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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2.2

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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

2.4

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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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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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

11.3

11.4

11.5

11.6

A

607 610 610 612 613 630 630 631 631 632

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

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 . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

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

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

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]

1.4 A brief history of sonar

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

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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.

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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 (<0.1 dB) for ranges up to 12 km.

Sec. 3.2]

3.2 Passive sonar 79

Figure 3.11. Signal excess [dB] vs. target range and depth for ﬁxed sonar depth zarr ¼ 30 m and frequency f ¼ 300 Hz. Other parameters are as Table 3.1 (INSIGHT).

its argument x, and Equation (3.79) simpliﬁes to 4 2 Dx W S zarr ztgt 2 3 ; 3 fR50 W N r2 Af

ð3:86Þ

where is the linear signal excess given by Equation (3.78) (ratio of the actual SNR to the minimum SNR required for detection). As already noted, Figures (3.8) and (3.9) show that the detection range is close to 2.5 km. A more precise calculation can be made by equating Rarr and R50 in Equation (3.86) and rearranging for the detection range r50 : ! 4 2 Dx W S 1=4 1=2 r50 ðzarr ztgt Þ : ð3:87Þ 3 3 fR50 W N Af Substituting numerical values into Equation (3.87) for a source power of 0.2 mW results in a detection range of 2.45 km. The main beneﬁt of solving the problem in this manner is the insight aﬀorded by Equation (3.87) into the sensitivity of r50 to sonar depth, array length, and processing bandwidth. In particular, using Equation (3.72) for R50 , Equation (3.87) can be written: 1=4 Dx=f 1=2 r50 / z arr : ð3:88Þ log2 ðNbeams Df =2nfa Þ

80 The sonar equations

[Ch. 3

According to Equation (3.88), the detection range can be increased by increasing the array depth or array length, or by decreasing the processing bandwidth. Treating nfa as ﬁxed, there is also a weak (logarithmic) dependence on the number of sonar beams and the total monitored bandwidth ðDf Þ. Dependence on the last two parameters arises through their inﬂuence on the rate of detection opportunities and hence the (acceptable) false alarm probability. 3.2.4

Incoherent processing: broadband passive sonar

In the case of conventional broadband (BB) passive sonar, the received pressure waveform is passed ﬁrst through a band-pass ﬁlter (the passband of which is the processing bandwidth of the sonar, denoted Df ). The output of this ﬁlter is squared and then integrated in time for a duration Dt. Such processing amounts to incoherent addition of energy over a wide frequency band. Thus, the knoll-shaped curve of Figure 3.2 now contributes to the signal, provided that it is within the ﬁlter passband, as illustrated by Figure 3.12. 3.2.4.1

Signal (single hydrophone)

The signal is now the total MSP integrated over the processing bandwidth Df , and is given for center frequency fm by ð fm þDf =2 QS ¼ Q Sf df : ð3:89Þ m Df =2

We deﬁne the BB propagation factor FBB where S0 is the source factor

QS ; S0

ð3:90Þ

ð

S0 ¼ ðS0 Þf df ;

ð3:91Þ

c S W : 4 f

ð3:92Þ

with associated spectral density ðS0 Þf ¼ It follows from Equation (3.90) that SBB ¼ S0 FBB :

ð3:93Þ

Now consider the behavior of FBB , which, for the assumed incoherent processing, can be written in the form ð Fcoh ðf ÞW Sf df

FBB

WS

;

ð3:94Þ

where W S denotes that part of the signal-radiated power spectrum within the sonarprocessing bandwidth. In this expression the integrand is an oscillatory function of frequency.

Sec. 3.2]

3.2 Passive sonar 81

Figure 3.12. Spectral density level of the transmitter source factor (upper) and mean square pressure at the receiver (lower). The sonar processing band is indicated by the solid red curve.

82 The sonar equations

[Ch. 3

As previously, we consider an idealized ocean of inﬁnitely deep isovelocity water. There are two possible ray paths from target to sonar. These two paths interfere, resulting for any single frequency in a sequence of fringes due to alternate constructive and destructive interference. The interference pattern can be written as 2kzarr ztgt e 2 ðf Þr Fcoh ðf ; r; zarr ; ztgt Þ ¼ 2 1 cos : ð3:95Þ r r2 This function is plotted vs. frequency and range in Figure 3.13 (upper graph) for a target at a depth of 150 m and a receiver at depth zarr ¼ 100 m. The integration in Equation (3.94) has the eﬀect of averaging over the interference fringes, so the integrand can be replaced by a smoothed version of itself (Figure 3.13, lower graph) without changing the result of the integration. Further, the smoothed integrand is approximately equal to the incoherent propagation factor (scaled by the source spectrum), provided that the frequency is high enough to neglect the systematic coherent cancellation (pressure node) at the sea surface , which becomes important below 1 kHz in this example. Therefore, above 1 kHz Equation (3.94) may be approximated by ð Finc ð f ÞW Sf df ð FBB ; ð3:96Þ W Sf df where Finc ð f ; r; zarr ; ztgt Þ ¼

2e 2 ðf Þr : r2

ð3:97Þ

Equation (3.96) can be further simpliﬁed by assuming that the source has a white power spectrum. In this situation, the propagation factor becomes ð 1 FBB ¼ Finc ð f Þ df ; ð3:98Þ Df where Df is the processing bandwidth. Using Equation (3.97) and assuming that varies linearly with frequency in the sonar passband ð f Þ ¼ m þ f ð f fm Þ;

ð3:99Þ

Equation (3.98) can be expressed in terms of the cardinal sinh function24 FBB ¼

3.2.4.2

2 expð2 m rÞ sinhcð f DfrÞ: r2

ð3:100Þ

Noise (single hydrophone)

As for the signal, the noise term is the total (noise) contribution to the MSP, in the 24

Denoted sinhc x; see Appendix A.

Sec. 3.2]

3.2 Passive sonar 83

Figure 3.13. Propagation loss [dB re m 2 ] vs. frequency and target range for array depth ¼ 100 m and target depth ¼ 150 m. Upper: coherent propagation loss PL ¼ 10 log10 Fcoh ; lower: smoothed version of the propagation loss from the upper graph (obtained by averaging in range) (INSIGHT).

84 The sonar equations

[Ch. 3

Figure 3.14. Spectral density level of signal 10 log10 Q Sf (red) and noise 10 log10 Q N f (cyan).

sonar-processing bandwidth Df

3.2.4.3

ð QN ¼ QN f df :

ð3:101Þ

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

It follows from the deﬁnition of Rhp (Equation 3.25) and of array gain (Equation 3.24) that QS Rarr ¼ N GA : ð3:102Þ Q Illustrative signal-and-noise spectra are shown in Figure 3.14. The signal term Q S is the area under the red curve, while the noise Q N is the area under the noise (cyan) curve in the same frequency band (between the vertical dashed lines). The array response25 is ð Y S ¼ Bð f ; OÞQ Sf O df dO ð3:103Þ for the signal, and

25

ð Y N ¼ Bð f ; OÞQ N f O df dO

For a hypothetical ﬂat-response ﬁlter in combination with a beamformer.

ð3:104Þ

Sec. 3.2]

3.2 Passive sonar 85

for the noise. The signal excess is, by deﬁnition, SEBB ðr; zarr ; ztgt Þ 10 log10

Rarr ðr; zarr ; ztgt Þ ¼ SL PL ðNL AGÞ DT; R50 ð3:105Þ

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

ð3:106Þ

ðpropagation loss: dB re m 2 Þ;

ð3:107Þ

ðnoise level: dB re mPa 2 Þ;

ð3:108Þ

AG ¼ 10 log10 GA

ðarray gain: dBÞ;

ð3:109Þ

DT ¼ 10 log10 R50

ðdetection threshold: dBÞ:

ð3:110Þ

SL ¼ 10 log10 S0 PLðr; zarr ; ztgt Þ ¼ 10 log10 FBB NLðzarr Þ ¼ 10 log10 Q N ðzarr Þ and

As for the NB case, the ﬁgure of merit is deﬁned as FOM PL50 : 3.2.4.4

ð3:111Þ

Array gain

From the deﬁnition of array gain (Equation 3.24) it follows that ð ð Y Sf df Q N f df ð GA ¼ ð : N S Y f df Q f df

ð3:112Þ

A useful approximation is obtained by assuming that the spectral densities of both the MSP (Qf ) and array response (Yf ) vary linearly with frequency within the sonar bandwidth. In this situation, GA may be approximated by a simple expression involving only the spectral densities evaluated at the center frequency fm . Speciﬁcally, GA Gm ;

ð3:113Þ

where Gm is the array gain at the center frequency Gm 3.2.4.5

Y Sf ðfm Þ Q N f ð fm Þ : S YN ð f Þ m Q f ð fm Þ f

ð3:114Þ

Probability of detection, detection threshold, and ROC curves

The probability of detection for incoherent processing, assuming Rayleigh statistics for both signal and noise amplitudes prior to incoherent processing (see Chapter 2) is " # pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 erfc 1 ð2pfa Þ M=2Rarr pd erfc ; ð3:115Þ 2 1 þ Rarr where M is the number of independent time samples added incoherently; and Rarr is the SNR after beamforming (more generally, at the point in the processing where a

86 The sonar equations

[Ch. 3

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Figure 3.15. ROC curves in the form 10 log10 ð M=2RT Þ vs. pfa for ﬁxed pT values, for a BB 1=2 1 signal in Rayleigh noise and M erfc ð2pT Þ.

‘‘target present’’ or ‘‘target absent’’ decision is made). Replacing pd with the chosen probability threshold (denoted pT ) and Rarr by its value at that threshold (i.e., Rarr ¼ RT ), and rearranging for RT gives rﬃﬃﬃﬃﬃ M erfc 1 ð2pfa Þ erfc 1 ð2pT Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ RT : 2 1 þ 2=M erfc 1 ð2pT Þ

ð3:116Þ

The right-hand side of Equation (3.116) simpliﬁes for large M to rﬃﬃﬃﬃﬃ M R erfc 1 ð2pfa Þ erfc 1 ð2pT Þ: 2 T

ð3:117Þ

This asymptotic value is converted to decibels and plotted vs. pfa in Figure 3.15. For the special case pT ¼ 0.5, Equation (3.116) simpliﬁes without further approximation to give rﬃﬃﬃﬃﬃ 2 R50 erfc 1 ð2pfa Þ; ð3:118Þ M resulting in the following useful approximation for the detection threshold DT50 10 log10 ½erfc 1 ð2pfa Þ 10 log10

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ M=2:

ð3:119Þ

Sec. 3.2]

3.2 Passive sonar 87

The number of independent samples M that can be obtained in time Dt is M¼

Dt ; tN

ð3:120Þ

tN ¼

1 ; 2Df

ð3:121Þ

where tN is the Nyquist interval

and therefore the value of M in Equation (3.119) is M ¼ 2Df Dt: 3.2.4.6

ð3:122Þ

Probability of false alarm

To complete the picture, a suitable false alarm probability is needed, which can be estimated from the desired false alarm rate nfa . It is usual to divide the total available bandwidth into smaller intervals of (say) an octave at a time. If the number of such intervals is Nintervals , the false alarm probability is pfa ¼ 3.2.4.7

nfa Dt : Nintervals Nbeams

ð3:123Þ

Special case: broadband target in the broadside beam of a horizontal line array

The purpose here is to simplify the BB sonar equation in order to express it in terms of familiar properties such as source power and array length as previously in the case of NB sonar. This is done for a similar situation as in Section 3.2.3.7, involving a horizontal line array. It has already been assumed (see Section 3.2.4.4) that the array response to signal and noise vary linearly with frequency so that ð Y Sf ð f Þ df S Y Sf ð fm Þ Y Rarr ¼ N ¼ ð N ; ð3:124Þ Y Y f ð fm Þ YN ð f Þ df f where it is understood that the signal and noise spectra are smoothly varying functions of frequency.26 It then follows that Rarr Gm

Q Sf ð fm Þ ; QN f ð fm Þ

ð3:125Þ

where Gm is given by Equation (3.114) and Q Sf ð fm Þ ¼ ðS0 Þf FBB ð fm Þ: 26

ð3:126Þ

If they are not smooth to start with, they can be locally averaged until they become so.

88 The sonar equations

[Ch. 3

Substituting Equation (3.126) in Equation (3.125) gives ðS0 Þf FBB ð fm Þ 1 Rarr Gm ; R50 R50 QN f ð fm Þ where ðS0 Þf ¼

c S W ðf Þ 4 f m

ð3:127Þ

ð3:128Þ

and, making the assumption as before that z 1, QN f ¼

3c N W Af : 2

ð3:129Þ

Converting Equation (3.127) to decibels provides a simpliﬁed form of the BB sonar equation: SEBB ¼ SLf PL ðNLf AGm Þ DT; ð3:130Þ where SLf ¼ 10 log10 ðS0 Þf ðsource spectrum level: dB re mPa 2 m 2 Hz 1 Þ ð3:131Þ ðpropagation loss: dB re m 2 Þ ð3:132Þ

PLðr; zarr ; ztgt Þ ¼ 10 log10 FBB

2 1 NLf ¼ 10 log10 Q N f ð fm Þ ðnoise spectrum level: dB re mPa Hz Þ ð3:133Þ

AGm ¼ 10 log10 Gm

ðcenter frequency array gain: dBÞ ð3:134Þ

and DT ¼ 10 log10

erfc 1 ð2pfa Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Df Dt

ðdetection threshold: dBÞ ð3:135Þ

For the case of a surface sheet (dipole) noise source we have (from Equation 3.63) Gm ¼

Dx ; 2 m

ð3:136Þ

c : fm

ð3:137Þ

where

m

Substituting Equation (3.136) for the array gain, the signal excess, deﬁned as ¼

Rarr R50

can be written in the following convenient form ! S Dx FBB W f ¼ ; R50 12 m W N Af

ð3:138Þ

ð3:139Þ

as used in the worked example below. 3.2.4.8

Worked example

Consider an omni-directional underwater communications transmitter at a depth of

Sec. 3.2]

3.2 Passive sonar 89

150 m. The carrier signal is a BB waveform with a white power spectrum. The signal is intercepted by a horizontal line array receiver of length 6 m, at depth 100 m from the surface, and oriented with the transmitter in its broadside beam. The interceptor uses incoherent processing in the frequency band 2 kHz to 4 kHz, with an integration time of 10 s. The power spectral density of the source in this frequency band is 100 nW/Hz. The wind speed is 5 m/s. It is useful to deﬁne the signal and noise spectrum levels, respectively, as L Sf SLf PL

ð3:140Þ

LN f BLf AG;

ð3:141Þ

and

so that the sonar equation becomes SE ¼ L Sf L N f DT:

ð3:142Þ

(i) Calculate the source spectrum level (SLf ) of the transmitter using Equation (3.131). Calculate the background noise spectrum level (NLf ) using Equation (3.133). (ii) Using Equations (3.134) and (3.135), calculate the center frequency array gain (AGm ) and detection threshold (DT) of the sonar. Assume that the direction and frequency band of the transmission are known, but not the waveform shape, and that a false alarm rate of one per hour is acceptable. (iii) Using the FOM method, or otherwise, calculate the maximum range at which the probability of detecting the communications transmission exceeds 50 % for a single (10 s) observation. (iv) The designers of the communications system wish to halve the intercept range. Assuming that all other parameters remain unchanged, by how much must the power be reduced? (v) How are the answers to parts (i) to (iv) affected if the transmitter depth is doubled to 300 m?

3.2.4.8.1 Part (i): source level SLf and noise spectrum level NLf First use Equation (3.128) to calculate the broadband source factor, giving ðS0 Þf ¼ 0.0122 Pa 2 m 2 Hz1 . The noise level is calculated from the sea surface–radiated power density, using Equation (3.129), with Equations (3.83) and (3.84) for the power spectral density. 2 The result for wind speed v10 ¼ 5 m/s is W N Hz 1 . The noise level Af ¼ 0:0918 pW m is then calculated using Equation (3.129), giving a noise spectral density of 2 1 QN f ¼ 0:209 mPa Hz . Conversions of the numerical values of ðS0 Þf and Q N f to micropascals and decibels are given in Table 3.2.

90 The sonar equations

[Ch. 3

Table 3.2. Sonar equation calculation for BB passive example. dB form

Description Symbol

Value

100.9 dB re mPa 2 m 2 Hz 1

Source spectrum level (Equation 3.131)

SLf

Noise spectrum level (Equation 3.133)

NLf

53.2 dB re mPa 2 Hz 1

Center frequency array gain (Equation 3.134)

AGm

12.8 dB re 1

Detection threshold (Equation 3.135)

DT

18.6 dB re 1

Linear form Expression

Numerical value in reference units

ðS0 Þf

1.22 10 þ10

3 N 2 cW Af

2.09 10 þ5

Dx 2 m

19.0

erfc 1 ð2pfa Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Df Dt

1.38 10 2

3.2.4.8.2 Part (ii): AGm and DT Equation (3.63) gives GA ¼ 19.0, and hence AGm ¼ 12.8 dB. The SNR threshold R50 is given by Equation (3.118) (see also Equation 3.119 for the detection threshold), which can be written in the form (using Equation 3.122 for M) R50

erfc 1 ð2pfa Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : Df Dt

ð3:143Þ

If the transmission direction is known, only one beam is of interest (Nbeams ¼ 1). If the frequency band is known, only one octave needs to be monitored (Nintervals ¼ 1). Thus, the desired maximum false alarm rate can be achieved with (Equation 3.123)27 pfa ¼ nfa Dt 0:0028:

ð3:144Þ

Putting Df ¼ 2 kHz and Dt ¼ 10 s in Equation (3.143) then gives R50 ¼ 0.0139. Answers to Parts (i) and (ii) are summarized in Table 3.2. 3.2.4.8.3 Part (iii): detection range Detection range depends on propagation loss (PL). It is usually not appropriate to use the same calculation method for NB and BB PL. For NB it was important to take into account the phase of various multipaths before adding these coherently. For BB, a legitimate approach is to repeat such a coherent calculation at every frequency and 27 Use of BB processing can tolerate a false alarm probability that is ﬁve orders of magnitude larger than for NB processing, without aﬀecting the false alarm rate. The diﬀerence is caused by the lower frequency resolution of the BB system, resulting in fewer detection opportunities.

Sec. 3.2]

3.2 Passive sonar 91

Figure 3.16. Propagation loss (blue) and ﬁgure of merit (cyan) vs. range. The probability of detection is 50 % when the propagation loss and ﬁgure of merit are equal, corresponding to the intersection between solid blue and cyan lines.

then average the result. However, this method is rarely used, because it is cumbersome and often unnecessary. Instead, it is more convenient to discard the phase at the beginning and add the contributions incoherently. This is justiﬁed if the averaging process would have destroyed the phase information anyway, although care is needed to retain those interference eﬀects that are persistent across the entire frequency band, and consequently remain even after averaging. (An example is the systematic cancellation or reinforcement that occurs close to a reﬂecting boundary, illustrated by Figure 3.13). If the target and receiver array are both deep enough for the boundary corrections described in Section 3.2.4.1 to be negligible, FBB may be approximated by Equation (3.100). A graph of PLðrÞ, calculated using this approximation, is plotted in Figure 3.16.28 Using Equations (3.111) and (3.113), the ﬁgure of merit is FOMBB ¼ SLf ðNLf AGm Þ DT;

ð3:145Þ

which for this example is 79.0 dB re m 2 . The corresponding detection range read from the graph is about 10 km. 28 The argument of the sinhc function in Equation (3.100), which is the change in r across the receiver bandwidth, is a small number (0.06 Np at 10 km with f Df ¼ 6:04 10 3 Np km 1 ). Consequently, the sinhcðxÞ function may be approximated by unity.

92 The sonar equations

[Ch. 3

Figure 3.17. Signal spectrum level (L Sf ) vs. range in blue (solid line), and in-beam noise spectrum level (L N f ) in red (dashed). The probability of detection is 50 % when the signal exceeds L N f by the detection threshold (DT ¼ 18.6 dB), corresponding to the intersection between solid blue and cyan lines.

An alternative method to calculate the detection range is to plot the signal spectrum level vs. range and compare the resulting curve with the in-beam noise level. A graph of L Sf ð¼ SLf PLÞ is shown in Figure 3.17 as a solid blue line. At the detection range (the intersection between L Sf and L N f þ DT, at 10.2 km) the signal 2 level is 21 dB re mPa 2 /Hz, 19 dB lower than the in-beam noise (L N f ) of 40 dB re mPa / Hz because, for this example, DT is negative. What this means is that a weak signal can be detected in the presence of a strong noise background (in this case the noise is nearly 100 times stronger than the signal). A negative detection threshold is a common characteristic of incoherent processing, the eﬀect of which is to reduce the detection threshold by removing ﬂuctuations, without altering the SNR.29 Detection probability is given by Equation (3.115) as (using Equations 3.118 and 3.122) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R50 Rarr 1 pd ¼ erfc Df Dt : 2 1 þ Rarr

ð3:146Þ

The result is plotted vs. range in Figure 3.18. 29

This contrasts with the purpose of coherent processing, which is to increase the SNR.

Sec. 3.2]

3.2 Passive sonar 93

Figure 3.18. Linear signal excess (Equation 3.138) and twice detection probability (Equation 3.146) vs. range for BB passive sonar (Gaussian statistics). The two curves cross at the detection range, where ¼ 2pd ¼ 1.

3.2.4.8.4

Part (iv): halving detection range

To answer Part (iv) it is convenient to rearrange the sonar equation in the form (see Equation 3.139) FBB ðrÞ ¼

12 m R50 W N Af ðrÞ; DxW Sf

ð3:147Þ

where is the signal excess in linear form, deﬁned by Equation (3.78). Using 50 to denote the ﬁgure of merit in linear form 50 10 FOM=10 ¼

1 Fðr50 Þ

ð3:148Þ

and substituting ¼ 1 in Equation (3.147), it follows that 50 ¼

DxW Sf : 12 m R50 W N Af

ð3:149Þ

In order for the intercept range to be halved (to r50 =2 ¼ 5:1 km), 50 must decrease by a factor of FBB ðr50 =2Þ=FBB ðr50 Þ ¼ 4:96, which means that W Sf must decrease by the same factor, to 20.1 nW/Hz.

94 The sonar equations

[Ch. 3

Figure 3.19. Propagation loss [dB re m 2 ] vs. range and depth for the BB passive worked example. Sonar depth zarr ¼ 100 m and frequency f ¼ 3 kHz (INSIGHT).

3.2.4.8.5

Part (v): doubling transmitter depth

For depths exceeding 20 m, PL is approximately independent of transmitter depth, as illustrated by Figure 3.19. None of the other terms vary with depth for this example, so the answers to Parts (i) to (iv) are unaﬀected by increasing the transmitter depth to 300 m. Notice the rapid increase in PL as the receiver approaches the surface for depths less than 20 m. In this region, a coherent calculation is necessary, even for BB processing, because the phase change at the sea surface results in a coherent cancellation occurring across the entire sonar bandwidth. This eﬀect does not aﬀect the worked example.

3.3 3.3.1

ACTIVE SONAR Overview

An active sonar system employs the same principle as radar: in order to detect the presence of an underwater object (the ‘‘target’’), it transmits a short burst (or pulse) of

Sec. 3.3]

3.3 Active sonar

95

Figure 3.20. Principles of active detection: the sonar transmitter (the dolphin) radiates a sound wave that travels through the sea and is scattered by the target (represented by a ﬁsh) after which it travels back to the sonar receiver.

energy (a sound wave in the case of sonar) and listens for an echo from that object, as illustrated by Figure 3.20. As with radar, the echo contains information about the position and nature of the target. Most obviously, the time delay between transmission and reception provides a direct measure of the target range, information that is not readily available from passive sonar data. Other practical advantages of active sonar (compared with passive sonar) include the following: — the transmitted waveform is known, leading to large signal-processing gains because it enables the receiving system to reject waveforms that do not resemble the transmitted one; — the signal level, and to some extent also the SNR, can be controlled by changing the source level and other properties of the transmitted pulse. The main performance disadvantages arising from active transmission are the loss of stealth (the transmitted pulse is much louder, and hence easier to detect than the target echo) and interference due to reverberation from boundaries and other underwater scatterers. In addition, there is a growing environmental concern about the impact on marine life (Southall et al., 2007) resulting from the use of loud underwater sound sources. Nevertheless, if there is no shortage of power and no need for stealth, then active sonar can oﬀer a performance advantage. Concepts and terminology speciﬁc to active sonar are introduced below. 3.3.2

Deﬁnition of standard terms (active sonar)

For active sonar (and also for transient waveforms generally) it is useful to express the sonar equation in terms of quantities proportional to total energy instead of power (or intensity). The reason for doing this is that energy is a conserved quantity, whereas power is not. If a signal of ﬁnite duration becomes stretched in time, its

96 The sonar equations

[Ch. 3

intensity is reduced, but the total energy, in the absence of any absorption, remains unchanged. In the following, the term ‘‘energy’’ is used asÐ a convenient shorthand for the time integral of the squared acoustic pressure (i.e., q 2 ðtÞ dt). 3.3.2.1

Signal energy, energy source level, and total path loss

Consider a sonar transmitter, comprising a single point source, whose transmitted waveform at a small distance s0 is q0 ðtÞ.30 The pulse propagates through the ocean to the target, from which it is reﬂected or scattered. The echo then propagates back31 through the same medium, returning to the sonar receiver position after a time delay equal to the two-way travel time, at which point the disturbance is denoted q S ðtÞ. These quantities are related through a time-integrated version of Equation (3.5)32 ð ð S 2 ½q ðtÞ dt ¼ ½q0 ðtÞ 2 s 20 dt F2 : ð3:150Þ Equation (3.150) serves to deﬁne F2 , the two-way energy propagation factor for a point source, as the ratio of energy (i.e., time-integrated squared pressure) at the receiver to that at unit distance from the transmitter. The time intervals are chosen to contain the whole of the transmitted pulse (in the case of transmitter) or target echo (for the receiver). It is assumed that the delay time is large compared with the pulse duration so that the transmitted pulse is easily separated from the target echo. Now deﬁne the received signal energy as ð ð3:151Þ Q SE ½q S ðx; tÞ 2 dt; and the energy source level as

SLE 10

log10 s 20

ð

2

q0 ðtÞ dt

dB re mPa 2 m 2 s:

ð3:152Þ

Further, deﬁning total path loss (TPL), in a manner analogous to propagation loss in the passive sonar equation, as TPL 10 log10 F2 30

dB re m 2 ;

ð3:153Þ

As in the case of passive sonar, this deﬁnition holds for a point source. The distance s0 must be small enough for distortions due to absorption, refraction, reﬂection, or diﬀraction to be Ð negligible. In this situation the ﬁeld is determined by spherical spreading. Thus, ½q0 ðtÞ 2 s 20 dt is a constant, independent of s0 . 31 If the receiver is in the same position as the transmitter then ‘‘back’’ just means back to the sonar. This situation is referred to as a ‘‘monostatic’’ geometry. In some sonars the transmitter and receiver are physically separate, in which case the geometry is ‘‘bistatic’’. For a bistatic geometry, the return path is always diﬀerent from the outward path. For a monostatic sonar the return path can either be the same or diﬀerent. 32 Recall that the real acoustic pressure is denoted ‘‘qðtÞ’’, while ‘‘pðtÞ’’ is reserved for the complex pressure ﬁeld.

Sec. 3.3]

3.3 Active sonar

97

it follows that 10 log10 Q SE ¼ SLE TPL:

ð3:154Þ

The left-hand side of Equation (3.154) is referred to henceforth as the signal energy level. If the transmitted power is constant over the pulse duration t, then Equation (3.152) simpliﬁes to SLE ¼ SL þ 10 log10 t

dB re mPa 2 m 2 s;

ð3:155Þ

where SL is the source level as deﬁned previously in terms of mean square pressure (Equation 3.42). More information about the source levels of man-made and biological sonar can be found in Chapter 10. 3.3.2.2

Background energy and background energy level

At any instant the total background (acoustic) pressure q B ðtÞ can be written as a sum of separate contributions from noise and reverberation33 q B ðtÞ ¼ q N ðtÞ þ q R ðtÞ:

ð3:156Þ

ðq B Þ 2 ¼ ðq N Þ 2 þ ðq R Þ 2 þ 2q N q R ;

ð3:157Þ

It follows that where the overbar indicates an average in time. Assuming that noise and reverberation are uncorrelated, the cross-term may be neglected, so that for a (transmission to echo) delay time , the total background becomes Q B ðÞ ¼ Q N þ Q R ðÞ:

ð3:158Þ

Henceforth the notation Q X is used to indicate the average value of the parameter ðq X Þ 2 over the duration of the received echo (assumed suﬃciently short that the background can be thought of as stationary in that time). The sonar is assumed to have a natural processing bandwidth, and the relevant part of the background (for both noise and reverberation) is that within this frequency band. For reverberation arriving after a given delay time , the longer the delay the further the sound has traveled and, in general, the more boundary reﬂections it has experienced. The sound becomes weaker at each reﬂection, so the reverberation Q R tends to decrease with increasing . For as long as the reverberation remains above the noise, the total background is therefore also a function of delay time. As with the signal, the background energy Q BE is deﬁned as the time integral of the corresponding squared pressure term: ð B Q E ½q B ðtÞ 2 dt; ð3:159Þ 33 Reverberation is the sound associated with scattering of the sonar pulse by any object other than the intended target. It can be deﬁned as sound pressure in the absence of ambient noise, self-noise, and the target.

98 The sonar equations

[Ch. 3

or in decibels, the background energy level is BLE 10 log10 Q BE 3.3.2.3

dB re mPa 2 s:

ð3:160Þ

Signal-to-background ratio and array gain

The array response to the signal energy can be written ð Y SE ¼ t Q SO BðOÞ dO;

ð3:161Þ

where the O subscript indicates a derivative with respect to solid angle. Similarly, the array response to the background spectral density is ð B Y f ðÞ ¼ Q Bf O ð; OÞBðOÞ dO; ð3:162Þ where the f subscript indicates a derivative with respect to frequency. A distinction is made between the signal-to-noise ratio (SNR) and signal-tobackground ratio (SBR), whereby SBR is the ratio of the signal power to the total background power (sum of noise plus reverberation). The SBR at the hydrophone is given by QS Rhp EB ; ð3:163Þ QE and after beamforming by Y SE Rarr ð : ð3:164Þ ðY BE Þf df The receiving array gain is then deﬁned in a similar way as for passive sonar. Speciﬁcally R GA arr ; ð3:165Þ Rhp such that AG 10 log10 GA ¼ 10 log10 ð

Y SE ðY BE Þf df

10 log10 ð

Q SE

dB:

ð3:166Þ

ðQ BE Þf df

Making the assumption (as in Section 3.2.2.5) of an incoming plane wave signal, arriving in the main beam of the receiving array, the signal parameters Y S and Q S are equal, and therefore ð ðQ BE Þf df AG ¼ 10 log10 ð : ð3:167Þ ðY BE Þf df

Sec. 3.3]

3.3.2.4

3.3 Active sonar

99

Target strength

Consider a plane wave incident on the target. Target strength (TS) is a measure of how much of the incident sound is scattered back in the direction of the sonar. Let tgt q in ðtÞ denote the instantaneous pressure due to the incident wave at the target, and q tgt 0 ðtÞ that of the scattered ﬁeld at a small distance s0 away from it. A deﬁnition that is valid for a point target is TSMSP 10 log10

2 s 20 ½q tgt 0 ðtÞ 2 ½q tgt in ðtÞ

dB re m 2 :

ð3:168Þ

The denominator and numerator can be written, respectively, in terms of the incident intensity Iin and scattered radiant intensity WO 2 ðq tgt in Þ ¼ cIin

ð3:169Þ

and 2 s 20 ðq tgt 0 Þ ¼ cWO :

ð3:170Þ

It follows from the deﬁnition of the backscattering cross section that34 back TSMSP ¼ 10 log10 dB re m 2 ; 4 where back ðÞ 4O ð; ; Þ and W ð ; Þ O ðin ; out ; Þ ¼ O out : Iðin Þ

back

(see Chapter 2) ð3:171Þ ð3:172Þ ð3:173Þ

A slight modiﬁcation is needed here to make the TS deﬁnition compatible with the other parameters introduced above, all of which are deﬁned in terms of energy. Consistency can be achieved by multiplying both the numerator and denominator of Equation (3.168) by the pulse duration t and replacing the resulting product with an integral: ð 2 s 20 ½q tgt 0 ðtÞ dt TSE 10 log10 ð 2 ½q tgt in ðtÞ dt

3.3.3

dB re m 2 :

ð3:174Þ

Coherent processing: CW pulse þ Doppler ﬁlter

A Doppler ﬁlter works by examining the spectrum of the received signal in small frequency bins. If the target is moving relative to the sonar, the echo is shifted in frequency relative to the transmitted pulse due to the Doppler eﬀect and might arrive in a diﬀerent Doppler bin to the reverberation. It is assumed below that a single 34 In some publications the factor 4 in the denominator of Equation (3.171) is included in the deﬁnition of back (see Chapter 5).

100 The sonar equations

[Ch. 3

frequency bin (or ‘‘Doppler bin’’) is found that contains all of the echo energy, but only a small fraction of the reverberation. For the purpose of Section 3.3.3, this fraction of the reverberation is assumed to be negligible compared with the ambient noise background. 3.3.3.1

Signal (single hydrophone)

The signal energy is the integral of the squared pressure over the duration of the pulse t (Equation 3.150). Assuming that the signal MSP does not change during this time interval, this integral may be replaced by the product Q SE ¼ Q S t;

ð3:175Þ

where, in terms of the two-way propagation factor F2 (see Equation 3.150), Q S ¼ S0 F2

ð3:176Þ

and S0 is the source factor given by Equation (3.10). 3.3.3.2

Background (single hydrophone)

For simplicity, Doppler processing is assumed to eliminate all reverberation, so the background consists of noise only. The relevant noise is that in the processing bandwidth (i.e., the width of each Doppler bin f ) Q BE ¼ ðQ N f f Þ t:

ð3:177Þ

Therefore (assuming for a CW pulse that f t 1) Q BE Q N f

ð3:178Þ

QN f ¼ 3cWAf E3 ð2 zÞ;

ð3:179Þ

and where E3 ðxÞ is an exponential integral of the third kind (Appendix A). 3.3.3.3

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

The signal-to-background ratio (SBR), after beamforming, is Rarr ¼ Rhp GA ;

ð3:180Þ

where Rhp ¼

Q SE : Q BE

ð3:181Þ

Using Equations (3.175) and (3.178) and dividing by R50 gives the sonar equation (in decibels) SE 10 log10

Rarr ¼ SLE TPL ðNLf AGÞ DT: R50

ð3:182Þ

Sec. 3.3]

3.3 Active sonar

101

From the deﬁnitions of TSE and TPL it follows (for a point-like target) that PL and TPL are related in the following way where PLTx

TPL ¼ PLTx þ PLRx TSE ; ð tgt 10 log10 FTx ¼ SLE 10 log10 ½q in ðtÞ 2 dt

and PLRx 10 log10 FRx

ð3:183Þ ð3:184Þ

ð tgt 2 2 ¼ 10 log10 s 0 ½q 0 ðtÞ dt 10 log10 Q SE ;

ð3:185Þ

where FRx is the one-way propagation factor from target to receiver (Rx), deﬁned such that ð S 2 2 Q t ¼ s 0 ½q tgt ð3:186Þ 0 ðtÞ dt FRx : The meaning of the ‘‘in’’ and ‘‘0’’ subscripts above is the same as for the deﬁnition of target strength, Equation (3.174). In other words, ‘‘in’’ indicates the acoustic ﬁeld incident on the target and ‘‘0’’ the scattered ﬁeld at a small distance s0 from the target. Similarly, FTx is the propagation factor from transmitter (Tx) to target ð 2 ð3:187Þ ½q tgt in ðtÞ dt ¼ S0 t FTx : Thus the NB active sonar equation, for a point target with coherent processing (and neglecting reverberation) is SE ¼ ½SLE ðPLTx þ PLRx TSE Þ ðNLf AGÞ DT;

ð3:188Þ

where SLE 10 log10 ðS0 tÞ

ðenergy source level: dB re mPa 2 s m 2 Þ;

ð3:189Þ

2

ð3:190Þ

2

ðpropagation loss target to Rx: dB re m Þ;

ð3:191Þ

ðtarget strength: dB re m 2 Þ;

ð3:192Þ

NLf ¼ 10 log10 Q N f

ðnoise spectrum level: dB re mPa 2 Hz 1 Þ;

ð3:193Þ

AG ¼ 10 log10 GA

ðarray gain: dBÞ;

ð3:194Þ

DT ¼ 10 log10 R50

ðdetection threshold: dBÞ:

ð3:195Þ

PLTx ¼ 10 log10 FTx PLRx ¼ 10 log10 FRx TSE ¼ 10 log10 back 4

ðpropagation loss Tx to target: dB re m Þ;

and For noise-limited active sonar, the ﬁgure of merit can be deﬁned as FOM 12 ðPLRx þ PLTx Þ50 ¼ 12 ðSLE þ TSE NLf þ AG DTÞ:

ð3:196Þ

The active sonar equation can then be written SE ¼ 2 FOM ðPLRx þ PLTx Þ:

ð3:197Þ

102 The sonar equations

3.3.3.4

[Ch. 3

Array gain

For a plane wave signal, the array gain is GA ¼ ð

QN

;

ð3:198Þ

QN O BðOÞ dO

where (for a continuous line array) BðOÞ ¼

sin 2 u u2

ð3:199Þ

and u¼

k Dx cos sin : 2

ð3:200Þ

The operating frequency of an active sonar is typically higher than for a passive one, making it necessary to consider the eﬀect of absorption on surface-generated noise, which then becomes 3 2 zarr N N QO ¼ cW A sin exp : ð3:201Þ 2 sin Near-horizontal contributions are penalized by the attenuation due to their long travel paths before reaching the array. Therefore, as z increases, the main noise contributions come increasingly from vertically above the receiving array. For a horizontal line array, the broadside beam is less eﬀective at ﬁltering out noise from directly above, so the resulting array gain is smaller than for the zero attenuation case. Speciﬁcally, the gain GA of a horizontal line array in the presence of absorption is (the ‘‘arr’’ subscript is implied) ð 2 ð =2 2 z d sin cos exp d sin 0 0 GA ¼ ð =2 ð3:202Þ ð 2 : 2 z sin uð; Þ 2 d sin cos exp d sin 0 uð; Þ 0 This equation is less daunting than it looks because the integrations over are unaﬀected by the introduction of attenuation, and can be evaluated in the same way as in Section 3.2.3.4, leaving ð =2 2 z 2 sin cos exp d sin 0 GA ¼ : ð3:203Þ ð 4 =2 2 z sin exp d kDx 0 sin The eﬀect of the exponential factors can be approximated by removing these and replacing the lower limit of both integrals of Equation (3.203) with an angle ", whose

Sec. 3.3]

3.3 Active sonar

value increases from 0 to =2 with increasing attenuation ð =2 2 sin cos d " GA ¼ : ð 4 =2 sin d kDx "

103

ð3:204Þ

The result is to multiply the numerator by cos 2 " and the denominator by cos ", leaving a net factor of cos " overall. Thus GA G0 cos "; ð3:205Þ 4 with G0 from Equation (3.66). The requirement for a good choice of " is that the equation ð =2 ð =2 2 z f ðÞ exp d ¼ f ðÞ d sin 0 "

ð3:206Þ

is satisﬁed for some appropriate choice of f ðÞ. A desirable choice, with Equation (3.203) in mind, would be one of either f ðÞ ¼ sin cos or f ðÞ ¼ sin . However, if f ðÞ varies slowly compared with the exponential term, its precise form does not matter much. In fact, it is convenient to choose a third form f ðÞ ¼

cos ; sin 2

ð3:207Þ

as this facilitates evaluation of the integrals. With this choice, Equation (3.206) can be solved to give 2 z " ¼ arcsin : ð3:208Þ 2 z þ expð2 zÞ This result can be used with Equation (3.205) to estimate the eﬀect of absorption on array gain.35 3.3.3.5

Probability of detection, detection threshold, and ROC curves

As for the case of NB passive sonar, Rayleigh statistics are assumed here for both noise and signal amplitudes. Thus, the corresponding ROC relationships are identical to those of Section 3.2.3.5 and are not repeated here. 3.3.3.6

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 35 There is a problem with the application of Equation (3.208) for large z. As " approaches =2 (meaning that the attenuation is so great that all sound is coming from directly overhead), the derivation fails. In this situation a lower limit of GA ¼ 1 is needed.

