 GðB; Þ; ð8:145Þ 1:196 where

8

< 2 B¼ > :3 4

50 B tan 2 



0 < deg < 40 40 < deg < 90

mud sand gravel rock

;

ð8:146Þ

ð8:147Þ

and deg is the grazing angle in degrees, given by Equation (8.50).

8.2

TARGET STRENGTH, VOLUME BACKSCATTERING STRENGTH, AND VOLUME ATTENUATION COEFFICIENT

This section deals with scattering either from individual objects that are small in comparison with a sonar beam so that it makes sense to treat them as point-like 6

The author is unaware of any published comparison demonstrating such a fit.

400 Sources and scatterers of sound

[Ch. 8

scatterers, or from large aggregations of objects occupying a large volume that, from the point of view of the sonar, is effectively infinite. The former are categorized by their target strength (TS) and the latter by their volume backscattering strength (VBS).7 Also considered is the attenuation due to distributed scatterers. TS is an important term in the active sonar equation. It directly controls the echo level and hence the signal-to-noise ratio. VBS can be thought of as the TS of a unit volume of a scatterer that is extended in three dimensions. Depending on the application, such a scatterer can either be the source of a masking background, or of the signal to be detected.

8.2.1

Target strength of point-like scatterers

The scattering properties of both natural and man-made objects are considered in this section. The objects concerned are assumed to be point-like in the sense that they are small by comparison with the footprint of a typical sonar beam. Such objects can be characterized by their target strength TS, which is related to the backscattering crosssection (BSX) (denoted  back ) by8 TS ¼ 10 log10

 back 4

dB re m 2 :

ð8:148Þ

The BSX is a measure of the power scattered by an object per unit solid angle power (its radiant intensity). This is a far-field concept so any direct measurement of TS must take place in the far field of the object (Morse and Ingard, 1968). BSX has dimensions of area and can be thought of as the apparent physical size of an object, in square meters, as perceived by an incident plane wave. For simple shapes, values of  back can be estimated using the results of Chapter 5. For more complicated shapes it is necessary to resort to measurements. TS measurements of natural and man-made objects are presented here. Where appropriate, comparison is made with theoretical expectation. The BSX of fish shoals (and hence their target strength through Equation 8.148) can be estimated from that of an individual fish as described in Chapter 5. 8.2.1.1

Marine organisms with a gas enclosure

The most important single factor in determining the likely TS of a marine animal is the presence or absence of a gas enclosure. This is because such an enclosure greatly enhances the acoustic scattering strength. Important examples of animals with a gas enclosure are bladdered fish (Section 8.2.1.1.1), marine mammals (Section 8.2.1.1.2), and human divers (Section 8.2.1.1.3). 7

Volume scattering strength is the volumic scattering cross-section, expressed in decibels. The 4 denominator is omitted by some authors, who incorporate it instead in the definition of  back (see Chapter 5 for details.) This alternative definition, denoted  back alt , can be converted to TS using TS ¼ 10 log10  back . The value of TS is unaffected because the 4 factors cancel out. alt 8

Sec. 8.2]

8.2 Target strength, volume backscattering strength

401

Table 8.6. Target strength measurements for bladdered fish. Species

Fish length Measurement TS  10 log10 L^2 Reference L=m frequency (kHz)

Gadoids (physoclists)

0.1–1.0 0.04–1.05

38 38–420

27.5 27.1  1.7

Foote (1997) MacLennan and Simmonds (1992) a

Clupeoids (physostomes)

0.1–0.3 0.06–0.34

38 30–120

31.9 31.8  1.2

Foote (1997) MacLennan and Simmonds (1992) b

a Unweighted average and standard deviation of 11 different species: blue whiting, cisco, cod, great silver smelt, haddock, Norway pout, Pacific whiting, redfish, saithe, sockeye salmon, and walleye pollock. b Unweighted average of 2 different species: herring and sprat.

8.2.1.1.1

Bladdered fish

A summary of available measurements of the TS of individual (bladdered) fish is provided in Table 8.6. All of these data are for high-frequency measurements in the sense that the product of bladder radius and acoustic wavenumber k is larger than unity. That is, kaS > 1, where aS is an equivalent radius—that of a sphere of surface area equal to Sbladder —defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sbladder aS
; ð8:149Þ 4 and Sbladder is given by Equation (8.172). Measurement frequencies are between 38 kHz and 420 kHz, and fish lengths range from 4 cm to about 1 m. These measurements can be compared with the theoretical TS using the highfrequency limit of  back from Chapter 5. For bladdered fish this is (neglecting the damping term, which vanishes at high frequency) TS ¼ 10 log10

Sbladder ; 4

ð8:150Þ

and hence TS ¼ 10 log10 L 2  26:4

dB re m 2 :

ð8:151Þ

In fact, the derivation of Equation (8.151) requires kaS  1, which is not compatible with the actual measurement frequencies of Table 8.6, and the true high-frequency TS can be up to 6 dB lower. Allowing for this uncertainty, the theoretical high-frequency target strength can be written TS  10 log10 L 2 ¼ 29:4  3:0

dB:

ð8:152Þ

Remarkably (though fortuitously), this revised value is within 0.2 dB of the average over all four TS measurements for gadoids and clupeoids from Table 8.6. According to theory this expression is dominated by the bladder term, while the flesh term contributes only 0.1 dB. The observed dependence on bladder type (there is a

402 Sources and scatterers of sound

[Ch. 8

difference of about 5 dB in TS between physoclists and physostomes) supports this case, at least for the physoclists. Nevertheless, there is evidence that for some species the bladder is less significant. Unusually low TS values, excluded from Table 8.6, are reported for orange roughy (McClatchie et al., 1999), capelin (Halldorson and Reynisson, 1983), yellowfish tuna (Bertrand and Josse, 2000), and horse mackerel (Axelsen et al., 2003), all for a frequency of 38 kHz. The dimensionless parameter TS  10 log10 L 2 on the left-hand side of Equation (8.152) is referred to henceforth as reduced target strength. Also excluded from the table are lanternfish. These are known to possess a gasfilled bladder (Yasuma et al., 2003), and are believed to make an important contribution to the deep scattering layer (Section 8.2.3.1). At the time of writing, no reliable TS measurements for an individual lanternfish are known to the author. However, the modeling results of Yasuma et al. (2003) suggest a reduced TS at 38 kHz of 29 dB for the Californian headlight fish and significantly lower value (about 46 dB) for the bigfin and Japanese lanternfishes. For some applications, an aspect average might not be suitable (e.g., if the aspect dependence of the TS is required). A computer model that takes into account the three-dimensional shape of fish flesh as well as the air-filled bladder is described by Au et al. (2004). For accurate work it might be necessary to allow for change in bladder volume due, for example, to changes in pressure with depth according to Boyle’s law (Feuillade and Nero, 1998; Nero et al., 2004). 8.2.1.1.2

Marine mammals

Target strength measurements, summarized in Table 8.7, have been made for the bottlenose dolphin and four species of whale, namely the gray, humpback, northern right, and sperm whales. Humpback and gray whales exhibit high values of reduced target strength, consistent with scattering from a large gas cavity, presumably their lungs. In general, it is likely that flesh also contributes to the total, especially at high frequency, for which sound reaching the lungs might be attenuated (Miller and Potter, 2001). The table includes measurements of aspect-averaged TS for the bottlenose dolphin (Au, 1996), northern right whale (Miller and Potter, 2001), and humpback whale (Love, 1973). Taking the average in decibels of these three (aspect-averaged) values gives the following estimate for the reduced target strength of mammals generally TS  10 log10 L 2 ¼ 25:1 dB: ð8:153Þ Recent measurements by Lucifredi and Stein (2007) for the gray whale, which are included in the table but not in the average, are significantly higher than predicted by Equation (8.153). 8.2.1.1.3 Human divers As for marine mammals, the TS of a human diver includes contributions from flesh (including skeleton) and lungs. In addition, the diver could be accompanied by paraphernalia such as a wet suit, breathing apparatus, and exhaled bubbles, all of

a

31.8 to 23.7

12.4 to 1.0 8.8 (unknown aspect)

8–15

12 a

Northern right whale (Eubalaena glacialis)

Sperm whale (Physeter macrocephalus)

Assumed value.

27.5 to 15.5 19.5

4.8 to +7.2 þ4.0 (side aspect)

9–14 15

Humpback whale (Megaptera novaeangliae)

30.4

23.7 to 8.0

2.9 (tail aspect) to þ12.8 (side)

11 a

Gray whale (Eschrichtius robustus)

21.8 to 17.8 30.8 to 24.8 45.8 to 24.8

Single aspect

26.6

19.1

29.7

Averaged over aspect

Reduced target strength TS  10 log10 L 2 (dB)

15 to 11 (side aspect) 24 to 18 (side aspect) 39 (tail aspect) to 18 (side)

Target strength (dB re m 2 )

2.2 2.2 2.2

Length L=m

Bottlenose dolphin (Tursiops truncatus)

Species

1

86

10–20 86

23

23–30 40–79 67

f/kHz

Table 8.7. Target strength measurements for whales, sorted by animal length.

Dunn (1969)

Miller and Potter (2001)

Love (1973) Miller and Potter (2001)

Lucifredi and Stein (2007)

Au (1996) Au (1996) Au (1996)

Reference

404 Sources and scatterers of sound

[Ch. 8

which complicate theoretical estimates and could make a significant contribution to the total. Hollett et al. (2006) report TS at 100 kHz between 25 dB and 20 dB re m 2 for the ‘‘diver’s body, suit, tanks’’. The measurements by the same authors of the TS of exhaled bubbles (for a single exhalation), also at a frequency of 100 kHz, are about 7 dB higher than this (i.e., 18 to 13 dB re m 2 ), and similar to the value suggested by Urick (1983, p. 324), although Urick does not state a measurement frequency. 8.2.1.2

Miscellaneous marine organisms, mostly without a gas enclosure

For the case of a convex object without a gas enclosure ensonified at random aspect, the BSX depends on surface area, so the shape becomes an important consideration. Many marine animals, including fish, are elongated in a well-defined direction, and such animals are considered first (in Section 8.2.1.2.1). This is followed in Section 8.2.1.2.2 by a discussion of animals with more complex shapes. A comprehensive discussion of scattering from zooplankton can be found in Lavery et al. (2007). 8.2.1.2.1

Animals with a pronounced elongated shape

Target strength measurements for euphausiids and bladder-less fish are shown in Table 8.8 for frequencies between 38 kHz and 2 MHz and animal lengths 2 cm to 35 cm. Table 8.8. Target strength measurements for euphausiids and bladder-less fish. Early measurements (before 1980) are excluded.

a

TS  10 log10 L^2 Reference

Species

Animal length L=cm

Measurement frequency (kHz)

Krill (Euphausia superba)

2.8–4.3

38–420

48.0 a

MacLennan and Simmonds (1992, Table 6.4) Pauly and Penrose (1998, Table I) Lawson et al. (2006)

Krill (Euphausia pacifica)

1.9–2.1

420

43.5

Simmonds and MacLennan (2005)

Unspecified fish species

3.0

1000–2000

45

Sandeel (Ammodytes spp.)

11–14

38

53.7

MacLennan and Simmonds (1992, Table 6.4)

Mackerel (Scomber scombrus)

31–35

38–120

46.9 a

MacLennan and Simmonds (1992)

Unweighted average over more than one data set.

Griffiths et al. (2002)

Sec. 8.2]

8.2 Target strength, volume backscattering strength

405

The high-frequency TS for bladder-less fish is determined by the surface area of fish flesh (Chapter 5)
 Sfish ðHFÞ 2 TS ¼ 10 log10 jR j ; ð8:154Þ 16 where jR ðHFÞ j 2  0:0045: ð8:155Þ Surface area Sfish can be related to fish length using the empirical formula (Chapter 4) Sfish ¼ 0:24L 2 :

ð8:156Þ

Substituting these parameters into Equation (8.154) yields TS  10 log10 L 2 ¼ 47

ð8:157Þ

dB:

This estimate is within 4 dB of the measured average value for four out of the five species included in Table 8.8.9 The exception is the sandeel, for which a low reduced TS can be expected because of its long, thin aspect ratio, giving it a smaller surface area than would be expected from its length alone. Comparison can be made with the theoretical expectation for a concave reflector of surface area S
 S ðHFÞ 2 TS ¼ 10 log10 jR j : ð8:158Þ 4 The surface area of a prolate spheroid is (from Chapter 4)
 arcsin e b S ¼ 2ab þ ; e a where ! b 2 1=2 e
1 2 : a

ð8:159Þ

ð8:160Þ

For the TS of Antarctic krill, Simmonds and MacLennan (2005, p. 279) recommend use of an expression that can be written: TS  10 log10 L 2 ¼ 47 þ 14:85 log10 ð42L^Þ

dB:

ð8:161Þ

Equation (8.157) predicts TS within 3 dB of this expression for animals of length between 15 mm and 40 mm. (Equality occurs for a length of 24 mm.) 8.2.1.2.2 Miscellaneous animals with irregular shapes Target strength measurements for squid, gastropods, and jellyfish are summarized in this section. 9 In the case of mackerel this agreement is to be expected, since the jRj 2 value of Equation (8.155) is chosen to match measurements of the target strength of mackerel.

406 Sources and scatterers of sound

[Ch. 8

Squid. Benoit-Bird et al. (2008) give a number of empirical equations fitting the target strength of live squid (Dosidicus gigas) to its mantle length L. The average of their three high-frequency equations (covering the range 70–200 kHz), is TS  10 log10 L 2 ¼ 27:6

dB;

ð8:162Þ

for animals of mantle length in the range 28 cm to 72 cm. This value of 27.6 dB for reduced target strength is 19 dB higher than for bladder-less fish of length L, and the cause of this discrepancy is not known. Part of the difference can be explained by the definition here of L as the mantle length, thus excluding the size of the tentacles, but on its own this seems unlikely to explain the full 19 dB difference. Benoit-Bird et al. find that the cranium scatters a disproportionately large amount of sound for its size, perhaps explaining a further part of the difference. At 38 kHz, the measured target strength is even higher (about 6 dB greater than Equation 8.162). At this frequency, the arms are identified as important scatterers, having ‘‘a stronger effect’’ than the beak or eyes (Benoit-Bird et al., 2008). Earlier measurements of reduced target strength reported by Kawabata (2005) and Simmonds and MacLennan (2005, Table 7.2), for frequencies between 28 kHz and 120 kHz, are lower than those of Benoit-Bird et al. (2008). Excluding one higher value of about 27 dB from Kajiwara et al. (1990), the unweighted average of the remaining four sets of measurements is TS  10 log10 L 2 ¼ 36:8

dB;

ð8:163Þ

about 9 dB lower than Equation (8.162). The animals involved in these earlier measurements, with mantle lengths between 8 cm and 42 cm, were smaller than those used by Benoit-Bird et al. (2008). Gastropods. Stanton et al. (1998a, Fig. 4) report TS measurements for the gastropod Limacina retroversa for frequencies in the range 370 kHz to 600 kHz. Their aspect averaged value can be written TS  10 log10 L 2 ¼ 19:5

dB;

ð8:164Þ

where L is the gastropod ‘‘length’’ (Stanton et al., 1998a), equal to 1.5 mm for this animal. In this case the most important difference compared with the measurements of Table 8.8 is probably not the shape, but the hardness of the gastropod’s shell. For example, putting R ðHFÞ ¼ 0.5 (from Greene et al., 1998) into Equation (8.154), retaining Equation (8.156) for the surface area, results in the formula TS  10 log10 L 2 ¼ 19:7

dB;

ð8:165Þ

thus predicting a value for the reduced target strength that is remarkably close to the measured value of Equation (8.164). Although shape appears not to be the determining factor here, in other circumstances it might be. Gastropods come in a variety of shapes, most but not all having a hard exterior shell.

Sec. 8.2]

8.2 Target strength, volume backscattering strength

407

Table 8.9. Target strength measurements for jellyfish (from Simmonds and MacLennan, 2005, Table 7.2). Disk diameter D=cm

f /kHz

TS (dB re m 2 )

TS 10 log10 D 2 (dB)

Aequorea victoria (crystal jelly)

4.2

420

64.8

37.3

Bolinopsis sp.

4.5

420

80.0

53.1

Aequorea aequorea

7.4

18–120

68.5 to 66.3

45.9 to 43.7

9.5–15.5

120–200

64.3 to 57.1

43.9 to 39.8

26.8

18–120

51.5 to 46.6

40.1 to 35.2

Aurelia aurita Chrysaora hysoscella

Table 8.10. Target strength measurements for siphonophores. TS (dB re m 2 )

Frequency/kHz

Pneumatophore size

75.0

200

a ¼ 0.3 mm (estimated)

Trevorrow et al. (2005)

69.1

120

a ¼ 0.6 mm (estimated)

Warren et al. (2001)

69.5

400–600

a ¼ 0.65 mm b ¼ 0:25 mm

Stanton et al. (1998a)

Reference

Jellyfish and siphonophores. TS measurements for various species of jellyfish are summarized in Table 8.9. The siphonophore is a colonial invertebrate resembling a jellyfish. An important feature is a small gas enclosure called a pneumatophore. The target strength of the siphonophore is dominated by scattering from the pneumatophore, which approximates in shape to a prolate spheroid. Measurements of siphonophore TS are summarized in Table 8.10. The size of the pneumatophore is characterized by the dimensions of its major and minor axes, denoted a and b, respectively. The reduced TS, based on measurements of Stanton et al. (1998a), is TS  10 log10 ð2aÞ 2 ¼ 11:0

dB:

ð8:166Þ

Substituting the measured values of a and b from Stanton et al. (1998a) in Equation (8.163) yields the theoretical TS value of TS ¼ 68:7

dB re m 2 :

ð8:167Þ

408 Sources and scatterers of sound

[Ch. 8

Table 8.11. Second World War measurements of the target strength of man-made objects (from Urick, 1983). TS (dB re m 2 ) Beam aspect

Bow aspect

Intermediate aspect

Submarine

24

9 (stern)

14

Surface ship

24

14

Mine

9

26 to þ9 21

Torpedo

Adjusting for the length 2a of the pneumatophore gives TS  10 log10 ð4a 2 Þ ¼ 10:2

ð8:168Þ

dB;

which is within 1 dB of Stanton’s measured value as given by Equation (8.166).

8.2.1.3

Man-made objects

Table 8.11 summarizes measurements of the TS of various surface ships, submarines, and underwater weapons made during and after the Second World War. The values are representative only, as the measurements are subject to high variability (Urick, 1983, p. 324). TS measurements of their modern equivalents are usually classified. An order of magnitude theoretical estimate of the TS of these and similar objects can be obtained using the expression from Chapter 5 for a smooth convex target of surface area S at random aspect TS ¼ 10 log10

jRj 2 S ¼ 17:0 þ 10 log10 S þ 10 log10 jRj 2 16

dB re m 2 ;

ð8:169Þ

where R is the reflection coefficient. For example, the aspect-averaged TS of a perfectly reflecting convex object of surface area S ¼ 100 m 2 is þ3 dB re m 2 . Equation (8.169) is not applicable to an object with sharp edges, as the TS might then be dominated by diffraction from these edges. Some modern military vessels are clad with special anechoic (literally ‘‘nonreflecting’’) materials. Their shapes might also be specially designed to deflect sound away from the expected receiver position (as is done for stealth aircraft designed to achieve a low radar cross-section). For such objects the random aspect equation is not applicable. As a result of the special materials or shapes used, comparable modern vessels of similar size to their Second World War (WW2) counterparts are likely to have a lower TS than the values quoted in Table 8.11.

Sec. 8.2]

8.2.2

8.2 Target strength, volume backscattering strength

409

Volume backscattering strength and attenuation coefficient of distributed scatterers

If there are many point-like objects forming an extended ‘‘cloud’’ of scatterers, it can be more useful to think of this cloud as a continuum instead of as a collection of discrete objects. In this situation the relevant quantity is BSX per unit volume (volumic10 BSX), which when converted to decibels gives the volume backscattering strength (VBS):  back VBS
10 log10 V dB re m 1 : ð8:170Þ 4 Low-frequency and high-frequency scattering effects are considered separately in Sections 8.2.2.1 and 8.2.2.2, respectively. Volume attenuation is discussed in Section 8.2.2.3. 8.2.2.1

Low-frequency VBS (mainly due to large fish)

The presence of large numbers of pelagic fish can result in high values of VBS. Measurements for a known or independently estimated fish population are very rare (see Love, 1993 for a notable exception). A theoretical estimate can be made using the expression for the volumic BSX  back from Chapter 5, giving the result for a V representative fish length Lgroup
 Sbladder VBS  10 log10 Qfish Qgroup þ 10 log10 ðL 2group NV Þ 4L 2  2 f0 ðLgroup Þ 10 2  Q 1 ; ð8:171Þ loge 10 group f where Sbladder is the bladder surface area (Chapter 4) Sbladder ðLÞ  0:0291L 2

ð8:172Þ

and the resonance frequency is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:1^ z þ 1:75 f0 ðLÞ  ð78:9 HzÞ : L^

ð8:173Þ

Assuming a nominal value of Qgroup  2, Equation (8.171) can be written  2 f0 ðLgroup Þ 2 VBS  23:3 þ 10 log10 ðQfish L group NV Þ  54:6  1 dB re m 1 : ð8:174Þ f Using this equation, a theoretical estimate of VBS can be made based on knowledge of the average population density of bladdered fish. An example prediction for the North Sea follows in Table 8.12, based on fish population data from Chapter 4. Fish depth used for the table is 25 m, one half of the assumed average water depth of 50 m. The main effect of changing this depth is in the resonance frequency, which is 10 Following Taylor (1995), the adjectives ‘‘areic’’ and ‘‘volumic’’ are used, respectively, to mean ‘‘per unit area’’ and ‘‘per unit volume’’.

410 Sources and scatterers of sound

[Ch. 8

Table 8.12. Predicted average night-time contribution to volume backscattering strength (VBS), column strength (CS, defined in Section 8.2.3.1), and attenuation due to pelagic fish in the North Sea; only those fish known to possess a swimbladder are included (thus, mackerel and sandeel are excluded from this table). Ntot is estimated North Sea population; VBS0 is average contribution to VBS at frequency f0 ; CS0 is average contribution to CS at f0 ; and a0 is average contribution to að f Þ at frequency f0 . Ntot =10 9 (Chap. 4)

NV = dam 3

L/m

VBS0 / (dB re m 1 )

CS0 /dB

a0 / (dB km1 )

f0 /kHz

Silvery pout (Gadiculus argenteus)

135.8

7.09

0.06

63

46

0.23

2.4

Norway pout (Trisopterus esmarkii)

64.1

3.34

0.13

59

42

0.51

1.1

Atlantic herring (Clupea harengus)

11.3

0.59

0.24

62

45

0.30

0.6

Whiting (Merlangius merlangus)

5.9

0.31

0.20

66

49

0.11

0.7

European sprat (Sprattus sprattus)

5.5

0.29

0.10

72

55

0.03

1.4

Haddock (Melanogrammus aeglefinus)

2.2

0.11

0.30

67

50

0.09

0.5

Horse mackerel (Trachurus trachurus)

1.9

0.10

0.24

69

52

0.05

0.6

Pollock (Pollachius virens)

0.4

0.02

0.45

71

54

0.04

0.3

Cod (Gadus morhua)

0.1

0.01

0.70

73

56

0.02

0.2

Species

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi proportional to z^ þ 17:5. The table further assumes a total North Sea volume of 19 200 km 3 . The night-time average nature of these estimates is emphasized (during the day the fish tend to aggregate in shoals). Local values can be significantly higher or lower, depending on the concentration of each species (Knijn et al., 1993). According to this table, the main contributions to VBS are due to Norway pout, herring, and silvery pout, the resonance frequencies for which are between 0.6 kHz and 2.4 kHz. These estimates are based on survey data that at the time of writing are 20 years old. Thus, they are not intended as a quantitative prediction for the North Sea in the

Sec. 8.2]

8.2 Target strength, volume backscattering strength

411

early 21st century, but rather as an estimate of typical values to be expected in regions sustaining a high density of fish. 8.2.2.2

High-frequency VBS (partly due to small fish)

For frequencies of 10 kHz to 60 kHz, APL-UW (1994) provides default values for VBS summarized in Table 8.13. For the Arctic region (under the ice cap and in the marginal ice zone), a separate average value of 75 dB re m1 is suggested. 8.2.2.3

Volume attenuation coefficient due to bubbles and bladdered fish

The attenuation of sound in pure seawater is described in Chapter 4. Here the additional contributions due to air bubbles and bladdered fish are considered, using results from Chapter 5. 8.2.2.3.1

Bubbles

The equations presented here are for the extinction coefficient in units of nepers per meter. This coefficient can be converted to decibels per meter by multiplying the numerical value by 20 log10 e. The expression for attenuation due to a cloud of bubbles is ð 1 ¼  ðaÞnðaÞ da; ð8:175Þ 2 e where   therm þ visc back e ðaÞ ¼  ða; !Þ 1 þ : ð8:176Þ !a=cm Expressions for the damping coefficients therm and visc for bubbles are given in Chapter 5. 8.2.2.3.2 Dispersed fish with swimbladder The equivalent expression for dispersed fish is ð 1 ¼  ðLÞnðLÞ dL; 2 e

ð8:177Þ

Table 8.13. Default advice for VBS between 10 kHz and 60 kHz for sparse, intermediate, and dense marine life (except for the Arctic region), from APL-UW (1994). VBS/(dB re m1 )

Deep water Shallow water

Depth

Sparse

Intermediate

Dense

0–300 m 300–600 m

94 81

87 74

79 66

Any

85

72

62

412 Sources and scatterers of sound

with

[Ch. 8

" e ¼

 back bladder ðL; !Þ

c 1þ m !

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 4aS ð þ flesh Þ : 3Vbladder therm

ð8:178Þ

where aS is the equivalent bladder radius given by Equation (8.149). Expressions for the damping coefficients therm and flesh for fish are given in Chapter 5. In the case of a large group of fish, the population distribution is likely to exhibit a peak around some value (say Lgroup ). Following Weston (1995), Equation (8.178) for the extinction cross-section can be replaced at resonance in the integrand of Equation (8.177) by e ðLÞ  4a 2S L0 ð f ÞQrad ðL  L0 ð f ÞÞ;

ð8:179Þ

where L0 ð f Þ is given by L0 

78:9 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:1^ z þ 1:75 f^

m;

ð8:180Þ

and ðL  L0 Þ is the Dirac delta function (see Appendix B). Using a Gaussian length distribution for nðLÞ gives the result (see Chapter 5) 
2  L0 ð f Þ 2 2  2a S Qrad Qgroup NV exp Q group 1 : ð8:181Þ Lgroup This expression is applicable to dispersed fish with a bladder, provided that the acoustic frequency is close to the resonance frequency of fish bladders. If the fish aggregate into shoals, their effect on attenuation is expected to be negligible. It is often the case that shoals form during the day and disperse at night. Thus, a simple rule, if no better information is available, is to use Equation (8.181) for bladdered fish at night-time, and to assume no effect for day-time (or if there is no bladder). Table 8.12 includes peak values of calculated for each species of fish, using Equation (8.181) with the average North Sea population density. Notice the relationship between the extinction coefficient and VBS (Equation 8.171): Q ¼ 2 rad 10 VBS=10 ; ð8:182Þ Qfish where Qrad ¼

8.2.3 8.2.3.1

55:2 : ð1 þ z^=17:6Þ 1=2

ð8:183Þ

Column strength and wake strength of extended volume scatterers Column strength and the deep scattering layer

If volume scatterers are distributed laterally in the two horizontal dimensions and confined to a finite extent in depth, it is useful to define the column strength of the

Sec. 8.2]

8.2 Target strength, volume backscattering strength

distribution as depth-integrated volumic BSX, expressed in decibels ð ð back  V ðzÞ dz dB: CS
10 log10 10 VBS=10 dz ¼ 10 log10 4

413

ð8:184Þ

Like SBS and BBS, CS is a dimensionless quantity. In deep water a region of high scattering strength, known as the deep scattering layer, is often found at depths of a few hundred meters. This layer contains many different species of myctophids (lanternfish) and euphausiids. The wide variety in the size of the different species leads to a broadband acoustic response. Some of the species stay at an almost constant depth, while others follow a diurnal migration pattern (Medwin and Clay, 1998). Order of magnitude estimates of CS for the North Sea due to pelagic fish are included in Table 8.12. These values are independent of the assumed water depth and can be compared with CS measurements in the Norwegian Sea and northeast Atlantic due to Love (1993). Love’s data show a peak value of up to 40 dB around 2 kHz, attributed to blue whiting and redfish of lengths between 20 cm and 40 cm. The fish in Love’s survey were at a depth of around 300 m. The increased pressure compared with the nominal depth of 25 m used to compile Table 8.12 might explain the higher resonance frequency observed by Love. Some of Love’s measurements in the northeast Atlantic exhibit a monotonically increasing CS with frequency from 2.5 kHz to 5 kHz, a feature that he attributes to the presence of lanternfish. At 5 kHz the CS is about 43 dB. The peak value is outside the measured frequency range. Estimates for high-frequency CS data can be derived from the VBS measurements summarized in Table 8.13. The resulting CS values for deep water are between 56 dB and 41 dB, similar to the spread of values predicted for fish from Table 8.12, which is applicable to frequencies of order 1 kHz. Similar values are shown in Urick (1983, p. 260, Figs. 8.15 and 8.16) for frequencies 3 kHz to 20 kHz, with a marked drop-off below 3 kHz. The overall impression is that, in areas that are densely populated with marine life, a reasonable default of CS in the frequency range 3 kHz to 60 kHz is about 45 dB, albeit with a large uncertainty. 8.2.3.2

Wake strength

Ships and submarines have extended wakes containing many millions of bubbles. The wake scatters sound, acting like an acoustic target that is confined in two dimensions and extended in a third. Such a target may be characterized by wake strength (WS), equal to the TS of a unit length of the wake. Wake strength has dimensions of area per unit length and is therefore expressed in units of decibels re 1 meter (dB re m). Measurements of WS dating to WW2 are listed for surface vessels in Table 8.14 and for submarines in Table 8.15. According to Anon. (1946), WS is approximately independent of frequency between 15 kHz and 60 kHz, and decreases with time at a rate of about 1 dB per minute. Modern measurements of volumic BSX of wakes of three different surface vessels at frequencies between 28 kHz and 400 kHz are presented by Trevorrow et al. (1994).

414 Sources and scatterers of sound

[Ch. 8

Table 8.14. Wake strength measurements for various WW2 surface ships (from Urick, 1983, p. 263). Surface ships

f /kHz

WS/dB re m

Aircraft carrier, escort (CVE)

24

9.4

Transport (AP)

24

9.4

Destroyer (DD)

24

11.4

Destroyer escort (DE)

24

11.4

Table 8.15. Wake strength for various WW2 submarines (from Anon., 1946). Speed/(m s1 )

Depth a /m

WS/(dB re m)

f /kHz

USS-S23 (SS-128)

4.9 3.1

0.0 27.4

13.4 21.4

60 60

USS-S34 (SS-139)

4.9 3.1

0.0 27.4

8.4 18.4

45 45

USS ‘‘Tilefish’’ (SS-307)

4.9 3.1

0.0 27.4

8.4 15.4

45 45

USS-S18 (SS-123)

3.1

13.7

28.4

45

Submarines

a

A depth of zero indicates that the submarine was surfaced at the time of the measurement.

8.3

SOURCES OF UNDERWATER SOUND

Underwater noise sources are an important consideration for sonar performance calculations, as they determine the minimum signal level required for successful detection. Useful reviews of the main sources of underwater sound are provided by Richardson et al. (1995), Anon. (2003), McDonald et al. (2006). The sounds are many and varied, and are caused, for example, by: — breaking gravity waves, either due to wind or surf (Wilson, 1983; Deane, 2000); — non-linear interactions between gravity waves passing through one another (known as ‘‘microseisms’’) (Longuet-Higgins, 1950; Webb, 1992); — precipitation (rain, snow, or hail) (Scrimger et al., 1987; McConnell et al., 1992; Nystuen, 2001; Ma et al., 2005); — violent geological or meteorological activity such as lightning (Hill, 1985), hurricanes (Bowen et al., 2003; Wilson and Makris, 2006), earthquakes or volcano eruptions (Dietz and Sheehy, 1954; Northrop, 1974); — other physical processes associated with the behavior of ice at the sea surface (Uscinski and Wadhams, 1999) or of gravel at the seabed (Thorne, 1986);

Sec. 8.3]

8.3 Sources of underwater sound 415

— marine mammals, crustacea, and other living organisms (Kelly et al., 1985; Cato, 1993); — anthropogenic sources such as sonar, shipping, or industrial activity (Richardson et al., 1995). A composite graph of typical ambient noise spectrum levels is shown in Figure 8.13, illustrating the varied nature of underwater noise sources. Different parts of the spectrum tend to be dominated by different, but specific noise sources. For example, between 300 Hz and 100 kHz the dominant source of noise is often wind-related, whereas at slightly lower frequency (30–300 Hz) the strongest component is usually due to distant shipping. Rain noise, when present, tends to peak at a few kilohertz. Biological noise can be broadband, but can also contain strong spectral peaks (e.g., around 20 Hz due to blue whales and fin whales—see McDonald et al., 2006). There is growing evidence that low-frequency sound levels in the sea (around 40 Hz) have increased on average by up to 3 dB per decade in the period between 1965 and 2003 (Andrew et al., 2002; McDonald et al., 2006).11 This increase is attributed to a doubling in the number of commercial ships during that period (from 41 900 to 89 900 ships) and a nearly four-fold increase in their gross tonnage (from 160 to 605 million tonnes). A comparable increase in levels has been observed in the spectral peaks associated with blue and fin whales (McDonald et al., 2006). In the same period the peak frequency of observed blue whale vocalizations dropped from 22 Hz to 16 Hz. There is a greater emphasis in Section 8.3 on measurements (as opposed to theory) than in the rest of this chapter, because the sound generation mechanisms are poorly understood compared with the mechanisms for reflection (Section 8.1) and scattering (Sections 8.1, 8.2). The modeling of propagation from the sound source to a receiver in the sea (i.e., to the sonar or animal listening to the sound) (Hamson, 1997) is addressed in Chapter 9. Measurements of ambient noise above 100 kHz are hampered by noise in the receiving equipment due to thermal agitation. This type of interference, called thermal noise, is described in Chapter 10. It is outside the scope of the present chapter because it is not caused by underwater sound. Also excluded here and included in Chapter 10 are sonar transmissions and other intentional man-made sounds, such as those produced by seismic survey or acoustic communications sources. This and subsequent chapters contain numerical values of source level and sound pressure level in water. These levels are expressed in decibels and it can be tempting to compare them with sound levels in air, also expressed in decibels. Such a comparison is fraught with difficulty because of the following differences (see also Chapman and Ellis, 1998): — The reference pressure is different: the standard reference pressure in air is 20 mPa, leading to a numerical difference of 26 dB in sound pressure level for the same RMS pressure in air and water. 11

An increase of 0.5 dB per year is reported by Ross (1974).

416 Sources and scatterers of sound

[Ch. 8

Figure 8.13. Typical ambient noise spectra. The x-axis covers five decades of frequency from 1 Hz to 100 kHz. The y-axis is the noise spectrum level from 0 dB to 140 dB re mPa 2 /Hz (adapted from Wenz, 1962, # American Institute of Physics, with permission).

Sec. 8.3]

8.3 Sources of underwater sound 417

— The medium is different: the characteristic impedance of water is 3600 times greater than that of air. This means that the intensity of a plane wave in water is 3600 times smaller than that of a plane wave in air of the same RMS pressure. Conversely, the RMS pressure of a plane wave in water is 60 times greater than that of a plane wave in air of the same acoustic intensity. — The hearing sensitivity, pain thresholds, or damage thresholds are different, even for the same species: most species live either only in air or only in water, so it only makes sense to consider their hearing in that medium. For a few amphibious species (mainly seals and human divers) it is known that the RMS pressure of a sound that is just audible in water is higher than the RMS pressure of a sound that is just audible in air. Little is known about injury thresholds in water (Southall et al., 2007). — Reporting conventions are different: in air, measures of sound reported in decibels are almost always in the form of a sound level (i.e., the sound pressure level or SPL, weighted according to the sensitivity of human hearing in air). In water, measurements are usually reported without adjustment for hearing sensitivity. Finally, it is common practice to characterize a source of underwater sound by its source level, which is a measure of transmitted power and not received intensity. Thus, it is rarely necessary, and almost always unwise, to compare sound levels in air and water. The natural human desire for comparison with known experience can be satisfied instead by invoking familiar sounds in water, such as that of rainfall or surf. See Figure 8.14 for some examples of underwater sounds and the corresponding sound pressure levels. The remainder of this section is structured as follows. Measurements of shipping source level spectra are summarized in Section 8.3.1, followed by a discussion of the source spectra for distributed sources at the sea surface (Section 8.3.2) and on the seabed (Section 8.3.3). Section 8.3.2 includes a description of shipping noise as a continuum of distant ships of given areic density.

8.3.1

Shipping source spectrum level measurements

Noise from distant ships, presumably in transit along commercial shipping lanes, is believed to dominate the underwater ambient noise spectrum at low frequency, from about 10 Hz to a few hundred hertz. A prediction of the likely received sound levels at a given location requires an approximate shipping density distribution and an estimate of the average source level of an individual ship. The latter is the subject of the present section. The measurement of source level is a difficult one, often involving the unwitting participation of passing vessels of opportunity, and in order to provide a meaningful average the measurement must be repeated a number of times with different ships. The following text presents measured spectra for individual vessels (Section 8.3.1.2), followed by some measurements of spectra averaged over many ships (Section 8.3.1.3). Only industrial and commercial shipping vessels are

418 Sources and scatterers of sound

[Ch. 8

Figure 8.14. Typical values of sound pressure level (left) and peak pressure level (right), in units of dB re mPa 2 and at a distance of 100 m from the source (except for the sound of rainfall, included as a reference). A typical range of hearing thresholds is also marked (# TNO, reprinted with permission).

considered. A summary of measurements for warships from WW2 is given by Urick (1983) and Collier (1997).

8.3.1.1

Conversion from far-field measurements

The source level of a ship is a measure of the amount of sound in the far field12 of that ship. Thus, any measurement of this quantity needs to be made in the far field, while at the same time be close enough to the source to ignore propagation effects. These two conflicting requirements are irreconcilable in the case of a surface ship, because the far field inevitably contains a contribution from sea surface reflected sound. Two quite different methods, described below, are in use for correcting for this separate contribution. The first method involves calculation of propagation loss (PL) for a point monopole source at some assumed, representative depth. The source level (SL mp ) 12

If a point in the radiated field is sufficiently distant from the sound radiator, the phase difference between sound paths arriving from different parts of the radiator is determined only by the bearing of the field point relative to the radiator and not by the distance between the field point and the radiator. The region in which this is satisfied is known as the far field of the radiator. The near field is where this condition is not satisified.

Sec. 8.3]

8.3 Sources of underwater sound 419

is then calculated from the measured SPL using SL mp ¼ SPL þ PL:

ð8:185Þ

This back-calculation is referred to in the following as the ‘‘monopole method’’ because the result is a conventional monopole source level. The advantage of the monopole method is that the measurement can be made in shallow water. A disadvantage is that the result is sensitive to the assumed value of source depth and to the accuracy of propagation loss predictions. The second method is a pragmatic one, involving measurements sufficiently far from the ship for spherical spreading to hold (i.e., not in the near field), while still close enough to neglect absorption (Ross, 1976; de Jong, 2009). Consider an ‘‘equivalent source level’’, SL eq , defined as the monopole source level that would result in the same SPL as the combined ship and surface image at the measurement distance, assuming free space propagation conditions apart from the inevitable presence of the sea surface. That is,13 SL eq ð; Þ
SPLð; Þ þ 10 log10 s 2 ;

ð8:186Þ

where s is the distance to the acoustic center of the ship. In general, this quantity is a function of elevation () and bearing ( ) aspect angles. If measured at keel aspect (i.e., with a hydrophone directly beneath the ship, at elevation  ¼ =2), the result at low frequency is the source level of the dipole created by the ship and its surface image. In the following, this keel aspect value is referred to as the ‘‘dipole source level’’ at all frequencies (even when the frequency is not low enough for a dipole to form) and denoted SL dp :14 SL dp ¼ SL eq ð ¼ =2Þ:

ð8:187Þ

This second method is referred to in the following as the ‘‘equivalent source method’’. An approximate conversion between SL dp and SL eq at other elevation angles, valid at frequencies low enough for the ship to behave as a point dipole, is SL eq ð; Þ  SL dp þ 10 log10 sin 2 :

ð8:188Þ

The result is independent of bearing for a true dipole, but departures from this idealized behavior can be expected in some bearings (e.g., directly ahead of or behind the ship—Arveson and Vendittis, 2000). The source levels denoted SL mp and SL dp defined above can take quite different values, especially at low frequency, for which the monopole and dipole source factors are related via dp 2 2 S mp 0 =S 0  1=ð4k d Þ: 13

ð8:189Þ

The equivalent source level defined in this way is not a property of the source only. It depends also on measurement distance (e.g., through absorption) and water depth (e.g., through reflections from the seabed). 14 This quantity is independent of .

420 Sources and scatterers of sound

[Ch. 8

At high frequency and neglecting absorption, they differ—on average—by a factor of 2. Thus, a more general conversion, incorporating both high-frequency and lowfrequency forms, is dp 2 2 1 S mp 0 =S 0  2 þ 1=ð4k d Þ:

ð8:190Þ

The advantage of the equivalent source method is that it does not require a choice to be made for the source depth. A disadvantage is that a complete characterization of the radiated noise of even a simple source requires measurements at many angles. 8.3.1.2

Industrial and commercial shipping (individual ships)

Measurements of the source spectra of individual ships are presented here in Table 8.16, in the form of third-octave source levels. The measurements are compiled from Richardson et al. (1995) and Arveson and Vendittis (2000). This table is not intended as a precise indication of expected source level of any given ship, as this depends not just on the ship type and speed, but also on its cargo and type of activity, and the condition of its propellers.15 Instead, it provides an indication of the spread of possible values to be expected. A further difficulty with the interpretation of this table is the absence in some cases of details of the measurement method. The measurements of Arveson and Vendittis (2000) are made using the equivalent source method, and include measurements at keel aspect ( ¼ =2), making it possibe to infer the dipole source level as given by Equation (8.187). In the absence of a statement to the contrary, the equivalent source method is assumed to have been used for the measurements from Richardson et al. (1995) as well, although for these the elevation angle is unknown. The term ‘‘third-octave level’’ means that the spectral density Qf is integrated over a third-octave band (i.e., one-third of an octave in frequency) before being converted to decibels. Thus, the third-octave sound pressure level L1=3 is given by L1=3
10 log10

ð 2 þ1=6 f0 2 1=6 f0

Qf df

dB re mPa 2 ;

ð8:191Þ

where the lower and upper limits of integration are respectively one-sixth of an octave below and above the center frequency f0 . If the spectral density varies approximately linearly with frequency, Equation (8.191) may be replaced by L1=3  10 log10 ½Df1=3 Qf ð f0 Þ

dB re mPa 2 ;

ð8:192Þ

where Df1=3 ¼ ð2 þ1=6  2 1=6 Þ f0  0:2316f0 :

ð8:193Þ

If the spectral density of Equation (8.192) is scaled to a nominal 1 m reference distance, then L1=3 becomes the third-octave source level, with units dB re mPa 2 m 2 . 15

The type of propulsion system can be an important consideration in its own right.

Sec. 8.3]

8.3 Sources of underwater sound 421

Table 8.16. Third-octave source levels of various commercial and industrial vessels, expressed in units of dB re mPa 2 m 2 . Entries are listed in descending order of the third-octave level at 500 Hz. Measurements for the data for cargo ship Overseas Harriette are taken from Arveson and Vendittis (2000) (dipole source levels). The remainder are from Richardson et al. (1995, Table 6.9) (equivalent source levels at unstated elevation). Type

Description

RMS source level (third octave)

Center frequency

50 Hz

100 Hz

200 Hz 500 Hz

1 kHz

2 kHz

Icebreaker

R Lemeur

177

183

180

180

176

179

Drillship

Kulluk

174

172

176

176

168

—

174

177

176

172

169

166

170

177

177

171

—

—

Modern cargo ship M/V Overseas Harriette 8.2 m/s (16 kn)

185

180

174

168

166

163

Supply ship

Kigoriak

162

174

170

166

164

159

Drillship

Canmar Explorer II

162

162

161

162

156

148

Modern cargo ship M/V Overseas Harriette 6.2 m/s (12 kn)

178

169

164

161

159

155

Tug and barge

5 m/s

143

157

157

161

156

157

Dredger

Beaver Mackenzie

154

167

159

158

—

—

Modern cargo ship M/V Overseas Harriette 4.1 m/s (8 kn)

163

154

156

157

156

152

Zodiac

128

124

148

132

132

138

Large tanker Dredger

Aquarius

5 m length

A third-octave level can be converted into a mean spectrum density level Lf using Lf  L1=3  10 log10 Df1=3 ;

ð8:194Þ

where the average is in frequency, across the third-octave band. Average equivalent source levels calculated in this way are plotted in Figure 8.15 (dotted blue curves). The red curves are explained in Section 8.3.1.3. 8.3.1.3

Commercial shipping (averaged source spectra)

Average source level spectra measured by Scrimger and Heitmeyer (1991) (hereafter abbreviated SH91) and by Wales and Heitmeyer (2002) (abbreviated WH02), both using the monopole method, are described below. SH91 estimates the source level

422 Sources and scatterers of sound

[Ch. 8

Figure 8.15. Measured equivalent source spectral density levels (SL eq f ), averaged over thirdoctave bands for commercial and industrial shipping: individual ships, selected from Table 8.16 (blue lines); averaged source spectra plotted in red (solid red line is from Wales and Heitmeyer, 2002; dashed red line is from Scrimger and Heitmeyer, 1991—see Section 8.3.1.3 for details).

spectra of 50 ships in the frequency band 70 Hz to 700 Hz. The vessels concerned were approaching or departing from the Mediterranean port of Genoa (Italy), with an average speed of 7.2 m/s (14 kn). WH02 describes the source spectra between 30 Hz and 1200 Hz of 272 ships in the Mediterranean Sea and eastern Atlantic Ocean, and proposes the following mean spectrum for the monopole source level spectrum 2 2 2 ^ ^ SL mp f ¼ 230:0  35:94 log10 f þ 9:17 log10 ½1þ ð f =340Þ  dB re mPa m =Hz: ð8:195Þ

Both SH91 and WH02 are plotted in Figure 8.16. The monopole source levels measured by SH91 (dashed line) are about 6 dB to 12 dB higher than those of WH02 (solid line). (The curves marked ‘‘Ov. Harriette’’ are explained in Section 8.3.1.4.) For both SH91 and WH02 measurements, the source spectrum was inferred using the monopole method by subtracting an estimate of propagation loss (PL) from the received spectrum at a distance of several kilometers from the source. Any bias in estimated PL would cause a bias in the inferred source spectrum, providing a possible explanation for the difference between reported source levels. From this point of view the WH02 data seem more reliable, because they involve a shorter measurement range and hence perhaps less uncertainty in PL, and the average is computed over a larger number of individual ships. However, a definitive statement cannot be made

Sec. 8.3]

8.3 Sources of underwater sound 423

Figure 8.16. Estimated third-octave monopole source levels SL mp for the cargo ship Overseas Harriette at various ship speeds, based on measurements from Arveson and Vendittis (2000); WH02 (solid red line) and SH91 (dashed red line) are included for comparison; the WH02 spectrum is extrapolated to 15 kHz, agreeing in the extrapolated region (dotted red line) with the measurements of Arveson–Vendittis for a ship speed close to 6.2 m/s (12 kn).

from the available measurements. SH91 and WH02 spectra are also shown in Figure 8.15 (red curves), where they are converted to dipole levels using Equation (8.190) assuming a monopole depth of 1.8 m. 8.3.1.4

Effect of ship speed and acceleration

A thorough investigation of the radiated noise characteristics of a single cargo ship was carried out by Arveson and Vendittis (2000) using the equivalent source method. Their measured spectra for different ship speeds are represented in Figure 8.16 by the blue and cyan curves. The Overseas Harriette measurements have been converted from a dipole to monopole source level using Equation (8.190), for an assumed monopole depth of 1.8 m, and may be compared with the WH02 and SH91 monopole source level spectra plotted in the same figure. The WH02 curve is extrapolated above 1200 Hz (see dotted red line) using an empirically determined gradient (SL mp ¼ constant  23 log10 F), chosen to match the Arveson and Vendittis (2000) f data at low ship speed. Figure 8.16 illustrates the effect of ship speed on the radiated noise of an individual ship traveling at constant velocity. The effect of turn rate is studied by Trevorrow et al. (2008). For the maximum turn rate considered of 4.5 deg/s, they

424 Sources and scatterers of sound

[Ch. 8

report an increase of between 6 dB and 18 dB in third-octave bands between 160 Hz and 4 kHz.