104 The sonar equations

[Ch. 3

eack look Dt (the time between successive pulses) ðnfa Þtrue ¼

Ntot pfa : Dt

ð3:209Þ

The number Ntot depends on the number of beams (Nbeams ), Doppler cells (NDoppler ), and range cells (Nranges ):36 Ntot ¼ Nbeams NDoppler Nranges ;

ð3:210Þ

where Nranges ¼

Dt : t

ð3:211Þ

The true pfa (and therefore also nfa ) depends on the threshold R50 . Assuming this threshold to have been chosen in advance to achieve an acceptable false alarm rate, one can estimate the false alarm probability by rearranging Equation (3.73) and removing the ‘‘true’’ qualiﬁer (i.e., assume that the acceptable rate is achieved, so that ‘‘true’’ and ‘‘acceptable’’ rates are equal), n nfa pfa fa Dt ¼ ; ð3:212Þ Ntot Nbeams Df where Df is the total system bandwidth Df ¼ NDoppler f : 3.3.3.7

ð3:213Þ

Special case: rigid spherical target in the broadside beam of a horizontal line array, CW pulse with Doppler processing

It is assumed that for a CW pulse with Doppler processing, reverberation may be neglected. In this situation, the signal-to-noise ratio (SNR) is Rarr ¼

Q SE Q S ðÞ G ¼ GA ; A QN QN E

ð3:214Þ

QN QN f f :

ð3:215Þ

where The signal MSP term is given by Equation (3.176), so that Rarr ¼ GA

S0 F2 : QN f f

ð3:216Þ

The two-way propagation factor may be written, assuming a monostatic geometry and single propagation path (with spherical spreading plus absorption) to the target37 36

The frequency of the received signal diﬀers from that of the transmitted one by an amount proportional to the radial component of the target velocity relative to that of the sonar platform. As this velocity is unknown, all reasonable possibilities need to be considered; hence the NDoppler factor. 37 The array and target are assumed here to be well separated from the sea surface, so that the direct path is clearly separated from later multipaths. This direct path contribution is treated as the signal.

Sec. 3.3]

3.3 Active sonar

105

and back to the sonar, F2 ¼ F 2

back ; 4

ð3:217Þ

where (neglecting surface reﬂections for simplicity)38 F¼

e 2 s : s2

ð3:218Þ

For a rigid sphere whose radius a is large compared with the acoustic wavelength, back is (see Chapter 2) back ðaÞ ¼ a 2 ; ð3:219Þ and therefore, using Equation (3.192) to give the target strength, TSE ¼ 10 log10

a2 4

dB re m 2 :

ð3:220Þ

The high-frequency target strength of a rigid sphere whose radius is 2 m is therefore 0 dB re m 2 . 3.3.3.8

Worked example

Consider a spherical steel target in deep water at a depth of 150 m from the sea surface. A CW sonar operating at 50 kHz, with a bandwidth of 500 Hz and source level of 200 dB re mPa 2 m 2 , is placed at a horizontal distance of 1.3 km from the sphere. The sonar receiver, a horizontal line array of length 2 m, is placed at a depth of 100 m. The target is in the broadside beam of the receiving array. It is convenient to introduce the signal energy level: L SE ¼ SLE ðPLTx þ PLRx TSE Þ

ð3:221Þ

and the in-beam noise spectrum level: LN f ¼ NLf AG:

ð3:222Þ

With these deﬁnitions, the sonar equation becomes SE ¼ L SE L N f DT:

ð3:223Þ

(i) If the wind speed is 15 m/s, what is the smallest sphere that could be detected by the sonar using a pulse of duration 0.2 s? (ii) For the sphere of Part (i), calculate the sonar ﬁgure of merit using Equation (3.196). (iii) Calculate pd ðrÞ for the same sphere. Calculate pd ðzarr Þ at a range of 1.3 km. 38

In reality there is always some sound reﬂected from the sea surface. However, at high frequency it is often heavily attenuated by near-surface bubbles and scattered due to sea surface roughness. If the pulse is suﬃciently short, the surface reﬂection can also be separated in time from the direct arrival, such that the two paths may be treated independently.

106 The sonar equations

[Ch. 3

(iv) From the answer to Part (iii), or otherwise, estimate the array depth at which this sonar is most likely to detect the sphere at a distance of 1.3 km. For the purpose of calculating the probability of false alarm, assume an acceptable false alarm rate of one per hour, and that 128 beams are formed. Assume that steel acts as a perfect reﬂector of sound so that Equation (3.220) may be used for the TS, and that the sphere is moving fast enough to justify neglecting the reverberation. For simplicity, assume further that any surface-reﬂected sound does not contribute to the signal (see Section 3.3.3.7).39 3.3.3.8.1

Part (i): smallest detectable sphere

The lowest value of the backscattering cross-section that results in a detection can be found by substituting Equation (3.217) in Equation (3.216) and then rearranging for back to give back ¼

N 4 Q f fRarr : GA S 0 F 2

ð3:224Þ

Using Equation (3.219) (assuming f t ¼ 1) and requiring Rarr > 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 <4 mm divisions of these sediment types are Silt 4 mm to 62.5 mm introduced in the discussion of mixed sizes below. For many applications, an Sand 62.5 mm to 2 mm unconsolidated sediment with a suﬃciently low shear speed may be Gravel >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<M<8

Silt

Siltstone

M>8

Clay

Claystone or shale

Sec. 4.4]

4.4 Properties of the seabed 181

support shear waves. The usual sound speed (i.e., the propagation speed of the longitudinal wave, denoted cp ) is known as compressional speed. Figure 4.21 shows measurements of cp and cs vs. density for all rocks, from Ludwig et al. (1970). For rocks of relatively low density (sedimentary rocks and

Figure 4.21. Compressional and shear speed (in km/s) vs. density of rocks (in g/cm 3 ) (reprinted from Ludwig et al., 1970, # Wiley).

182 Sonar oceanography

[Ch. 4

some igneous rocks, of density rock in the range 1900 kg/m 3 to 2400 kg/m 3 ), the variation of both wave speeds with density is approximately quadratic: 2 9 cp rock rock > > 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

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Stanton, T. K., Nash, R. D. M., Eastwood R. L., and Nero, R. W. (1987) A ﬁeld examination of acoustical scattering from marine organisms at 70 kHz, IEEE J. Oceanic Eng., OE-12, 339–348. Stephens, R. W. B. and Bate, A. E. (1966) Acoustics & Vibrational Physics, Edward Arnold, London. Sternlicht, D. D. and de Moustier, C. P. (2003) Time-dependent seaﬂoor acoustic backscatter (10–100 kHz), J. Acoust. Soc. Am., 114, 2709–2725. Su¨ndermann, J. (1986) Landolt–Bo¨rnstein Numerical Data and Functional Relationships in Science and Technology (New Series, editors-in-chief K.-H. Hellwege and O. Madelung, Group V: Geophysics and Space Research, Vol. 3a, Oceanography), Springer-Verlag, Berlin. Tang, D. (2004) Fine-scale measurements of sediment roughness and subbottom variability, IEEE J. Oceanic Eng., 29, 929–939. Tang, D., Briggs, K. B., Williams, K. L., Jackson, D. R., Thorsos E. I., and Percival, D. B. (2002) Fine-scale volume heterogeneity measurements in sand, IEEE J. Oceanic Eng., 27, 546–560. Taylor, B. N. (1995) Guide for the Use of the International System of Units (SI) (NIST Special Publication 811, 1995 Edition), U.S. Department of Commerce, National Institute of Standards and Technology, U.S. Government Printing Oﬁce, Washington, D.C. Ternon, J. F., Oudot, C., Gourlaouen, V., and Diverres, D. (2001) The determination of pHT in the equatorial Atlantic Ocean and its role in the sound absorption modeling in seawater, J. Marine Systems, 30, 67–87. Thorpe, S. (1982) On the clouds of bubbles formed by breaking wind-waves in deep water, and their role in air–sea gas transfer, Phil. Trans. Roy. Soc. A, 304, 155–210. Trevorrow, M. V. (2003) Measurements of near-surface bubble plumes in the open ocean with implications for high-frequency sonar performance, J. Acoust. Soc. Am., 114, 2672–2684. Udden, J. A. (1914) Mechanical composition of clastic sediments, Bull. Geol. Soc. Am., 25, 655–744. van Moll, C. A. M., Ainslie, M. A., and van Vossen, R. (2009) A simple and accurate formula for the absorption of sound in seawater, IEEE J. Oceanic Eng., 34(4), 610–616. Wedborg, M., Turner, D. R., Anderson, L. G., and Dryssen, D. (1999) Determination of pH, in Methods of Seawater Analysis (edited by K. Grasshoﬀ, K. Kremling, and M. Ehrhardt), Wiley, New York. Weisstein (www) E. W. Weisstein, Ellipsoid, http://mathworld.wolfram.com/Ellipsoid.html, last accessed May 19, 2006. Wentworth, C. K. (1922) A scale of grade and class terms for clastic sediments, J. Geology, 30, 377–392. Weston, D. E. (1967) Sound propagation in the presence of bladder ﬁsh, in V. M. Albers (Ed.), Underwater Acoustics, Vol II: Proceedings 1966 NATO Advanced Study Institute, Copenhagen (pp 55–88), Plenum Press, New York. Weston, D. E. (1995) Assessment Methods for Biological Scattering and Attenuation in Ocean Acoustics (BAeSEMA Report C3305/7/TR-1, April), BAeSEMA, Esher, U.K. WMO (1970) The Beaufort Scale of Wind Force: Technical and Operational Aspects (reports on marine science aﬀairs, Report No. 3, submitted by the President of the Commission for Maritime Meteorology to the WMO Executive Committee at its 22nd Session), Secretariat of the World Meteorological Organization, Geneva. WOA (1999) World Ocean Atlas 1998 (WOA98) (CD-ROM documentation version 1.0), Ocean Climate Laboratory, National Oceanographic Data Center (NODC), Silver Springs, MD.

190 Sonar oceanography

[Ch. 4

Worthington, L. V. (1981) The water masses of the world ocean: Some results of a ﬁne-scale census, in B. A. Warren and C. Wunsch (Eds.), Evolution of Physical Oceanography (pp. 42–69), MIT Press, Cambridge, MA. Yang, J. (1982) An estimate of the ﬁsh biomass in the North Sea, J. Cons. Int. Explor. Mer, 40, 161–172.

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

<

Rigid object of arbitrary size

A simple approximate expression for an object of arbitrary size can be obtained by combining the low-frequency result from Section 5.4.2.1 with the high-frequency one from Section 5.4.2.2, using 1 1 1 back þ back : ð5:126Þ back ð f Þ LF HF

214 Underwater acoustics

[Ch. 5

As an example, for a rigid object of arbitrary convex shape of volume V and radii of curvature A and B, this is " !# 1 2 2 1 back þ : ð5:127Þ AB 5V In particular, for a sphere of radius a: a 2

back 1þ

5.4.2.4

3 2 20 2 a 2

! : 2

ð5:128Þ

Sand grains of irregular shape and arbitrary size

Thorne and Meral (2008) propose the following simple semi-empirical expression for the form factor of irregular sand particles in water ðkaÞ 2 ka 1:5 2 ka 1:8 2 j f ðkaÞj 1 0:35 exp 1þ 0:5 exp : 0:7 2:2 1 þ 0:9ðkaÞ 2 ð5:129Þ 5.4.3 5.4.3.1

Backscattering from ﬂuid objects Small ﬂuid object of arbitrary shape

For a small ﬂuid object of volume V, density , and bulk modulus B, and immersed in water, the diﬀerential scattering cross-section is ! 2 2V 2 w c 2w 3ð=w 1Þ O ðÞ ¼ 1 cos ; ð5:130Þ B 1 þ 2=w 4 where is the acoustic wavelength in water. Substituting ¼ gives the monostatic result for the backscattering cross-section (BSX) ! 2 2 2 V c 3ð= 1Þ w w w 3 back 1 þ : ð5:131Þ LF ¼ 4 B 1 þ 2=w 4 This expression reduces trivially to Equation (5.121) in the limit w = ! 0. Another important special case is that of a gas-like object (i.e., one with negligible density and high compressibility): ! 2 w c 2w 2 back 3V LF ¼ 4 4 : ð5:132Þ B

5.4.3.2

Large ﬂuid object

In general, some of the energy incident on any real object is reﬂected and some is transmitted into the interior of the object. Assuming that the transmitted part does

Sec. 5.4]

5.4 Scattering of plane waves

215

Table 5.3. Backscattering cross-sections of large ﬂuid objects. back HF 4

Shape Arbitrary convex object with principal radii of curvature A1 and A2

jRj 2

A1 A2 4

Cylinder of radius a and length L, aL 2 sinðkL sin Þ 2 2 ensoniﬁed at an angle to the normal jRj 2 cos 2 kL sin to the cylinder axis Arbitrary convex object with surface area S, ensoniﬁed at random aspect (ensemble average cross-section)

jRj 2

S 16

Conditions

kA1 1; kA2 1

ka 1 kl 1, where l is the smallest radius of curvature

not contribute to the scattered ﬁeld, the BSX may be approximated by multiplying the rigid body result by jRj 2 , where 1 R¼ ; ð5:133Þ þ1 and c ¼ : ð5:134Þ w c w The applicable results are summarized in Table 5.3. 5.4.3.3

Fluid object of arbitrary size

A simple approximation for BSX, of the same form as Equation (5.126), is 1 back ð f Þ

1 back LF

þ

1 back HF :

ð5:135Þ

This expression is not applicable to cases featuring a strong resonance such as scattering from a gas bubble, which is considered next. Approximate results for a ﬂuid sphere, prolate spheroid, straight cylinder, and bent cylinder are given by Stanton (1989). 5.4.3.4

Gas bubble

A gas bubble in water, once disturbed from its rest state by a compression or rarefaction, will pulsate at its natural frequency (Morfey, 2001), provided that it is not subject to further forcing. If forced to pulsate at this frequency (by an incoming acoustic wave) the bubble resonates, meaning that it undergoes high-amplitude oscillations or pulsations in response to the force.12 12 At resonance, the bubble’s shape is likely to depart from spherical symmetry, but the radiated ﬁeld is nevertheless well approximated by considering only the changes in volume (Leighton, 1994, p. 203).

216 Underwater acoustics

[Ch. 5

The frequency of maximum response is known as the resonance frequency of the bubble and denoted !res . Whether or not a bubble is resonating, its motion is damped, meaning that some of the vibrational energy of the bubble is lost to its surroundings. If the amount of damping is small, the resonance frequency is approximately equal to natural frequency. The concept of resonance is essential for understanding the response of a gas bubble to an acoustic signal. The approach used here to analyze bubble response is a low-frequency one in the sense that kw a 1 is required, where kw is the acoustic wavenumber in water. This is not a big constraint because most bubbles are small compared with the acoustic wavelengths of interest. For a single resonating air bubble close to atmospheric pressure the kw a product is always small—a consequence of the high compressibility of air compared with that of water. For a spherical bubble of radius a, the BSX (for kw a 1) can be approximated in the form (Weston, 1967)13 back

4a 2 : ½1 !res ðaÞ =! 2 2 þ ða; !Þ 2 2

ð5:136Þ

Resonance occurs when the term in square brackets vanishes and the BSX becomes very large, limited only by the damping coeﬃcient . Full expressions for the resonance frequency and damping coeﬃcient are rather involved and thus deferred to the treatment of bubble resonance in Section 5.5.3.4. Here, simpliﬁed expressions are presented that are valid for moderately large bubbles (radius exceeding about 100 mm). For such bubbles, resonance frequency is equal to the Minnaert frequency (Minnaert, 1933): 3 a Pw 1=2 !0 ðaÞ ¼ : ð5:137Þ w a 2 The product of wavenumber and bubble radius at resonance, as determined by the Minnaert frequency at atmospheric pressure, is a constant that appears repeatedly in the discussion concerning bubble acoustics. This constant is denoted D0 and deﬁned by the formula 3 a PSTP 1=2 D0 0:01367: ð5:138Þ w c 2w If the value of in Equation (5.136) is small, a more precise condition for resonance is that ! and !res be equal. Thus, at resonance, Equation (5.136) becomes 2 2 back res ¼ 4a Q ;

ð5:139Þ

where Q is the Q-factor, deﬁned as 1=ð!res Þ. In practice, the damping coeﬃcient may often be replaced in Equation (5.136) by 1=Q even away from resonance because 2 is 13 Ainslie and Leighton (2009) point out that Equation (5.136) is ambiguous without explicit deﬁnitions of !res and . An alternative expression without this ambiguity is given in Section 5.5.3.6.

Sec. 5.4]

5.4 Scattering of plane waves

217

then small compared with other terms in the denominator of Equation (5.136). In other words back

4a 2 : ½1 !res ðaÞ 2 =! 2 2 þ 1=Q 2

ð5:140Þ

The simplest form of damping is caused by re-radiation: as the bubble pulsates it acts as a source of sound in its own right, the radiated sound carrying energy away from the bubble. Damping can also be caused by absorption (i.e., conversion of sound energy to heat or, in some special situations, to light, see Leighton, 1994).14 Close to resonance, for a gas bubble in water, the dominant form of absorption is due to the transport of heat inside the bubble. Absorption due to the viscosity of water provides additional damping for small bubbles. The total reciprocal Q-factor can be written as the sum of three separate contributions due to re-radiation, thermal conduction, and viscous losses: 1 1 1 1 ¼ þ þ ; Q Qrad Qtherm Qvisc

ð5:141Þ

where radiation and thermal Q-factors are given by 1 ! a ¼ 0 Qrad cw and 1 Qtherm

3ð a 1Þ Da 1=2 ¼ : a 2!0

ð5:142Þ

ð5:143Þ

Parameters Da and a are the thermal diﬀusivity and speciﬁc heat ratio of air, respectively. The third Q-factor, for viscous losses, is given by Leighton (1994, p. 190) for the case of negligible bulk viscosity via 1 4S ¼ ; Qvisc w !0 a 2

ð5:144Þ

where S is the shear viscosity of seawater. With the introduction of bulk viscosity, which in seawater contributes at least as much to the damping as shear viscosity (see Chapter 4), this expression for Qvisc becomes (Love, 1978, Eq. (9)) 1 3 ¼ ; Qvisc w !0 a 2

ð5:145Þ

where is a viscosity parameter, deﬁned as 13 S þ B :

ð5:146Þ

14 A ﬂash of light is sometimes observed associated with the sound creation mechanism of the snapping shrimp (see Chapter 8).

218 Underwater acoustics

[Ch. 5

The low-frequency and high-frequency limits of Equation (5.136) (but always within the frequency regime satisfying ka 1) are: 2 4 back LF ¼ 4a ð!=!0 Þ

ð5:147Þ

2 back HF ¼ 4a :

ð5:148Þ

and

5.4.3.5

Dispersed bubbles

Consider a cloud of bubbles of diﬀerent sizes, spread uniformly in space. Let nðaÞ da be the number of bubbles per unit volume whose radius is between a and a þ da, such that the backscattering cross-section per unit volume of the bubble cloud (the volumic15 BSX) ð1 d back back ¼ back ðaÞnðaÞ da; ð5:149Þ V dV 0 where back is given by Equation (5.136). For bubbles large enough to pulsate adiabatically, but still small in the sense of ka 1, this simpliﬁes to ð1 a2 back 4 nðaÞ da; ð5:150Þ V 2 2 2 0 f1 ½!0 ðaÞ=! g þ where !0 is given by Equation (5.137) and may be replaced by 1=Q from Equation (5.141). Corrections to !0 and for smaller bubbles are described in Section 5.5.3.4. A simplifying approximation that is sometimes made to Equation (5.150) is to assume that the contribution from near-resonant bubbles dominates the integral. The result is (Medwin and Clay, 1998) back 2 2 Qa 30 nða0 Þ; V

ð5:151Þ

where a 20 ¼

3 a Pw : w ! 2

ð5:152Þ

This approximation, though appealing, is not valid for situations involving a wide spectrum of bubble sizes. Nevertheless, it can still be useful to calculate the contribution to BSX from resonant bubbles in this way. 5.4.3.6

Single ﬁsh (with bladder)

Horne and Clay (1998) review methods for computing the BSX from ﬁsh and other marine organisms, including a sophisticated model of scattering from ﬁsh ﬂesh and bladder due to Clay and Horne (1994). A simpler approach is adopted here by ﬁrst 15 Following Taylor (1995), the adjectives ‘‘areic’’ and ‘‘volumic’’ are used, respectively, to mean ‘‘per unit area’’ and ‘‘per unit volume’’.

Sec. 5.4]

5.4 Scattering of plane waves

219

expressing total BSX as a sum of contributions from ﬂesh and bladder back back back total ¼ bladder þ flesh :

ð5:153Þ

In practice, if a bladder is present, the contribution from the bladder usually dominates the scattering, so the BSX may then be approximated as back back total bladder :

ð5:154Þ

The scattering cross-section of a ﬁsh bladder can be calculated in much the same way as for a gas bubble, modiﬁed due to the additional tension from ﬂesh elasticity (Pe ) and corrected for the non-spherical shape of the bladder. The result can be approximated as (Weston, 1967) 4a 2S back ; ð5:155Þ bladder ½1 ð!0 =!Þ 2 2 þ Q 2 where

½P ðzÞ þ Pe ! 20 ¼ 4aS a w ; ð5:156Þ w Vbladder and aS is an equivalent bladder radius deﬁned as rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Sbladder aS ; 4

ð5:157Þ

such that the surface area of a sphere of that radius, 4a 2S , is equal to the bladder surface area. The resonance frequency f0 in cycles per unit time (i.e., f0 ¼ !0 =2) is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ K0 PðzÞ f0 ðLÞ ¼ ; ð5:158Þ L where P is the dimensionless pressure inside the bladder16 PðzÞ

Pw ðzÞ þ Pe PSTP

and K0 is a constant, with dimensions of length times frequency, equal to ! 1 aS L 2 1=2 K0 ¼ D0 cw 78:9 Hz m: 3 Vbladder

ð5:159Þ

ð5:160Þ

The physical signiﬁcance of this constant is the resonance frequency of a ﬁsh multiplied by its length, at the sea surface, with a (hypothetical) completely limp bladder (i.e., whose shear modulus is negligible). The measurements of f0 by Løvik and Hovem (1979) are consistent with a total bladder pressure ðPw þ Pe Þ of about 250 kPa. 16

The term Pe was introduced in Chapter 4 as the ‘‘pressure Pe exerted by the bladder wall on the gas contents.’’ More precisely, Pe is the sum of two terms: one equal to the increase in bladder pressure due to the bladder wall tension at a ﬁxed bladder size (denoted E); the other describing the variation of this pressure with the volume of the cavity (dE=dV). The details are described in Section 5.5.3.4.2.

220 Underwater acoustics

[Ch. 5

In Equation (5.159), the pressures PSTP , Pw , and Pe are, respectively, one standard atmosphere (approximately 101 kPa), the hydrostatic pressure at depth z, and the contribution to the pressure inside the bladder due to the elasticity of the bladder membrane. The elasticity term is uncertain, with values between 50 kPa and 300 kPa used in diﬀerent publications. A value of 75 kPa is adopted here (see Chapter 4). The square root dependence on pressure in Equation (5.158) assumes that changes in depth are slow compared with the time required for the ﬁsh to adjust to new conditions (such that the parameter K0 is a constant). Assuming a constant bladder mass for rapid changes (instead of the constant volume implied by Equation 5.160) results in a 5/6 power law dependence on pressure (Weston, 1995). The time required for the ﬁsh to make this adjustment is greater for the physoclist than for the physostome17 because the latter is able to exchange gas relatively quickly with its gut (or by releasing bubbles if rising; Weston, 1995). According to Løvik and Hovem (1979), the adjustment time for coalﬁsh (Pollachius virens, a physoclist) is between 12 and 24 hours. Equation (5.158) can be written in the alternative form pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ K0 PðzÞ L0 ð f Þ ¼ ; ð5:161Þ f where L0 is the length of ﬁsh whose bladder resonance frequency is f . The (reciprocal) Q-factor for a single ﬁsh can be written 1 1 1 1 ¼ þ þ ; Qfish Qrad Qtherm Qflesh

ð5:162Þ

where the ﬁrst two terms take the same form as for large bubbles: 1 ¼ Krad PðzÞ 1=2 Qrad

ð5:163Þ

and 1 Qtherm

¼

Ktherm PðzÞ 1=4 ; L 1=2

ð5:164Þ

where the constants Krad and Ktherm are (see Equation 5.138) Krad ¼ D0 0:0137 and Ktherm ¼

3ð a 1Þ Da 1=2 0:0089 m 1=2 : 2aS =L K0

ð5:165Þ ð5:166Þ

The third term in Equation (5.162) contains the Q-factor for ﬁsh ﬂesh, replacing that for water viscosity (Weston, 1967, p. 69; 1995, p. 10): 1 ¼ Kflesh PðzÞ 1 ; Qflesh 17

See Appendix C for an explanation of these terms.

ð5:167Þ

Sec. 5.4]

5.4 Scattering of plane waves

221

where (assuming Im ¼ 90 kPa) Kflesh ¼

Im ðaS =LÞ 2 0:61: K 20 w 2

ð5:168Þ

At resonance, ﬂesh damping is an order of magnitude larger than thermal and radiation terms. It is also the most uncertain of the three in magnitude, because of the uncertainty in the value of ImðÞ. From Equation (5.162), the overall Q-factor for a single ﬁsh can be written 1 K ¼ Krad P 1=2 þ therm P 1=4 þ Kflesh P 1 : Qfish L 1=2

ð5:169Þ

A typical value for Qfish at atmospheric pressure, obtained by substituting P ¼ 2 and L ¼ 0.1 m, is 3. The theoretical (reciprocal) Q-factor is dominated by the ﬂesh term, which is inversely proportional to pressure. If the value of P is increased to 10, Qfish increases to about 9. According to Andreeva (1964), acoustic radiation dominates the damping at depths exceeding about 200 m. 5.4.3.7

Single ﬁsh (without bladder)

In the absence of a gas-ﬁlled bladder,18 scattering is from ﬂesh alone and can be written 1 back ; ð5:170Þ flesh 1 1 ½ back

þ ½ back LF HF where the low-frequency limit (from Equation 5.131 and assuming the diﬀerence in impedance between water and ﬂesh to be small) is 2 back 3 8Vfish LF ¼ jR ðLFÞ j 2 ð5:171Þ 2 and the high-frequency limit (using the result from Table 5.3 for a convex shape at random aspect) is Sfish ðHFÞ 2 back jR j : ð5:172Þ HF ¼ 4 The low-frequency reﬂection coeﬃcient is jR ðLFÞ j 2

1 c 2 þ ; 4 cw w

ð5:173Þ

where c and are the diﬀerence in sound speed and density between average ﬁsh ﬂesh properties and those of water. For a bladderless ﬁsh it is reasonable to assume 18 Appendix C contains a list of species of ﬁsh indicating whether or not they have a bladder. Common examples of bladdered ﬁsh are cod and herring. An example of a bladderless species is mackerel.

222 Underwater acoustics

[Ch. 5

that the density contrast is small. Speciﬁcally, using Love’s values of c ¼ 0:033 cw

ð5:174Þ

¼ 0:022 w

ð5:175Þ

jR ðLFÞ j 2 0:00076:

ð5:176Þ

and

the reﬂection coeﬃcient is The high-frequency reﬂection coeﬃcient can be determined empirically from measurements of the target strength of bladderless ﬁsh. A typical value, inferred from measurements of the target strength of mackerel (see Chapter 8), is jR ðHFÞ j 2 0:0045: 5.4.3.8

ð5:177Þ

Dispersed ﬁsh (with bladder)

For ﬁsh that are dispersed (i.e., uniformly distributed in space), the contributions from individual ﬁsh can be integrated in the same way as for bubbles. Speciﬁcally, let nðLÞ dL be the number of ﬁsh per unit volume whose length is between L and L þ dL. The total volumic BSX is then ð1 back V ¼ back ð5:178Þ bladder ðLÞnðLÞ dL; 0

where the BSX is given by Equation (5.155). Compared with a cloud of gas bubbles, a group of ﬁsh is likely to have a relatively narrow size distribution, centered on some value of ﬁsh length, say Lgroup . This distribution is represented in terms of a Gaussian of width Lgroup /Qgroup , that is, following Weston (1995) 2 Qgroup L nðLÞ ¼ NV exp Q 2group 1 ; ð5:179Þ Lgroup Lgroup such that NV is the total population density ð1 NV ¼ nðLÞ dL:

ð5:180Þ

0

Thus, the presence of nðLÞ in the integrand has the eﬀect of broadening the resonance by an amount that depends on Qgroup . The resonance peak can be approximated by a Dirac delta function 2 back bladder ðLÞ 4a S L0 ð f ÞQfish ðL L0 ð f ÞÞ;

ð5:181Þ

where the length L0 is the length of ﬁsh whose bladder would resonate at a given frequency f (see Equation 5.161). Substituting Equations (5.179) and (5.181) into Equation (5.178) yields an approximation that relies on the ﬁsh being close to

Sec. 5.4]

5.4 Scattering of plane waves

223

resonance back V

4a 2S Qfish Qgroup NV

2 L0 ð f Þ L0 ð f Þ 2 exp Q group 1 : Lgroup Lgroup

ð5:182Þ

If no better information is available concerning ﬁsh length distribution, Weston (1995) suggests a default value of Qgroup ¼ 2. 5.4.3.9

Dispersed ﬁsh (without bladder)

The volumic BSX for dispersed bladderless ﬁsh can be written ð1 back V ¼ back flesh ðLÞnðLÞ dL;

ð5:183Þ

0

with O from Equation (5.170). If all ﬁsh are of approximately the same length Lgroup , such that nðLÞ NV ðL Lgroup Þ; ð5:184Þ it follows that back ¼ back ð5:185Þ V flesh ðLgroup ÞNV : 5.4.3.10

Aggregated ﬁsh (with bladder)

If, instead of being distributed uniformly in space (the dispersed model of Section 5.4.3.8), the ﬁsh form discrete aggregations, or ‘‘shoals’’, an estimate for the BSX of such a shoal of bladdered ﬁsh can be obtained from the volumic BSX, back V . Speciﬁcally, if ﬁsh density within the shoal is suﬃciently low, total shoal BSX may be approximated as the product of back and shoal volume Vshoal . For a high-density V shoal, it seems reasonable to propose an upper limit on the scattering cross-section determined by its surface area, resulting in the following tentative expression 1 back shoal

4 Sshoal

þ

1 ; Vshoal back V

ð5:186Þ

valid for both low and high ﬁsh density, with back from Section 5.4.3.8. V 5.4.3.11

Aggregated ﬁsh (without bladder)

The reasoning leading to Equation (5.186) applies just as well to a shoal of bladderless ﬁsh. Thus, 1 back shoal

4 1 þ ; Sshoal Vshoal flesh V

ð5:187Þ

with flesh equal to back from Section 5.4.3.9. V V 5.4.4

Scattering from rough boundaries

For scattering from a rough boundary, the main parameter of interest is the scattering coeﬃcient AO , which is the diﬀerential scattering cross-section O per unit area (i.e.,

224 Underwater acoustics

[Ch. 5

the areic diﬀerential scattering cross-section), and sometimes denoted mS . The behavior close to the direction of specular reﬂection is very diﬀerent from other directions, so this case is treated separately. 5.4.4.1

Non-specular term

For incident and scattered grazing angles 0 and , the scattering coeﬃcient away from the specular direction is (Brekhovskikh and Lysanov, 2003) AO ð0 ; ; Þ ¼ 4k 4 G2 ðs0 Þ sin 2 0 sin 2 ;

ð5:188Þ

where G2 is the spatial roughness spectrum (see Section 5.3.4), evaluated here for the Bragg scattering vector s0 . For simplicity, attention is limited to isotropic spectra, so that only the magnitude of the vector s is important, and G2 ðsÞ may be replaced by the 1D spectrum G1 ðÞ, where ¼ jsj:

ð5:189Þ

In particular, for the backscattering direction, AO ðÞ AO ð; ; Þ ¼ 4k 4 G1 ð2k cos Þ sin 4 ;

ð5:190Þ

where the use of a single subscript in AO ðÞ is used hereafter to indicate evaluation of the scattering coeﬃcient in the backscattering direction ( ¼ 0 ; ¼ ). The parameter AO ðÞ is known as the backscattering coeﬃcient. No distinction is made between and 0 in this situation because they are identical. Equation (5.188) is a general result, applicable to any roughness spectrum. For the special case of a Gaussian roughness spectrum of correlation radius L (see Equation 5.103), it follows from Equation (5.188) that ! 1 2L2 2 2 2 2 AO ð0 ; ; Þ ¼ ðkÞ ðkLÞ sin 0 sin exp : ð5:191Þ 4 Specializing to the backscattering case, we have ¼ 2k cos 0

ð5:192Þ

and hence AO ðÞ ¼ 5.4.4.2

1 ðkÞ 2 ðkLÞ 2 sin 4 exp½ðkLÞ 2 cos 2 :

ð5:193Þ

Near-specular term

Brekhovskikh and Lysanov (2003, pp. 207–209) consider rough boundary scattering for the case of isotropic roughness with a normally distributed slope and at high frequency. For this situation they derive an expression for the near-specular scattering coeﬃcient, which can be written in the form (Ellis and Crowe, 1991) DO 2 AO ð0 ; ; Þ ¼ ð1 þ DOÞ exp 2 ; ð5:194Þ 2

Sec. 5.5]

5.5 Dispersion in the presence of impurities

225

where (in this section) refers to the RMS roughness slope of the seabed, DO ¼

cos 2 0 þ cos 2 2 cos 0 cos cos ðsin 0 þ sin Þ 2

ð5:195Þ

and ¼

Rð0 Þ 2 : 8 2

For in-plane scattering, Equation (5.194) becomes 1 exp ; AO ð0 ; ; Þ ¼ sin 4 2 2 tan 2 where ð0 þ Þ=2 ¼ ¼ ð0 þ Þ=2 ¼ 0.

ð5:196Þ

ð5:197Þ

ð5:198Þ

Finally, the backscattering coeﬃcient is AO ðÞ ¼

5.5

1 RðÞ 2 1 exp : 8 2 sin 4 2 2 tan 2

ð5:199Þ

DISPERSION IN THE PRESENCE OF IMPURITIES

The speed and attenuation of sound in seawater can be aﬀected by impurities, especially if these contain any gas. The simplest situation, considered in Section 5.5.1, involves a dilute suspension of solid particles. The situation is complicated signiﬁcantly if there is the possibility of contact between solid grains, as considered in Section 5.5.2. The eﬀects of bubbles and bladdered ﬁsh are considered in Section 5.5.3. All of these eﬀects take as a baseline the sound speed and attenuation of pure seawater, covered in Chapter 4. 5.5.1

Wood’s model for sediments in dilute suspension

Consider a mixture of a substance x, of density x , and bulk modulus Bx , with water. If the fraction of x by volume is U, the density of the mixture is m ¼ ð1 UÞw þ Ux :

ð5:200Þ

Similarly, the bulk modulus Bm of the mixture is given by (Wood, 1941) 1 1U U ¼ þ ; Bm Bw Bx

ð5:201Þ

where Bw is the bulk modulus of water Bw ¼ w c 2w :

ð5:202Þ

226 Underwater acoustics

The speed of sound in the mixture is then given by sﬃﬃﬃﬃﬃﬃﬃ Bm cm ¼ m or, equivalently, 1 1 c 2 m ¼ ½ð1 UÞw þ Ux ½ð1 UÞB w þ UB x :

[Ch. 5

ð5:203Þ

ð5:204Þ

The last form is known as Wood’s equation. If one or both of Bw and Bx are complex, indicating the presence of an attenuation mechanism,19 the right-hand side of Equation (5.204) becomes complex. This complex sound speed is denoted c~m to distinguish it from the real value cm . The two variables are related to each other according to cm c~m ; ð5:205Þ 1 þ icm =! so that 1 cm ¼ ð5:206Þ Reð1=~ cm Þ and ¼ ! Imð1=~ cm Þ: ð5:207Þ The corresponding complex wavenumber is then ! k¼ : c~m

ð5:208Þ

Though derived originally with a mixture of two ﬂuids in mind, Wood’s equation is applied above without modiﬁcation to the case of a solid in dilute suspension. A generalization that allows for the presence of a solid frame is derived by Gassmann and extended further by Biot and Stoll. The interested reader is referred to Buchanan et al. (2004) or Jackson and Richardson (2007) for details of these models. An alternative approach that continues to treat sediment grains as individual particles, but allows contact between them, is described in Section 5.5.2. A further complication, not considered here, is the possible presence of gas in the sediment (Anderson and Hampton 1980a, b; Leighton, 2007b). 5.5.2

Buckingham’s model for saturated sediments with intergranular contact

Application of Wood’s model requires that sediment particles be in a dilute suspension. The acoustical behavior of a material in which the individual grains are in contact, as when they are deposited on the seabed, is described by Buckingham (2000, 2005). An important parameter in determining the acoustical properties of marine sediments is the sediment porosity . Assuming the sediment to be saturated with 19 The attenuation of sound due to suspended sediment is described by Richards (1998) and Richards et al. (2003).

Sec. 5.5]

5.5 Dispersion in the presence of impurities

227

water, porosity is the ratio by volume of water to the total (water þ sediment grain) medium. The density of the water–grain mixture is m ¼ w þ ð1 Þg ;

ð5:209Þ

where g is the density of solid grain. Buckingham shows that for grain–grain interactions, complex compressional and shear wave speeds (in the sense of Equation 5.205) are, respectively PðzÞ þ 43 SðzÞ in=2 1=2 c~p ðzÞ ¼ c0 1 þ ! ^n e ð5:210Þ B0 and n ^ SðzÞ 1=2 in=4 ! c~s ðzÞ ¼ c0 e ; ð5:211Þ B0 where ! ^ 2f^ ð5:212Þ is the angular frequency in radians per second; c0 is the sound speed of a noninteracting mixture as given by Wood’s formula (Equation 5.203) c 20 ¼ B0 =m

ð5:213Þ

1 1 ¼ þ : B0 Bw Bg

ð5:214Þ

and

The exponent n in Equations (5.210) and (5.211) is described as the ‘‘strain-hardening index’’. The parameters P and S (denoted p and s by Buckingham, 2005) are related to the bulk modulus and shear modulus of the combined porous medium comprising sediment grain and water, and are referred to by Buckingham as the compressional and shear rigidity coeﬃcients. The main depth dependence arises through these rigidity coeﬃcients. The equations that describe this dependence on depth z in the sediment are PðzÞ ¼ P0 ðd; zÞ ð5:215Þ and SðzÞ ¼ S0 ðd; zÞ; ð5:216Þ where is a dimensionless function of grain diameter d, porosity , and depth z 1 ðd; zÞ d z 1=3 ðd; zÞ ¼ ; ð5:217Þ 1 0 d0 z0 where parameters 0 , d0 , and z0 are reference values of the variables , d, and z. These and other parameters required for Buckingham’s model are listed in Table 5.4. 5.5.3

Eﬀect of bubbles or bladdered ﬁsh

Despite the reference to ﬁsh in the title of this section, it is mostly about the properties of sound in bubbly water. However, much of the theory can be applied with little

228 Underwater acoustics

[Ch. 5

Table 5.4. Water and solid grain sediment parameter values needed for Buckingham’s grainshearing model. Values for the density and bulk modulus of water, and the density of solid grain are from Chapter 4. All other parameter values are from Buckingham (2005). Symbol

Description

Value 1027 kg/m 3

w

Density of water

Bw

Bulk modulus of water

2.28 GPa

g

Density of solid grain

2680 kg/m 3

Bg

Bulk modulus of solid grain

P0

Compressional rigidity coeﬃcient of porous medium at depth z0 , of grain diameter d0 , and porosity 0

388.8 MPa

Shear rigidity coeﬃcient of porous medium at depth z0 , of grain diameter d0 , and porosity 0

45.88 MPa

S0

36 GPa

n

Strain-hardening index

0.0851

z0

Reference depth for P0 and S0

30 cm

d0

Reference grain diameter for P0 and S0

1 mm

0

Reference porosity for P0 and S0

0.377

modiﬁcation to dispersed populations of bladdered ﬁsh as demonstrated in Weston’s classic 1967 article and illustrated in Section 5.5.3.6.

5.5.3.1

Dispersion in bubbly water

Commander and Prosperetti (1989) and Hall (1989) derive equations for the complex eﬀective sound speed of a mixture of water and bubbles. Hall’s approach, based on Wood’s equation, results in the following expression for c~m in terms of the densities of air (g ) and water (w ) and gas volume fraction U ð1=~ cm Þ 2 ¼ ½ð1 UÞw þ Ua ½ð1 UÞKw þ DK ;

ð5:218Þ

where Kw is the compressibility of bubble-free seawater such that Kw ¼

1 1 ¼ : Bw w c 2w

ð5:219Þ

Sec. 5.5]

5.5 Dispersion in the presence of impurities

229

The contribution to compressibility from the air fraction (denoted DK) is given by20 ð 4 anðaÞ DK ¼ da; ð5:220Þ 2 w ! 0 ! 2 þ 2i! where !0 , a parameter closely related to bubble resonance frequency, can be written in terms of its bulk modulus Bb ! 20 ¼

3 Re Bb ða; !Þ ; w a 2

ð5:221Þ

and the damping factor21 is given by 2

3 Im Bb ða; !Þ !a 4 ¼ þ þ : 2 2 ! cm w a 2 ! w a !

ð5:222Þ

The bulk modulus Bb is given (anticipating a result from Section 5.5.3.2) by Bb ¼ GPa

2 : 3a

ð5:223Þ

Neglecting the density of air ( a w ), and assuming the gas fraction to be small (U 1), it follows from Equation (5.218) that 2 cw DK 1þ : ð5:224Þ c~m Kw The validity of Equation (5.224) does not depend on DK=Kw being small, provided that U is small. It is convenient to deﬁne the apparent void fraction as Pw DK; so that Equation (5.224) can be written 2 cw ðzÞ 1þ : c~m Kw Pw ðzÞ

ð5:225Þ

ð5:226Þ

A useful simpliﬁcation is obtained by approximating the denominator of the integrand of Equation (5.220) by ! 20 , requiring frequency ! to be small. Using Equation (5.221) for !0 , the parameter is then ð Pw ¼ VðaÞnðaÞ da; ð5:227Þ Re Bb where VðaÞ is the volume of a single spherical bubble of radius a. Before considering the bulk modulus in detail in the next section, for isothermal pulsations (consistent with the low-frequency approximation already made), and neglecting surface tension, 20 The same result follows by linearizing the non-linear theory of bubble dynamics (Leighton et al., 2004). 21 The damping factor and damping coeﬃcient are related according to Equation (5.308).

230 Underwater acoustics

[Ch. 5

it can be replaced in Equation (5.227) by Pw , so that is real and equal to the gas fraction U, and hence 2 cw U 1þ : ð5:228Þ cm Kw P w Substituting numerical values for seawater gives, for the stated conditions, 2 cw 1 þ 2:250 10 4 U: ð5:229Þ cm At very low frequency (see Section 5.5.3.4.1 for details) all bubbles behave isothermally. A more common situation, at slightly higher frequency, is a mixture of diﬀerent types of behavior, with the smallest bubbles pulsating isothermally and the larger ones adiabatically. If the frequency is increased further, eventually the approximation leading to Equation (5.227) breaks down as the resonance frequency of the larger bubbles is approached. 5.5.3.2

Bulk modulus Bb ða; !Þ

The bulk modulus of a bubble of volume V is deﬁned as Bb V

dQw ; dV

ð5:230Þ

where dQw denotes the change in externally applied pressure required to eﬀect an inﬁnitesimal change dV in the bubble volume. The notation Qw denotes instantaneous pressure, and is used to avoid possible confusion with the hydrostatic pressure Pw , which is equal to the equilibrium value of Qw . Similarly, let Qa ðRÞ be the pressure inside the bubble at the moment its instantaneous radius is R. This internal pressure is related to the instantaneous applied pressure Qw ðRÞ via the surface tension according to Qw ðRÞ ¼ Qa ðRÞ

2 : R

ð5:231Þ

Diﬀerentiating Qw and substituting the result into Equation (5.230), it follows that Bb ¼ VðaÞ

dQa 2 : dV 3a

ð5:232Þ

Gas pressure is assumed to vary according to the ideal gas law, which for adiabatic conditions is Qa V a ¼ constant; ð5:233Þ where a is the speciﬁc heat ratio of air. More generally, the exponent can take a value between 1 and a depending on the speed of the change. In equation form, this statement can be written Qa V G ¼ constant; ð5:234Þ where the exponent G is known as the polytropic index. For small bubbles and low frequency (leading to isothermal conditions), G is close to unity, whereas for large

Sec. 5.5]

5.5 Dispersion in the presence of impurities

231

bubbles and high frequency there is no opportunity for heat transfer, and G approaches a . With G deﬁned in this way, Equation (5.223) follows from Equation (5.232). More generally, using E to denote the diﬀerence in pressure between gas and liquid, that is, EðRÞ ¼ Qa ðRÞ Qw ðRÞ; ð5:235Þ regardless of the physical mechanism responsible for the pressure diﬀerence, and therefore a dE Bb ¼ GPa þ : ð5:236Þ 3 dR R¼a This expression becomes relevant if the bubble is replaced by a ﬁsh bladder kept under pressure by an elastic membrane. 5.5.3.3

Eﬀect of surface tension on small bubbles at low frequency

If surface tension is not negligible, its eﬀect is to decrease the compressibility of an air–water mixture. This decrease in compressibility can be quantiﬁed by making the assumption that the frequency is low enough for the bubbles to pulsate isothermally, the bubble bulk modulus Bb is related to the static pressure inside and outside the bubble according to 2 4 Re Bb ! Pa ¼ Pw þ : ð5:237Þ 3a 3a It is useful to deﬁne a distance a as the bubble radius for which the contributions to the internal bubble pressure from surface tension and hydrostatic pressure are equal. In other words 2 a ; ð5:238Þ Pw equal to 1.4 mm for clean bubbles at atmospheric pressure. Substituting for Bb in Equation (5.227) then gives for the apparent void fraction ð 1 ¼ VðaÞnðaÞ da: ð5:239Þ 1 þ 2a =3a Using the Hall–Novarini bubble model for nðaÞ, this can be written (following Chapter 4) ¼ IðzÞn0 uðv10 ÞDðz; v10 Þ; ð5:240Þ where ð 1 IðzÞ ¼ VðaÞGða; zÞ da: ð5:241Þ 1 þ 2a =3a The depth distribution is described by the function Gða; zÞ, given by Gða; zÞ ¼

a pða;zÞ ref

a

;

ð5:242Þ

232 Underwater acoustics

[Ch. 5

where

pða; zÞ ¼

4

a aref

xðzÞ

a > aref

ð5:243Þ

;

and x ¼ 4:37 þ

z 2 : 2:55 m

ð5:244Þ

Equation (5.241) can be written as the sum of two separate integrals IðzÞ ¼

4 4 a 3 ref

ð aref

1 4 da þ a xref 2 3 amin a þ 3 a

ð amax aref

a 4x da: a þ 23 a

ð5:245Þ

The integration runs from amin (the lower limit of the ﬁrst integral, equal to 10 mm for the Hall–Novarini model) to amax (the upper limit of the second, equal to 1000 mm). The value of aref is intermediate between these and varies with depth. The second integral (from aref to amax ) can be expressed formally in terms of the hypergeometric function 2 F1 (see Appendix A). Alternatively, a useful approximation is obtained by using the ﬁrst-order expansion ! 1 2a a 2 ¼1 þO 2 ; 1 þ 2a =3a 3a a

ð5:246Þ

valid for suﬃciently large bubbles, satisfying a =a 1. Using this expansion in the second integral of Equation (5.245) only,22 it follows that ! aref þ 23 a 1ðaref =amax Þ x4 3 2a 1ðaref =amax Þ x3 a 2 IðzÞ ¼ loge þ þO 2 : ðx 4Þ 3aref ðx 3Þ 4a 4ref a ref amin þ 23 a ð5:247Þ

5.5.3.4

Bubble resonance

It is sometimes useful to be able to express resonance frequency as a function of bubble radius, or the resonant bubble radius as a function of ensoniﬁcation frequency. The Minnaert relationship (Equation 5.137) may be used for large bubbles and low resonance frequency, but for small bubbles (high resonance frequency) there are important corrections caused by heat conduction and surface tension. The purpose of this section is to quantify these corrections. The equation describing the motion of a spherical bubble of (instantaneous) radius R in an incompressible medium of density w and shear viscosity S , known 22 The approximation is not made in the ﬁrst term because, although for the Hall–Novarini bubble population model the ratio a =amin is small, in general it might not be.

Sec. 5.5]

5.5 Dispersion in the presence of impurities

as the Rayleigh–Plesset equation, is (Leighton, 1994, p. 305)23 ! _2 3 R R_ w RR€ þ ¼ Qw ðRÞ Pw 4S p0 ðtÞ; 2 R

233

ð5:248Þ

where p0 ðtÞ is the acoustic pressure at inﬁnity (the forcing term); and Qw indicates total (static plus acoustic) pressure. Using the gas law—Equation (5.234) (with Equation 5.231)—this becomes ! a 3G 2 3R_ 2 R_ € w R R þ ¼ Pa Pw 4S p0 ðtÞ; ð5:249Þ 2 R R R where Pa is the equilibrium value of Qa Pa ¼ Qa ðaÞ:

ð5:250Þ

Equation (5.249) is a non-linear equation that is diﬃcult to solve without further approximation. It simpliﬁes if it is assumed that the amplitude of the oscillations is small compared with the bubble radius. Speciﬁcally, writing RðtÞ ¼ a þ "ðtÞ;

ð5:251Þ

w a 2 "€ 4S "_ 3Bb " ap0 ðtÞ;

ð5:252Þ

if j"j a it follows that

where Bb is the bulk modulus. Thus, for the simple harmonic motion of frequency !, with p0 ¼ A expði!tÞ and " ¼ "A expði!tÞ, the response "A is related to the forcing amplitude A according to Aa ¼ w a 2 ! 2 þ 4i!S 3Bb : "A

ð5:253Þ

It is convenient to deﬁne the variables Z

pi ðx ¼ 0Þ A ¼i _ !" R A

ð5:254Þ

and O 2u ! 2 þ i

!Z ; w a

ð5:255Þ

so that Equation (5.253) takes the form w að! 2 O 2u Þ"A ¼ A:

ð5:256Þ

Using Equation (5.253), Equation (5.255) can be written where

24

O 2u ¼ ! 2u 2i!; ! 2u Re O 2u ¼

23 24

3 Re Bb w a 2

The eﬀects of vapor pressure are neglected here. The variable !u is approximately equal to the bubble resonance frequency.

ð5:257Þ ð5:258Þ

234 Underwater acoustics

[Ch. 5

and is the damping factor25

Im O 2u 4S ! 3 Im Bb ¼ : 2! 2w a 2 !

ð5:259Þ

Thus, ! 2u

¼

3Pa Re G w a 2

2 a ;

ð5:260Þ

or more generally, using Equation (5.236) for the bulk modulus,

! 2u

¼

3Pa Re G þ a w a 2

dE dR :

ð5:261Þ

The condition for resonance is a maximum in the magnitude of the ratio "A =A, which (if is suﬃciently small) occurs when ! and !u are equal. For the remainder of this chapter, the term resonance frequency, denoted !res , is used to mean the frequency at which !u is equal to !. If !u were a constant (independent of frequency), that constant would equal !res . For large bubbles this is indeed the case, with the polytropic index then equal to the speciﬁc heat ratio. The resonance frequency derived in this approximation is the Minnaert frequency !0 of Equation (5.137). For smaller bubbles, G and hence also !u is a function of frequency, thus complicating the calculation of !res . For a hypothetical case with negligible thermal diﬀusivity Da , the eﬀect of surface tension can be taken into account without diﬃculty in the adiabatic limit. In this situation, Equation (5.258) simpliﬁes to ! 2u 1 a ¼1þ 1 ; ð5:262Þ 3 a a ! 20 where a is given by Equation (5.238). Using the subscript ‘‘ad’’ to denote the resonance frequency (or bubble radius) for adiabatic conditions, it follows from Equation (5.262) that ! 2ad ðaÞ 1 a ¼ 1 þ 1 : ð5:263Þ 3 a a ! 20 Equation (5.263) is useful for gas bubbles that are large enough to neglect thermal conduction but small enough for surface tension to become signiﬁcant. For air bubbles in seawater, this combination arises rarely, if ever.26 Nevertheless, Equation (5.263) serves to deﬁne the parameter !ad , used later. 25

In general, there is an additional contribution to due to acoustic radiation. No such term appears in Equation (5.259) due to the assumption here of an incompressible medium. 26 For large bubbles (satisfying a a ), Equation (5.263) may still be used, but there is little point as the Minnaert frequency is usually a better approximation.

Sec. 5.5]

5.5 Dispersion in the presence of impurities

235

A more realistic scenario involves very small bubbles such that the pulsations are isothermal. In this situation Equation (5.260) becomes !2 2a ! 2u ¼ 0 1 þ ; ð5:264Þ

a 3a which is also independent of frequency. The isothermal resonance frequency is therefore !0 2 a 1=2 !iso ðaÞ ¼ pﬃﬃﬃﬃﬃ 1 þ : ð5:265Þ

a 3a It is often the case that the pulsations are neither purely adiabatic nor purely isothermal. In this situation the bubble resonance frequency is higher than !iso and lower than !ad . 5.5.3.4.1 Polytropic index G The polytropic index is a dimensionless complex number whose real part takes values between unity for isothermal oscillations to the speciﬁc heat ratio a for adiabatic ones. It is a function of both ! and a.27 Speciﬁcally (Devin, 1959; Hall, 1989)

a Gð!Þ ¼ ; ð5:266Þ ð1 iÞx 3ið a 1Þ 1þ 1 tanh½ð1 iÞx 2x 2 where ! 1=2 xða; !Þ ¼ a : ð5:267Þ 2Da The parameter x can be written x¼

a ; aD ð!Þ

where aD is the thermal diﬀusion length, given by 2Da 1=2 aD ¼ : !

ð5:268Þ

ð5:269Þ

At atmospheric pressure the diﬀusion length is equal to about 80 mm at a frequency of 1 kHz, and 8 mm at 100 kHz. A pulsating bubble develops an isothermal layer of gas, at the edge of the bubble and in contact with the surrounding water. The thickness of this layer is approximately one thermal diﬀusion length aD . If aD is small compared with the bubble radius (corresponding to large x) there is negligible heat transfer, and oscillations are adiabatic so that G a . If it is large (small x), heat is transferred to the interior of the bubble, and the oscillations are isothermal. 27 For large bubbles, the polytropic index is a function of the product a! 1=2 ; for small ones there is a more complicated dependence because of the eﬀect of surface tension on pressure (and hence on gas thermal diﬀusivity).

236 Underwater acoustics

[Ch. 5

An alternative form, showing the frequency dependence of x explicitly, is ! 1=2 x¼ ; !D ðaÞ

ð5:270Þ

where !D , by analogy with aD , is referred to as the thermal diﬀusion frequency and is given by 2D !D ¼ 2a : ð5:271Þ a Diﬀusion frequency is a transition or threshold frequency between isothermal and adiabatic behavior. It is the frequency at which the diﬀusion length is equal to the bubble radius. Its value varies by four orders of magnitude between 6 Hz for large bubbles (a 1 mm) and about 60 kHz for small ones (a 10 mm). A useful approximation to Equation (5.266) is obtained by assuming that x is large:

a : G ð5:272Þ 3ð a 1Þ 1 1þ 1þi 1 2x x From this it follows that

and

a 2 ð1 1=xÞ 2 1þ þ Re G x x 1 þ =x

ð5:273Þ

2 2 1 2 1þ þ 2 1

a x x x ; 1 Im G 1 x x

ð5:274Þ

¼ 32 ð a 1Þ 0:602:

ð5:275Þ

where

5.5.3.4.2 Resonance frequency Given a bubble of known size, at what frequency does it resonate? If the bubble radius exceeds about 1 mm, the answer is at the Minnaert frequency, given by Equation (5.137). For smaller bubbles, two corrections are needed, one arising from thermal conduction inside the bubble, which reduces the resonance frequency, and another due to surface tension, which increases it. Because the two mechanisms work in opposite directions, their net eﬀect rarely exceeds a 10 % correction except for microscopic bubbles. General solution for arbitrary bubble radius. For a given arbitrary bubble radius a, the frequency ! at which !u ða; !Þ is equal to ! is the resonance frequency !res . This statement can be written in the following equation form, !u ða; !res Þ ¼ !res :

ð5:276Þ

Sec. 5.5]

5.5 Dispersion in the presence of impurities

237

In the general case, bubble resonance frequency can be found by writing Equation (5.276) as: !2 a a

a res ¼ 1 þ Re Gð!res Þ : ð5:277Þ 2 a 3a !0 The parameter Re G varies between its isothermal and adiabatic values (1 and a ), which means that the right-hand side of Equation (5.277) depends only weakly on the frequency variable (!res ). This suggests an iterative solution with ﬁrst guess equal to one of !iso or !ad . In other words ! 2 h a a i ð jþ1Þ ð jÞ ð! res Þ 2 ¼ 0 1 þ Re Gð! res Þ ; j 0 ð5:278Þ

a a 3a with either ð0Þ

ð5:279Þ

ð0Þ

ð5:280Þ

! res ¼ !ad ; or ! res ¼ !iso :

The choice of !iso as a seed tends to work best for small bubbles (a < 30 mm) whereas !ad works better for larger ones. The solid blue line in Figure 5.4 shows the converged result after repeated application of Equation (5.278), normalized by dividing by the Minnaert frequency. Also shown are the adiabatic and isothermal resonance frequencies !ad and !iso , and an alternative approximation derived below (Equation 5.283). The parameter values used are listed in Table 5.5. The thermal diﬀusivity of air is (see Chapter 4) Ka : ð5:281Þ Da a ðCP Þa Thermal conductivity Ka and speciﬁc heat capacity ðCP Þa are treated as constants, given by Table 5.5. However, the density of air ( a ) is a function of pressure and hence, because of surface tension, of bubble radius. Although the eﬀect of this change in diﬀusivity with bubble radius on the resonance frequency is small, the correction to the damping coeﬃcient is signiﬁcant (see Section 5.5.3.5.3 and Leighton, 1994). Approximation for bubbles exceeding radius 20 m. For bubbles whose radius is larger than about 20 mm, the diﬀerence between !ad and !0 is small, so in this situation the iteration can be simpliﬁed by using !0 as a seed instead of !ad . With this simpliﬁcation, the ﬁrst iteration of Equation (5.278) gives ! 2res ! 20

1 þ ð1 1=ð3 a ÞÞa =a : 1 þ ð!D =!0 Þ 1=2

ð5:282Þ

The second iteration, dropping all terms involving a =a except the ﬁrst-order one, gives 1 þ ð1 1=ð3 a ÞÞa =a ! 2res ! 20 ; ð5:283Þ 1 þ ð!D =!0 Þ 1=2 þ ð5 2 =4Þ!D =!0

238 Underwater acoustics

[Ch. 5

Figure 5.4. Approximations to the resonance frequency (normalized by dividing by the Minnaert frequency !0 ) for air bubbles in water at atmospheric pressure (upper graph) and at a depth of 90 m, corresponding to 10 atmospheric pressure (lower): ‘‘pert.’’ ¼ Equation (5.283); ‘‘conv.’’ ¼ converged iterative solution to Equation (5.278); ‘‘Minnaert’’ ¼ Equation (5.137); ‘‘adiab.’’ ¼ Equation (5.263); ‘‘isoth.’’ ¼ Equation (5.265).

Sec. 5.5]

5.5 Dispersion in the presence of impurities

Table 5.5. Values of physical constants (from Chapter 4) used for the evaluation of the bubble resonance characteristics in Figures 5.4 and 5.5.

where !D is the thermal diﬀusion frequency; and is given by Equation (5.275). The largest omitted terms are of order ð!D =!0 Þ 3=2 and ða =aÞð!D =!0 Þ 1=2 and may be neglected if the bubble radius is larger than about 20 mm, as illustrated by Figure 5.4. Also apparent from Figure 5.4 is that, for the conditions considered and bubbles of radius greater than 3 mm, the error incurred by using the Minnaert frequency is less than 10 %. Finally, a simple approximation for resonance frequency, with an error of less than 3 % across the entire range of bubble sizes considered, is obtained by taking the larger of !iso and Equation (5.283).

Parameter

Value used

a

1.4011

Ka

24.9 mW m1 K1

ðCP Þa

1.005 J g1 K1

0.072 N m1

239

Fish bladder resonance. A ﬁsh bladder can be modeled acoustically as a large deformed bubble under tension. The formula for resonance is found by writing the bulk modulus in the form dE Bb ¼ a ðPw þ EÞ þ Ve ; ð5:284Þ dV V¼Ve where Ve denotes the equilibrium bladder volume; and EðVÞ is the diﬀerence in static pressure across the bladder wall (cf. Equation 5.236). Substituting the formula from Chapter 4 Vbladder ¼ 3:40 10 4 L 3 ð5:285Þ into Equation (5.258) gives resonance frequency as ! 2res ¼

3 a ðPw þ Pe Þ ; w a 2

ð5:286Þ

where Pe ¼ E þ 2:43 10 4

dE 3 L : dV

ð5:287Þ

5.5.3.4.3 Resonant bubble radius A more likely scenario than a single bubble is one involving a cloud of bubbles of diﬀerent sizes. The question addressed here is, if such a cloud is ensoniﬁed with a plane wave of arbitrary frequency, which bubbles will resonate? If the frequency is lower than about 10 kHz, the answer is those whose radius is equal to the Minnaert radius, given by rearranging Minnaert’s equation (Equation 5.137) in the form sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 a Pw a0 ¼ a0 ð!Þ ; ð5:288Þ w ! 2

240 Underwater acoustics

[Ch. 5

(i.e., the adiabatic resonant radius if the restoring force is due to water pressure only). For higher frequency, two corrections are needed, one arising from thermal conduction inside the bubble, which reduces resonant bubble radius, and another due to surface tension, which increases it. Because the two mechanisms work in opposite directions, their net eﬀect rarely exceeds a 10 % correction for sonar frequencies. The answer is those whose radius ares satisﬁes the equation !u ðares ; !Þ ¼ !:

ð5:289Þ

Limits for adiabatic and isothermal conditions. If the conditions are adiabatic, the equation for the resonant radius (substituting G ¼ a in Equation 5.289) is a 3ad a 20 aad ð1 1=ð3 a ÞÞa 20 a ¼ 0;

ð5:290Þ

where a0 is the Minnaert radius given by Equation (5.288). The solution to Equation (5.290) can be written ( "pﬃﬃﬃﬃﬃ #) 4a 20 27 1 a 2 2 1 a ad ð!Þ ¼ cos arccos 1 : ð5:291Þ 3 3 2 3 a a0 The equations for isothermal conditions can be solved in exactly the same way as the adiabatic case, except with G ¼ 1. The resulting cubic equation for resonant radius is a 3iso

a 20 2 a2 aiso 0 a ¼ 0;

a 3 a

ð5:292Þ

whose solution is a 2iso ð!Þ ¼

pﬃﬃﬃﬃﬃﬃﬃ a 4a 20 1 cos 2 arccos 3 a : 3 a 3 a0

ð5:293Þ

General solution for arbitrary frequency. To solve for the bubble radius at resonance, Equation (5.289) can be written in the form a2 a a

a res Re Gðares Þ : ¼ 1 þ ð5:294Þ 2 ares 3ares a0 The solution is between aiso and aad . A general iterative solution to Equation (5.294) is " ! # a 20 a a ð jþ1Þ 2 ð jÞ ða res Þ ¼ 1 þ ð jÞ Re Gða res Þ j ; j 0; ð5:295Þ

a 3a res a res with either ð0Þ

ð5:296Þ

ð0Þ

ð5:297Þ

a res ¼ aad ; or a res ¼ aiso

as seed. Resonant bubble radius calculated in this way is shown in Figure 5.5 as a function of frequency (dark blue line). Also shown are the adiabatic and isothermal

Sec. 5.5]

5.5 Dispersion in the presence of impurities

241

Figure 5.5. Approximations to the resonant radius (normalized by dividing by the Minnaert radius a0 ) for air bubbles in water at atmospheric pressure (upper graph) and at a depth of 90 m, corresponding to about 10 atmospheric pressure (lower): ‘‘pert.’’ ¼ Equation (5.299); ‘‘conv.’’ ¼ converged iterative solution to Equation (5.295); ‘‘Minnaert’’ ¼ Equation (5.288); ‘‘adiab.’’ ¼ Equation (5.291); ‘‘isoth.’’ ¼ Equation (5.293).

242 Underwater acoustics

[Ch. 5

values aad and aiso , and an alternative approximation valid for frequencies up to about 100 kHz (Equation 5.299). The Minnaert radius (dashed cyan line) is surprisingly accurate up to frequencies of several megahertz. Approximation for frequencies up to 100 kHz. The use of the general solution is often unnecessary. If the frequency is not too high, an accurate solution can be obtained using one or two iterations. A single iteration, using a0 as a seed, yields a 2res a 20

1 þ ð1 1=ð3 a ÞÞa =a0 : 1 þ aD =a0

ð5:298Þ

The second iteration, dropping all terms involving a =a0 except the ﬁrst-order one, gives28 1 þ ð1 1=ð3 a ÞÞa =a0 a 2res a 20 : ð5:299Þ 1 þ aD =a0 þ ð3 2 =2ÞðaD =a0 Þ 2 The largest omitted terms are of order ðaD =a0 Þ 3 and ða =a0 ÞðaD =a0 Þ and may be neglected if the frequency is lower than about 100 kHz. The ﬁrst-order surface tension term contributes order 1 % to the numerator at 20 kHz and 10 % at 200 kHz. Finally, a simple approximation, with an error of less than 3 % across the entire frequency range considered, is obtained by taking the larger of aiso and Equation (5.299). 5.5.3.5 5.5.3.5.1

Damping factor Thermal and viscous damping

Equation (5.257) can be written in the form O 2u ¼ ! 2u 2iðvisc þ therm Þ!;

ð5:300Þ

3 Im Bb 2w a 2 !

ð5:301Þ

where therm ¼ therm ð!Þ ¼ and visc ¼

2S ; w a 2

ð5:302Þ

where therm and visc are the thermal and viscous damping factors. These equations are derived for the Rayleigh–Plesset model, which assumes that the liquid medium containing the bubble is incompressible. An incompressible medium does not support acoustic waves, so there is no mechanism to carry sound away from the bubble and thus no radiation damping. A more subtle point is that, by deﬁnition, an incompressible medium does not support changes in volume. As a result, only shear viscosity (i.e., no bulk viscosity) is included in the viscous damping term. For a more complete description that includes the contribution to damping from acoustic radiation and from volume viscosity, see Section 5.5.3.5.3. 28 In Equation (5.299), the parameter aD is a function of pressure and hence of bubble radius. The error incurred by evaluating aD at atmospheric pressure is usually small.

Sec. 5.5]

5.5 Dispersion in the presence of impurities

243

5.5.3.5.2 Radiation and thermal damping Consider the response of a bubble of equilibrium radius a, centered at the origin, to a plane wave traveling parallel to the x-axis, such that the incident acoustic pressure is p0 ¼ A exp iðkx !tÞ;

ð5:303Þ

where A is the amplitude, assumed constant. The scattered wave ps is assumed to have spherical symmetry: C ps ¼ exp iðkr !tÞ ðr aÞ; ð5:304Þ r where r is the distance from the origin; and C is a constant with dimensions pressure times distance. The scattering cross-section can then be expressed in the form ¼ 4jC=Aj 2 :

ð5:305Þ

An expression for the ratio C=A, derived by Ainslie and Leighton (2009), is C A ! 2u !2 where ¼

ae ika ; Im G ka 1 i Re G

! 2u Im G ka : Re G !2

ð5:306Þ

ð5:307Þ

Substituting Equation (5.306) in Equation (5.305) gives the scattering cross-section with radiation and thermal damping. A more complete expression is given in Section 5.5.3.6. 5.5.3.5.3 Total damping The total damping coeﬃcient tot for a gas bubble in water is related to the damping factor tot according to ! 2 ! 2u tot ¼ tot þ 1 ka: ð5:308Þ ! !2 For the special case of radiation damping only (i.e., if 2tot =! ¼ ka), it follows that rad ¼

! 2u ka; !2

ð5:309Þ

2 ka; ! tot

ð5:310Þ

and therefore tot rad ¼

providing a conversion between damping coeﬃcient and damping factor for nonacoustic forms of damping. Equation (5.308) can be written in the form tot ð!Þ ¼ rad ð!Þ þ therm ð!Þ þ visc ð!Þ;

ð5:311Þ

244 Underwater acoustics

[Ch. 5

in which the two main contributions are losses due to re-radiation of sound29 (rad , Equation 5.309), and dissipation of heat due to the ﬁnite thermal conductivity of air (therm ), which is (see Equation 5.307) therm ¼

! 2u Im G : ! 2 Re G

ð5:312Þ

The third contribution, resulting from the viscosity of water (visc ), is important only for bubbles smaller than about 10 mm (or very high–frequency sound). The basic form of this term for the case of an incompressible liquid can be derived from the damping factor (Equation 5.302) as visc ¼

2 4S ¼ : ! visc w !a 2

ð5:313Þ

For a compressible liquid there is an additional contribution due to bulk viscosity (Love, 1978), such that visc ¼

3ðB þ 43 S Þ : w !a 2

ð5:314Þ

The variation of the damping coeﬃcient with frequency is described by Medwin (1977), except with the opposite frequency dependence for the radiation damping term rad to that presented here. The discrepancy is explained by Ainslie and Leighton (2009). 5.5.3.5.4

Q-factors

It is sometimes convenient to express the damping coeﬃcient in terms of its reciprocal at resonance, the so-called Q-factor. Viscous and radiation loss terms are inversely proportional to frequency and hence can be written visc ¼

1 !res ; Qvisc !

ð5:315Þ

rad ¼

1 !res Qrad !

ð5:316Þ

and

where 3ðB þ 43 S Þ 1 ¼ Qvisc w !res a 2

ð5:317Þ

1 ! a ¼ res : Qrad cw

ð5:318Þ

and

29

That is, the release as an acoustic wave of the energy invested in bubble pulsations.

Sec. 5.5]

5.5 Dispersion in the presence of impurities

245

The frequency dependence of therm is more complicated. For suﬃciently large bubbles:30 1 !res 5=2 therm ¼ ; ð5:319Þ Qtherm ! where 1 Qtherm

¼

3ð a 1Þ !D 1=2 : 2 !res

ð5:320Þ

At resonance it is usually the case that !=!D is larger than 3, so that the approximation of Equation (5.272) may be used. In the low-frequency limit (isothermal conditions), Equation (5.266) may be written Gð!Þ ¼ 1

2i a 1 ! : 15 a !D

ð5:321Þ

Combining the low-frequency and high-frequency forms, it follows from Equation (5.312) that 8 2 ! ! > > 1 > < 5 ! ! a D D ð a 1ÞPa ðaÞ therm ¼ ð5:322Þ > w ! 2 a 2 9 a !D 1=2 ! > > : 1. 2 ! !D 5.5.3.6

Scattering, extinction, and absorption cross-sections

The scattering cross-section s of an object relates the total power scattered by the object to the intensity of an incident plane wave, and can be written ð s ¼ O dO; ð5:323Þ where the integral is over all scattered solid angles O; and O is the diﬀerential scattering cross-section. Some of the incident energy is absorbed (i.e., converted to heat rather than scattered), as quantiﬁed by the absorption cross-section a . Ainslie and Leighton (2009) derive a general purpose expression for s . Converting to the present notation, their Eq. (43) is s ¼

4a 2 ½! 2u =! 2 1 ðtot rad Þ" 2 þ 2tot ;

ð5:324Þ

with the Andreeva–Weston model for the damping coeﬃcient (Ainslie and Leighton, 2009). 30 Equation (5.319) is valid if the bubble radius is large compared with both the diﬀusion length aD (Equation 5.269) and the parameter a , related to the surface tension (Equation 5.238).