8.3.2

Distributed sources on the sea surface

The ubiquitous ocean noise caused by wind is considered next, followed by rain noise, which is itself also sensitive to local wind speed. Empirical relations providing dipole source level as a function of wind speed and rain rate are given in Sections 8.3.2.1 and 8.3.2.2. A uniform distribution of distant ships can also be regarded as a distributed source at the sea surface, as described in Section 8.3.2.3. 8.3.2.1

Wind noise source level

The physical origin of wind noise is thought to be associated with the natural pulsations of gas bubbles created by breaking waves or similar surface activity. Because of the close proximity of such bubbles to the sea surface, a dipole radiation pattern is usually assumed, for which it is convenient to define a parameter K wind as K wind ¼

3c wind W ; 2 Af

ð8:196Þ

wind where W wind is the spectral density of Af is the areic power spectral density. Thus, K wind the areic dipole source factor. The quantity 10 log10 K is known as the ‘‘dipole source spectrum level’’ or sometimes just ‘‘dipole source level’’. The term ‘‘areic dipole source spectrum level’’ (i.e., the dipole source factor per unit area, expressed in decibels), is suggested for a sheet source, to distinguish it from the source level of a single dipole. It follows from Chapter 2 that the noise spectral density due to such a source, for a receiver in a uniform half-space, is given by

Qf  32 cWAf ;

ð8:197Þ

and hence, eliminating the power spectral density from Equations (8.196) and (8.197), 10 log10 Qf ¼ 10 log10 ðK wind Þ

dB re mPa 2 Hz 1 :

ð8:198Þ

8.3.2.1.1 High-frequency wind noise (APL model) At frequencies above a few kilohertz, wind noise decreases monotonically with increasing frequency. A useful parameterization from APL-UW (1994), intended for the frequency range 10 kHz to 100 kHz, can be written K wind APL ¼ ðDTÞ ¼

10 4:12^v 2:24 APL F 1:59 10 0:1 ( 0

mPa 2 Hz 1

0:26ðDT  1:0Þ

DT < 1 2

DT  1

ð8:199Þ ð8:200Þ

Sec. 8.3]

8.3 Sources of underwater sound 425

where DT is the temperature difference in degrees Celsius DT ¼ T^air  T^water ;

ð8:201Þ

vAPL is related to wind speed v10 according to ^vAPL ¼ maxð^v10 ; 1Þ

ð8:202Þ

and F is frequency in kilohertz. At high frequency and sufficiently high wind speed (above about 30 kHz for a wind speed of 10 m/s, or above 10 kHz for 15 m/s) special attention needs to be given to the absorbing effect of near-surface bubbles. While largely responsible for the generation of wind-related noise in the first place, if present in sufficient numbers, such bubbles also absorb some of the sound before it can escape the bubble layer. The effect can be modeled by computing the attenuation along each ray path as described by APL-UW (1994). An alternative, more pragmatic approach is to cap the dipole source level so that it does not exceed the following frequency-dependent value (obtained by inspection of Fig. 17 from APL-UW, 1994, p. II-43): 10 log10 ðKmax Þ ¼ 79  20 log10 F:

ð8:203Þ

8.3.2.1.2 Low-frequency wind noise (Kuperman–Ferla measurements) The behavior of the wind noise source level at frequencies of order 1 kHz and below is more difficult to measure, and hence less well established than at higher frequency. One reason for this is masking from shipping noise. Another is that low-frequency sound can travel further, tending to complicate propagation effects, making it more difficult to separate changes in propagation loss from those in the source level. The measurements of Kuperman and Ferla (1985) exhibit a similar wind speed dependence to that of the APL formula, with a spectrum that flattens off at frequencies less than 400 Hz. The asymptotic low-frequency value can be approximated (purposefully mimicking the wind speed dependence of Equation 8.199) by K wind LF ¼

10 4:12 2:24 ^v 1:5

mPa 2 Hz 1 ;

ð8:204Þ

where the value of the constant in the denominator (1.5) is chosen to match the measured source level at 400 Hz. The level and frequency dependence of this low-frequency wind noise are not well established, with some measurements showing decreasing wind noise with decreasing frequency below about 1 kHz (Ingenito and Wolf, 1989; Cato and Tavener, 1997). An alternative wind noise model is proposed by Ma et al. (2005) for frequencies in the range 1 kHz to 50 kHz. Using an approximate relationship relating sound pressure level to source level (Equation 8.198), their Eqs. (3) and (4) can be written
 ð53:91^v10  104:5Þ 2 8 1:57 wind mPa 2 Hz 1 ð1 < F < 50; ^v10 > 2Þ: ð8:205Þ K Ma ¼  F

426 Sources and scatterers of sound

[Ch. 8

8.3.2.1.3 Proposed composite wind noise model A smooth transition between the Kuperman–Ferla and APL wind noise source spectrum models is obtained by using the following composite formula for the dipole source factor 10 4:12^v 2:24 APL K wind ¼ mPa 2 Hz 1 ; ð8:206Þ ð1:5 þ F 1:59 Þ10 0:1 which is plotted in Figure 8.17. 8.3.2.2

Rain noise source level

In the same way as for wind, rain-related noise sources—also attributed to the creation of tiny air bubbles close to the sea surface (Leighton, 1994)—are commonly assigned a dipole radiation pattern, with Equation (8.207) defining the dipole source factor K for rain 3c rain K rain ¼ W : ð8:207Þ 2 Af Examples of measurements of rain noise in the open ocean are Scrimger et al. (1989), McConnell et al. (1992), Nystuen (2001), and Ma et al. (2005). In the absence of wind, the rain noise spectrum has a strong peak at a few kilohertz. With wind, the peak, though still present, is less pronounced (Scrimger et al., 1989; Ma et al., 2005). The measurements of Nystuen show an additional dependence on the type of rain. Thus,

Figure 8.17. Wind noise areic dipole source spectrum level vs. frequency: ‘‘composite’’ ¼ evaluated using Equation (8.206); ‘‘saturated’’ ¼ composite model, capped using Equation (8.203).

Sec. 8.3]

8.3 Sources of underwater sound 427

while the most important single parameter is the rainfall rate, the rain noise spectrum also depends on drop size distribution and wind speed. An ideal predictive model would take all three parameters into consideration. Different rain noise models of varying complexity are given by APL-UW (1994), Nystuen (2001), and Ma et al. (2005). Nystuen (2001) gives a detailed algorithm with a claimed accuracy of 1 dB, but with no explicit dependence on wind speed. The simpler model of APL, described below, is based on the measurements of Scrimger et al. (1989) and includes a dependence on wind speed, but not on drop size. The measurements of Ma et al. (2005) show a dependence on wind speed for light rain, but not for heavy rain. In the APL model, valid between 1 kHz and 100 kHz, the dipole source level can be written 8 10 log10 F 1  F < 10 > > > > > < 49 log10 F  59:0 10  F < 16 rain 10 log10 K rain v10 Þ þ APL ¼ 10 log10 K 20 ðRrain ; ^ > 0 16  F  24 > > > > : 23 log10 F þ 31:7 24 < F  100, ð8:208Þ the value at 20 kHz is given by the following function of rain rate Rrain and wind speed
 Rrain rain 10 log10 K 20 ðRrain ; UÞ ¼ bðUÞ þ aðUÞ log10 min ; 10 ð8:209Þ 1 mm/h and F is the frequency in kilohertz as before. Finally, the functions aðUÞ and bðUÞ are 8 25:0 U  1:5 > < aðUÞ ¼ 5:0 þ 5:7ð5:0  UÞ 1:5 < U < 5:0 ð8:210Þ > : 5:0 U  5:0 and 8 41:6 U  1:5 > < bðUÞ ¼ 50:0  2:4ð5:0  UÞ 1:5 < U < 5:0 ð8:211Þ > : 50:0 U  5:0. In this model the dipole source level is independent of U in the limits of both high and low wind speed. These limiting cases are plotted in Figure 8.18 as dashed and solid lines, respectively, for rain rates between 2 mm/h and 10 mm/h. 8.3.2.3

Shipping noise source level

In Section 8.3.1, individual ships were considered as discrete sources of background noise. It can be useful to think of distant shipping lanes as continuous (sheet or line) sources. In the following a group of ships is represented first by a sheet of monopole sources and then by a sheet of dipoles.

428 Sources and scatterers of sound

[Ch. 8

Figure 8.18. Rain noise areic dipole source spectrum level vs. frequency, evaluated using Equation (8.208) for wind speed up to 1.5 m/s (solid lines) and exceeding 5.0 m/s (dashed lines); rain rates are 2 mm/h (lowest) to 10 mm/h (highest) in steps of 2 mm/h.

8.3.2.3.1

Monopole density

Imagine a distribution of distant ships with an areic number density N ship A . Each individual ship can be characterized as a point (monopole) source a few meters beneath the surface. If the (average) power spectral density of each monopole is W ship , the average areic spectral density due to this distribution is given by f ship ship W ship ; Af ¼ N A W f

ð8:212Þ

or, in terms of a source level in decibels mp SLAf ¼ 10 log10 N ship A þ SL f ;

where SL mp is the monopole source spectrum level of an individual ship f  c  ship SL mp ¼ 10 log W : 10 f 4 f

ð8:213Þ

ð8:214Þ

8.3.2.3.2 Dipole density The field radiated by each monopole source interferes with its surface reflection in such a way as to create a dipole radiation pattern at low frequency. The spectral radiant intensity at grazing angle  due to this dipole is related to the power spectral density of the original monopole at depth d (i.e., the power that the monopole would radiate in an infinite uniform medium of the same characteristic impedance as the

Sec. 8.3]

8.3 Sources of underwater sound 429

true medium) in the following manner: W dp fO ¼

k 2 d 2 sin 2  W mp f 

ðkd < =4Þ:

ð8:215Þ

In Chapter 2 a relationship is derived between the power and radiant intensity of a dipole, which in spectral form can be written W dp fO ¼

3 sin 2  dp Wf : 2

ð8:216Þ

dp Eliminating W dp f O from Equations (8.215) and (8.216), rearranging for W f , and substituting into Equation (8.196) gives an expression for the corresponding dipole source factor mp

SL f K ship ¼ 4k 2 d 2 N ship A 10

=10

:

ð8:217Þ

Databases containing estimates of shipping density for the main global shipping lanes are described by Hamson (1997), Etter (2003), and Anon. (2003).

8.3.3

Distributed sources on the seabed (crustacea)

There are times when sound sources located on the seabed drown out the waves, especially in habitats sustaining crustacea colonies. Some species of crustacea, and snapping shrimp in particular, can cause very loud and persistent broadband noise. The sound is caused by many individuals clicking (or ‘‘snapping’’) in unison. Because of the large number of individuals involved, the net result is that of an extended source on the seabed. 8.3.3.1

Snapping shrimp

Snapping shrimp are a ubiquitous source of underwater sound in shallow water with a rock or coral bottom of depth less than 60 m, and in warm latitudes within about 35 deg latitude from the equator16 (Johnson et al., 1947; Cato and Bell, 1992). Typical reported spectral levels at 5 kHz are 60 dB to 70 dB re mPa 2 /Hz (Anon., 2003), but significantly higher and lower values are sometimes encountered, perhaps depending on the proximity of the hydrophone to the seabed. Shrimp noise is subject to up to 8 dB diurnal variation, with highest levels occurring just after sunset and before sunrise. Cato and Bell (1992) observed no significant seasonal variation. The sound creation mechanism involves the creation and subsequent collapse of a large cavitation bubble. The temperature and pressure reached inside the bubble during its collapse are so high17 that a flash of light is sometimes emitted (Lohse et al., 2001). 16 17

More precisely, within latitudes whose winter temperature does not fall below 11  C. The estimated maximum temperature inside the cavitation bubble exceeds 5000 K.

430 Sources and scatterers of sound

[Ch. 8

According to a laboratory experiment by Au and Banks (1998), a single shrimp snap has an energy source level of between 127 dB and 135 dB re mPa 2 m 2 s. Also reported is the peak-to-peak source level SLp-p (see Box), the values of which are between 183 dB and 189 dB re mPa 2 m 2 , depending on the size of the claw. Ferguson and Cleary (2001) obtain similar values from in situ measurements. Peak acoustic pressures of up to 80 kPa are reported by Versluis et al. (2000) at a distance of 4 cm, making snapping shrimp one of the loudest animals in the sea. The frequency spectrum of shrimp noise covers a very wide frequency band. The spectral density falls off slowly from its peak at about 2 kHz, with significant contributions remaining even up to 200 kHz, as illustrated by Figure 8.19.

8.3.3.2

Other crustaceans

Other species of crustacean known as sources of underwater sound, though less well studied than snapping shrimp, include mussels (APL-UW, 1994) and spiny lobsters (Latha et al., 2005; Patek et al., 2009). Sounds made by crustaceans are reviewed by Schmitz (2002).

Figure 8.19. Measured waveform and frequency spectrum of a single shrimp snap. The peak in the spectrum at 2 kHz is due to the 400 ms delay between the precursor and the main arrival (reprinted with permission from Au and Banks, 1998, # American Institute of Physics).

Sec. 8.4]

8.4 References 431

Peak-to-peak, zero-to-peak, and peak-equivalent RMS source levels The term peak-to-peak (p-p) source level, abbreviated SLp-p , is used to mean 10 times the base-10 logarithm of the squared difference between the maximum and minimum pressure in a short impulse-like wave form, measured in the far field of the source and scaled to a standard reference distance from the source of rref ¼ 1 m. If the far-field (and free-field) measurement distance is s0 , it is common practice to obtain the source level by multiplying the measured pressure by a factor of s0 =rref . The assumptions implied by this conversion are that spherical spreading holds and that the waveform does not change in shape or duration. With these assumptions, SLp-p can be written as18 SLp-p ¼ 10 log10 ðs 20 fmax½q0 ðtÞ  min½q0 ðtÞg 2 Þ

dB re mPa 2 m 2 : ð8:218Þ

For the special case of a sinusoidal wave form, SLp-p is related to the definition introduced in Chapter 3, which is based on RMS pressure and denoted SLRMS here for clarity (elsewhere it is simply SL), according to SLp-p ¼ SLRMS þ 10 log10 8  SLRMS þ 9:0

dB re mPa 2 m 2 : ð8:219Þ

For this reason, peak pressures are sometimes reported as peak-equivalent RMS values, denoted SLpeRMS and defined as (Møhl et al., 2000) SLpeRMS
SLp-p  10 log10 8  SLp-p  9:0

dB re mPa 2 m 2 : ð8:220Þ

Also used, especially in the context of explosive or seismic survey sources, is the zero-to-peak source level, SLz-p , which, given the same assumptions as above can be defined as SLz-p ¼ 10 log10 ½s 20 maxjq0 ðtÞj 2  dB re mPa 2 m 2 :

ð8:221Þ

Conversions between these different measures of source level are discussed in Chapter 10 for a selection of representative pulse shapes.

8.4

REFERENCES

Ainslie, M. A. (1992) The sound pressure field in the ocean due to bottom interacting paths, PhD thesis, University of Southampton. Ainslie, M. A. (1995) Plane wave reflection and transmission coefficients for a three-layered elastic medium, J. Acoust. Soc. Am., 97, 954–961. [Erratum, ibid., 105, 2053 (1999).] Ainslie, M. A. (2003) Conditions for the excitation of interface waves in a thin unconsolidated sediment layer, J. Sound Vib., 268, 249–267. Ainslie, M. A. (2005a) The effect of centimetre scale impedance layering on the normalincidence seabed reflection coefficient, in N. G. Pace and P. Blondel (Eds.), Boundary 18

There is some ambiguity in Equation (8.218) unless a time window is specified. It is normal practice to interpet the right-hand side as meaning the difference between the largest positive excursion and the negative peak immediately following it (or, if greater, the difference between the largest negative excursion and the positive peak following that).

432

Sources and scatterers of sound

[Ch. 8

Influences in High Frequency, Shallow Water Acoustics (pp. 423–430), University of Bath, Bath, U.K. Ainslie, M. A. (2005b) Effect of wind-generated bubbles on fixed range acoustic attenuation in shallow water at 1–4 kHz, J. Acoust. Soc. Am., 118, 3513–3523. Ainslie, M. A. (2007) Observable parameters from multipath bottom reverberation in shallow water, J. Acoust. Soc. Am., 121, 3363–3376. Ainslie, M. A. and Harrison, C. H. (1989) Reflection of sound from deep water inhomogeneous fluid sediments, in A. J. Cohen (Ed.), Proc. Annual Meeting of the Canadian Acoustical Association, Halifax, Nova Scotia, October 16–19 (pp. 12–17). Andrew, R. K., Howe, B. M., Mercer, J. A., and Dzieciuch, M. A. (2002) Ocean ambient sound: Comparing the 1960s with the 1990s for a receiver off the Californian coast, Acoustics Research Letters Online, 3, 65–70. Anon. (1946) Physics of Sound in the Sea (NAVMAT P-9675, Summary Technical Report of Division 6, Volume 8), National Defense Research Committee, Washington, D.C. [Reprinted by Department of the Navy Headquarters Naval Material Command, Washington. D.C., 1969.] Anon. (2003) Ocean Noise and Marine Mammals (National Research Council of the National Academies), The National Academies Press, Washington, D.C. APL-UW (1994) APL-UW High-frequency Ocean Environmental Acoustic Models Handbook (APL-UW TR9407, AEAS 9501 October), Applied Physics Laboratory, University of Washington, Seattle, WA, available at http://staff.washington.edu/dushaw/epubs/ APLTM9407.pdf Arveson, P. T. and Vendittis, D. J. (2000) Radiated noise characteristics of a modern cargo ship, J. Acoust. Soc. Am., 107, 118–129. Au, W. W. L. (1993) The Sonar of Dolphins, Springer Verlag, New York. Au, W. W. L. (1996) Acoustic reflectivity of a dolphin, J. Acoust. Soc. Am., 99, 3844–3848. Au, W. W. L. and Banks, K. (1998) The acoustics of the snapping shrimp Synalpheus parneomeris in Kaneohe Bay, J. Acoust. Soc. Am., 103, 41–47. Au. W. L., Ford, J. K. B., Horne, J. K., and Newman Allman, K. A. (2004) Echolocation signals of free-ranging killer whales (Orcinus orca) and modeling of foraging for chinook salmon (Oncorhynchus tshawytscha), J. Acoust. Soc. Am., 115, 901–909. Axelsen, B. E., Bauleth-d’Almedia, G., and Kanandjembo, A. (2003) In situ measurements of the acoustic target strength of Cape horse mackerel Trachurus trachurus capensis off Namibia, Afr. J. Mar. Sci., 25, 239–251. Benoit-Bird, K. J., Gilly, W. F., Au, W. W. L., and Mate, B. (2008) Controlled and in situ target strengths of the jumbo squid Dosidicus gigas and identification of potential acoustic scattering sources, J. Acoust. Soc. Am., 123, 1318–1328. Bertrand, A. and Josse, E. (2000) Tuna target-strength related to fish length and swimbladder volume, ICES Journal of Marine Science, 57(4), 1143–1146. Bowen, S. P., Richard, J. C., Mancini, J. D., Fessatidis, V., and Crooker, B. (2003). Microseism and infrasound generation by cyclones, J. Acoust. Soc. Am., 113, 2562–2573. Boyle, F. A. and Chotiros, N. P. (1995) A model for high-frequency acoustic backscatter from gas bubbles in sandy sediments at shallow grazing angles, J. Acoust. Soc. Am., 98, 531– 541. Brekhovskikh, L. M. and Lysanov, Yu. P. (2003) Fundamentals of Ocean Acoustics (Third Edition), AIP Press Springer-Verlag, New York. Cato, D. (1993) The biological contribution to the ambient noise in water near Australia, Acoust. Australia, 20, 76–80.

Sec. 8.4]

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9 Propagation of underwater sound

So if the physics is necessarily complicated it can pay to keep the mathematics simple, giving a better chance of seeing the wood despite the trees. David E. Weston (1971) No book on sonar would be complete without a chapter on underwater sound propagation, and this is it. The subject is central to sonar performance modeling because all sound, whether contributing to the signal, ambient noise, or reverberation, must propagate through the sea before arriving at the sonar. Thus, the scope includes not only propagation loss (PL), but also the effect of sound propagation on noise level (NL), and for active sonar the reverberation level (RL) and target echo level (EL). These terms were all introduced in Chapter 3, where they were applied to simplified sonar problems. The present chapter adds more realism by describing the effects of a reflecting seabed and a sound speed profile. Since the pioneering work of Lichte (1919),1 modeling of underwater sound propagation has increased steadily in sophistication, such that today a variety of reliable computational models exists (oalib, www). The interested reader is referred to Jensen et al. (1994) for a thorough description of the different computational techniques used by these models and to Brekhovskikh and Lysanov (2003) for the theoretical foundations of ocean acoustics. It is inevitable that some material from these books is duplicated here, but an attempt is made to keep such duplication to a minimum. (For example, the reader is assumed to be familiar with the basic concepts of image theory, ray theory, and normal-mode theory—Jensen et al., 1994). The emphasis here is placed on providing physical explanations for the effects, with simple 1 Lichte was the first scientist to recognize the effect that pressure, temperature, and salinity gradients would have on the propagation of sound in the sea (see Chapter 1).

440 Propagation of underwater sound

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estimates of their magnitude where feasible, drawing heavily in so doing from the ideas of Weston (see, e.g., Weston, 1960, 1979, 1980, 1994). In the passive sonar equation there are two terms, namely PL and NL, that are strongly affected by propagation. These two terms are considered first (Sections 9.1 to 9.2). Both are relevant also to the active sonar equation, with the EL (Section 9.3) and RL (Section 9.4) terms also influenced by propagation effects. The signal-toreverberation ratio is considered in Section 9.5.

9.1

PROPAGATION LOSS

The purpose of this section is to illustrate the influence on the propagation of underwater sound of important mechanisms omitted from the simpler description presented in Chapter 2. The main new effects considered are reflection of sound from the seabed and refraction in the water due to variations of sound speed with depth. Also relevant are horizontal gradients in sound speed and water depth. Although horizontal sound speed gradients are small compared with vertical ones, their effects become increasingly important at increasing distance from the source. These longrange effects (see, e.g., Jensen et al. 1994, p. 36, 323ff, 397ff ) are outside the present scope. Also excluded are time-dependent effects due to the motion of surface waves, currents, eddies, and internal waves (the Doppler shift associated with a moving target is described in Chapter 6). The emphasis here is on simple analytical formulas rather than exact solutions, whether analytical or numerical. The approximate analytical solutions are not intended to replace those of numerical models, but to complement them by explaining trends and promoting insight. 9.1.1 9.1.1.1

Effect of the seabed in isovelocity water Deep water

Compared with the sea surface, the seabed is a poor reflector of sound whose reflection coefficient depends strongly on angle. The proportion of sound energy reflected increases from around 1% to 10% at normal incidence to 80% to 100% at angles close to grazing incidence. The influence of the reflected sound is especially important in shallow water and for sound traveling close to the horizontal direction, which becomes trapped between the sea surface and seabed. The geometry of the problem is illustrated by Figure 9.1. The deep-water assumption implies that, in relative terms, both source and receiver are close to the sea surface. It then becomes convenient to collect ray paths in groups of four with an identical number of bottom reflections, as the individual rays in such a group follow very similar trajectories. Using m to denote the number of bottom interactions, the first such group (m ¼ 1) follows a V-shaped path, corresponding

Sec. 9.1]

9.1 Propagation loss

441

Figure 9.1. Diagrammatic ray paths illustrating the geometry for bottom reflections in deep water (reprinted with permission from Ainslie, 1993, # American Institute of Physics).

to the left-hand picture of Figure 9.1, and the second one (m ¼ 2) a W shape (righthand picture).2 If the pressure field is written as a sum over image contributions, the similarity between paths simplifies the analysis considerably. An example calculation is presented in Figure 9.2 with the density of the seabed equal to 1.222 relative to that of water, and no change in sound speed. These values are deliberately chosen to provide a weak reflection (only 1 % of the incident energy is reflected for this problem), so that the second and subsequent reflections are heavily damped, making it easier to study the first reflection separately from the others. The water depth is 1000 m, which while less deep than the main oceans is deep enough to illustrate the effects of interest here. The upper graph of Figure 9.2 (solid line) shows PLðrÞ calculated using the fastfield program SAFARI (Schmidt, 1988; Jensen et al., 1994) for this case. PL increases systematically with increasing range, and superimposed on this trend is a beat pattern of increasing period and amplitude. This result can be understood in terms of

2

The water is considered here to be sufficiently deep that only a small number of bottomreflected paths contributes to the total field, making the effects of the seabed easier to understand.

442 Propagation of underwater sound

[Ch. 9

Figure 9.2. Propagation loss [dB re m 2 ] vs. range for reflecting seabed (
sed =
w ¼ 1.222) at f ¼ 250 Hz. Upper: propagation loss (SAFARI) and BL, LM components; lower: components BL (blue curve: Equation 9.15), LM (green curve: Equation 9.1) and their sum (INSIGHT).

Sec. 9.1]

9.1 Propagation loss

443

interference between Lloyd mirror paths (abbreviated LM) and bottom-reflected ones (abbreviated BL), plotted separately and color-coded green (LM) and blue (BL) in the lower graph, calculated using the INSIGHT model (Ainslie et al., 1996). It can be seen that the dominant contributions come from LM at short range and from BL at long range. The beat pattern can be understood as resulting from interference between these. For example, its dynamic range is greatest when the separate LM and BL contributions are equal (e.g., at 4 km or 5.2 km). The shape of the individual LM and BL components is explained in Sections 9.1.1.1.1 and 9.1.1.1.2, respectively. 9.1.1.1.1 Lloyd mirror Neglecting attenuation and assuming a perfectly reflecting sea surface, the complex LM pressure can be written as the following sum of two images (Chapter 2) ! pffiffiffi e iks e iksþ  e i!t ; ð9:1Þ pLM ðr; zÞ ¼ 2s0 p0 s sþ where s ¼ s ðr; zÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ ðz  z0 Þ 2 :

ð9:2Þ

If the depths z and z0 are both small compared with the range r, then s  r þ

ðz  z0 Þ 2 ; 2r

ð9:3Þ

and hence p i kzz0 iðkr!tÞ pffiffiffiLM  2 sin e : r r 2p0 s0

ð9:4Þ

Thus, the propagation factor (the squared modulus of Equation 9.4) is FLM 

4 kzz0 sin 2 : 2 r r

ð9:5Þ

At long range, the sine function can be replaced by its argument, giving an r 4 dependence on range (40 log10 r in dB), and explaining the shape of the LM curve of Figure 9.2. 9.1.1.1.2 Bottom-reflected paths For each value of m, in addition to the obvious straight-there-and-back bottomreflected (BL) ray path that does not interact with the sea surface, there are two others that reflect once from the surface and one that does so twice. These four multipaths are illustrated in the close-up of Figure 9.1. A convenient expression for the sum of images, derived by Harrison (Harrison, 1989; Ainslie and Harrison, 1990), is

444 Propagation of underwater sound

[Ch. 9

reproduced below. The mth BL term can be written   pm ðr; zÞ exp ik0 sþ exp ik0 sþ m m1 exp ik0 s 2 exp ik0 sþþ pffiffiffi ¼ RB RS þRS þRS þR S e i!t ; s sþ sþ sþþ 2p0 s0 ð9:6Þ where s 2 ¼ ð2mH  z0  zÞ 2 þ r 2

ð9:7Þ

s 2þ ¼ ð2mH þ z0  zÞ 2 þ r 2 :

ð9:8Þ

and Putting RS ¼ 1 gives (correcting a sign error in Eq. A1.5 of Ainslie and Harrison, 1990) pm ðr; zÞ cos 0 pffiffiffi ¼ 4iðRB Þ m ðcos z cos z0 sin  þi sinz sinz0 cos Þ expðik0 SÞ e i!t ; r 2p0 s0 ð9:9Þ where S¼

r þ rþ þ rþ þ rþþ ; 4

 ¼ k0 sin 0 ; z z ¼ 0 r 2mH tan 0 ¼ r

ð9:10Þ ð9:11Þ ð9:12Þ ð9:13Þ

and ¼ k0 cos 0 :

ð9:14Þ

From Equation (9.9) it follows that (Harrison, 1989) FBL ¼ 16R 2m B

cos 2 0 ðsin 2 z0 sin 2 z cos 2  þ cos 2 z0 cos 2 z sin 2 Þ: r2

ð9:15Þ

A simpler version, valid for near-surface source and receiver, and more convenient for comparison with later expressions for bottom-refracted contributions, follows by assuming  is negligible. The result is   pm ðr; zÞ cos 0 k0 r pffiffiffi  4ð1Þ m R m sin z sin z exp i e i!t ; ð9:16Þ B 0 r cos 0 2p0 s0 and hence FBL  16R 2m B

cos 2 0 2 sin z0 sin 2 z: r2

ð9:17Þ

The blue curve of Figure 9.2 (lower graph) is PLBL ¼ 10 log10 FBL for the case m ¼ 1. The beating pattern in the blue curves is due to interference between the four multi-paths. The deep nulls are places at which either z or z0 is an integer multiple

Sec. 9.1]

9.1 Propagation loss

445

of , so that the right-hand side of Equation (9.16) vanishes. For higher order paths (m 2) the propagation loss exceeds 110 dB re m 2 in the graph, and these paths are consequently too weak to be visible. Finally, the uppermost line is the incoherent sum of both contributions (i.e., 10 log10 ðFLM þ FBL Þ), color-coded according to the larger of the individual propagation factors. 9.1.1.1.3 Bottom-refracted paths If 1 % of the energy is reflected in the above example, consider now the fate of the remaining 99 %. If the seabed were an infinite uniform half-space with density and sound speed independent of depth, the sound would continue unimpeded on its downward path forever. In practice there are changes in impedance, some abrupt and some gradual, that cause some of the sound to be reflected. Further the speed of sound tends to increase with increasing depth in the seabed. This sound speed gradient, typically of order 1 s1 (Chapter 4), has an important effect on low-frequency sound because it refracts the sound upwards, eventually returning it to the water if the initial angle is not too steep, after following a U-shaped path in the sediment as illustrated by Figure 9.3. High-frequency sound is refracted in exactly the same way, but the effects are less important due to the increased attenuation (see graphs of reflection loss vs. angle and frequency in Chapter 8). Notice the shadow near the middle of the ray trace (Figure 9.3), and the region to the right of this filled by bottom-refracted (BR) paths. These two regions are separated by a line called a caustic, along which an infinite ray density is reached. The sound field has a maximum close to this line, and a dramatically different character either side of it. For a numerical example (see Figure 9.4), we choose a sediment sound speed gradient c 0 ¼ 1/s and attenuation sed ¼ 0.03 decibels per wavelength. Other parameters are as Figure 9.2. The expected range to the caustic, using Equation (9.20) below, is 4.9 km. At short range, before the caustic, there is little difference

(a)

(b)

Figure 9.3. Bottom-refracted (BR) ray paths travel through the sediment and form a caustic in the reflected (i.e., bottom-refracted) field (reprinted with permission from Ainslie, 1993, # American Institute of Physics).

446 Propagation of underwater sound

[Ch. 9

Figure 9.4. Propagation loss [dB re m 2 ] vs. range for a reflecting and refracting seabed. Upper: propagation loss (SAFARI) (reprinted with permission from Ainslie, 1993, # American Institute of Physics); lower: LM and BL components (from Figure 9.2), BR (red curve: Equation 9.19—in this example, Equation 9.29 is not needed for the field through the caustic itself because the steep caustic paths are absorbed during their transit through the sediment), and their sum (INSIGHT). (c 0 ¼ 1/s, sed ¼ 0.03 dB/ ; other parameters as Figure 9.2.)

Sec. 9.1]

9.1 Propagation loss

447

between the two graphs. Beyond this point, however, the total PL in Figure 9.4 is strongly affected by the arrival of BR rays, contributions from which are shown in red in the lower graph of Figure 9.4. These paths completely dominate the field between 5 km and 10 km, to the right of the caustic. Ensonified region. The sum of images used for BL cannot be applied to BR paths because of the sound speed gradient in the sediment. Suitable alternative methods such as ray or mode theory are described by Jensen et al. (1994). Ainslie (1993) uses normal mode theory with the stationary phase approximation (Appendix A) to derive the result pm ðr; zÞ ¼ pþ þ p ; ð9:18Þ where (for r > mrc )

 1=2   p m m cos   þ 1 i!t pffiffiffi ¼ 4ð1Þ ½RB ð Þ   sin  z0 sin  z exp ið  =4Þ e r  1  2p0 s0  ð9:19Þ

and rc is the caustic range

 rc ¼ 4

c0 H c0

1=2 :

ð9:20Þ

Comparing Equation (9.19) with Equation (9.16), the main difference is the factor containing terms of the form ð  1Þ 1=2 . This factor quantifies the bunching up of rays in the vicinity of the caustic. At the caustic itself the denominator vanishes and the factor becomes infinite. The true pressure field, which must be finite, then needs to be calculated a different way, as explained in the section entitled caustic and shadow region below. The amplitude reflection coefficient RB is given by    2!" 2!

RB ðÞ ¼ exp  0 YðÞ þ i 0 ½YðÞ  sin   ; ð9:21Þ c c 2 where  

 YðÞ ¼ loge tan þ ð9:22Þ 4 2 and " is the fractional imaginary part of the sediment wavenumber "¼

sed ; 40 log10 e

ð9:23Þ

with sed in decibels per wavelength. The phase term is given by  ¼  r þ 2m H þ =4;

ð9:24Þ

 ¼ k0 sin 

ð9:25Þ

 ¼ k0 cos 

ð9:26Þ

where and

448 Propagation of underwater sound

[Ch. 9

are the vertical and horizontal wavenumbers; and  is the ray grazing angle of each of the two branches of the caustic  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c0 ð9:27Þ tan  ¼ r  r 2  mr 2c : 4mc0 The phase of Equation (9.19) also includes the term  =4. The origin of this term is related to the transition of individual rays through the caustics. Each ray passes through one caustic per cycle, and each traversal results in a phase change of =2. The phase difference arises because the steeper ray has traversed one fewer caustic than the shallow one, with a consequent =2 lag in phase (Boyles, 1984, p. 235). One final parameter, needed for the evaluation of Equation (9.19), is c  ¼ 0 0 tan 2  : ð9:28Þ cH Caustic and shadow region. Equation (9.19) is valid in the ensonified region to    þ 1  in the right of the caustic in Figure 9.3, but not at the caustic itself. The ratio     1 Equation (9.19) is a measure of ray density. At the caustic itself, this term becomes infinite and the method used to derive Equation (9.19) breaks down. It can be shown using second-order stationary phase theory that the field in the immediate vicinity of the caustic is described by an Airy function (see Appendix A). Through the caustic itself, the field is given by (Ainslie, 1993)  1=2 pm 2

m mQ pffiffiffi ¼ ð1Þ ½RB ðc Þ sin c z0 sin c z AiðÞ e iðc !tÞ ; ð9:29Þ r H 2p0 s0 c where the phase term is c ¼ c r þ 2mc H þ =4

ð9:30Þ

and the wavenumber components are c ¼ k0 sin c

ð9:31Þ

c ¼ k0 cos c ;

ð9:32Þ

and where c is the angle of the caustic grazing ray  0 1=2 cH tan c ¼ : c0

ð9:33Þ

The Airy function argument, which determines the position and width of the caustics, is ! r  mrc 4H cos 2 c ! 2 c 0 1=3 ¼ : ð9:34Þ rc c0 2m This Airy function provides a smooth transition between the oscillatory field in the

Sec. 9.1]

9.1 Propagation loss

449

ensonified region to the right of the caustic ( > 0) and the shadow to the left of it ( < 0). Equation (9.29) is valid for ranges close to the integer multiples of the caustic range. There is an assumption in its derivation that the density is uniform everywhere and the sound speed is continuous across the water–sediment boundary. The small step in density for the present example is neglected on the grounds that only 1 % of the energy is reflected, so its effect on the result is minor.

Sensitivity to seabed parameters. The sensitivity of the sound field to seabed parameters is considered next (see Figure 9.5). The first two graphs (upper row) are simple to understand: increasing the sediment attenuation (upper left graph) turns off BR paths (in red), while decreasing the density ratio
sed =
w (upper right) turns off BL (in blue) because of the decreasing reflection coefficient, while leaving the BR contribution approximately unchanged. The second row is more subtle. The effect of csed =cw on BL is similar to that of
sed =
w , and it also influences the angles of the BR paths through Snell’s law, and hence the interference patterns associated with these. The main effect of increasing c 0 (lower right) is to reduce the caustic range (Equation 9.20), in this example to about 4 km.

9.1.1.2

Shallow water

In the world’s oceans there is a clear distinction between the deep ocean of depth 3 km to 5 km (most of it) and the continental shelf, of depth 20 m to 200 m, separated by regions of relatively steep slopes. The close proximity of the seabed in shallow water means that its acoustical properties play a central role in shallow-water propagation. The reason why the seabed is so important is the ability of the sea to trap sound between the seabed and sea surface, forming a waveguide able to carry sound over many kilometers. Each time sound is reflected from the seabed a little energy is lost, eventually limiting low-frequency propagation in shallow water. The proportion of energy lost at each reflection depends on the bottom type as well as the grazing angle and acoustic frequency.3 Propagation in shallow water can also be strongly frequency-dependent and the main reason for this is the existence of a minimum propagation frequency, known as the cut-off frequency, below which waveguide propagation is not supported. The discussion below assumes initially that the frequency is above this cut-off frequency, which is the subject of Section 9.1.1.2.5. A second assumption is that the field is not influenced by coherent interference effects due to cancellation between upwardtraveling and downward-traveling paths due to phase reversal at the sea surface. This second assumption is addressed in Section 9.1.1.2.6. 3 Thus, a complicating feature of shallow-water propagation is the great variety of bottom types encountered, with mud, sand, rock, and gravel all common, sometimes in close proximity to one another.

450 Propagation of underwater sound

[Ch. 9

Figure 9.5. Propagation loss [dB re m 2 ] vs. range/km for a reflecting and refracting seabed: sensitivity to sed (upper left),
sed =
w (upper right), csed =cw (lower left), and c 0 (lower right). Other parameters as Figure 9.4 (INSIGHT).

Sec. 9.1]

9.1 Propagation loss

451

452 Propagation of underwater sound

[Ch. 9

9.1.1.2.1 Multipath propagation An important consequence of the factor-40-or-so difference in depth between deep and shallow water is that, at any given range, there are many more ray arrivals to consider in shallow water. For calculations in shallow water, it is helpful to express the total field as a sum over the energy contribution from each multipath (MP), in the form 1 X cos 2 m FMP ¼ 4 jRðm Þj 2m ; ð9:35Þ 2 r m¼1 where r tan m : 2H

m¼

ð9:36Þ

The validity of the energy sum in Equation (9.35) requires the phase of the multipaths to be randomly related to one another, in such a way that there is no systematic constructive or destructive interference. The approximation breaks down close to a smooth boundary (such as the sea surface), where pairs of rays tend to arrive with almost identical pathlengths—their phase differing only by the phase change at the reflecting boundary. A rule of thumb for the use of Equation (9.35) is that the distance of both source and receiver from the sea surface must exceed =sin , where is the acoustic wavelength and  the grazing angle of the corresponding ray arrival. Approximating the sum as an integral over a continuum in m and changing the integration variable to  using dm ¼

r d; 2H cos 2 

it follows that FMP 

2 rH

ð =2

jRðÞj 2m d:

ð9:37Þ

ð9:38Þ

0

More generally, it is convenient to write F in terms of the differential propagation factor GðÞ ð =2 F¼ GðÞ d; ð9:39Þ 0

where GðÞ d is the contribution to the propagation factor from ray paths at elevation angles between  and  þ d. For this example, it is given by GðÞ 

2 jRðÞj 2m : rH

ð9:40Þ

9.1.1.2.2 Spherical and cylindrical spreading regions If the reflection coefficient is approximated as a Heaviside step function, such that  1 < c jRðÞj  ð9:41Þ 0  > c,

Sec. 9.1]

9.1 Propagation loss

453

Equation (9.38) simplifies to FMP 

2 c 1 r : H

ð9:42Þ

This 1=r behavior (10 log10 r in dB) is known as cylindrical spreading (CS) because of the cylindrical geometry that leads to it, as clarified below. For spherical spreading (SS), the area into which the sound spreads is a sphere of radius r ASS ¼ 4 r 2 :

ð9:43Þ

In shallow water, sound cannot spread into an indefinitely large sphere, but is limited instead to a cylinder of height H and radius r, so that ACS ¼ 2 rH;

ð9:44Þ

leading to the 1=r dependence in Equation (9.42). 9.1.1.2.3 Mode-stripping region A modification to cylindrical spreading is needed for long ranges once the reflection losses due to multiple bottom reflections begin to accumulate. An improved approximation for FMP can be obtained by using a more realistic approximation for RðÞ, taking into account reflection losses near grazing incidence, of the form  expðÞ  < c jRðÞj  ð9:45Þ 0  > c, where  is the rate of increase of reflection loss with the angle in units of nepers per radian. It is referred to below as the ‘‘reflection loss gradient’’. Substitution of Equation (9.45) into Equation (9.38) yields FMP ¼

2eff

1=2 c erf ; rH 2eff

where

 eff ¼

H 4r

ð9:46Þ

1=2 :

ð9:47Þ

Equation (9.46) contains the cylindrical spreading result (Equation 9.42) as a special case in the short-range limit ( c  eff ). At long range ( c  eff ) it becomes FMP 

2eff ; rH



1=2

or, equivalently, FMP 

H

r 3=2 :

ð9:48Þ

ð9:49Þ

This 15 log10 r dependence on range is known as mode stripping because it results from the gradual erosion of steep ray paths (high-order modes) after multiple bottom reflections.

454 Propagation of underwater sound

[Ch. 9

The transition between Equation (9.42) and Equation (9.49) (the range at which cylindrical-spreading and mode-stripping contributions are equal) occurs at a range rCS given by

H rCS ¼ : ð9:50Þ 4 2c To illustrate the transition from cylindrical spreading to mode stripping we now consider shallow-water propagation for two different bottom types, sand and mud, the relevant properties of which are summarized in Table 9.1. The reflection loss gradient for sand or coarse silt is given by (see Chapter 8) sand ¼ 2"


sed cos 2
w sin 3

c

ð9:51Þ

;

c

with a typical value of between 0.1 Np/rad and 1.0 Np/rad. For mud (clay or fine silt) there is no critical angle, so Equation (9.51) is not appropriate. Instead the reflection coefficient for a refracting sediment can be used from Section 9.1.1.1.3. For small , Equation (9.21) implies   2!" jRðÞj  exp  0  ð9:52Þ c and hence 2!" mud ¼ 0 : ð9:53Þ c According to this result, if " is a constant the reflection loss for the mud case is proportional to frequency, as indicated by the corresponding entry in Table 9.1. The Table 9.1. Characteristic properties from Chapter 4 of medium sand (Mz ¼ 1.5) and mud (Mz ¼ 8). a Sand

Mud

Grain size Mz

1.5

8

csed =cw

1.20

1.00


sed =
w

2.1

1.4

sed =(dB/ )

0.88

0.09

0.0161

0.00165

0.0

1.0

0.28 (Equation 9.51)

0.021f^ (Equation 9.53)

" ¼ sed =40 log10 e (Equation 9.23) c 0 =s1 =Np rad1

a The mud sediment (Mz ¼ 8) has properties intermediate between those of very fine silt and coarse clay.

Sec. 9.1]

9.1 Propagation loss

455

precise frequency dependence of " is the subject of ongoing research (Buchanan, 2006). The work of Hamilton (1980, 1987), and Kibblewhite (1989) demonstrates that the general trend is consistent with the assumption of attenuation being proportional to frequency (consistent with constant ") over a frequency range of about five decades between 0.01 kHz and 1000 kHz. However, the existence of such a trend does not preclude departures from linearity across a more limited frequency range. The accuracy of these expressions for  (Equation 9.51 for sand and Equation 9.53 for mud) should not be taken too seriously. Both are approximations intended to illustrate the difference in behavior between sand and mud at low frequency in a qualitative manner. For example, Equation (9.51) tends to overestimate the reflection loss for sand sediments at grazing angles close to c , as illustrated by Figure 9.6. In practice, the effect is less serious than it seems because it is the contribution from near-grazing angles, for which the error is small, that provides most of the energy at long range. A more precise calculation of reflection loss can be found in Chapter 8 for sand (Mz ¼ 2.5) and mud (Mz ¼ 8.5). The single most important parameter of Table 9.1 is the sound speed ratio. It is this parameter that determines the presence or absence of a critical angle, its magnitude if present, and hence in broad terms the overall reflectivity of the seabed. If there is no critical angle, the most important parameter then becomes the ratio sed =c 0 , which determines the loss per cycle due to absorption in the sediment. The importance of the seabed for shallow-water propagation is illustrated by Figure 9.7, which shows propagation loss vs. range and bottom reflection loss vs. angle for sand and mud sediments at a frequency of 250 Hz. In all other respects

Figure 9.6. Reflection loss [dB] vs. angle for sand (1.5) comparing the Rayleigh reflection coefficient (solid red) with the approximation of Equation 9.51 (dotted blue).

456 Propagation of underwater sound

[Ch. 9

Figure 9.7. Propagation loss [dB re m 2 ] vs. range (upper) and reflection loss [dB] vs. angle (lower) for sand (thick solid lines) and mud (thin lines) in shallow water at frequency 250 Hz (INSIGHT).