246 Underwater acoustics

[Ch. 5

Total acoustic attenuation (due to scattering plus absorption) is determined by the extinction cross-section e (Weston, 1967; Ainslie and Leighton, 2009): e ¼ 2 tot " 1 ; s ! m

ð5:325Þ

where "m ¼

!a : cm

ð5:326Þ

Parameters and are related via Equation (5.310). Hence, the extinction and absorption cross-sections can be calculated, respectively, as rad e ¼ 1 þ tot s ð5:327Þ "m and rad a e s ¼ tot s : ð5:328Þ "m At low frequency, a small object like a bubble scatters sound equally in all directions. Thus, O may be approximated by O

back ; 4

ð5:329Þ

where back is the backscattering cross-section (BSX) of a single bubble, equal in this situation to the total scattering cross-section s . Using this equation for O together with Equation (5.327) (and Equation 5.323), the extinction cross-section for a single bubble can be written tot rad back e ¼ 1þ : ð5:330Þ "m Thus, with thermal and viscous absorption: therm þ visc back e ¼ 1þ ; "m

ð5:331Þ

where back can be calculated using Equation (5.136). Thermal and viscous damping terms are given by Equations (5.301) and (5.314). The extinction cross-section of a ﬁsh bladder can be calculated in the same way as for a gas bubble, by replacing the viscosity term with the appropriate damping coeﬃcient for ﬂesh damping and making allowance for non-sphericity. The result, following Weston (1995), is " # ! 20 therm þ flesh back e ¼ bladder ðL; !Þ 1 þ 2 ; ð5:332Þ rad ! where !0 is the resonance frequency (see Equation 5.156); and back may be approximated by Equation (5.155). The damping terms can be expressed in terms

Sec. 5.6]

5.6 References

247

of their respective Q-factors !0 ðLÞ 5=2 therm ðL; !Þ ¼ ; Qtherm ðLÞ ! 1 !0 ðLÞ 2 flesh ðL; !Þ ¼ ; Qflesh ðLÞ ! 1

ð5:333Þ ð5:334Þ

and rad ðL; !Þ ¼

1 !0 ðLÞ : Qrad ðLÞ !

ð5:335Þ

The three Q-factors are given, respectively, by Equations (5.164), (5.167), and (5.163).

5.6

REFERENCES

Achenbach, J. D. (1975) Wave Propagation in Elastic Solids, North-Holland, Amsterdam. Ainslie, M. A. (1995) Plane-wave reﬂection and transmission coeﬃcients for a three-layered elastic medium, J. Acoust. Soc. Am., 97, 954–961. [Erratum, J. Acoust. Soc. Am., 105, 2053 (1999).] Ainslie, M. A. (1996) Reﬂection and transmission coeﬃcients for a layered ﬂuid sediment overlying a uniform solid substrate, J. Acoust. Soc. Am., 99, 893–902. Ainslie, M. A. and Burns, P. W. (1995) Energy-conserving reﬂection and transmission coeﬃcients for a solid–solid boundary, J. Acoust. Soc. Am., 98, 2836–2840. Ainslie, M. A. and Leighton, T. G. (2009) Near resonant bubble acoustic cross-section corrections, including examples from oceanography, volcanology, and biomedical ultrasound, J. Acoust. Soc. Am., 126, 2163–2175. Anderson, A. L. and Hampton, L. D. (1980a) Acoustics of gas-bearing sediment. I: Background, J. Acoust. Soc. Am., 67, 1865–1889. Anderson, A. L. and Hampton, L. D. (1980b) Acoustics of gas-bearing sediment. II: Measurements and models, J. Acoust. Soc. Am., 67, 1890–1905. Andreeva, I. B. (1964) Scattering of sound by air bladders of ﬁsh in deep sound-scattering ocean layers, Akust. Zh., 10, 20–24 [English translation in Sov. Phys. Acoust., 10, 17–20 (1964)]. Anon. (1946) Physics of Sound in The Sea (NAVMAT P-9675, p. 462). National Defense Research Committee, Washington, D.C. 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. Brekhovskikh, L. M. and Godin, O. A. (1990). Acoustics of Layered Media I: Plane and Quasi-Plane Waves, Springer-Verlag, Berlin. Brekhovskikh, L. M. and Lysanov, Yu. P. (2003) Fundamentals of Ocean Acoustics (Third Edition). AIP Press Springer-Verlag, New York. Buchanan, J. L., Gilbert, R. P., Wirgin A., and Xu, Y. S. (2004) Marine Acoustics: Direct and Inverse Problems, Society for Industrial and Applied Mathematics, Philadelphia. Buckingham, M. J. (2000) Wave propagation, stress relaxation, and grain-to-grain shearing in saturated, unconsolidated marine sediments, J. Acoust. Soc. Am., 108, 2796–2815. Buckingham, M. J. (2005) Compressional and shear wave properties of marine sediments: Comparisons between theory and data, J. Acoust. Soc. Am., 117, 137–152.

248 Underwater acoustics

[Ch. 5

Chapman, C. H. (2004) Fundamentals of Seismic Wave Propagation, Cambridge University Press, Cambridge. Clay, C. S. and Horne, J. K. (1994) Acoustic models of ﬁsh: The Atlantic cod (Gadus morhua), J. Acoust. Soc. Am., 96, 1661–1668. Clay, C. S. and Medwin, H. (1977) Acoustical Oceanography: Principles and Applications, Wiley, New York. Commander, K. W. and Prosperetti, A. (1989) Linear pressure waves in bubbly liquids: Comparison between theory and experiments, J. Acoust. Soc. Am., 85, 732–746. Crocker, M. J. (Ed.) (1997) Encyclopedia of Acoustics, Wiley, New York. Deschamps, M. and Changlin, C. (1989) Re´ﬂexion-re´fraction de l’onde plane he´te´roge`ne: lois de Snell-Descartes et continuite´ de l’e´nergie, J. Acoust., 2, 229–240 [in French]. Devin, C., Jr. (1959) Survey of thermal, radiation and viscous damping of pulsating air bubbles in water, J. Acoust. Soc. Am., 31, 1654–1667. Dragonette, L. R. and Gaumond, C. F. (1997) Transient and steady-state scattering and diﬀraction from underwater targets, in M. J. Crocker (Ed.), Encyclopedia of Acoustics (pp 469–482), Wiley, New York. Eckart, C. (1953) The scattering of sound from the sea surface, J. Acoust. Soc. Am., 25, 566– 570. Ellis, D. D. and Crowe, D. V. (1991). Bistatic reverberation calculations using a threedimensional scattering function, J. Acoust. Soc. Am., 89, 2207–2214. Fortuin, L. (1973) The sea surface as a random ﬁlter for underwater sound waves, Ph.D. thesis, Technological University of Twente, Uitgeverij Waltman, Delft, The Netherlands (SACLANTCEN Report SR-7). Hall, M. V. (1989) A comprehensive model of wind-generated bubbles in the ocean and predictions of the eﬀects on sound propagation at frequencies up to 40 kHz, J. Acoust. Soc. Am., 86, 1103–1117. Horne, J. K. and Clay, C. S. (1998) Sonar systems and aquatic organisms: Matching equipment and model parameters, Can. J. Fish. Aquat. Sci., 55, 1296–1306. Jackson, D. R. and Richardson, M. D. (2007) High-Frequency Seaﬂoor Acoustics, SpringerVerlag, New York. Jensen, F. B., Kuperman, W. A., Porter M. B., and Schmidt, H. (1994) Computational Ocean Acoustics, AIP Press, New York. Kennett, B. L. N. (1974) Reﬂections, rays, and reverberations, Bull. Seis. Soc. Am., 64, 1685– 1696. Kolsky, H. (1963) Stress Waves in Solids, Dover, New York. Leighton, T. G. (1994) The Acoustic Bubble, Academic Press, London. Leighton, T. G. (2007a) What is ultrasound?, Progress in Biophysics and Molecular Biology, 93, 3–83. Leighton, T. G. (2007b) Theory for acoustic propagation in marine sediment containing gas bubbles which may pulsate in a non-stationary nonlinear manner, Geophys. Res. Lett., 34, L17607. Leighton, T. G., Meers, S. D. and White, P. R. (2004). Propagation through nonlinear timedependent bubble clouds and the estimation of bubble populations from measured acoustic characteristics, Proc. R. Soc. Lond. A, 460(2049), 2521–2550. 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.

Sec. 5.6]

5.6 References

249

MacLennan, D. N., Fernandes, P. G., and Dalen, J. (2002) A consistent approach to deﬁnitions and symbols in ﬁsheries acoustics, ICES J Marine Science, 59, 365–369. Medwin, H. (1977) Counting bubbles acoustically: A review, Ultrasonics, 7–13, January. Medwin, H. and Clay, C. S. (1998) Fundamentals of Acoustical Oceanography, Academic Press, Boston. Miklowitz, J. (1978) Elastic Waves and Waveguides, North-Holland, Amsterdam. Minnaert, M. (1933) On musical air-bubbles and the sounds of running water, Phil. Mag., 16, 235–248. Morfey, C. L. (2001) Dictionary of Acoustics, Academic Press, San Diego. Morse, P. M. and Ingard, K. U. (1968) Theoretical Acoustics, Princeton University Press, Princeton. Neubauer, W. G. (ca. 1982) Acoustic Reﬂection from Surfaces and Shapes, Naval Research Laboratory, Washington, D.C. Pierce, A. D. (1989) Acoustics: An Introduction to Its Physical Principles and Applications, American Institute of Physics, New York. Richards, S. D. (1998) The eﬀect of temperature, pressure and salinity on sound attenuation in turbid seawater, J. Acoust. Soc. Am., 103, 205–211. Richards, S. D., Leighton, T. G., and Brown, N. R. (2003) Visco-inertial absorption in dilute suspensions of irregular particles, Proc. R. Soc. Lond. A, 459(2038), 2153–2167. Robins, A. J. (1991) Reﬂection of a plane wave from a ﬂuid layer with continuously varying density and sound speed, J. Acoust. Soc. Am., 89, 1686–1696. Schmidt, H. (1988) SAFARI, Seismo-Acoustic Fast-ﬁeld Algorithm for Range-Independent environments (SACLANTCEN Report SR-113). SACLANT Undersea Research Centre, La Spezia, Italy. Schmidt, H. (ca. 2000) OASES Version 3.2 User Guide and Reference Manual, Massachusetts Institute of Technology. Stanton, T. K. (1989) Simple approximate formulas for backscattering of sound by spherical and elongated objects, J. Acoust. Soc. Am., 86, 1499–1510. Taylor, B. N. (1995) Guide for the Use of the International System of Units (SI) (NIST Special Publication 811, 1995 Edition), United States Department of Commerce, National Institute of Standards and Technology. Thorne, P. D. and Meral, R. (2008) Formulations for the scattering properties of suspended sandy sediments for use in the application of acoustics to sediment transport processes, Continental Shelf Research, 28, 309–317. Urick, R. J. 1983) Principles of Underwater Sound, Peninsula, Los Altos, CA. Weston, D. E. (1967) Sound propagation in the presence of bladder ﬁsh, in V. M. Albers (Ed.), Underwater Acoustics, Vol II: Proceedings of the 1966 NATO Advanced Study Institute, Copenhagen (pp. 55–88), Plenum Press, New York. Weston, D. E. (1995) Assessment Methods for Biological Scattering and Attenuation in Ocean Acoustics (Report C3305/7/TR-1, April), BAeSEMA, Esher, U.K. Wood, A. B. (1941) A Textbook of Sound, Bell, London.

6 Sonar signal processing

Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuﬀ of any degree of ﬁtness; but, nevertheless, what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat-ﬂour from peascods, so pages of formulae will not get a deﬁnite result out of loose data. Thomas Henry Huxley (ca. 1894) Once a pressure disturbance has been converted to an electric current, it can be processed in various ways to enhance the signal-to-noise ratio before being presented to an operator for interpretation. In a modern sonar system, the processing is mostly carried out by a digital computer, which means that one of the ﬁrst steps must be a conversion from an analogue signal to a digital one. Even before this, an analogue low-pass ﬁlter, known as an anti-alias ﬁlter, is used to remove frequencies that exceed the speciﬁcation of the analogue-to-digital converter (ADC). The purpose of digital signal processing is partly to ﬁlter out as much unwanted noise as possible. Both active and passive sonars are designed to ﬁnd signals in a predetermined frequency band, so noise outside this band is not normally considered to contribute to the noise level term in the sonar equation (NL). The noise left after this ﬁltering is referred to as the noise ‘‘in the sonar bandwidth’’. Assuming that the sonar has more than one receiving hydrophone, the next step after this initial ﬁltering, for both active and passive systems, is to ﬁlter out unwanted angles by beamforming as summarized in Chapter 2. The calculation of array gain for passive sonar is the main subject of Section 6.1. For active sonar a further gain can be achieved by means of a special process that exploits knowledge of the shape of the transmitted pulse, enabling rejection of any noise that does not resemble that pulse. This process, known as matched ﬁltering, and the resulting gain, is the main subject of Section 6.2.

252 Sonar signal processing

6.1

[Ch. 6

PROCESSING GAIN FOR PASSIVE SONAR

Time domain and spatial domain ﬁlters are both used for passive sonar processing. The gain from time domain ﬁltering is by convention considered as a reduction in the noise level. The subject of this section is the gain from spatial ﬁltering, which is largely determined by the beam pattern of the receiving array, described in Section 6.1.1. This is followed by a discussion of the directivity index, which is a function of the beam pattern, in Section 6.1.2. The array gain, discussed in Section 6.1.3, depends on the beam pattern in a similar way as the directivity index, and on the directional properties of the signal and noise ﬁelds. Array gain (AG) is arguably the most diﬃcult term of the passive sonar equation to calculate precisely. However, often there is no need to do so, because it is straightforward to approximate its eﬀect by replacing it with the directivity index (DI), which is equal to AG in certain idealized circumstances.

6.1.1

Beam patterns

Angular discrimination using omni-directional hydrophones can be achieved by constructing an array with a horizontal or vertical aperture, or both, with the individual hydrophones placed a fraction of a wavelength apart. Such an array has a beam pattern BðOÞ, which is the squared magnitude of the array output in response to an acoustic plane wave, normalized by dividing by its maximum value in angle. 6.1.1.1

Steered line array

The beamformer output from Chapter 2 can be generalized by multiplying each term in the sum by a scaling factor wðxÞ, so that (using H here for the beamformer output instead of G, to avoid confusion with the array gain and directivity index) H

N1 X

wðxn ÞFðxn Þ expðikxn sin Þ;

ð6:1Þ

n¼0

where FðxÞ is the incident pressure ﬁeld;1 and is the steering angle (denoted m in Chapter 2), deﬁned as k arcsin m ; ð6:2Þ k and km is the steer direction in wavenumber space. The eﬀect of the scaling factor, known as a shading function or window function2 is considered in Sections 6.1.1.1.2 to 6.1.1.1.4. Before that, the beam pattern of an unshaded array is described. 1

In a modern sonar, the hydrophone signal is converted to voltage and then digitized. If all processes are linear the ﬁnal digital representation is proportional to the original pressure. 2 Alternative names are weighting function or taper function. See Harris (1978) and Nuttall (1981) for two in-depth reviews of the properties of many diﬀerent shading functions.

Sec. 6.1]

6.1 Processing gain for passive sonar 253

6.1.1.1.1 Unshaded Before considering the eﬀects of shading, it is useful to revisit the properties of an unshaded array, equivalent to the trivial case wðxÞ ¼ 1 in Equation (6.1). More precisely wðxÞ ¼ Pðx=LÞ; ð6:3Þ where PðÞ is the rectangle function, equal to unity if jj < 1=2 and zero otherwise (see Appendix A). This window is known as a rectangular or Dirichlet window. For a plane wave of unit amplitude, the beamformer output is then (Chapter 2)3 Hðkm Þ N sincðuÞ;

ð6:4Þ

where u ¼ kðsin sin Þ

L : 2

ð6:5Þ

From Equation (6.5) it can be seen that is the value of the look angle at which the sinc argument u passes through zero. In other words, it is the look direction4 in which the beamformer output is greatest. The broadside direction corresponds to ¼ 0 (i.e., to an unsteered array) and the fore and aft endﬁre directions are given by ¼ =2. Recall that the beam pattern is the normalized squared magnitude of the beamformer output, so that BðÞ sinc 2 uðÞ:

ð6:6Þ

The function BðÞ is plotted in Figure 6.1 for an array of length ﬁve wavelengths, with steering angles of 0 deg, 30 deg, 60 deg, and 90 deg (solid lines). The corresponding graphs for negative are obtained from these curves by taking their mirror images in the ¼ 0 axis. If the array axis is aligned horizontally (in which case it is known as a horizontal line array), it is convenient to replace the single angle , representing the look direction, with the elevation and azimuth by writing sin ¼ cos sin

ð6:7Þ

L ðcos sin sin Þ;

ð6:8Þ

and hence u¼

where is the acoustic wavelength at the array. The reason for doing so is that and

are sometimes more natural co-ordinates for describing the search geometry (e.g.,

might be the bearing of a distant sonar contact). From Figure 6.1 it is apparent that the main beam gets broader as the steering angle is increased from 0 deg to 90 deg. The width in wavenumber is given by 3

The notation ‘‘sinc’’ is shorthand for the sine cardinal function (sinc x ¼ sin x=x) (see Appendix A). 4 That is, the direction of maximum sensitivity to an incoming plane wave, relative to the array axis.

254 Sonar signal processing

[Ch. 6

Figure 6.1. Sinc beam patterns for L= ¼ 5 and steering angles 0, 30, 60, 90 deg anticlockwise from upper left (solid). The dashed curves illustrate the eﬀect of shading (see Section 6.1.1.1.2).

Sec. 6.1]

6.1 Processing gain for passive sonar 255

256 Sonar signal processing

[Ch. 6

(Chapter 2) kfwhm ¼

4 1 sinc 1 pﬃﬃﬃ : L 2

ð6:9Þ

This is the full-width at half-maximum (fwhm). The expression holds for all beams, but is of limited use in this form. To understand its implications for the width of the beams in real physical space, an angular width is more appropriate, as derived below. For non-zero steering angles the beams lose their symmetry, becoming increasingly asymmetrical as they approach endﬁre. In this situation it is appropriate to talk of two half-widths rather than a single full-width, as the beams are unequal in extent to either side of the main peak. From Equation (6.9), the half-power half-width5 (i.e., half-width at half-maximum) to broadside of the peak, denoted , is determined by the equation sin m sinðm Þ ¼ Y; ð6:10Þ where pﬃﬃﬃ sinc 1 ð1= 2Þ Y 0:4430 : ð6:11Þ L L Similarly, the half-width to endﬁre ( þ ) is found from sinðm þ þ Þ sin m ¼ Y:

ð6:12Þ

For the endﬁre beam itself (m ¼ =2), Equation (6.10) simpliﬁes to ð Þef ¼ arccos Y:

ð6:13Þ

The array of Figure 6.1 has a length of 5, so at endﬁre its half-width ( ) from Equation (6.13) is 24 deg, consistent with the graph. Similarly for the broadside beam (m ¼ 0), Equation (6.10) or Equation (6.12) gives ð Þbs ¼ arcsin Y:

ð6:14Þ

Equations (6.10) and (6.12) can be combined into a single more compact expression to give ¼ arcsinðY sin m Þ m : ð6:15Þ Because the angle is deﬁned as the half-power half-width, for it to exist at all the beam pattern must drop below 0.5 somewhere in the real range of angles, and for this it is necessary for the array to exceed a certain minimum length. A necessary condition for the existence of þ (and a suﬃcient one for ) is that Y be less than unity. Necessary and suﬃcient conditions for a continuous line array are Y < 1 sin m

ðfor þ Þ

ð6:16Þ

Y < 1 þ sin m

ðfor Þ:

ð6:17Þ

and For an array whose length exceeds =2, the existence of (half-width to broadside) 5 The half-power full-width (i.e., f.w.h.m.) is sometimes referred to as the ‘‘3 dB’’ width, because 10 log10 ð1=2Þ 3.01 dB is approximately equal to 3 dB.

Sec. 6.1]

6.1 Processing gain for passive sonar 257

is ensured, but the half-width to endﬁre exists only for beams that are not too close to the endﬁre direction. The sinc function has a main peak at u ¼ 0 and secondary ones close to odd multiples of =2. The secondary peaks, called sidelobes, are unwanted because they result in undesirable sensitivity to sound from directions other than the signal direction. They are caused by the abrupt start and end of the array, and can be reduced by smoothing these edges. That is, by reducing the contribution from hydrophones close to the two ends of the array relative to those at the center. This procedure is known as array shading or array weighting. To illustrate how shading helps control sidelobe levels, consider the choice of function wðxÞ in Equation (6.1). The simplest way of removing the step at x ¼ L=2 is to introduce a linear variation in amplitude, meaning that wðxÞ decreases linearly from 1 at x ¼ 0 to 0 at L=2. The resulting function is known as a triangular window because of the triangular shape of the taper. Thus, 2jxj wtri ðxÞ ¼ 1 Pðx=LÞ; ð6:18Þ L with the resulting beam pattern, u btri ðuÞ ¼ sinc 4 : 2

ð6:19Þ

A lower case ‘‘b’’ is used here for the beam pattern expressed as a function of the variable u (Equation 6.5). This is to distinguish it from the upper case ‘‘B ’’ for the beam pattern as a function of angle . The functions bðuÞ and BðÞ are therefore related via BðÞ ¼ b½uðÞ : ð6:20Þ 6.1.1.1.2 Cosine shading (cos n ) Another function that tapers in a simple way to zero is a half-cycle of a cosine, perhaps raised to a power n such that x wcos n ðxÞ ¼ cos n Pðx=LÞ: ð6:21Þ L Beam patterns for the special cases n ¼ 1 and n ¼ 2 are 2 h i2 sinc u þ bcos ðuÞ ¼ þ sinc u 16 2 2 and bcos 2 ðuÞ ¼ fsinc u þ 12 ½sincðu þ Þ þ sincðu Þ g 2 :

ð6:22Þ ð6:23Þ

The cosine squared window (i.e., Equation 6.21 with n ¼ 2) is also known as a Hann window.6 The associated beam pattern, given by Equation (6.23), is shown in Figure 6.1 (dashed lines) for various steer angles. 6 The window is named after Julius von Hann. The term ‘‘Hann window’’ is preferred here over its synonym ‘‘Hanning window’’ to avoid possible confusion with ‘‘Hamming window’’.

258 Sonar signal processing

[Ch. 6

Figure 6.2. Beam patterns 10 log10 BðÞ for continuous line array of length L= ¼ 10 for unshaded array (cyan), cosine window (dashed red), and Hann window (blue).

Beam patterns for rectangular, cosine, and cosine-squared (Hann) windows are plotted in Figure 6.2, this time for a 10-wavelength array. As the severity of shading increases (corresponding to increasing the value of n in Equation 6.21), the height of the sidelobes decreases (the desired eﬀect) while the width of the main lobe increases (an unwanted side-eﬀect, as it decreases the angular resolution of the beam). The nulls for rectangular and Hann shading occur at identical angles, whenever the argument u is equal to a non-zero integer multiple of . For cosine shading the nulls are shifted by =2. The width of the main beam and the number of sidelobes depend on the length of the array in wavelengths. Speciﬁcally, the longer the array in wavelengths, the narrower the beam and the larger the number of sidelobes.

6.1.1.1.3

Cosine on a pedestal (Hamming family)

The cos 2 window of Section 6.1.1.1.2 can be generalized by placing it on a pedestal of height ". Because of the simple relationship between cos 2 and cos 2 , this type of window is also known as a raised cosine. The general case taper function for this window is wðxÞ ¼ "Pðx=LÞ þ ð1 "Þwcos 2 ðxÞ;

ð6:24Þ

where wcos 2 ðxÞ is given by Equation (6.21) with n ¼ 2. The corresponding beamformer output is H ¼ "Hrect þ ð1 "ÞHcos 2 ;

ð6:25Þ

Sec. 6.1]

6.1 Processing gain for passive sonar 259

where Hrect ¼ N sinc u

ð6:26Þ

and Hcos 2 ¼

N ½sinc u þ 12 sincðu þ Þ þ 12 sincðu Þ : 2

The resulting beam pattern is

2 1" bðuÞ ¼ sinc u þ ½sincðu þ Þ þ sincðu Þ : 2ð1 þ "Þ

ð6:27Þ

ð6:28Þ

This family of windows includes a special case known as the Hamming window,7 obtained with " ¼ 0.08: wHamming ðxÞ ¼ 0:08Pðx=LÞ þ 0:92wcos 2 ðxÞ; with corresponding beam pattern

2 23 bHamming ¼ sinc u þ ½sincðu þ Þ þ sincðu Þ : 54

ð6:29Þ

ð6:30Þ

The generic window described by Equation (6.24), with arbitrary " between 0 and 1, is referred to below as the ‘‘Hamming family’’. In addition to the Hamming window itself, the Hamming family includes as members the rectangular and cos 2 windows (obtained with " ¼ 1 and " ¼ 0, respectively). Figure 6.3 shows BðÞ for the Hamming family with various values of the pedestal height ", including the Hamming window. The eﬀect of reducing " from 1.00 to 0.08 is to reduce the peak sidelobe level at the expense of an increased beam width. When " is reduced further than this the sidelobe levels increase. Thus, the Hamming window (" ¼ 0.08) is close to an optimum from this point of view. The precise value of " that minimizes the highest sidelobe level is 0.076711, the beam pattern for which is shown as a solid cyan line. Graphs of bðuÞ for selected " values are plotted in Figure 6.4, alongside their respective shading functions. 6.1.1.1.4

Tukey shading (raised cosine spectrum)

The Tukey family of windows provides an alternative generalization of the rectangular and Hann windows, with the cosine function displaced laterally instead of vertically. The two halves of a Hann window are compressed and pulled out towards each of the two ends of the window, leaving a gap in the middle. The Tukey window is obtained by padding this gap with ones. This approach removes the undesirable discontinuities associated with a rectangular window (thus reducing sidelobe levels), while improving the resolution compared with the Hann window. The Tukey shading function (correcting a typographical error in Eq. (38) of 7

Named after Richard Hamming.

260 Sonar signal processing

[Ch. 6

Figure 6.3. Beam patterns 10 log10 BðÞ for continuous line array of length L= ¼ 10 with raised cosine shading and " values as marked.

Harris, 1978) is

8 > > 1 > > > > > < ðjxj "L=2Þ wTukey ðxÞ ¼ cos 2 > 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 sin ¼ : 1

> 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<

2f0 : j0 j

ð6:201Þ

292 Sonar signal processing

[Ch. 6

The instant of stationary phase is to ¼

f f0 ; f þ f 0 0

ð6:202Þ

from which it follows that

and

2f 20 f þ f0 f

ðto Þ ¼ loge 0 f0 f þ f0

ð6:203Þ

20 f þ f0 2

ðto Þ ¼ ¼ 20 : f0 ð1 0 to =f0 Þ 2

ð6:204Þ

00

The frequency spread is the change in instantaneous frequency during the time interval deﬁned by eff < t < þ eff : ð6:205Þ 2 2 For an HFM pulse, the instantaneous frequency at times eff =2 is f ¼ f0 where ¼

; 1

0 eff : 2f0

ð6:206Þ ð6:207Þ

Therefore the change in frequency in this time interval (assuming eff to be less than 2f0 =j0 j) is fþ f ¼ 2f0 ; ð6:208Þ 1 2 the magnitude of which is the frequency spread B ¼ 2f0 19

j j : 1 2

ð6:209Þ

The spectrum is

f Mð f Þ 0 f þ f0

sﬃﬃﬃﬃﬃﬃﬃ 1 aðt Þ e i o ð f Þ Pð f Þ; j0 j o

ð6:210Þ

where o ð f Þ is deﬁned as the phase at the moment of stationary phase o ð f Þ ðto Þ; such that o ð f Þ ¼ 2

f0 f þ f0 f0 loge f : 0 f0

ð6:211Þ ð6:212Þ

19 A similar result, except with jPð f Þj approximated by a rectangle function, is derived by Kroszczyn´ski (1969, Eq. (32a)).

Sec. 6.2]

6.2 Processing gain for active sonar 293

The function Pð f Þ is given by Equation (6.179), with pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f þ f0 T f u ¼ 2j0 j f 0 2 0 and s ¼ sgn 0 :

ð6:213Þ ð6:214Þ

6.2.2.3.1 Gaussian envelope Application of the stationary phase method for a Gaussian envelope gives the result ! h f0 2 1=4 t 2o i pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 2 Mð f Þ exp i o ð f Þ þ si ; ð6:215Þ f þ f0 j0 jeff 4 eff where to is the instant of stationary phase, given by Equation (6.202). 6.2.2.3.2 Rectangular envelope For a rectangular envelope of duration T, the instantaneous frequency runs from f at T=2 to fþ at þT=2, and Equation (6.210) becomes Mð f Þ

f0 e i o ð f Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pð f Þ; f þ f0 j0 jT

ð6:216Þ

where o ð f Þ is given by Equation (6.212). The variables u and s, needed for Pð f Þ through Equation (6.179), are given by Equations (6.213) and (6.214). If the time bandwidth product is large enough, the Fresnel integrals may be approximated as step functions, in which case jPð f Þj is a rectangle function (see Equation 6.183). In other words f fc jPð f Þj P ; ð6:217Þ B where B and fc are the frequency spread (Equation 6.209) and center frequency fc

f þ fþ 2 ¼ f0 ; 2 1 2

ð6:218Þ

respectively, with from Equation (6.207). While the change in frequency during the pulse (Equation 6.208) can be positive or negative depending on the sign of 0 , for an HFM pulse the center frequency fc is always positive. The eﬀective bandwidth can be calculated from Equation (6.159) as eff ¼ f 40

ð þ1

20 T 2

:

ð6:219Þ

jPð f Þj 4 ð f þ f0 Þ 4 df

1

Use of Equation (6.217) simpliﬁes the evaluation of eff to give eff

12 2 : f 20 jð fþ þ f0 Þ 3 ð f þ f0 Þ 3 j

ð6:220Þ

294 Sonar signal processing

[Ch. 6

The start and end frequencies are given by Equation (6.206), so that f þ f0 ¼ f =

ð6:221Þ

and hence j0 jT

eff

1 þ 13 2

ð6:222Þ

:

6.2.2.3.3 Synthesis of HFM envelopes The procedure used previously for the LFM case (Section 6.2.2.2.3) can be applied also to HFM modulation, resulting in the equations HFM ¼ 2

0 t !t 1 0 t=f0

ð6:223Þ

and to ð f Þ ¼

f f0 : f þ f0 0

ð6:224Þ

The spectral and envelope amplitudes are SHFM ð f Þ and

f0 a½to ð f Þ ; f þ f0 a0

a0 0 t aHFM ðtÞ ¼ S : 1 0 t=f0 1 0 t=f0

ð6:225Þ

ð6:226Þ

The third column of Table 6.6 (labeled ‘‘HFM’’) is derived using Equation (6.226). 6.2.2.4

Hybrid spectra

A phase modulation suggested by Rosenbach and Ziegenbein (1993) is

ðtÞ t þ T=2 ðt þ T=2Þ þ1 ¼ þ 2b 2 ð þ 1ÞT

ðT=2 < t < þT=2Þ;

ð6:227Þ

where is a constant to be chosen in the interval ½0; 1 , with the extremes ¼ 0 and 1 corresponding to the CW and LFM cases, respectively. The idea is that can be tuned within this interval to explore the properties of modulations that are intermediate between CW and LFM, in order to obtain both range and frequency resolution with a single pulse. The time origin is chosen to coincide with the moment of zero instantaneous frequency. The pulse duration is T and the constant b is a measure of its bandwidth. The ﬁrst two time derivatives of the phase are

0 ðtÞ 1 ðt þ T=2Þ ¼ þ 2b 2 T

ð6:228Þ

00 ðtÞ T ¼ : 2b ðt þ T=2Þ 1

ð6:229Þ

and

Sec. 6.2]

6.2 Processing gain for active sonar 295

The instant of stationary phase is given by to f 1 1= 1 ¼ þ T b 2 2

ð6:230Þ

and hence Mð f Þ ðBÞ 1=2 where

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Tð f =b þ 1=2 Þ ð1Þ= aðto Þ e i o ð f Þ Pð f Þ;

o ð f Þ f f 1 ð1þÞ= ¼ þ b: 2T 2 1þ b 2

ð6:231Þ

ð6:232Þ

The function Pð f Þ depends on u and s through Equation (6.179). These are rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ j 00 ðto Þj T u ¼ to ð6:233Þ 2 and s sgn 00 ðto Þ ¼ sgn b;

ð6:234Þ

where

00 ðto Þ ¼

2b : T ð f =b þ 1=2 Þ ð1Þ=

ð6:235Þ

For the case of a rectangular envelope, the true and eﬀective pulse durations are equal. Thus, from Equation (6.228), the instantaneous frequency finst runs from f ¼ b=2

ð6:236Þ

fþ ¼ f þ b

ð6:237Þ

at time T=2 to

at time þT=2. Hence, the frequency spread and eﬀective bandwidth are B ¼ jbj

ð6:238Þ

eff ¼ ð2 ÞB:

ð6:239Þ

and

The LFM limit (Equation 6.227, with ¼ 1) is an optimum pulse in the sense that, for ﬁxed B, eff has a maximum of eff ¼ B:

ð6:240Þ

It is shown in Section 6.2.5 that, for a ﬁxed pulse duration, the SNR is proportional to bandwidth. Thus, maximizing the SNR requires setting to 1, which means that Doppler resolution can only be bought at the expense of a reduced SNR. A compromise value of ¼ 0:5 is suggested by Norrmann and Ziegenbein (1995).

296 Sonar signal processing

[Ch. 6

Table 6.7. Autocorrelation functions for CW and LFM pulses. Envelope

Autocorrelation function

Gaussian (Burdic, 1984)

Rectangular

CW 2 exp 2 2 eff

LFM 2 exp 2 ð1 þ 20 4eff Þ 2 eff

T jj ½Hð þ TÞ Hð TÞ T

T jj sinc½0 ðT jjÞ T (Russo and Bartberger, 1965)

6.2.3 6.2.3.1

Autocorrelation and cross-correlation functions and the matched ﬁlter Autocorrelation function

The autocorrelation function for the envelope ðtÞ is ð þ1 AðÞ ðtÞ ðt Þ dt:

ð6:241Þ

1

A useful property of the autocorrelation function is that it forms a Fourier transform pair with the power spectrum.20 Thus, jMð f Þj 2 ¼ I½AðÞ

ð6:242Þ

AðÞ ¼ I 1 ½jMð f Þj 2 :

ð6:243Þ

and Autocorrelation functions for simple (CW and LFM) pulses are listed in Table 6.7. For more complicated pulses, evaluation of Equation (6.241) becomes surprisingly diﬃcult, and an alternative is to use Equation (6.243) instead, with the stationary phase approximation for Mð f Þ. In particular, using the approximation of Equation (6.217) in Equation (6.189) gives 2 f fc jaðto Þj 2 P : ð6:244Þ jMð f Þj 2 00 j ðto Þj B To understand the nature of this approximation, consider its application to the LFM case, for which use of Equation (6.244) yields ( sincð0 TÞ rectangular ALFM ðÞ ð6:245Þ exp 20 2eff 2 Gaussian. 2 Comparison with the exact results of Table 6.7 suggests that the approximation 20 Hence the alternative name ‘‘autospectral density’’ for the power spectrum jMð f Þj 2 (see Section 6.2.2.2.1).

Sec. 6.2]

6.2 Processing gain for active sonar 297

requires that T be large compared with jj, implying a large bandwidth. Speciﬁcally, given that the width of AðÞ is of order 1=ðj0 jTÞ, this requirement amounts to j0 jT 2 1: Applying the same method to a rectangular HFM pulse gives ð !þ i! 2f 20 e AHFM ðÞ expð2if0 Þ d!; 2 0 T ! ! where ! ¼ 2ð f þ f0 Þ:

ð6:246Þ

ð6:247Þ ð6:248Þ

This can be written (cf. Lin, 1988) " !# if0 e i!þ e i! AHFM ðÞ expð2if0 Þ E1 ði! ÞE1 ði!þ Þþi ; ð6:249Þ !þ ! where the deﬁnition of the exponential integral E1 ðzÞ (see Appendix A) is such that, for an imaginary argument, the following relationship is satisifed ð b iu e du: ð6:250Þ E1 ðiaÞ E1 ðibÞ ¼ a u 6.2.3.2

Cross-correlation and the matched ﬁlter

Given two functions of time f ðtÞ and gðtÞ, their cross-correlation function is given (following Burdic, 1984—see also Appendix A) by ð þ1 Cf g ðÞ f ðtÞg ðt Þ dt: ð6:251Þ 1

The cross-correlation function is a measure of how similar the functions f and g are to each other. If they are similar in shape (though perhaps shifted by time t 0 ) the crosscorrelation function has a maximum at ¼ t 0 and not otherwise. If the two functions are identical, the result is the autocorrelation function C ðÞ ¼ AðÞ:

ð6:252Þ

The cross-correlation operation is of importance for sonar processing because it is used to separate a target echo from random noise in active sonar returns. This is done by cross-correlating the received signal with a delayed replica of the transmitted pulse. The expectation is that the echo will resemble the transmitted pulse, resulting in a peak in the cross-correlation output, whereas the noise will produce no such peak. This process is known as matched ﬁltering and the processor in which it is implemented is called a matched ﬁlter. The success of a matched ﬁlter relies on receiving an undistorted echo. 6.2.3.3

Doppler processing

A common source of echo distortion, and fortunately one that is straightforward to correct for, is a Doppler shift relative to the transmitted pulse. A Doppler-shifted

298 Sonar signal processing

[Ch. 6

echo, if cross-correlated with the original undistorted pulse, will suﬀer a reduction in the peak of the matched ﬁlter output, resulting in a loss in the ﬁlter’s ability to discriminate between signal and noise. However, if the echo is correlated instead with a Doppler-shifted replica with the same oﬀset in frequency, the full performance is recovered. This process of Doppler-shifting the replica is described below, following Russo and Bartberger (1965). The main eﬀect of target or sonar motion21 is a shift (denoted D ) in the carrier frequency of D ¼ ð 1Þf0 ; ð6:253Þ where cV ¼ ð6:254Þ cþV and V is the relative target velocity, deﬁned as the rate at which the distance between the target and sonar increases with time. The Doppler factor describes the compression or stretching of the time axis due to the Doppler eﬀect. If V=c is small then may be approximated as 2V c

ð6:255Þ

2V f: c 0

ð6:256Þ

1 and hence D

In general, the target Doppler is a priori unknown, so Doppler processing involves a bank of replica echoes covering a range of guessed Doppler factors, say fn g. Each replica can be written yreplica ðtÞ ¼ 1=2 n ðn tÞ expði!0 n tÞ:

ð6:257Þ

The Doppler factor stretches not just the carrier but also the envelope function ðtÞ. Because of this, to understand the impact of a Doppler shift on a cross-correlation receiver it is necessary to consider the full waveform yðtÞ given by Equation (6.148). In general, the Doppler-shifted echo can be written yecho ðtÞ ¼ 1=2 ðtÞ expði!0 tÞ: The cross-correlation function of the echo and replica is ð þ1 ð; ; n Þ yreplica ðtÞy echo ðt Þ dt:

ð6:258Þ

ð6:259Þ

1

Substituting Equation (6.257) and Equation (6.258) in the right-hand side gives ð þ1 1=2 1=2 i!0 ð; ; n Þ ¼ n e ðn tÞ ½ðt Þ e i!0 ðn Þt dt: ð6:260Þ 1 21 If the sonar transmitter and receiver move in unison, and assuming the water to be stationary, the Doppler shift is determined only by the relative motion between target and sonar.

Sec. 6.2]

6.2 Processing gain for active sonar 299

To understand the eﬀect of the mismatch it is suﬃcient to consider the special case n ¼ 1. Thus, the Doppler autocorrelation function (DACF) can be deﬁned as ð þ1 i!0 1=2 ð; Þ e ð; ; 1Þ ¼ ðtÞ ½ðt Þ e i!0 ð1ÞðtÞ dt: ð6:261Þ 1

The normalization of ðtÞ ensures that ð; Þ has a peak value when ¼ 0 and ¼ 1 of ð0; 1Þ ¼ 1: ð6:262Þ If the Doppler shift is small ( 1), the DACF may be approximated by putting ¼ 1 everywhere except in the ﬁnal (exponential) term in the integrand. The result, known as the ‘‘narrowband approximation’’, is ð þ1 NB ð; D Þ ¼ e 2iD ðtÞ ðt Þ e 2iD t dt: ð6:263Þ 1

Examples of narrowband DACFs for an LFM-modulated pulse are: NB ð; D Þ ¼ e iD

T jj sinc½ðD 0 ÞðT jjÞ T

for a rectangular envelope, and

NB ð; D Þ ¼ e iD exp 2 ½ 2 þ ðD 0 Þ 2 4eff 2 eff

(LFM, rect.)

ð6:264Þ

(LFM, Gauss)

ð6:265Þ

for a Gaussian one. In both cases the DACF for CW modulation is obtained in the limit of small 0 . For example, in the case of a Gaussian this is iD 2 2 4 NB ð; D Þ ¼ e exp 2 ð þ D eff Þ (CW, Gauss): ð6:266Þ 2 eff The applicability of the NB approximation is limited to low Doppler shift. If the low Doppler requirement is not met, the full BB DACF is needed, which for a rectangular envelope gets surprisingly complicated (Russo and Bartberger, 1965). For the Gaussian case the BB DACF (with LFM modulation) is 1=2

2 ð; Þ ¼ e 2iD exp 2 ½ 2 2 ð 2 iD 2eff Þ 2 ; ð6:267Þ eff where and , both complex variables, are given by ¼ þ 2

ð6:268Þ

¼ 1 þ i0 2eff :

ð6:269Þ

¼1

ð6:270Þ

¼ 2;

ð6:271Þ

and For zero Doppler shift, such that and

300 Sonar signal processing

[Ch. 6

Equation (6.267) simpliﬁes to ! 2 2 ð; 1Þ ¼ exp 2 jj ; 2 eff

ð6:272Þ

as given by Table 6.7. As an example, consider the BB DACF for a CW pulse, derived by putting ¼1

ð6:273Þ

¼ 2 þ 1

ð6:274Þ

and in Equation (6.267). The result is " !# 2 2 1=2 2 2 2 CW ð; Þ ¼ exp D eff 2 þ 2iD : eff 1 þ 2 ð1 þ 2 Þ

ð6:275Þ

The idea of BB processing for a CW pulse sounds like a contradiction, and is of interest only for fast targets, moving at a relative speed (i.e., range rate) comparable with that of sound. This is because only a very fast–moving object would cause a diﬀerence between the BB and NB calculations for this case. Thus, for a CW pulse, the requirement is more one of low Doppler than of narrow band. The NB approximation of Equation (6.266) follows if 1. The DACF is an important result. From it can be obtained the autocorrelation function (the zero Doppler case) AðÞ ¼ ð; 1Þ;

ð6:276Þ

the power spectrum (Equation 6.242) and the ambiguity function (see Equation 6.277). 6.2.4

Ambiguity function

The ambiguity function is deﬁned as the squared magnitude of the Doppler autocorrelation function22 Xð; Þ jð; Þj 2 :

ð6:277Þ

It provides a convenient way of representing the resolution properties of the pulse in delay Doppler space. For simplicity, attention is limited in the following to the socalled NB approximation XNB ð; D Þ jNB ð; D Þj 2 :

ð6:278Þ

The NB ambiguity volume is deﬁned as the integral of XNB over all time and frequency ð þ1 ð þ1 VNB ¼ XNB ð; D Þ d dD ð6:279Þ 1 22

1

Alternative deﬁnitions of the term ‘‘ambiguity function’’ are ð; Þ and jð; Þj.

Sec. 6.2]

6.2 Processing gain for active sonar 301

and this integral is equal to unity VNB ¼ 1: The BB version of Equation (6.279) is ð þ1 ð þ1 VBB ¼ Xð; Þ d dD ; 1

ð6:280Þ

ð6:281Þ

1

which satisﬁes the inequality (Rihaczek, 1967; Sibul and Titlebaum, 1981) VBB 1:

ð6:282Þ

Henceforth the subscripts NB and D are omitted, but they are implied wherever the notation Xð; Þ is used. 6.2.4.1

CW pulse

To illustrate the time and Doppler resolution properties of a CW pulse, consider the ambiguity function for a Gaussian envelope. This case is chosen because it has the particularly simple form Xð; Þ ¼ exp½ð 2 þ 2 Þ ;

ð6:283Þ

where and are dimensionless time delay and frequency (Doppler) shift variables deﬁned by ¼ =eff

ð6:284Þ

¼ eff :

ð6:285Þ

and A graph of the ambiguity function Xð; Þ, known as an ambiguity surface, is shown in Figure 6.11, converted to decibels. The width in delay and Doppler of this surface indicates the resolution of the pulse. In the – plane, a locus of equal ambiguity is a circle. In absolute time and frequency co-ordinates (say in the – plane), this circle becomes an ellipse, and the ellipse whose ambiguity is expðÞ is known as the ambiguity ellipse.23 This ellipse has width eff in the delay axis and 1=eff in Doppler, and area . Thus, a long CW pulse is well suited for discriminating between diﬀerent frequencies (i.e., target Doppler) but poorly able to measure target range (delay time). It is said that such a pulse has high ‘‘Doppler resolution’’ and low ‘‘range resolution’’. By convention the Doppler axis is often converted to speed V. If the target and platform speeds are small compared with the speed of sound, the Doppler and dimensionless frequency are related through the expression V

c : 2f0 eff

ð6:286Þ

23 The choice of expðÞ to deﬁne the ambiguity ellipse, from Burdic (1984), is not universal (see Russo and Bartberger, 1965 for other possibilities) but it is a convenient one for a Gaussian envelope.

302 Sonar signal processing

[Ch. 6

Figure 6.11. Generic ambiguity surface for Gaussian CW pulse, 10 log10 X plotted vs. ¼ eff and ¼ =eff . Lines of constant ambiguity are circles in these co-ordinates.

Similarly, the time delay is converted to a range oﬀset R. If the sound paths are assumed to be conﬁned to the horizontal plane, the range variable R and dimensionless time are related according to R

ceff : 2

ð6:287Þ

The eﬀect on the ambiguity surface of changing the pulse duration is shown in Figure 6.12. In R–V co-ordinates, the shape of the ellipse, though not its area, depends on the value of the pulse duration eff . Speciﬁcally, the ellipse is described by the equation R2 c2 þ ð f0 eff Þ 2 V 2 ¼ : 2 4 eff

ð6:288Þ

In R–V space, the width of the ellipse (its semi-axis length) is ceff =2 in the range direction and =ð2eff Þ in Doppler. The area of the ellipse is c 2 =4f0 , which for a center frequency of 1 kHz is about 1750 m 2 /s. Thus, to achieve a Doppler resolution of 1 m/s or better the best achievable range resolution is 1750 m, whereas a range resolution of 100 m would imply a Doppler resolution no better that 17.5 m/s.

Sec. 6.2]

6.2 Processing gain for active sonar 303

Figure 6.12. Ambiguity surfaces 10 log10 X plotted vs. V ðc=2f0 eff Þ and R ðceff =2Þ for Gaussian CW pulses with duration 0.5 s (upper) and 2.0 s (lower). Increasing the pulse duration increases the Doppler resolution at the expense of range resolution.

304 Sonar signal processing

6.2.4.2

[Ch. 6

LFM pulse

For an LFM pulse, still for a Gaussian envelope, the (NB) ambiguity function is24 Xð; Þ ¼ expf½ 2 þ ð 0 2eff Þ 2 g:

ð6:289Þ

This is a function of the dimensionless time and frequency variables as before, plus the frequency rate in the form 0 2eff . The eﬀect of changing the frequency rate is illustrated by Figure 6.13. Increasing the magnitude of this parameter turns the circle into an increasingly elongated ellipse in – space. As the ellipse elongates, it rotates at the same time, starting at 45 deg with its major axis along the line =eff ¼ eff , aligning itself eventually with the Doppler axis ( ¼ 0), meaning that high-bandwidth pulses have high range resolution and low Doppler resolution. The graph shows results for 0 0. For negative 0 , the ellipse rotates in the same manner but in the opposite direction. For suﬃciently large values of j0 j 2eff , eventually an ambiguity surface reminiscent of that of a short CW pulse is recovered (cf. Figure 6.12). Thus, range resolution is achieved at the expense of Doppler, so what is the advantage? The answer is an improved SNR associated with the high bandwidth. For a pulse of ﬁxed duration, the higher the bandwidth the greater the control over the shape of the waveform, which in turn gives greater discrimination over a random background, as explained in Section 6.2.5. The ambiguity ellipse (from Equation 6.289) can be written pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 sgnð0 Þ 2 1 þ 2 ¼ 1; ð6:290Þ where is the time bandwidth product eff eff :

ð6:291Þ

Equation (6.290) describes an ellipse whose axes are rotated through some angle

relative to the and axes. The rotation angle can be found by ﬁrst deﬁning rotated coordinates (; ) such that cos

sin ¼ : ð6:292Þ sin cos The ellipse is then, for Cartesian co-ordinates rotated at an arbitrary angle

A 2 þ B þ C 2 ¼ 1; where

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A ¼ 2 cos 2 þ sin 2 2 1 sin 2 ; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B ¼ ð1 2 Þ sin 2 2 2 1 cos 2 ;

and C ¼ 2 sin 2 þ cos 2 þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 sin 2 :

ð6:293Þ ð6:294Þ ð6:295Þ ð6:296Þ

24 The (BB) DACF for a rectangular envelope is given by Russo and Bartberger (1965). See also Lin (1988).

Sec. 6.2]

6.2 Processing gain for active sonar 305

Figure 6.13. Generic ambiguity surfaces for Gaussian LFM pulse. Increasing the frequency rate for a ﬁxed pulse duration increases range resolution at the expense of Doppler resolution. (The frequency rate increases anticlockwise from the upper left panel.)

The natural co-ordinates for an ellipse are those aligned with the axes of that ellipse. The rotation angle associated with this natural co-ordinate system is found by requiring the cross term to vanish (i.e., B ¼ 0 in Equation 6.293) so that 2 tan 2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 2 1

ð6:297Þ

With this rotation angle, the ellipse is given by 2 tan 2 þ 2 =tan 2 ¼ 1:

ð6:298Þ

The area of this ellipse is equal to , irrespective of the rotation angle . 6.2.4.3

HFM pulse

The Doppler autocorrelation function for an HFM pulse25 is calculated by Lin (1988). An important property of HFM processing is that it is robust to small Doppler shifts, making it easier to maintain performance against a moving target. 25

For a rectangular envelope.

306 Sonar signal processing

[Ch. 6

Table 6.8. Derivation of matched ﬁlter gain for pulse duration T and sample interval T. The function f ðtÞ has unit amplitude and mean square value 12. xðtÞ

Signal sðtÞ

Random noise (or reverberation) SNR

Input ð x 2 dt

af ðt 0 Þ

a2 T 2

nðtÞ

n2T

Output (squared magnitude of cross-correlation peak) jCð0 Þj 2 2 aT 2

2n 2

T 2

T 2

T 2

a2 T 2n 2 T

T T

n2T

a2

Gain ¼ output/ input

This is because the eﬀect of a small shift in frequency can be approximated by a time delay. The price paid for this robustness is a reduced Doppler resolution and a small error in range estimation. 6.2.5

Matched ﬁlter gain for perfect replica

The gain due to a matched ﬁlter is equal to the SNR at the output of the ﬁlter divided by that at the input, expressed either as a ratio or in decibels. This gain is a measure of the ﬁlter’s ability to reject noise that does not resemble the transmitted waveform. In order to quantify this gain, the signal and background are given both before and after the matched ﬁlter in Table 6.8 for a rectangular pulse of duration T. The gain in SNR is MG 10 log10 G;

ð6:299Þ

where G¼

T : T

ð6:300Þ

If samples are taken at the Nyquist rate for bandwidth B (such that T ¼ 1=B), the gain is then G ¼ BT: ð6:301Þ There are some practical considerations that place an upper limit on the achievable value of the BT product as follows: — The bandwidth of the transducer limits the maximum achievable value of B. — Many sonars are not able to receive sound during the time they are transmitting, resulting in a blind period that lasts at least as long as the transmitted pulse. The duration of such a blind period would limit the maximum achievable value of T, depending on the required (minimum) detection range.

Sec. 6.2]

6.2 Processing gain for active sonar 307

One way of understanding the origin of the matched ﬁlter gain is by means of the following thought experiment, involving a reversal of the propagation and matched ﬁlter operations. Imagine that, instead of an FM pulse, a sonar were to transmit the pﬃﬃﬃﬃﬃﬃﬃ autocorrelation function of that pulse, with its amplitude increased by a factor BT with respect to the original pulse, in order to maintain the same total transmitted energy.26 The noise level is unaﬀected by this change because the original pulse and its autocorrelation function have the same bandwidth. The reverberation level would also be unchanged because the reduced scattering area caused by the shorter duration precisely balances the increased intensity of the transmitted pulse. Thus, the SNR increases by a factor BT, entirely due to the increased signal intensity. In practice, the correlation ﬁlter is invariably applied after reception in order to minimize the amplitude of the transmitted pulse for a given signal energy.

6.2.6

Matched ﬁlter gain for imperfect replica (coherence loss)

However large the value of BT, the theoretical gain is achieved only if the echo is an identical replica of the transmitted pulse, or departs from it in a known or predictable way (a Doppler shift is an example of a predictable departure). Unpredictable departures result in a reduction in the processing gain known as coherence loss. Possible causes of coherence loss include: — multipaths (e.g., due to target highlights or multiple boundary reﬂections); — short-term ﬂuctuations in the environment, causing changes in the propagation conditions on a timescale shorter than the duration of the pulse and thereby distorting its shape; — scattering from a rough boundary; — changes to the pulse shape due to dispersion (frequency-dependent sound speed or attenuation, frequency-dependent target spectrum, or frequency-dependent propagation loss) Coherence loss can be illustrated by means of a simple example involving two scaled replicas of the transmitted pulse, identical in every respect except for their amplitudes and arrival times. If the replicas arrive with delay times 1 ; 2 and amplitudes a; b, the signal can be written sðtÞ ¼ af ðt 1 Þ þ bf ðt 2 Þ:

ð6:302Þ

The precise form of both input (s 2 ) and output (jCð0 Þj 2 ) depend in general on the time separation. Results are presented in Table 6.9 for the case of small separation compared with the duration of the original pulse (T) and large compared with that of the compressed one (1=B). Coherence loss CL for this situation (two scaled replicas in 26 In the thought experiment, the reversal of the order of propagation and cross-correlation means that the cross-correlation output becomes the autocorrelation function of the actual transmitted pulse.

308 Sonar signal processing

[Ch. 6

Table 6.9. Eﬀect of multipath on matched ﬁlter gain. sðtÞ

Input s2

Gain

2

GSNR

Signal

af ðt 1 Þ þ bf ðt 2 Þ

a2 þ b2 2

jCð0 Þj 2 aT 2

Noise

nðtÞ

n2

2 T n 2

SNR a

Output a

a2 þ b2

a2

2n 2

2 Tn 2

T

T 2 a2 2 a2 þ b2 T T 2 T a2 T a 2 þ b 2

Squared magnitude of cross-correlation peak.

random noise) may be estimated as CL 10 log10

T 10 log10 GSNR ; T

ð6:303Þ

where GSNR is the SNR gain from Table 6.9. Hence, adopting the convention that jaj jbj a2 þ b2 CL ¼ 10 log10 : ð6:304Þ a2 In this situation, the worst case degradation, which arises when the replicas have equal amplitude, is 3 dB. More generally, the worst case coherence loss for N identically shaped replicas that are closely spaced in time (but resolved after compression) is 10 log10 N.

6.2.7

Array gain and total processing gain (active sonar)

A modern sonar processing chain is likely to incorporate both beamforming and matched ﬁltering. If the output SNR (after both processes) is Rout , the combined processing gain is PG 10 log10

Rout ¼ AG þ MG; Rhp

ð6:305Þ

where Rhp is the input SNR at the hydrophone. As both operations are linear ones, the output of the combined beamformer plus matched ﬁlter processing does not depend on the order in which the individual operations are carried out. Assuming arbitrarily that the beamformer comes ﬁrst the array gain is R AG ¼ 10 log10 arr ; ð6:306Þ Rhp

Sec. 6.3]

6.3 References 309

where Rarr is the SNR at the beamformer output, which in turn is the input SNR for the matched ﬁlter. Thus, the matched ﬁlter gain is MG ¼ 10 log10

Rout : Rarr

ð6:307Þ

For active sonar the array gain (and hence also the total processing gain) depends on whether the background is dominated by noise or reverberation, which depends on the processing applied and the distance to the target. A general rule is that a beamformer is more eﬀective (has a larger AG) against ambient noise than against reverberation, because the noise tends to originate in directions other than that of the target. An exception to this rule is the use of a monostatic sonar with a horizontal receiving array and an omni-directional transmitter, for which the gain is comparable for both noise and reverberation.

6.3

REFERENCES

Barger, J. E. (1997) Sonar systems, in M. J. Crocker (Ed.), Encyclopedia of Acoustics (pp. 559– 579), Wiley, New York. Burdic, W. S. (1984) Underwater Acoustic Systems Analysis, Prentice-Hall, Englewood Cliﬀs NJ. Cheston, T. C. and Frank, J. (1990) Phased array radar antennas, in M. Skolnik (Ed.), Radar Handbook (Second Edition, pp. 7.1–7.82), McGraw-Hill, New York. Crocker, M. J. (Ed.) (1997) Encyclopedia of Acoustics, Wiley, New York. Farnett, E. C. and Stevens, G. H. (1990) Pulse compression radar, in M. Skolnik (Ed.), Radar Handbook (Second Edition, pp. 10.1–10.39), McGraw-Hill, New York. Harris, F. J. (1978) On the use of windows for harmonic analysis with the discrete Fourier transform, Proc. IEEE, 66(1) 51–83. Kroszczyn´ski, J. J. (1969) Pulse compression by means of linear-period modulation, Proc. IEEE, 57(7), 1260–1266. Lin, Zhen-biao (1988) Wideband ambiguity function of broadband signals, J. Acoust. Soc. Am., 83, 2108–2116. McDonough, R. N. and Whalen, A. D. (1995) Detection of Signals in Noise (Second Edition), Academic Press, San Diego. Norrmann, J. and Ziegenbein, J. (1995) Investigation of the ambiguity function of a special kind of sonar signals, Proc. IOA, 17(8), 259–268. Nuttall, A. H. (1981) Some windows with very good sidelobe behavior, IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSSP-29(1). Proakis, J. G. (1995) Digital Communications (Third Edition), McGraw-Hill, Boston. Rihaczek, A. W. (1967) Delay-Doppler ambiguity function for wideband signals, IEEE Transactions on Aerospace and Electronic Systems, AES-3(4), 705–711. Rosenbach, K. and Ziegenbein, J. (1993) About the eﬀective Doppler sensitivity of certain nonlinear chirp signals (NLFM), paper presented at the LFAS Symposium, SACLANTCEN, May 1993, p. H/24. Russo, D. M. and Bartberger, C. L. (1965) Ambiguity diagram for linear FM sonar, J. Acoust. Soc. Am., 38, 183–190.

310 Sonar signal processing

[Ch. 6

Sibul, L. H. and Titlebaum, E. L. (1981) Volume properties of the wideband ambiguity function, IEEE Transactions on Aerospace and Electronic Systems, AES-17(1), 83–87. Skolnik, M. (Ed.) (1990) Radar Handbook (Second Edition), McGraw-Hill, New York. Tucker, D. G. and Gazey, B. K. (1966) Applied Underwater Acoustics, Pergamon, Oxford. Woodward, P. M. (1964) Probability and Information Theory with Applications to Radar (Second Edition), Pergamon Press, Oxford.

7 Statistical detection theory

While the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will be up to, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician. Arthur Conan Doyle (1890)1

Natural statistical ﬂuctuations in both signal and noise mean that it is not possible to state with certainty what particular signal-to-noise ratio (SNR) will result in a successful detection. Instead, it is necessary to consider the likelihood of an event occurring in percentage terms. Thus, in this chapter we deal in the currencies of probabilities of detection and of false alarm. The ﬁrst problem considered, in Section 7.1, is the probability of detecting a single pulse of known shape as a function of the SNR and the probability of false alarm. In Section 7.2 the results are generalized to the reception of multiple pulses whose shape is still known, but whose initial phase varies randomly from one pulse to the next. Much of detection theory applied to sonar was developed originally for radar, and the material presented in Sections 7.1 and 7.2 is largely based on a book entitled Radar Detection (DiFranco and Rubin, 1968). The language used might sometimes convey the impression that the application is primarily for active sonar, but this is not intended. Some derivations can be found, where not provided here, in the various source references—primarily DiFranco and Rubin (1968) and McDonough and Whalen (1995).2 Some readers will prefer to skip these relatively 1 These words, from The Sign of Four, are spoken by the ﬁctional character Sherlock Holmes, misquoting Winwood Reade. 2 See also Rice (1948) and Kay (1998).

312 Statistical detection theory

[Ch. 7

mathematical sections initially and jump instead to Section 7.3, which explains how to apply the main results to passive and active sonar, before reading the relevant sections in more detail. The methods and results of Sections 7.1–7.3 can be said to describe a single observation or ‘‘look’’. If a given observation is repeated under the same conditions, the probability of detection for the second observation is unchanged, but the information from the two observations can be combined, at least in principle, in such a way as to improve the overall performance. The eﬀect on detection performance of combining information from multiple observations is the subject of Section 7.4.

7.1

SINGLE KNOWN PULSE IN GAUSSIAN NOISE, COHERENT PROCESSING

Consider a narrowband signal pulse with amplitude AS at the receiver and duration Dt, such that the signal pressure (or voltage) can be represented by the function sðtÞ ¼ AS sinð!t þ Þ;

ð7:1Þ

0 < t < Dt:

Adding random noise nðtÞ, the total received signal plus noise becomes rSþN ðtÞ sðtÞ þ nðtÞ ¼ AS sinð!t þ Þ þ nðtÞ;

0 < t < Dt:

ð7:2Þ

It is assumed that the continuous function rSþN ðtÞ is sampled at discrete times ti and that the individual noise samples nðti Þ follow a Gaussian distribution. The term , equal to the phase at initial time t ¼ 0, is assumed to be constant for all samples within the pulse. Its value is taken from a random uniform distribution in ½0; 2. The signal amplitude (AS ) is also assumed constant within the pulse, with the value of this constant assumed either ﬁxed (Section 7.1.2.1) or taken from one of three random distributions speciﬁed in Sections 7.1.2.2 to 7.1.2.4. The special case with no signal (AS ¼ 0) is considered ﬁrst, in Section 7.1.1, in which the false alarm probability for Gaussian noise is derived. In Section 7.1.2, detection probability is given as a function of SNR for a sinusoidal signal with various amplitude distributions. The case of a more general, but still known pulse shape is considered in Section 7.1.4.

7.1.1

False alarm probability for Gaussian-distributed noise

False alarm probability is deﬁned as the probability that a given amplitude threshold is exceeded in the absence of any signal. Thus, the distribution of interest is that of the noise term rN ðtÞ ¼ nðtÞ ð7:3Þ alone. The assumption of a (zero-mean) Gaussian distribution for nðtÞ means that the probability density function (pdf ) of the noise amplitude distribution, after coherent

Sec. 7.1]

7.1 Single known pulse in Gaussian noise, coherent processing 313

processing,3 is (McDonough and Whalen, 1995) ! fRayleigh ðA=Þ A A2 fN ðAÞ ¼ 2 exp 2 ¼ 2

ðA > 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 kHz)

If the frequency is suﬃciently low, the layering in the uppermost meter or so of sediment may be ignored. Instead it becomes necessary to consider deeper layering on a depthscale of 10 m to 100 m, depending on the acoustic frequency. It might also be necessary to consider interaction with the solid rock beneath the sediment layer. Before proceeding to a layered model, it is useful to consider the reﬂective properties of a half-space of solid rock. Figure 8.6 shows reﬂection loss vs. angle for four representative rock types from Chapter 4. The curves for sandstone and limestone (not shown) are similar to the one for basalt. As in the case of an unconsolidated sediment previously, the shape of each of these curves depends on the corresponding values of vp , except that here the requirement for small vs is lifted. Consider ﬁrst the case vs < 1, corresponding to the two softest rocks, chalk and mudstone. For these two cases there exists a critical angle (denoted p ) at arccosð1=vp Þ, equal to 52 deg and 43 deg for chalk and mudstone, respectively. Because of the low shear speed, a shear wave is always generated in the rock, irrespective of the grazing angle in water, so that total internal reﬂection is not possible. The loss of energy associated with this mechanism, proportional to v 3s (see Equation 8.96), can be very high.5 For the remaining rock types, all satisfying vs > 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 <1

GðB; Þ ¼ 2 : 1 þ 125 exp 2:64ðB 1:75Þ 8 1 > < 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

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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).

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Sec. 8.4]

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Norton, G. V. and Novarini, J. C. (2001) On the relative role of sea-surface roughness and bubble plumes in shallow-water propagation in the low-kilohertz region, J. Acoust. Soc. Am., 110, 2946–2955. Nystuen, J. A. (2001) Listening to raindrops from underwater: An acoustic disdrometer, J. Atmospheric and Oceanic Technology, 18(10), 1640–1657. Ogden, P. M. and Erskine, F. T. (1994) Surface scattering measurements using broadband explosive charges in the Critical Sea Test experiments, J. Acoust. Soc. Am., 95, 746–761. Pace, N. G. and Blondel P. (eds.) (2005) Boundary Inﬂuences in High Frequency, Shallow Water Acoustics, University of Bath, Bath, U.K., September 5–9. Patek, S. N., Shipp, L. E., and Staaterman, E. R. (2009) The acoustics and acoustic behavior of the California spiny lobster (Panulirus interruptus), J. Acoust. Soc. Am., 125, 3434–3443. Pauly, T. and Penrose, J. D. (1998) Laboratory target strength measurements of free-swimming Antarctic krill (Euphausia superba), J. Acoust. Soc. Am., 103, 3268–3280. Pouliquen, E. and Lurton, X. (1994) Identiﬁcation de la nature du fond de la mer a` l’aide de signaux d’e´cho-sondeurs, I: Mode´lisation d’e´chos re´verbe´re´s par le fond, Acta Acustica, 2, 113–126. Richardson, W. J., Greene, C. R., Malme, C. I., and Thomson, D. H. (1995) Marine Mammals and Noise, Academic Press, San Diego. Robins, A. J. (1991) Reﬂection of a plane wave from a ﬂuid layer with continuously varying density and sound speed, J. Acoust. Soc. Am., 89, 1686–1696. Ross, D. (1976) Mechanics of Underwater Noise, Pergamon Press, New York. Ross, D. (2005) Ship sources of ambient noise, Proc. International Workshop on Low-Frequency Propagation and Noise, October. [Reprinted in IEEE J. Ocean. Eng., 30, 257–261 (2005).] Schmitz, B. (2002) Sound production in crustacea with special reference to the Alpheidae, in K. Wiese (Ed.), The Crustacean Nervous System (pp. 536–547), Springer Verlag, Berlin. Scrimger, P. and Heitmeyer, R. M. (1991) Acoustic source level measurements for a variety of merchant ships, J. Acoust. Soc. Am., 89, 691–699. Scrimger, J. A., Evans, D. J., McBean, G. A., Farmer, D. M., and Kerman, B. R. (1987) Underwater noise due to rain, hail, and snow, J. Acoust. Soc. Am., 81, 79–86. Scrimger, J. A., Evans, D. J., and Yee, W. (1989) Underwater noise due to rain: Open ocean measurements, J. Acoust. Soc. Am., 85, 726–731. Simmonds, E. J. and MacLennan, D. N. (2005) Fisheries Acoustics (Second Edition), Blackwell, Oxford. Simons, D. G., Snellen, M., and Ainslie, M. A. (2007) A multivariate correlation analysis of high frequency bottom backscattering strength measurements with geo-technical parameters, IEEE J. Oceanic Eng., 32, 640–650. 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. Stanton, T. K., Chu, D., Wiebe, P. H., Martin, L. V., and Eastward, R. L. (1998) Sound scattering by several zooplankton groups, I: Experimental determination of dominant scattering mechanisms, J. Acoust. Soc. Am., 103, 225–235. Sternlicht, D. D. and de Moustier, C. P. (2003) Time-dependent seaﬂoor acoustic backscatter (10–100 kHz), J. Acoust. Soc. Am., 114, 2709–2725. Stockhausen, J. H. (1963) Scattering from the Volume of an Inhomogeneous Half-space (Report No. 63/9), Naval Research Establishment, Dartmouth, Canada. Taylor, B. N. (1995) Guide for the Use of the International System of Units (SI) (NIST Special Publication 811, 1995 Edition), U.S. Department of Commerce/National Institute of Standards & Technology.

438

Sources and scatterers of sound

[Ch. 8

Thorne, P. D. (1986) Laboratory and marine measurements on the acoustic detection of sediment transport, J. Acoust. Soc. Am., 80, 899–910. Tollefsen, D. (1998) Thin-sediment shear-induced eﬀects on low-frequency broadband acoustic propagation in a shallow continental sea, J. Acoust. Soc. Am., 104, 2718–2726. Trevorrow, M. V., Vagle, S., and Farmer, D. M. (1994) Acoustical measurements of microbubbles within ship wakes, J. Acoust. Soc. Am., 95, 1922–1930. Trevorrow, M. V., Mackas, D. L., and Benﬁeld, M. C. (2005) Comparison of multifrequency acoustic and in situ measurements of zooplankton abundances in Knight Inlet, British Columbia, J. Acoust. Soc. Am., 117, 3574–3588. Trevorrow, M. V., Vasiliev, B., and Vagle, S. (2008) Directionality and maneuvering eﬀects on a surface ship underwater acoustic signature, J. Acoust. Soc. Am., 124, 767–778. Urick, R. J. (1954) The backscattering of sound from a harbor bottom, J. Acoust. Soc. Am., 26, 231–235. Urick, R. J. (1983) Principles of Underwater Sound (Third Edition), Peninsula, Los Altos. Uscinski, B. J. and Wadhams, P. (1999) Ice-ocean acoustic energy transfer: Ambient noise in the ice-edge region, Deep-Sea Research II, 46, 1319–1333. Versluis, M., Schmitz, B., von der Heydt, A., and Lohse, D. (2000). How snapping shrimp snap: Through cavitation bubbles, Science, 289, 2114–2117. Wales, S. C. and Heitmeyer, R. M. (2002) An ensemble source spectra model for merchant ship-radiated noise, J. Acoust. Soc. Am., 111, 1211–1231. Warren, J. D., Stanton, T. K., Benﬁeld, M. C., Wiebe, P. H., Chu, D., and Sutor, M. (2001) In situ measurements of acoustic target strength of gas-bearing siphonophores, ICES Journal of Marine Science, 58, 740–749. Webb, S. C. (1992) The equilibrium oceanic microseism spectrum, J. Acoust. Soc. Am., 92, 2141–2158. Wenz, G. M. (1962) Acoustic ambient noise in the ocean: Spectra and sources, J. Acoust. Soc. Am., 34, 1936–1956. Weston, D. E. (1971) Intensity-range relationships in oceanographic acoustics, J. Sound Vib., 18, 271–287. Weston, D. E. (1995) Assessment Methods for Biological Scattering and Attenuation in Ocean Acoustics (BAeSEMA Report C3307/7/TR-1, April), BAeSEMA, Esher, U.K. Weston, D. E. and Ching, P. A. (1989) Wind eﬀects in shallow-water transmission, J. Acoust. Soc. Am., 86, 1530–1545. Wiese, K. (Ed.) (2002) The Crustacean Nervous System, Springer Verlag, Berlin. Wille, P. (1986) Landolt–Bo¨rnstein Acoustical Properties of the Ocean (Group V: Geophysics, Vol. 3a: Oceanography and Space Research, pp. 265–382), Springer-Verlag, Berlin. Williams, K. L. and Jackson, D. R. (1998) Bistatic bottom scattering: Model, experiments, and model/data comparison, J. Acoust. Soc. Am., 103, 169–181. Williams, K. L., Jackson, D. R., Thorsos, E. I., Tang D., and Briggs, K. B. (2002) Acoustic backscattering in a well characterized sand sediment: Data/model comparisons using sediment ﬂuid and Biot models, IEEE J. Oceanic Eng., 27, 376–387. Wilson, J. D. and Makris, N. C. (2006) Ocean acoustic hurricane classiﬁcation, J. Acoust. Soc. Am., 119, 168–181. Wilson, J. H. (1983) Wind-generated noise modeling, J. Acoust. Soc. Am., 73, 211–216. Yasuma, H., Sawada, K., Ohshima, T., Mihashita, K., and Aoki, I. (2003) Target strength of mesopelagic lanternﬁshes (family Myctophidae) based on swimbladder morphology, ICES J. Marine Sci., 60, 584–591.

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

[Ch. 9

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]

10.1 Transmitter characteristics

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]

10.1 Transmitter characteristics

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<0 t

pﬃﬃﬃ 2A sinð!tÞ exp t 0 2 2A 2 sin 2 2 exp 2 !

where 2 ¼ arcsin

Exponential (

0

t

A exp

t<0 t 0

A2

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !2 2 1 þ !2 2

ðS0 Þp-p

h i2 ðS0 Þz-p 1 þ exp !

A2

ðS0 ÞE

A2 !2 2 2 1 þ !2 2

A2 2

averaged over a time interval that is often left unspeciﬁed, making the reported values ambiguous. Madsen (2005) discusses this problem and suggests two qualitatively diﬀerent ways of choosing an appropriate averaging time. The ﬁrst and simplest method is based on an amplitude threshold. With this approach, pulse duration is deﬁned as the time during which a speciﬁed threshold (e.g., 3 dB relative to the maximum amplitude) is exceeded. The MSP averaged over this time interval is denoted Xð Þ in Table 10.18, expressed as a fraction of the maximum MSP value,2 where is the chosen threshold, such that ¼ 12 corresponds to the 3 dB pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ point. The RMS pressure calculated using a zero dB threshold (i.e., A Xð1:0Þ), is also known as the peak-equivalent RMS (peRMS) pressure. The peRMS pressure is the RMS pressure of a hypothetical sine wave whose peak sound pressure is equal to the peak sound pressure of the true pulse. In Table 10.18 and for the remainder of the present section it is assumed that the frequency is high enough for the sinesquared terms to be replaced by their mean value of 12. The ﬁrst of the three cases considered is the trivial one involving a top-hat or unweighted pulse, followed by the Gaussian-weighted pulse from Table 10.16 and the exponentially modulated sine wave from Table 10.17.

2

For oscillatory pulses, this statement involves an implied assumption that the frequency is high enough for the ‘‘mean square’’ operation to be over several cycles. In the case of exponentially decaying acoustic pressure the MSP is equal to the square of instantaneous pressure.

532 Transmitter and receiver characteristics

[Ch. 10

Figure 10.3. Exponentially damped sine (upper) and decaying exponential (lower) pulses from Table 10.17.