Sec. 9.1]

9.1 Propagation loss

457

the two environments are identical. The reflection loss for mud is much higher than that for sand, and this manifests itself as a correspondingly higher propagation loss. 9.1.1.2.4

Single-mode region

Up to this point the field has been described without taking into account the discrete nature of the normal-mode spectrum. Pairs of plane waves traveling in the ocean waveguide (one in the upward direction and one downward) combine to form ‘‘modes’’ if their phases are aligned in such a way as to match the boundary conditions of the waveguide (Jensen et al., 1994). The alignment only occurs for certain preferred directions that depend on these boundary conditions. The density of modes (i.e., the number of preferred directions per unit angle) increases with increasing frequency such that at sufficiently high frequency the discrete nature of the modes may be disregarded. At low frequency, however, there are some features of propagation that cannot be explained without invoking individual modes, and the single-mode region is one such feature, as follows. The process of mode stripping has the effect of gradually reducing the number of modes contributing to the field, starting by removing the highest order modes and continuing until only a few low-order modes remain. Eventually only one mode is left and at this point the mode-stripping regime ends—there are no more modes to strip away—and the single-mode regime begins. Beyond this point, the field is dominated by the lowest order mode and can be approximated by the formula ! 4  2 2 z0 2 z sin exp  r ; ð9:54Þ FMP  2 sin He He H er 4H 3e where He is the effective water depth, which is the depth at which a pressure release boundary appears to exist, a short distance beneath the true seabed (Weston, 1960, 1994), given by4
sed =
w He ¼ H þ : ð9:55Þ ð!=cw Þ sin c The transition from Equation (9.49) to Equation (9.54) occurs when the effective angle (Equation 9.47) falls to a value between the propagation angle of the first and second modes. The propagation angle for the nth mode is approximately n

n  : ð9:56Þ ð!=cw ÞHe The transition range between mode-stripping (MS) and single-mode regions can be estimated by equating n and eff with n ¼ 3=2 (halfway between integers 1 and 2) rMS 

k 2 H 3e : 9

ð9:57Þ

4 For an extension of this concept to include the effects of sediment shear waves, see Chapman et al. (1989).

458 Propagation of underwater sound

[Ch. 9

The single-mode region is usually a feature of low-frequency propagation only, say below 100 mHz in deep water and about 10 Hz in shallow water. In very shallow water, however, the single-mode region can be important to higher frequencies, up to about 1 kHz for a water depth of 10 m. 9.1.1.2.5

Cut-off frequency

An important feature of shallow-water propagation, mentioned at the start of Section 9.1.1.2, is the existence of a waveguide cut-off frequency, below which ducted propagation does not occur. The condition for a cut-on duct is that at least one propagating mode exists. The total number of propagating modes N can be estimated by requiring that the product of effective water depth He and wavenumber be an integer multiple of , that is, ð!=cw ÞHe sin

c

¼ N :

ð9:58Þ

Thus, the requirement for at least one cut-on mode (i.e., N 1) translates to f > fc ;

ð9:59Þ

where fc is the cut-off frequency fc ¼


sed =
w c : 2 sin c H

ð9:60Þ

An alternative form is H 
sed =
w > : 2 sin c

ð9:61Þ

9.1.1.2.6 Depth dependence If the receiver is close to the sea surface, a coherent interference effect occurs between an upward-traveling path and the corresponding downward-traveling surface reflected path. As the receiver approaches the surface the path difference tends to zero and, because of the phase reversal at the surface, the phase difference to . In this situation (perfect reflection with phase reversal) the total field is proportional to the quantity WðzÞ ¼ 2 sin 2 z: ð9:62Þ The assumptions made in the flux derivation of Sections 9.1.1.2.1 to 9.1.2.1.3 amount to replacing this function by its average value, an approximation that works well almost everywhere except at the sea surface. A pragmatic version that retains the correct depth dependence at the surface without fussing about detail elsewhere is WðzÞ 

1 : 1 þ ð2 2 z 2 Þ 1

ð9:63Þ

The same logic applies at the source, such that the combined dependence on both

Sec. 9.1]

9.1 Propagation loss

459

source and receiver depth (assuming these are not coincident) is Wðz0 ÞWðzÞ 

1 1 : 1 þ ð2 2 z 20 Þ 1 1 þ ð2 2 z 2 Þ 1

ð9:64Þ

This behavior is known as ‘‘surface decoupling’’ because the pressure at the sea surface is decoupled from the rest of the medium. 9.1.2

Effect of a sound speed profile

An important characteristic of the world’s oceans is that the speed of sound in the sea is not uniform, but varies with temperature T and salinity S, both of which vary in space (especially with depth) and time. It also increases with increasing pressure P. The result of these variations in depth is called a sound speed profile. The sound speed profile can have a profound influence on the propagation of underwater sound. Horizontal gradients in S and T can occasionally be important (e.g., across fronts and eddies), but horizontal gradients are usually much smaller than vertical gradients. 9.1.2.1

Deep water

The acoustic consequences of vertical gradients are considered below. For example, assuming uniform T and S, the inexorable increase of P with depth leads to a positive sound speed gradient. This isothermal behavior is characteristic of the deep ocean at depths exceeding 2 km, as illustrated by the upper graph of Figure 9.8. This graph shows two different profiles, differing mainly in the top 75 m. The two profiles are for winter and summer conditions, and calculated using Mackenzie’s formula (see Chapter 4). An isothermal layer can also arise close to the sea surface, where it is typically caused by wind mixing, and this is illustrated by the uppermost 75 m of the winter profile (see lower graph of Figure 9.8). The uniform temperature results in a mildly increasing sound speed with depth caused by the pressure gradient. In this situation, sound rays are refracted upwards in accordance with Snell’s law, in this case forming a near-surface waveguide known as a surface duct. Surface heating can reverse this situation because an increase in surface temperature leads to a negative sound speed gradient and downward refraction for the summer profile (Figure 9.8, upper graph). In this situation, sound is then deflected away from the surface and an acoustic shadow zone forms there. These two situations, and more complicated phenomena caused by a combination of both upward and downward refraction, are described in subsequent sections. More generally, any variation of the speed of sound with depth can result in a subtle but important change in direction of ray paths through refraction. Depending on the details of the sound speed profile, regions of particularly high or low density of ray paths can form, resulting in correspondingly high or low acoustic intensity. Sound speed minima are particularly important features. Sound becomes trapped in these minima by refraction in the same way as for the surface duct. For this reason,

460 Propagation of underwater sound

[Ch. 9

Figure 9.8. Upper: sound speed profile vs. depth for (19 N, 150 E) in the northwest Pacific (see Chapter 4 and Table 9.2): summer profile (red solid) and winter profile (blue dashed); lower: zoomed sound speed profile (top 300 m).

Sec. 9.1]

9.1 Propagation loss Table 9.2. Sound speed profiles for the northwest Pacific location, as plotted in Figure 9.8 (calculated using temperature and salinity profiles from Chapter 4). Depth/m

cðzÞ/m s1 Summer

Winter

0.

1543.503

1536.640

10.

1543.551

1536.664

20.

1543.583

1536.761

30.

1543.483

1536.848

50.

1542.269

1536.999

75.

1538.729

1536.931

100.

1535.677

1536.078

125.

1532.834

1533.442

150.

1530.063

1530.022

200.

1523.705

1523.186

250.

1518.034

1517.543

300.

1513.976

1513.821

400.

1504.685

1504.478

500.

1494.562

1493.563

600.

1487.147

1486.780

700.

1484.019

1483.651

800.

1482.513

1482.765

900.

1481.934

1482.070

1000.

1482.083

1482.393

1100.

1482.300

1482.410

1200.

1482.737

1483.290

1300.

1483.392

1484.274

1400.

1484.199

1484.278

1500.

1485.211

1485.479

1750.

1488.094

1488.326

2000.

1490.909

1490.945

2500.

1498.109

1498.281

3000.

1506.224

1506.069

3500.

1514.722

1514.350

4000.

1523.093

1522.983

4500.

1531.909

1531.800

5000.

1540.885

1541.012

461

462 Propagation of underwater sound

[Ch. 9

the region around a sound speed minimum is known as a sound channel (or acoustic waveguide) and the minimum itself is the channel axis. Of particular importance is the behavior in the top few hundred meters, where seasonal dependence is greatest. The sound channel permits underwater sound to travel long distances, sometimes without contact with the ocean boundaries. A well-known example is associated with the formation of convergence zones in deep water, caused by the monotonic increase in sound speed at depths exceeding ca. 1 km. In the present example, the corresponding channel axis (that of the deep sound channel ) occurs at a depth of about 900 m. For the winter profile there is a second minimum at the sea surface, which is the axis of the surface duct.5 9.1.2.1.1 Examples for the northwest Pacific Ocean Despite the apparent similarity of winter and summer profiles (Figure 9.8), the small differences between them are sufficient to cause very different acoustical behavior. For the situation considered below, the main effect of refraction in the summer case is associated with the negative sound speed gradient between 30 m and 100 m depth, which has the effect of directing sound downwards towards the seabed. This downward refraction introduces shadow zones that are filled in by steep (bottom-reflected) paths, leading to increased propagation loss compared with the isovelocity case because the steeper paths experience greater reflection losses (upper graph of Figure 9.9). At ranges up to about 70 km in this graph, the predicted field is dominated by paths involving a single bottom reflection (BL1 ). Beyond this distance, only paths suffering two or more bottom reflections can reach the receiver—hence the step increase at that point.6 The seabed parameters used are the mud values from Table 9.1, which are representative of a deep-water sediment. The winter case (Figure 9.9, lower graph) involves a surface duct in the top 50 m, where a positive sound speed gradient exists, and a deep sound channel with its axis at depth 900 m (see Figure 9.8). The deep sound channel results in convergence zones appearing at integer multiples of 65 km. The difference between the summer and winter cases, which consistently exceeds 30 dB beyond 70 km, is caused by the small changes in the uppermost 100 m of the sound speed profile (see Figure 9.8). 9.1.2.1.2

Surface duct (upward refraction)

If the sound speed gradient is positive (i.e., if the sound speed increases with increasing depth), sound is refracted upwards and can become trapped near the surface. This situation is typical of winter conditions, with an isothermal near-surface layer resulting from wind-driven mixing. The positive sound speed gradient is mainly due to increasing pressure with depth. Such conditions typically result in long-range 5

The summer profile also has a minimum at the sea surface, but its consequences are minor because the gradient and thickness of the resulting duct are too small to have a significant effect on propagation in the circumstances considered. 6 Although the sharpness of this step, as predicted here by the INSIGHT model, is artificial, its presence, position, and approximate magnitude are real effects.

Sec. 9.1]

9.1 Propagation loss

463

Figure 9.9. Propagation loss [dB re m 2 ] vs. range for NWP summer (upper) and winter (lower). The thin line, computed for isovelocity water of the same depth, is the same curve in both cases. The acoustic frequency is 1.5 kHz (INSIGHT).

464 Propagation of underwater sound

[Ch. 9

propagation, limited mainly by surface scattering, as illustrated in Figure 9.10 for the winter profile of Figure 9.8. With the exception of the near-vertical stripes7 at 0 km, 65 km, and 130 km, sound is restricted to the uppermost 50 meters, the depth at which the sound speed reaches a maximum. (The second and third of these stripes, the convergence zones, as seen previously in Figure 9.9, are considered further in Section 9.1.2.1.3.) At a given range (other than at convergence zones), propagation loss first decreases with increasing depth, reaching a minimum when the receiver depth passes through the source depth (30 m) and then increasing again to the edge of the duct (see Figure 9.10, upper graph, calculated by applying the simple flux concepts described in this chapter). At any fixed depth in the duct, propagation loss tends to increase with increasing range in accordance with cylindrical spreading. The lower graph is calculated for the same case using a coherent ray-tracing method. It shows that the trends illustrated by the upper graph are accompanied by fluctuations on a finer scale, caused by interference between the different multipaths contributing to the total field. Weston’s flux theory. Weston has developed a powerful and elegant method for analyzing the distribution of sound intensity with range and depth in a situation like that of Figure 9.10. The method involves calculation of range-averaged energy flux (Weston, 1980). Denoting the cycle distance r0 , the propagation factor so calculated can be expressed as the following integral over ax , the ray grazing angle at the duct axis (the sound speed minimum),

ð 4 0 tan ax jRj 2m dax ; ð9:65Þ F¼ r 2 r0 tan s tan r where s ; r are the ray angles evaluated at the source and receiver depth, respectively. The upper limit 0 is the grazing angle (measured at the duct axis) of the steepest ray trapped within the duct (i.e., the one that is horizontal at the depth h, where the sound speed has a maximum). The lower limit 2 is the grazing angle of the shallowest ray that traverses both source and receiver depth. This ray is horizontal at the depth z2 , defined as the deeper of the two, z2 ¼ maxðz; z0 Þ:

ð9:66Þ

The depth z1 is similarly defined as the smaller of source and receiver depths, z1 ¼ minðz; z0 Þ: The jRj 2m scaling factor due to m surface reflections is addressed later. Equation (9.65) can be written

ð 4 0 tan ax F¼ jRj 2m dax ; r 2 r0 tan 1 tan 2

ð9:67Þ

ð9:68Þ

7 Although the stripes seem vertical in this graph, they appear so only because of the distortion caused by the stretched depth axis. In reality the ray paths are close to horizontal.

Sec. 9.1]

9.1 Propagation loss

465

Figure 9.10. Effect of upward refraction: propagation loss [dB re m 2 ] vs. range and depth for source depth z0 ¼ 30 m at 2,000 Hz. Upper: INSIGHT; lower: BELLHOP (oalib, www).

466 Propagation of underwater sound

[Ch. 9

where cos 1 ¼

cðz1 Þ cos ax c0

ð9:69Þ

cos 2 ¼

cðz2 Þ cos ax : c0

ð9:70Þ

and

In an isovelocity channel the term in square brackets of Equation (9.68) would be 1=ð2hÞ, in which case the propagation factor becomes ð 2 0 F¼ jRj 2m ðÞ d; ð9:71Þ rh 0 consistent with Equation (9.38). For the isovelocity case, sound reflects from both boundaries, so R is then the product of sea surface and seabed reflection coefficients. Returning to the surface duct, the profile of interest here is a linear one, with gradient c 0 > 0, for which it is convenient to introduce the radius of curvature c ðax Þ ¼ 0 0 : ð9:72Þ c cos ax Absolute differences in sound speed are small across the duct, so rays trapped in the duct must be nearly horizontal. Therefore, for these rays8 c ðax Þ  ð0Þ ¼ 00 : ð9:73Þ c In order to proceed with evaluation of Equation (9.68) the cycle distance is needed, given by r0 ¼ 20 tan ax  20 ax ; ð9:74Þ where 0 is shorthand for ð0Þ, and hence m

r : 20 ax

ð9:75Þ

For a surface duct there are no reflections from the seabed, while the surface reflection coefficient is assumed to take the form jRj ¼ jRS j ¼ expðS ax Þ:

ð9:76Þ

jRj 2m ¼ expð2S rÞ;

ð9:77Þ

It then follows that where S ¼

S : 20

ð9:78Þ

The right-hand side of Equation (9.77) is independent of angle, which means that the integrand of Equation (9.71) may be factored out of the integral for the propagation factor. 8 For isothermal conditions (i.e., if c 0 ¼ 0.016/s), the radius of curvature is approximately equal to 90 km.

Sec. 9.1]

9.1 Propagation loss

467

Following Weston (1980), the depth factor can be defined (allowing here for attenuation due to surface reflection losses) as rh 2S r e ; 2 0

ð9:79Þ

2 0  Dðz0 ; zÞ e 2S r : rh

ð9:80Þ

DF so that FSD ¼

Substituting Equation (9.68) in Equation (9.79) and assuming small angles,9 the depth factor is

ð 2h 0 ax D dax : ð9:81Þ r0 1 2 0 2 The result is (Weston, 1980, Eq. (17)) Dðz0 ; zÞ ¼

0

2

Fð I Þ;

ð9:82Þ

2

where Fð I Þ is an incomplete elliptic integral of the first kind (Appendix A) ð Fð I Þ  ð1  sin 2  sin 2 Þ 1=2 d: ð9:83Þ 0

The arguments of Equation (9.83) are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 0 2  ¼ arcsin 2 2  0 1

ð9:84Þ

and  ¼ arcsin

1

;

ð9:85Þ

2

where the angles 0 , 1 , and 2 are the grazing angles at the duct axis of the rays whose turning depths are equal to h, z1 , and z2 , respectively. Thus, 2 0



2 h; 0

ð9:86Þ

2 1



2 z ; 0 1

ð9:87Þ

2 2



2 z : 0 2

ð9:88Þ

and

The arguments  and  can also be written in terms of depth variables, that is, z sin 2  ¼ 1 ð9:89Þ z2 9

tan  is approximated by its argument.

468 Propagation of underwater sound

[Ch. 9

and sin 2  ¼

h  z2 : h  z1

ð9:90Þ

The result of evaluating Equation (9.82) is shown in the upper graph of Figure 9.11. Weston’s flux formula provides useful insight into depth dependence that is difficult to gain in other ways, although quantitative corrections to it are sometimes needed. For example, when the source and receiver depths are equal, the right-hand side of Equation (9.82) becomes infinite. This singularity is a remnant of the infinities associated with ray theory caustics, which are lines along which the density of ray paths becomes infinite, illustrated by Figure 9.12. (The caustics are the thick bunches of rays that, for example, intersect the sea surface at 9 km, 16 km, and so on.) Flux theory smooths out these features by averaging in range, such that most of them disappear. However, caustics that form at the source depth, known as cusps, are of a particularly resilient variety, and these can be seen in Figure 9.12 at ranges of 6 km, 13 km, 20 km and so on, at a depth of 30 m. The infinities associated with these cusps survive the range-averaging process, albeit weakened (Weston, 1980), manifesting themselves as the singularities at the source depth in Figure 9.11. Unless z2 (the larger of source and receiver depth) is small, the cusp singularities are logarithmic in nature (after range averaging) and their effects are confined to a very small region either side of the source depth. For small z2 there is an additional enhancement caused by the rays being confined to a smaller and smaller proportion of the duct. This focusing behavior is analogous to the trapping of sound against a curved reflecting surface, known as a whispering gallery (Weston, 1979). One solution to the infinity at the cusp is to use wave theory to calculate the maximum value of the depth factor at depth z ¼ z1 ¼ z2 as a function of frequency. The result is (Weston, 1980; Harrison, 1989)   1 h 1=2 4z D! loge ðz1 ! z2 Þ ð9:91Þ 2 2z " where ! 0 c 20 1=3 "¼ : ð9:92Þ 2! 2 The parameter " can be interpreted as a sort of ‘‘ray thickness’’ into which the energy associated with the cusp spreads. For 0 ¼ 90 km and wavelength about 1 m, " is about 20 wavelengths. Unless z=" is very large (or z=h is small), the right-hand side of Equation (9.91) is always of order unity. An alternative approach for dealing with the singularity is to find an approximation to Equation (9.82) that does not exhibit one. Unless  and  are both close to =2, the integrand is of order 1 and the integral is of order . Thus, a very simple approximation is Fð I Þ  : ð9:93Þ

Sec. 9.1]

Figure 9.11. Depth factor vs. receiver depth. Upper: evaluated using Equation (9.82) without approximation for z0 ¼ 0, h=2, and 9h=10 (reprinted with permission from Weston, 1980, # American Institute of Physics); lower: evaluated using various approximations, solid line (——): Equation (9.94); dashed line (– – –): Equation (9.93); dotted line (    ): Equation (9.95).

9.1 Propagation loss

469

470 Propagation of underwater sound

[Ch. 9

Figure 9.12. Ray trace illustrating the formation of caustics and cusps up to a range of 40 km, for a source depth of 30 m, and for the same case as Figure 9.10 (BELLHOP).

A better approximation, derived in Appendix A, is  

2 Fð I Þ  arcsin sin  : 2 sin 

ð9:94Þ

An alternative to Equation (9.93), motivated also by simplicity, is Fð I Þ  sin ; which results in a depth factor of the form   1 h=z2  1 1=2 D : 2 1  z1 =h

ð9:95Þ

ð9:96Þ

This last expression is deceptively simple. It tends to underestimate the field whenever z1 and z2 are equal, but otherwise captures the main features of the elliptic integral, as illustrated by the lower graph of Figure 9.11 (dotted line). For example, in the limit of small z2 =h, near-surface behavior is readily found to be   1 h 1=2 D : ð9:97Þ 2 z2 This depth factor tends to infinity as z2 approaches zero. The physical reason for this is that sound rays whose turning depth is less than z2 are confined to a smaller and

Sec. 9.1]

9.1 Propagation loss

471

pffiffiffiffiffi smaller area (/ z2 ). The available energy also decreases, but at a slower rate ð/ z2 Þ pffiffiffiffiffi so the resulting depth factor is proportional to 1= z2 (ratio of energy to area), consistent with Equation (9.97).10 Surface decoupling. A surface duct ray reflecting from the sea surface suffers a

phase change, resulting in near-surface cancellation the same way as described in Section 9.1.1.2.6, except that here we are dealing with curved rays due to the sound speed gradient. In this situation, Equation (9.62) can be written WðzÞ ¼ 2 sin 2 ðzÞ; where, using the WKB approximation (Boyles, 1984), ðz ðzÞ ¼ ðÞ d:

ð9:98Þ

ð9:99Þ

0

In the same vein, Equation (9.63) generalizes to WðzÞ 

1 : 1 þ ð2 2 Þ 1

ð9:100Þ

If the sound speed gradient is c 0 , the integral of Equation (9.99) for a ray whose grazing angle at the sea surface 0 is ðzÞ ¼ ð!=c 0 ÞfYð0 Þ  Y½1 ðzÞ þ sin½1 ðzÞ  sin 0 g;

ð9:101Þ

where 1 is the corresponding angle at depth z according to Snell’s law. The function YðÞ is defined in Equation (9.22). Surface-scattering loss. Another important property of the sea surface is its wavy nature, which has the effect of scattering high-frequency sound. The role played by near-surface bubbles in this process is described in Chapter 8. An empirical approach is adopted here, assuming linear variation of reflection loss with angle regardless of the mechanism. A suitable value of S for use in Equation (9.78), based on the measurements of (Weston and Ching, 1989), is   v10 4 3=2 S ¼ 3:8F Np/rad; ð9:102Þ 10 m/s where F is the numerical value of the acoustic frequency when expressed in units of kilohertz; and v10 is the wind speed at 10 m height. Volume attenuation. For frequencies f between 200 Hz and 10 kHz, the attenuation coefficient for the representative conditions considered previously11 10

This behavior is known as the ‘‘whispering gallery’’ effect. The name originates from a focusing effect associated with multiple reflections from a hard curved surface such as occurs in churches or chambers of oval or spherical design. 11 The conditions (see Chapters 2 and 4) are T ¼ 10  C, S ¼ 35, and K ¼ 1:0.

472 Propagation of underwater sound

[Ch. 9

can be approximated by V ¼ 0:0140

F2 þ 0:00102F 2 F 2 þ 1:32

Np km 1 ;

ð9:103Þ

or, converting to decibels, F2 þ 0:0088F 2 F 2 þ 1:32

dB km 1 ;

ð9:104Þ

aV =ðdB kmÞ 1 ¼ ð20 log10 eÞV =ðNp km 1 Þ;

ð9:105Þ

aV ¼ 0:122 where

Duct cut-off frequency. In the same way as for a shallow-water waveguide (Section 9.1.1.2.5), there exists for a surface duct a cut-off frequency below which waveguide propagation is not supported. The cut-off frequency fc can be calculated using 1=2 9c  fc ¼ 0 0 3=2 ; ð9:106Þ 8 ð2hÞ and hence rffiffiffiffiffi 0 fc  ð590 m/sÞ : ð9:107Þ h3 Assuming a nominal radius of curvature of 0 ¼ 90 km (corresponding to isothermal conditions), the cut-off frequency is   100 m 3=2 fc  ð180 HzÞ : ð9:108Þ h Irrespective of the sound speed gradient, the ‘‘ray thickness’’ (Equation 9.92) evaluated at the cut-off frequency of Equation (9.106) is 43 % of duct thickness. The frequency of 2 kHz chosen for Figure 9.10 is close to the optimum for longrange propagation in this duct. The reason there is an optimum at all is that at higher frequency the sound is scattered or absorbed, and at lower frequency (below the duct cut-off frequency, equal to 800 Hz for this case) the energy leaks out by means of the tunneling effect. The decay rate due to low-frequency tunneling, in nepers per unit distance, can be estimated using (Packman, 1990, Eq. (1)) where ð f Þ ¼

T ¼ 2 5=2 ð0 hÞ 1=2 e ð f Þ ; ( 3 ð f Þ f > f1 3 ½ 0 ð f1 Þð f  f1 Þ þ ð f1 Þ

ð f Þ ¼ ½ð f =fc Þ 2=3  1 3=2

f  f1 , f fc

ð9:109Þ ð9:110Þ ð9:111Þ

and f1 ¼ 1:15fc :

ð9:112Þ

Optimum propagation frequency. Frequency dependence at a fixed receiver depth of 10 m is illustrated by Figure 9.13 for two different wind speeds. Without wind (see

Sec. 9.1]

9.1 Propagation loss

473

Figure 9.13. Propagation loss [dB re m 2 ] vs. frequency and range for a surface duct with v10 ¼ 0 (upper) and v10 ¼ 15 m/s (lower). Source and receiver depths are 30 m and 10 m (INSIGHT).

474 Propagation of underwater sound

[Ch. 9

upper graph), the channel acts as a filter with a passband of 1 kHz to 10 kHz and a peak response close to 2 kHz (the horizontal lines at 65 and 130 km are convergence zones). The effectiveness of the channel is sensitive to wind speed, as can be seen by comparing the upper graph of Figure 9.13 (for v ¼ 0) with the lower one (v ¼ 15 m/s). Simple surface duct formula. The various effects described above can be combined to provide a simple formula that gives a reasonable approximation to the behavior of Figures 9.10 and 9.13: FSD ðzÞ ¼

2 0 Dðz0 ; zÞWðz0 ÞWðzÞ expð2SD rÞ: rh

ð9:113Þ

Attenuation comprises three components SD ¼ S þ T þ V ;

ð9:114Þ

representing contributions due to surface scattering, tunneling, and volume attenuation. They are given by Equations (9.78), (9.109), and (9.104), respectively. An implicit assumption of Equation (9.113) is that the individual effects of cutoff, surface decoupling, and surface scattering may be combined multiplicatively, but this is not always the case. For example, one complication arises from a non-linear variation of surface loss with angle, as this changes the depth dependence of the propagation factor, which then becomes a function of range. 9.1.2.1.3

Convergence zones

We now return to the convergence zone features in Figures 9.10 and 9.13, which are characteristic of long-range propagation in deep water. To understand them it is necessary to consider the behavior of the entire sound speed profile (see Figure 9.8), and the impact that it has on relevant ray paths. The positive sound speed gradient in the lower half of the ocean (2–5 km) refracts sound upwards, and ‘‘the convergence zone’’ is the name given to the region where this sound returns to the surface, in this case at a distance of some 65 km, as illustrated by the ray trace of Figure 9.14 (upper graph), Rays then reflect from the sea surface (or refract from the thermocline if the surface is warm enough) and the process repeats itself at about 130 km, 195 km, and so on. This yo-yo-like behavior can continue over hundreds or even thousands of kilometers if the conditions are right. Within the convergence zone region, sound pressure levels can be much higher than in its immediate surroundings, as illustrated by the lower graph of Figure 9.14, showing propagation loss calculated from the ray paths shown in the upper graph. The same convergence zone features are clearly visible in the graphs of Figures 9.10 and 9.13. 9.1.2.1.4

Lloyd mirror with downward refraction

Now consider a sound speed profile with a negative gradient instead of the positive one considered so far. A negative gradient means that sound speed decreases with increasing distance from the sea surface, typical of summer conditions with solar

Sec. 9.1]

9.1 Propagation loss

475

Figure 9.14. Upper: ray trace for source depth 30 m, illustrating convergence zones at the sea surface at intervals of 65 km; lower: propagation loss [dB re m 2 ] vs. range and depth for the same case, with an acoustic frequency of 2 kHz (BELLHOP). The surface duct is visible as a horizontal stripe across the top of each graph. Source depth is 30 m.

476 Propagation of underwater sound

[Ch. 9

heating. The negative gradient results in sound being deflected away from the surface as illustrated by the lower graph of Figure 9.15. There is a parallel here with a well-studied problem in radar, namely propagation in an isovelocity medium (the atmosphere) close to a spherical boundary (the earth’s surface). A transformation can be made to a co-ordinate system in which the earth is flat and the rays are curved due to an effective refractive index profile that has the effect of refracting radio waves away from the flat surface representing the earth– atmosphere boundary. In the transformed co-ordinate system, acoustic and electromagnetic problems are equivalent. Radar scientists have solved this problem (Fishback, 1951; Freehafer, 1951) and their result is given below, in the form quoted by Ainslie and Harrison (1990). Consider the wavenumber profile   !2 2 k2 ¼ 2 1 þ z ; ð9:115Þ 0 c0 where the radius of curvature 0 is related to the sound speed c0 and its gradient c 0 at the sea surface according to 0 ¼ c0 =jc 0 j:

ð9:116Þ

In this situation, the propagation factor is similar to Equation (9.5) except that the depths z; z0 are replaced by the transformed co-ordinates 1 ; 2 (neglecting surface reflection loss) 4 !  FLM ¼ 2 expð2V rÞ sin 2 1 2 ; ð9:117Þ c0 r r where r2 1 ¼ z1  1 ð9:118Þ 20 and r2 2 ¼ z2  2 : ð9:119Þ 20 As previously (see Equations 9.66 and 9.67), depths z1 and z2 are the smaller and larger of the source and receiver depths, respectively. The ranges r1 and r2 are distances from either the source or receiver to the point of reflection. They satisfy the condition r2 > r1 and can be found by solving the following cubic equation for r2 (see Appendix A) 2r 32  3rr 22 þ ½r 2  20 ðz1 þ z2 Þ r2 þ 20 rz2 ¼ 0;

ð9:120Þ

r1 ¼ r  r2 :

ð9:121Þ

and then applying The accuracy of Equation (9.117) is demonstrated by Ainslie and Harrison (1990) for a frequency of 50 Hz.

Sec. 9.1]

9.1 Propagation loss

477

Figure 9.15. Effect of downward refraction (dc=dz ¼ 0:03/s) on propagation loss [dB re m 2 ] for LM at 900 Hz: isovelocity (upper), downward refracting (lower). The source depth is 15 m (INSIGHT).

478 Propagation of underwater sound

9.1.2.2

[Ch. 9

Shallow water

A characteristic feature of shallow-water propagation is that any sound that has traveled a few kilometers horizontally is likely to have suffered several interactions with either the sea surface or the seabed, or both. The importance of the sound speed profile in shallow water arises largely from the way it influences these boundary interactions. Assuming a uniform sound speed gradient there are two possible types of ray path: one that is steep enough to interact with both boundaries (surface– bottom multipaths); and one that is not, trapped instead by refraction in the water before it reaches the high-speed boundary. In the latter case, a surface duct is formed if the sound speed gradient is positive (upward-refracting) and a bottom duct if it is negative (downward-refracting). In the following, propagation from surface–bottom multipaths is referred to as ‘‘V-duct’’ (or ‘‘VD’’) propagation, because each cycle of a ray path follows a V shape as for the isovelocity case, albeit slightly curved due to refraction. Similarly, a surface duct or bottom duct is referred to as a ‘‘U-duct’’ (or ‘‘UD’’) because the ray paths follow a U shape. If wind speed is low, corresponding to a smooth sea surface, the surface is a good reflector of sound (see Chapter 8), which means that conditions of upward refraction (surface duct) can lead to long-range propagation. This point is illustrated by Figure 9.16. Theoretical expressions follow for U-duct and V-duct behavior, starting with the V-duct. 9.1.2.2.1 Surface–bottom multipaths (‘‘V-duct’’) For rays steep enough to reflect from both boundaries (the condition for a V-duct), the propagation factor of Equation (9.68) can be written ð max F¼ Gðax Þ dax ; ð9:122Þ min

where the integrand GðÞ is the differential propagation factor, which, neglecting volume attenuation, is given by (see Equation 9.65)

4 tan ax Gðax Þ ¼ jRj 2m ; ð9:123Þ r r0 tan 1 tan 2 where m is the number of ray cycles mðÞ ¼ r=r0 ðÞ:

ð9:124Þ

It is convenient to introduce the subscripts ‘‘hi’’ and ‘‘lo’’ to denote the properties of high-speed and low-speed boundaries, respectively, while ‘‘B’’ and ‘‘S’’ denote the properties of the seabed and sea surface. Thus, chi ¼ maxðcS ; cB Þ;

ð9:125Þ

clo ¼ minðcS ; cB Þ;

ð9:126Þ

hi ¼ minðS ; B Þ;

ð9:127Þ

lo ¼ maxðS ; B Þ:

ð9:128Þ

and

Sec. 9.1]

9.1 Propagation loss

479

Figure 9.16. The thick solid curve shows propagation loss [dB re m 2 ] vs. range for shallow water with a mud bottom for two different sound speed profiles. Upper: upward refraction (isothermal profile); lower: downward refraction (thermocline). The thin line is a reference curve for isovelocity water from Figure 9.7 (INSIGHT).

480 Propagation of underwater sound

[Ch. 9

With this notation, the integration limits are: sffiffiffiffiffiffiffi 2H min  0 and  c max ¼ arccos lo cos cB

ð9:129Þ  :

c

This maximum angle is the sediment critical angle cw ; c ¼ arccos cB

ð9:130Þ

c,

ð9:131Þ

corrected for refraction between the seabed and the duct axis according to Snell’s law. If the critical angle is large, max may be approximated by max 

c:

ð9:132Þ

The main effect of refraction is to change the functional form of r0 ðÞ compared with the isovelocity case: r0 ðÞ ¼ 2ðlo Þðsin lo  sin hi Þ;

ð9:133Þ

where ðÞ is the radius of curvature given by Equation (9.72), simplifying for angles close to horizontal to r0 ðÞ  20 ðlo  hi Þ:

ð9:134Þ

The term RðÞ is the product of both surface and bottom reflection coefficients: RðÞ ¼ RS ðS ÞRB ðB Þ:

ð9:135Þ

Equation (9.123) can be written GðÞ 



2 2H tan ax jRðÞj 2r=r0 ðÞ ; rH r0 tan 1 tan 2

ð9:136Þ

where the factor in square brackets is a dimensionless ratio of order unity. Defining the angle difference D  lo  hi ;

ð9:137Þ

jRðÞj ¼ expðlo lo Þ exp½hi ðlo  DÞ :

ð9:138Þ

Equation (9.135) becomes

The cumulative reflection coefficient is then jRðÞj 2m  e 2hi r e 2ðhi þlo Þrlo =D ; where

ð9:139Þ

hi ¼

hi ; 20

ð9:140Þ

lo ¼

lo : 20

ð9:141Þ

and

Sec. 9.1]

9.1 Propagation loss

Using Snell’s law it can be shown that, if lo and D are both small sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!  2 1 lo  1  1  min : D  2lo

481

ð9:142Þ

In principle, all of the ingredients needed for calculation of the propagation factor using Equation (9.122) are now in place. The recipe involves substitution of YVD ðÞ and jRðÞj in Equation (9.136) (using Equation 9.139) and carrying out the integral over angle. One could stop here, but it is instructive to simplify the integral—with care to avoid losing the baby with the proverbial bathwater—to facilitate its evaluation. The important point is to keep a careful track of the exponent of Equation (9.139), as small errors there will be amplified by exponentiation. The exponent can nevertheless be simplified by replacing Equation (9.142) with the approximation (Ainslie, 1992) lo 2  2 2lo  1: D  min

ð9:143Þ

For small lo (i.e., close to min ), the right-hand sides of Equations (9.142) and (9.143) are both approximately equal to 1. For large lo they approach the same asymptotic result of 2 2lo = 2min . The behavior for intermediate values is examined later. The final step in the simplification process is to recognize that the term in square brackets in Equation (9.136) may be approximated by unity without incurring a large error.12 With these simplifications, and substituting Equation (9.139) into Equation (9.136), it follows from Equation (9.122) that ð max 2 2hi r 2ðhi þlo Þrlo =D FVD  e e dax ð9:144Þ min rH and hence rffiffiffiffiffiffiffiffiffi1=2 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tot r FVD  eff e 2ð2hi þlo Þr erf max  erfð2 ðhi þ lo ÞrÞ ; ð9:145Þ rH H where  2eff ¼

H 4tot r

ð9:146Þ

and tot ¼ hi þ lo :

ð9:147Þ

At short range, Equation (9.145) simplifies to give the cylindrical spreading result FVD  12

2ðmax  min Þ rH

ðeff  max Þ:

ð9:148Þ

This approximation works best at short range, where ray angles are steep, because for this situation the three angles 1 , 2 , and ax are equal and r0 ðÞ ¼ 2H=tan . At long range, where the angles are small, although a small error (a few dB) from this approximation is likely, it is more important to keep tabs on the exponential terms in Equation (9.139). In any case, the VD term is eventually exceeded in importance by the UD contribution (Section 9.1.2.2.2).

482 Propagation of underwater sound

[Ch. 9

Figure 9.17. Approximation to D= for values of min (in radians) as stated. Cyan solid line: exact; dashed line: Equation (9.143); dotted line: Equation (9.142).

At longer range, mode stripping sets in, with an important correction factor compared with the isovelocity case, equal to the term in curly brackets in Equation (9.149) FVD 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2eff ferfc½2 ðhi þ lo Þr e 2ð2hi þlo Þr g rH

ðeff  max Þ:

ð9:149Þ

For extremely long ranges, satisfying eff  min , the correction factor simplifies, and the propagation factor can then be written13 (Ainslie, 1992) 1 e 2lo r FVD  pffiffiffiffiffiffiffiffiffiffiffiffi ; 2H0 tot r 2

ð9:150Þ

resulting in exponential decay if lo is non-zero. Because of the importance of accurately representing the exponent, the approximations of Equations (9.142) and (9.143) are compared in Figure 9.17. The magnitude of the exponent is correct for both small and large angles as expected. For intermediate angles it is underestimated slightly. 13

Equation (9.150) follows from Equation (9.149) using (see Appendix A) erfc x 

expðx 2 Þ

1=2 x

ðx  1Þ:

Sec. 9.2]

9.2 Noise level

483

9.1.2.2.2 Surface or bottom duct propagation (‘‘U-duct’’) Equation (9.145) describes the contribution to sound propagation from ray paths that are steep enough to reflect from both upper and lower boundaries. A complete description must also include a contribution FUD due to paths that reflect from the low-speed boundary but are not steep enough to reach the high-speed one. Such paths form a surface duct in the upward-refracting case and a bottom duct in the downward-refracting case. These waveguides are referred to henceforth as ‘‘U’’-ducts to distinguish them from the ‘‘V’’-ducts involving reflections from both boundaries. The U-duct contribution can be written (using Equation 9.113 and neglecting surface decoupling)14 2 FUD ðr; zÞ ¼ 0 Dðh0 ; hÞ expð2lo rÞ; ð9:151Þ rH where h and h0 are the distances from receiver and source, respectively, to the low sound speed boundary. Notice the resemblance to Equation (9.148), associated with typical cylindrical spreading behavior. 9.1.2.2.3 Total (VD þ UD) The total propagation factor is the sum of Equations (9.145) and (9.151) F ¼ FVD þ FUD :

9.2

ð9:152Þ

NOISE LEVEL

Sound traveling in the sea is subject to many different propagation effects that need to be taken into account when estimating the propagation loss term of the sonar equation. The noise arriving at the receiver has traveled through the same medium, so the same propagation effects can also be important in determining the level of ambient noise, which is considered next. Typical sources of ambient noise are described in Chapter 8, the main ones being associated with shipping, wind, and precipitation. These sources are characterized in terms of their areic15 source spectrum level, which, combined with an ambient noise model16 (a computer program designed to calculate the field from a continuous or discrete distribution of noise sources), enables prediction of the noise level term in the sonar equation. 14 The angle 0 is the same as min used previously for V-duct. A change in notation is appropriate because here it is not the minimum angle, but the maximum one. 15 Following Taylor (1995), the adjectives ‘‘areic’’ and ‘‘volumic’’ are used, respectively, to mean ‘‘per unit area’’ and ‘‘per unit volume’’. 16 A review of ambient noise models available in 1997 is given by Hamson (1997). See also Jensen et al. (1994) and Etter (2003).

484 Propagation of underwater sound

[Ch. 9

Figure 9.18. Predicted deep ocean noise spectra [dB re mPa 2 /Hz] (INSIGHT): shipping noise, wind noise, and thermal noise as marked. See text for details.

9.2.1

Deep water

Figure 9.18 shows predicted noise spectra for two cases, one for a high-noise situation and the other for low noise. The high-noise case involves heavy shipping (see Table 9.3) and a wind speed of 15 m/s, and the low-noise case is for light shipping and 2.5 m/s. A third contribution to sonar noise is thermal noise, which is the Table 9.3. Nomenclature used for shipping name given to random pressure fluctuadensities. tions at the hydrophone due to thermal Shipping category Shipping density agitation of water molecules. Mm 2 ð a Þ

9.2.1.1

Typical spectra for wind, shipping, and thermal noise

Wind noise occupies a large part of the ambient noise spectrum in the frequency range of interest to sonar. This is illustrated by the prediction of Figure 9.18 showing wind noise dominating the spectrum roughly between 1 kHz and 100 kHz. At lower frequency, the contribution due to (distant) shipping is

Very heavy Heavy Moderate Light Very light

50 000

nmi 2 ð a Þ 0.171

5000

0.0171

500

0.00171

50

0.000171

5

0.0000171

a 1 Mm (one megameter) ¼ 1000 km; 1 nmi (one nautical mile) ¼ 1.852 km (see Appendix B).

Sec. 9.2]

9.2 Noise level

485

important, whereas at high frequency (above 300 kHz) it is thermal noise that dominates. 9.2.1.1.1

Shipping noise

At frequencies between about 10 Hz and 100 Hz, the noise from distant shipping dominates the spectrum. This component of background noise depends on distance to the ships, their source levels, and the density of distant ships.17 Typical values of areic shipping density are suggested in Table 9.3. (The author is unaware of any standard definition for terms like ‘‘heavy’’ and ‘‘light’’ shipping). In addition to shipping density, a prediction of shipping noise requires an estimate of the average source level for a single ship. 9.2.1.1.2 Thermal noise The thermal noise spectrum is included in Figure 9.18. Though not acoustic in origin,18 it is nevertheless noise that interferes with the detection of high-frequency sound. Thermal noise increases sharply at high frequency, as described by the equation (see Chapter 10) 2 10 log10 Q N f ¼ 14:7 þ 10 log10 F

dB re mPa 2 =Hz;

ð9:153Þ

where F is the frequency in kilohertz. 9.2.1.2

Effect of rain rate and wind speed

Rainfall is an important, though intermittent, source of broadband noise, centered around 10 kHz, as illustrated by Figure 9.19, showing the predicted sensitivity of ambient noise to rain rate for two different wind speeds. Similarly, Figure 9.20 shows sensitivity to wind speed with and without rainfall. These graphs are both calculated for light shipping (50/Mm 2 ), as defined by Table 9.3. The reader will notice a difference between (say) Figure 9.20 and the corresponding wind and rain source spectra in Chapter 8, and this is partly because different physical quantities are considered in each chapter. Specifically, the above graphs show received noise spectral density (Q N f ) at a depth of 30 m, whereas the Chapter 8 graphs show the areic dipole source factor K (e.g., K wind or K rain ) which is a measure of radiated acoustic power per unit area of the sea surface. Though conceptually different, these two quantities are closely related. They are easily confused because they share the same dimensions and units (both are reported in dB re mPa 2 / Hz) and are similar in numerical value. For example, in isovelocity water (and infinite water depth), the received noise spectrum is given by (see Chapter 2) QN f ¼ 2 E3 ð2zÞK 17

ð9:154Þ

Nearby ships need to be treated not as a continuum, but as discrete entities. That is, the pressure fluctuations are not caused by traveling acoustic waves, but by thermal agitation of the water molecules in direct contact with the hydrophone. 18

486 Propagation of underwater sound

[Ch. 9

Figure 9.19. Predicted sensitivity of deep-water noise spectra [dB re mPa 2 /Hz] to rain rate, for a wind speed of 2.5 m/s (upper) and 7.5 m/s (lower) (INSIGHT).

Sec. 9.2]

9.2 Noise level

487

Figure 9.20. Predicted sensitivity of deep-water noise spectra [dB re mPa 2 /Hz] to wind speed. Upper: no rain; lower: rain rate ¼ 10 mm/h (INSIGHT).

488 Propagation of underwater sound

[Ch. 9

Figure 9.21. Predicted ambient noise spectral density level [dB re mPa 2 /Hz] vs. frequency and depth (v10 ¼ 5 m/s) (INSIGHT).

and hence 10 log10 Q N f ¼ 10 log10 þ 10 log10 K þ 10 log10 ½2E3 ð2zÞ :

ð9:155Þ

If there is no attenuation, the factor 2E3 is equal to 1. The constant 10 log10 is approximately 5 dB, so Equation (9.155) then simplifies to 10 log10 Q N f  5 þ 10 log10 K:

ð9:156Þ

Thus, there is a systematic 5 dB offset in these conditions. 9.2.1.3

Depth dependence of surface-generated noise

The variation of ambient noise with depth, illustrated by Figure 9.21, is considered next. The high-frequency component of surface-generated noise decays exponentially with increasing depth, whereas thermal noise is independent of depth. The depth effect due to absorption can be quantified by means of Equation (9.155) and using the approximation (Appendix A) E3 ðxÞ 

e x : x þ 3  e 0:434x

ð9:157Þ

Sec. 9.2]

9.2 Noise level

489

Figure 9.22. Predicted effect of the seabed on the ambient noise spectrum [dB re mPa 2 /Hz] in isovelocity water of depth H ¼ 100 m, for wind speed v10 ¼ 2.5 m/s. Thick solid line: sand; dashed and dotted line: clay (INSIGHT).

providing an estimate for depth dependence at high frequency or in deep water19 10 log10 Q N f  10 log10

2 K  ð20 log10 eÞz: 2z þ 3  expð0:868zÞ

ð9:158Þ

At low frequency, or in shallow water, it is necessary to consider the additional propagation effects due to refraction and reflection from the seabed. Equations (9.155) and (9.158) apply to a uniform distribution of (dipole) surface sources, such as wind or rain noise, but not to thermal noise (which is described by Equation 9.153) or shipping (which is not uniformly distributed).

9.2.2

Shallow water

Ambient noise in shallow water is highly variable compared with deep water. This is partly due to the presence of location-dependent noise sources that are absent in deep water (e.g., surf, fauna) and partly because of the influence of the seabed, which introduces a dependence on water depth and bottom type. Figure 9.22 illustrates the 19 The requirement for the validity of Equation (9.158) is that the noise field be dominated by direct contributions from the sea surface, and not from other paths (e.g., via the seabed).

490 Propagation of underwater sound

[Ch. 9

Figure 9.23. Predicted effect of the sound speed profile on the ambient noise spectrum [dB re mPa 2 /Hz] for a clay seabed. Water depth H ¼ 100 m and wind speed v10 ¼ 2.5 m/s. Thick solid line: c 0 ¼ 0.02/s; dashed line (from Figure 9.22): c 0 ¼ 0 (INSIGHT).

effect of bottom type on ambient noise in the presence of rain and heavy shipping. The sand seabed (thick solid curve) has a higher critical angle, which has the effect of enhancing distant contributions at moderate frequencies (100 Hz to 10 kHz). At low frequencies (below the waveguide cut-off frequency) it is the clay seabed that is better able to support long-distance propagation, due to low attenuation in the sediment (see Table 9.1). Hence the crossover close to 30 Hz. The effect of refraction in the water is illustrated by Figure 9.23 for a clay seabed. The presence of a surface duct enhances the contribution from distant shipping at 300 Hz.