Sec. 10.1]

10.1 Transmitter characteristics

533

Table 10.18. Relative MSP, denoted Xð Þ, deﬁned as the mean square pressure relative to its maximum value (peRMS). The average is over the time window during which the local average (mean square averaged over a small number of cycles) exceeds the threshold . The parameter x used in the table is given by xð Þ ¼ loge . Pulse description

Unweighted sine or cosine (Table 10.16) 1 2

Gaussian-weighted sine (Table 10.16) Exponentially damped sine (Table 10.17)

Xð Þ

Xð1:0Þ [0 dB] peRMS

Xð0:5Þ [3.0 dB]

Xð0:1Þ [10 dB]

1

1

1

1

1.000 (0 dB)

0.810 (0.9 dB)

0.565 (2.5 dB)

1.000 (0 dB)

0.721 (1.4 dB)

0.391 (4.1 dB)

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ erf xð Þ xð Þ 1 xð Þ

Applying the concepts leading to Table 10.18, the 0 dB and 10 dB values are related, for the exponentially damped sinusoidal pulse, by SL10 dB ¼ SL0 dB 4:1 dB;

ð10:9Þ

where the subscript indicates the amplitude threshold, and SL0 dB SLpeRMS ¼ SLp-p 9:0

dB:

ð10:10Þ

In this case, the diﬀerence between SL10 dB and SLp-p is therefore 13.1 dB. The second method described by Madsen (2005) for choosing the averaging time is to deﬁne the duration as the time interval during which a speciﬁed proportion (say 90 %) of the total pulse energy arrives. Table 10.19 lists the MSP calculated in this way, denoted YðRÞ, as a function of the chosen energy fraction R, for Gaussianweighted and exponentially weighted pulses as well as for the trivial unweighted case. In this second table a fourth pulse shape is added (bottom row), made by combining a rising exponential for t < 0 with a decaying one for t > 0. For the exponentially damped sine wave, the MSP based on R ¼ 0 is 3.0 dB relative to the peRMS value, which in turn is 9.0 dB less than the p-p value. Thus, for this case ð10:11Þ SL0% ¼ SLp-p 12:0 dB: Further, the MSP based on 0 % and 97 % energy fractions diﬀer by 3.3 dB, which means that (for the same exponential damping) SL97% ¼ SLp-p 15:3 dB:

ð10:12Þ

It is clear from either Table 10.18 or Table 10.19 that, for any of the pulses except the trivial unweighted one, if there is any leeway in choosing the averaging time the result is an undesirable ambiguity in the RMS pressure of up to 6.3 dB for the examples considered. Larger diﬀerences are possible if longer averaging times are used, up to a

534 Transmitter and receiver characteristics

[Ch. 10

Table 10.19. Relative MSP, denoted YðRÞ, deﬁned as the mean square pressure relative to its maximum value (peRMS). The average is over the time window during which the pulse energy accumulates to a proportion R of the total. Thus, the notation YðRÞ means that a proportion ð1 RÞ=2 of the total energy arrives before the start (and the same proportion arrives after the end) of the averaging window. Description Unweighted sine or cosine Gaussianweighted sine

YðRÞ

Yð0:00Þ

Yð0:50Þ

Yð0:90Þ

Yð0:97Þ

YðYÞ

1

1

1

1

1

1

pﬃﬃﬃ R 2 erf 1 R

1.000 (0 dB)

pﬃﬃﬃ 0.929 0.686 0.560 erf 0:790 2 (0.3 dB) (1.6 dB) (2.5 dB) (1.0 dB)

R Exponentially 0.500 0.455 0.306 0.232 log ½ð1 þ RÞ=ð1 RÞ (3.0 dB) (3.4 dB) (5.1 dB) (6.3 dB) damped sine e Double-sided exponentially weighted sine

R jloge ð1 RÞj

1.000 (0 dB)

0.721 0.390 0.277 (1.4 dB) (4.1 dB) (5.6 dB)

e1 0:462 eþ1 (3.4 dB) e1 0:632 e (2.0 dB)

maximum diﬀerence when the averaging time reaches the pulse repetition interval. For this reason, any quantitative statement of an RMS pressure for a transient sound is incomplete unless it is accompanied by a description of the averaging time used. The preceding discussion begs the question of which MSP deﬁnition is used, or should be used, in practice. The answer depends on circumstances. The choice Xð Þ has the beneﬁt of simplicity. The disadvantage is the ambiguity caused by possible non-monotonic behavior to either side of the chosen peak. Further, an amplitude threshold takes no account of the energy present in the low-amplitude region, which might sometimes be a signiﬁcant proportion of the total. Use of the energy fraction solves both of these problems, but requires a measurement of total pulse energy, resulting in a need for more complicated processing. Given a value of R, the deﬁnition in terms of YðRÞ is unambiguous, but suﬀers from the problem that, for lopsided pulses like the exponential one, the peak value might be excluded from the averaging time. This concern can be mitigated by choosing the value of R satisfying YðRÞ ¼ R, forcing a short averaging time for the (single-sided) exponential pulse, while requiring a longer time for the symmetrical pulses. The result of this choice, denoted YðYÞ, is shown in the last column of Table 10.19, and illustrated by Figure 10.4. 10.1.1.2.2

Seismic survey sources

The term ‘‘seismic survey’’ sources refers to a general category of low-frequency sources designed to transmit sound deep into the seabed, with the objective of determining its structure or layering. Compared with a search sonar designed to

Sec. 10.1]

10.1 Transmitter characteristics

535

Figure 10.4. MSP Y vs. energy fraction R. The various functions YðRÞ from Table 10.19 are plotted. The intersections with the diagonal deﬁne YðYÞ. The parameter YðYÞ is less sensitive to the precise pulse shape than is YðRÞ for ﬁxed R.

transmit sound horizontally, a relatively small proportion of the acoustic energy from seismic survey sources remains in the water column. For an overview of seismic sources see usgs (www). The choice of frequency is a compromise between the desire for high resolution (i.e., high bandwidth and hence some energy at high frequency) and the need for low attenuation (i.e., low frequency). The solution adopted is typically a BB signal with most energy at order 100 Hz (Caldwell and Dragoset, 2000). Some early survey work was done with explosive sources (Hersey, 1977), but this is now rare and the high farﬁeld pressures required are achieved instead by means of large air gun arrays, which work by injecting compressed air into the water. The air bubbles so formed expand rapidly, creating a high-amplitude pressure pulse in the required frequency range. The properties of air guns are summarized in Table 10.20. These are dipole source levels in the sense that they are based on the combined eﬀect of the air gun itself and its sea surface image, resulting in a dipole radiation pattern at low frequency. Source levels are often reported in the seismic literature in the form of a source product (pressure multiplied by distance) in units of bar meters (bar m). The bar is a unit of pressure equal to 100 kPa. Thus, a bar meter can be converted to the present units by means of the relation 1 bar m ¼ 10 5 Pa m ¼ 10 11 mPa m;

ð10:13Þ

536 Transmitter and receiver characteristics

[Ch. 10

Table 10.20. Summary of dipole source levels from air guns and air gun arrays. a Sorted by source level. Type

Description

Zero–peak dipole source Reference level SL dp z -p = ( dB re mPa 2 m 2 )

Small gun

0.7 L (40 in3 )

Small gun

0.2–1.0 L (12–60 in3 )

223–229

sercel (www)

Medium gun

2.5–4.1 L (150–250 in3 )

229–230

sercel (www)

Sub-array

GECO 594 sub-array

235

Two-gun array

2-17 L (120–1040 in3 )

233–238

sercel (www)

Small array

Various

242–246

Richardson et al. (1995)

Medium array

Various

248–252

Richardson et al. (1995)

Large array

56 L (3397 in3 ) array

254

Caldwell and Dragoset (2000)

Large array

66 L array

255

Richardson et al. (1995)

Large array

Western geophysical 37 L (2250 in3 )

255

Dragoset (2000)

Large array

ARCO 4000

255

Richardson et al. (1995)

Large array

LDEO array b 137 L (8385 in3 )

257

ldeo (www)

Large array

GSC 7900

259

Richardson et al. (1995)

224

Dragoset (2000)

Richardson et al. (1995)

a The inch (1 in) is a non-metric unit of distance, deﬁned as 25.4 mm. The cubic inch (1 in3 , approximately equal to 16 cm 3 or 0.016 L) is often used to quantify the volume of air used by an air gun. One liter (1 L) ¼ 1 dm 3 . Similarly, pressure is sometimes reported in units of pounds-force per square inch (psi). One psi is approximately equal to 6.9 kPa (see Appendix B). b Lamont Doherty Earth Observatory.

or, equivalently, 1 bar 2 m 2 ¼ 10 22 mPa 2 m 2 :

ð10:14Þ

Taking logarithms of this last equation it is seen that a z-p source product of 1 bar meter is equivalent to a z-p source level of 220 dB re mPa 2 m 2 . It is customary to deploy air guns in a synchronized array, thus achieving higher source levels than would be possible with a single device. As for the SURTASS and HIFT sources mentioned previously, air gun arrays span several wavelengths, and the source level of such an array is not related in a simple way to the sound pressure level at 1 m distance. The sound-producing mechanism of an air gun is associated with the sudden release of compressed air into the water. After the initial release of air, an expanding

Sec. 10.1]

10.1 Transmitter characteristics

537

Table 10.21. Summary of zero-to-peak dipole source levels for generator–injector (GI) air guns. Total Volume volume generator ðG þ IÞ gun ðGÞ

Volume injector gun ðIÞ

Zero–peak dipole source level SL dp z-p = ( dB re mPa 2 m 2 )

Reference

Notes

1.47 L

0.74 L

0.74 L

223.5

sercel (www) GI gun in harmonic mode

2.46 L

1.23 L

1.23 L

226.0

sercel (www) GI gun in harmonic mode

3.44 L

1.72 L

1.72 L

226.5

sercel (www) GI gun in harmonic mode

2.46 L

0.74 L

1.72 L

223.5

sercel (www) GI gun in true GI mode

2.46 L

2.46 L

N/A

228.5

sercel (www) Not a GI gun, but a single air gun of the same total volume, included for comparison from Table 10.20

bubble forms which reaches a maximum volume and then begins to contract. On contraction, the bubble overshoots its equilibrium radius and continues to contract until a minimum volume is reached, ready to start expanding again and repeat the cycle. The pulsating bubble emits a series of so-called ‘‘bubble pulses’’ (or ‘‘bubble oscillations’’) of successively smaller amplitudes. Much eﬀort is put by the developers of air guns into reducing the eﬀect of these bubble pulses, as they interfere with the main signal and degrade performance (Dragoset, 2000). One method used to minimize the bubble pulse is to arrest the implosion of the initial bubble by ﬁring a second air gun a short time after the ﬁrst. When used in this way the ﬁrst gun is called a ‘‘generator’’ (G) gun and the second one an ‘‘injector’’ (I) gun. The combination is called a ‘‘GI gun’’. A GI gun can operate in a so-called ‘‘harmonic mode’’, meaning that equal volumes of air are ﬁred by the separate G and I guns, or ‘‘true GI mode’’, for which the volume ratio is optimized to minimize the bubble pulse (sercel, www). Source levels for these combinations are listed in Table 10.21, where they are compared with the source level of an air gun with the same total volume from Table 10.20. The properties of seismic survey sources other than air guns are summarized in Table 10.22. These include water guns, sleeve exploders, sparkers, and boomers. (Explosives are not described well by a zero-to-peak pressure and are discussed separately in Section 10.1.1.2.3.) A water gun operates in a similar manner to an air gun except that the expanding air, instead of being released directly into the water, is used to accelerate a piston into a cylinder ﬁlled with seawater. The water is ﬁrst pushed out into the sea and then returns suddenly, creating a negative pressure peak as the piston returns to its original position. A sleeve exploder (or gas gun) ignites an explosive mixture of gases inside a rubber tube. The explosion causes the rubber tube

538 Transmitter and receiver characteristics

[Ch. 10

Table 10.22. Summary of zero-to-peak source levels of seismic survey sources other than air guns, sorted by type. SLz-p = ( dB re mPa 2 m 2 )

Type

Description

Reference

Water gun

0.9 L

217

Richardson et al. (1995)

Water gun

1.3 L

239

Finneran et al. (2002)

Water gun array

24 L

245

Richardson et al. (1995)

Sleeve exploder

One sleeve

217

Richardson et al. (1995)

Boomer

500 J electric discharge

212

Richardson et al. (1995)

Boomer (Huntec)

340 J electric discharge

213

CCC (2000)

Minisparker (SQUID 2000)

1.5 kJ electric discharge

217

CCC (2000)

Sparker array (Geo-spark 200)

100 J electric discharge

206

geospark (2003)

Sparker array (Geo-spark 200)

1 kJ electric discharge

225

geospark (2003)

to expand, thus creating a pressure pulse in the water. In the case of a water gun or sleeve exploder, no air is released into the water so these systems do not suﬀer from the pulsating bubble. Finally, sparkers and boomers work by creating an electrical discharge across two electrodes placed in water. The resulting spark creates a pulsating bubble. Source levels for these systems are indicative only. The actual value depends on the conductivity of the seawater, and hence on its salinity and temperature.

10.1.1.2.3

Explosives

Pulse amplitude, duration, and energy. The acoustic eﬀects of explosives are summarized by Weston (1960, 1962). The main feature is a short shock wave comprising a sharp—almost instantaneous—rise in pressure followed by an exponential decay with a time constant of a few hundred microseconds. The explosion leaves behind a large pulsating bubble whose successive expansions and contractions give rise to a series of weaker, more symmetrical bubble pulses, in a similar manner to an air gun (Weston, 1960; Cole, 1965). Only the initial shock is considered here.

Sec. 10.1]

10.1 Transmitter characteristics

539

If the peak pressure of the shock wave is denoted P, the initial shock can be characterized by the expression ( 0 t<0 t

qðtÞ ¼ ð10:15Þ P exp t > 0. Arons (1954) provides the following numerical values for P and , as a function of distance s from the explosion: ^s 1:13 P ¼ ð52:4 MPaÞ ð10:16Þ ^ 1=3 W and ^s 0:22 1=3 ^ ¼ ð92:5 msÞW ; ð10:17Þ ^ 1=3 W where W is the mass of explosive.3 These equations apply to a spherical TNT charge4 of density exp ¼ 1520 kg/m 3 . The signiﬁcance of the density is that it determines the radius of a sphere, given its mass. The parameter W 1=3 is proportional to the charge radius, with a constant of proportionality equal to ð4exp =3Þ 1=3 . The repeated appearance of the ratio s=W 1=3 results from application of the similarity theory of Kirkwood and Bethe, as described by Cole (1965). Equations (10.16) and (10.17) are valid for values of this ratio, referred to henceforth as the scaled charge distance, in the range 0.5 m kg 1=3 to 800 m kg 1=3 . The amplitude P does not obey a simple spherical spreading law (i.e., the exponent of Equation 10.16 diﬀers from minus 1). The main reason for this is that as the pulse spreads out from the explosive source, its leading edge (the shock front) expands more rapidly than its exponential tail, resulting in a pulse duration that increases with increasing distance traveled. Conservation of energy demands that the amplitude of the pulse must decrease more quickly than spherical spreading to compensate for the increased duration. Let the ‘‘pulse energy’’ be characterized by the squared pressure integrated over all time and denote this quantity E: ð1 E qðtÞ 2 dt: ð10:18Þ 0

If Equation (10.18) is applied, using Equation (10.16) for the amplitude and Equation (10.17) for the duration, the result (for TNT) is ^s 2:04 ^ 1=3 E ¼ ð0:127 MPa 2 sÞW : ð10:19Þ ^ 1=3 W The exponent is close to minus 2, the value expected for spherical spreading, but there is nevertheless a small departure. This departure is associated with the dissipation of high-frequency components at the shock front. 3 4

^ is the numerical value of W, expressed in units of kg. W The same value of the time constant applies also for pentolite.

540 Transmitter and receiver characteristics

[Ch. 10

Table 10.23. Summary of peak pressure and pulse energy (Equation 10.18) for three types of explosive (from Cole, 1965). P s^ 1:13 ð52:4 MPaÞ ^ 1=3 W

^ 1=3 E=W ^ s 2:05 ð0:126 MPa 2 sÞ ^ 1=3 W

Loose tetryl (930 kg m 3 )

s^ 1:15 ð51:0 MPaÞ ^ 1=3 W

s^ 2:10 ð0:150 MPa 2 sÞ ^ 1=3 W

Pentolite (1600 kg m 3 )

s^ 1:13 ð54:6 MPaÞ ^ 1=3 W

s^ 2:12 ð0:161 MPa 2 sÞ ^ 1=3 W

Explosive (density) TNT (1520 kg m 3 )

Arons’s equations are based partly on results published previously by Cole (1965). Cole’s original results are listed separately in Table 10.23, as these include expressions for the pulse energy as well as data for explosives other than TNT. Cole (1965, p. 242) also gives expressions for the impulse (the time-integrated magnitude of the acoustic pressure). The formulas quoted in Table 10.23 are applicable over at least the range of Cole’s data, which in the case of TNT covers one order of magnitude of the scaled charge distance, between 0.5 m kg 1=3 and 11 m kg 1=3 . The measurements of Arons for pentolite suggest that the region of validity could extend signiﬁcantly farther than this, to scaled distances around 250 m kg 1=3 . For TNT, notice the consistency between the coeﬃcient and exponent of Cole’s measurements (parameter E in Table 10.23) and those derived from the amplitude and time constant of Arons’s data (Equation 10.19).

Energy source level. Seismic sources other than explosives were described (in Section 10.1.1.2.2) in terms of their zero-to-peak source level SLz-p . As explained in Chapter 8, this parameter—and its close cousin the peak-to-peak source level SLp-p —apply to a pulse whose shape does not vary with distance from the source. For explosive pulses, whose shape does vary with distance, a more robust characterization can be made by means of the energy source level SLE ¼ 10 log10 ½EðsÞs 2 dB re mPa 2 m 2 s:

ð10:20Þ

Because of the non-linear eﬀects mentioned previously, Equation (10.20) provides an apparent source level that decreases slightly with increasing distance from the explosion (see Equation 10.19). The variation with scaled charge distance of the apparent source level deﬁned in this way is shown in Table 10.24 for distances up to 5000 charge radii. (The adjective ‘‘speciﬁc’’ is used to indicate that these values are per unit

Sec. 10.1]

10.1 Transmitter characteristics

541

Table 10.24. Variation with range of speciﬁc pulse energy and apparent speciﬁc SLE for pentolite using Equation (10.21). The ratio s=aexp is the number of charge radii, where aexp is the charge radius aexp ¼ ð3W=4exp Þ 1=3 . Distance in charge radii

Scaled charge distance

Scaled speciﬁc pulse energy

s=aexp

sW 1=3

Es 2 =W

(m kg 1=3 ) (MPa 2 m 2 s kg1 )

Apparent speciﬁc explosion energy 4 2 Es =W c (MJ kg1 )

Apparent Notes speciﬁc energy source level SLE ( dB re mPa 2 m 2 s kg1 )

10

0.5

0.166

1.36

232.2

Lower-limit total explosion energy

200

10.6

0.143

1.17

231.5

Representative value

5000

265.2

0.122

1.00

230.8

Upper-limit acoustic SLE

mass of explosive.) The pulse energy (for pentolite) is calculated using (Arons, 1954)5 ^s 2:06 2 1=3 ^ EðsÞ ¼ ð0:152 MPa sÞW : ð10:21Þ ^ 1=3 W Thus, the energy lost from the pressure wave as the shock front expands from 10 to 5000 charge radii is 232.2 230.8 ¼ 1.4 dB (see column 5). The value at distance s ¼ 5000aexp provides an upper limit for the speciﬁc acoustic source level of 230.8 dB re mPa 2 m 2 s kg1 (column 5). Using the value at 5000 charge radii as an estimate of the source level (substituting Equation 10.21 in Equation 10.20) gives ^ SLE 231 þ 10 log10 W

dB re mPa 2 m 2 s:

ð10:22Þ

A useful rule of thumb is that this value, if converted to energy in joules (see column 4), is approximately equivalent to one megajoule of acoustic energy per kilogram of explosive. Depth dependence. The equations presented so far are for shock waves, whose energy and spectrum are independent of depth. Much of the low-frequency energy from an explosion is carried by the ﬁrst bubble pulse, and a complete characterization of the source spectrum, omitted here, needs to include this contribution, which is a function of the explosive depth. Weston (1960) gives a thorough and insightful account of these eﬀects. For more recent measurements, see Chapman (1988). 5

Valid for pentolite at a distance between 10 and 5000 charge radii.

542 Transmitter and receiver characteristics

10.1.2

[Ch. 10

Of marine mammals

Marine mammals are responsible for a remarkable repertoire of sounds in the sea. As with man-made sounds, in the following text a distinction is made between continuous and impulsive sources. Reviews of the sounds made by marine mammals can be found in Richardson et al. (1995) and Wartzok and Ketten (1999). 10.1.2.1

Continuous vocalizations

Marine mammals make a bewildering variety of continuous sounds (Au, 1993; Richardson et al., 1995; Wartzok and Ketten, 1999; Tyack, 1999; Anon., 2003). Only a very brief summary of these is provided below. The diversity is illustrated here by the assortment of often onomatopoeic descriptions of the vocalizations that appear in the scientiﬁc literature, many suggestive of more familiar sounds made by terrestrial animals, such as barks, bleats, roars, whinnies, and yelps. The sounds made by baleen whales are summarized by Richardson et al. (1995) in their Table 7.1. The highest source level quoted for a non-impulsive sound, equal to 189 dB re mPa 2 m 2 , is that for bowhead whale song in the frequency range 20 Hz to 500 Hz. Collectively, baleen whales make a noticeable contribution to ambient noise at low frequency (Watkins et al., 1987; Anon., 2003, pp. 28, 44). Richardson et al. (1995, Table 7.4) also lists source levels for pinnipeds6 and sirenians,7 for which the highest value quoted (for the Weddell seal call) is 193 dB re mPa 2 m 2 . The toothed whales, widely known for their echolocation clicks, also produce a variety of whistles, chirps. and screams with source levels up to about 180 dB re mPa 2 m 2 (for whistles of the short-ﬁnned pilot whale—Richardson et al., 1995, Table 7.2). 10.1.2.2

Impulsive sources

Impulsive sounds made by marine mammals (including non-vocal sounds such as tail slaps) are described variously in the literature as clicks, gunshots, knocks, slaps, taps, and thumps. For example, Parks and Tyack (2005) describe loud ‘‘gunshot’’ sounds with (RMS) source level up to 192 dB re mPa 2 m 2 , based on a 90 % energy averaging time (i.e., Yð0:9Þ in the notation of Table 10.19). Summaries of the characteristics of echolocation clicks are given by Au (1993), Richardson et al. (1995), Rasmussen et al. (2002), and Zimmer et al. (2005). Of most obvious relevance here are the echolocation clicks used by mammals for navigation, hunting, or inspection. Table 10.25 summarizes the most important properties of such pulses for seven diﬀerent species. (Short pulses considered suitable for echolocation are included in the table whether or not it has been established they are used by the animal for this purpose.) Though of short duration, the impulsive sounds can be very loud (the highest reported peak-to-peak source level for the sperm whale Physeter macrocephalus is higher than that of most of the man-made sonars listed 6 7

Seals, sea lions, and walruses. For example, manatees.

Table 10.25. Summary of echolocation pulse parameters for selected animals. Species a

Peak freq./ kHz

10 dB band width/ kHz

3 dB band width/ kHz

t= ms

Physeter macrocephalus (sperm whale)

15

10

5

100

243

193

Zimmer et al. (2005) Møhl et al. (2003)

Ziphius cavirostris (Cuvier’s beaked whale)

40

23

12

160

214

164

Zimmer et al. (2005)

Monodon monoceros (narwhal)

40

35

20

50

227

174

Møhl et al. (1990)

Grampus griseus (Risso’s dolphin)

49

66

27

40

220

164

Madsen et al. (2004b)

Pseudorca crassidens (false killer whale)

40

63

35

30

220

163

Madsen et al. (2004b)

Tursiops truncatus (Atlantic bottlenose dolphin)

120

100

30

25

225

167

Zimmer et al. (2005) Au (1980)

Phocoena phocoena

140

14–46

6–26

100

205

151

Villadsgaard et al. (2007)

a Thumbnail images (except narwhal) reprinted from Wikipedia.

SLp-p SLE = Reference (dB re (dB re mPa 2 m 2 ) mPa 2 m 2 sÞ

# Garth Mix, GMIX Designs, reprinted with permission. Narwhal image

544 Transmitter and receiver characteristics

[Ch. 10

in Tables 10.1 to 10.9), leading to speculation that the sound might sometimes be used not just for echolocation, but also for stunning prey (Møhl et al., 2000).8 As with man-made sonar, the source level of a marine mammal is usually measured at a distance greater than 1 m from the source, and scaled back to the standard reference distance by correcting for propagation loss. Some pulses are very short (less than a millisecond) so there is scope for some distortion of the pulse due to dispersion, leading to variability in measured p-p source level. Variability can be reduced by characterizing such sounds by their energy source level SLE , which is independent of the shape of the pulse. Nevertheless, it is common for quantitative measurements of the strength of mammal echolocation clicks to be reported in the form of peak-topeak levels, as summarized in Table 10.26 (high-frequency clicks) and Table 10.27 (short, low-frequency pulses with otherwise similar characteristics to high-frequency echolocation clicks). If SLp-p and t are both known, it is possible to estimate SLE either using Equation (10.6) or Equation (10.8), depending on the pulse shape. Echolocation clicks are characterized by their high intensity and short duration (typically fewer than 20 cycles). Large cetaceans are also known to produce short, high-intensity pulses, albeit at a much lower frequency (below 100 Hz). Regardless of their precise function,9 which might or might not include echolocation (Mellinger and Clark, 2003), the resemblance in shape of these low-frequency pulses to their highfrequency counterparts makes it useful to characterize them in a similar way (see Table 10.27). The echolocation pulses of the killer whale (Orcinus orca) and harbor porpoise (Phocoena phocoena) are compared in Figure 10.5. Despite its higher center frequency of 130 kHz, the Phocoena pulse has a signiﬁcantly longer duration and smaller bandwidth, resulting in a recognizable oscillatory behavior, whereas Orcinus uses a characteristic broadband click. The Orcinus pulse is also louder (see Table 10.26). For some species, there exists evidence that the click intensity is adjusted according to the distance s from the target. The observed relationships are (in units of dB re mPa 2 m 2 ) SLp-p ¼ 151:59 þ 19:37 log10 ^ s (finless porpoise, Li et al., 2006)

ð10:23Þ

SLp-p ¼ 181:4 þ 20 log10 ^ s

(killer whale, Au et al., 2004, s < 25 mÞ

ð10:24Þ

SLp-p ¼ 177:8 þ 20 log10 ^ s

(dusky dolphin, Au and Wursig, 2004) ¨

ð10:25Þ

and SLp-p ¼ 192 þ 16 log10 ^ s

(white-beaked dolphin, Rasmussen et al., 2002) ð10:26Þ

In each case, the decreasing distance approximately cancels the eﬀect of spherical spreading at short distances, so that, while the SNR at the sonar receiver increases as 8

Benoit-Bird et al. (2006) investigate this hypothesis, ﬁnding no evidence of stunning or disorientation of ﬁsh by simulated odontocete echolocation clicks. 9 Their probable function for blue and ﬁn whales is described by Sˇirovic´ et al. (2007) as ‘‘communication during mating and feeding’’.

Sec. 10.2]

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545

the target is approached, the sound levels at the target remain relatively stable. Such a strategy is likely to make it more diﬃcult for the animal’s prey to discern the threat associated with an approaching predator.

10.2

RECEIVER CHARACTERISTICS

10.2.1

Of man-made sonar

The maximum eﬀectiveness of a simple sonar receiver is determined by its sensitivity and its self-noise voltage. In the presence of an external source of noise, this maximum will not be reached unless the external noise can be ﬁltered out. 10.2.1.1

Hydrophone sensitivity and non-acoustic noise

10.2.1.1.1 Sensitivity Hydrophone sensitivity can be deﬁned as the open-circuit voltage at the output of the hydrophone per unit of pressure in the water (Stansﬁeld, 1991). Sensitivity is a function of frequency, so it is convenient to deﬁne it as a ratio of the spectral density of the output voltage (mean square voltage per unit frequency band), denoted Uf , to that of the pressure, Qf . For the present purpose, the following deﬁnition of sensitivity M is adopted for an omni-directional hydrophone Mð f Þ

Uf ð f Þ : Qf ð f Þ

ð10:27Þ

The usual reference values of voltage and pressure are one volt and one micropascal, so hydrophone sensitivity expressed in decibels (denoted HS) is HSð f Þ 10 log10 Mð f Þ

dB re V 2 =mPa 2 :

ð10:28Þ

Mean square signal voltage at the hydrophone output is given by U S ¼ MQ S :

ð10:29Þ

Mean square noise voltage, including electrical noise Nelec , is U N ¼ MQ N þ Nelec :

ð10:30Þ

The signal-to-noise ratio at the hydrophone output (restricting attention here to a single frequency) is " # ðNelec Þf 1 U Sf Q Sf SNRout N ¼ N 1 þ : ð10:31Þ Uf Qf Mð f ÞQ N f The mean square noise pressure Q N includes all pressure ﬂuctuations at the hydrophone, whether or not they are acoustic in origin. Acoustic sources are described in Chapter 8. The main non-acoustic pressure ﬂuctuations, considered brieﬂy below, are ﬂow noise, caused by a turbulent boundary layer, and thermal noise, caused by thermal agitation of individual water molecules.

Table 10.26. Summary of maximum peak-to-peak source levels of high-frequency marine mammal clicks with peak frequency exceeding 10 kHz. Sort order is by scientiﬁc name. Peak b frequency/ kHz

Approx. pulse duration t=ms

SLp-p = (dB re mPa 2 m 2 )

112–130

140

151

Au (1993, Table 7.2)

40–60

—

202

100–115

50–80

225

Au et al. (1985) (San Diego Bay) Au (1993, Table 7.2)

45 & 117

25

223

Madsen et al. (2004a)

30–60

—

180

Au (1993, Table 7.2)

Grampus griseus (Risso’s dolphin)

49

40

220

Table 10.25

Lagenorhynchus albirostris (whitebeaked dolphin)

120

10–30

219

Rasmussen et al. (2002, table I)

Lagenorhynchus obliquidens (Paciﬁc white-sided dolphin)

59

34–52

170

Rasmussen et al. (2002, table I)

Lagenorhynchus obscurus (dusky dolphin)

80–110

—

210

Au and Wu¨rsig (2004)

Lipotes vexillifer (Chinese river dolphin)

100–120

—

156

Au (1993, Table 7.2)

Monodon monoceros (narwhal) c

40

29–45

227

Table 10.25

Neophocoena phocoenoides (ﬁnless porpoise) c

125

185

Li et al. (2006)

Species

Cephalorhynchus hectori (Hector’s dolphin) Delphinapterus leucas (beluga whale) Feresa attenuata (pygmy killer whale) Globicephala melaena (pilot whale)

a b c

Images a

Reference

Unless otherwise stated, thumbnail images are # Garth Mix, GMIX Designs. Reprinted with permission. Further information on spectral content (e.g., bandwidth) is available from many of the source references. Narwhal and ﬁnless porpoise images reprinted from Wikipedia.

Peak b frequency/ kHz

Approx. pulse duration t=ms

SLp-p = (dB re mPa 2 m 2 )

45–80

30

218

Au et al. (2004)

Phocoena phocoena (harbor porpoise)

140

100

205

Table 10.25

Phocoenoides dalli (Dall’s porpoise)

120–160

180–400

170

Au (1993, Table 7.2)

15

100

243

Table 10.25

40 100–130

30 100–120

220 228

Table 10.25 Au (1993,Table 7.2)

Stenella attenuata (pantropical spotted dolphin)

69

43

220

Rasmussen et al. (2002, table I)

Stenella longirostris (longsnouted spinner dolphin)

70

31

222

Rasmussen et al. (2002, table I)

Tursiops truncatus (Atlantic bottlenose dolphin

52

50–250

170

120

25

225

Rasmussen et al. (2002, table I) Table 1025

110–130

50–80

228

Au (1993, Table 7.2)

40

160

214

Table 10.25

Species

Orcinus orca (killer whale)

Physeter macrocephalus (sperm whale) Pseudorca crassidens (false killer whale)

Tursiops gilli (Paciﬁc bottlenose dolphin) Ziphius cavirostris (Cuvier’s beaked whale) a b

Images a

Reference

Unless otherwise stated, thumbnail images are # Garth Mix, GMIX Designs. Reprinted with permission. Further information on spectral content (e.g., bandwidth) is available from many of the source references.

548 Transmitter and receiver characteristics

[Ch. 10

Table 10.27. Summary of peRMS and p-p source levels of low-frequency marine mammal pulses with peak frequency less than 100 Hz. The sort order is by scientiﬁc name. Images a

Peak frequency/ Hz

Approx. pulse duration t=ms

SLpeRMS = (dB re mPa 2 m 2 )

SLp-p = (dB re mPa 2 m 2 )

Balaenoptera musculus (blue whale)

25–29

1000

189 3

198

Sˇirovic´ et al. (2007)

Balaenoptera physalis (ﬁnback whale)

15–28

1000

189 4

198

Sˇirovic´ et al. (2007)

Megaptera novaeangliae (humpback whale)

25–80

300–400

178

187

Thompson et al. (1986)

Species

a

Thumbnail images are

Reference

# Garth Mix, GMIX Designs, reprinted with permission.

Waveform

Animal

Harbor porpoise (Phocoena phocoena) Duration: 100 ms Center frequency: 140 kHz Source factor: ðS0 Þz-p ¼ 80 kPa 2 m 2

Killer whale (Orcinus orca) Duration: 40 ms Center frequency: 50 kHz Source factor: ðS0 Þz-p ¼ 1600 kPa 2 m 2 Figure 10.5. Comparison of echolocation pulses made by the harbor porpoise (upper) and killer whale (lower). The killer whale pulse has a higher peak sound pressure, lower center frequency, larger bandwidth, and shorter duration (reprinted from Au, 1993 and Au et al., 2004; porpoise pulse originally from Kamminga and Wiersma, 1981). Porpoise and orca thumbnail images # Garth Mix, GMIX Designs, reprinted with permission.

Sec. 10.2]

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549

10.2.1.1.2 Molecular thermal noise Molecular noise is caused by the thermal agitation of water molecules at the sensitive face of a hydrophone. The resulting bombardment contributes to RMS pressure and is thus converted to electricity in the same way as an acoustic pressure would be. Mellen (1952) shows that the equivalent acoustic spectral density10 associated with this signal is given by ðQmolec Þf ¼ Atherm f 2 ; ð10:32Þ where the constant of proportionality is Atherm ¼ 4

KT w ¼ 0:0338 cw

nPa 2 =Hz 3 ;

ð10:33Þ

where T is absolute temperature; and K is Boltzmann’s constant, equal to 1.381 10 23 J/K. The equivalent noise spectrum level is 10 log10 Qf ¼ 14:7 þ 10 log10 F 2

dB re mPa 2 =Hz:

ð10:34Þ

It is clear that thermal noise increases with increasing frequency. At frequencies below about 100 kHz it is usually negligible. 10.2.1.1.3 Flow noise As with molecular noise, ﬂow noise is caused by non-acoustic pressure ﬂuctuations in the immediate vicinity of the receiving hydrophone (Dowling, 1998). In the case of ﬂow noise, these ﬂuctuations only arise if sonar and water are in relative motion (e.g., if the hydrophone is towed, or attached to the hull of a moving vessel). In this situation, a hydrodynamic boundary layer forms between the moving sonar and the stationary water at some distance away. The pressure ﬂuctuations are caused by turbulence in this boundary layer. A characteristic feature of ﬂow noise is that it is inversely proportional to frequency cubed (Urick, 1983). 10.2.1.2

Array directivity

Modern sonars use arrays of receiving hydrophones in order to ﬁlter out sound from unwanted directions. Some relevant properties of such arrays are described in Chapter 6.

10.2.2

Of marine mammals, amphibians, human divers, and ﬁsh

Hearing is not the only sensor available to animals living in the sea, but it is the most important one for some species (Wartzok and Ketten, 1999). Audiograms of selected aquatic animals are shown in Section 10.2.2.2, including the underwater hearing capability of human divers. 10 The equivalent acoustic spectral density is the spectral density that would be required to trigger the same RMS response in the hydrophone as caused by the thermal bombardment.

550 Transmitter and receiver characteristics

[Ch. 10

In principle, the thermal noise limit is the same for biological receivers as for man-made equipment (see Block, 1992). However, the sensitivity and self-noise of bio-sonar cannot easily be measured separately. The combined eﬀect of both determines the hearing threshold (i.e., the minimum sound pressure level that can be heard by the animal, in the absence of external noise), and for a speciﬁed success rate, the value of which depends on the design of the experiment (Au, 1993, p. 33). An audiogram is a graph of hearing threshold as a function of acoustic frequency (Morfey, 2001). The measurement is a laborious one, requiring an animal specially trained to respond in a prescribed way to an auditory stimulus. Thus, data are limited to a small number of species traditionally kept in captivity. In most cases, the measurements are limited to one or two individuals per species. 10.2.2.1

The intensity of underwater sound: typical orders of magnitude

Before describing the audiograms themselves it is useful to get a feel for the orders of magnitude involved in typical values of underwater acoustic pressure and the associated intensity and particle velocity. Consider a plane wave whose RMS pressure (denoted pRMS ) is 1.5 mPa. The magnitude of the mean sound intensity of this plane wave is I¼

p 2RMS ¼ 1:5 pW m 2 ; Z

ð10:35Þ

where Z is the characteristic impedance of seawater, equal to 1.5 MPa s/m. The magnitude of the corresponding RMS particle velocity is uRMS ¼

pRMS ¼1 Z

nm s 1 :

ð10:36Þ

These are truly microscopic values. For example, the velocity is of order 1 atomic diameter per second, while the intensity corresponds to that of a 100-watt light bulb at a distance s of sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 100 W s¼ 2300 km: 4 1:5 pW m 2 This is approximately the distance, for example, between Amsterdam and Moscow, or between Los Angeles and Houston. The calculation is idealized in assuming a bulb of 100 % eﬃciency and no attenuation of radiated light. Nevertheless, the point is that light intensity at the receiver position is very weak and would be invisible to the unaided eye.11 This minuscule intensity of sound could nevertheless, depending on the frequency, be audible to human divers (see Section 10.2.2.2.4) and to several species of dolphin and other small whales (Section 10.2.2.2.1). 11 The acoustic intensity of 1.5 pW/m 2 can be compared with the optical intensity threshold of a dark-adapted harp seal, which is about 1,000 times greater (Wartzok and Ketten, 1999).

Sec. 10.2]

10.2.2.2 10.2.2.2.1

10.2 Receiver characteristics

551

Measured audiograms Of cetaceans

Hearing threshold data are available for some odontecetes (toothed whales, dolphins, and porpoises), but at the time of writing none are known to the author for the larger baleen whales. Based on available data, species that appear to be particularly sensitive are the harbor porpoise (Phocoena phocoena) (see Figure 10.6) and killer whale (Orcinus orca) (Figure 10.7), with a mimumum threshold of between 30 dB and 35 dB re mPa 2 measured at 15 kHz (orca) and 100 kHz (porpoise). For other species, see Table 10.28 and reviews by Wartzok and Ketten (1999) and Nedwell et al. (2004).

10.2.2.2.2

Of pinnipeds (seals, sea lions, and walruses)

In general the hearing of pinnipeds in water is less sensitive than that of odontecetes. An example (for the harbor seal) is shown in Figure 10.8. For other examples see Richardson et al. (1995), Wartzok and Ketten (1999), Kastelein et al. (2002b). Pinnipeds are amphibious, and can hear sound in both air and water. The hearing of a given pinniped in water can be compared with the hearing of the same pinniped in air. For example, the threshold of the northern fur seal (Callorhinus ursinus) in air is about 9 dB re (20 mPa) 2 , which converts to 35 dB re 1 mPa 2 . This means that for a sound to be audible to Callorhinus in air, the mean square pressure (MSP) must

Figure 10.6. Underwater audiograms for harbor porpoise. The y-axis is the hearing threshold in units of dB re mPa 2 . Sources as marked.

552 Transmitter and receiver characteristics

[Ch. 10

Figure 10.7. Underwater audiograms for killer whale. The y-axis is the hearing threshold in units of dB re mPa 2 . Sources as marked.

exceed 0.003 mPa 2 (10 3:5 mPa 2 ), which is more than 100 times less than the MSP threshold in water (0.4 mPa 2 ) for the same species. Some prefer to compare thresholds of equivalent plane wave intensity12 (EPWI), deﬁned as MSP divided by the characteristic impedance of the medium. When expressed in this way it is the air threshold that exceeds the one in water, this time by a factor of order 20. In Table 10.29 the MSP and EPWI thresholds in both air and water are listed for ﬁve pinnipeds, including Callorhinus, and for humans. This comparison illustrates the need to specify which of the two ﬁeld variables (MSP or EPWI) is being considered when expressing a quantity in decibels. Complete audiograms measured in air and water are compared by Hemila¨ et al. (2006) for four species of pinnipeds. Two of these four species, the northern elephant seal (Mirounga angurirostris) and common seal (Phoca vitulina) are phocids (earless seals), and the other two, the California sea lion (Zalophus californianus) and north-

12 The EPWI of a sound is the intensity of a plane-propagating wave of the same MSP. This quantity can depart signiﬁcantly from the true intensity. For example, the intensity of a standing wave is identically zero, whereas the MSP (and hence EPWI also) is determined by the amplitudes of individual traveling waves. A similar situation arises in an isotropic noise ﬁeld. Thus, it is misleading to abbreviate EPWI as ‘‘intensity’’ unless there is a single well-deﬁned ﬁeld direction.

Sec. 10.2]

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Table 10.28. Hearing thresholds and sensitive frequency bands of selected cetaceans in order of decreasing sensitivity. Images a

Species

a

Minimum Frequency hearing of best threshold/ hearing/ (dB re mPa 2 ) kHz

Upper frequency limit/ kHz

Phocoena phocoena (harbor porpoise) (Kastelein et al., 2002)

31

20–150

200

Orcinus orca (killer whale) (Au, 1993) (Szymanski et al., 1999)

34

15–30

120

Pseudorca crassidens (false killer whale) (Au, 1993)

39

17–74

115

Delphinapterus leucas (beluga whale) (Au, 1993)

40

11–105

120

Tursiops truncatus (Atlantic bottlenose dolphin) (Au, 1993)

42

15–110

150

Stenella coeruleoalba (striped dolphin) (Kastelein et al., 2003)

42

32–120

160

Tursiops gilli (Paciﬁc bottlenose dolphin) (Au, 1993)

47

30–80

135

Inia geoﬀrensis (Amazon river dolphin) (Au, 1993)

51

12–64

100

Thumbnail images are

# Garth Mix, GMIX Designs, reprinted with permission.

ern fur seal (Callorhinus ursinus) are otariids (eared seals). The lowest thresholds in water are, for the most sensitive of both phocids (northern elephant seal) and otariids (northern fur seal), between 57 dB and 67 dB re mPa 2 in the approximate frequency range 5 kHz to 30 kHz. The audiograms for nine pinniped species are reviewed by Nedwell et al. (2004). For many of these, a threshold in air is also quoted.

554 Transmitter and receiver characteristics

[Ch. 10

Figure 10.8. Underwater audiograms for harbor seal. The y-axis is the hearing threshold in units of dB re mPa 2 . Sources as marked.

10.2.2.2.3

Of sirenians

The audiograms for two species of manatee (Trichechus inunquis and Trichechus manatus) are reviewed by Nedwell et al. (2004). The lowest threshold reported, for Trichechus manatus, is 50 dB re mPa 2 at frequencies of 16 kHz and 18 kHz (Gerstein et al., 1999). 10.2.2.2.4

Of human divers

Underwater audiograms for human divers measured by Al-Masri (1993) and Parvin et al. (2002) are shown in Figure 10.9, where they are compared with audiograms measured in air (see upper graph—notice the unconventional use, in order to facilitate quantitative comparisons, of 1 mPa as the reference pressure in air). The lowest threshold measured, that of Al-Masri at 500 Hz, is 55 dB re mPa 2 . The lower graph zooms in on the most sensitive region, with the mean, mode, and median of Al-Masri’s measurements all plotted. Al-Masri (1993, pp. 139–140) also measured the eﬀect of a neoprene hood on the diver’s hearing, reporting an increase in threshold of between 9 dB (at 250 Hz) and 30 dB (8 kHz). At 500 Hz the diﬀerence is about 20 dB. Fothergill et al. (2004) observe a smaller diﬀerence of about 10 dB at 500 Hz as well as a higher threshold at that frequency (85 dB re mPa 2 , without wet suit hood). Table 10.29 compares human hearing in air and water with that of amphibious mammals. Taken at face value, the table appears to suggest that human hearing sensitivity is greater than that of even the most sensitive of the seals. In practice,

Sec. 10.2]

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555

Table 10.29. MSP and EPWI hearing thresholds in air and water for four pinnipeds plus human subjects, sorted by increasing reported threshold in water. Except for human hearing in air, for which a standard value of 0 dB re (20 mPa) 2 is used, the thresholds are from the publications listed in the ﬁrst column. Image a

Species

a b

Threshold in water

Threshold in air

MSP/ mPa 2

EPWI/ pW m 2

MSP/ mPa 2

EPWI/ pW m 2

Human (Al-Masri, 1993) Homo sapiens

0.32

0.21

0.00040

0.95

Northern elephant seal (Hemila¨ et al., 2006) Mirounga angustirostris

0.48

0.32

5.4

13 000

Northern fur seal (Hemila¨ et al., 2006) Callorhinus ursinus

0.61

0.40

0.0013

3.2

Harbor seal b (Hemila¨ et al., 2006) Phoca vitulina

1.9

1.3

0.027

63

Harp seal (Richardson et al., 1995) Phoca groenlandica

3.2

2.1

1.0

2400

Californian sea lion (Hemila¨ et al., 2006) Zalophus californianus

122

79

0.103

32

Unless otherwise stated, reprinted from Wikipedia. Harbor seal thumbnail, # Garth Mix, GMIX Designs, reprinted with permission.

such a conclusion cannot be justiﬁed on the strength of this table because of the measurement uncertainties and diﬀerences in measurement procedure.

10.2.2.2.5

Of ﬁsh

Fish are sensitive to sound in the frequency range 10 Hz to 1000 Hz (Hawkins and Myrberg, 1983; Tyack, 1999; Popper and Hastings, 2009). Audiograms for many ﬁsh

556 Transmitter and receiver characteristics

[Ch. 10

Figure 10.9. Underwater audiograms for human divers, measured by Al-Masri (1993) and Parvin et al. (2002). The y-axis is the hearing threshold in units of dB re mPa 2 . The same reference pressure of 1 mPa is used for both air and water. The horizontal line (upper graph) is the standard reference pressure in air, expressed in decibels (i.e., 0 dB re (20 mPa) 2 26.0 dB re mPa 2 ).

Sec. 10.2]

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Table 10.30. Hearing thresholds in water for 10 species of ﬁsh (all hearing specialists) (from reviews by Hawkins and Myrberg, 1983 and Nedwell et al., 2004). Common name (species)

Threshold of hearing (sound pressure level [dB re mPa 2 ])

Frequency of best hearing/Hz

Goldﬁsh (Carassius auratus)

50

500

Squirrelﬁsh (Holocentrus ascensionis)

50

1000

Soldier ﬁsh (Myripristis kuntee)

50

500–2000

Mexican blind cave ﬁsh (Astianax jordani)

52

1000

Carp (Cyprinus carpio)

58

500

Mexican river ﬁsh (Astianax mexicanus)

60

1000

Cubbyu (Equetus acuminatus)

64

600

65–75

20–300

Elephant nose ﬁsh (Gnathonemus petersii)

67

500

Clown knifeﬁsh (Notopterus chitala)

67

500

Cod (Gadus morhua)

species are summarized by Nedwell et al. (2004). The most sensitive of these (those species whose hearing threshold is less than 70 dB re mPa 2 ) are listed in Table 10.30. It is conventional to express hearing thresholds in terms of the minimum audible sound pressure level. However, some species of ﬁsh are thought to respond to particle motion rather than pressure (Hawkins and Johnstone, 1978; Hawkins and Myrberg, 1983; Wysocki et al., 2009), especially for infrasound (Enger et al., 1993), and for these species it might be more appropriate to quote the hearing threshold in terms of a particle velocity or particle acceleration level. The lowest velocity level threshold quoted by Hawkins and Myrberg (1983), for Limanda, is 66 dB re nm 2 /s 2 . 10.2.2.3

Discrimination against background noise

The terms critical ratio and critical bandwidth are both measures of an animal’s ability to detect a tone in white broadband noise. Both measures have dimensions of bandwidth and, in both cases, the smaller the value of this bandwidth the greater the discrimination ability. The diﬀerence between them is in the way the bandwidth is measured, as described by Au (1993) and summarized below. 10.2.2.3.1 Critical bandwidth The concept of a critical bandwidth,13 denoted Bc , is a simple one. It is based on the 13

Also known as the aural critical band (ASA, 1994).

558 Transmitter and receiver characteristics

[Ch. 10

premise that, for the purpose of ﬁltering out broadband noise, the ear acts like a passband ﬁlter of bandwidth equal to Bc . Thus, if the noise bandwidth BN exceeds Bc , the excess can be ﬁltered out, and not otherwise. The following procedure is used for the measurement of Bc . For a given value of S BN , the masked audibility threshold of the ratio Q S =Q N f is measured, where Q is the N signal MSP and Q f is the noise spectral density. The value of this threshold is denoted WB ðBN Þ in the following discussion. This measurement of WB is repeated for successive BN values. If the premise is correct, two diﬀerent regimes are expected: one for BN > Bc , in which WB is independent of BN ; the other for BN < Bc , where WB increases linearly with increasing BN . Thus, Bc is the value of BN at which the transition takes place between these two regimes. 10.2.2.3.2

Critical ratio

The critical ratio, denoted Wc , is an alternative measure of passband bandwidth that is easier to determine experimentally than the critical bandwidth. It is deﬁned as the value of the threshold WB for the case of white BB noise, that is, Wc lim WB ðBN Þ:

ð10:37Þ

BN !1

This large BN limit of WB is a natural by-product of the measurement of Bc . Typical values of Wc at 10 kHz for odontecetes and pinnipeds are between 100 Hz and 1 kHz (Richardson et al., 1995, Fig. 8.6). The critical ratio is expressed sometimes as a proportion of the passband center frequency (e.g., in octaves) and sometimes in decibels: CR ¼ 10 log10 Wc dB re Hz: ð10:38Þ According to Richardson et al. (1995), the critical ratio of many odontecetes and pinnipeds at frequencies between 2 kHz and 30 kHz is less than one-sixth of an octave. The critical ratio of Tursiops exceeds its critical bandwidth by 7.5 dB (i.e., about a factor of 6 in bandwidth) (Au, 1993). Kastelein et al (2009b) reports measurements of the critical ratio of the harbor porpoise in the frequency range 0.3 kHz to 150 kHz, comparing these with other measurements for the bottlenose dolphin, beluga, and false killer whale. At low frequency the harbor porpoise critical ratio is approximately constant, with a value of 18 dB re Hz up to 4 kHz. At higher frequency the critical ratio increases to 39 dB re Hz at 150 kHz, approximately following the relationship CR ¼ 12:1 þ 12:1 log10 F 10.2.2.4

dB re Hz

ð4 < F < 150Þ:

ð10:39Þ

Hearing impairment and behavioral eﬀects

Compared with hearing thresholds, little is known about levels of sound that might disturb or harm an animal in water. For example, a loud sound might impair an animal’s hearing ability by increasing its hearing threshold. Such impairment is known as a temporary threshold shift (TTS) if the animal eventually recovers normal hearing and a permanent threshold shift (PTS) if not. The pulse energy E (deﬁned by Equation 10.18), is a commonly used measure to

Sec. 10.2]

10.2 Receiver characteristics

559

quantify the biological impact of sound, and in the context of its possible eﬀect on an animal is known as sound exposure (Southall et al., 2007). The impact of a given sound depends not only on the sound itself, but also on the hearing of the animal perceiving the sound. For this reason, sound exposure is sometimes weighted according to how strongly a particular animal senses each frequency present in the sound spectrum. Thus, it is standard practice to weight sounds in air according to the sensitivity of human hearing using a process known as ‘‘A-weighting’’. A measure of underwater sound that is used to assess the impact on hearing is the sound exposure level, which is given by sound exposure, converted to decibels in the obvious way LE ¼ 10 log10 E dB re mPa 2 s: ð10:40Þ An equivalent form is ð LE ¼ 10 log10 QðtÞ dt dB re mPa 2 s,

ð10:41Þ

where QðtÞ is the MSP averaged on a timescale much shorter than the duration of the sound. In some cases, a weighted sound exposure level is used, meaning that the spectral density is modiﬁed by multiplying by a dimensionless weighting function in frequency. Thus, denoting weighted variables by means of the subscript ‘‘w’’, the weighted sound exposure level is ð ðLE Þw ¼ 10 log10 Qw dt; ð10:42Þ where Qw is the weighted MSP

ð Qw ¼ Qf ð f Þ10 W ð f Þ=10 df ;

ð10:43Þ

and Wð f Þ is the weighting function, in decibels, applied to the spectral density Qf ð f Þ. No internationally accepted standard exists for the choice of weighting in water, so any weighting function applied needs to be speciﬁed. For example, the comprehensive review paper by Southall et al. (2007) proposes a number of ﬁlter functions intended for applications involving exposure of marine mammals to sound. Southall’s ﬁlter functions take the form Wð f Þ ¼ 20 log10

f 2high f 2 ; ð f 2low þ f 2 Þð f 2high þ f 2 Þ

ð10:44Þ

diﬀering only in the choice of low-frequency and high-frequency limits flow and fhigh . Southall et al. (2007) provide four pairs of values of these two frequencies for marine mammals, for each of four ‘‘functional hearing groups’’ (Table 10.31), with the resulting ﬁlter shapes plotted in Figure 10.10. The members of these four groups are listed in Table 10.32. The process of applying Southall’s ﬁlters is known as ‘‘Mweighting’’.

560 Transmitter and receiver characteristics

[Ch. 10

Table 10.31. Parameters of bandpass ﬁlter used in Mweighting (Equation 10.44) (Southall et al., 2007). flow /kHz

fhigh /kHz

Low-frequency cetaceans (lf )

0.007

22

Mid-frequency cetaceans (mf )

0.150

160

High-frequency cetaceans (hf )

0.200

180

Pinnipeds in water (pw)

0.075

75

Functional hearing group

Southall et al. (2007) provide advice on sound exposure levels—and other related parameters—that they consider likely to result in either hearing impairment or behavioral response in marine mammals. Thresholds for marine mammals quoted in this section are based on Southall et al. (2007), with bold text indicating thresholds14 taken directly from that review. All other thresholds quoted from this publication involve some interpretation by the present author. The situation for ﬁsh is less well established. A review of present knowledge of the eﬀects of sound exposure on ﬁsh, with particular emphasis on pile-driving noise, is given by Popper and Hastings (2009). In June 2008 the Fisheries Hydroacoustic Working Group (FHWG) reached an Agreement in Principle that states agreed limits on the levels of sound to which ﬁsh may be exposed in terms of peak pressure and sound exposure level (FHWG, 2008). Whether for mammals or ﬁsh, readers intending to use thresholds from this section are advised to consult more recent literature, where available. 10.2.2.4.1 Sound exposure thresholds for hearing impairment to mammals and ﬁsh The thresholds of sound exposure level proposed by Southall et al. (2007) applicable to marine mammals are reproduced here in Table 10.33. Sounds of two diﬀerent types are considered, namely pulses and nonpulses.15 Pulses are further sub-divided into 14 Southall et al. (2007) uses the term ‘‘criterion’’ rather than ‘‘threshold’’. The term is interpreted here as a threshold above which an eﬀect is considered likely for a functional hearing group, based on the (limited) available evidence. 15 ‘‘Pulses’’ are deﬁned by Southall et al. (2007) as ‘‘brief, broadband, atonal, transients’’ such as explosions, gunshots, sonic booms, seismic air gun pulses, and pile-driving strikes. They are ‘‘characterized by a relatively rapid rise from ambient pressure to a maximal pressure value followed by a [possibly oscillatory] decay period’’ and ‘‘generally have an increased capacity to induce physical injury as compared with sounds that lack these features.’’ This special use of the word ‘‘pulse’’, which diﬀers from that elsewhere in this chapter (the present special use excludes, for example, many sonar transmissions), is indicated here by means of italics. ‘‘Nonpulses’’ are deﬁned by Southall et al. (2007) as intermittent or continuous sounds that lack the essential properties of pulses. They ‘‘can be tonal, broadband or both’’ such as machinery noise, wind noise, communications signals, and many active sonar sources.

Sec. 10.2]

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561

Figure 10.10. Underwater sound level weighting curves for three groups of cetaceans plus pinnipeds, as proposed by Southall et al. (2007).

Table 10.32. Genera represented by the four groups considered by Southall et al. (2007), from their Table 2. Functional hearing group

Genera represented

Low-frequency (LF) cetaceans

Baleana, Caperea, Eschrictius, Megaptera, and Balaeoptera

Mid-frequency (MF) cetaceans

Steno, Sousa, Tursiops, Stenella, Delphinus, Lagenodelphis, Lagenorynchus, Lissodelphis, Grampus, Peponocephala, Feresa, Pseudorca, Orcinus, Globicephala, Orcaella, Physeter, Delphinapterus, Monodon, Ziphius, Berardius, Tasmacetus, Hyperoodon, and Mesoplodon

High-frequency (HF) cetaceans

Phocoena, Neophocoena, Phocoenoides, Platanista, Inia, Kogia, Lipotes, Pontoporia, and Cephalorhynchus

Pinnipeds

Arctocephalus, Callorhinus, Zalophus, Eumetopias, Neophoca, Phocarctos, Otaria, Erignathus, Phoca, Pusa, Halichoerus, Histriophoca, Pagophilus, Cystophora, Monachus, Mirounga, Leptonychotes, Ommatophoca, Lobodon, Hydrurga, and Odobenus

562 Transmitter and receiver characteristics

[Ch. 10

Table 10.33. Proposed thresholds of M-weighted sound exposure level [dB re 1 mPa 2 s] for permanent and (in brackets) temporary auditory threshold shift in cetaceans and pinnipeds; exposure is integrated over a time interval of 24 h. Thresholds for permanent threshold shift (PTS) (in bold text) are taken directly from Southall et al. (2007); thresholds for temporary threshold shift (TTS) are inferred from the caption of Southall’s Table 3. a Single pulse

Multiple pulses

Nonpulses

Weighting

LF cetaceans

198 (183)

198 (183)

215 (195)

Mlf

MF cetaceans

198 (183)

198 (183)

215 (195)

Mmf

HF cetaceans

198 (183)

198 (183)

215 (195)

Mhf

Pinnipeds (in water b )

186 (171)

186 (171)

203 (183)

Mpw

Functional hearing group

a Speciﬁcally, the TTS thresholds are: PTS threshold minus 15 dB in the cases of a single pulse or multiple pulses; PTS threshold minus 20 dB in the case of nonpulses. b See Southall et al. (2007) for corresponding information relating to pinnipeds in air.

single or multiple pulses. For each case, two thresholds are quoted in the table. The ﬁrst (in bold type) is a threshold for the onset of PTS. The second (in brackets) is a threshold for the onset of TTS. The TTS thresholds are not taken directly from Southall et al. (2007), but are inferred from the information provided with their Table 3. The notation Mlf , Mmf , etc., in the last column of Table 10.33 refers to one of the four weighting functions deﬁned by Equation (10.44) and Table 10.31. For example, the entry for pinnipeds means that injury (PTS) might result from a sound exposure level LE (weighted with Mpw ) in excess of 186 dB re mPa 2 s, whereas a TTS might result if LE exceeds 171 dB re mPa 2 s. The thresholds presented in Table 10.33 and other similar tables have a provisional status as they are based on the small number of measurements that existed at the time of the review published by Southall et al. (2007). The ﬁndings of two recent publications that are not included in Table 10.33 (Mooney et al., 2009; Lucke et al., 2009) are considered brieﬂy below. Mooney et al. (2009) examine the eﬀect on a bottlenose dolphin of exposure to octave band noise in the frequency range 4 kHz to 8 kHz. They ﬁnd that TTS occurs for SEL thresholds between 190 dB and 198 dB re mPa 2 s, depending on the duration of exposure T. The lowest SEL thresholds correspond to the longest exposure durations and vice versa. Speciﬁcally, the following relationship is observed, applicable for duration T approximately in the range 100 s to 1800 s: SELTTS ¼ 200:21 6:17 log10

T 60 s

dB re mPa 2 s.

ð10:45Þ

Lucke et al. (2009) examine the eﬀect on a harbor porpoise of exposure to an air gun pulse. They ﬁnd that TTS occurs for a single-pulse SEL of 164.3 dB re mPa 2 s and above, suggesting that this animal might be more sensitive than implied by Table 10.33.

Sec. 10.2]

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563

Table 10.34. Proposed peak sound pressure thresholds for PTS in cetaceans and pinnipeds for single pulses, multiple pulses, and nonpulses; pressures are unweighted (bracketed values are thresholds for TTS, inferred from the footnote of Southall’s Table 3). a Peak sound pressure threshold for PTS (TTS) LF cetaceans

316 kPa (158 kPa)

MF cetaceans

316 kPa (158 kPa)

HF cetaceans

316 kPa (158 kPa)

Pinnipeds in water

79 kPa (40 kPa)

a Speciﬁcally, the TTS thresholds for peak sound pressure are calculated from the PTS thresholds by dividing these by 10 0:3 1.995.

For ﬁsh, the FHWG Agreement in Principle speciﬁes a SEL threshold for TTS of 187 dB re mPa 2 s if the mass of an individual exceeds 2 g.16 This threshold is reduced for smaller ﬁsh to 183 dB re mPa 2 s (FHWG, 2008). Although no weighting is mentioned, according to Reyﬀ (2009) these thresholds are of unweighted cumulative SEL for an exposure duration of 24 hours. 10.2.2.4.2

Peak sound pressure thresholds for hearing impairment to mammals and ﬁsh PTS thresholds of peak sound pressure of marine mammals are quoted by Southall et al. (2007) in decibels. These thresholds, converted here to linear pressure17 (and rounded to the nearest integer value in kilopascals), are quoted in Table 10.34. As previously, TTS thresholds are inferred from the information provided with Southall’s Table 3. For ﬁsh, the FHWG Agreement in Principle speciﬁes a peak sound pressure threshold for the onset of TTS of 20 kPa (FHWG, 2008). The same threshold applies to all ﬁsh, regardless of mass. 10.2.2.4.3

Thresholds for behavioral eﬀects

Sound pressure level (nonpulses and multiple pulses). Behavioral eﬀects are more diﬃcult to assess quantitatively than threshold shifts. Southall et al. (2007) deal with this diﬃculty by establishing a qualitative severity scale of 10 points, ranging from ‘‘no observable response’’ (response score 0) through various intermediate responses 16 The agreed limit is described by FHWG (2008) as an ‘‘injury’’ threshold, but Stadler and Woodbury (2009) clarify that what is meant in this context by ‘‘injury’’ is the onset of TTS. 17 This is done to avoid use of the inherently ambiguous term ‘‘peak sound pressure level’’, which can mean either ‘‘peak (sound pressure level)’’, which would be the maximum value of the (RMS) sound pressure level or ‘‘(peak sound pressure) level’’, meaning the peak sound pressure expressed as a level.

564 Transmitter and receiver characteristics

[Ch. 10

Table 10.35. Outline of the severity scale from Southall et al. (2007). The text is paraphrased by the present author for brevity. For a complete description of Southall’s severity scale, the reader is referred to the original in Table 4 of Southall et al. (2007). Response score

Observed behavior

0

No observable response

3

Minor changes in behavior; no avoidance of sound source

6

Minor or moderate avoidance of sound

9

Outright panic; predator avoidance reaction

Table 10.36. Spread of sound pressure level (SPL) values [dB re mPa 2 ] resulting in the speciﬁed behavioral responses in cetaceans and pinnipeds for nonpulses (from Southall et al., 2007). The modal group (the range of SPL values resulting in the largest number of occurrences of the speciﬁed response) is given in brackets; the median is the same as the modal in every case except for mid-frequency cetaceans with response score 6–9, for which the median group is shown separately in square brackets. The severity scale on which the ‘‘response score’’ is measured is summarised in Table 10.35. Response score 0 to 2 Response score 3 to 5 Response score 6 to 9 LF cetaceans a

80–150 (90–100)

100–150 (100–110)

90–150 (110–120)

MF cetaceans b

100–200 (110–120)

80–130 (110–120)

90-200 (100–110) [120–130]

HF cetaceans c

80–130 (90–100)

Sparse data

80–170 (140–150)

Sparse data

Sparse data

Sparse data

Pinnipeds in water d a

Based on Southall et al. (2007, Table 15). Based on Southall et al. (2007, Table 17). c Based on Southall et al. (2007, Table 19) (all entries in this table are for the harbor porpoise). d Based on Southall et al. (2007, Table 21). b

(see Table 10.35) to ‘‘outright panic’’ (response score 9). Simpliﬁed dose-response relationships based on Southall et al. (2007), applicable to marine mammals and using (unweighted) sound pressure level as a metric, are provided in Table 10.36 for exposure to nonpulses and in Table 10.37 for multiple pulses. The reader’s attention is drawn to the uncertainty associated with the wide spread of values presented in Tables 10.36 and 10.37, and to the warning from Southall et al. (2007) that ‘‘The available data on marine mammal behavioral responses to multiple pulse and nonpulse sounds are simply too variable and context-speciﬁc to justify proposing single disturbance criteria for broad categories of taxa and sounds.’’ The purpose of including the two tables here is partly to illustrate this uncertainty and partly to promulgate what little information there is available.

Sec. 10.3]

10.3 References 565

Table 10.37. Spread of sound pressure level a (SPL) values [dB re mPa 2 ] resulting in the speciﬁed behavioral responses in cetaceans and pinnipeds for multiple pulses (from Southall et al., 2007). The modal group is given in brackets; the median is the same as the modal in every case except for mid-frequency cetaceans with response score 0–2, for which the median group is shown separately in square brackets. Response score 0 to 2 Response score 3 to 5 Response score 6 to 9 LF cetaceans b

110–180 (110–120)

Sparse data

110–180 (120–130)

MF cetaceans c

100–180 (170–180) [130–140]

No data

Sparse data

HF cetaceans

No data

No data

No data

150–200 (170–180)

No data

160–200 (190–200)

Pinnipeds in water d

a The SPL values quoted are based on the mean square pressure averaged ‘‘over pulse duration’’ (see p. 452 and Appendix A of Southall et al., 2007). b Based on Southall et al. (2007, Table 7). c Based on Southall et al. (2007, Table 9). d Based on Southall et al. (2007, Table 11).

Sound exposure level and peak sound pressure (single pulse). Very little is known about thresholds of SEL and peak pressure likely to cause a behavioral change. (No systematic review is available comparable to that leading to Table 10.36.) For the case of a single pulse only, Southall et al. (2007) suggest adopting the TTS threshold as a surrogate for a behavioral threshold on the grounds that a sound loud enough to cause TTS has the potential to cause a change in the animal’s behavior, even if the only change is a short-term adjustment for the temporary loss of hearing.

10.3

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Sec. 10.3]

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Li, S., Wang, D., Wang, K., and Akamatsu, T. (2006) Sonar gain control in echolocating ﬁnless porpoises (Neophocaena phocaenoides) in an open water, J. Acoust. Soc. Am., 120, 1803– 1806. LIPI (2006) Research Equipment, Research and Development Center for Oceanology, Indonesian Institute of Sciences (LIPI), available at http://www.bj8.or.id/alat.htm (last accessed January 12, 2009). Lo¨vgren, B. (2007) Rapid underwater charting using an AUV-based MBES, in Second International Conference and Exhibition: Underwater Acoustic Measurements, Technologies and Results, Foundation for Research and Technology, Crete, Greece, June 25–29. Lucke, K., Lepper, P. A., Hoeve, B., Everaarts, E., Van Elk, N., and Siebert, U. (2007) Perception of low-frequency acoustic signals by a harbour porpoise (Phocoena phocoena) in the presence of simulated oﬀshore wind turbine noise, Aquatic Mammals, 33(1), 55–68. Lucke, K., Siebert, U., Lepper, P. A., and Blanchet, M.-A. (2009) Temporary shift in masked hearing thresholds in a harbor porpoise (Phocoena phocoena) after exposure to seismic airgun stimuli, J. Acoust. Soc. Am., 125, 4060–4070. Lurton, X. (2002) An Introduction to Underwater Acoustics: Principles and Applications, Springer/Praxis, Heidelberg, Germany/Chichester, U.K. Madsen, P. T. (2005) Marine mammals and noise: Problems with root mean square sound pressure levels for transients, J. Acoust. Soc. Am., 117, 3952–3957. Madsen, P. T., Kerr, I., and Payne, R. (2004a) Source parameter estimates of echolocation clicks from wild pygmy killer whales (Feresa attenuata), J. Acoust. Soc. Am., 116, 1909– 1912. Madsen, P. T., Kerr, I., and Payne, R. (2004b) Echolocation clicks of two free-ranging, oceanic delphinids with diﬀerent food preferences: False killer whales Pseudorca crassidens and Risso’s dolphins Grampus griseus, J. Experimental Biology, 207, 1811–1823. massa (www) http://www.massa.com/underwater_comericaloceanographic.htm (last accessed December 15, 2008). McCauley, R. D., Jenner, C., Bannister, J. L., Burton, C. L. K., Cato, D. H., and Duncan, A. (2001) Blue Whale Calling in the Rottnest Trench—2000 (Project CMST 241, Report R2001-6), Center for Marine Science & Technology, Curtin University of Technology, Perth, Western Australia. Mellen, R. H. (1952) The thermal-noise limit in the detection of underwater acoustic signals, J. Acoust. Soc. Am., 24, 478–480. Mellinger, D. K. and Clark, C. W. (2003) Blue whale (Balaenoptera musculus) sounds from the North Atlantic, J. Acoust. Soc. Am., 114, 1108–1119. Møhl, B. (1968) Auditory sensitivity of the common seal in air and water, J. Aud Res., 8, 27–38. Møhl, B., Surlykke, A., and Miller, L. A. (1990) High intensity narwhal clicks, in J. Thomas and R. Kastelein (Eds.), Sensory Ability of Cetaceans (pp. 294–303). Møhl, B., Wahlberg, M., Madsen, P. T., Miller, L. A., and Surlykke, A. (2000) Sperm whale clicks: Directionality and source level revisited, J. Acoust. Soc. Am., 107, 638–648. Møhl, B., Wahlberg, M., Madsen, P. T., Heerfordt, A., and Lund, A. (2003) The monopulsed nature of sperm whale clicks, J. Acoust. Soc. Am., 114, 1143–1153. Mooney, T. A., Nachtigall, P. E., Breese, M., Vlachos, S., and Au, W. W. L. (2009) Predicting temporary threshold shifts in a bottlenose dolphin (Tursiops truncatus): The eﬀects of noise level and duration, J. Acoust. Soc. Am., 125, 1816–1826. Morfey, C. L. (2001) Dictionary of Acoustics, Academic Press, San Diego. Morse, P. M. and Ingard, K. U. (1968) Theoretical Acoustics, Princeton University Press, Princeton, NJ.

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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. Nedwell, J. R., Edwards, B., Turnpenny, A. W. H., and Gordon, J. (2004) Fish and Marine Mammal Audiograms: A Summary of Available Information (Report Ref: 534R0214), Subacoustech, Bishop’s Waltham, U.K. neptune (www) http://www.neptune-sonar.co.uk (last accessed December 15, 2008). ore (www) Model BRT6000 Burn Wire Release/Transponder, Operating and Maintenance Manual (Rev. B, October 26, 2007), available at http://www.ore.com www.ore.com (last accessed October 19, 2009). Parks, S. E. and Tyack, P. L. (2005) Sound production by North Atlantic right whales (Eubalaena glacialis) in surface active groups, J. Acoust. Soc. Am., 117, 3297–3306. Parvin, S. J., Cudahy, E. A., and Fothergill, D. M. (2002) Guidance for diver exposure to underwater sound in the frequency range 500 to 2500 Hz, Undersea Defence Technology, La Spezia, Italy (conference and exhibition), Nexus Ltd. Popper, A. N. and Hastings, M. C. (2009) The eﬀects of anthropogenic sources of sound on ﬁshes, Journal of Fish Biology, 75, 455–489. rafos (www) http://www.whoi.edu/instruments/viewInstrument.do?id=1061 (last accessed January 12, 2009). Rasmussen, M. H., Miller, L. A., and Au, W. W. L. (2002) Source levels of clicks from freeranging white-beaked dolphins (Lagenorhynchus albirostris Gray 1846) recorded in Icelandic waters, J. Acoust. Soc. Am., 111, 1122–1125. Reyﬀ, J. A. (2009) Underwater sounds from marine pile driving, Inter-Noise 2009, August 23– 26 2009, Ottawa, Canada. Reynolds, III, J. E. and Rommel, S. A. (Eds.) (1999) Biology of Marine Mammals, Smithsonian Institution Press, Washington, D.C. Richardson, W. J., Greene, C. R., Malme, C. I., and Thomson, D. H. (1995). Marine Mammals and Noise, Academic Press, San Diego. Rossing, T. D. (2007) Springer Handbook of Acoustics, Springer Verlag. SCAR (2002) Impacts of Marine Acoustic Technology on the Antarctic Environment (Version 1.2, July), SCAR Ad Hoc Group on Marine Acoustic Technology and the Environment, available at http://www.geoscience.scar.org/geophysics/acoustics_1_2.pdf (last accessed December 15, 2008). Schusterman, R. J., Balliet, R. J., and Nixon, J. (1972) Underwater audiogram of the California sea lion by the conditioned vocalization technique, J. Exp. Anal. Behav., 17, 339–350. seabeam (www) http://www.seabeam.com (last accessed December 15, 2008. sercel (www) Marine Sources, available at ftp://ftp.sercel.com/pdf/brochures/Marinesources.pdf (last accessed January 15, 2009). simrad (www) http://www.km.kongsberg.com (last accessed January 20, 2009). Sˇirovic´, A., Hildebrand, J. A., and Wiggins, S. M. (2007) Blue and ﬁn whale call source levels and propagation range in the Southern Ocean, J. Acoust. Soc. Am., 122, 1208–1215. sonardyne (www) http://www.sonardyne.com (last accessed October 19, 2009). 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. Stadler, J. H. and Woodbury, D. P. (2009) Assessing the eﬀects to ﬁshes from pile driving: Application of new hydroacoustic criteria, Inter-Noise 2009, August 23-26, Ottawa, Canada.