9.2.3

Noise maps

Given a suitable noise model and the necessary inputs, it is possible to predict maps of the geographical distribution of underwater sound due to natural or anthropogenic noise sources. An example follows for dredger noise in the North Sea, from Ainslie et al. (2009). Figure 9.24 shows the predicted broadband noise distribution due to dredging activity close to the Port of Rotterdam (see figure caption for details). The corresponding bathymetry is shown in Figure 9.25.

Sec. 9.3]

9.3 Signal level (active sonar) 491

Figure 9.24. Prediction of broadband radiated noise level (10 Hz to 10 kHz) [dB re mPa 2 ] for a hypothetical dredging operation involving two dredgers operating in the vicinity of Rotterdam harbor. The assumed broadband source level of each dredger is about 188 dB re mPa 2 m 2 (# TNO, 2009, reprinted with permission).

9.3

SIGNAL LEVEL (ACTIVE SONAR)

In this section the effect of propagation is considered on the signal term in the active sonar equation (i.e., on the target echo level). Consider an active sonar with an omnidirectional transmitter (Tx). A transmitted pulse travels through the sea until it reaches a submerged object, at which point some of the sound scattered by the object travels back through the sea to the sonar receiver (Rx). It is assumed that the transmitted pulse is long enough to ensure that the duration of the received signal is equal to those transmitted and reflected. With this assumption, for a monostatic geometry, the mean square pressure (MSP) at the receiver is  Q S ¼ S0 FTx FRx ; ð9:159Þ 4

where S0 is the source factor;  is the backscattering cross-section of the target (assumed independent of elevation angle); and the propagation factors are FTx (for the outward path from sonar to target) and FRx (return path from target to receiver). An important property of solutions to the wave equation is that the field at a point B due to a monopole source at point A is the same as the field at A due to an

492 Propagation of underwater sound

[Ch. 9

Figure 9.25. Bathymetry used for Figure 9.24. The contours show (minus) water depth in meters (# TNO, 2009, reprinted with permission).

‘‘identical’’ monopole source at B. This principle is usually interpreted as meaning that, for a monostatic geometry, FTx and FRx in Equation (9.159) are equal. This interpretation is correct if the medium density at A is equal to that at B, but not otherwise. The precise relationship between FTx and FRx , for the case when the densities differ, is derived below.

9.3.1

The reciprocity principle

The reciprocity relationship relating the acoustic pressure at rB due to a point source at rA , denoted pðrB ; rA Þ, to that at rB due to a point source at rB , can be written (Pierce, 1989) pðrB ; rA ÞUB ¼ pðrA ; rB ÞUA ;

ð9:160Þ

where UA and UB are the respective source strengths of the sources at A and B. The monopole source strength (the amplitude of volume velocity) of each source is related to its source factor S and frequency f according to pffiffiffiffi 2 S : ð9:161Þ U¼ i
f

Sec. 9.3]

9.3 Signal level (active sonar) 493

Therefore, Equation (9.161) can be written pðrB ; rA Þ pðrA ; rB Þ
B pffiffiffiffiffiffi ¼ pffiffiffiffiffiffi : SA SB
A

ð9:162Þ

Squaring both sides and interpreting point B as the sonar position and A as that of the target gives


ðz0 Þ 2 FRx ¼ FTx : ð9:163Þ
ðztgt Þ

9.3.2

Calculation of echo level

Substituting Equation (9.163) into Equation (9.159) gives for the received sonar signal


ðz0 Þ 2 Q S ¼ S0 F 2Tx ; ð9:164Þ 4


ðztgt Þ which, converted to decibels, results in the echo level EL ¼ SL þ TS þ 2PLTx þ 10 log10


ðz0 Þ 2 ;
ðztgt Þ

ð9:165Þ

where SL is the sonar source level; and TS is the target strength, related to the target backscattering cross-section according to (see Chapter 8)20  TS ¼ 10 log10 : ð9:166Þ 4

Given the (sonar to target) propagation loss PLTx , Equation (9.165) can be used to calculate EL. The density ratio is an important correction21 if the target is in a different medium to the sonar (e.g., if it is buried in sand). For the monostatic case, and assuming henceforth that the target is not buried (i.e., that
ðz0 Þ and
ðztgt Þ are equal), it follows from Equation (9.152) that  Q S ¼ S0 ðFVD þ FUD Þ 2 : ð9:167Þ 4

Equation (9.167) can be extended to a bistatic geometry by rewriting it in the form Q S ¼ S0 O ðFVD þ FUD ÞTx ðFVD þ FUD ÞRx ; 20

ð9:168Þ

The 4 denominator is omitted by some authors, who incorporate it instead into the definition of . See Chapter 5 for details. 21 The precise form of this correction depends on the chosen definition of propagation loss (PL). In early work on underwater acoustics (before about 1980), it was customary to define PL as a ratio, in decibels, of the equivalent plane wave intensity rather than MSP (see Appendix B). If the early PL definition is used, the density ratio
ðz0 Þ=
ðztgt Þ in Equation (9.165) is replaced by the sound speed ratio cðztgt Þ=cðz0 Þ (Ainslie, 2008).

494 Propagation of underwater sound

[Ch. 9

where the shorthand ‘‘Tx’’, ‘‘Rx’’ is used to indicate propagation from ‘‘transmitter (Tx) to target’’ and ‘‘target to receiver (Rx)’’, respectively; and O is the differential scattering cross-section evaluated at the (azimuthal) bistatic angle.22 The results of Section 9.1 can be used to calculate FVD and FUD in Equation (9.168). Two special cases are considered below. First, an isovelocity profile is considered, for which only FVD is relevant. This is followed by long-range propagation in a duct with a uniform non-zero sound speed gradient, for which only FUD is relevant. The effects of surface decoupling and tunneling are neglected.

9.3.3

V-duct propagation (isovelocity case)

The term ‘‘V-duct’’ (abbreviated VD) is introduced in Section 9.1.2.2 to describe propagation in a shallow-water waveguide that is bounded by reflection from both upper and lower boundaries, for which the appropriate propagation factor is FVD from Equation (9.46). Substituting this expression in Equation (9.168), with FUD ¼ 0, gives for the signal MSP23  rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi

O

rTx

rRx 3=2 3=2 Q S ðrTx ; rRx Þ ¼ S0 r Tx erf r Rx erf ; ð9:169Þ tot H 4rCS 4rCS where rCS is the transition range between cylindrical-spreading and mode-stripping regimes, given by Equation (9.50); and tot is given by Equation (9.147). Equation (9.169) simplifies at short and long range to ( 1=rTx rRx rTx  rCS ; rRx  rCS

O S Q ðrTx ; rRx Þ ¼ S0 ð9:170Þ 3=2 Htot rCS rCS =ðrTx rRx Þ rTx  rCS ; rRx  rCS . There is no dependence on the depth of transmitter, receiver, or target in Equation (9.169). This is because of the assumption leading to Equation (9.46) that neither source nor receiver are close to the sea surface.

9.3.4

U-duct propagation (linear profile)

For long-range propagation in a U-duct (a waveguide bounded on one side by reflection and on the other by refraction), the propagation factor is given by Equation (9.151), so that Equation (9.168) becomes Q S ðrTx ; rRx Þ ¼ S0 where 22

0

4O 20 DTx e 2lo rTx DRx e 2lo rRx ; rTx rRx H2

ð9:171Þ

is given by Equation (9.86). Dependence on depth arises through the two

The cross-section is assumed to be independent of elevation angle. An absorption factor of the form expð2V rÞ, though omitted here, is always implied. A cuton duct is further assumed, with neither source nor receiver close to the sea surface. 23

Sec. 9.4]

9.4 Reverberation level 495

depth factors, which are defined as DTx  DðzTx ; ztgt Þ

ð9:172Þ

DRx  Dðztgt ; zRx Þ;

ð9:173Þ

and where Dðz0 ; zÞ is calculated as described in Section 9.1.2.1.2.

9.4

REVERBERATION LEVEL

The physics and hence calculation of reverberation level (RL) has much in common with that of EL. Both are specific to active sonar, and for both there is propagation to a scattering region and then back to the sonar. The main difference is that the target echo is assumed to originate from a single point and arrive after a well-defined delay time, whereas reverberation originates from an extended region of scatterers, arriving continuously from a short time after transmission, determined by the distance to the nearest boundary. Although there are plenty reverberation models to choose from Etter (2003), these have not yet reached the level of maturity of one-way propagation models. Reverberation modeling requires the computation and summation of contributions to the received pressure field of scattered paths from many different locations and directions. As such it is one of the most challenging applications of propagation theory in sonar performance modeling, and is the subject of intensive ongoing research (see, e.g., Nielsen et al., 2008). Low-frequency reverberation typically decays to a level below the noise background after a few seconds or tens of seconds after transmission, depending on the source level, although this decay is not necessarily monotonic. Of particular interest is reverberation at the arrival time of the target echo, as it is this that might limit the sonar’s ability to discriminate the echo from the background. Scattered paths that contribute to reverberation at a given time t originate from a scattering annulus at a distance rðtÞ from the source24 c rðtÞ  t; ð9:174Þ 2 whose width r is determined by the pulse duration T c rðtÞ  T: ð9:175Þ 2 Consider a narrow beam radiated by the source that propagates to some distant point on the seabed after multiple boundary reflections, at which point the sound is scattered into a wide range of angles for the return path. Let the grazing angles of the radiated beam be between in and in þ in , and of the scattered sound consider the contribution to reverberation from a narrow return beam, between angles out 24 This and subsequent equations assume that the sound is traveling close to the horizontal direction.

496 Propagation of underwater sound

[Ch. 9

and out þ out . Applying the continuum flux approach used in Section 9.1.2.1, for a monostatic geometry the contribution  2 Q R to the reverberation MSP due to these two beams combined can be written25  2 Q R ðtÞ ¼ S0

AðtÞ GTx Sðin ; out ÞGRx in out ; 4

ð9:176Þ

where S0 is the source factor; Sðin ; out Þ is the seabed scattering coefficient; and G is the differential propagation factor from Equation (9.123), to be expressed here as a function of time (see Ainslie, 2007). The area of the scattering annulus is 2 r r:

c 2 T t: ð9:177Þ 2 The reverberation due to the narrow transmitted beam at angle in , integrating over all out , is ð S0 R Q ðtÞ ¼ in GTx AðtÞSðin ; out ÞGRx dout : ð9:178Þ 4 AðtÞ 

Reverberation from an omni-directional source is then found by integrating over a continuum of such transmitted beams: ð S Q R ðtÞ ¼ 0 AðtÞGTx Sðin ; out ÞGRx din dout : ð9:179Þ 4 Assuming that Sðin ; out Þ is a separable function of its two arguments such that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sðin ; out Þ ¼ SB ðin ÞSB ðout Þ; ð9:180Þ Equation (9.179) simplifies to ð

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðtÞ Gðax Þ SB ðB Þ dax ; 4

4 tan ax Gðax Þ ¼ jRj 2rðtÞ=r0 : rðtÞ r0 tan 1 tan 2

Q R ðtÞ  S0 where

ð9:181Þ

ð9:182Þ

The angles ax and B are the ray grazing angles at the channel axis (sound speed minimum) and seabed, respectively. It is useful to introduce the dimensionless reverberation coefficient ð pffiffiffiffiffiffiffiffiffiffiffiffi Hcw t X¼ G SB ðÞ d; ð9:183Þ 4 defined in such a way that (for an omni-directional transmitter)26 Q R ¼ S0 25

2 T 2 X ; H 2t

ð9:184Þ

Assuming, as for the echo, no change in medium density between sonar and scatterer, and that the received pulse is not stretched or compressed in time relative to the transmitted one. 26 For a directional transmitter, the angle 2 is replaced by the horizontal beamwidth of the transmitter.

Sec. 9.4]

9.4 Reverberation level 497

where T is the pulse duration. More generally, allowing the transmitter and receiver to be located at different depths, this becomes, using Equation (9.177) Q R ¼ S0

2 T XTx XRx ; H 2t

ð9:185Þ

where XTx and XRx are given by Equation (9.183) with the depth dependence of transmitter and receiver, respectively. This result and subsequent ones in this section are for a monostatic geometry in range and a bistatic one in depth. Some simple cases are considered below. For treatment of a fully bistatic geometry, see Harrison (2005a, b).

9.4.1

Isovelocity water

For an isovelocity channel, Equation (9.182) becomes GðÞ ¼

9.4.1.1

4 jRj ðcw t=4HÞ sin  : Hcw t cos 

ð9:186Þ

General power law scattering coefficient

Consider a scattering coefficient of the form Sðin ; out Þ ¼  sin q in sin q out ;

ð9:187Þ

where  and q are constants, such that the backscattering coefficient is SB ðÞ  Sð; Þ ¼  sin 2q :

ð9:188Þ

It follows from Equation (9.183) that X¼

ð Hcw t 1=2 sin q GðÞ d; 4

ð9:189Þ

and therefore, using a small-angle approximation in Equation (9.186) (Ainslie, 2007), X¼

 1=2 ð; u 2

2  ; c Þu

ð9:190Þ

where c is the seabed critical angle, given by Equation (9.131). Here  is the lower incomplete gamma function (Appendix A), the parameter  is given by qþ1 2

ð9:191Þ

tot cw t ; 2H

ð9:192Þ

¼ and u is a dimensionless time variable u¼

498 Propagation of underwater sound

[Ch. 9

where tot is given by Equation (9.147). The reverberation MSP follows from Equation (9.184) Q R ¼ S0

cw Ttot ð; u 4H 3 u 2þ1

2 2 cÞ :

ð9:193Þ

For reverberation, the transition between cylindrical-spreading (CS) and modestripping (MS) behavior occurs roughly at time 2rCS =cw , where rCS is given by Equation (9.50). The asymptotic limits of Equation (9.193) are ( 4 2

 c T tot w c =u c u  1 (CS) Q R  S0  ð9:194Þ 2 3 2 2 2þ1 2 4 H  GðÞ =u c u  1 (MS): Equation (9.185) can be applied to scattering from either boundary, depending on whether interest is in surface or bottom reverberation. Total reverberation from both boundaries can be found by calculating Q R for each one and then adding the two separate Q R values. This statement relies on the absence of multiple scattering, and is therefore valid for a slightly rough boundary. Similar expressions are derived by Zhou (1980), who approximates the gamma function by GðÞ  1 : ð9:195Þ 9.4.1.2

Application to a reference problem with Lambert’s rule (RMW11)

An important special case is obtained with q ¼ 1 in Equation (9.188), as this corresponds to the widely used Lambert rule. For this case SB ðÞ ¼  sin 2 ;

ð9:196Þ

and  then becomes the Lambert parameter. It follows by substituting q ¼ 1 in Equation (9.193) that (Zhou, 1980, Eq. (4.1); Harrison, 2003, Eq. (28)) " !# 2 2 T tot cw 2c R Q ZH ðtÞ ¼ S0 2 2 3 1  exp  t : ð9:197Þ 2H  tot c w t Converting to decibels and including absorption explicitly, Equation (9.197) becomes " !# 2  tot cw 2c RLZH ðtÞ ¼ SLE þ 10 log10 2 2 3 þ 20 log10 1  exp  t  aV cw t; 2H  tot c w t ð9:198Þ where SLE ¼ SL þ 10 log10 T

ð9:199Þ

and aV is the volume attenuation coefficient, given by Equation (9.105). Reverberation for the case in hand, namely isovelocity water in combination with Lambert’s rule for seabed scattering, is plotted vs. delay time in Figure 9.26, for one

Sec. 9.4]

9.4 Reverberation level 499

Figure 9.26. Model comparison for problem RMW11 and frequency 3.5 kHz (SLE ¼ 0 dB re mPa 2 m 2 s; aV ¼ 0.2397 dB/km; cw aV ¼ 0.3596 dB/s). Water depth is 100 m and source and receiver depths are 30 m and 50 m, respectively. The speed of sound in water is cw ¼ 1500 m/s. See Table 9.4 for further details. INTEGRAL: Equation (9.189) (see Ainslie, 2007 for details); ZHOU-HARRISON: Equation (9.198), with B  0.274 Np/rad; HARRISON 2006: improved approximation from Harrison (2006).

of the test cases from a reverberation modeling workshop (RMW) held in November 2006 at the University of Texas at Austin. At the time of writing, the results of this workshop are available online from an ftp site (rmw, 2006). The case considered is based on workshop problem XI and referred to here as ‘‘RMW11’’.27 The water depth is 100 m and the frequency chosen for the present comparison is 3.5 kHz. The seabed parameters correspond to fine sand (Mz  2.5, see Chapter 4). A complete description of the seabed properties is provided in Table 9.4. The sea surface is treated as a perfect reflector (S ¼ 0), so that tot ¼ B . The value of B can be estimated using Equation (9.51), giving B  0.274 Np/rad, which is the value used for the Zhou–Harrison formula in Figure 9.26. Also plotted is an improved approximation from Harrison (2006).

27 RMW11 is identical to Problem XI from the RMW except for the choice of source level, which here is chosen to be SLE ¼ 0 dB re mPa 2 m 2 s. The source level used by workshop participants for the chosen frequency of 3.5 kHz is SLE ¼ 28:72 dB re mPa 2 m 2 s, which means that the reverberation level shown in Figure 9.26 is about 29 dB higher than the corresponding workshop results (rmw, 2006). The latest workshop results available at the time of writing are available through the Solutions.html link at rmw (2006).

500 Propagation of underwater sound

[Ch. 9

Table 9.4. Seabed parameters for problems RMW11 (see Figure 9.26) and RMW12 (Figure 9.28). Description

Symbol

Value

Water depth

H

100 m

Sediment sound speed

csed

1700 m/s

Sediment attenuation

sed

0.5 dB/


sed =
w

2



10 2:7

Density ratio Lambert parameter

9.4.2

Comments

c

 0.4900 rad

sed =csed  0.294 dB/(m kHz)

10 log10  ¼ 27 dB

Effect of refraction

In the presence of refraction there is a need to distinguish between two different cases, depending on whether the scattering occurs at the low-speed or high-speed boundary. Denoting the respective contributions Xlo and Xhi , these can be written ð pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hct Xlo ¼ Gðlo Þ SB ðlo Þ dlo ð9:200Þ 4 and ð pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hct Xhi ¼ Gðlo Þ SB ðhi Þ dlo : ð9:201Þ 4 If scattering occurs from both boundaries, total reverberation can be found by calculating the contribution from each boundary separately using Equation (9.185) and adding them. The asymmetry in the integrand makes the second integral (for Xhi ) more difficult to evaluate. This asymmetry arises because the argument  in GðÞ is the grazing angle at the channel axis (i.e., lo ) regardless of where the scattering occurs. Further attention is restricted to evaluation of Xlo . One justification for this is that the effect of refraction is to steer the sound towards, and hence enhance scattering from, the low-speed boundary. Further, conditions of downward refraction are usually associated with a calm sea surface, whereas upward refraction often arises with a rough one. 9.4.2.1

V-duct propagation

For a V-duct (waveguide bounded by reflection at both boundaries), the differential propagation factor can be approximated by (see Section 9.1.2.2.1) G

2 2ð2hi þlo Þr 4ðhi þlo Þr 2 = 2min e e ; rH

ð9:202Þ

where the decay rates hi and lo are given by Equations (9.140) and (9.141); and min is from Equation (9.129). The reverberation coefficient Xlo can therefore be calculated

Sec. 9.4]

9.4 Reverberation level 501

as28 ðXlo ÞVD ¼  1=2 e ð2hi þlo Þcw t

ð max

sin q lo exp½2ðhi þ lo Þcw t 2lo = 2min dlo ;

ð9:203Þ

min

where max is given by Equation (9.130). It is convenient to make the approximation (accurate to order  qþ3 ) sin q    q e q

2

=6

ð9:204Þ

;

from which it follows that ðXlo ÞVD 

 1=2 ð2hi þlo Þct e ½ð; K 2max Þ  ð; K 2min Þ ; 2K 

where K¼

q þ 2ðhi þ lo Þcw t 6

ð9:205Þ ð9:206Þ

and  is given by Equation (9.191). 9.4.2.2

U-duct propagation

The reverberation coefficient for propagation in a U-duct (waveguide bounded by reflection at one boundary and refraction at the other), for the outward path between source and scatterer (using the subscript ‘‘Tx’’ here instead of ‘‘s’’ to indicate properties at the sonar transmitter), is ð  1=2 20 0 sin q  ðXlo ÞUD ¼ d: ð9:207Þ 2 tan  tan Tx Tx Assuming small angles  and making the change of variable  ¼ ðXlo ÞUD 

 1=2 2

qþ1 0 Dq

where Dq ¼ Dq ðxÞ  cos q1 x

e lo ct ;

ðx 0

and

 x ¼ arccos

Tx

du cos q u

Tx =cos

u results in ð9:208Þ ð9:209Þ

 :

ð9:210Þ

0

For q ¼ 0, Equation (9.209) gives D0 ðxÞ ¼ x=cos x; and for q ¼ 1, D1 ðxÞ ¼ loge tan

 x þ ¼ artanhðsin xÞ: 4 2

ð9:211Þ ð9:212Þ

28 The precise value used for cw in Equation (9.203) is not critical. Any value in the range clo to chi will give accurate results, provided that lo and min are correctly calculated.

502 Propagation of underwater sound

[Ch. 9

Recall that Tx (previously denoted 2 ; see Equation 9.68) is the grazing angle at the channel axis (low-speed boundary) of the ray that is horizontal at the source depth. It follows from this that rffiffiffiffiffiffiffi hTx x  arccos ; ð9:213Þ H where H is the water depth; and hTx is the distance between the source and the lowspeed boundary. For the return path from the scatterer back to the receiver, the same equations apply except that hTx is replaced with hRx , the distance between the receiver and the low-speed boundary. The reverberation MSP is ðQ R lo ÞUD ¼ S0

T 2H 2 t

2qþ2 ðDq ÞTx ðDq ÞRx 0

e 2lo cw t :

ð9:214Þ

For integer q ¼ N, the following recursion equation for DN is obtained using integration by parts: ðN  1ÞDN ðxÞ ¼ sin x þ ðN  2Þðcos 2 xÞDN2 ðxÞ ðN 2Þ:

ð9:215Þ

Similar expressions for reverberation are derived for the downward refracting case by Zhou (1980), whose Eq. (11) implies use of the approximation q Dq  D 1q 0 D 1;

ð9:216Þ

which is exact for q ¼ 0 and q ¼ 1, and permits approximate interpolation between these two values. By comparison, Equation (9.215) is exact for any integer N greater than 1. All of the depth dependence is contained within the reverberation depth factor Dq ðxÞ. This function is plotted for various q in Figure 9.27. The first two (for q ¼ 0; 1) are from Equations (9.211) and (9.212). The remainder (q ¼ 2; 3) are evaluated using Equation (9.215). Thus, for example, D2 ðxÞ ¼ sin x

ð9:217Þ

2D3 ðxÞ ¼ sin x þ ðcos 2 xÞD1 ðxÞ:

ð9:218Þ

and

For q ¼ 2, the reverberation MSP varies linearly with depth, as it is proportional to sin 2 x, which is sin 2 x ¼ 1  hTx =H:

ð9:219Þ

The reverberation varies more quickly with depth for small values of q than for large ones (see Figure 9.27). This sensitivity of the depth factor to q makes it possible, in principle, to determine the angle dependence of sea surface backscattering strength by measurements of the depth dependence of reverberation in a surface duct.

Sec. 9.4]

9.4 Reverberation level 503

Figure 9.27. Reverberation depth factor Dq for integer q between 0 and 3. Height h ¼ distance from the low-speed boundary.

9.4.2.3

Application to a reference problem with Lambert’s rule (RMW12)

The reverberation for a second case based on the 2006 reverberation modeling workshop (problem XII) is considered next. The modified problem is referred to here as ‘‘RMW12’’.29 The details are the same as for RMW11 (see Section 9.4.1.2) except for a uniform negative gradient of magnitude 0.3/s. (The sound speed decreases linearly from 1530 m/s at the sea surface to 1500 m/s at the seabed.) Reverberation vs. time for this case is shown in Figure 9.28, predicted using three different methods as described in the caption, for a frequency of 3.5 kHz. For the first 10 s, this calculation is dominated by the VD term, for which there is little difference in reverberation level compared with RMW11 (see Figure 9.26). After about 30 s, the UD term dominates, resulting in an exponential decay that becomes apparent at later times in Figure 9.28. (For example, by 90 s, the RMW12 reverberation is about 10 dB lower than for RMW11). Between 10 s and 30 s there is a transition region in which both UD and VD contribute significantly to the total. Apart from this exponential decay, another difference between RMW11 and RMW12 is a series of regularly spaced features that appears in the coherent mode 29

‘‘RMW12’’ is identical to ‘‘Problem XII’’ from the RMW except for the choice of source level, which here is chosen to be SLE ¼ 0 dB re mPa 2 m 2 s. The source level used by workshop participants for Problem XII at the chosen frequency is SLE ¼ 28:72 dB re mPa 2 m 2 s, so the present predictions are about 29 dB higher than the corresponding RMW results (rmw, 2006).

504 Propagation of underwater sound

[Ch. 9

Figure 9.28. Model comparison for problem RMW12 and frequency 3.5 kHz (SLE ¼ 0 re mPa 2 m 2 s; aV ¼ 0.2397 dB/km). Water depth is 100 m and source and receiver depths are 30 m and 50 m, respectively. For the values of other parameters see text and Table 9.4 (the parameters of Figure 9.17—with min ¼ 0:2—are chosen to match those of RMW12). MODES (coh): coherent mode sum (Ellis, 1995); INTEGRAL: Equation (9.200); VD þ UD: Equation (9.229), with B  0.195 Np/rad.

sum of Figure 9.28. For example, three peaks between 10 s and 15 s are clearly visible in the close-up (Figure 9.29). The presence of these maxima can be explained in terms of a sequence of caustics associated with propagation from the source to the seabed and back to the receiver. This point is illustrated by Figure 9.30, which shows propagation loss vs. range and depth for this case, with a source at depth 30 m, including the first five caustics intersecting the seabed at 2:4; 4:1; . . . ; 9:2 km. These distances can be predicted using the simple formula pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sn ðhÞ  8nðn þ 1Þ0 h; ð9:220Þ where n is a positive integer; and h is the height from the seabed to either the source or the receiver. The ray radius of curvature 0 is given by Equation (9.73). More generally, Equation (9.220) translates to arrival times given by tn ðhÞ 

2sn ðhÞ : cw

ð9:221Þ

The derivation of Equation (9.220) follows. Consider a ray with launch angle (i.e., grazing angle at the source) Tx . This ray reflects multiple times from the seabed with a cycle distance of 2ðlo Þ sin lo , where lo is the corresponding grazing angle at

Sec. 9.4]

9.4 Reverberation level 505

Figure 9.29. Model comparison for problem RMW12 and frequency 3.5 kHz (close-up). See Figure 9.28 caption for details.

the low-speed boundary and ðÞ is the radius of curvature given by Equation (9.72). The precise ranges (horizontal distances) at which the ray intersects the seabed are therefore rn ðTx Þ ¼ ðlo Þ½ð2n þ 1Þ sin lo  sin Tx : ð9:222Þ The position of each caustic is determined by the condition drn =dTx ¼ 0, which gives sin lo ¼ ð2n þ 1Þ sin Tx :

ð9:223Þ

Substituting this result into Equation (9.222), and applying Snell’s law, gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sn ¼ 4nðn þ 1Þðc 2Tx =c 2lo  1Þ0 : ð9:224Þ Equation (9.220) follows using the fact that clo and cTx are approximately equal. The caustic ranges and corresponding two-way travel times for the first seven caustics are listed for each of source and receiver height in Table 9.5. For a source depth of 30 m (i.e., h ¼ 70 m), caustics arise at the seabed at distances 2.4 km, 4.1 km, 5.8 km, and so on from the source. The shaded entries of Table 9.5 provide an explanation for why the set of caustics between 10 s and 15 s stand out prominently in Figure 9.29. The arrival times corresponding to the formation of these caustics on the outward path (caustics 4 to 6 for h ¼ 70 m) are approximately equal to those on the return path (caustics 5 to 7

506 Propagation of underwater sound

[Ch. 9

Figure 9.30. Upper: ray trace illustrating formation of caustics and cusps (source depth ¼ 30 m) for a water depth of 100 m and sound speed gradient of 0.3/s (BELLHOP). Lower: propagation loss at 3.5 kHz, corrected for spherical spreading [dB] (the color axis runs from 16 dB (white) to 29 dB (black)) (SCOOTER, see oalib (www)).

for h ¼ 50 m). This numerical coincidence gives enhancement in both directions, resulting in a reinforcement of these three caustics. An approximation to the total reverberation MSP for this case can be derived from the results of Sections 9.4.2.1 and 9.4.2.2 using Q R ¼ S0

2 T ½XVD þ XUD ðzTx Þ ½XVD þ XUD ðzRx Þ : H 2t

ð9:225Þ

Putting q ¼ 1 (Lambert’s rule), hi ¼ 0, and lo ¼ B in Equation (9.205) gives the VD contribution XVD ¼

 1=2 B ct e ½expð2HK=0 Þ  expð 2K

2 c KÞ ;

ð9:226Þ

Sec. 9.4]

9.4 Reverberation level 507 Table 9.5. Caustic ranges and corresponding two-way travel arrival times for a source at depth 30 m (height h from seabed 70 m) (left) and receiver at depth 50 m (height h from seabed 50 m) (right). See Figure 9.30. Source zTx ¼ 30 m ðh ¼ 70 mÞ

Receiver zRx ¼ 50 m ðh ¼ 50 mÞ

n

Range/km

Time/s

Range/km

Time/s

1

2.4

3.2

2.0

2.7

2

4.1

5.5

3.5

4.6

3

5.8

7.7

4.9

6.5

4

7.5

10.0

6.3

8.4

5

9.2

12.2

7.7

10.3

6

10.8

14.5

9.2

12.2

7

12.5

16.7

10.6

14.1

where K ¼ KðtÞ ¼

1 B cw t þ : 6 2H

ð9:227Þ

The UD equivalent is (Equation 9.208)  1=2 XUD ðzÞ ¼ 2

2 0

rffiffiffiffiffi z B cw t artanh e : H

ð9:228Þ

Converting to decibels and including the absorption term explicitly, Equation (9.225) becomes

 40  ðaB þ aV Þcw t 2H 2 t pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 10 log10 f½ f ðtÞ þ artanh zTx =H ½ f ðtÞ þ artanh zRx =H g;

RLðtÞ  SLE þ 10 log10

ð9:229Þ

where f ðtÞ ¼

e 2B cw t fexp½ð2H=0 ÞKðtÞ  exp½ 2 0 KðtÞ

2 c KðtÞ g

ð9:230Þ

and (see Equation 9.141) aB ¼ ð10 log10 eÞ

B : 0

ð9:231Þ

The coefficient B can be evaluated using Equation (9.51) as previously for RMW11, but this approximation tends to overestimate the exponential decay term aB and hence underestimate the reverberation level. A better approximation is obtained

508 Propagation of underwater sound

[Ch. 9

by matching the reflection loss at the steepest angle sustained by the bottom duct, such that 1 B ¼  loge jRð 0 Þj: ð9:232Þ 0

The curve labeled ‘‘VD þ UD’’ in Figure 9.28 is calculated using Equation (9.232) for B , giving a value of 0.195 Np/rad for this parameter.

9.5

SIGNAL-TO-REVERBERATION RATIO (ACTIVE SONAR)

If the source level is sufficiently high, detection performance is determined by the difference between the echo and reverberation levels (i.e., the signal-to-reverberation ratio). This quantity is considered in the closing section of this chapter. The echo level is given vs. range in Section 9.3, whereas reverberation in Section 9.4 is found vs. delay time. For the purpose of determining the signal-to-reverberation ratio (SRR), what matters is the reverberation arriving at the same time as the target echo, so in the following the reverberation is evaluated at the echo arrival time, given by tE ðrÞ 

2r : cw

ð9:233Þ

First, the isovelocity V-duct is considered (Section 9.5.1), followed by the long-range U-duct case (Section 9.5.2). The geometry considered is bistatic in depth and monostatic in range. These SRR calculations do not take into account the effects of beamforming or matched filtering and are therefore relevant to the SRR at the hydrophone, before any processing. 9.5.1

V-duct (isovelocity case)

From Equation (9.169) the signal and reverberation can be written (hereafter including absorption explicitly)  ferff c ðr=HÞ 1=2 g 2 expð4V rÞ ð9:234Þ Q S ðrÞ ¼ S0 4tot Hr 3 and (see Equation 9.193) Q R ðtE Þ ¼ S0



cw  qþ1  ;u 2 4H 3 u qþ2

2 c

 2 expð4V rÞ;

ð9:235Þ

where the variable u (see Equation 9.192) is evaluated at time tE . The SRR is therefore ( ) Q S ðrÞ  erf½ c ðr=HÞ 1=2 2 q1 ¼ u : ð9:236Þ Q R ðtE Þ Hcw  ½ðq þ 1Þ=2; u 2c The absorption cancels in Equation (9.236), which simplifies for short and long

Sec. 9.5]

ranges to

9.5 Signal-to-reverberation ratio (active sonar)

8 2qþ4 > c > > u 1 > > ðq þ 1Þ 2 > < S Q ðrÞ   2

R Q ðtE Þ Hcw  >   2 u q1 > > > q þ 1 > > : G 2

2 cu

1

2 cu

 1.

509

ð9:237Þ

The value q ¼ 1 is critical, resulting in an SRR that is independent of target range (Zhou et al., 1997; Harrison, 2003). For q > 1, in this simple model (which neglects refraction) the SRR actually increases with increasing target range.

9.5.2

U-duct (linear profile)

Putting r ¼ rTx ¼ rRx in Equation (9.171) gives Q S ðrÞ ¼ S0

4O 20 e 4ðlo þV Þr D D : Tx Rx H2 r2

ð9:238Þ

UD reverberation is given by Equation (9.214) Q R ðtE Þ ¼ S0

c 4H 2 r

2qþ2 ðDq ÞTx ðDq ÞRx 0

e 4ðlo þV Þr ;

ð9:239Þ

Figure 9.31. SRR depth factor YðzÞ for the same three different target depths as Figure 9.11 (U-duct propagation).

510 Propagation of underwater sound

[Ch. 9

and therefore Q S ðrÞ 4 ¼ R 2 Q ðtE Þ c

2q 0 r

YTx ðzTx ÞYRx ðzRx Þ:

ð9:240Þ

Notice the simple (1=r) range dependence. The complexity here is in the depth dependence, and it is convenient to define the SRR depth factor YðzÞ such that YðzTx Þ ¼

DTx ðDq ÞTz

ð9:241Þ

YðzRx Þ ¼

DRx : ðDq ÞRx

ð9:242Þ

and

This depth factor is plotted in Figure 9.31 for q ¼ 1; 2. This function is relevant to the long-range U-duct problem, with no V-duct paths.

9.6

REFERENCES

Ainslie, M. A. (1992) The sound pressure field in the ocean due to bottom interacting paths, Ph.D. thesis, ISVR, University of Southampton, U.K. Ainslie, M. A. (1993) Stationary phase evaluation of the bottom interacting field in isovelocity water, J. Acoust. Soc. Am., 94, 1496–1509. [Erratum, J. Acoust. Soc. Am., 95, 3670 (1994).] Ainslie, M. A. (1994) Caustics and beam displacements due to the reflection of spherical waves from a layered half-space, J. Acoust. Soc. Am., 96, 2506–2515. Ainslie, M. A. (2007) Observable parameters from multipath bottom reverberation in shallow water, J. Acoust. Soc. Am., 121, 3363–3376. Ainslie, M. A. (2008) The sonar equations: Definitions and units of individual terms, Acoustics 08, Paris, June 29–July 4 (pp. 119–124).30 Ainslie, M. A. and Harrison, C. H. (1990) Diagnostic tools for the ocean acoustic modeller, in D. Lee, A. Cakmak, and R. Vichnevetsky (Eds.), Computational Acoustics (Vol. 3, pp. 107130), Elsevier. Ainslie, M. A., Harrison C. H., and Burns, P. W. (1996) Signal and reverberation prediction for active sonar by adding acoustic components, IEE Proc.-Radar, Sonar Navig., 143(3), 190– 195. Ainslie, M. A., de Jong, C. A. F., Dol, H. S., Blacquie`re, G., and Marasini, C. (2009) Assessment of Natural and Anthropogenic Sound Sources and Acoustic Propagation in the North Sea (TNO-DV 2009 C085, February), TNO, The Hague, The Netherlands. [M. A. Ainslie, C. A. F. de Jong, H. S. Dol, and G. Blacquie`re, Errata (TNO-DV 2009 C085, May 13), available at http://www.noordzeeloket.nl/overig/bibliotheek.asp (last accessed June 24, 2009).] Boyles, C. A. (1984) Acoustic Waveguides: Applications to Oceanic Science, Wiley, New York. Brekhovskikh, L. M. and Lysanov, Yu. P. (2003) Fundamentals of Ocean Acoustics (Third Edition), Springer Verlag, New York. 30 Available at http://intellagence.eu.com/acoustics2008/acoustics2008/cd1 April 12, 2010).

(last

accessed

Sec. 9.6]

9.6 References

511

Buchanan, J. L. (2006) A comparison of broadband models for sand sediments, J. Acoust. Soc. Am., 120, 3584–3598. Chapman, D. M. F., Ward P. D., and Ellis, D. D. (1989) The effective depth of a Pekeris ocean waveguide, including shear wave effects, J. Acoust. Soc. Am., 85, 648–653. Ellis, D. D. (1995) A shallow-water normal-mode reverberation model, J. Acoust. Soc. Am., 97, 2804–2814. Etter, P. C. (2003) Underwater Acoustics Modeling and Simulation: Principles, Techniques and Applications, Spon Press, New York. Fishback, W. T. (1951) Methods for calculating field strength with standard refraction, in D. E. Kerr (Ed.), Propagation of Short Radio Waves (p. 113), McGraw-Hill. Freehafer, J. E. (1951) The linear modified-index profile, in D. E. Kerr (Ed.), Propagation of Short Radio Waves (p. 99), McGraw-Hill. Hamilton, E. L. (1980) Geoacoustic modeling of the sea floor, J. Acoust. Soc. Am., 68, 1313– 1340. Hamilton, E. L. (1987) Acoustic properties of sediments, in A. Lara Sa´enz, C. Ranz Guerra, and C. Carbo´ Fite (Eds.) (1987) Acoustics and Ocean Bottom, II: F.A.S.E. Specialized Conference, June 18–20, Madrid (pp. 3–58), Consejo Superior de Investigaciones Cientı´ ficas, Madrid. Hamson, R. M. (1997) The modelling of ambient noise due to shipping and wind sources in complex environments, Applied Acoustics, 51(3), 251–287. Harrison, C. H. (1989) Simple techniques for estimating transmission loss in deep water, 13th ICA, Satellite Symposium on Sea Acoustics, Dubrovnik, September 4–6 (pp. 169–172). Harrison, C. H. (2003) Closed-form expressions for ocean reverberation and signal excess with mode stripping and Lambert’s law, J. Acoust. Soc. Am., 114, 2744–2756. Harrison, C. H. (2005a) Closed form bistatic reverberation and target echoes with variable bathymetry and sound speed, IEEE J. Oceanic Eng., 30, 660–675. Harrison, C. H. (2005b) Fast bistatic signal-to-reverberation-ratio calculation, J. Comp. Acoust., 13, 317–340. Harrison, C. H. (2006) An Approximate Form of the Rayleigh Reflection Loss and Its Phase: Application to Reverberation Calculation (NURC-FR-2006-21), NATO Undersea Research Centre, La Spezia, Italy. Jensen, F. B., Kuperman, W. A., Porter, M. B., and Schmidt, H. (1994) Computational Ocean Acoustics, AIP Press, New York. Kerr, D. E. (Ed.) (1951) Propagation of Short Radio Waves, McGraw-Hill. Kibblewhite, A. C. (1989) Attenuation of sound in marine sediments: A review with emphasis on new low-frequency data, J. Acoust. Soc. Am., 86, 716–738. Lara Sa´enz, A., Ranz Guerra, C., and Carbo´ Fite´, C. (1987) Acoustics and Ocean Bottom, II: F.A.S.E. Specialized Conference, June 18–20, Madrid, Consejo Superior de Investigaciones Cientı´ ficas, Madrid. LePage, K. and Thorsos, E. (2006) Final LePage and Thorsos Gaussian Waveform Expression 103006, October 30, available at ftp://ftp.ccs.nrl.navy.mil/pub/ram/RevModWkshp_I/ Final_LePage_and_Thorsos_Gaussian_Waveform_Expression_103006.pdf (last accessed December 15). Lichte, H. (1919) U¨ber den Einfluß horizontaler Temperaturschichtung des Seewassers auf die Reichweite von Unterwasserschallsignalen, Physikalische Zeitschrift, 17, 385–389 [in German]. Nielsen, P. L., Harrison, C. H., and Le Gac, J.-.C. (2008) Proc. International Symposium on Underwater Reverberation and Clutter, September 9-12, NATO Undersea Research Center, La Spezia, Italy.

512 Propagation of underwater sound

[Ch. 9

oalib (www) Ocean Acoustics Library, available at http://oalib.hlsresearch.com/ (last accessed December 15, 2009). Packman, M. N. (1990) A review of surface duct decay constants, Proc. IOA, Vol. 12, Acoustics ’90, IOA Spring Conference, Southampton, Institute of Acoustics, St. Albans, U.K., pp. 139–146. Pierce, A. D. (1989) Acoustics: An Introduction to Its Physical Principles and Applications, American Institute of Physics, New York. rmw (2006) First Reverberation Modeling Workshop, University of Texas at Austin, November, avilable at ftp://ftp.ccs.nrl.navy.mil/pub/ram/RevModWkshp_I/ (last accessed June 10). rmw (2008) Second Reverberation Modeling Workshop, University of Texas at Austin, May, avilable at ftp://ftp.ccs.nrl.navy.mil/pub/ram/RevModWkshp_II/ (last accessed June 10). Schmidt, H. (1988) Seismo-acoustic Fast Field Algorithm for Range-independent Environments: User’s Guide (SACLANTCEN Report SR-113), SACLANT Undersea Research Centre, La Spezia, Italy. Taylor, B. N. (1995) Guide for the Use of the International System of Units (SI) (NIST Special Publication 811), U.S. Department of Commerce/National Institute of Standards & Technology. Weston, D. E. (1960) A Moire´ fringe analog of sound propagation in shallow water, J. Acoust. Soc. Am., 32, 647–654. Weston, D. E. (1979) Guided acoustic waves in the ocean, Reports on Progress in Physics, 42, 347–387. Weston, D. E. (1980) Acoustic flux formulas for range-dependent ocean ducts, J. Acoust. Soc. Am., 68, 269–281. Weston, D. E. (1994) Wave shifts, beam shifts, and their role in modal and adiabatic propagation, J. Acoust. Soc. Am., 96, 406–416. Weston, D. E. and Ching, P. A. (1989) Wind effects in shallow-water transmission, J. Acoust. Soc. Am., 86, 1530–1545. Weston, D. E. and Rowlands, P. B. (1979) Guided acoustic waves in the ocean, Rep. Prog. Phys., 42, 347–387. Winokur, R. and Herr, F. L. (2006) Reverberation Modeling Workshops (Office of the Oceanographer of the Navy (N84), 3140, Ser. N84/875014 and Office of Naval Research 3140, Ser. 321OA/032/06, joint memorandum), available at rmw (2006). Zhou, Jixun (1980) The analytical method of angular power spectrum, range and depth structure of echo-reverberation ratio in shallow water sound field, Acta Acustica, 5, 86–99 [in Chinese].31 Zhou, J., Zhang X. Z., and Luo, E. (1997) Shallow-water reverberation and small angle bottom)scattering, International Conference on Shallow-Water Acoustics, Beijing, China, April 21–25.

31

Translated into English by Zhou in 2007 (Jixun Xhou, pers. commun., August 7 2008).

10 Transmitter and receiver characteristics

I like Wagner’s music much better than anybody’s. It’s so loud that one can talk the whole time without people hearing you. Bob Marley (ca. 1970) Before returning to the sonar equation in Chapter 11, there is one remaining piece to be fitted in the puzzle, namely the characteristics of the sonar itself. Some sonar properties, such as transmitter power through the source level and receiver directivity through the array gain, are incorporated explicitly in the sonar equations. Also important are the frequency, bandwidth, and pulse duration, which affect terms like processing gain, detection threshold, and propagation loss. In this chapter those properties that are intrinsic to sonar systems are collected and tabulated, with particular emphasis on the transmitter source level for active sonar, whether man-made or biological. Receiver sensitivity and self-noise are also considered, represented in the case of biological sonar by the animal’s audiogram. Finally, thresholds are given for possible impact on marine life in the form of behavioral effects and hearing threshold shifts. In the case of man-made equipment, the scope is extended to include all deliberate use of underwater sound. The directivity index of the receiving array is considered in Chapter 6. The information is compiled from many different sources, including Internet sites and commercial literature from equipment manufacturers, as acknowledged in the individual tables. While reasonable attempts have been made to exclude erroneous values, the extended use of non peer–reviewed sources makes it likely that some errors remain. In any case, the systems themselves are subject to review and modification by manufacturers, so that some of the data are likely to be superseded within a few years of publication. In addition, the source level of many sound transmitters is not fixed, but depends on other parameters such as beamwidth and pulse duration. Finally,

514 Transmitter and receiver characteristics

[Ch. 10

regarding the use of sound by, or impact of sound on marine mammals and other aquatic animals, new publications appear continually in the scientific literature, providing new data or questioning old assumptions. Whether for man-made or for biological systems, the reader is therefore advised to check the information by comparing it with up-to-date sources. Transmitters are described first in Section 10.1, which comprises mainly tables of source level and frequency for search sonars, seismic survey sources, explosives, acoustic deterrents, and other miscellaneous sound-producing devices. This is followed in Section 10.2 by a discussion of the sensitivity properties of sonar receivers, including hearing and behavioral thresholds for biological systems.

10.1

TRANSMITTER CHARACTERISTICS

Two of the most important parameters determining the suitability of an acoustic transmitter for a given application are its frequency and its source level. These two properties are listed below for a number of different sonar types. Also important, for the assessment of both sonar performance and environmental impact, are pulse duration, bandwidth, and beamwidth. Though not presented here, in many cases these can be found in the references provided, or obtained from the manufacturer. A useful distinction can be made between devices whose objective it is to measure some property of the seafloor (echo sounders, bottom profilers, and seismic survey sources), generally designed to direct energy down in the vertical direction, and search sonar or communications equipment, which is generally directed in a horizontal direction. A third category, which includes acoustic deterrents and transponding or communication equipment, generally uses omni-directional transmitters. The source level of a transducer depends on the supplied voltage and its sensitivity.1 Alternatively, the source level can be written as a function of total transmitted power and sonar directivity. Either way, transmitter sensitivity and directivity are excluded from the present scope (the interested reader is referred to publications by Tucker and Gazey, 1966; Stansfield, 1991; and Blue and Van Buren, 1997). Thus, for the remainder of this chapter the terms ‘‘sensitivity’’ and ‘‘directivity’’ refer exclusively to the sonar receiver. A common misunderstanding that arises about the term source level arises from its definition by the American National Standards Institute (ANSI) and the International Electrotechnical Commission (IEC) as the sound pressure level at the standard reference distance (1 m) from the source. Neither of these definitions mentions the need for the measurement to be in the far field (Morse and Ingard, 1968), as required by the more complete descriptions of Kinsler et al. (1982) and Urick (1983), thus limiting their applicability to compact sources. The details are deferred to Chapter 11, but the important point here is that in general the source level is not even approximately equal to the sound pressure level at 1 m, making the ANSI and IEC definitions at best misleading. Briefly, the role of the source level is to provide a 1

The sensitivity of a sonar transmitter is the source factor divided by the mean square voltage.

Sec. 10.1]

10.1 Transmitter characteristics

515

prediction of the sound pressure level in the far field of the source. It is not a useful measure of the field close to an array of transducers spanning several wavelengths.

10.1.1

Of man-made systems

Man-made equipment for the transmission of underwater sound is considered here. Applications of such transmissions include conventional sonar (i.e., equipment whose primary purpose is the detection and localization of underwater objects), acoustic deterrents, communication and positioning equipment, and oceanographic measurement systems such as used for seismic or bathymetric surveys, acoustic thermometry, or current measurement. It is useful to distinguish between a continuous source, whose RMS pressure field remains approximately unchanged during transmission, and an impulsive source, which either decays without oscillation or whose amplitude changes significantly from one cycle to the next. These are introduced in the following two sub-sections. More specialized information about sonar transducers is provided by Hunt (1954) and Stansfield (1991). 10.1.1.1

Continuous sources

By a continuous source is meant one that transmits a signal whose amplitude remains unchanged after many cycles, such that an associated RMS pressure field can be defined unambiguously, as is the case for a tonal source or a frequency sweep of constant amplitude. Typical examples include echo sounders, sidescan sonar, communications equipment, and military search sonar. 10.1.1.1.1

Single-beam echo sounders

Perhaps the most widely used man-made sonar is the basic single-beam echo sounder, designed to measure the travel time (and hence distance) to an object (usually the seabed or a shoal of fish) beneath the vessel carrying the sonar. The frequency and source level of some single-beam echo sounders are listed in Table 10.1. Based on this table, a typical value for the (maximum) source level of a single-beam echo sounder is about 214
6 dB re mPa 2 m 2 , with little dependence on frequency between 12 kHz and 200 kHz. The values quoted are maximum source levels for each sonar. The actual source level depends on pulse duration and beamwidth. The highest source levels are usually associated with narrow beams and short pulses. 10.1.1.1.2 Sidescan sonar A sidescan sonar works on a similar principle as an echo sounder, except that the sound is projected sideways as well as vertically downwards (Blondel, 2009). Successive echoes are recorded and used to build up a high-resolution image of the surroundings. Table 10.2 summarizes source level data for sidescan sonars. A typical (maximum) value is about 225 dB re mPa 2 m 2 at 100 kHz, decreasing with increasing frequency.

516 Transmitter and receiver characteristics

[Ch. 10

Table 10.1. Summary of single-beam echo sounder source levels, sorted by frequency. Frequency/ Max. SL/ kHz (dB re mPa 2 m 2 Þ

Manufacturer

System

Reference

Notes

Submarine Signal Co. (now Raytheon)

Fessenden oscillator

ca. 1

204

Hackmann (1984)

Early echo sounder, included for historical interest

Massa

TR-1073A

12

216

massa (www)

Simrad

38/200 combi w

38

208

simrad (www)

Simrad

HTL 430D

42

200

kongsberg (www)

Simrad

EK 500

57

214

LIPI (2006)

Simrad

SD 570

57

220

LIPI (2006)

Simrad

38/200 combi w

200

208

simrad (www)

Biosonics

DT4000

208

221

Churnside et al. (2003)

Also operates at 200 kHz

Also operates at 38 kHz

10.1.1.1.3 Multibeam echo sounders More sophisticated echo sounders exist that are capable of beamforming the echo and thus distinguish between vertical arrival angles of seabed returns (Lurton, 2002). Source levels and related properties of such multibeam echo sounders are listed in Table 10.3. The trend of decreasing source level with increasing frequency is similar to that for a sidescan sonar. Figure 10.1 shows a graph of source levels for sidescan and multibeam sonars taken from Tables 10.2 and 10.3, plotted as a function of frequency. The graph shows a downward trend with increasing frequency for frequencies between 12 kHz and 675 kHz. The regression curve is a straight line fit in log F SL ¼ 259:5  17:43 log10 F

dB re mPa 2 m 2 ;

ð10:1Þ

where F is the frequency in kilohertz.

10.1.1.1.4 Sub-bottom profilers A depth profiler or sub-bottom profiler is similar to an echo sounder, except that a lower frequency is used in order to probe more deeply into the sediment. A list of source levels is given in Table 10.4.

Table 10.2. Summary of sidescan sonar source levels, sorted by frequency. Frequency/ Max. SL/ kHz (dB re mPa 2 m 2 Þ

Manufacturer

System

Reference

Notes

Racal

SeaMARC SB12

12

233

Funnell (1998, p. 121)

Simrad

AMS 36/120SI

35

228

Watts (2000, p. 453) Funnell (1998, p. 124)

Racal

SeaMARC SB50

50

200

Funnell (1998, p. 121)

Simrad

AMS 60SI

57.6

227

Watts (2000) Funnell (1998, p. 124)

Neptune

990 Tow Fish

59

227

neptune (www)

Ultra

Deepscan 60

60

231

Watts (2000, p. 446) Funnell (1998, p. 136)

Massa

TR-1101

97

223

massa (www)

Innomar

Sidescan ses-2000

100

220

innomar (www)

Neptune

422 Tow Fish

100

228

neptune (www)

Also operates at 500 kHz

GEC Marconi Bathyscan

100

220

Funnell (1998, p. 122)

Also operates at 300 kHz

Neptune

272 Tow Fish (dual beam)

105

234

neptune (www)

Second beam operates with a source level of 229 dB re mPa 2 m 2

Neptune

272 Tow Fish (dual frequency)

105

229

neptune (www)

Also operates at 500 kHz

Geoacoustics

DSSS

114

223

geoacoustics (www)

Also operates at 410 kHz

Simrad

AMS 120SI AMS 120SP

120

224

Watts (2000, p. 453) Funnell (1998, p. 124)

Simrad

AMS 36/120SI

120

224

Watts (2000, p. 453) Funnell (1998, p. 124)

Also operates at 35 kHz

Benthos

C3D

200

224

benthos (www)

Also operates at 100 kHz

GEC Marconi Bathyscan

300

220

Funnell (1998, p. 122)

Also operates at 100 kHz

Tritech

SeaKing

325

208

Funnell (1998, p. 104)

Also operates at 675 kHz

Geoacoustics

DSSS

410

223

geoacoustics (www)

Also operates at 114 kHz

Neptune

272 Tow Fish (dual frequency)

500

223

neptune (www)

Also operates at 105 kHz

Neptune

422 Tow Fish

500

220

neptune (www)

Also operates at 100 kHz

Tritech

SeaKing

675

208

Funnell (1998, p. 104)

Also operates at 325 kHz

Actual frequencies are 33.3 and 36.0 kHz; also operates at 120 kHz

Table 10.3. Summary of multibeam echo sounder source levels, sorted by increasing frequency. Frequency/ Max. SL/ kHz (dB re mPa 2 m 2 Þ

Manufacturer

System

Reference

Simrad

EM 120

12

245

Hammerstad (www)

Simrad

EM 121A

12

238

Watts (2005) Funnell (1998, p. 126)

ELAC

Sea Beam 2000

12

234

Watts (2000, p. 456) Funnell (1998, p. 131)

ELAC

Sea Beam 3012

12

239

seabeam (www)

Thomson TSM 5265 Marconi Sonar (Thales)

12

235

Watts (2000, p. 441) Funnell (1998)

Simrad

EM 12

13

238

Funnell (1998, p. 125)

ELAC

Sea Beam 2120

20

247

seabeam (www)

Simrad

EM 300

30

241

Hammerstad (www)

ELAC

Sea Beam 1050

50

234

Kvitek et al. (1999)

Simrad

EM 710

85

232

Hammerstad (www)

Simrad

EM 1000

95

225

Funnell (1998, p. 127) Hammerstad (www)

Simrad

EM 1002

95

226

Kvitek et al. (1999)

Simrad

EM 950

95

225

Kvitek et al. (1999) Funnell (1998, p. 127)

Simrad

EM 952

95

226

Kvitek et al. (1999)

Atlas

Fansweep 20

100

227

Kvitek et al. (1999)

Thomson TSM 5260 Marconi Sonar (Thales)

100

210

Watts (2000, p. 441) Funnell (1998, p. 135)

Triton

ISIS 100

117

219

Kvitek et al. (1999)

ELAC

Sea Beam 1185

180

217

Kvitek et al. (1999)

ELAC

Sea Beam 1180

180

220

Funnell (1998, p. 130)

Atlas

Fansweep 15

200

227

Kvitek et al. (1999)

Atlas

Fansweep 20

200

227

Kvitek et al. (1999)

Reson

Seabat 7125

200

220

Lo¨vgren (2007)

Reson

Seabat 8124

200

210

Kvitek et al. (1999)

Simrad

EM 2000

200

218

Hammerstad (www)

200

225

Kvitek et al. (1999)

ECHOSCAN Triton

ISIS 100

234

219

Kvitek et al. (1999)

Reson

Seabat 8101

240

217

Kvitek et al. (1999)

Simrad

EM 3000

300

215

Kvitek et al. (1999)

Reson

Seabat 9001

455

210

Kvitek et al. (1999) Watts (2005)

Notes

70–100 kHz

Also operates at 200 kHz

Also operates at 234 kHz

Also operates at 100 kHz

Also operates at 117 kHz

Sec. 10.1]

10.1 Transmitter characteristics

519

Figure 10.1. Maximum multibeam echo sounder and sidescan sonar source levels vs. transmitter frequency.

10.1.1.1.5 Fisheries sonar Some echo sounders are adapted to look for fish by tilting them away from the vertical direction, giving them increased area coverage by scanning at oblique angles. Examples are listed in Table 10.5.

10.1.1.1.6

Military search sonar

For some military applications there is a requirement to detect objects at very long ranges in order to respond early to a potential threat. By contrast, some systems are designed to achieve high resolution, working by necessity at much higher frequency and hence limited to short range, leading to a wide range of sonar specifications. The following tables summarize the source levels and frequency ranges for hull-mounted sonar (Table 10.6), dipping sonar (Table 10.7), towed array sonar (Table 10.8), and other sonars (Table 10.9). Recall that the source level is a measure of the power (more precisely the radiant intensity) projected by a sonar transmitter into its far field. This point is of special significance if the extent of the sonar transmitter is large compared with the acoustic wavelength, such as for an array of two or more synchronized transducers (an example from Table 10.8 is SURTASS). In this situation, the source level is not related in a simple way to the sound pressure level at 1 m.

520 Transmitter and receiver characteristics

[Ch. 10

Table 10.4. Summary of depth profilers, sorted by increasing maximum source level. Manufacturer System

Center Max. SL/ frequency/ (dB re kHz mPa 2 m 2 Þ

Reference

Notes

Massa

TR-1061A

5

199

massa (www)

Massa

TR-1075A

4

201

massa (www)

Geoacoustics

geochirp

7

205

geoacoustics (www)

Ultra

Deepscan 60

10

212

Watts (2000, p. 446) 7.5–12.5 kHz Funnell (1998, p. 136)

Geoacoustics

T135 transducer

5.5

214

geoacoustics (www)

3–7 kHz

Geoacoustics

Array of 16 T135 transducers

5.5

225

geoacoustics (www)

3–7 kHz

Geoacoustics

geopulse

7

227

geoacoustics (www)

2–12 kHz

Innomar

compact

7

236

innomar (www)

2–12 kHz

0.5–13 kHz

Table 10.5. Summary of fisheries sonar source levels, sorted by maximum frequency. Manufacturer System

Frequency/ Max. SL/ kHz (dB re mPa 2 m 2 Þ

Reference

Notes

Simrad

SX90

20–30

219

simrad (www)

Simrad

SP70

26

222

kongsberg (www)

Simrad

SP90

26

223

kongsberg (www)

Simrad

SH80

116

210

kongsberg (www)

Simrad

SH80

110–122

210

simrad (www)

Simrad

ES60 single beam

12–200

>206

simrad (www)

4 kW at 38 kHz

Simrad

ES60 split beam

18–200

>206

simrad (www)

4 kW at 38 kHz

10.1.1.1.7 Minesweeping sonar Modern navies seek to reduce the threat of sea mines by a number of different measures. One way is to use high-frequency search sonar, called ‘‘minehunting’’ sonar (see Tables 10.6 and 10.9), to find the mines before avoiding or deactivating them. An alternative strategy consists of transmitting a signal that reproduces the acoustic signature of a passing ship, thus neutralizing the mine by precipitating its premature

Sec. 10.1]

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521

Table 10.6. Summary of hull-mounted search sonars, sorted by maximum frequency. Model

Frequency/ kHz

USN AN/SQS-53C

2.6, 3.3, 3.5

235

USN AN/SQS-56/DE 1160

6.7, 7.5, 8.4

218–232

Watts (2005, p. 127ff )

7.5, 12.0

227

Watts (2005, p. 161) See also DE 1167 VDS (Table 10.8)

DE 1167 HM

Improved DE 1160

Source level/ Reference (dB re mPa 2 m 2 ) Anon. (2001) Anon. (2003, p. 66) Kuperman and Roux (2007, Table 5.3) AGISC (2005)

3.75, 5.0, 7.5, 232 (7.5 kHz) Watts (2005, p. 127ff ) 12 238 (3.75 kHz)

SS 2450

24

213

Watts (2005, p. 148)

CTS-24 ASW OMNI sonar

24

223

Watts (2005, p. 135)

CMAS-36/39 mine detection and avoidance sonar

36, 39

223

Watts (2005, p. 437)

CTS-36/39 OMNI sonar

36, 39

223

Watts (2005, p. 135)

95

220

Watts (2005)

SS 9500 mine detection and avoidance sonar

Table 10.7. Summary of helicopter dipping sonars, sorted by maximum frequency. Model

Frequency/ kHz

Helras

1.31–1.45

218

Watts (2005)

AN/AQS-13F

9.2–10.7

216

Watts (2005, p. 193)

AN/AQS-18

9.2–10.7

216

Watts (2005, p. 194)

AN/AQS-18A

9.2–10.7

216

Watts (2005, p. 194)

13

212

Watts (2005)

HS 12

Source level/ Reference (dB re mPa 2 m 2 )

522 Transmitter and receiver characteristics

[Ch. 10

Table 10.8. Summary of active towed array sonars, sorted by maximum frequency. Model

Frequency/ kHz

USN SURTASS-LFA a projector (single LFA frequency projector)

0.1–0.5

215

USN SURTASS-LFA projector (array of up to 18 LFA projectors)

0.1–0.5

221–240

Hildebrand (2004) dosits (www)

1.38

219–222

Watts (2005, p. 163)

DE 1167 VDS

12

217

Watts (2005, p. 161) See also DE 1167 HM sonar (Table 10.6)

ST 2400

24

213

Watts (2005, p. 149)

LFATS (derived from Helras technology b )

a b

Source level/ Reference (dB re mPa 2 m 2 ) Anon. (2003, p. 66) Kuperman and Roux (2007) AGISC (2005)

SURTASS-LFA: U.S. Navy Surveillance Towed Array System—Low Frequency Active. See Table 10.7.

Table 10.9. Summary of miscellaneous search sonar (including coastguard and risk mitigation sonar), sorted by maximum frequency. Sonar type

Model

Active sonobuoy

RASSPUTIN

Active sonobuoy

AN/SSQ-62RO

Coastguard search sonar

SS105

Marine mammal risk HFM3 mitigation sonar Minehunting ROV a scanning sonar a

SeaBat 6012

Frequency/ kHz

Source level/ Reference (dB re mPa 2 m 2 )

1.5

205

Watts (2005)

6.5–9.5

>199

Watts (2005)

14

230

Watts (2000, p. 99)

30–40

220

Ellison and Stein (2001)

455

210 (nominal)

Watts (2005)

ROV: remotely operated vehicle.

explosion. The transmitting device is called a ‘‘minesweeping’’ or ‘‘influence sweep’’ sonar. An example from Watts (2000) is the Sterne 1 system of Thomson Marconi (now Thales Underwater Systems), with a source level of 160 dB re mPa 2 m 2 in the frequency range 10 Hz to 200 Hz.

Sec. 10.1]

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523

10.1.1.1.8 Acoustic deterrent devices Sound transmitters are used or have potential for use underwater to protect: — fish farms by deterring predators; — sensitive fauna by warning them away from fishing nets, explosions, pile driving, or other equipment posing a potential danger; — valuable or sensitive harbor facilities by deterring malicious human divers or trained animals. Such acoustic deterrents can be grouped into two broad classes. The first class includes relatively low power devices, sometimes known as ‘‘pingers’’ or ‘‘alarms’’, intended to deter mammals from approaching fishing gear, in order to prevent them becoming entangled in the nets. A list of the source levels of these low-amplitude deterrents is given in Table 10.10. The second class of deterrents, with a higher source level (sometimes known as ‘‘acoustic harassment devices’’), is used to clear a larger area, either to protect the animals from exposure to loud sounds that might be anticipated (such as an explosion) or to protect fish farms from predators. The properties of some higher amplitude deterrents are listed in Table 10.11. The source levels in this table all exceed 190 dB re mPa 2 m 2 , compared with up to 179 dB re mPa 2 m 2 in Table 10.10. 10.1.1.1.9

Underwater communications systems and transponders

In the same way that sonar is used as an alternative to radar for the detection of underwater objects, sound provides an alternative to radio waves for the transmission of underwater signals. The source levels of transducers used in underwater communications systems (Table 10.12) and transponders (Table 10.13) are summarized below. 10.1.1.1.10

High-frequency imaging sonar

Very high frequency sonars, operating at frequencies around 1 MHz, are able to produce high-resolution images that resemble photographs. For this reason they are sometimes known as ‘‘acoustic cameras’’. The properties of such systems are listed in Table 10.14. 10.1.1.1.11 Research instruments (global oceanography) Sonar technology is increasingly used for exploration and monitoring of the world’s oceans. The source levels of a selection of research sonars are summarized in Table 10.15, including the source used for the Heard Island Feasibility Test (HIFT), a global-scale test transmission carried out in 1991 from the Indian Ocean to both Pacific and Atlantic Oceans (Munk et al., 1994). Because the HIFT source spans several wavelengths, the source level is not related in a simple way to the sound pressure level at 1 m.

Table 10.10. Summary of low-amplitude acoustic deterrents, sorted by maximum source level. BB: broadband. Manufacturer or originator

System

Frequency/ kHz

Source Reference level/ (dB re mPa 2 m 2 Þ

Notes

Gearin

BB

122–125

Gearin et al. (2000)

Peaks at 3 and 20 kHz

Lien

2.5

110–132

Fullilove (1994)

McPherson

3.5

110–132

Gordon & Northridge (2002, table 2)

FMP 332

10

130–134

Gordon & Northridge (2002, table 2)

Airmar

Airmar gillnet

9.8

134

Kastelein et al. (2007)

SaveWave

Endurance

Aquatec Sub-sea

Aquamark 200

SaveWave

White high impact

Fumunda

FMDP 2000

SaveWave

Black high impact

5–110

134
1.3 Kastelein et al. (2007)

BB

BB

134
1.3 Kastelein et al. (2007)

Frequency sweeps

5–95

140
0.6 Kastelein et al. (2007)

BB

9.6 33–97

141

Kastelein et al. (2007)

143
0.7 Kastelein et al. (2007)

BB 3 tonals

Loughborough PICE University

55 83 100

137–145 133–138 95–120

Culik et al. (2001)

Aquatec Sub-sea

Aquamark 300

10

145

Gordon & Northridge (2002, table 2)

Dukane

NetMark 1000

10–12

120–146

Barlow and Cameron (2003)

Aquatec Sub-sea

Aquamark 100

20–160

148
3.7 Kastelein et al. (2007)

Frequency sweeps

Dukane

NetMark 2000

10

130–150

Gordon & Northridge (2002, table 2)

Discontinued

Dukane

NetMark 2MP

9–15

127–152

Kastelein et al. (2001)

16 tonals

Dukane

NetMark XP-10

9–15

133–163

Kastelein et al. (2001)

16 tonals

Ocean Engineering Enterprise

DRS-8

0.6

172

Kastelein et al. (2007)

TERECOS

Type DSMS-4

4.9

179

Lepper et al. (2004)

Discontinued

10.1 Transmitter characteristics

525

Table 10.11. Summary of high-amplitude acoustic deterrents, sorted by maximum source level.

a

Manufacturer

System

Frequency/ kHz

Source level/ (dB re mPa 2 m 2 Þ

Simrad

Reference

Notes

fishguard

15

191

Gordon & Northridge (2002, table 1)

Airmar

dB Plus II

10.3

192

Lepper et al. (2004)

Ace Aquatec

Silent scrammer

10 16

193 194

Lepper et al. (2004) ace (www) a

Multi-tone (19 frequencies from 3.3 to 20 kHz)

FerrantiThomson

Mk. 2 seal scrammer

8–30

194

Gordon & Northridge (2002, table 1)

Multi-tone

FerrantiThomson

Mk. 3 seal scrammer

25

194

Taylor et al. (1997)

Ocean Engineering Enterprise

DRS-8

3.0

202

Kastelein et al. (2007)

Universal scrammer AA-01-048V2.

10.1.1.2 10.1.1.2.1

Impulsive sources General characteristics

The continuous sources described above are characterized by their mean square pressure (MSP), or SLMSP . Pulses whose amplitudes vary with time, sometimes rapidly, are considered next. For such pulses it is not obvious how to define MSP because the result of averaging depends critically on the extent in time during which the averaging takes place. The following comments apply whether the source is natural or man-made. Peak-to-peak pressure, zero-to-peak pressure, and integrated pressure squared. It is common to characterize impulsive sources in terms of their peak-to-peak (p-p) or zero-to-peak (z-p) pressure. For short pulses, changes in the shape of the pulse can occur over time (e.g., due to multipath propagation in shallow water) so that care is needed in the interpretation of reported peak values. For this reason, the source energy (characterized by the energy source level SLE ), which is not affected by changes in pulse shape, is a more robust measure than p-p or z-p pressure for the characterization of short pulses. Peak sound pressure values are often converted to corresponding source levels in decibels, denoted SLp-p or SLz-p (these parameters are defined in Chapter 8). Such conversion is sometimes questioned on the grounds that the decibel should be reserved for expressing ratios of power or energy, whereas the peak sound pressure

526 Transmitter and receiver characteristics

[Ch. 10

Table 10.12. Summary of acoustic communications systems, sorted by increasing maximum source level. (LF: low frequency; HF: high frequency). Manufacturer

System

Sparton Corporation Mk. 84 SUS (expendable air to submarine communications device)

Frequency/ kHz

Max. SL/ Reference (dB re mPa 2 m 2 )

3.3, 3.5

160

Watts (2005, p. 268)

Orcatron

Scubaphone

30

171

Watts (2000, p. 395)

Fugro UDI

Subcom 3400

25

171

Funnell (1998, p. 262)

Nautronix

Secure Hellephone

8, 25

174

Funnell (1998, p. 264)

Ocean Technology Systems

Aquacom SSB-2010

30–35

176

Funnell (1998, p. 265)

Ocean Technology Systems

Aquacom SSB-1001B

22–35

178

(Funnell 1998, p. 266)

Tritech

AM-300

8–16, or 16–24

184

tritech (www)

Orca Instrumentation MATS 12

10–14

185

Funnell (1998, p. 257)

Orca Instrumentation MATS 53

50–58

185

Funnell (1998, p. 257)

MARCOM Defence

Type 185/ G732 Mk. II

8.4–11.3

186

Watts (2005, p. 265)

Kongsberg Simrad

SPT 319

24.5–32.5

195

ashtead (www)

L3 Communications ELAC Nautik

UT 2000

1–60

196 (LF) 188 (HF)

Massa

TR-1036D

8

198

massa (www)

Massa

TR-1055C

12

199

massa (www)

Harris Products Corporation

AN/WQC-2A

1.45–3.10 (LF) 8.3–11.1 (HF)

199 (LF) 198 (HF)

Kongsberg Simrad

MPT 331DTRDUB

24.5–32.5

206

Watts (2005, p. 257)

Watts (2005, p. 268) ashtead (www)

is neither of these, even when squared. The practical reality is that the decibel often does get used for ratios that are not powers or energies, and this is one example. The use here of p-p and z-p source levels in decibels, while not intended as an endorsement of this practice, acknowledges its widespread adoption in the literature.

Sec. 10.1]

10.1 Transmitter characteristics

527

Table 10.13. Summary of selected acoustic transponders and alerts, sorted by increasing maximum source level. Frequency/ Max. SL/ kHz (dB re mPa 2 m 2 )

Manufacturer

System

Reference

IXSEA

RP402E

37.5
1

157

ixsea (www)

IXSEA

RP162E

37.5
1

160

ixsea (www)

Sonardyne

Type 7097

36–44

185

sonardyne (www)

InterOcean

1090ET

7.5–9

190

interocean (www)

Ore Offshore

BRT6000

12

190

ore (www)

Kongsberg

MST 319

ca. 30

190

kongsberg (www)

Sonardyne

Type 7815

35–55

190

sonardyne (www)

Sonardyne

Type 8014

50–110

190

sonardyne (www)

IXSEA

MF range

20–30

191
4

ixsea (www)

IXSEA

LF range

8-16

192
4

ixsea (www)

Sonardyne

Type 8065

19–36

193

sonardyne (www)

Sonardyne

Type 8011

7.5–15

195

sonardyne (www)

Kongsberg

MST 324

ca. 30

197

kongsberg (www)

Sonardyne

Type 8106

14–19

197

sonardyne (www)

Kongsberg

MST 342

ca. 30

203

kongsberg (www)

Sonardyne

Type 8129

18–36

207

sonardyne (www)

Table 10.14. Summary of acoustic cameras. Manufacturer

System

Fugro-UDI

Sonavision 2000

Tritech Tritech

Frequency/ Max. SL/ kHz (dB re mPa 2 m 2 )

Reference

500

208

Funnell (1998, p. 101)

SeaKing DFS

325, 675

212

Funnell (1998, p. 103)

SeaKing DFP

580, 1200

212

Funnell (1998, p. 103)

528 Transmitter and receiver characteristics

[Ch. 10

Table 10.15. Summary of miscellaneous oceanographic sonar. Frequency/ Hz

Max. SL/ (dB re mPa 2 m 2 )

Global thermometry ATOC a source

75

195

Anon. (2003, p. 70) Kuperman and Roux (2007)

Global thermometry HIFT source (single transducer)

57

206

Munk et al. (1994)

Global thermometry HIFT source (array of five transducers)

57

221

Munk et al. (1994)

300–400 (sweep), or 185–310 (CW)

195

Hildebrand (2004, p. 9)

Application

Oceanography

System

RAFOS b

Reference

a

ATOC: Acoustic Thermometry of Ocean Cllimate (atoc, www). RAFOS (for SOFAR, spelt backwards) is a system of ocean floats designed to collect oceanographic data. The floats rely for navigation on a network of moored transmitters (rafos, www). b

It is convenient to define peak-to-peak (p-p) and zero-to-peak (z-p) source factors ðS0 Þp-p and ðS0 Þz-p in terms of the corresponding source levels as ðS0 Þp-p  10 ðSLp-p =10Þ

mPa 2 m 2

ð10:2Þ

ðS0 Þz-p  10 ðSLz-p =10Þ

mPa 2 m 2 :

ð10:3Þ

and

Similarly, the energy source factor can be characterized in terms of the timeintegrated squared pressure (a measure of total transmitted energy) in the form ðS0 ÞE  10 ðSLE =10Þ

mPa 2 m 2 s:

ð10:4Þ

Expressions for these three parameters are listed in Table 10.16 for two pulse shapes, both of which are symmetrical about the origin. The first one is a cosine wave with uniform amplitude between times  and þ, and zero outside this range; the other is a Gaussian-modulated cosine wave with the same maximum amplitude as the unweighted pulse. In both cases, the parameter  is the time at which the pulse amplitude decays to 1=e of its peak value. Thus, the pulse extends approximately between  and þ in time. The two functions are plotted in Figure 10.2. In both cases the z-p and p-p source levels are related via SLp-p ¼ SLz-p þ 6:0:

ð10:5Þ

Similarly, the ‘‘pulse energy’’ (strictly, the energy source factor, which for a point source is the product of time-integrated squared pressure and the square of the

Sec. 10.1]

10.1 Transmitter characteristics

529

Table 10.16. Relationships between different source level definitions for two symmetrical wave forms. The expressions for q0 ðtÞ (acoustic pressure at distance s0 ) are valid for a point source in free space. Tx descriptor

Unweighted cosine

s0 q0 ðtÞ

pffiffiffi 2A cosð!tÞHð  tÞHð þ tÞ The duration 2 is assumed equal to an odd number of half-cycles such that 2! ¼ ð2n þ 1Þ, where n is an integer

Gaussian weighted cosine  2 pffiffiffi t 2A cosð!tÞ exp  2  The duration (i.e., the width of the Gaussian) is chosen to ensure the same pulse energy, in the highfrequency limit, as the unweighted cosine

ðS0 Þz-p

2A 2

2A 2

ðS0 Þp-p

8A 2

  2 2 2A 2 1  cos 1 exp  2 1 2 ! 

ðS0 ÞE

2A 2 

Here 1 is the first non-zero solution to the transcendental equation 21 þ ! 2  2 tan 1 ¼ 0. The highfrequency limit is 1 !  and ðS0 Þp-p ! 8A 2 . rffiffiffi     2 !2  2 A  1 þ exp  2 2

measurement distance, close to the source) is related to the p-p source level via
SLp-p þ 10 log10 ð2 Þ  9:0 unweighted SLE ¼ ð10:6Þ SLp-p þ 10 log10 ð2 Þ  11:0 Gaussian modulated. Next consider a pressure pulse that is switched on at its maximum amplitude at time t ¼ 0, followed by an exponential decay with time constant . Table 10.17 lists the properties of two such asymmetrical pulses, the first a damped sine wave and the second a simple exponentially decaying pressure (see Figure 10.3). The relationships between SLE , SLz-p , and SLp-p corresponding to Table 10.17 are
SLz-p þ 6:0 damped sine SLp-p ¼ ð10:7Þ SLz-p exponential and
SLp-p þ 10 log10   12:0 damped sine SLE ¼ ð10:8Þ SLp-p þ 10 log10   3:0 exponential. RMS pressure. Source levels or received levels of transient fields are sometimes reported as RMS values. What this means is that the squared pressure has been

530 Transmitter and receiver characteristics

[Ch. 10

Figure 10.2. Unweighted (upper) and Gaussian-weighted (lower) cosine pulses from Table 10.16.

Sec. 10.1]

10.1 Transmitter characteristics

531

Table 10.17. Relationships between different source level definitions for two asymmetrical wave forms. The expressions for q0 ðtÞ (acoustic pressure at distance s0 ) are valid for a point source in free space. Tx descriptor

Exponentially damped sine (

s0 q0 ðtÞ

ðS0 Þz-p

0 t 445:2F kHz  344:3 0:5  FkHz < 11:3 > < 0:7578 1:076 HTð f Þ ¼ 242:9F kHz þ 0:5643F kHz 11:3  FkHz < 46:2 ð11:159Þ > > : 2:792F 0:7537 46:2  FkHz  80, kHz  2:064 where FkHz is the frequency in kilohertz f : ð11:160Þ 1 kHz The audiogram calculated using Equation (11.159) is plotted in Figure 11.20, including the original measurements on which it is based. Using the same formula, the minimum hearing threshold, of 39.0 dB re mPa 2 , occurs at 22.6 kHz. FkHz ¼

11.4.6.1.3

Maximum audibility range

Strictly speaking, what is needed for a calculation of audibility is the hearing threshold for a broadband pulse. Because such an audiogram is not available, a rough estimate is made instead using the hearing threshold at the pulse center frequency. The threshold using Equation (11.159) at the pulse center frequency (50 kHz) is 51.2 dB re mPa 2 . Armed with this information, the figure of merit (the value of one-way PL for which the unmasked sound is just audible) can be calculated as (see Table 11.6) FOMHT ¼

SL þ TS  HT ¼ 59:0 2

dB re m 2 ;

ð11:161Þ

which intersects PLð fm Þ at about 400 m (see Figure 11.19). This is the maximum distance at which the echo could be heard if the frequency were precisely 50 kHz and

620 The sonar equations revisited

[Ch. 11

Figure 11.20. Orca audiogram due to Wensveen and Van Roij (2007) and individual hearing threshold measurements of Hall and Johnson (1972) and Szymanski et al. (1999).

Table 11.6. Sonar equation calculation for active sonar example (orca vs. salmon)— limited by hearing threshold. Description

Symbol

Source level

SL

198.2 dB

re mPa 2 m 2

Target strength

TS

29.0 dB

re m 2

Hearing threshold at 50 kHz

HT

51.2 dB

re mPa 2

FOMHT

59.0 dB

re m 2

Figure of merit (hearing threshold limited)

Value

there were no masking noise or reverberation. The actual broadband propagation loss is slightly lower than the mid-frequency value PLð fm Þ, and the broadband hearing threshold is also lower than at fm (Figure 11.20). Both factors favor even longer ranges, illustrating this animal’s remarkable potential in a quiet environment.27 The real-world limitations of the orca’s detection capability caused by the presence of background noise are considered next. 27 Most of the energy is in the lower half of the frequency range, to which the animal is most sensitive.

Sec. 11.4]

11.4 Active sonar with coherent processing: matched filter 621

Figure 11.21. Echo level (EL) vs. distance between orca and salmon. Also shown (horizontal lines) are noise level (NL), and the quantities (NL  AG) and (NL  AG þ DT), all for wind speed v10 ¼ 2 m/s. The detection range (i.e., the intersection of EL with NL  AG þ DT) is approximately 240 m.

11.4.6.2

Part (ii) detection range for low wind speed (noise-limited)

At what distance s is the orca able to detect the salmon using its active sonar? To answer this question, it is necessary to consider the background against which the echo is to be detected, which for low wind is determined by ambient noise. Signal excess is given by Equation (11.91), which, neglecting reverberation, simplifies to SE ¼ EL  ðNL  AGÞ  DT:

ð11:162Þ

The four terms on the right-hand side are now considered in turn. First, echo level (EL) is plotted in Figure 11.21 (red curve, calculated using Equation 11.141). The calculation of noise level (NL), plotted as a dashed blue line, is described in Section 11.4.6.2.1, followed by discussions of array gain (AG) in Section 11.4.6.2.2 and detection threshold (DT) in Section 11.4.6.2.3. Finally, signal excess and detection range are the subject of Section 11.4.6.2.4.

11.4.6.2.1

Noise level (NL)

In the frequency range of interest here it is appropriate to use the APL formula for the wind noise source level from Chapter 8. The noise MSP at the sea surface is obtained

622 The sonar equations revisited

[Ch. 11

by integrating the dipole source spectrum over the sonar bandwidth and multiplying by . Thus,28 ð Q N ¼ K wind ð11:163Þ APL df ; where K wind APL ¼

ðvÞ F 1:59 kHz

mPa 2 kHz 1

ð11:164Þ

and (assuming unstable conditions, with water temperature exceeding that of air) ðvÞ ¼ 10 7:12^v 2:24 :

ð11:165Þ

Evaluating the integral gives QN ¼

ðvÞ 0:59 ðF min  F 0:59 max Þ 0:59

mPa 2 :

ð11:166Þ

The integrated noise level 10 log10 Q N is plotted in Figure 11.21 as the dashed blue horizontal line at 75 dB re mPa 2 .29 11.4.6.2.2

Array gain (AG)

Consider array gain in linear form, defined as GBB  10 AG=10 ;

ð11:167Þ

AG ¼ SG  NG:

ð11:168Þ

where The signal is assumed to originate from a single direction (no multipaths are involved), which means that SG ¼ 0. In this situation, array gain is determined by noise gain alone GBB ¼ 10 NG=10 : ð11:169Þ The (broadband) in-beam noise is ð QN 1 ¼ K wind df ; APL GBB GD ð f Þ

ð11:170Þ

where the directivity factor GD ð f Þ is30 GD ð f Þ ¼ ð D=Þ 2 : 28

ð11:171Þ

The orca is assumed sufficiently close to the surface for the effect of absorption on ambient noise to be negligible. 29 Following Au et al. (2004), it is assumed here that the receiver bandwidth is equal to that of the transmitter pulse. 30 The use of the directivity factor GD ð f Þ in the integrand of Equation (11.170) instead of the noise gain at frequency f is made for simplicity only. In the model problem, the sonar is pointing down (away from the sea surface and hence also from the noise source), so a more careful calculation of AG is expected to result in a value greater than DI.

Sec. 11.4]

11.4 Active sonar with coherent processing: matched filter 623

It follows that QN c 2 ¼ ðF 2:59  F 2:59 max Þ GBB 2:59 D 2 min

mPa 2 : kHz 2

ð11:172Þ

Rearranging Equation (11.172) for GBB gives GBB ¼

0:59 2 2 2:59 f 0:59 min  f max D : 2:59 0:59 f 2:59 c2 min  f max

ð11:173Þ

Inserting numerical values, array gain is found to be 16.5 dB.31 11.4.6.2.3

Detection threshold (DT)

Assuming 1D þ R statistics for the signal, the detection threshold (DT) can be approximated by Equation (11.22). Putting pfa ¼ 10 3 in this equation gives DT ¼ 8.7 dB. 11.4.6.2.4 Signal excess (SE) and detection range Signal excess is given by SE ¼ EL  ðNL  AG þ DTÞ:

ð11:174Þ

The condition for detection is SE > 0 dB, so the detection range is the intersection between EL and NL  AG þ DT in Figure 11.21, which occurs at a range of ca. 240 m. An alternative calculation of the detection range is by means of the figure of merit FOM ¼

SL þ TS  NL þ AG  DT ; 2

ð11:175Þ

which, using Table 11.7, is found to be 51.0 dB re m 2 . This value can be used in combination with Figure 11.19 for broadband propagation loss. In this way, the same result for the detection range is found by equating FOM with ðPLTx þ PLRx Þ=2. 11.4.6.3

Part (iii) detection range for high wind speed (noise-limited)

The result of Figure 11.21 for wind speed 2 m/s is repeated in Figure 11.22 for wind speeds of 6 m/s and 10 m/s. The noise-limited ranges for the three wind speeds are stated in the figure caption. The above calculations assume that wind is the only source of ambient noise. Au et al. (2004) point out that the increase in ambient noise due to rain has a detrimental effect on the orca’s detection performance. It is left as an exercise for the reader to show this. 31 For sufficiently large GBB  4:4GD ð fmin Þ.

bandwidth

( fmax  fmin ),

Equation (11.173)

simplifies

to

624 The sonar equations revisited

[Ch. 11

Table 11.7. Sonar equation calculation for active sonar example (orca vs. salmon)—limited by wind noise. Description

Symbol

Source level

SL

198.2 dB

re mPa 2 m 2

Target strength

TS

29.0 dB

re m 2

Noise level

NL

75.0 dB

re mPa 2

Array gain

AG

16.5 dB

re 1

Detection threshold

DT

8.7 dB

re 1

FOMNL

51.0 dB

Figure of merit (noise-limited)

11.4.6.4

Value

re m 2

Part (iv) effect of reverberation (for high wind speed)

As wind speed increases, so do roughness of the sea surface and the near-surface bubble population density, and consequently so too does surface reverberation. The purpose of this last exercise is to illustrate the effect of this increased reverberation on predicted sonar performance.

Figure 11.22. Echo level (red curve) vs. distance between orca and salmon. The intersections between echo level (EL) and the cyan horizontal lines (NL  AG þ DT) are the predicted detection ranges of 117, 149, and 241 m, respectively, for v10 ¼ 10, 6, and 2 m/s.

Sec. 11.4]

11.4 Active sonar with coherent processing: matched filter 625

Signal excess is SE ¼ EL  ðBL  PG þ DTÞ:

ð11:176Þ

Echo level is not affected by the presence of reverberation and it is assumed for simplicity that the detection threshold is also unaffected. Of the four terms on the right-hand side, this leaves BL and PG to be determined. 11.4.6.4.1

Background level (BL)

Background level (see Equation 11.113) is BL ¼ 10 log10 ½Q R ðÞ þ Q N :

ð11:177Þ

The noise term Q N has already been considered (see Section 11.4.6.2). Contribution to the background from reverberation is calculated below. As previously for echo and ambient noise, reverberation needs to be integrated over the entire frequency band of the animal’s sonar. There are several frequencydependent effects, associated with the source spectrum ðS0 Þf , propagation factor, sea surface backscattering coefficient s ð f Þ, and beam pattern bTx ðuÞ. Taking all of these into account, the contribution to reverberation from a sea surface area element dA of azimuthal width d is   expð2 sÞ 2 dQ R ðÞ ¼ ðS Þ bTx ðuÞ s ðS ; f Þ dA; ð11:178Þ 0 f f s2 where the area element is dA ¼ s

c t d 2

ð11:179Þ

and s is the one-way pathlength corresponding to a delay time  s ¼ c=2:

ð11:180Þ

Reverberation arriving at this delay time is caused by scattering at the sea surface from paths whose elevation angle is S ¼ arcsin

dO : s

ð11:181Þ

Following Au et al. (2004), the beam pattern is assumed to be that of a circular disk (Chapter 6)   2J1 ðuÞ 2 bTx ðuÞ ¼ ; ð11:182Þ u where uðÞ ¼ ð D=Þ sin  ð11:183Þ and  is the angle measured from the projector axis ( ¼ 0 corresponds to the center of the beam). Approximating the beam pattern by a top-hat function of horizontal

626 The sonar equations revisited

[Ch. 11

width FðÞ and integrating Equation (11.178) over azimuth gives   expð2 sÞ 2 s c t QR ðÞ  Fð þ  ÞðS Þ

ðS ; f Þs ; S F 0 f f 2 s2

ð11:184Þ

where F is the elevation angle of a straight line between fish and orca F ¼ arcsin

dF  dO : s

ð11:185Þ

Assuming that the ray angles of interest are close to horizontal (i.e., if the distance s is large compared with the fish depth), the angle F in Equation (11.184) may be approximated using FðS þ F Þ 2  F0 ð f Þ 2  ðdF =sÞ 2 ;

ð11:186Þ

where F0 is the full width F0 ð f Þ ¼

4c :

Df

ð11:187Þ

Integrating the spectral density over frequency and approximating the right-hand side of Equation (11.186) by its first term only32 then gives   ð c t expð2 sÞ 2 s Q R ðÞ  s F0 ð f ÞðS0 Þf

ðS ; f Þ df : ð11:188Þ 2 s2 The integrand factors are now considered in turn. First, the beamwidth F0 is given by Equation (11.187). Second, for a center frequency of fm and bandwidth B, the assumed source spectrum is flat from fm  B=2 to fm þ B=2, equal to ðS0 Þf ¼

S0 B

ð11:189Þ

inside this range and zero outside it. Next is the propagation factor, which varies with frequency via the frequency-dependent attenuation , in the same way as the echo (Section 11.4.6.1.1). The only remaining term is the scattering coefficient s . If the orca swims close to the surface, the grazing angle S at the sea surface of ray paths contributing to surface reverberation is close to zero. Because of the small angle, rough surface scattering is negligible and consequently the surface scattering coefficient can be approximated by the contribution from bubble scattering alone. The bubble scattering coefficient varies strongly with wind speed when the wind speed is low and only weakly when it is high. The high wind speed limit is a simple function of grazing angle  and 32

This approximation is consistent with the assumption that the orca is looking ahead rather than down, so that dF =s is small. Furthermore, the high-frequency contributions to the integral are filtered out by the exponential decay in the propagation factor, so the second term needs only to be negligible at low frequency. A necessary condition is s > ð fm  B=2Þ DdF =2c, which is satisfied for distances exceeding 30 m.

Sec. 11.4]

11.4 Active sonar with coherent processing: matched filter 627

frequency f given by (Chapter 8)33

s ð; f Þ ¼ bubble  0:214 f^1=3 sin : Using this expression for the surface scattering coefficient gives  ð fþ    c t 1 expð4 sÞ fm 4=3 Q R ðÞ ¼ S0 Fð fm Þ

bubble ð fm ; S Þ df f 2 B f s4

ð11:190Þ

ð fþ > f Þ: ð11:191Þ

The limits of integration are determined on the one hand by the bandwidth, and on the other by the condition that the right-hand side of Equation (11.186) be positive. This second condition imposes an upper bound on fþ equal to fD ¼

2c s :

D d F

ð11:192Þ

The lower-frequency and upper-frequency limits are therefore f ¼ fm  B=2

ð11:193Þ

fþ ¼ minð fm þ B=2; fD Þ:

ð11:194Þ

and

The frequency fD is that at which the edge of the main beam intersects the sea surface precisely at a distance s from the animal. At frequencies higher than this, reverberation enters only through sidelobes and is disregarded from the present calculation. It is convenient to define a (one-way) propagation factor, weighted in frequency according to the various frequency-dependent mechanisms and denoted FW , as the square root of the quantity in curly brackets in Equation (11.191): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   ð 1 fþ expð4 sÞ fm 4=3 FW ðsÞ ¼ df : ð11:195Þ B f f s4 With this definition, the expression for reverberation becomes QR ¼

0:214 dO c t S 0 F0 ð f m Þ F ðsÞ 2 1=3 2 W f^m s

ð fþ > f Þ:

ð11:196Þ

The integral for FW can be simplified by assuming a linear variation of attenuation with frequency (see Equation 11.48)  4=3 ð exp½4ð m  0 fm Þs fþ fm 0 FW ðsÞ 2 ¼ expð4 fsÞ df : ð11:197Þ f s 4B f Using the change of variable x ¼ 4 0 sf , this integral can be evaluated in terms of the 33

If the wind speed is low, the problem becomes noise-limited.

628 The sonar equations revisited

[Ch. 11

Figure 11.23. Background level vs. distance between orca and salmon (v10 ¼ 10 m/s).

incomplete gamma function ða; xÞ:34 exp½2ð m  0 fm Þs

FW ðsÞ ¼ ð4 0 sfm Þ 1=6 s2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fm ½ð 13 ; 4 0 sfþ Þ  ð 13 ; 4 0 sf Þ : B ð11:198Þ

Figure 11.23 shows the result of evaluating Equation (11.196) with FW given by Equation (11.198) and F0 from Equation (11.187). The graph shows NL (blue) and RL (cyan) separately vs. distance. The combined noise plus reverberation is also shown as the dashed blue line. Because reverberation is much weaker than (unprocessed) noise, this dashed line almost coincides with noise alone. Processing gain is considered next. 11.4.6.4.2 Processing gain (PG) The gain from any hypothetical matched filter used by the orca is neglected. This is justifiable because the duration of the transmitted pulse is close to the theoretical minimum for the bandwidth used, so no gain is possible from further compression. Thus PG  AG: ð11:199Þ 34

From Appendix A: ða; xÞ 

ðx 0

e t t a1 dt:

Sec. 11.4]

11.4 Active sonar with coherent processing: matched filter 629

Figure 11.24. Array gain (equal to processing gain) vs. distance between orca and salmon (v10 ¼ 10 m/s).

It is appropriate to use GR ¼ 1 in Equation (11.122) because reverberation arrives predominantly in the main beam.35 Assuming for simplicity that GN  1=GBB (see Equation 11.173) it follows that

GA ðÞ ¼

Q N þ Q R ðÞ : Q N =GBB þ Q R ðÞ

ð11:200Þ

Figure 11.23 shows the effect of processing on background noise. The difference between the curves ‘‘BL’’ and ‘‘BL  PG’’, plotted in Figure 11.24, is processing gain (in this case equal to array gain), which is a function of distance to the target. It passes a minimum at a distance of 60 m, corresponding to the peak in RL at that range.

11.4.6.4.3 Signal excess and detection range The level that must be exceeded by signal level for a detection is BL  PG þ DT (the cyan curve in Figure 11.25). Thus, the detection range, which is reduced from 120 m in Figure 11.22 to about 70 m in this case, is determined by the intersection between this curve and EL (solid red line). 35 This is a consequence of the assumption that the same transducer is used for both transmission and reception.

630 The sonar equations revisited

[Ch. 11

Figure 11.25. Signal and background levels vs. distance between orca and salmon (v10 ¼ 10 m/s).

11.5

THE FUTURE OF SONAR PERFORMANCE MODELING

Nearly a century after the invention of sonar, sound provides the only practical window into the sea and its contents. Much of the world’s oceans remains unexplored, so the need for sonar will increase well into the 21st century. As Niels Bohr once said prediction is very difficult, especially of the future, but a brief look ahead into the future of sonar and sonar performance modeling seems an appropriate way to end this account.

11.5.1

Advances in signal processing and oceanographic modeling

The future can be expected to bring — increasingly sophisticated sonar hardware and processing software, leading to a need for increasingly detailed modeling of linear or non-linear pressure and particle velocity fields; — increasingly sophisticated knowledge and understanding of the oceanographic features responsible for fluctuations in both signal and background (surface waves, internal waves, multiscale seabed roughness, etc.); — increasingly powerful computing facilities, leading to an increasing ability to model this detail, including closer integration of sonar models with oceanographic databases and ocean forecasting models.

Sec. 11.5]

11.5 The future of sonar performance modeling

631

These trends point towards increasingly sophisticated sonar performance prediction models, capable of modeling not just fluctuations in the signal and background waveforms themselves, but also the oceanographic features that cause them, and their consequences for sonar processing gain and detection probability. 11.5.2

Autonomous platforms

It seems reasonable to expect an increase in the autonomy of underwater vehicles generally and hence also of sonar platforms. A possible scenario involves a group of small autonomous platforms working together or separately to execute a designated task. Specific applications for such platforms might include — Ocean survey: In this scenario the vehicles gather data independently, exchanging findings with any neighboring platforms within acoustic communication distance, and surfacing occasionally to upload data, recharge batteries, and perhaps receive new instructions. Such surveys would take place initially in benign conditions of temperature and pressure and then in increasingly hostile ones, perhaps eventually to support planetary exploration.36 Less exotic applications might include a seabed survey, monitoring of gas exchange processes with the atmosphere for climate modeling, and a census of protected species. — Military reconnaissance or marine archeology: The vehicles rendezvous at a predetermined location and carry out co-ordinated, possibly covert measurements and upload data to a network or mother platform. Common to these applications is the reduced scope for human intervention compared with the more traditional use of sonar on manned (or unmanned, remotely operated) platforms. The challenge to sonar performance modeling is to provide a robust solution to the effective deployment and co-ordination of multiple autonomous platforms and their sensors. 11.5.3

Environmental impact of anthropogenic sound

Many sea animals, especially marine mammals, rely on underwater sound to carry out routine tasks such as foraging, communication, and navigation, in much the same way as humans rely on light in air. There is a growing concern that such animals might be adversely affected by anthropogenic sources of sound such as sonar (including echo sounders, seismic survey sources, and communications equipment), acoustic deterrents, underwater explosions (such as arising from the controlled disposal of unexploded ordnance), and radiated sound from shipping vessels (Richardson et al., 36

The potential for acoustic remote sensing is demonstrated by Collins et al (1995, 1997) in their analysis of the impact of Comet Shoemaker-Levy on Jupiter. Leighton and others make a case for acoustic monitoring in suspected lakes on Titan (liquid methane/ethane lake) (Leighton et al., 2005) and Europa (ice-covered liquid water) (Lee et al., 2003; Leighton et al., 2008).

632

The sonar equations revisited

[Ch. 11

1995; Southall et al., 2007; Dolman et al., 2009; Popper and Hastings, 2009). The acoustic modeling tools that have been developed for predicting sonar performance (and similar modeling tools developed for the offshore prospecting industry) are well suited to the task of estimating the sound pressure field and its spectrum at a given location. What these models are not yet able to do is assess the impact on individual animals exposed to the sound, on groups of animals, or on the ecosystem as a whole. The need to develop models with this capability will lead to increasing co-operation between biologists and sonar scientists.

11.6

REFERENCES

Abraham, D. A. (2003) Signal excess in K-distributed reverberation, IEEE J. Oceanic Eng., 28, 526–536. Ainslie, M. A. (2004) The sonar equation and the definitions of propagation loss, J. Acoust. Soc. Am., 115, 131–134. Ainslie, M. A., Harrison, C. H., and Burns, P. W. (1996) Signal and reverberation prediction for active sonar by adding acoustic components, IEE Proc.-Radar, Sonar Navig., 143(3), 190–195. ASA (1994) American National Standard: Acoustical Terminology, ANSI S1.1-1994 (ASA 1111994, revision of ANSI S1.1-1960 (R1976)), Acoustical Society of America, New York. Au, W. W. L., Ford, J. K. B., Horne J. K., and Newman Allman, K. A. (2004) Echolocation signals of free-ranging killer whales (Orcinus orca) and modelling of foraging for chinook salmon (Oncorhynchus tshawytscha), J. Acoust. Soc. Am., 115, 901–909. Clark, C. A. (2007) Vertical directionality of midfrequency surface nois in downwardrefracting environments, IEEE J. Oceanic Eng., 32, 609–619. Collins, M. D., McDonald, B. E., Kuperman W. A., and Siegmann, W. L. (1995) Jovian acoustics and Comet Shoemaker–Levy 9, J. Acoust. Soc. Am., 97, 2147–2158. Collins, M. D., McDonald, B. E., Kuperman, W. A., and Siegmann, W. L. (1997) Jovian acoustic matched-field processing, J. Acoust. Soc. Am., 102, 2487–2493. Dolman, S. J., Weir, C. R., and Jasny, M. (2009) Comparative review of marine mammal guidance implemented during naval exercises, Marine Pollution Bulletin, 58, 465–477. Etter, P. C. (2003) Underwater Acoustics Modeling and Simulation: Principles, Techniques and Applications, Spon Press, New York. Hall, J. D. and Johnson, C. S. (1972) Auditory thresholds of a killer whale Orcinus orca Linnaeus, J. Acoust. Soc. Am., 106, 1134–1141. Harrison, C. H. (1996) Formulas for ambient noise level and coherence, J. Acoust. Soc. Am., 99, 2055–2066. Harrison, C. H. and Harrison, J. A. (1995) A simple relationship between frequency and range averages for broadband sonar, J. Acoust. Soc. Am., 97, 1314–1317. IEC (www) Electropedia, Acoustics and Electroacoustics/IEV 801 (International Electrotechnical Commission), available at http://www.electropedia.org/iev/iev.nsf (last accessed June 23, 2009). Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V. (1982) Fundamentals of Acoustics (Third edition), Wiley, New York. Lee, S., Zanolin, M., Thode, A. M., Pappalardo, R. T., and Makris, N. C. (2003) Probing Europa’s interior with natural sound sources, Icarus, 165, 144–167.

Sec. 11.6]

11.6 References 633

Leighton, T. G., White, P. R., and Finfer, D. C. (2005) The sounds of seas in space, Proc. International Conference on Underwater Acoustic Measurements: Technologies and Results, Heraklion, Crete, Greece, June 28–July 1, 2005 (edited by J. S. Papadakis and L. Bjørnø, pp. 833–840). Leighton, T. G., Finfer, D. C., and White, P. R. (2008) The problems with acoustics on a small planet, Icarus, 193(2), 649–652. Nielsen, P. L., Harrison, C. H., and Le Gac, J. C. (2008) International Symposium on Underwater Reverberation and Clutter, September 9–12, 2008, NATO Undersea Research Center, La Spezia, Italy. oalib (www) Ocean Acoustics Library, available at http://oalib.hlsresearch.com/ (last accessed February 9, 2009). Popper, A. N. and Hastings, M. C. (2009) The effects of anthropogenic sources of sound on fishes, Journal of Fish Biology, 75, 455–489. Richardson, W. J., Greene, C. R., Malme, C. I., and Thomson, D. H. (1995) Marine Mammals and Noise, Academic Press, San Diego. Southall, B. L., Bowles, A. E., Ellison, W. T., Finneran, J. J., Gentry, R. L., Greene Jr., C. R., Kastak, D., Ketten, D. R., Miller, J. H., Nachtigall, P. E., Richardson, W. J., Thomas, J. A., and Tyack, P. L. (2007) Marine mammal noise exposure criteria: Initial scientific recommendations, Aquatic Mammals, 33(4), 411–521. Szymanski, M. D., Bain, D. E., Kiehl, K., Pennington, S., Wong, S., and Henry, K. R. (1999) Killer whale (Orcinus orca) hearing: Auditory brainstem response and behavioral audiograms, J. Acoust. Soc. Am., 106, 1134–1141. Urick, R. J. (1983) Principles of Underwater Sound, Peninsula Publishing, Los Altos, CA. Wensveen, P. J. and Van Roij, Y. A. L. (2007) Exposure Data Analysis: 3S-2006 Trial (TNO report TNO-DV 2007 SV291). TNO, The Hague, The Netherlands.

Appendix A Special functions and mathematical operations

The purpose of this appendix is to define the special functions and mathematical operations used in the main text, and to describe their most important properties. The material draws heavily from two valuable resources: the Handbook of Mathematical Functions edited by Abramowitz and Stegun (1965) and Weisstein’s MathWorld (Weisstein, www). Unless stated otherwise, the symbols x and z denote real and complex variables, respectively.

A.1 A.1.1

DEFINITIONS AND BASIC PROPERTIES OF SPECIAL FUNCTIONS Heaviside step function, sign function, and rectangle function

Three closely related functions are the Heaviside step function 8 x 0 and the rectangle function

8 jxj < 1=2 >

: 0 jxj > 1=2.

ðA:1Þ

ðA:2Þ

ðA:3Þ

636 Appendix A

It follows from these definitions that sgnðxÞ ¼ 2½HðxÞ  12

ðA:4Þ

PðxÞ ¼ Hðx þ 12 Þ  Hðx  12 Þ:

ðA:5Þ

and

Table A.1. Integrals of integer powers of the sine cardinal function (Weisstein, 2006). ð1 dx sinc N x N 0

A.1.2

Sine cardinal and sinh cardinal functions

1

=2

2

=2

ðA:6Þ

3

3=8

some integrals of which are included in Table A.1. Similarly, the sinh cardinal function is (Weisstein, 2003a)

4

=3

5

115=384

The sine cardinal, or ‘‘sinc’’, function is sincðxÞ 

sinhcðxÞ  A.1.3

sin x ; x

sinh x : x

ðA:7Þ

Dirac delta function

Dirac’s delta function has zero magnitude everywhere except the origin, and unit area. It can be defined in terms of a limiting form of, for example, the rectangle function Pðx="Þ ðxÞ ¼ lim ; ðA:8Þ "!0 " or the Gaussian exp½ðx="Þ 2 pffiffiffi ðxÞ ¼ lim : ðA:9Þ "!0 "

A.1.4

Fresnel integrals

The Fresnel integrals are CðxÞ 

ðx 0

and SðxÞ 

lim CðxÞ ¼

x!1

and lim SðxÞ ¼

x!1

ðA:10Þ

  u 2 du: 2

ðA:11Þ

ðx sin 0

Asymptotic properties are

  cos u 2 du 2

ð1

  1 u 2 du ¼ 2 2

ðA:12Þ

  1 u 2 du ¼ : 2 2

ðA:13Þ

cos 0

ð1 sin 0

Appendix A

A.1.5

637

Error function, complementary error function, and right-tail probability function

The error function is 2 erfðxÞ  pffiffiffi 

ðx

2

e t dt:

ðA:14Þ

0

Its limiting value for large x is lim erfðxÞ ¼ 1:

ðA:15Þ

x!1

The complementary error function, plotted in Figure ð A.1 (cyan line of upper graph), is 2 1 t 2 erfcðxÞ  1  erfðxÞ ¼ pffiffiffi e dt: ðA:16Þ  x A simple approximation to erfcðxÞ, shown as ‘‘approx 1’’ in Figure A.1 and valid for large x, is 2

e x erfcðxÞ pffiffiffi : x

ðA:17Þ

A slightly more accurate version (‘‘approx 2’’) is (Abramowitz and Stegun, 1965) 2

2 e x pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : erfcðxÞ pffiffiffi  x þ x2 þ 2

ðA:18Þ

At the expense of a little more complication, a very accurate value can be obtained using the approximation 2

2 e x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; erfcðxÞ pffiffiffi  x þ x 2 þ 2  ð1  2=Þ2 11:2117x

ðA:19Þ

shown as ‘‘approx 3’’. The fractional errors for all three approximations are also plotted (lower graph). For Equation (A.19), the fractional error is less than 0.1 % for all x  0. For negative arguments, the following symmetry property can be used erfcðxÞ ¼ 2  erfcðxÞ:

ðA:20Þ

The erfc function is closely related to the right-tail probability function (Kay, 1998, p. 21), defined as ! ð 1 1 u2 exp  du: ðA:21Þ FðxÞ  pffiffiffiffiffiffi 2 2 x The precise relationships between these two functions and their inverses are 1 x FðxÞ ¼ erfc pffiffiffi ðA:22Þ 2 2 and pffiffiffi F 1 ðxÞ ¼ 2 erfc 1 ð2xÞ: ðA:23Þ

638 Appendix A

Figure A.1. The complementary error function erfcðxÞ and approximations 1 to 3 (upper graph) and fractional error (lower). The approximations are indicated by ‘‘approx 1’’ (Equation A.17), ‘‘approx 2’’ (Equation A.18), and ‘‘approx 3’’ (Equation A.19).

Appendix A

A.1.6 A.1.6.1

639

Exponential integrals and related functions Definition of the exponential integral

The exponential integral of order n is (Abramowitz and Stegun, 1965) ð 1 zt e En ðzÞ  n dt: 1 t

ðA:24Þ

The recursion relation between En ðzÞ and Enþ1 ðzÞ, for positive integers n  1, is nEnþ1 ðzÞ ¼ e z  zEn ðzÞ:

ðA:25Þ

For real positive arguments, n > 0, the function is bounded by the inequality (Abramowitz and Stegun, 1965, Eq. (5.1.19)) 1 1 < e x En ðxÞ < : xþn xþn1 A.1.6.2

ðA:26Þ

Exponential integral of first order (imaginary argument)

An example of particular interest is the first-order exponential integral (i.e., Equation A.24 with n ¼ 1) with a purely imaginary argument ð 1 ixt e E1 ðixÞ  dt: ðA:27Þ t 1 This can be written in the equivalent form E1 ðixÞ ¼ þ loge x þ

ðx 0

e iu  1 du  i=2; u

ðA:28Þ

where is the Euler–Mascheroni constant 0:57722: A.1.6.3

ðA:29Þ

Exponential integral of third order (real argument)

The third-order exponential integral (Equation A.24 with n ¼ 3), this time with a real argument, is ð 1 xt e E3 ðxÞ  dt: ðA:30Þ t3 1 This function is introduced in Chapter 2, for calculation of the radiated noise field of an infinite sheet. An approximation to it, for all x  0 (based on Equation A.26) is e x E3 ðxÞ : ðA:31Þ x þ 3  e 0:434x For values of x in the range ½0; 2 , the largest fractional error in E3 ðxÞ incurred by the use of Equation (A.31) is 2 %.

640 Appendix A

A.1.6.4

Sine and cosine integral functions

The sine integral and cosine integral functions are, respectively, ðx sin u SiðxÞ  du 0 u and ðx cos u  1 CiðxÞ  þ loge x þ du: u 0

ðA:32Þ

ðA:33Þ

These two functions are related to the exponential integral via (Abramowitz and Stegun, 1965, p. 232) SiðxÞ ¼ and

 1 þ ½E ðixÞ  E1 ðixÞ 2 2i 1

1 CiðxÞ ¼  ½E1 ðixÞ þ E1 ðixÞ : 2

ðA:35Þ

E1 ðixÞ ¼ CiðxÞ  i½SiðxÞ  =2 :

ðA:36Þ

It follows that

Asymptotic values are ð1 sin u  lim SiðxÞ ¼ du ¼ x!1 u 2 0

ðA:37Þ

and lim CiðxÞ ¼ 0:

x!1

A.1.7 A.1.7.1

ðA:38Þ

Gamma function and incomplete gamma functions Gamma function

A.1.7.1.1 Definition and important values The gamma function is ð1 GðzÞ  t z1 e t dt; ðA:39Þ 0

which for real arguments satisfies the property Gðx þ 1Þ ¼ xGðxÞ ðx > 0Þ:

ðA:40Þ

Important values of GðxÞ are listed in Table A.2. It follows from Equation (A.39) and the result Gð1Þ ¼ 1 that, for integer n Gðn þ 1Þ ¼ n!

ðA:34Þ

ðn  1Þ:

ðA:41Þ

Table A.2. Selected values of the gamma function GðxÞ for 0 < x  1. Values outside this range can be calculated using Gðx þ 1Þ ¼ xGðxÞ. All GðxÞ values in the table are approximate except Gð1Þ. The exact value of Gð1=2Þ is  1=2 . x

GðxÞ

1/5

4.5908

1/4

3.6256

1/3

2.6789

2/5

2.2182

1/2

1.7725

3/5

1.4892

2/3

1.3541

3/4

1.2254

4/5

1.1642

1

1

Appendix A

641

A.1.7.1.2 Approximations Stirling’s formula can be used to estimate the value of n! for large arguments (Abramowitz and Stegun, 1965): n! lim pffiffiffiffiffiffi nþ1=2 n ¼ 1: n!1 2 n e

ðA:42Þ

The assumption that Equation (A.42) may be generalized to non-integer n (through use of Equation A.41) results in the approximation pffiffiffiffiffiffi GðxÞ GStirling ðxÞ ¼ 2 x x1=2 e x ; ðA:43Þ where Equation (A.43) serves to define the function GStirling ðxÞ. A more general version is obtained using Stirling’s series (Weisstein, 2004a) pffiffiffiffiffiffi 1 1 1 loge GðxÞ ¼ loge 2 þ ðx  1=2Þ loge x  x þ  þ O 5 ; ðA:44Þ 3 12x 360x x from which it follows that 

 1 1 1 GðxÞ ¼ GStirling ðxÞ 1 þ þ þO 3 : 12x 288x 2 x

ðA:45Þ

A convenient approximation is obtained by retaining the first two terms of this expansion   1 GðxÞ GStirling ðxÞ 1 þ ; ðA:46Þ Kx with K ¼ 12:

ðA:47Þ

Alternative values of K for Equation (A.46) are now considered. Insisting that Equation (A.46) should give the correct value of GðxÞ at x ¼ 1 (i.e., Gð1Þ ¼ 1) results in 1 K ¼ pffiffiffiffiffiffi

11:843: ðA:48Þ e= 2  1 When substituted in Equation (A.46), Equations (A.47) and (A.48) both give good accuracy for large x, but result in large errors in the region 0 < x < 1, especially at the lower end of this range. This problem can be remedied by applying Equation (A.40) for x < 1. Thus, 8

1 > > > 1þ G ðxÞ x1 < Kx Stirling

ðA:49Þ GðxÞ > 1 1 > > 1þ G ðx þ 1Þ 0 < x < 1. : x Kðx þ 1Þ Stirling

642 Appendix A

In general, there is a small discontinuity through x ¼ 1, which can be removed by choosing pffiffiffi e 2 K ¼ pffiffiffi

11:840: ðA:50Þ 8e Figure A.2 shows the gamma function with various approximations (upper graph) and the fractional error incurred by these (lower). The approximation obtained using Equation (A.49) (with Equation A.50 for K) is not shown in the upper graph because it cannot be distinguished from the exact function GðxÞ on this scale. The largest fractional error incurred by use of this approximation (shown as a cyan curve in the lower graph) is about 0.01%, and occurs when x 3:5. A.1.7.1.3

Use of the gamma function

Integrals of the form

ð1

x p expðBx q Þ dx

ðA:51Þ

0

appear in several chapters of this book. It follows from the definition of the gamma function (Equation A.39) that this integral can be written

ð1 B ðpþ1Þ=q pþ1 x p expðBx q Þ dx ¼ G : ðA:52Þ q q 0

A.1.7.2

Incomplete gamma functions

Two incomplete gamma functions are of interest here. The first, known as the lower incomplete gamma function, is defined as (Abramowitz and Stegun, 1965, p. 260) ðx ða; xÞ  e t t a1 dt: ðA:53Þ 0

The second is the upper incomplete gamma function (Abramowitz and Stegun, 1965; Weisstein, 2002) ð 1

Gða; xÞ 

e t t a1 dt:

ðA:54Þ

x

These two functions are complementary in the sense that their sum gives an ordinary (i.e., complete) gamma function ða; xÞ þ Gða; xÞ ¼ GðaÞ:

ðA:55Þ

Gða; 0Þ ¼ lim ða; xÞ ¼ GðaÞ

ðA:56Þ

Important properties include x!1

and (Weisstein, 2002)  Gð0; xÞ ¼

E1 ðxÞ  i E1 ðxÞ

x 0.

ðA:57Þ

Appendix A

643

Figure A.2. Upper graph: the gamma function GðxÞ defined by Equation (A.39) and approximations ‘‘Stirling1’’ (Equation A.43), ‘‘Stirling3’’ (Equation A.45), ‘‘K ¼ 12’’ (Equation A.49 þ Equation A.47); lower graph: fractional error incurred by the three approximations from the upper graph, plus a fourth approximation, labeled ‘‘K ¼ 11.840’’ (Equation A.49 þ Equation A.50).

644 Appendix A

The asymptotic behavior of ða; xÞ is  ða; xÞ

x a =a

x1

GðaÞ

x  1.

ðA:58Þ

An alternative form, used in some textbooks devoted to detection theory, is Pearson’s incomplete gamma function Iðu; pÞ, defined as (Abramowitz and Stegun, 1965) ð upffiffiffiffiffiffi pþ1 1 Iðu; pÞ  e t t p dt: ðA:59Þ Gðp þ 1Þ 0 This function is related to the lower incomplete gamma function of Equation (A.55) via pffiffiffiffiffiffiffiffiffiffiffi ð p þ 1; u p þ 1Þ ¼ Gð p þ 1ÞIðu; pÞ: ðA:60Þ

A.1.8

Marcum Q functions

The ordinary Marcum Q function is ! ð1 x2 þ 2 Qð ; Þ  x exp  I0 ð xÞ dx; 2 

ðA:61Þ

where I0 is the modified Bessel function of order zero. Helstrom (1968, p. 219) defines the generalized Marcum function as ! ð 1  M1 x x2 þ 2 QM ð ; Þ  x exp  IM1 ð xÞ dx; ðA:62Þ

2  where IN is a modified Bessel function of order N. To simplify the notation and to reinforce the point that Q1 ð ; Þ ¼ Qð ; Þ, the ordinary Marcum Q function is denoted Q1 ð ; Þ in Chapter 7.

A.1.9

Elliptic integrals

Elliptic integrals of the first and second kind, introduced in Chapter 9, are described below. The elliptic integral of the first kind is defined as (Abramowitz and Stegun, 1965, p. 589) ð’ Fð’ I Þ  ð1  sin 2 sin 2 Þ 1=2 d : ðA:63Þ 0

The integrand of Equation (A.63) is always greater than or equal to unity, so the integral must be greater than or equal to ’. If sin in the integrand is approximated by 2 =, the integral becomes Fð’ I Þ

 ð’; Þ; 2 sin

ðA:64Þ

Appendix A

where

ð’; Þ  arcsin

2’ sin : 

The right-hand side of Equation (A.64) satisfies the inequality  ’ ð’; Þ  Fð’ I Þ: 2 sin

645

ðA:65Þ

ðA:66Þ

The function Fð’ I Þ has a singularity at  ¼ ¼ =2. Use of Equation (A.64) avoids this singularity, while still providing a useful approximation away from it. The elliptic integral of the second kind is ð’ Eð’ I Þ  ð1  sin 2 sin 2 Þ þ1=2 d : ðA:67Þ 0

A similar approximation to that leading to Equation (A.64) gives  Eð’ I Þ ð þ sin  cos Þ; 4 sin

ðA:68Þ

where  ¼ ð’; Þ is given by Equation (A.65). This approximation satisfies the inequality  ’ ð þ sin  cos Þ  Eð’ I Þ: ðA:69Þ 4 sin

A.1.10 A.1.10.1

Bessel and related functions Bessel function of the first kind

Bessel functions of the first kind are solutions to the ordinary differential equation (Abramowitz and Stegun, 1965, p. 358) z2

d2w dw þz þ ðz 2   2 Þw ¼ 0: 2 dz dz

ðA:70Þ

The solutions to this equation, denoted J ðzÞ, are Bessel functions (of the first kind) of order . The normalization (for positive integer n) is (Weisstein, 2004b) ð1 ½Jn ðxÞ 2 dx ¼ 1: ðA:71Þ 0

Related integrals are (Wolfram, www) ð1 1 1 J ðxÞ 2 dx ¼ 2 0 x

ðRe  > 0Þ

ðA:72Þ

and (Weisstein, 2004b) ð1 0

 J1 ðxÞ 2 4 dx ¼ : x 3

ðA:73Þ

646 Appendix A

A series expansion is (Abramowitz and Stegun, 1965, p. 360) J ðxÞ ¼

1 x X ðx 2 =4Þ n : 2 n¼0 n! Gð þ n þ 1Þ

ðA:74Þ

The asymptotic behavior of J ðxÞ for small and large x is given by (Abramowitz and Stegun, 1965) 8 x 1 > x1 > < Gð þ 1Þ 2 J ðxÞ rffiffiffiffiffiffi ðA:75Þ   > > : 2 cos x     x  1, x 2 4 valid for x > 0 and real, non-negative . A.1.10.2

Modified Bessel function

Modified Bessel functions of the first kind, denoted I ðzÞ, are solutions to the ordinary differential equation (Abramowitz and Stegun, 1965) z2

d2w dw þz  ðz 2 þ  2 Þw ¼ 0: dz dz 2

ðA:76Þ

They are related to J ðzÞ according to (Abramowitz and Stegun, 1965, p. 375):  expði=2ÞJ ðizÞ  < arg z  =2 I ðzÞ ¼ ðA:77Þ expð3i=2ÞJ ðizÞ =2 < arg z  . Other important properties include In ðzÞ ¼ In ðzÞ; I ðzÞ ¼ and

ðA:78Þ

1 z  X

2

ðz 2 =4Þ k ; k! Gð þ k þ 1Þ k¼0

" # ez 4 2  1 2 I ðzÞ  pffiffiffiffiffiffiffiffi 1  þ Oðz Þ 8z 2z

jarg zj < =2:

ðA:79Þ

ðA:80Þ

Levanon (1988) suggests the approximation pffiffiffi ! 1 1 x 3x I0 ðxÞ ð1 þ cosh xÞ þ cosh þ cosh : 6 3 2 2

ðA:81Þ

The modified Bessel function is plotted in Figure A.3 (upper graph), together with the approximation of Equation (A.81). The fractional error increases with increasing argument (lower graph). For the range 0 < x < 15 the error is less than 2 %.

Appendix A

647

Figure A.3. Upper graph: the modified Bessel function I0 ðxÞ and Levanon’s approximation (Equation A.81); lower graph: fractional error incurred by use of Levanon’s approximation.

648 Appendix A

A.1.10.3

Airy functions

The second-order differential equation d 2w dw z ¼0 dz dz 2

ðA:82Þ

has two independent solutions, known as Airy functions, one of which, denoted AiðzÞ, vanishes for large real values of its argument, while the other, BiðzÞ, is unbounded in this limit. They are related to the Bessel functions J1=3 and I1=3 via (Abramowitz and Stegun, 1965, p. 446) pffiffiffi z AiðzÞ ¼ ½I ðÞ  Iþ1=3 ðÞ ðA:83Þ 3 1=3 and rffiffiffi z BiðzÞ ¼ ½I ðÞ þ Iþ1=3 ðÞ ðA:84Þ 3 1=3 where  ¼ 23 z 3=2 : ðA:85Þ Alternative expressions that are more convenient to use for negative arguments are pffiffiffi z AiðzÞ ¼ ½J ðÞ þ J1=3 ðÞ ; ðA:86Þ 3 þ1=3 and rffiffiffi z BiðzÞ ¼ ½J ðÞ  Jþ1=3 ðÞ : ðA:87Þ 3 1=3 The value and gradient of the Airy functions at the origin are given by Bið0Þ 3 2=3 Aið0Þ ¼ pffiffiffi ¼

0:35503 Gð2=3Þ 3

ðA:88Þ

Bi 0 ð0Þ 3 1=3 Ai 0 ð0Þ ¼ pffiffiffi ¼

0:25882: Gð1=3Þ 3

ðA:89Þ

and

A.1.11 A.1.11.1

Hypergeometric functions Gauss’s hypergeometric function

Gauss’s hypergeometric function (sometimes abbreviated as the ‘‘hypergeometric function’’) is (Weisstein, 2004c) ð 1 b1 GðcÞ t ð1  tÞ cb1 dt: ðA:90Þ 2 F1 ða; b; c; zÞ ¼ GðbÞGðc  bÞ 0 ð1  tzÞ a This function is a solution of the differential equation zð1  zÞ

d 2u du þ ½c  ða þ b þ 1Þz  abu ¼ 0 dz dz 2

ðA:91Þ

Appendix A

649

that is regular at the origin, and normalized such that 2 F1 ða; b; c; 0Þ

¼ 1:

ðA:92Þ

If jxj < 1, Equation (A.90) may be expanded as a power series: 2 F1 ða; b; c; xÞ ¼

1 GðcÞ X Gða þ nÞGðb þ nÞ n x : GðaÞGðbÞ n¼0 Gðc þ nÞ

ðA:93Þ

Of particular interest (for Chapter 5, in connection with the bulk modulus of bubbly water) is the special case for b ¼ c  1 ¼ a ð1 t a1 F ða; a; a þ 1; zÞ ¼ a ðA:94Þ 2 1 a dt: 0 ð1  tzÞ A.1.11.2

Confluent hypergeometric function of the first kind

The confluent hypergeometric function of the first kind, denoted 1 F1 ða; b; zÞ, is (Weisstein, 2003b) ð1 GðbÞ e zt t a1 dt: ðA:95Þ 1 F1 ða; b; zÞ ¼ Gðb  aÞGðaÞ 0 ð1  tÞ 1þab Of particular interest (for Chapter 7, in connection with the third and higher moments of the Rician probability distribution function) is the special case b ¼ 1 ð 1 zt a1 1 e t dt: ðA:96Þ 1 F1 ða; 1; zÞ ¼ GðaÞGð1  aÞ 0 ð1  tÞ a

A.2 A.2.1

FOURIER TRANSFORMS AND RELATED INTEGRALS Forward and inverse Fourier transforms

The Fourier transform of the function f ðxÞ is written I½ f ðxÞ . The outcome of this operation, denoted FðkÞ, is defined as: ð þ1 FðkÞ ¼ I½ f ðxÞ  f ðxÞ expðikxÞ dx: ðA:97Þ 1

The inverse Fourier transform is f ðxÞ ¼ I 1 ½FðkÞ 

1 2

ð þ1 FðkÞ expðþikxÞ dk:

ðA:98Þ

1

An equivalent alternative form used in Table A.3 is ð þ1 Gð f Þ ¼ I½gðtÞ  gðtÞ expð2iftÞ dt; 1

ðA:99Þ

650 Appendix A Table A.3. Examples of Fourier transform pairs (based on Weisstein, 2004d). Function

gðtÞ

Gð f Þ

Constant

1

ð f Þ

Cosine

cosð2f0 tÞ

1 2 ½ð f

Sine

sinð2f0 tÞ

1 ½ð f  f0 Þ  ð f þ f0 Þ 2i

ðt  t0 Þ

expð2ift0 Þ

Dirac delta function

expð2f0 jtjÞ

Exponential

exp½ða þ ibÞt 2

Complex Gaussian

Hðt  t0 Þ

Shifted Heaviside step function Rectangle Symmetrical ramp

 f0 Þ þ ð f þ f0 Þ

1 f0  f 2 þ f 20 rffiffiffiffiffiffiffiffiffiffiffiffi

 f 2 exp  a þ ib a þ ib   1 i ð f Þ  expð2ift0 Þ 2 f

Pðt=TÞ

T sincðfTÞ

ð1  jtj=TÞPðt=2TÞ

T sinc 2 ðfTÞ

sincðt=aÞ

aPð faÞ

1=t

i½2Hðf Þ  1

Sine cardinal Reciprocal (Cauchy principal value)

with gðtÞ ¼ I 1 ½Gð f Þ ¼

ð þ1

Gð f Þ expðþ2iftÞ df :

ðA:100Þ

1

A.2.2

Cross-correlation

The cross-correlation operation between two complex functions hðtÞ and gðtÞ, denoted here by the operator s, is defined by Weisstein (wwwa) as ð þ1 hðtÞsgðtÞ  h  ð Þgðt  Þ d; ðA:101Þ 1 

where h ðtÞ denotes the complex conjugate of hðtÞ. From this definition it follows that ð þ1 hðtÞsgðtÞ ¼ h  ðÞgðt þ Þ d: ðA:102Þ 1

An important result, known as the cross-correlation theorem, is (Weisstein, wwwb) hsg ¼ I 1 ½H  ð f ÞGð f Þ ;

ðA:103Þ

Appendix A

651

where Hð f Þ ¼ I½hðtÞ

ðA:104Þ

Gð f Þ ¼ I½gðtÞ :

ðA:105Þ

and The special case with h ¼ g, known as the Wiener–Khinchin theorem, relates the autocorrelation function hsh to the Fourier transform of the power spectrum: hsh ¼ I 1 ½jHð f Þj 2 :

ðA:106Þ

An alternative definition, used in Chapter 6 (following Burdic, 1984; McDonough and Whalen, 1995), is ð þ1 Chg ðtÞ  hðÞg  ð  tÞ d: ðA:107Þ 1

The two definitions are related according to hðtÞsgðtÞ ¼ C hg ðtÞ: A.2.3

ðA:108Þ

Convolution

The convolution operation between functions hðtÞ and gðtÞ is denoted here by the operator  and defined as (Weisstein, 2003c) ð þ1 hðtÞ  gðtÞ  hðÞgðt  Þ d: ðA:109Þ 1

It follows from Equations (A.101) and (A.109) that hðtÞsgðtÞ ¼ h  ðtÞ  gðtÞ:

ðA:110Þ

The Fourier transform of the product hðtÞgðtÞ is equal to the convolution of the individual transforms Hð f Þ and Gð f Þ (i.e., Weisstein, 2003c) I½hðtÞgðtÞ ¼ Hð f Þ  Gð f Þ:

ðA:111Þ

Equation (A.111) is known as the convolution theorem. Alternative forms of the theorem are (Weisstein, wwwc) I½h  g ¼ FG;

ðA:112Þ

I 1 ½HG ¼ h  g;

ðA:113Þ

I 1 ½H  G ¼ hg:

ðA:114Þ

and

A.2.4

Discrete Fourier transform

The discrete Fourier transform (DFT) of the function xðnÞ is

N1 X 2m XðmÞ  xðnÞ exp i n ; N n¼0

ðA:115Þ

652 Appendix A

the inverse transform of which is (Oppenheim and Schafer, 1989)

X 1 N1 2mn xðnÞ ¼ XðmÞ exp þi ; n ¼ 0; 1; 2; . . . ; N  1: N m¼0 N

ðA:116Þ

A common application of the DFT is for a continuous function of time, say FðtÞ, that has been sampled at discrete time intervals tn ¼ t0 þ n t:

ðA:117Þ

In the analysis of signals of this form, it is common to evaluate expressions of the form N1 X Gð!Þ  Fðtn Þ expði!tn Þ; tn ¼ t0 þ n t: ðA:118Þ n¼0

The inverse transform that follows from Equation (A.116) is Fðtn Þ ¼

X 1 N1 Gð!m Þ expðþi!m tn Þ; N m¼0

n ¼ 0; 1; 2; . . . ; N  1;

ðA:119Þ

where !m ¼ A.2.5

2 m: N t

ðA:120Þ

Plancherel’s theorem

The Fourier transform pair gðtÞ and Gð f Þ are related according to Plancherel’s theorem (Weisstein, wwwd) ð þ1 ð þ1 2 jgðtÞj dt ¼ jGð f Þj 2 df : ðA:121Þ 1

1

2

Thus, jGð f Þj is the energy spectral density of the time series gðtÞ. The corresponding relationship for the discrete transform pair is N1 X

jxðnÞj 2 ¼ f t

n¼0

A.3

A.3.1

N1 X

jXðmÞj 2 :

ðA:122Þ

n¼0

STATIONARY PHASE METHOD FOR EVALUATION OF INTEGRALS Stationary phase approximation

The stationary phase method is a way of approximating integrals of the form ðb Iða; bÞ ¼ f ðxÞ exp½iðxÞ dx; ðA:123Þ a

where f ðxÞ is a slowly varying function; and ðxÞ is a phase term. It is one of a more

Appendix A

653

general class of approximations known as saddle point methods (Skudrzyk, 1971; Chapman, 2004). The basic requirement is for f ðxÞ to vary slowly compared with , in such a way that the amplitude f does not change significantly during a period of e i . There is also a requirement that the phase approaches a maximum or minimum either within or close to the integration interval. If there is only one such point of stationary phase, the integral is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Iða; bÞ f ðx0 ÞEs ð ; Þ e iðx0 Þ ; ðA:124Þ j 00 ðx0 Þj where x0 is the point of stationary phase such that and

 0 ðx0 Þ ¼ 0

ðA:125Þ

s ¼ sgn½ 00 ðx0 Þ :

ðA:126Þ

The variables and  are related to a and b according to

¼ gðaÞ;

ðA:127Þ

 ¼ gðbÞ

ðA:128Þ

and where

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j 00 ðx0 Þj gðxÞ  ðx  x0 Þ: 

Finally, the function Es ð ; Þ is defined as ð    1  Es ð ; Þ  pffiffiffi exp si x 2 dx; 2 2

ðA:129Þ

ðA:130Þ

which in terms of Fresnel integrals becomes Es ð ; Þ ¼

CðÞ  Cð Þ þ si½SðÞ  Sð Þ pffiffiffi : 2

ðA:131Þ

If there is more than one stationary phase point, and if these are not too close together, their individual contributions may be added. A.3.2

Derivation

The derivation of Equation (A.124) follows. It is convenient to write the integration limits as x such that ð xþ

I¼

f ðxÞ exp½iðxÞ dx

ðA:132Þ

x

and expand ðxÞ around some point x0 (to be specified) ðxÞ ¼ ðx0 Þ þ  0 ðx0 Þðxx0 Þ þ 12  00 ðx0 Þðxx0 Þ 2 þ 16  000 ðx0 Þðxx0 Þ 3 þ   

ðA:133Þ

If ðxÞ is a rapidly varying function, the exponential is oscillatory and the net contribution to the integral averaged over many cycles is small. However, if the

654 Appendix A

phase slows down, the contributions can build up quickly. For this reason it is useful to expand about points at which the first derivative vanishes (known as points of ‘‘stationary phase’’). Thus, the value of x0 is chosen to ensure that  0 ðx0 Þ ¼ 0, and therefore ðxÞ ¼ ðx0 Þ þ 12  00 ðx0 Þðx  x0 Þ 2 þ 16  000 ðx0 Þðx  x0 Þ 3 þ   

ðA:134Þ

and I ¼ e iðx0 Þ

ð xþ x

f ðxÞ expfi½12  00 ðx0 Þðx  x0 Þ 2 þ 16  000 ðx0 Þðx  x0 Þ 3 þ    g dx:

ðA:135Þ

So far no approximation has been made, other than the assumptions that a point of stationary phase exists and the function ðxÞ may be replaced by a Taylor expansion about that point. To proceed further, the third and higher order derivatives are assumed to make a negligible contribution to the phase in the vicinity of x0 , such that the phase of Equation (A.135) is approximated by its first term only ð xþ I e iðx0 Þ f ðxÞ expfi½12  00 ðx0 Þðx  x0 Þ 2 g dx: ðA:136Þ x

If the variation in the amplitude term is assumed to be negligible in the region of interest, f ðx0 Þ may then be factored out of the integral ð xþ iðx0 Þ I f ðx0 Þ e expfi½12  00 ðx0 Þðx  x0 Þ 2 g dx: ðA:137Þ x

Changing the integration variable to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j 00 ðx0 Þj u¼ ðx  x0 Þ;  Equation (A.137) can be written (without further approximation) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I f ðx0 Þ e iðx0 Þ Es ðu ; uþ Þ; j 00 ðx0 Þj where

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j 00 ðx0 Þj u ¼ ðx  x0 Þ 

ðA:138Þ

ðA:139Þ

ðA:140Þ

and s ¼ sgn½ 00 ðx0 Þ : Thus,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I f ðx0 ÞEs ðu ; uþ Þ e iðx0 Þ ; 00 j ðx0 Þj

ðA:141Þ

ðA:142Þ

which is equivalent to Equation (A.124). The function Es ðu ; uþ Þ is a linear combination of Fresnel integrals (see Equation A.131). If the limits of integration in Equation (A.137) are extended to infinity it

Appendix A

655

becomes lim Es ðu ; uþ Þ ¼ e is=4 :

ju j!þ1

ðA:143Þ

Therefore (in this limit) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I f ðx0 Þ e i½ðx0 Þþs=4 ; 00 j ðx0 Þj

ðA:144Þ

which is the standard stationary phase result quoted in many textbooks and is valid when the point of stationary phase is well within the range of integration. Equation (A.142) is a generalization that retains its accuracy for situations with a stationary phase point close to the integration limits.

A.4 A.4.1

SOLUTION TO QUADRATIC, CUBIC, AND QUARTIC EQUATIONS Quadratic equation

Readers will be familiar with the quadratic equation Ax 2 þ Bx þ C ¼ 0

ðA:145Þ

and its solution in the form x¼ A.4.2

B 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 2  4AC : 2A

ðA:146Þ

Cubic equation

There are times when the solution to a third-order polynomial (a cubic equation) is needed and this is given below. Any cubic equation can be written in the form x 3 þ Ax 2 þ Bx þ C ¼ 0:

ðA:147Þ

There are three solutions to Equation (A.147), given by (Archbold, 1964; Weisstein, 2004e) A xn ¼ y n ¼ ; ðA:148Þ 3 where Q yn ¼ bn  ; ðA:149Þ 3bn sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3ffi!1=3 R R Q bn ¼ e 2in=3    ; ðA:150Þ 2 2 3 Q¼

A2 þ B; 3

ðA:151Þ

656 Appendix A

and R¼

2A 3 AB  þ C: 27 3

ðA:152Þ

The three solutions to Equation (A.147) are obtained using n ¼ 0, 1, and 2 (or any three consecutive integers) in Equation (A.150). The choice of sign in Equation (A.150) is arbitrary,1 but once made it must remain the same for all three values of n. A.4.3

Quartic and higher order equations

Sometimes a fourth-order polynomial (quartic equation) is encountered. The solution to such an equation is described by Archbold (1964) and Weisstein (2004f ). The visionary 19th-century mathematician E´variste Galois proved that no general purpose formula, comparable with the algorithm given above for the solution to the cubic equation, exists for polynomials of order 5 or higher. In doing so he also laid the foundations of modern group theory, all before a tragic death at the age of just 20. Livio (2005) gives a fascinating historical account of the events leading up to this proof.

A.5

REFERENCES

Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions, U.S. Government Printing Office, Washington, D.C., available at http://www.math.sfu.ca/cbm/aands/ (last accessed March 23, 2009). Archbold, J. W. (1964) Algebra (Third Edition), Pitman, London. Burdic, W. S. (1984) Underwater Acoustic Systems Analysis, Prentice Hall, Englewood Cliffs, NJ. Chapman, C. H. (2004) Fundamentals of Seismic Wave Propagation (Appendix D: Saddle-point Methods), Cambridge University Press, Cambridge, U.K. Helstrom, C. W. (1998) Statistical Theory of Signal Detection, Pergamon Press, Oxford, U.K. Kay, S. M. (1998) Fundamentals of Statistical Signal Processing: Detection Theory, Prentice Hall, Upper Saddle River, NJ. Levanon, N. (1988) Radar Principles, Wiley, New York. Livio, M. (2005) The Equation that Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry, Simon & Schuster, New York. McDonough, R. N. and Whalen, A. D. (1995) Detection of Signals in Noise (Second Edition), Academic Press, San Diego, CA. Oppenheim, A. V. and Schafer, R. W. (1989) Discrete-Time Signal Processing, Prentice Hall, Englewood Cliffs, NJ. Skudrzyk, E. (1971) The Foundations of Acoustics: Basic Mathematics and Basic Acoustics, Springer Verlag, Vienna. 1 Although in theory the two roots give identical answers, any practical implementation is subject to rounding errors. These can be reduced by choosing the larger of the two roots in magnitude.

Appendix A

657

Weisstein, E. W. (2002) Incomplete gamma function, available at http://mathworld.wolfram. com/IncompleteGammaFunction.html (last accessed August 28, 2008). Weisstein, E. W. (2003a) Sinhc function, available at http://mathworld.wolfram.com/Sinhc Function.html (last accessed August 28, 2008). Weisstein, E. W. (2003b) Confluent hypergeometric function of the first kind, available at http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html (last accessed August 28, 2008). Weisstein, E. W. (2003c) Convolution, available at http://mathworld.wolfram.com/ Convolution.html (last accessed August 28, 2008). Weisstein, E. W. (2004a) Stirling’s series, available at http://mathworld.wolfram.com/Stirlings Series.html (last accessed August 28, 2008). Weisstein, E. W. (2004b) Bessel function of the first kind, available at http://mathworld. wolfram.com/BesselFunctionoftheFirstKind.html (last accessed August 28, 2008). Weisstein, E. W. (2004c) Hypergeometric function, available at http://mathworld.wolfram.com/ HypergeometricFunction.html (last accessed August 28, 2008). Weisstein, E. W. (2004d) Fourier transform, available at http://mathworld.wolfram.com/ FourierTransform.html (last accessed August 28, 2008). Weisstein, E. W. (2004e) Cubic formula, available at http://mathworld.wolfram.com/Cubic Formula.html (last accessed August 28, 2008). Weisstein, E. W. (2004f ) Quartic equation, available at http://mathworld.wolfram.com/Quartic Equation.html (last accessed August 28, 2008). Weisstein, E. W. (2006) Sinc function, available at http://mathworld.wolfram.com/Sinc Function.html (last accessed August 28, 2008). Weisstein, E. W. (www) Wolfram MathWorld, available at http://mathworld.wolfram.com/ (last accessed April 12, 2007). Weisstein, E. W. (wwwa) Cross-correlation, available at http://mathworld.wolfram.com/CrossCorrelation.html (last accessed July 10, 2007). Weisstein, E. W. (wwwb) Cross-correlation theorem, available at http://mathworld.wolfram. com/Cross-CorrelationTheorem.html (last accessed July 10, 2007). Weisstein, E. W. (wwwc) Convolution theorem, available at http://mathworld.wolfram.com/ ConvolutionTheorem.html (last accessed July 10, 2007). Weisstein, E. W. (wwwd) Plancherel’s theorem, available at http://mathworld.wolfram.com/ PlancherelsTheorem.html (last accessed November 28, 2008). Wolfram (www) Wolfram functions, available at http://functions.wolfram.com/BesselAiry StruveFunctions/BesselJ/21/02/02/] (last accessed April 11, 2007).

Appendix B Units and nomenclature

B.1 B.1.1

UNITS SI units

The International System of Units (abbreviated SI, from the French Syste`me Internationale d’Unite´s) is used throughout this book (bipm, www; Taylor and Thompson, 2008; Anon., 2008). For example, energy is expressed in joules (symbol J), pressure in pascals (symbol Pa), and intensity in watts per square meter (W/m 2 ). Further, standard SI prefixes are used to denote multiples of integer powers of 1000, such as ‘‘mega’’ for one million and ‘‘milli’’ for one thousandth, as indicated by Table B.1. Also in use are prefixes for integer powers of 10 between 10
2 and 10 þ2 , the most common being centi for 10
2 (as in centimeter). These are listed in Table B.2.

B.1.2

Non-SI units

For mainly historical reasons, units that are not part of SI are sometimes encountered in underwater acoustics, especially for units of distance or pressure. Some common non-SI units are listed in Table B.3, together with a conversion to their SI equivalents. For the definition of many other units see Rowlett (www).

B.1.3

Logarithmic units

Logarithmic units form a special category of (non-SI) units that are typically used to quantify ratios of parameters that might vary by many orders of magnitude. Special names are typically given to such logarithmic units to help remind us of the physical quantity they represent. Common examples are the octave (a base-2 logarithmic unit used to quantify frequency ratios), the decibel (a base-10 logarithmic unit used to

660 Appendix B Table B.1. SI prefixes for indices equal to an integer multiple of 3. One terajoule (10 12 J) is written 1 TJ. The range of prefixes most likely to be encountered is in the white (unshaded) region. Those least likely to be encountered are shaded dark gray. Prefix name

Symbol

Index

Example

yotta-

Y

24

1 YJ ¼ 10 24 J

zetta-

Z

21

1 ZJ ¼ 10 21 J

exa-

E

18

1 EJ ¼ 10 18 J

peta-

P

15

1 PJ ¼ 10 15 J

tera-

T

12

1 TJ ¼ 10 12 J

giga-

G

9

1 GJ ¼ 10 9 J

mega-

M

6

1 MJ ¼ 10 6 J

kilo-

k

3

1 kJ ¼ 10 3 J

—

—

0

1 J ¼ 10 0 J

milli-

m


3

1 mJ ¼ 10
3 J

micro-

m


6

1 mJ ¼ 10
6 J

nano-

n


9

1 nJ ¼ 10
9 J

pico-

p


12

1 pJ ¼ 10
12 J

femto-

f


15

1 fJ ¼ 10
15 J

atto-

a


18

1 aJ ¼ 10
18 J

zepto-

z


21

1 zJ ¼ 10
21 J

yocto-

y


24

1 yJ ¼ 10
24 J

quantify power ratios) and the neper (a base-e logarithmic unit used to quantify amplitude ratios). These and other relevant logarithmic units are described below.

B.1.3.1 B.1.3.1.1

Base-10 logarithmic units Bel and decibel

Relative levels. The bel is a logarithmic unit of power or energy ratio. A physical parameter that is proportional to power or energy is referred to in the following as a

Appendix B 661 Table B.2. SI prefixes for indices equal to an integer between þ3 and
3. One decijoule (10
1 J) is written 1 dJ. Name

Symbol

Index

Example

kilo

k

3

1 kJ ¼ 10 3 J

hecto

h

2

1 hJ ¼ 10 2 J

deca

da

1

1 daJ ¼ 10 1 J

—

—

0

1 J ¼ 10 0 J

deci

d


1

1 dJ ¼ 10
1 J

centi

c


2

1 cJ ¼ 10
2 J

milli

m


3

1 mJ ¼ 10
3 J

‘‘power-like’’ quantity. The level of a power-like quantity W2 is N bels higher than that of W1 if (Morfey, 2001) W N ¼ log10 2 : ðB:1Þ W1 The symbol for the bel is B. The decibel is defined as one tenth of a bel. Thus, the same two power levels differ by M decibels if W ðB:2Þ M ¼ 10 log10 2 : W1 The symbol for the decibel is dB. Neither the bel nor the decibel are recognized as SI units, but use of the decibel is permitted alongside SI units by the International Committee for Weights and Measures (CIPM) and at least one national standards body (Taylor and Thompson, 2008). For example, the decibel is used to express ratios of mean squared acoustic pressure (MSP) of statistically stationary pressure signals pðtÞ in this way using MMSP ¼ 10 log10

hp 2 i2 : hp 2 i1

ðB:3Þ

It is sometimes argued that the MSP in both the numerator and denominator of Equation (B.3) must first be divided by the characteristic acoustic impedance, in order to convert to the equivalent plane wave intensity (EPWI).1 In other words MEPWI ¼ 10 log10

hp 2 i2 =ðcÞ2 ; hp 2 i1 =ðcÞ1

ðB:4Þ

1 The EPWI is the intensity of a propagating plane wave whose MSP is equal to that of the true acoustic field.

Table B.3. Frequently encountered non-SI units (in alphabetical order). Unit

Symbol

SI equivalent

atmosphere

See standard atmosphere

bar

bar

100 kPa

dyne

dyn

10 mN

dyn/cm 2

0.1 Pa

erg

0.1 mJ

dyne per square centimeter erg erg per square centimeter

Notes

erg/cm

2

fathom

1 mJ/m

1 Pa ¼ 1 N/m 2 1 dyn ¼ 1 g cm/s 2 ; 1 N ¼ 1 kg m/s 2

1 erg ¼ 1 dyn cm; 1 J ¼ 1 N m 2

1.8288 m

1 fathom ¼ 6 ft (international fathom)

foot

ft

304.8 mm

hour

h

3600 s

inch

in

25.4 mm

1 ft ¼ 12 in. The capacity of air guns (see Chapter 10) is sometimes expressed in cubic inches (1 in 3  16:39 cm 3 )

international nautical mile

nmi

1.852 km

There is no internationally recognized symbol or abbreviation for this unit. The abbreviation ‘‘nmi’’ is adopted (preferred over ‘‘nm’’ to avoid a conflict with the SI symbol for a nanometer)

knot

kn

(1852/3600) m/s  0.5144 m/s

The knot is defined as one nautical mile per hour (1 nmi/h), such that 9 kn ¼ 4.63 m/s, exactly

liter

L

1000 cm 3

The uppercase ‘‘L’’ is preferred to the alternative (lowercase) letter ‘‘l’’ to avoid possible confusion with the number ‘‘1’’

microbar

mbar

0.1 Pa

millimeter per hour

mm/h

1 mm/(3600 s) 0.2778 mm/s

1 mbar ¼ 10
6 bar Used as a unit of rainfall rate

MKS rayl

See rayl

nautical mile

See international nautical mile

poise pound-force per square inch rayl

0.1 Pa s psi

6.895 kPa

dyn s/cm 2

10 Pa s/m

standard atmosphere

yard

101.325 kPa

yd

0.9144 m

1 poise ¼ 1 dyn s/cm 2

One pascal second per meter (1 Pa s/m) is sometimes known as an ‘‘MKS rayl’’. The rayl is not an SI unit. Pressure under standard conditions of temperature and pressure, denoted PSTP (see Section 14.2.2) 1 yd ¼ 3 ft

Appendix B 663

where ðcÞn is the characteristic impedance at the measurement location indicated by the value of the subscript n. Often the impedance is the same at locations 1 and 2, in which case Equations (B.3) and (B.4) are equivalent. In all other cases it is important to state which of the two is being used. Throughout this book the convention of Equation (B.3) (MSP ratio) is adopted, partly to conform to the de facto definition of propagation loss used in underwater acoustics, which since 1980 omits the impedance ratio (Ainslie and Morfey, 2005) and partly to avoid the ambiguities associated with the EPWI definition in the absence of an agreed standard reference value for the impedance (Ainslie, 2004, 2008).

Absolute levels. It is common practice to specify absolute power levels by replacing the denominator W1 in Equation (B.2) with an agreed standard reference value. Thus, a power W may be expressed as an absolute level by defining the power level LW in decibels, relative to a reference value Wref , as LW 10 log10

W : Wref

ðB:5Þ

When the decibel is used in this way, to avoid ambiguity both the reference value and the nature of the quantity W (in this case power) must be stated. Internationally accepted reference values for power and energy levels are 1 pW and 1 pJ, respectively. For example, a sound source of acoustic power (one watt) has a power level of 10 log10 ð1=10
12 Þ ¼ 120 dB re pW.2 The sound pressure level Lp is defined in terms of the MSP (Morfey, 2001) Lp 10 log10

hp 2 i ; p 2ref

ðB:6Þ

where the reference pressure pref is equal to 1 mPa, making the MSP reference value equal to 1 mPa 2 . Thus, the sound pressure level of an acoustic field whose RMS pressure is one pascal (MSP ¼ 1 Pa 2 ) is 10 log10 ð1=10
12 Þ ¼ 120 dB re mPa 2 . The same quantity is often written 120 dB re mPa. The squared unit is adopted here to avoid inconsistencies that otherwise arise when this quantity is combined with other ratios in decibels.3 For example, it seemspmore ffiffiffiffiffiffiffi natural to express the spectral density level in dB re mPa 2 /Hz than in dB re mPa/ Hz. Other physical parameters relevant to acoustics are energy density and intensity. When expressed as levels, their standard reference values are, respectively, 1 pJ/m 2 and 1 pW/m 2 (Morfey, 2001). When used in a spectral density, the reference unit for frequency is one hertz. For example, the power spectral density level has the unit dB re W/Hz. 2 3

Or, equivalently, 120 dB re 1 pW. It is p 2ref and not pref that appears in the denominator of Equation (B.6).

664 Appendix B

B.1.3.1.2 pH (acidity measure) The pH of a solution is a logarithmic measure of the reciprocal concentration of hydrogen ions dissolved in the solution. pH ¼
log10 ½H þ ;

ðB:7Þ

where ½H þ denotes the molar concentration of hydrogen (H þ ) ions. The precise definition depends on convention. For example, it might include only the concentration of free protons (the free proton scale) or might also include that of protons associated with other ions. Chapter 4 mentions four different pH scales: the U.S. National Bureau of Standards4 scale ( pHNBS ), the ‘‘seawater scale’’ ( pNSWS ), the ‘‘total proton scale’’ ( pHT ), and the ‘‘free proton scale’’ ( pHF ). As there is no single universally adopted convention, a choice is necessary between these. The NBS scale is considered unsuitable for modern use in seawater (Brewer et al., 1995; Millero, 2006). The other three are defined below (following Millero, 2006). The free proton scale is given by pHF
log10 ½H þ F ;

ðB:8Þ

where the notation ½X indicates the concentration of ion X, defined as the number of moles of that ion per kilogram of solution. Thus, ½H þ F is the concentration of free hydrogen ions in units of moles per kilogram (Brewer et al., 1995). The total proton scale is given by pHT
log10 ½H þ T ;

ðB:9Þ

where ½H þ T includes hydrogen sulfate ions ½H þ T ¼ ½H þ F þ ½HSO
4 :

ðB:10Þ

Finally, the SWS scale, recommended by UNESCO for use in seawater (Dickson and Millero, 1987), also includes the concentration of hydrogen associated with fluoride ions. Thus pHSWS
log10 ½H þ SWS ; ðB:11Þ where ½H þ SWS ¼ ½H þ T þ ½HF : ðB:12Þ B.1.3.1.3 Decade The decade is a logarithmic unit of frequency ratio. The frequency f2 is N decades higher than f1 if (Pierce, 1989) N ¼ log10

f2 : f1

ðB:13Þ

If N is negative then it is more conventional to say that f2 is jNj decades lower than f1 . 4

Now the National Institute of Standards and Technology (NIST).

Appendix B 665

B.1.3.2

Base-e logarithmic unit (neper)

The neper is a logarithmic unit of amplitude ratio. Consider a sinusoidal oscillation of amplitude A2 . The amplitude level of this oscillation is N nepers higher than that of another of amplitude A1 if (Morfey, 2001) A2 : A1

N ¼ loge

ðB:14Þ

The symbol for the neper is Np. A change in amplitude level of 1 Np is associated with a change in power level of 20 log10 e decibels. However, it is not correct to say that 1 Np is equal to 20 log10 e decibels unless the neper is redefined in terms of (the square root of ) a power ratio (Mills and Morfey, 2005). B.1.3.3

Base-2 logarithmic units

B.1.3.3.1 Octave The octave is a logarithmic unit of frequency ratio. The frequency f2 is N octaves higher than f1 if (Pierce, 1989) N ¼ log2

f2 : f1

ðB:15Þ

If N is negative then it is more conventional to say that f2 is jNj octaves lower than f1 . B.1.3.3.2

Phi

The phi unit is a logarithmic unit of reciprocal grain diameter. A spherical sediment grain of diameter5 d has a grain size of N phi units if (Krumbein and Sloss, 1963) d ; dref

ðB:16Þ

mm:

ðB:17Þ

N ¼
log2 where the reference diameter is dref 1

The symbol for the phi unit is . For example, if d ¼ 0.25 mm, the grain size expressed in phi units is written 2.

B.2 B.2.1

NOMENCLATURE Notation

A concerted effort has been made to employ a consistent notation throughout this book. While there is no separate list of symbols, the notation used is defined as and 5 The ‘‘diameter’’ of non-spherical grains is defined implicitly in terms of the mesh sizes of sieves able to separate them.

666 Appendix B

where it is introduced. The following notation conventions are used: — variable names are italic: frequency f ; — two- or three-letter abbreviations for sonar equation terms are upright and upper case: detection threshold is DT, whereas DT would mean a product of the variables D and T; — other abbreviations are also upright, though often lower case: ‘‘fa’’ in ‘‘pfa ’’ is an abbreviation of ‘‘false alarm’’; — symbols for some standard functions are upright: sin x; — non-standard function names are italic: f ðxÞ or FðkÞ; — differential operators are upright: dðsin xÞ=dx ¼ cos x; pffiffiffiffiffiffiffi — mathematical constants are upright: e ¼ expð1Þ; i ¼
1;  ¼ 2 arccos(0); — variable names with a circumflex denote the numerical value of that variable when expressed in the corresponding (base) unit in the SI system. For example, if the frequency f is 3 kHz, then f^ is a dimensionless number equal to (3 kHz)/(1 Hz) ¼ 3000. Thus f^ f f gHz and c^ fcgm=s . The following conventions are used for subscripts. Subscripts are used for a variety of purposes, indicating, for example: (1) the medium to which the subscripted parameter corresponds: air is the density of air (if no medium is specified, water is usually implied); (2) a derivative with respect to the subscript variable: Wf is the power spectral density (power W per unit frequency f ; i.e., dW=df ); higher order derivatives are indicated in the same way, so that the power spectral density per unit area A is denoted WAf , meaning d 2 W=dA df ; (3) a calculation method: ‘‘inc’’ in Finc stands for ‘‘incoherent’’, indicating that the propagation factor F is evaluated without regard for phase information; (4) evaluation for particular conditions: the ‘‘50’’ in DT50 means that the detection threshold corresponds to a 50 % detection probability. B.2.2

Abbreviations and acronyms

The abbreviations and acronyms used are listed in Table B.4. Abbreviations with multiple meanings (e.g., BL) are further qualified with an integer in brackets: BL (2), meaning ‘‘bottom reflection loss’’, is the second of three uses of the abbreviation ‘‘BL’’. B.2.3

Names of fish and marine mammals

Many animals have more than one common name, and a small number have more than one scientific name. Where the author has found more than one name in use he has followed Froese and Pauly (2007) for fish and Read et al. (2003) for marine mammals.

Appendix B 667 Table B.4. List of abbreviations and acronyms, and their meanings. Abbreviation Meaning AG

array gain

ANSI

American National Standards Institute

APL

Applied Physics Laboratory (University of Washington)

arr

array

atm

atmospheric

ATOC

acoustic thermometry of ocean climate

BB

broadband

BBS

bottom backscattering strength

BIPM

Bureau International des Poids et Mesures (International Bureau of Weights and Measures)

BL (1)

background level

BL (2)

bottom reflection loss

BL (3)

bottom reflected (path)

BR

bottom refracted (path)

BSS

bottom scattering strength

BSX

backscattering cross-section

BW

the quantity BW ¼ 10 log10 B^, where B^ is the numerical value of the bandwidth in hertz

CIPM

Comite´ International des Poids et Mesures (International Committee for Weights and Measures)

coh

coherent

CS

column strength

CW

continuous wave

dB

decibel (see Section B.1.3)

deg

degree (angle)

DFT

discrete Fourier transform (continued)

668 Appendix B Table B.4 (cont.) Abbreviation Meaning DI

directivity index

DT

detection threshold

EPWI

equivalent plane wave intensity

FFT

fast Fourier transform

FG

filter gain

FL

fork length (of fish)

FM

frequency modulation

FOM

figure of merit

FRF

flat response filter

ft

foot (see Table B.3)

ftp

file transfer protocol

fwhm

full width at half-maximum

GEOSECS

Geochemical Ocean Sections Study

GI

generator injector (air gun)

h

hour (see Table B.3)

HF

high frequency

HFM

hyperbolic frequency modulation

HIFT

Heard Island feasibility test

hp

hydrophone

IEC

International Electrotechnical Commission

in

inch (see Table B.3)

inc

incoherent

kn

knot (see Table B.3)

L

liter (see Table B.3)

LF

low frequency

LFM

linear frequency modulation

Appendix B 669

Abbreviation Meaning LPM

linear period modulation

MKS

meter kilogram second system of units (predecessor to SI)

MSP

mean square (acoustic) pressure

NB

narrowband

NBS

National Bureau of Standards (now NIST)

NIST

National Institute of Standards and Technology

NL

noise level

nmi

international nautical mile (see Table B.3)

Np

neper (see Section B.1.3)

pdf (1)

probability density function

pdf (2)

portable document format

peRMS

peak equivalent RMS

PG

processing gain

pH

logarithmic measure of acidity (see Section B.1.3)

PL

propagation loss

p-p

peak to peak

psi

pound-force per square inch (see Table B.3)

RAFOS

‘‘SOFAR’’ spelt backwards

RL

reverberation level

RMS

root mean square

ROC

receiver operating characteristic

Rx

receiver

SBR

signal to background ratio

SBS

surface backscattering strength

SE

signal excess (continued)

670 Appendix B Table B.4 (cont.) Abbreviation Meaning SI

Syste`me Internationale d’Unite´s (International System of Units)

SL (1)

source level

SL (2)

surface reflection loss

SL (3)

standard length (of fish)

SNR

signal to noise ratio

SOFAR

sound fixing and ranging

SPL

sound pressure level

SRR

signal to reverberation ratio

SSS

surface scattering strength

stat

static

STP

standard temperature and pressure; note: at STP the temperature and pressure are YSTP ¼ 273:15 K and PSTP ¼ 101:325 kPa (one standard atmosphere), respectively

SWS

seawater scale (of pH)

tgt

target

tot

total

TL

total length (of fish)

TPL

total path loss

TS

target strength, the quantity TS ¼ 10 log10

Tx

transmitter

UNESCO

United Nations Educational, Scientific and Cultural Organization

VBS

volume backscattering strength

vs.

versus

WMO

World Meteorological Organization

WS

wake strength

^back , where ^ back is the 4 backscattering cross-section in square meters

Appendix B 671

Abbreviation Meaning WW1

First World War

WW2

Second World War

yd

yard (see Table B.3)

z-p

zero to peak

B.3

REFERENCES

Ainslie, M. A. (2004) The sonar equation and the definitions of propagation loss, J. Acoust. Soc. Am., 115, 131–134. Ainslie, M. A. (2008) The sonar equations: Definitions and units of individual terms, Acoustics ’08, Paris, June 29–July 4, 2008, pp. 119–124. This article is missing from the search index of the CD version of the Acoustics ’08 Proceedings. The paper can be located on the CD by means of its identification number (475), at /data/articles/2008/000475.pdf It is also available at http://intellagence.eu.com/acoustics2008/acoustics2008/cd1 (last accessed April 12, 2010). Ainslie, M. A. and Morfey, C. L. (2005) ‘‘Transmission loss’’ and ‘‘propagation loss’’ in undersea acoustics, J. Acoust. Soc. Am., 118, 603–604. Anon. (2008) The Little Big Book of Metrology, National Physical Laboratory, Teddington, U.K. bipm (www) The International System of Units (SI), Bureau International des Poids et Mesures, available at http://www.bipm.org/en/si (last accessed September 21, 2008). Brewer, P. G., Glover, D. M., Goyet, C., and Shafer, D. K. (1995) The pH of the North Atlantic Ocean: Improvements to the global model of sound absorption, J. Geophysical Res., 100(C5), 8761–8776. Crocker, M. J. (Ed.) (1997) Encyclopedia of Acoustics, Wiley, New York. Dickson, A. G. and Millero, F. J. (1987) A comparison of the equilibrium constants for the dissociation of carbonic acid in sea water media, Annex 3 of Thermodynamics of the Carbon Dioxide System in Seawater (report by the Carbon Dioxide Sub-panel of the Joint Panel on Oceanographic Tables and Standards, Unesco Technical Papers in Marine Science 51, Unesco, Paris. Froese, R. and Pauly D. (Eds.), FishBase, version (01/2007), available at http://www.fishbase.org/search.php (last accessed March 23, 2009). Jensen, F. B., Kuperman, W. A., Porter, M. B., and Schmidt, H. (1994) Computational Ocean Acoustics, AIP Press, New York. Krumbein, W. C. and Sloss, L. L. (1963) Stratigraphy and Sedimentation (Second Edition), Freeman, San Francisco. Kuperman, W. A. and Roux, P. (2007) Underwater Acoustics, in T. D. Rossing (Ed.), Springer Handbook of Acoustics (pp. 149–204), Springer Verlag, New York.

672 Appendix B Kuperman, W. A. (1997) Propagation of sound in the ocean, in M. J. Crocker (Ed.), Encyclopedia of Acoustics (pp. 391–408), Wiley, New York. Millero, F. J. (2006) Chemical Oceanography (Third Edition), CRC/Taylor & Francis. Mills, I. and Morfey, C. L. (2005). On logarithmic ratio quantities and their units, Metrologia, 42, 246–252. Morfey, C. L. (2001) Dictionary of Acoustics, Academic Press, San Diego, CA. Pierce, A. D. (1989) Acoustics: An Introduction to its Physical Principles and Applications, American Institute of Physics, New York. Read, A. J., Halpin, P. N., Crowder, L. B., Hyrenbach, K. D., Best, B. D., and Freeman S. A. (Eds.) (2003) OBIS-SEAMAP: Mapping Marine Mammals, Birds and Turtles, World Wide Web electronic publication, available at http://seamap.env.duke.edu/species (last accessed October 22, 2009). Rossing, T. D. (Ed.) (2007) Springer Handbook of Acoustics, Springer Verlag, New York. Rowlett (www) R. Rowlett, A Dictionary of Units, available at http://www.unc.edu/rowlett/ units/ (last accessed April 2, 2007). Taylor, B. N. and Thompson, A. (2008) The International System of Units (SI) (NIST Special Publication 330, 2008 Edition), U.S. Department of Commerce, National Institute of Standards & Technology.

Appendix C Fish and their swimbladders

C.1

TABLES OF FISH AND BLADDER TYPES

The scattering properties of fish generally are sensitive to the presence or absence of a gas enclosure, or ‘‘swimbladder’’. The main purpose of this appendix is to enable the reader to assess the likelihood that a particular order, family, or species of fish is equipped with such a bladder, and where a bladder is present to provide further information about its relevant properties. General rules are described in Table C.3 (by order) and Table C.4 (by family). Where known to the author, information about fish length is also provided. Table C.7 presents a long list of information by individual species, but despite its length it is not a complete list. In fact it is not even close to complete. Rather, it comprises relevant information collected by the author over a number of years. Regardless of its shortcomings, its existence at all owes itself partly to David Weston, who impressed upon the author the importance of bladdered fish in underwater acoustics, and partly to FishBase (Froese and Pauly, 2007), from which much of the information is gleaned. Table C.1 describes abbreviations used to describe types of fish in terms of whether or not a bladder is present, and if so whether a duct is present connecting it to the gut of the fish (in which case the fish is known as a physostome) or not (a physoclist). The shape of the bladder varies between different species. Each time the bladder code is used, it is accompanied by a lower case suffix indicating the source of the information, and these suffixes are listed in Table C.2. For example, ‘‘Sw’’ means that the fish is a physostome according to Whitehead and Baxter (1989), whereas ‘‘Nb’’ means that it has no swimbladder according to Froese and Pauly (2007). Two more keys are presented below to aid the interpretation of the main list of species in Table C.7. The first (Table C.5) describes a list of categories, referred to here as ‘‘Yang groups’’, which describe the likely behavior of the fish. The groups are

674 Appendix C Table C.1. Bladder presence and type key used in Tables C.3, C.4, and C.7. Bladder code

a

Means

J

Bladder missing in adults ( juveniles physoclist or physostome)

L

Physoclist

M

With bladder (bladder sometimes partly or completely filled with fat; uncertain air fraction) a

N

No bladder

P

With bladder (physoclist or physostome)

S

Physostome

The ‘‘M’’ stands for ‘‘Myctophidae’’, a family representative of this category.

Table C.2. Reference key. Reference code Means b

Froese and Pauly (2007)

e

Egloff (2006)

f

Foote (1997)

i

Iversen (1967)

k

Kitajima et al. (1985)

m

Simmonds and MacLennan (2005)

r

Bertrand et al. (1999)

w

Whitehead and Baxter (1989)

used by Yang (1982) to describe the relative ‘‘catchability’’ of the different species for his population estimates. The reason they are useful here is that catchability is influenced by the fish’s behavior which in turn affects its likely acoustical properties, its environment, or both. For example, groups B and C are demersal fish, which means that their properties are easily confused with (and might be affected by) the properties of the seabed. The other groups are pelagic. For Yang’s group C, the terms ‘‘sandeels’’ and ‘‘gobies’’ are interpreted here, respectively, as Ammodytidae and Gobidae. The second key (Table C.6) defines the abbreviations used to describe the fish length information (last column of Table C.7).

Table C.3. Bladder type by order for ray-finned fishes (Actinopterygii). See Tables C.1 and C.2 for bladder and reference codes used in the last column. Order

Families

Bladder present (bladder code)

Relevant extract

Anguilliformes

Anguillidae, Chlopsidae, Colocongridae, Congridae, Derichthyidae, Heterenchelyidae, Moringuidae, Muraenesocidae, Muraenidae, Myrocongridae, Nemichthyidae, Nettastomatidae, Ophichthidae, Serrivomeridae, Synaphobranchidae

Yes (Sb)

‘‘Swim bladder present, duct usually present’’

Clupeiformes

Chirocentridae, Clupeidae, Denticipitidae, Yes (Sw) Engraulidae, Pristigasteridae

‘‘Clupeoids . . . are physostomes with one, or often two, ducts between the swimbladder and the exterior: a pneumatic duct from the stomach, and an anal duct to the ‘cloaca’,’’ p. 300. ‘‘[Pneumatic duct] is invariably present,’’ p. 346

Gadiformes

Bregmacerotidae, Euclichthyidae, Yes (Lb) Gadidae, Lotidae, Merluccidae, Moridae, Muranolepididae, Phycidae

‘‘Swim bladder without pneumatic duct’’

Gadiformes

Macrouridae, genus Squalogadus

No (Nb)

‘‘The swim bladder is absent in Melanomus and Squalogadus’’

Gadiformes

Melanonidae

No (Nb )

‘‘The swim bladder is absent in Melanomus and Squalogadus’’

Myctophiformes

Myctophidae

Yes (Mb)

‘‘Swim bladder usually present’’

Myctophiformes

Neoscopelidae, genus Scopelengys

No (Nb)

‘‘Swim bladder present in all but Scopelengys’’

Myctophiformes

Neoscopelidae, except Scopelengys

Yes (Mb)

‘‘Swim bladder present in all but Scopelengys’’

Notacanthiformes

Halosauridae, Notacanthidae

Yes (Pb)

‘‘Swim bladder present’’

Perciformes

Sciaenidae

Yes (Pb)

‘‘Swim bladder usually having many branches and used as a resonating chamber’’

Perciformes

Ammodytidae

No (Nb)

‘‘No swim bladder’’

Pleuronectiformes

Achiridae, Achiropsettidae, Bothidae, Citharidae, Cynoglossidae, Paralichthyidae, Pleuronectidae, Psettodidae, Samaridae, Scophthalmidae, Soleidae

Only in juveniles (Jb)

‘‘Adults almost always without swim bladder’’

Table C.4. Bladder type by family; see Table C.3 for details. Family

Order

Achiridae

Pleuronectiformes

Bladder code Jb

Achiropsettidae

Pleuronectiformes

Jb

Ammodytidae

Perciformes

Nb

Anguillidae

Anguilliformes

Sb

Bothidae

Pleuronectiformes

Jb

Bregmacerotidae

Gadiformes

Lb

Chirocentridae

Clupeiformes

Sw

Chlopsidae

Anguilliformes

Sb

Citharidae

Pleuronectiformes

Jb

Clupeidae

Clupeiformes

Sw

Colocongridae

Anguilliformes

Sb

Congridae

Anguilliformes

Sb

Cynoglossidae

Pleuronectiformes

Jb

Denticipitidae

Clupeiformes

Sw

Derichthyidae

Anguilliformes

Sb

Engraulidae

Clupeiformes

Sw

Euclichthyidae

Gadiformes

Lb

Gadidae

Gadiformes

Lb

Halosauridae

Notacanthiformes

Pb

Heterenchelyidae

Anguilliformes

Sb

Lotidae

Gadiformes

Lb

Macrouridae, genus Squalogadus

Gadiformes

Nb

Melanonidae

Gadiformes

Nb

Merluccidae

Gadiformes

Lb

Moridae

Gadiformes

Lb

Moringuidae

Anguilliformes

Sb

Muraenesocidae

Anguilliformes

Sb

Muraenidae

Anguilliformes

Sb

Muranolepididae

Gadiformes

Lb

Myctophidae

Myctophiformes

Mb

Myrocongridae

Anguilliformes

Sb

Nemichthyidae

Anguilliformes

Sb

Neoscopelidae, except Scopelengys

Myctophiformes

Mb

Neoscopelidae, genus Scopelengys

Myctophiformes

Nb

Nettastomatidae

Anguilliformes

Sb

Notacanthidae

Notacanthiformes

Pb

Table C.4. (cont.) Family

Order

Ophichthidae

Anguilliformes

Bladder code Sb

Paralychthyidae

Pleuronectiformes

Jb

Pleuronectidae

Pleuronectiformes

Jb

Phycidae

Gadiformes

Lb

Pristigasteridae

Clupeiformes

Sw

Psettodidae

Pleuronectiformes

Jb

Samaridae

Pleuronectiformes

Jb

Sciaenidae

Perciformes

Pb

Scophthalmidae

Pleuronectiformes

Jb

Serrivomeridae

Anguilliformes

Sb

Soleidae

Pleuronectiformes

Jb

Synaphobranchidae

Anguilliformes

Sb

Table C.5. ‘‘Catchability’’ key (Yang groups) used in Table C.7. Yang group

Means

A

Cod-like

B

Flatfish

C

Eels

D

Herring-like

E

Mackerel-like

Table C.6. Length key used in Table C.7. Length code

Name

FL

Fork length

SL

Description Distance from tip of snout to end of middle caudal rays (Froese and Pauly, 2007)

Standard length Distance from tip of snout to end of vertebral column (roughly the start of the caudal fin) (Froese and Pauly, 2007)

TL

Total length

L50

—

Distance from tip of snout to end of caudal fin (Froese and Pauly, 2007) The length at which 50% of females have reached sexual maturity (Knijn et al., 1993)

Family

Labridae (wrasses) Acipenseridae (sturgeons) Agonidae (poachers) Clupeidae (herrings, shads, sardines, menhadens) Ammodytidae (sand lances) Ammodytidae (sand lances) Anarhichadidae (wolf-fishes) Anarhichadidae (wolf-fishes) Anarhichadidae (wolf-fishes) Anguillidae (freshwater eels) Stichaeidae (pricklebacks) Anoplogastridae Moridae (morid cods) Trichiuridae (cutlassfishes) Gobiidae (gobies) Gadidae (cods and haddocks)

Common name

Scale-rayed wrasse

Sturgeon

Hooknose

Alewife

Lesser sand-eel

Small sand-eel

Northern wolffish

Wolf-fish

spotted wolffish

European eel

Stout eelblenny

Common fangtooth

Blue hake

Black scabbardfish

Transparent goby

Arctic cod

Species (scientific name)

Acantholabrus palloni

Acipenser sturio

Agonus cataphractus

Alosa pseudoharengus

Ammodytes marinus

Ammodytes tobianus

Anarhichas denticulatus

Anarhichas lupus

Anarhichas minor

Anguilla anguilla

Anisarchus medius

Anoplogaster cornuta

Antimora rostrata

Aphanopus carbo

Aphia minuta

Arctogadus glacialis

Sm

C

A

C

C

B

32.5 (TL)

7.9 (TL)

110 (SL)

15.2 (SL)

30.0 (TL)

133 (TL)

180 (TL)

150 (TL)

180 (TL)

20.0 (SL)

25.0 (TL)

40.0 (SL)

21.0 (TL)

500 (TL)

25.0 (TL)

Bladder Yang Max. length/cm code group (TL, SL, or FL); L50 /cm

Table C.7. Fish and their bladders, sorted by scientific name. Keys: for bladder code see Tables C.1 and C.2; for Yang group see Table C.5. Maximum length is from Froese and Pauly (2007) (see Table C.6); L50 is from Knijn et al. (1993).

678 Appendix C

Sternoptychidae Sternoptychidae

Half-naked hatchetfish

Hatchet-fish

Argyropelecus hemigymnus

Argyropelecus olfersii

Sciaenidae (drums or croakers) Bothidae (lefteye flounders) Cottidae (sculpins) Triglidae (sea-robins) Stomiidae (barbeled dragonfishes) Belonidae (needlefishes) Trichiuridae (cutlassfishes) Myctophidae (lanternfishes) Myctophidae (lanternfishes) Myctophidae (lanternfishes) Myctophidae (lanternfishes) Myctophidae (lanternfishes) Berycidae (alfonsinos)

Meagre

Scaldfish

Atlantic hookear sculpin

East Atlantic red gurnard

Snaggletooth

Garpike

Elongate frostfish

Spinycheek lanternfish

Glacier lanternfish

Lamp fish

Skinnycheek lanternfish

Smallfin lanternfish

Alfonsino

Argyrosomus regius

Arnoglossus laterna

Artediellus atlanticus

Aspitrigla cuculus

Astronesthes gemmifer

Belone belone

Benthodesmus elongatus

Benthosema fibulatum

Benthosema glaciale

Benthosema panamense

Benthosema pterotum

Benthosema suborbitale

Beryx decadactylus

Sciaenidae (drums or croakers)

Argentinidae (argentines or herring smelts)

Argentine

Argentina sphyraena

Argyrosomus hololepidotus Madagascar meagre

Argentinidae (argentines or herring smelts)

Greater argentine

Argentina silus

Le

Lm

B

B

D

D

(continued)

3.9 (SL)

7.0

5.5

10.3 (SL)

10.0

100.0 (TL)

17.0 (SL)

50.0 (TL)

15.0 (SL)

25.0 (SL)

200 (TL)

3.9 (SL)

70.0

Appendix C 679

D

Gadidae (cods and haddocks) Bramidae (breams) Clupeidae (herrings, shads, sardines, menhadens) Lotidae (hakes and burbots) Gobiidae (gobies)

Macrouridae (grenadiers or rattails) Callionymidae (dragonets) Callionymidae (dragonets) Labridae (wrasses) Centrolophidae

Polar cod

Atlantic pomfret

Atlantic menhaden

Tusk

Jeffrey’s goby

Solenette

Hollowsnout grenadier

Dragonet

Spotted dragonet

Rock cook

Blackfish

Boreogadus saida

Brama brama

Brevoortia tyrannus

Brosme brosme

Buenia jeffreysii

Buglossidium luteum

Caelorhinchus caelorhinchus

Callionymus lyra

Callionymus maculatus

Centrolabrus exoletus

Centrolophus niger

B

B

C

11.7 (SL)

Myctophidae (lanternfishes)

Bolinichthys supralateralis

50.0 (TL)

40.0 (TL)

7.3 (SL)

Myctophidae (lanternfishes)

Bolinichthys photothorax

Le

5.0 (SL)

Bolinichthys indicus Myctophidae (lanternfishes)

9.0 (SL)

Bladder Yang Max. length/cm code group (TL, SL, or FL); L50 /cm

Bolinichthys longipes

Myctophidae (lanternfishes)

Family

4.5 (SL)

Lanternfish

Common name

Myctophidae (lanternfishes)

Bolinichthys distofax

Species (scientific name)

Table C.7 (cont.)

680 Appendix C

Mugilidae (mullets) Chimaeridae (shortnose chimaeras or ratfishes) Stichaeidae (pricklebacks) Lotidae (hakes and burbots) Lotidae (hakes and burbots) Clupeidae (herrings, shads, sardines, menhadens) Clupeidae (herrings, shads, sardines, menhadens) Clupeidae (herrings, shads, sardines, menhadens) Congridae (conger and garden eels) Salmonidae (salmonids) Macrouridae (grenadiers or rattails) Macrouridae (grenadiers or rattails) Psychrolutidae (fatheads) Psychrolutidae (fatheads) Gobiidae (gobies) Labridae (wrasses) Cyclopteridae (lumpfishes)

Thicklip grey mullet

Rabbit fish

Yarrel’s blenny

Fivebeard rockling

Northern rockling

Atlantic herring

Baltic herring

Pacific herring

European conger

Cisco

Abyssal grenadier

Roundnose grenadier

Polar sculpin

Pallid sculpin

Crystal goby

Goldsinny-wrasse

Lumpsucker

Chelon labrosus

Chimaera monstrosa

Chirolophis ascanii

Ciliata mustela

Ciliata septentrionalis

Clupea harengus harengus

Clupea harengus membras

Clupea pallasii pallasii

Conger conger

Coregonus artedi

Coryphaenoides armatus

Coryphaenoides rupestris

Cottunculus microps

Cottunculus thomsonii

Crystallogobius linearis

Ctenolabrus rupestris

Cyclopterus lumpus

Sm

Sfm

A

C

A

D

A

(continued)

35.0 (SL)

30.0 (SL)

110 (TL)

102 (TL)

57.0 (TL)

300 (TL)

46.0 (TL)

24.2 (TL)

45.0 (SL); 24 (L50 )

Appendix C 681

Family

Gonostomatidae (bristlemouths) Myctophidae (lanternfishes) Moronidae (temperate basses) Trachinidae (weeverfishes) Carapidae (pearlfishes) Lotidae (hakes and burbots) Engraulidae (anchovies) Engraulidae (anchovies) Engraulidae (anchovies) Engraulidae (anchovies) Engraulidae (anchovies) Engraulidae (anchovies) Engraulidae (anchovies) Sygnathidae (pipefishes and seahorses) Dalatiidae (sleeper sharks) Scombridae (mackerels, tunas, bonitos) Scombridae (mackerels, tunas, bonitos)

Common name

Garrick

Californian headlightfish

European seabass

Lesser weever

Pearlfish

Fourbeard rockling

Argentine anchoita

Australian anchovy

European anchovy

Silver anchovy

Japanese anchovy

Californian anchovy

Anchoveta

Snake pipefish

Velvet belly lantern shark

Kawaka

Little tunny

Species (scientific name)

Cyclothone braueri

Diaphus theta

Dicentrarchus labrax

Echiichthys vipera

Echiodon drummondii

Enchelyopus cimbrius

Engraulis anchoita

Engraulis australis

Engraulis encrasicolus

Engraulis eurystole

Engraulis japonicus

Engraulis mordax

Engraulis ringens

Entelurus aequoreus

Etmopterus spinax

Euthynnus affinis

Euthynnus alleteratus

Table C.7 (cont.)

Nb

Nbi

Sm

Le

A

A

A

B

122 (TL)

100.0 (FL)

20.0 (SL)

24.8 (SL)

18.0 (TL)

15.5 (TL)

20.0 (SL)

15.0 (SL)

17.0 (SL)

103 (TL)

11.4 (TL)

3.8 (SL)

Bladder Yang Max. length/cm code group (TL, SL, or FL); L50 /cm

682 Appendix C

Scombridae (mackerels, tunas, bonitos) Triglidae (sea-robins) Gadidae (cods and haddocks) Gadidae (cods and haddocks) Gadidae (cods and haddocks) Lotidae (hakes and burbots) Triakidae (houndsharks) Gasterosteidae (sticklebacks and tubesnouts) Pleuronectidae (righteye flounders) Gobiidae (gobies) Gobiidae (gobies) Ammodytidae (sand lances) Zoarcidae (eelpouts) Moridae (morid cods) Sebastidae (rockfishes, rockcods, and thornyheads) Syngnathidae (pipefishes and seahorses) Pleuronectidae (righteye flounders)

Black skipjack

Grey gurnard

Silvery cod

Silvery pout

cod

Three-bearded rockling

Tope shark

Three-spined stickleback

Witch

Black goby

Two-spotted goby

Smooth sand-eel

Aurora unernak

Slender codling

Blackbelly rosefish

Long-snouted seahorse

American plaice

Euthynnus lyneatus

Eutrigla gurnardus

Gadiculus argenteus argenteus

Gadiculus argenteus thori

Gadus morhua

Gaidropsarus vulgaris

Galeorhinus galeus

Gasterosteus aculeatus aculeatus

Glyptocephalus cynoglossus

Gobius niger

Gobiusculus flavescens

Gymnammodytes semisquamatus

Gymnelus retrodorsalis

Halargyreus johnsonii

Helicolenus dactylopterus dactylopterus

Hippocampus guttulatus

Hippoglossoides platessoides

Le

Le

Lfm

Nb

B

C

C

C

B

A

A

A

D

B

(continued)

82.0 (TL); 17 L50

16.0 (TL)

47.0 (TL)

56.0 (TL)

14.0 (TL)

200 (TL); 70 L50

15.0 (TL)

15.0 (TL)

60.0 (TL); 19 L50

84.0 (FL)

Appendix C 683

Trachichthyidae (slimeheads) Myctophidae (lanternfishes) Ammodytidae (sand lances) Ammodytidae (sand lances) Scombridae (mackerels, tunas, bonitos) Labridae (wrasses)

Orange roughy

Benoit’s lanternfish

Greater sandeel

Great sandeel

Skipjack tuna

Ballan wrasse

Hoplostethus atlanticus

Hygophum benoiti

Hyperoplus immaculatus

Hyperoplus lanceolatus

Katsuwonus pelamis

Labrus bergylta

Myctophidae (lanternfishes) Latidae (lates, perches) Gobiidae (gobies) Gobiidae (gobies) Moridae (morid cods) Trichiuridae (cutlassfishes)

Rakery beaconlamp

Nile perch

Guillet’s goby

Diminutive goby

North Atlantic codling

Silver scabbardfish

Lampanyctus macdonaldi

Lates niloticus

Lebetus guilleti

C

Lebetus scorpioides

Lepidion eques

Lepidopus caudatus

C

193 (TL)

16.0 (SL)

20.0 (SL)

Myctophidae (lanternfishes)

Lampanyctus intricarius

30.0 (SL)

Myctophidae (lanternfishes)

Jewel lanternfish

Lampanyctus crocodilus

15.3 (SL)

Myctophidae (lanternfishes)

18.0 (SL)

108 (FL)

5.5 (SL)

75.0

Mirror lanternfish

Sm

Nbi

C

C

B

Bladder Yang Max. length/cm code group (TL, SL, or FL); L50 /cm

Lampadena speculigera

Myctophidae (lanternfishes)

Pleuronectidae (righteye flounders)

Atlantic halibut

Hippoglossus hippoglossus

Lampadena anomala

Family

Common name

Species (scientific name)

Table C.7 (cont.)

684 Appendix C

Zoarcidae (eelpouts)

Doubleline eelpout

Lycodes eudipleurostictus

Zoarcidae (eelpouts) Zoarcidae (eelpouts)

Scalebelly eelpout

Vahl’s eelpout

Lycodes squamiventer

Lycodes vahlii

Macquaria novemaculeata

Australian bass

Zoarcidae (eelpouts)

Longear eelpout

Lycodes seminudus

Percichthyidae (temperate perches)

Zoarcidae (eelpouts)

Zoarcidae (eelpouts)

Arctic eelpout

Lycodes reticulatus

Lycodonus flagellicauda

Zoarcidae (eelpouts)

Pale eelpout

Lycodes pallidus

Zoarcidae (eelpouts)

Zoarcidae (eelpouts)

Greater eelpout

Lycodes esmarkii

Lycodes frigidus

Zoarcidae (eelpouts)

Sars’s wolf eel

Lycenchelys sarsi

Le

A

A

50.0 (TL)

(continued)

60.0 (TL)

19.9 (SL)

26.0 (TL)

51.7 (TL)

36.0 (TL)

69.0 (TL)

22.6 (SL)

Stichaeidae (pricklebacks)

Snakeblenny

Lumpenus lampretaeformis

A

Zoarcidae (eelpouts)

Lophiidae (goosefishes)

Angler

Lophius piscatorius

40.0 (SL); 12 L50

Lycenchelys muraena

Pleuronectidae (righteye flounders)

Dab

Limanda limanda

C

26.7 (SL)

Gobiidae (gobies)

Fries’s goby

Lesuerigobius friesii

B

Zoarcidae (eelpouts)

Scopthalmidae (turbots)

Megrim

Lepidorhombus whiffiagonis

40.0 (SL)

Lycenchelys alba

Scopthalmidae (turbots)

Fourspotted megrim

Lepidorhombus boscii

Appendix C 685

Family

Macrouridae (grenadiers or rattails) Merlucciidae (merluccid hakes) Osmeridae (smelts) Sternoptychidae Gadidae (cods and haddocks) Gadidae (cods and haddocks) Gadidae (cods and haddocks) Gadidae (cods and haddocks) Merlucciidae (merluccid hakes) Merlucciidae (merluccid hakes) Merlucciidae (merluccid hakes) Merlucciidae (merluccid hakes) Merlucciidae (merluccid hakes) Merlucciidae (merluccid hakes)

Gadidae (cods and haddocks) Gadidae (cods and haddocks)

Common name

Onion-eyed grenadier

Blue grenadier

Capelin

Pearlsides

Haddock

Pelagic cod

Arrowtail

Whiting

Offshore hake

Southern hake

South Pacific hake

Peruvian hake

European hake

North Pacific hake

Thickback sole

Atlantic tomcod

Southern blue whiting

Species (scientific name)

Macrourus berglax

Macruronus novaezelandiae

Mallotus villosus

Maurolicus muelleri

Melanogrammus aeglefinus

Melanonus gracilis

Melanonus zugmayeri

Merlangius merlangus

Merluccius albidus

Merluccius australis

Merluccius gayi gayi

Merluccius gayi peruanus

Merluccius merluccius

Merluccius productus

Microchirus variegatus

Microgadus tomcod

Micromesistius australis

Table C.7 (cont.)

Lm

Le

Lm

Lm

Lm

Lem

Lm

B

A

A

A

90.0 (TL)

38.0 (TL)

91.0 (TL)

68.0 (TL)

87.0 (TL)

126 (TL)

40.0 (TL)

70.0 (TL); 20 L50

28 (TL)

18.7 (SL)

100.0 (TL); 30 L50

120 (TL)

Bladder Yang Max. length/cm code group (TL, SL, or FL); L50 /cm

686 Appendix C

Lotidae (hakes and burbots) Lotidae (hakes and burbots) Moronidae (temperate basses)

Blue ling

Ling

Striped bass

Red mullet

Molva dipterygia

Molva molva

Morone saxatilis

Mullus surmuletus

120 (TL)

Common Atlantic grenadier Macrouridae (grenadiers or rattails) Notacanthidae (spiny eels) Myctophidae (lanternfishes) Myctophidae (lanternfishes) Salmonidae (salmonids) Salmonidae (salmonids)

Deep-sea spiny eels

Japanese lanternfish

Lancet fish

Pink salmon

Sockeye salmon

Nezumia aequalis

Notacanthus chemnitzii

Notoscopelus japonicus

Notoscopelus kroyeri

Oncorhynchus gorbuscha

Oncorhynchus nerka

(continued)

84.0 (TL)

14.3 (SL)

36.0 (TL)

Gempylidae (snake mackerels)

Black gemfish

Nesiarchus nasutus

130 (SL)

Sygnathidae (pipefishes and seahorses)

Straight-nosed pipefish

30.5 (SL)

Neoscopelidae

Nerophis ophidion

Neoscopelus microchir

Large-scaled lantern fish

Neoscopelus macrolepidotus

25.0 (SL)

B

Hagfish

Myxine glutinosa

200 (TL)

65.0 (TL); 20 L50

Neoscopelidae

B

A

A

A

B

Bull-rout

Sm

Le

Lm

Myoxocephalus scorpius

Myctophidae (lanternfishes)

Pleuronectidae (righteye flounders)

Lemon sole

Microstomus kitt

Myctophum punctatum

Gadidae (cods and haddocks)

Blue whiting

Micromesistius poutassou

Appendix C 687

Scombridae (mackerels, tunas, bonitos) Adrianichthyidae (ricefishes) Osmeridae (smelts) Osmeridae (smelts) Sparidae (porgies) Percidae (perches) Pholidae Pholidae

Gadidae (cods and haddocks) Pleuronectidae (righteye flounders) Gadidae (cods and haddocks) Gadidae (cods and haddocks) Gobiidae (gobies) Gobiidae (gobies)

Plain bonito

Japanese rice fish

Arctic rainbow smelt

Atlantic rainbow smelt

Red seabream

European perch

Rock gunnel

Crescent gunnel

Norwegian (topknot)

Forkbeard

European plaice

Pollack

Pollock

Common goby

Sand goby

Orcynopsis unicolor

Oryzias latipes

Osmerus mordax dentus

Osmerus mordax mordax

Pagrus major

Perca fluviatilis

Pholis gunnellus

Pholis laeta

Phrynorhombus norvegicus

Phycis blennoides

Pleuronectes platessa

Pollachius pollachius

Pollachius virens

Pomatoschistus microps

Pomatoschistus minutus

Gadidae (cods and haddocks)

Gadidae (cods and haddocks)

Arctic rockling

Onogadus argentatus

Onogadus ensis

Family

Common name

Species (scientific name)

Table C.7 (cont.)

Lm

Le

Pk

Sm

Le

Nb

C

C

A

A

B

B

A

130 (TL)

100.0 (SL); 33 L50

25.0 (TL)

25.0 (SL)

51.0 (TL)

100.0 (SL)

35.6 (TL)

32.4 (TL)

4.0 (TL)

130 (FL)

Bladder Yang Max. length/cm code group (TL, SL, or FL); L50 /cm

688 Appendix C

B B B

Gadidae (cods and haddocks) Cyprinidae (minnows or carps)

Spotted ray

Cuckoo ray

Starry ray

Tadpole fish

Silver cyprinid

Raja montagui

Raja naevus

Raja radiata

Raniceps raninus

Rastrineobola argentea

Salmonidae (salmons, trouts) Salmonidae (salmons, trouts) Salmonidae (salmons, trouts) Scombridae (mackerels, tunas, bonitos) Scombridae (mackerels, tunas, bonitos)

Atlantic salmon

Sea trout

Charr

Australian bonito

Eastern Pacific bonito

Salmo trutta trutta

Salvelinus alpinus

Sarda australis

Sarda chiliensis chiliensis

Gadidae (cods and haddocks)

Salmo salar

Rhinonemus cimbrius

B

Shagreen ray

Raja fullonica

Nb

Nb

Sm

B

Roker

Raja clavata

Pc

B

Sandy ray

Raja circularis

Rajidae (skates)

B

Skate

Raja batis

Bramidae (breams)

Silver pomfret

C

C

Pterycombus brama

Myctophidae (lanternfishes)

Gobiidae (gobies)

Painted goby

Pomatoschistus pictus

Protomyctophum arcticum

Gobiidae (gobies)

Norway goby

Pomatoschistus norvegicus

(continued)

102 (TL)

180 (FL)

140 (SL)

150 (TL)

9.0 (SL)

47 L50

80.0 (TL)

40.0

Appendix C 689

B B

Scombridae (mackerels, tunas, bonitos) Clupeidae (herrings, shads, sardines, menhadens) Clupeidae (herrings, shads, sardines, menhadens) Sciaenidae (drums or croakers) Scombridae (mackerels, tunas, bonitos) Scombridae (mackerels, tunas, bonitos) Scomberesicidae (sauries) Neoscopelidae

Sebastidae (rockfishes, rockcods, and thornyheads) Sebastidae (rockfishes, rockcods, and thornyheads)

Atlantic bonito

European pilchard

South American pilchard

Red drum

Chub mackerel

Atlantic mackerel

Skipper

Pacific blackchin

Turbot

Brill

Acadian redfish

Ocean perch

Sarda sarda

Sardina pilchardus

Sardinops sagax

Sciaenops ocellatus

Scomber japonicus

Scomber scombrus

Scomberesox saurus

Scopelengys tristis

Scopthalmus maximus

Scopthalmus rhombus

Sebastes fasciatus

Sebastes marinus

Lm

Nb

Nf

Sm

Nb

A

E

Scombridae (mackerels, tunas, bonitos)

Striped bonito

Sarda orientalis

Nb

Scombridae (mackerels, tunas, bonitos)

Pacific bonito

Sarda chiliensis lineolata

100.0 (TL)

30.0 (TL)

20.0 (SL)

60.0 (FL); 30 L50

64.0 (TL)

155 (TL)

39.5 (SL)

25.0 (SL)

91.4 (FL)

102 (FL)

102 (FL)

Bladder Yang Max. length/cm code group (TL, SL, or FL); L50 /cm

Family

Common name

Species (scientific name)

Table C.7 (cont.)

690 Appendix C

Gasterosteidae (sticklebacks and tubesnouts) Clupeidae (herrings, shads, sardines, menhadens) Clupeidae (herrings, shads, sardines, menhadens)

Fifteen-spined stickleback

Baltic sprat

European sprat

Spurdog

Spinachia spinachia

Sprattus sprattus balticus

Sprattus sprattus sprattus

Squalus acanthias

Synaphobranchidae (cut-throat eels) Sygnathidae (pipefishes and seahorses) Bramidae (breams)

Kaup’s arrowtooth eel

Greater pipefish

Rough pomfret

Walleye pollock

Synaphobranchus kaupii

Syngnathus acus

Taractes asper

Theragra chalcogramma

Labridae (wrasses)

Sparidae (porgies)

Gilthead seabream

Sparus aurata

Symphodus melops

Soleidae (soles)

Common sole

Solea solea

Myctophidae (lanternfishes)

Sillaginidae (smelt-whitings)

Sand sillago

Sillago ciliata

Symbolophorus californiensis Bigfin lanternfish

Sebastidae (rockfishes, rockcods, and thornyheads)

Lfm

Sm

Le

Le

A

D

B

A

(continued)

100.0 (TL)

11.0 (SL)

16.0 (SL); 10 L50

16.0 (TL)

70.0 (TL)

70.0; 27 L50

51.0 (TL)

65.0 (TL)

Sebastidae (rockfishes, rockcods, and thornyheads)

Sm

55.0 (TL)

Sebastidae (rockfishes, rockcods, and thornyheads)

Norway haddock

Deepwater redfish

Sebastes viviparus

Sebastes schlegelii

Sebastes mentella

Appendix C 691

B A

Scombridae (mackerels, tunas, bonitos) Scombridae (mackerels, tunas, bonitos)

Carangidae ( jacks and pompanos) Carangidae ( jacks and pompanos) Carangidae ( jacks and pompanos) Carangidae ( jacks and pompanos) Macrouridae (grenadiers or rattails)

Pacific albacore

Bigeye tuna

Northern bluefin tuna

Greater weever

Cape horse mackerel

Blue jack mackerel

Pacific jack mackerel

Atlantic horse mackerel

Roughnose grenadier

Tub gurnard

Moustache sculpin

Norway pout

Bib

Thunnus germo

Thunnus obesus

Thunnus thynnus

Trachinus draco

Trachurus capensis

Trachurus picturatus

Trachurus symmetricus

Trachurus trachurus

Trachyrinchus murrayi

Trigla lucerna

Triglops murrayi

Trisopterus esmarkii

Trisopterus luscus

Gadidae (cods and haddocks)

B

Lr

Scombridae (mackerels, tunas, bonitos)

Blackfin tuna

Thunnus atlanticus

Lm

Lm

Lm

Lm

Pb

A

E

B

Pi

Scombridae (mackerels, tunas, bonitos)

Yellowfin tuna

Thunnus albacares

Lr

Scombridae (mackerels, tunas, bonitos)

Albacore

Thunnus alalunga

35.0 (TL); 13 L50

37.0 (TL)

70.0 (TL); 24 L50

60.0 (TL)

60.0 (FL)

458 (TL)

250 (TL)

108 (FL)

140 (FL)

Bladder Yang Max. length/cm code group (TL, SL, or FL); L50 /cm

Family

Common name

Species (scientific name)

Table C.7 (cont.)

692 Appendix C

Gadidae (cods and haddocks) Gadidae (cods and haddocks) Sternoptychidae Xyphiidae

Poor cod

White hake

Constellationfish

Swordfish

Dory

Trisopterus minutus

Urophycis tenuis

Valenciennellus tripunctulatus

Xiphias gladius

Zeus faber

A

A

3.1 (SL)

Appendix C 693

694 Appendix C

C.2

REFERENCES

Bertrand, A., Josse, E., and Masse´, J. (1999) In situ acoustic target-strength measurement of bigeye (Thunnus obesus) and yellowfin tuna (Thunnus albacares) by coupling split-beam echosounder observations and sonic tracking, ICES J. Marine Science, 56, 51–60. Crocker, M. J. (Ed.) (1997) Encyclopedia of Acoustics, Wiley, New York. Egloff, M. (2006) Failure of swim bladder inflation of perch, Perca fluviatilis L. found in natural populations, Aquatic Sciences, 58(1), 15–23. Foote, K. G. (1997) Target strength of fish, in M. J. Crocker (Ed.), Encyclopedia of Acoustics (pp. 493–500), Wiley, New York. Froese, R. and Pauly, D. (Eds.), FishBase, version (01/2007), available at http://www. fishbase.org/search.php (last accessed March 23, 2009). Iversen, R. T. B. (1967) Response of yellowfin tuna (Thunnus albacares) to underwater sound, in W. N. Tavolga (Ed.), Marine Bio-acoustics (Vol. 2, pp. 105–121), Proceedings Second Symposium on Marine Bio-Acoustics, American Museum of Natural History, New York, Pergamon Press, Oxford, U.K. Kitajima, C., Tsukashima, Y., and Tanaka, M. (1985) The voluminal changes of swim bladder of larval red sea bream Pagrus major, Bull. Japanese Soc. Scientific Fisheries, 51, 759–764. Knijn, R. J., Boon, T. W., Heessen, H. J. L., and Hislop, J. R. G. (1993) Atlas of North Sea Fishes: Based on Bottom-trawl Survey Data for the Years 1985–1987 (ICES Cooperative Research Report No. 194), International Council for the Exploration of the Sea, Copenhagen, 1993. Simmonds, E. J. and MacLennan D. N. (2005) Fisheries Acoustics (Second Edition), Blackwell, Oxford, U.K. Tavolga, W. N. (Ed.) (1967) Marine Bio-acoustics (Vol. 2), Proceedings Second Symposium on Marine Bio-Acoustics, American Museum of Natural History, New York, Pergamon Press, Oxford, U.K. Whitehead, P. J. P. and Baxter, J. H. S. (1989) Swimbladder form in clupeoid fishes, Zoological Journal of the Linnean Society, 97, 299–372. Yang, J. (1982) An estimate of the fish biomass in the North Sea, J. Cons. int. Explor. Mer, 40, 161–172.

Index (bold indicates main entry)

absorption cross-section 245 see also extinction cross-section; scattering cross-section acidity see pH acoustic deterrents 523, 524, 525, 632 acoustic intensity 32 ff, 56 ff, 95 ff, 209, 245, 417, 550, 663 acoustic power 31 ff, 37 ff, 56 ff, 80 ff, 97, 209, 245 acoustic pressure 31 ff, 41, 58, 96 ff, 192 ff, 233 ff, 430, 492, 525 ff, 548 ff, 563 ff, 661 see also pressure (RMS) acoustic sensors communications equipment 88 ff, 523, 526, 575, 599 ff, 632 echo sounder see echo sounder fisheries sonar 5, 22, 519, 520, 575 minesweeping sonar 520, 522, 575 navigation sonar 22, 528, 575 oceanographic sensors 5, 22, 528, 575 search sonar 519, 522, 575 seismic survey sensors 534 ff, 575 sidescan sonar 515 ff, 575 acoustic waveguide 462 bottom duct 478, 483, 508 channel axis 21, 138, 462 ff, 496 ff convergence zone 474 cut-off frequency 449, 458, 472, 490

deep sound channel 148, 462 multipath propagation 308, 452, 525 SOFAR channel 20 21, 23 surface duct 459 ff, 462 ff, 471 ff, 478 ff, 502 adiabatic pulsations (of air bubble) 230 ff, 240, 367 see also isothermal pulsations; polytropic index Airy functions 204, 448, 648 Albersheim’s approximation 315 316, 330 ff, 597 ff ambiguity ellipse 301, 304 function 300 301, 304 ff surface 301 ff volume 300 amplitude threshold 51, 63, 312, 313, 326 ff, 346, 531 ff see also detection threshold (DT); energy threshold analogue to digital converter (ADC) 251 analytic signal 281 282 see also envelope function; Hilbert transform APL-UW High-Frequency Ocean Environmental Acoustic Models Handbook 175, 364 ff, 372 ff, 392 ff, 411, 424 ff, 622

696 Index array gain (AG) 62 ff, 69 ff, 76, 85 ff, 90, 98 ff, 102 ff, 107, 114 ff, 122, 252, 271 ff, 308, 580 ff, 594 ff, 611 ff, 622 ff, 629 asdics 12, 16, 17 asdivite 12 ATOC 528 attenuation coefficient of compressional wave 197, 199 see also attenuation coefficient of shear wave; volume attenuation coefficient in pure seawater see attenuation of sound in seawater in rocks 183 in sediments 172 ff, 377 ff, 604 in whale tissue 156 attenuation coefficient of shear wave 197, 199 see also attenuation coefficient of compressional wave in rocks 183 in sediments 180 attenuation of sound in seawater 18, 28 ff, 146 148, 471 audibility of sound in seawater 29, 615 ff see also attenuation of sound in seawater; visibility of light in seawater audiogram 550 see also hearing threshold of cetaceans 551 ff, 619 ff of fish 555 ff of human divers 554 ff of pinnipeds 551 ff of sirenians 554 autocorrelation function 296 ff, 651 narrowband approximation 299 autospectral density 287, 296 background energy level 97, 98, 113 background level (BL) 610, 625 ff backscattering cross-section (BSX) 41, 106, 209, 400, 491, 493, 607 see also scattering cross-section; target strength (TS) of fish 219, 223, 246 of fluid objects 214 215 of gas bubble 216, 246 of metal spheres 211 of rigid objects 210 ff

backscattering strength bottom 391 ff Chapman Harris model 372 Ellis Crowe model 224, 396, 398 Ogden Erskine model 371 372 surface 371 ff, 502 Bacon, Francis 3 Ballard, Robert see historical vessels (Titanic) Balls, R. (Captain) see historical sonar equipment (fish finder) bandwidth 42, 61 ff, 68, 73, 76, 80 ff, 104, 112 ff, 279, 283 ff, 306, 345, 346, 577, 579, 587, 604, 612 ff see also critical bandwidth; critical ratio; effective bandwidth Batchelder, L. 17 see also historical institutions (Submarine Signal Company); sound speed profile (thermocline) bathymetry 126 ff, 142 bathythermograph see also conductivity temperature depth (CTD) probe expendable (XBT) 129 ff Spilhaus 17 beamformer 44 ff, 252 ff array response 45, 61 ff, 84 ff, 98, 114 array shading see shading function beam pattern 45 ff, 61, 252 ff, 272 ff, 576 ff, 602, 607 ff, 625 beamwidth 46, 69 ff, 261, 264, 265, 496, 513, 626 broadside beam 46, 71 ff, 87 ff, 102 ff, 115 ff, 253 ff, 267 ff endfire beam 47, 114, 253 ff, 267 ff, 580 sidelobe 257 ff, 264, 265, 627 steering angle 46, 252 ff Beaufort wind force 159 ff da Silva et al. 160, 162, 165 Lindau 160 WMO CMM IV 165 WMO code 1100 159, 164, 165 Beauvais, G. A. see historical sonar equipment (Brillouin Beauvais amplifier) Behm, Alexander 16 Bessel function 261, 316, 645 646, 648 see also modified Bessel function

Index Beudant, Franc¸ois see historical events (speed of sound in water, first measurement of ) binary integration see M out of N detection bistatic sonar 96, 493 ff, 508, 587 Blake, L. I. 14 Boltzmann constant 126, 549 bottom reflected path 443 ff, 462 bottom refracted path 444 ff Boyle, Robert William 10 ff see also historical institutions (Applied Research Laboratory); historical institutions (Board of Invention and Research) Bragg scattering vector 206, 224 Bragg, W. H. (Professor) see historical institutions (Board of Invention and Research) Brillouin, Le´on see historical sonar equipment (Brillouin Beauvais amplifier) bubble pulse 537 ff bulk modulus 193, 194 ff adiabatic 367 see also polytropic index of air 367 of dilute suspension 225 of gas bubble 229, 230 ff isothermal 367 see also polytropic index of saturated sediment 227 of water 8, 32, 192, 225, 228, 649 carrier wave 280 caustic 445 ff, 468 ff, 504 ff characteristic impedance 58, 429, 552, 576, 663 see also impedance of air 37, 417 of seabed 172 of water 417, 550 chemical relaxation 18, 146 boric acid 18, 147 magnesium carbonate 18, 147 magnesium sulfate 18 Chilowski, Constantin 10 ff see also Langevin, Paul chi-squared distribution 51, 328

697

coherent addition 35 ff coherent processing 51, 64 ff, 99 ff, 279, 312 ff, 346, 574 ff, 606 ff Colladon, Daniel see historical events (speed of sound in water, first measurement of ) column strength (CS) 410, 412 413 complementary error function (erfc) 49 ff, 85 ff, 339 ff, 482, 597, 637 638 compressibility see bulk modulus compressional wave 179, 193 ff, 379 see also attenuation coefficient of compressional wave speed of compressional wave Conan Doyle, Arthur 311 conductivity temperature depth (CTD) probe 129, 134 see also bathythermograph convergence zone (CZ) 474 see also acoustic waveguide convolution 281, 344, 651 theorem 651 correlation function 206 length 205, 206, 362, 370 radius 207, 224 cosine integral function (Ci) 640 Cox Munk surface roughness slope see roughness slope (surface) critical angle see reflection coefficient critical bandwidth 557 ff critical ratio 557 ff cross-correlation function 297, 298 cross-correlation theorem 650 CTD see conductivity temperature depth (CTD) probe cubic equation, roots of 240, 476, 655 Curie, Jacques and Pierre see historical events (piezoelectricity, discovery of ) cusp 468 da Vinci, Leonardo 18, 53 damping coefficient 216, 229, 237, 243 ff, 373 see also damping factor damping factor 229 ff see also damping coefficient

698

Index

decibel (dB) 29, 58, 175, 525 526, 661 663 see also logarithmic units deep scattering layer 402, 412 density 192 ff, 492 ff of air 30, 151, 237 of fish flesh 153, 155, 222 of metals 210, 212 of rocks 180 ff of seawater 8, 28, 127 ff, 233 of sediments 172 ff, 176, 178, 203, 227, 377 ff, 393, 441, 449, 500, 604 of whale tissue 156 of zooplankton 156, 157 detection area 590 detection probability 47 ff, 71 ff, 85 ff, 92 ff, 103 ff, 107 ff, 115 ff, 313 ff, 329 ff cumulative 354 detection range 77 ff, 90 ff, 107 ff, 117 ff, 585 ff, 605, 614 ff detection theory 21, 47 ff, 311 detection threshold (DT) 63 ff, 74 ff, 85 ff, 89 ff, 103 ff, 107 ff, 115 ff, 279, 315 ff, 326 ff, 347 ff, 355 ff, 581 ff, 597 ff, 612 ff detection volume 587 ff DFT see discrete Fourier transform (DFT) dilatation 193 ff dilatational viscosity see viscosity (bulk) dipole source 38, 69, 419 ff, 424 ff, 485, 535 ff, 621 see also monopole source Dirac delta function 62, 222, 314 ff, 412, 591, 636, 650 Dirac distribution 314 ff directivity factor 115, 266 ff, 580 ff, 611 ff, 622 see also directivity index (DI) directivity index (DI) 62, 69, 266, 580 ff, 594 ff, 611 see also directivity factor Dirichlet window see shading function (rectangular window) discrete Fourier transform (DFT) 43 ff, 651 ff Doppler autocorrelation function (DACF) 299 ff Doppler effect 99, 298 Doppler resolution 294, 295, 301 ff

see also frequency resolution; range resolution dose response relationship 563 duct axis see acoustic waveguide (channel axis) echo energy level 606 echo level (EL) 400, 493, 508, 607 ff echo sounder 5, 16, 22, 516, 575 see also acoustic sensors multi-beam 516, 518, 519, 575 single beam 515, 516 effective angle 457 effective bandwidth 283 ff effective pulse duration 282 ff effective water depth 457 ff electromagnetic wave radar 17, 21, 311 ff, 476 visibility of light 10, 29, 163 ellipsoid 212 surface area 155 volume 155 elliptic integrals 155, 467 ff, 644 energy density 32, 663 kinetic energy density 32 potential energy density 32 energy threshold 51, 63, 328 see also amplitude threshold; detection threshold (DT) envelope function 282 ff, 298 see also analytic signal; Hilbert transform equivalent plane wave intensity (EPWI) 58, 493, 552 ff, 661 ff equivalent target strength 607 ff see also target strength (TS) error function (erf ) 453, 481, 494, 508, 533 ff, 637 Ewing, Maurice see historical events (SOFAR channel, discovery of ) explosives 431, 538 ff scaled charge distance 539 ff shock front 539 ff similarity theory of Kirkwood and Bethe 539 exponential integrals 39, 66, 100, 297, 639 extinction cross-section see also absorption cross-section; scattering cross-section

Index of fish 246 of gas bubble 246 facet strength 399 false alarm 7, 48, 54 false alarm probability 50 ff, 72, 87, 103, 115, 312, 328, 345, 350 ff, 582, 597 ff, 613 false alarm rate 104, 115, 582, 613 far field 209, 400, 418, 431, 514 ff, 576, 608 Fay, H. J. W. 13 Fessenden, Reginald 9 ff, 516 see also historical sonar equipment (fathometer); historical sonar equipment (Fessenden oscillator) figure of merit (FOM) 69, 75 ff, 85, 91 ff, 101, 113, 121, 585, 619 ff filter anti-alias 42, 251, 594 band-pass 80 Doppler 99, 297 see also discrete Fourier transform (DFT); Fourier transform flat response 62, 84, 594 ff, 610 high-pass 42 low-pass 42, 251, 289 matched 280, 296 ff, 345, 508, 606, 612 passband 42, 43, 62, 80, 264, 474, 558 pre-whitening 595 spatial see beamformer temporal 42, 594 filter gain (FG) 593 ff FishBase 152, 673 Fisheries Hydroacoustic Working Group (FHWG) 560, 563 form function 210 see also scattering cross-section Fourier transform 206, 281, 286, 289, 296, 649, 651, 652 Franklin, Benjamin 13, 18, 573 frequency modulation (FM) 22 hyperbolic (HFM) 283 ff, 305 linear (LFM) 283 ff, 304 ff frequency resolution 44, 90 see also Doppler resolution frequency spread 285 ff Fresnel integrals 288, 293, 636, 653 full width at half-maximum (f.w.h.m.) 44 ff, 256 ff, 287

699

fusion gain 350 ff gamma function 498, 640 incomplete 291, 328, 342, 497, 628, 642 Stirling’s formula 641 Gaussian distribution 47 ff, 71, 208, 312, 316, 322 Gerrard, Harold 10, 14 see also historical institutions (Board of Invention and Research) grain size 172 ff, 180, 377, 392 ff, 454, 583, 599, 665 Gray, Elisha 14 see also historical institutions (Submarine Signal Company) grazing angle 38, 114, 116, 198 ff, 205 ff, 224, 362 ff, 376 ff, 428, 448 ff, 464 ff, 495 ff, 607, 626 Hall Novarini bubble population density model 169, 231 ff, 367 Hamming, Richard 259 see also shading function (Hamming window) Hayes, Harvey 13 ff see also historical institutions (Naval Experimental Station); historical institutions (Naval Research Laboratory) Heard Island Feasibility Test (HIFT) 22, 528 hearing threshold 418, 550 ff, 619 see also audiogram; permanent threshold shift; temporary threshold shift Heaviside step function 281, 452, 635, 650 HFM see frequency modulation (FM) HIFT see Heard Island Feasibility Test (HIFT) Hilbert transform 281 historical events 1918 Armistice 12 Cold War 4, 21 echolocation, first demonstration of 12 echo ranging, conception of 8, 13 First World War (WW1) 4, 7, 10 ff piezoelectricity, discovery of 10, 13 Roswell incident 21

700

Index

historical events (cont.) Second World War (WW2) 4, 12 ff, 408 ff, 418 SOFAR channel, discovery of 20 ‘‘sonar’’, coining of 17 speed of sound in water, first measurement of 8 historical institutions Anti-Submarine Division 12 see also asdics Applied Research Laboratory (ARL) 16 Board of Invention and Research (BIR) 10, 14 British Admiralty 12, 13 California, University of 17 Columbia, University of 17 Lighthouse Board, U.S. 14 Manchester, University of 10 Marine Studios, Florida 23 National Defense Research Committee (NDRC) 17 Naval Experimental Station, New London 14 Naval Research Laboratory (NRL) 16, 17 Oxford University Press 12 Public Instruction and Inventions, Ministry of 13 Submarine Signal Company 14, 16, 17 Woods Hole Oceanographic Institution (WHOI) 17 historical sonar equipment Brillouin Beauvais amplifier 12, 13 ‘‘eel’’ 15 fathometer 15 Fessenden oscillator 10, 11, 516 fish finder 15 gruppenhorchgera¨t (GHG) 17 JK projector 16 M B tube 14, 15 M V tube 15 QB 16 recording echo sounder 16 rho-c rubber 16 Rochelle salt 10, 16 sound fixing and ranging (SOFAR) 21 see also RAFOS sound surveillance system (SOSUS) 21 towed fish 14

U-3 tube 15 underwater bell 8, 10, 14 historical vessels Glen Kidston 16 Nautilus, USS 22 Prinz Eugen 17 Titanic, RMS 4, 10, 22 Hooke’s law 193 Hunt, F. V. 5, 7, 17, 515 see also historical events (‘‘sonar’’, coining of ) Huxley, Thomas Henry 251 hydrophone sensitivity 54, 514, 545, 594 hydrophone array horizontal line array 55, 69 ff, 87 ff, 102 ff, 116, 253, 267 ff line array 44, 114, 252 ff, 267 ff, 580 planar array 261, 266 ff vertical line array 602 hypergeometric functions 232, 320, 648 in-beam noise level 74 ff, 92, 107, 584 ff, 601 ff in-beam noise spectrum level 92, 105 in-beam signal level 584 ff, 601 ff incoherent addition 51, 80, 335, 341, 343, 578 incoherent processing 51, 80, 112, 327, 591 instantaneous frequency 283, 285 ff, 291 ff integration time 72, 89, 316, 345, 346, 597, 599, 602 Iselin, Columbus see historical institutions (Woods Hole Oceanographic Institution) isothermal pulsations (of air bubble) 229 ff, 367 see also adiabatic pulsations; polytropic index K distribution 348 Kirchhoff approximation 208, 212 Lame´ parameters 195 Langevin, Paul 10 ff see also historical events (echo location, first demonstration of ) LFM see frequency modulation (FM)

Index 701 Lichte, H. 19, 20, 439 see also historical events (SOFAR channel, discovery of ) Liebermann, L. 18, 139 Lippmann, Gabriel see historical events (piezoelectricity, discovery of ) Lloyd mirror 36, 64, 443, 474, 592 logarithmic units see also pH bel 660 see also decibel (dB) decade 664 neper (Np) 29, 30, 660, 665 octave 264, 420, 558, 562, 595, 596, 665 phi unit () 173, 665 see also grain size third octave 420 ff longitudinal wave see compressional wave M out of N detection 356 Marcum function 21 generalized Marcum function 330, 644 Marcum Q function 314 ff, 644 Marcum, J. 21 Marley, Bob 513 Marti, P. see historical sonar equipment (recording echo sounder) matched filter gain (MG) 306 ff, 612 mean square pressure (MSP) see pressure (mean square) see also pressure (RMS) Mersenne see historical events (echo ranging, conception of ) Michel, Jean Louis see historical vessels (Titanic) Minnaert, Marcel 191 modified Bessel function 314, 320, 326, 329, 644, 646 647 see also Bessel function monopole source 31 ff, 418 ff, 428, 491 ff, 576 see also dipole source Mundy, A. J. 14 see also historical institutions (Submarine Signal Company) M-weighting 559

Nash, G. H. see historical sonar equipment (towed fish) natural frequency 215, 216 see also resonance frequency near field see far field Neptunian waters 146 noise ambient 55, 66, 73, 309, 415 ff background 37 ff, 55, 61, 67, 427, 485, 557, 578, 614, 629 colored 596 dredger 490 flow 545, 549 foreground 578, 579 gain (NG) see array gain (AG) isotropic 61 ff, 269 ff level (NL) 61, 483, 585, 593, 605, 621, 624 non-acoustic 549, 578, 579 platform 579 precipitation 414, 415, 426, 489, 578, 596 self 55, 545, 550, 579 shipping 425, 427, 484, 485, 599 spectrum level 75 thermal 415, 484, 485, 488, 489, 545, 549, 578, 579 wind 115, 424 426, 484, 560, 614, 621, 624 non-SI units 659, 662 see also logarithmic units; SI units Nyquist frequency 42 Nyquist interval 87, 345, 346 Nyquist rate 306, 345 Ockham, William of 27 one-dominant-plus-Rayleigh distribution 313, 318, 322, 342 Painleve´, Paul 13 see also historical institutions (Public Instruction and Inventions, Ministry of ) particle velocity 32, 192, 209, 550, 557, 630 permanent threshold shift (PTS) 558 ff pH 664 free proton scale 664 National Bureau of Standards (NBS) scale 138, 147, 664 of seawater 28, 138, 147

702 Index pH (cont.) seawater scale 138, 664 total proton scale 138, 664 Physics of Sound in the Sea 17 physoclist 152, 158, 220, 401, 402, 619, 673, 674 physostome 152, 157, 158, 220, 401, 402, 673 ff Pichon, Paul 12 Pierce, G. W. 13 Plancherel’s theorem 652 Planck, Max 361 plane propagating wave 58, 552 Poisson’s ratio 195, 196 polytropic index 230, 234 ff pressure acoustic 31 ff, 58, 96, 97, 192, 198, 233, 243, 430, 492, 529, 531, 540, 550, 661 atmospheric 31, 60, 126, 127, 139, 151, 177, 216 ff complex 32, 35, 96 gauge 31 hydrostatic 30, 127, 151, 220, 230, 231 peak 431, 539, 540, 560, 565 peak to peak 525 peak-equivalent RMS (peRMS) 431, 531, 533, 534, 548, 617 RMS 32, 59, 415, 417, 431, 515, 529, 531, 533, 534, 549, 550, 576, 663 static 30, 31, 127, 231, 239, 367 zero to peak 525, 537 Principles of Underwater Sound 5, 18, 514, 576 prior knowledge 357, 583, 589, 591 probability of detection see detection probability probability of false alarm see false alarm probability processing gain (PG) 308, 593, 602, 610, 628 propagation factor 33, 59, 80, 608 see also propagation loss (PL) coherent 64, 577 cylindrical spreading 452, 454, 481, 483, 494, 498 differential 452, 478, 496, 500, 576, 607, 608 incoherent 36, 82, 593 Lloyd mirror 443, 476

mode stripping 453, 494, 498 multipath propagation 443 ff, 452 ff, 478 ff, one-way 101, 106, 107, 116, 491, 494, 616, 619, 625, 627 single mode 457 spherical spreading 452 two-way 96, 100, 104, 113 Weston’s flux method 464 ff, 478 ff propagation loss (PL) 58, 60, 66 ff, 83 ff, 96, 101 ff, 113 ff, 307, 365, 418 ff, 440, 483, 493, 504, 506, 544, 576, 583 ff, 592 ff, 607 ff, 663 see also propagation factor PTS see permanent threshold shift (PTS) pulse duration 96 ff, 115, 285, 287, 291, 294, 295, 302, 303, 305, 306, 345, 346, 495, 497, 531, 539, 565 see also effective pulse duration p-wave see compressional wave Q-factor 216 ff, 244 ff quadratic equation, roots of 655 quartic equation, roots of 656 radiant intensity 33, 34, 60, 428, 429, 576, 592, 608 scattered 40, 99, 209, 396, 397, 400, 608 radiation damping 242, 243, 244, 373 see also damping coefficient; damping factor radius of curvature 466, 472, 476, 480, 504, 505 RAFOS 528 see also SOFAR raised cosine spectrum see shading function (Tukey window) range resolution 115, 301 ff, 613 see also doppler resolution Rayleigh distribution 51, 71, 317, 323, 340, 345, 347, 352, 355, 612 Rayleigh fading 317, 319, 582 Rayleigh parameter 205, 208, 373 Rayleigh Plesset equation 232, 242 receiver operating characteristic (ROC) curve 71, 85, 103, 115, 315 ff, 344 ff, 581 reciprocity principle 492

Index 703 rectangle function 70, 253, 280, 285 ff, 635, 636 reduced target strength 402, 406 see also target strength (TS) reference distance 59, 60, 420, 431, 514, 544 reference pressure 59, 415, 554, 556, 663 reflection coefficient 408 see also reflection loss amplitude 198, 201, 221, 222 angle of intromission 378 bottom 172, 177, 202 ff, 375 ff, 447 ff, 452 ff, 480 coherent 205, 207, 209 critical angle 378 cumulative 480 energy 200 Rayleigh 199, 375, 455 surface 30, 35, 37, 362 ff, 466 total internal reflection 377 reflection loss see also reflection coefficient bottom 375 ff, 445, 454 ff, 508 surface 364 ff, 467 ff relaxation frequency 29, 147 resonance frequency 152, 216, 219 ff, 232 ff, 238, 239, 246, 409, 412, 413 see also resonant bubble radius adiabatic 234 isothermal 235 Minnaert frequency 216, 232, 234, 236, 237, 238, 239 resonant bubble radius 232 ff, 241 see also resonance frequency reverberation level (RL) 495, 508 Reverberation Modeling Workshop 498, 503 Rice, Stephen 21 see also Rician distribution Richardson, Lewis 10 see also historical vessels (Titanic) Rician distribution 21, 317, 318, 319, 322 Rician fading 318, 319, 321 right-tail probability function 329, 637 rigidity modulus see shear modulus RMS pressure see pressure (RMS) see also acoustic pressure ROC curve see receiver operating characteristic (ROC) curve rock 180

igneous 179, 180, 182, 183 metamorphic 180, 182 sedimentary 179, 180, 181, 182, 183, 184 roughness slope bottom 225 surface 374 roughness spectrum 206, 224, 369, 392, 398 see also wave height spectrum Gaussian 207, 224 Rutherford, Ernest (Lord) 10, 11, 14, 125 see also historical institutions (Manchester, University of ) Ryan, C. P. (Captain) 14 see also historical institutions (Board of Invention and Research) salinity 20, 128, 129, 133, 139, 146, absolute 129 practical 129 profile 134, 136, 439, 461 surface 135 scattering coefficient 41, 223, 224 see also scattering strength backscattering coefficient 224, 225, 371 bottom 391 ff, 496, 497 surface 42, 116, 369 ff, 625 ff scattering cross-section see also absorption cross-section; backscattering cross-section (BSX); extinction cross-section differential 40, 41, 209, 210, 214, 494, 607 of gas bubble 216, 243 total 209, 245 scattering strength see also scattering coefficient backscattering strength 371, 391 bottom 391 ff Ellis Crowe model 398 Lambert’s rule 396 McKinney Anderson model 399 surface 371 ff sea state 165 ff search sonar see also acoustic sensors coastguard sonar 522 helicopter dipping sonar 521, 575 hull-mounted sonar 15, 521, 575 sonobuoy 522, 575 towed array sonar 15, 522, 575, 579

704

Index

sediment biogenic 172 chemical 172 clastic 172 consolidated 180, 385 unconsolidated 172 ff, 375 ff, 583 seismic survey sources see also acoustic sensors air gun 535 ff, 560, 562, 575, 662 boomer 537, 538, 575 sleeve exploder 537, 538 sparker 537, 538, 575 sub-bottom profiler 514, 516, 520, 575 water gun 537, 538, 575 shading degradation 264, 269, 270 shading function 252, 259 cosine window 257, 264 Hamming window 259, 202, 264 Hann window 254, 258, 270, 583, 585 raised cosine window 258, 260 rectangular window 253, 254, 264 Taylor window 261, 264 triangular window 264 Tukey window 259, 264 shadow zone 459, 462 shear modulus 153, 192 ff, 219, 227 see also bulk modulus shear speed see speed of shear wave shear viscosity see viscosity (shear) shear wave 172, 179 ff, 194 ff, 227, 379 ff, 457 see also attenuation coefficient of shear wave; speed of shear wave SI units 39, 128, 141, 164, 659, 661 see also logarithmic units; non-SI units sign function 635 signal energy level 97, 105 signal excess (SE) 63, 67, 84, 100, 112, 121, 322, 346, 357, 583 ff, 602 ff, 621 ff see also detection threshold (DT); figure of merit (FOM) signal gain (SG) 63, 69, 273, 584, 602 see also array gain (AG) signal level 74 ff, 92 ff, 95, 109 ff, 117, 414, 491, 585, 629 signal to background ratio (SBR) 55, 98, 100, 112, 116 signal to noise ratio (SNR) 5, 41, 51, 62, 67, 84, 104, 271, 272, 314, 324 ff,

332, 340 ff, 345 ff, 400, 545, 595, 598, 599 signal to reverberation ratio (SRR) 508 sine cardinal function (sinc) 43 ff, 253 ff, 296 ff, 636, 650 sine integral function (Si) 267, 640 sinh cardinal function (sinhc) 82, 91, 203, 383, 636 Smith, B. S. see historical institutions (Applied Research Laboratory) snapping shrimp 429 Snell’s law 19, 171, 199, 366, 377, 449, 459, 471, 480, 481, 505 SOFAR see acoustic waveguide (SOFAR channel); historical sonar equipment (SOFAR); see also historical sonar equipment (SOSUS) sonar equation 5, 6, 53, 573, 666 active (Doppler filter) 100 active (energy detector) 112 active (matched filter) 606 broadband passive (incoherent) 84, 279, 591 narrowband passive (coherent) 67, 279, 574 use of (worked examples) 74, 88, 105, 117, 583, 599, 613 sonar oceanography 27, 125 sound exposure level 559 ff sound pressure level (SPL) 58, 417, 418, 663 sound speed see speed of compressional wave; speed of sound in seawater sound speed profile 145, 383, 459 ff, 474 ff, 490 afternoon effect 16, 17 downward refracting 459, 462, 474 ff, 478 ff, 500, 502 isothermal layer 599 solar heating 459, 474 sound speed gradient 20, 28, 177 ff, 389, 440, 445 ff, 459, 471 ff, 478, 494, 506 summer 20, 459 ff, 474 thermocline 129, 474, 479 see also Batchelder, L.; bathythermograph upward refracting 20, 462 ff, 478 ff, 500 wind mixing 365, 459, 462 winter 459 ff, 583

Index source factor 60, 65, 74, 80, 81, 89, 100, 106, 419 ff, 424, 426, 429, 485, 491, 492, 496, 528, 548, 575, 576, 592, 607, 615 source level (SL) 60, 68, 85, 96, 97, 101, 113, 417, 493, 514, 525, 529, 531, 575, 592, 608 of acoustic cameras 523, 527 of acoustic communications systems 523, 526 of acoustic deterrent devices 523, 524, 525 of acoustic transponders 523, 527 dipole 419 ff, 424 ff, 535 ff of echo sounders 515, 516, 518, 519 energy 96, 430, 525, 540, 544 of explosives 541 of fisheries sonar 519, 520 of marine mammals 542 ff, 616 ff of military search sonar 519, 521, 522 of minesweeping sonar 520 monopole 419 ff, 428 of oceanographic research sonar 523, 528 peak to peak 430, 431, 540 ff, 616 peak-equivalent RMS (peRMS) 431, 533, 548, 617 of seismic survey sources 534 ff of sidescan sonars 515, 517, 519 of sub-bottom profilers 516, 520 zero to peak 431, 540 source spectrum level 88 ff, 417, 424 ff, 483, 599, 604 spatial filter see beamformer specific heat ratio of air 150, 217, 230, 234, 235 spectral density level 57, 61, 65, 67, 68, 81, 84, 488, 602, 663 power 66, 75, 89, 287, 424, 428, 595 speed of compressional wave 193 see also speed of shear wave; Wood’s equation in air 30, 148 in bubbly water 228, 365 see also Wood’s equation in dilute suspension 226 see also Wood’s equation in fish flesh 153, 155, 221 in metals 212 in rocks 181 183

705

in seawater see speed of sound in seawater in sediments 172 ff, 176, 177, 178, 196, 203, 227, 377 ff, 389, 393, 445 ff, 455, 500 in whale tissue 156 in zooplankton 156, 157 speed of shear wave see also speed of compressional wave in metals 212 in rocks 181 183 in sediments 179, 379 speed of sound in seawater 8, 13, 19, 28, 126, 139, 145, 379 Leroy et al. formula 144 Mackenzie’s formula 140, 459 spherical wave 31 34 spheroid oblate 155 prolate 153, 154, 155, 215, 405, 407 Spilhaus, Athelstan 17 see also bathythermograph; historical institutions (Woods Hole Oceanographic Institution) Spitzer Jr., Lyman see Physics of Sound in the Sea SPL see sound pressure level (SPL) standard atmosphere 126, 220, 662 see also standard temperature and pressure (STP) standard gravity 126 standard temperature and pressure (STP) 126, 128, 151, 216, 219, 220, 662, 670 stationary phase approximation 284, 289, 290, 291 295, 296, 447, 448, 652 statistical detection theory 21, 47, 311 Stirling’s formula see gamma function Stokes, G. 18 STP see standard temperature and pressure (STP) Sturm, Charles-Franc¸ois see historical events (speed of sound in water, first measurement of ) surface area of ellipsoid 155, 405 of fish 153, 405 of fish bladder 153, 401, 409 surface decoupling 459, 471, 474 surface tension 151, 230 ff

706 Index surface wave spectrum 362 Neumann Pierson 166, 167, 363 Pierson Moskowitz 166, 168, 362, 364, 369, 370 SURTASS 519, 522 s-wave see shear wave Swerling distributions 21 Swerling I 328 Swerling II 327, 340, 341, 344, 357 Swerling III 328 Swerling IV 327, 342, 343, 344, 347, 357 swim bladder 675, 676, 678 taper function see shading function target strength (TS) 99, 101, 105, 113, 400, 493, 607, 610 of cetaceans 402, 403 of euphausiids 404 of fish 222, 401, 404, 619, 620, 624 of fish shoal 400 of gastropods 406 of human diver 402 of jellyfish 407 of marine mammals 402, 403 of mine 408 of siphonophore 407 of squid 406 of submarine 408 of surface ship 408 of torpedo 408 temperature profile 127, 129, 131, 134, 135 see also sound speed profile (thermocline) potential 133 surface 20, 128, 129, 130, 459 temporary threshold shift (TTS) 558 ff Texas at Austin, University of see Reverberation Modeling Workshop thermal conductivity of air 151, 237, 244 thermal damping 243 see also damping coefficient; damping factor thermal diffusion frequency 236, 239 thermal diffusion length 235 thermal diffusivity 151 of air 151, 217, 235, 237 thermohaline circulation 128 third octave see logarithmic units

total internal reflection see reflection coefficient total path loss (TPL) 96, 97, 100, 101, 113 transmission coefficient amplitude 199, 202 energy 200 transmission loss 60 transverse wave see shear wave triangulation 15, 21 TTS see temporary threshold shift (TTS) tunneling 472, 474 Udden Wentworth sediment classification scheme 173, 174 see also grain size underwater acoustics 30, 191 Urick, R. J. 13, 19 see also Principles of Underwater Sound viscosity 18, 146 see also attenuation of sound in seawater; viscous damping bulk 139, 155, 217, 244 shear 139, 155, 217, 232, 242, 244 viscous damping 242, 246 see also damping coefficient; damping factor visibility of light in seawater 29 see also audibility of sound in seawater volume see also surface area of arthropods 152 of euphausiids 152 of fish 153 of fish bladder 153 volume attenuation coefficient see also attenuation coefficient of compressional wave; attenuation of sound in seawater of bubbly water 411 of dispersed fish 411 volume backscattering strength 399, 409 ff volume viscosity see viscosity (bulk) von Hann, Julius 257 see also shading function (Hann window) wake strength 413 Washington, University of see APL UW High-Frequency Ocean

Index Environmental Acoustic Models Handbook wave equation 192, 193, 194, 200, 491 wave height 37 spectrum 166, 367 see also roughness spectrum mean peak-to-trough 167 RMS 167, 168, 169 significant 166, 167, 168 waveguide see acoustic waveguide Weibull distribution 348 Wells, A. F. 16 Weston, David E. 245, 439, 464, 468 see also propagation factor (Weston’s flux method) whispering gallery 468 Wiener Kinchin theorem 651 wind speed 40, 159, 162 165, 166 ff, 367 ff, 424 ff, 471 ff, 478, 484 ff, 585, 621, 623, 624 window function see shading function WMO see World Meteorological Organization (WMO)

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WOA see World Ocean Atlas (WOA) Wood, Albert Beaumont 10, 11, 13, 14, 16 see also historical institutions (Applied Research Laboratory); historical institutions (Board of Invention and Research); historical sonar equipment (recording echo sounder); Wood’s equation Wood’s equation 226, 228, 366 World Meteorological Organization (WMO) 159, 160, 162, 163, 164, 165, 166, 168 World Ocean Atlas (WOA) 129, 130, 131, 133, 135, 136, 137, 145 XBT see bathythermograph (expendable) Young’s modulus 196 Zacharias, J. (Professor) see historical sonar equipment (SOSUS)