Sec. 10.3]

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Stansﬁeld, D. (1991) Underwater Electroacoustic Transducers, Bath University Press, Bath, U.K. [Now available from Peninsula Publishing.] 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. Taylor, V. J., Johnston, D. W., and Verboom, W. C. (1997) Acoustic harassment device (AHD) use in the aquaculture industry and implications for marine mammals, Proc. Symposium on Bio-Sonar and Bioacoustics, Loughborough University. Terhune, J. M. (1988) Detection thresholds of a harbour seal to repeated underwater highfrequency, short-duration sinusoidal pulses, Can. J. Zool., 66, 1578–1582. Thompson, P. O., Cummings, W. C., and Ha, S. J. (1986) Sounds, source levels, and associated behavior of humpback whales, Southeast Alaska, J. Acoust. Soc. Am., 80, 735–740. tritech (www) http://www.tritech.co.uk (last accessed December 15, 2008). Tucker, D. G. and Gazey, B. K. (1966) Applied Underwater Acoustics, Pergamon Press, Oxford, U.K. Turnbull, S. D. and Terhune, J. M. (1990) White noise and pure tone masking of pure tone thresholds of a harbour seal listening in air and underwater, Can. J. Zool., 68, 2090–2097. Turnbull, S. D. and Terhune, J. M. (1993) Repetition enhances hearing detection thresholds in a harbour seal (Phoca vitulina), Can. J. Zool., 71, 926–932. Tyack, P. L. (1999) Communication and cognition, in J. E. Reynolds, III and S. A. Rommel (Eds.), Biology of Marine Mammals (pp. 287–323), Smithsonian Institution Press, Washington, D.C. Urick, R. J. (1983) Principles of Underwater Sound, Peninsula Publishing, Los Altos, CA. usgs (www) Seismic Proﬁling Systems: Seismic Equipment, Woods Hole Science Center, http:// woodshole.er.usgs.gov/operations/sfmapping/equipment.htm (last accessed January 19, 2009). Villadsgaard, A., Wahlberg, M., and Tougaard, J. (2007) Echolocation signals of wild harbour porpoises, Phocoena phocoena, J. Experimental Biology, 210, 56–64. Wartzok, D. and Ketten, D. R. (1999) Marine mammal sensory systems, in J. E. Reynolds, III and S. A. Rommel (Eds.), Biology of Marine Mammals (pp. 117–175), Smithsonian Institution Press, Washington, D.C. Watkins, W. A., Tyack, P., and Moore, K. E. (1999) The 20-Hz signals of ﬁnback whales (Balaenoptera physalus), J. Acoust. Soc. Am., 82, 1901–1912. Watts, A. J. (Ed.) (2000) Jane’s Underwater Warfare Systems (12th Edition 2000–2001), Jane’s Information Group, Coulsdon, U.K. Watts, A. J. (Ed.) (2005] Jane’s Underwater Warfare Systems (17th Edition 2005–2006), Jane’s Information Group, Coulsdon, U.K. Weston, D. E. (1960) Underwater explosions as acoustic sources, Proc. of the Physical Society, LXXVI, 233–249. Weston, D. E. (1962) Explosive sources, Underwater Acoustics (pp. 51–66), Plenum Pess, New York. Wysocki, L. E., Codarin, A., Ladich, F., and Picciulin, M. (2009) Sound pressure and particle acceleration audiograms in three marine ﬁsh species from the Adriatic Sea, J. Acoust. Soc. Am., 126, 2100–2107. Zimmer, W. M. X., Johnson, M. P., Madsen P. T., and Tyack, P. L. (2005) Echolocation clicks of free-ranging Cuvier’s beaked whales (Ziphius cavirostris), J. Acoust. Soc. Am., 117, 3919–3927.

11 The sonar equations revisited

Were it oﬀered to my choice, I should have no objection to a repetition of the same life from its beginning, only asking the advantages authors have in a second edition to correct some faults in the ﬁrst. Benjamin Franklin (ca. 1771)

11.1

INTRODUCTION

The objective of this chapter is to reverse (or at least to mitigate) the preference in some previous chapters for simplicity over realism. The sonar equations and selected worked examples of Chapter 3 are revisited, with the aim of introducing the necessary realism to become not only didactic but also practical. To make this possible, use is made of a computer model for the calculation of propagation loss, noise level, and reverberation level. Selected sonar equation terms are redeﬁned to establish a more rigorous basis for the revised worked examples. Relevant material introduced in the intervening Chapters 4 to 10, a selection of which is considered in Chapter 11, includes — the eﬀect of boundaries and sound speed proﬁle on sound propagation and reverberation (Chapters 4, 8, and 9); — the eﬀect of bubbles and suspensions on sound propagation and scattering (Chapter 5); — the eﬀect of a matched ﬁlter and coherence loss on processing gain (Chapter 6); — the eﬀect of steering and shading on the sonar beam pattern and array gain (Chapter 6); — departures of signal statistics from idealized Gaussian or Rayleigh distributions (Chapter 7);

574 The sonar equations revisited

[Ch. 11

— the presence of ambient noise sources other than wind (Chapters 8 and 9);1 — typical measured values of target strength (Chapter 8); — typical properties of natural and man-made sonar transmitters and receivers (Chapter 10). Some generic examples of man-made sonar and other underwater acoustic sensors are listed in Table 11.1. The wide range of frequencies involved (from a few tens of hertz for low-frequency passive sonar to several megahertz for high-resolution acoustic cameras or Doppler proﬁlers), and also of water depths (1 m to 10 km) and ranges (from less than a meter to thousands of kilometers) means that there exists no single computer model that is suitable for all sonar performance problems. Hence, there will always be reason to check the output of any one model. Obtaining a second opinion is a vital part of the process of increasing (or in some cases, decreasing) conﬁdence in the model predictions. To bring this point home, the reader is encouraged to run his or her preferred performance model on the worked examples provided (starting with the simpler ones in Chapter 3), and compare the individual sonar equation terms with the values given here. The author is conﬁdent that diﬀerences will be found, and that the magnitude of these diﬀerences will often not be negligible. Identifying and understanding such diﬀerences and their causes leads to a better understanding of the problem and eventually to an improved capability to model it correctly. Perhaps the single most unrealistic feature of the examples in Chapter 3 is the assumption of inﬁnite water depth, and this is remedied here by considering depths in the range 30 m to 5,000 m. A realistic sound speed proﬁle is also introduced in order to demonstrate its eﬀect on acoustic propagation. Passive sonar is discussed ﬁrst, with coherent processing in Section 11.2, followed by incoherent processing in Section 11.3. For active sonar (Section 11.4), only coherent processing is considered. Finally, chapter and book are rounded oﬀ with a brief indulgence in crystal ball gazing (Section 11.5).

11.2

11.2.1

PASSIVE SONAR WITH COHERENT PROCESSING: TONAL DETECTOR Sonar equation

The passive sonar equation with (narrowband) coherent processing is SENB ¼ ðSL PLÞ ðNLf þ BW AGÞ DT:

ð11:1Þ

The six individual terms on the right-hand side are described in turn below, followed by a worked example in Section 11.2.8. 1

Wind was the only noise source considered for the examples in Chapter 3.

Sec. 11.2]

11.2 Passive sonar with coherent processing: tonal detector

575

Table 11.1. List of applications of man-made active and passive underwater acoustic sensors, with order-of-magnitude indication of frequency range for sound transmitters. An asterisk (*) indicates a broadband source. Application

Examples of active sensors [frequency range in kHz]

Examples of passive sensors

Military surveillance

Active sonobuoy [1–30] Active towed array sonar [0.3–10] Helicopter dipping sonar [1–30] Hull-mounted search sonar [3–300]

Bottom-mounted array Hull-mounted passive sonar (e.g., ﬂank array) Intercept sonar Passive sonobuoy Passive towed array

Weapons and countermeasures

Active homing sonar [10–100] Minesweeping equipment*

Acoustic sensing mine Passive homing sonar

Navigation

Navigation echo sounder [10–100] Obstacle avoidance sonar [30–300]

Fisheries

Fisheries sonar [10–300]

Seismic survey

Air gun array* Boomer* Sparker array* Water gun array*

Underwater positioning and communications

Acoustic transponders [0.3–100] Underwater telephone and telemetry [3–100]

Oceanographic survey, weather observation, and scientiﬁc research

Acoustic camera [300–3000] Acoustic Doppler proﬁler [30–3000] Inverted echo sounder [10–100] Multibeam echo sounder [10–1,000] Sub-bottom proﬁler* Sidescan sonar [10–1000]

11.2.2

Environmental sensing equipment

Source level (SL)

The source level SL is deﬁned as SL 10 log10 S0

dB re mPa 2 m 2 ;

ð11:2Þ

where S0 is the source factor, which was deﬁned in Chapter 3 for a point source as2 S0 ¼ lim ðp 20 s 20 Þ s0 !0

ð11:3Þ

2 The deﬁnition in Chapter 3 is not written formally as a limit in this way, but the limiting process is implied.

576 The sonar equations revisited

[Ch. 11

and p0 is the RMS pressure at distance s0 from the source. This Chapter 3 deﬁnition does not make physical sense for a source of ﬁnite size because the limit of small s0 would take us ﬁrst into the near ﬁeld of the source and eventually inside the object itself. This weakness is addressed below. As pointed out in Chapters 8 and 10, the source level is a property of the far ﬁeld3 radiated by a sound source, which means that it must be deﬁned in terms of other properties of that far ﬁeld. A suitable far-ﬁeld property is radiant intensity J (the radiated power per unit solid angle, denoted WO in Chapter 3), which for a point monopole source is equal to S0 =ðcÞ. A non-monopole source will, in general, radiate sound with diﬀerent intensity in diﬀerent directions, making radiant intensity a function of angle, which is indicated by writing J ¼ Jð; Þ, or alternatively J ¼ JðOÞ, where O ¼ Oð; Þ represents a direction deﬁned by the elevation and azimuth . If J0 is the maximum value of JðOÞ as is varied, with ﬁxed in the direction towards the target, the source factor of a directional source can be deﬁned as S0 cJ0 ; ð11:4Þ replacing Equation (11.3). In the case of an idealized (uniform, lossless) medium, the far-ﬁeld radiant intensity JðOÞ is a well-deﬁned property of a distributed source. To complete the deﬁnition of source level it is also necessary to consider a real medium (in this case water), for which the parameter J0 in Equation (11.4) is interpreted as the radiant intensity that the same source would produce in an ideal (uniform lossless) medium of the same characteristic impedance, and if driven with the same velocity on its active surfaces.

11.2.3

Narrowband propagation loss (PL)

For realistic situations, it is often necessary to calculate PL with the aid of a computer model. There are many models to choose from, either in the form of a standalone propagation model (oalib, www) or embedded within a complete sonar performance modeling package (Etter, 2003). For an omni-directional source, PL is related to the narrowband (NB) propagation factor FNB according to PL ¼ 10 log10 FNB ;

ð11:5Þ

where FNB can be written in terms of the diﬀerential propagation factor GNB as follows ð FNB ¼ GNB ðÞ d: ð11:6Þ For a directional source whose vertical beam pattern is bðÞ, Equation (11.5) 3 The far-ﬁeld nature of the source level, missing from the deﬁnitions of national (ASA, 1994) and international (IEC, www) standards bodies, is essential for its correct use in the sonar equation—Kinsler et al. (1982) and Urick (1983, pp. 75 and 328).

Sec. 11.2]

11.2 Passive sonar with coherent processing: tonal detector

generalizes to PL ¼ 10 log10

ð

bðÞGNB ðÞ d ;

577

ð11:7Þ

where bðÞ

Jð; arr Þ ; J0

ð11:8Þ

and arr is the bearing of the sonar receiver relative to the target, such that the beam pattern bðÞ is evaluated in the plane of propagation (i.e., the vertical plane containing both target and sonar receiver). Notice the change in notation compared with Chapter 6, where the same function was denoted BðÞ. The narrowband propagation factor may be approximated by FNB ¼ Fcoh ð f0 Þ;

ð11:9Þ

where f0 is the nominal center frequency of the narrowband tonal (and of the processing band). A better approximation is ð 1 f0 þB=2 FNB ð f0 Þ ¼ F ð f Þ df ; ð11:10Þ B f0 B=2 coh where B is the eﬀective processing bandwidth, deﬁned here as the smaller of the tonal width ftonal and true processing bandwidth fFFT B minðftotal ; fFFT Þ:

ð11:11Þ

For narrowband sonar, the bandwidth B is usually very small, and the approximation of Equation (11.9) is often adequate. An alternative, should greater accuracy be required, is to evaluate Equation (11.10) using the method of Harrison and Harrison (1995), which replaces the frequency average with a proportional range average at a single frequency4 ; 5 ð f0 r0 ð1þB=2f0 Þ FNB ðr0 ; f0 Þ F ðr; f0 Þ dr: ð11:12Þ Br0 r0 ð1B=2f0 Þ coh The main eﬀect of this averaging, whether in frequency or in range, is to smooth over any interference nulls whose position is sensitive to the precise acoustic frequency. An important consideration for application to coherent processing is that the propagation model must be able to account for the phase diﬀerence between diﬀerent propagation paths. A model that does so is sometimes called a ‘‘coherent’’ model, or, alternatively, might permit the user to select a ‘‘coherent’’ option. An incoherent model (i.e., one that does not take account of such phase diﬀerences) might miss important features due to constructive or destructive interference between the diﬀerent paths. It is also necessary to distinguish between the incoherent addition 4

An additional Gaussian-weighting function is applied by Harrison and Harrison (1995). The beneﬁt of converting an average over frequency to one over range is one of computational eﬃciency. This is because, for many solution methods, the required computation time is proportional to the number of frequencies considered. 5

578 The sonar equations revisited

[Ch. 11

of rays and that of modes, as qualitatively diﬀerent information is lost in the two cases. 11.2.4

Noise spectrum level (NLf )

Background noise, foreground noise, and non-acoustic noise are considered separately below. 11.2.4.1

Background noise

Background noise is sound arriving at a sonar receiver from distant sources that cannot be resolved by the sonar as coming from spatially distinct sources. The term ‘‘background’’ is used here as an adjective to distinguish from ‘‘foreground’’ noise, described below.6 In Chapter 3, only wind noise—an example of background noise—is considered. In reality, at low frequency (below 1 kHz), wind noise is supplemented by contributions from shipping, and at high frequency (above 100 kHz) by thermal noise. Occasional noise sources that sometimes contribute include precipitation (rain or snow) and some fauna (especially crustaceans and marine mammals). See Chapter 8 for a general review of sound sources and Chapter 9 for examples of predicted noise levels. 11.2.4.2

Foreground noise

Foreground noise comprises all interfering sound other than background noise; that is, sound from those sources that can be resolved by the sonar. Potential foreground noise contributions include — acoustic noise radiated by the sonar platform, either via a direct path or indirectly, after one or more boundary reﬂections; — acoustic noise radiated by vessels accompanying the sonar platform, either via a direct path or indirectly, after one or more boundary reﬂections; — mechanical vibrations from the platform structure or tow cable; — interference from active sonar transmitters operating nearby, including echo sounders; — sounds made by nearby animals such as marine mammals. Some examples of foreground noise are localized in time as well as in space. These are known as ‘‘transient sources’’ or sometimes just ‘‘transients’’. 11.2.4.3

Non-acoustic noise

Non-acoustic noise is sonar noise that is not caused by sound waves reaching the hydrophone. It is useful to distinguish between the following two categories of 6 Compare the alternative use of ‘‘background’’ as a noun to mean the combination of noise and reverberation.

Sec. 11.2]

11.2 Passive sonar with coherent processing: tonal detector

579

non-acoustic noise: — non-acoustic pressure ﬂuctuations at the hydrophone position, converted to an electrical signal by the hydrophone’s transducer action;7 — voltage ﬂuctuations in the receiving equipment other than those present in the hydrophone output. 11.2.4.4

Self-noise and ambient noise

11.2.4.4.1 Self-noise, including platform noise Background, foreground, and non-acoustic noise together amount to a complete description of noise sources relevant to passive sonar. The term ‘‘self-noise’’ is sometimes used to mean all noise that is caused by the presence of the sonar itself or its platform. Thus, it is partly non-acoustic noise and partly (acoustic) foreground noise. That part of self-noise associated with the sonar platform is known as ‘‘platform noise’’. This can be structure-borne noise if the sonar is mounted on the hull of its platform, or acoustic noise propagated through the sea if it is towed. A special kind of (non-acoustic) self-noise arises in the context of animal hearing, for which self-noise manifests itself in the form of an auditory threshold. Speculation into the nature of the noise that gives rise to this threshold is outside the present scope, other than to mention the ubiquitous presence of thermal noise. 11.2.4.4.2 Ambient noise All acoustic noise not included in the deﬁnition of self-noise can be considered to be ambient noise.

11.2.5

Bandwidth (BW)

A perfect tonal has no width in frequency. For such a tonal, BW is the width of the processing band, expressed in decibels. In reality the tonal always has a ﬁnite width that might exceed that of the processing band, in which case a proportion of the signal energy is lost, leading to a degradation relative to the ideal. This degradation can be modeled by deﬁning BW in terms of the processing bandwidth fFFT as Rhp BW 10 log10 f ; ð11:13Þ RFFT FFT where Rhp is the SNR at the hydrophone in the processing bandwidth before narrowband processing; and RFFT is the SNR at the hydrophone in the processing bandwidth after narrowband processing and before beamfoming.8 In the ideal case, 7

Examples of non-acoustic pressure ﬂuctuations are: — ﬂow noise (hydrodynamic noise from ﬂow around dome or towed array); — molecular thermal noise (usually unimportant for passive sonar; see Chapter 10). 8 The assumed processing chain is a Fourier transform followed by a beamformer.

580 The sonar equations revisited

[Ch. 11

the entire tonal is included in the processing band, Rhp and RFFT are equal, and the Chapter 3 expression for BW is then recovered. More generally, allowing for the possibility that the tonal is over-resolved B=ftonal B < ftonal RFFT Rhp ð11:14Þ 1 B > ftonal , where B is given by Equation (11.11). Thus, Equation (11.13) can be written BW 10 log10 maxðfFFT ; ftonal Þ:

11.2.6

ð11:15Þ

Array gain (AG) and directivity index (DI)

Array gain (AG) is the gain from spatial ﬁltering (beamfoming). It can be deﬁned in terms of the ratio of Rarr (SNR after beamfoming) to RFFT GA

Rarr : RFFT

ð11:16Þ

The AG term, equal to 10 log10 GA , is rarely calculated in full because of the complications introduced (for example) by anisotropic noise (Harrison, 1996; Clark, 2007) and departures from a plane wave signal. The calculation can be simpliﬁed by writing Equation (11.1) in the form SENB ¼ ðSL PLÞ ðNLf þ BW DIÞ DT þ ½AG DI ;

ð11:17Þ

and neglecting the term in square brackets, so that SENB ðSL PLÞ ðNLf þ BW DIÞ DT:

ð11:18Þ

The neglected term can be positive or negative and in an average sense (for a random geometry) can be expected to cancel. For any given geometry, a bias is likely, as illustrated by the examples of Chapter 3. A simple calculation method for the directivity index (DI) of an unsteered line array, derived in Chapter 6, is summarized below. It is convenient to write DI in the form DI ¼ 10 log10 GD ; ð11:19Þ where GD is the directivity factor (or directivity gain), which can be written !

2 GD 1 þ G0 tanh G ; ð11:20Þ 36 0 where G0 ¼ 2L=:

ð11:21Þ

The directivity index is plotted in Figure 11.1 as a function of G0 using Equation (11.20) (dashed line). Except when steered close to the endﬁre direction, the eﬀect of steering on the directivity index of a line array is small (see Chapter 6).

Sec. 11.2]

11.2 Passive sonar with coherent processing: tonal detector

581

Figure 11.1. Directivity index DI ¼ 10 log10 GD for an unsteered continuous line array vs. G0 , where G0 ¼ 2L=. Solid blue line: full integral of Chapter 6; dashed red line: approximation of Equation (11.20).

11.2.7 11.2.7.1

Detection threshold (DT) Calculation of DT for given pfa

If the target statistics (in the form of the distribution of signal amplitude, after all processing) are known, a sensible choice of distribution can be made from the options described in Chapter 7. If nothing is known about the distribution, it makes sense to minimize the maximum error by selecting one of an intermediate character, such as one dominant plus Rayleigh (1D þ R). An ROC curve for this case is shown in Figure 11.2. If pfa is known or given, the detection threshold for this situation can be estimated using9 DT50 ð pfa Þ 10 log10

1 log2 2pfa

0:8 dB;

ð11:22Þ

which is accurate to 0.1 dB for pfa < 10 2 with 1D þ R statistics. For other signal amplitude distributions, the error incurred by assuming 1D þ R anyway is no more than 0.8 dB even for extreme situations of a stable signal (no ﬂuctuations) on the one 9 Unless otherwise speciﬁed, the detection threshold is that corresponding to a 50 % detection probability.

582 The sonar equations revisited

[Ch. 11

Figure 11.2. ROC curve in the form 10 log10 R vs. log10 pfa for 1D þ R amplitude signal in Rayleigh noise (the relevant curve is the one for pd ¼ 12 ).

hand (column 2 of Table 11.2) or strong ﬂuctuations (Rayleigh fading) on the other (column 3). 11.2.7.2

Estimation of pfa

Table 11.2. Error in DT (i.e., the value of DTtrue DT1DþR in decibels) incurred by assuming 1D þ R if the true signal amplitude statistics follow a Dirac (nonﬂuctuating) or Rayleigh (strongly ﬂuctuating) distribution.

In order to use Equation (11.22) it might be necessary to estimate the value of pfa from knowledge of sonar parameters. For example, if the number of beams is not known, the directivity factor GD (i.e., 10 DI=10 ) can be used instead of Nbeams to estimate pfa . Speciﬁcally, use of the approximation Nbeams GD in (see Chapter 3) pfa ¼

pfa

Dirac

Rayleigh

10 4

0.7

þ0.8

10 8

0.8

þ0.8

10 12

0.8

þ0.8

nfa ; Df Nbeams

ð11:23Þ

nfa : Df GD

ð11:24Þ

yields pfa ¼

In some circumstances there might be special interest in just one beam. In that case it could be that contacts in all but that one beam are ignored, so that Nbeams ¼ 1 would be used in Equation (11.23). The false alarm probability can then be increased (e.g., by increasing the detection threshold) without aﬀecting the false alarm rate nfa . The beneﬁt of higher false alarm probability is a lower detection threshold and hence higher detection probability for a given SNR in the chosen beam. The disadvantage is that a signal in any other beam would be missed.

Sec. 11.2]

11.2 Passive sonar with coherent processing: tonal detector

583

The same logic justiﬁes a reduced value of Df for use in Equation (11.24) if there is interest in only a narrow range of frequencies. A possible application is for a tracking system, with prior knowledge about the expected target bearing and frequency obtained from a previous contact.

11.2.8

Worked example

The narrowband passive sonar worked example of Chapter 3 (henceforth abbreviated NBp) is used here as a basis to develop a more realistic scenario. The extra realism makes it necessary to make use of an automated software tool to carry out the calculations, with the implied acceptance of assumptions and approximations made by that tool. Where necessary, some accuracy is sacriﬁced to achieve the desired realism. Perhaps the most unrealistic assumption of all made in Chapter 3 (NBp) was that of inﬁnite water depth. For the present example, a ﬁne-grain unconsolidated sediment10 seabed is placed at a depth of 5 km, representative of the deep basins of the Paciﬁc Ocean. Further, a winter sound speed proﬁle, corresponding to the northwest Paciﬁc (NWP) case from Chapter 4 (see also Chapter 9), is considered, and the wind speed is increased from 5 m/s to 7 m/s. Further minor changes compared with the Chapter 3 example include — the array gain AG is approximated by the directivity index DI; — Hann shading is applied to the receiver array, resulting in a small reduction in DI; — the signal amplitude is assumed to follow a 1D þ R distribution (instead of a Rayleigh distribution), resulting in a reduction of 0.8 dB in the detection threshold; — loss mechanisms due to sound absorption (chemical relaxation) and sea surface scattering are introduced; — a moderate shipping density of 500 Mm 2 (i.e., 5 10 4 km 2 , or ﬁve ships per 100 km square) is considered for ambient noise prediction.

11.2.8.1

Propagation loss and signal excess

The eﬀect of these changes on propagation is demonstrated in Figure 11.3. The most noticeable changes in the ﬁrst 50 km (compared with NBp) are associated with the reﬂection of sound from the seabed. The contributions from bottom-interacting paths are labeled ‘‘bottom reﬂection’’ and ‘‘bottom refraction’’ in Figure 11.3. At greater distances, the sound speed proﬁle makes itself felt through the appearance of convergence zones (CZs) at intervals of 65 km. 10 The parameters correspond to a grain size of 8 (see Chapter 4), which is intermediate between coarse clay and ﬁne silt.

584 The sonar equations revisited

[Ch. 11

Figure 11.3. Propagation loss [dB re m 2 ] vs. range for NWP winter case. The frequency is 300 Hz (INSIGHT—Ainslie et al., 1996).

For the calculation of SNR it is useful to introduce the concepts of in-beam signal and noise levels IBSL 10 log10 Y S ¼ SL PL þ SG

ð11:25Þ

IBNL 10 log10 Y N ¼ NLf þ BW þ NG:

ð11:26Þ

and The sonar equation can then be written SENB ¼ IBSL IBNL DT: S

ð11:27Þ

N

These concepts are closely related to the levels L and L introduced in Chapter 3, as follows: IBSL ¼ L S þ SG ð11:28Þ and IBNL ¼ L N þ SG; ð11:29Þ such that IBSL IBNL ¼ L S L N : ð11:30Þ In Chapter 3, L N was also referred to as ‘‘in-beam noise level’’. The change in nomenclature is intended to emphasize the reﬁnement in the deﬁnition of this term, which here distinguishes between the contributions from noise gain (NG) and signal gain (SG) to total array gain (AG). From the above equations, the diﬀerence between

Sec. 11.2]

11.2 Passive sonar with coherent processing: tonal detector

585

IBNL and L N (and also between IBSL and L S ) is equal to SG, which is normally a small correction. In addition to large corrections to PL, changes in the scenario also lead to a smaller, but still signiﬁcant, change to NL. The combined eﬀect of increasing the wind speed and the more realistic sound speed proﬁle increases NL by about 4 dB. Hann shading reduces DI by 1.8 dB, resulting in an overall increase in in-beam noise of about 6 dB, as illustrated by Figure 11.4. In general, wind speed might also aﬀect propagation loss through increased surface scattering, but on this occasion the eﬀect is negligible. The upper graph of Figure 11.4 shows signal level as a function of target range, calculated using propagation loss from Figure 11.3. Also shown for comparison (lower graph) is the corresponding prediction for the NBp example. In both graphs the horizontal line is the noise level. By deﬁnition, detection is likely ( pd > 12 ) if the signal exceeds the noise by at least the detection threshold, which for this case is 13 dB (see Table 11.3). Threshold crossings occur for ranges up to 2 km, and at the ﬁrst CZ. The maximum signal excess at the second CZ, though negative, is close to zero, meaning that the detection probability is close to 50 %. A complete description of the scenario is provided in Figure 11.5. 11.2.8.2

What is the detection range?

Calculation of either detection probability or detection range can be simpliﬁed by ﬁrst calculating the ﬁgure of merit (FOM). The FOM calculated using the numerical values from Table 11.3 is FOMNB ðzarr Þ ¼ SL þ ðAG BWÞ NLf ðzarr Þ DT ¼ 73:8

dB re m 2 :

ð11:31Þ

The range at which PL is equal to FOM, denoted r50 , is known as the detection range. If the crossing is unambiguous, the concept is a useful one, but this is not always the case. There can be several FOM crossings, making the deﬁnition of r50 unclear. For the present situation, the oscillatory nature of PL results in multiple crossings. Figure 11.6 shows signal excess vs. range and depth for distances up to and including the ﬁrst CZ. In this graph the gray shading indicates regions where the threshold is either just exceeded or nearly exceeded (2 dB < SE < þ2 dB). Black indicates that the threshold is exceeded by at least 2 dB. For a target depth of 10 m, unambiguous detections occur during the ﬁrst 2 km, but on the basis of Figure 11.6 it seems misleading to declare a detection range of only 2 km. Further threshold crossings occur also at the ﬁrst CZ (and even at the second CZ for a slightly deeper target), but, because of the gaps between them, it would be similarly misleading to declare a detection of range of 65 km or 130 km. A close-up of the ﬁrst CZ is shown in Figure 11.7, illustrating the intermittent nature of CZ detections. The purpose of examining signal excess behavior in this level of detail is to raise and address the issue of highly variable signal excess. Rapid variations through the CZ region provide a natural laboratory in which to do so. No statement is made here about the accuracy in detail of Figures 11.6 and 11.7, but it is expected that other

586 The sonar equations revisited

[Ch. 11

Figure 11.4. In-beam signal and noise levels vs. range for NWP winter (upper) and Chapter 3 NBp worked example (lower). The frequency is 300 Hz. Small diﬀerences in level between the lower graph and its equivalent from Chapter 3 are due to the use here of a linear range scale and a diﬀerent computer model. The reason for including the comparison is to enable the reader to identify changes due to the scenario alone.

Sec. 11.2]

11.2 Passive sonar with coherent processing: tonal detector

587

Table 11.3. Sonar equation calculation for NWP winter. Description

Symbol

Value

Source level

SL

133.9

dB re mPa 2 m 2

Noise spectrum level

NLf

64.2

dB re mPa 2 Hz1

Array gain

AG

11.1

dB re 1

Detection threshold

DT

13.0

dB re 1

Analysis bandwidth

BW

6.0

dB re Hz

sonar performance models would predict a comparable variation in signal excess through the CZ for the present worked example. 11.2.8.3

Alternative performance measures

Despite diﬃculties with an unambiguous deﬁnition of r50 , it remains desirable to represent sonar performance by means of a single variable. Can the deﬁnition of detection range be modiﬁed in a simple way to achieve this? This question leads to the concepts of detection volume Vd and detection area Ad introduced below, and associated ranges rV and rA . These measures are extremely robust and applicable to bistatic or multistatic sonar geometries. 11.2.8.3.1 Detection volume (radius of equivalent volume sphere) Consider ﬁrst the total volume of space ensoniﬁed by the sonar (i.e., the total volume within which a target, if present, is likely to be detected) þ Vd ¼ pd ðxÞ dV; ð11:32Þ where the integration is over all space. The simplicity of Equation (11.32) is appealing. It oﬀers an elegant measure of performance as a single parameter, but there are two complications. The ﬁrst is that pd pfa everywhere, which means that the integral is inﬁnite. This is a consequence of including (in the deﬁnition of ‘‘detection’’) those threshold crossings due to random ﬂuctuation in noise alone. Such crossings are treated as ‘‘detections’’ if a target is present, regardless of whether or not sound from the target is responsible for that crossing. Crossings that are not caused by the target can be removed in a pragmatic fashion by subtracting pfa from the integrand: þ Vd ðpd pfa Þ dV; ð11:33Þ rendering the integral ﬁnite. In cylindrical co-ordinates this becomes (for water

588 The sonar equations revisited

[Ch. 11

OPTIONS

Coherence Profile mode Signal Processing Passive Beam Pattern Wind Noise Shipping Noise Rain Noise Self Noise No. of bottom reflections No. of bottom refractions

PASSIVE PARAMETERS

Analysis frequency Array depth Array length Electronic steering angle HLA heading clockwise from North Analysis bandwidth Detection threshold (narrowband)

SOUND SPEED PROFILE

USER-DEFINED LAW: Water depth 5000 m c (m/s) z (m) 1536.64 0 1536.66 10 1536.76 20 1536.85 30 1537 50 1536.93 75 1536.08 100 1533.44 125 1530.02 150 1523.19 200 1517.54 250 1513.82 300 1504.48 400 1493.56 500 1486.78 600 1483.65 700 1482.77 800 1482.07 900 1482.39 1000 1482.41 1100 1483.29 1200 1484.27 1300 1484.28 1400 1485.48 1500 1488.33 1750 1490.94 2000 1498.28 2500 1506.07 3000 1514.35 3500 1522.98 4000 1531.8 4500 1541.01 5000

Coherent Value vs depth Narrow band Horizontal steered—Cosine squared Wind noise model Shipping noise model Rain noise off Self noise off 5 5 300. Hz 30. m 45. m 0. degrees 90. degrees 0.25 Hz 13. dB

dc/dz (/s) 2.e-3 1.e-2 9.e-3 7.5e-3 -2.8e-3 -3.4e-2 -0.1056 -0.1368 -0.1366 -0.113 -7.44e-2 -9.34e-2 -0.1092 -6.78e-2 -3.13e-2 -8.8e-3 -7.e-3 3.2e-3 2.e-4 8.8e-3 9.8e-3 1.e-4 1.2e-2 1.14e-2 1.044e-2 1.468e-2 1.558e-2 1.656e-2 1.726e-2 1.764e-2 1.842e-2

H (m) 10 10 10 20 25 25 25 25 50 50 50 100 100 100 100 100 100 100 100 100 100 100 100 250 250 500 500 500 500 500 500

VOLUME LOSS

ALTERNATIVE LAW:

Thorp

SURFACE LOSS

ALTERNATIVE LAW: Wind speed

Marsh-Schulkin-Kneale 7. m/s

BOTTOM LOSS

GEO-ACOUSTIC PARAMETERS: Sediment depth cs/cw Sediment velocity gradient Maximum sediment velocity Sediment density Sediment attenuation coefficient Sediment attenuation gradient

500. m 1. 1. /s 1900. m/s 1.4 6.e-002 dB/m/kHz 1.e-006 dB/m/kHz/m

TARGET PARAMETERS

Target range Target depth Narrow-band radiated noise level at 1kHz Radiated noise gradient Target bearing clockwise from North

* 10. m 133.9 dB 0. dB/octave 0. degrees

BACKGROUND PARAMETERS

Ship length Ship speed Shipping density Wind speed

100. m 5. m/s 5.e-004 /km2 7. m/s

Figure 11.5. Input parameters for northwest Paciﬁc (NWP) problem.

Sec. 11.2]

11.2 Passive sonar with coherent processing: tonal detector

589

Figure 11.6. Signal excess [dB] vs. range and depth for NWP winter (30 m sonar depth). The frequency is 300 Hz (INSIGHT).

depth H) Vd ¼

ð 1 ð 2 ð H 0

0

½ pd ðr; z; Þ pfa r dr d dz:

ð11:34Þ

0

Given the volume Vd , one can imagine an equivalent sphere of the same volume, whose radius is 3Vd 1=3 rV : ð11:35Þ 4

In contrast to r50 , the detection volume and hence the equivalent sphere radius rV also, are robust and unambiguous measures of performance. They are not overly sensitive to small changes in SNR or DT, and the equivalent radius rV provides a useful feel for sonar coverage in terms of distance, which readers might ﬁnd more intuitive than volume. A good measure of performance needs to recognize the value of providing new information. If it is known in advance that no target of interest is present at a particular location, there is little value to be added by a sonar conﬁrming that known fact. In other words, a good measure would include only the proportion of Vd in which elementary prior knowledge would not already rule out the possibility of target presence. The second complication is that the measure rV fails this test.

590 The sonar equations revisited

[Ch. 11

Figure 11.7. Signal excess [dB] vs. range and depth for NWP winter (30 m sonar depth) (closeup of ﬁrst CZ). The frequency is 300 Hz (INSIGHT).

11.2.8.3.2

Detection area (radius of equivalent area circle)

Regardless of any speciﬁc local knowledge, common sense tells an archeologist looking for a sunken wreck that he can safely limit his search to targets satisfying the condition ztgt H, whereas a pelagic ﬁshing trawler is interested in targets whose depths are in the range 0 < ztgt < H. The ability to detect targets that do not meet these elementary criteria has little value and would be excluded from any good performance measure. Thus, an improvement to rV is obtained by weighting the integrand according to the prior probability of target presence. A suitable weighting factor is provided by the depth probability distribution function WðzÞ (the prior probability of the target being within unit depth, given that there is one and only one target present), the application of which results in a measure with dimensions of area11 þ Ad ¼ WðzÞ½ pd ðr; z; Þ pfa r d dr dz: ð11:36Þ Consider ﬁrst the case of the ﬁsherman who wishes to search for targets in the depth 11 In general, the probability distribution W ðzÞ could also depend on the bearing and range coordinates, leading naturally to a probability per unit area or volume, but this line of thinking is not explored further.

Sec. 11.3]

11.3 Passive sonar with incoherent processing: energy detector

591

range ½0; H . Assume for simplicity that the ﬁsh have no preference within this range, so that the appropriate weighting function is 1 z 1 WðzÞ ¼ P : ð11:37Þ H H 2 Assuming cylindrical symmetry, the integral becomes ð ð 2 1 H Ad ¼ ½ p ðr; zÞ pfa r dr dz: H 0 0 d

ð11:38Þ

In terms of a plan view, Ad is the area coverage of sonar for the assumed depth probability distribution. This can be related to a distance by calculating the radius of an equivalent circle of the same area: 1=2 Ad rA : ð11:39Þ

This distance is closely related to sweep width, which, if multiplied by platform speed, provides a simple estimate of the area coverage rate. Another possibility is the depth being known precisely (the case of the archeologist) WðzÞ ¼ ðz HÞ; ð11:40Þ where ðxÞ is the Dirac delta function, and hence ð1 Ad ¼ 2 ½ pd ðr; HÞ pfa r dr:

ð11:41Þ

0

The incorporation of prior knowledge in rA makes it a more useful measure of detection performance than rV . Depth knowledge need not be sophisticated to be useful—in the above examples it is almost trivial.

11.3

11.3.1

PASSIVE SONAR WITH INCOHERENT PROCESSING: ENERGY DETECTOR Sonar equation

The passive sonar equation with incoherent broadband processing is SEBB ¼ SL PL ðNL PGÞ DT:

ð11:42Þ

The ﬁve individual terms on the right-hand side are described below. To avoid the complication of a beam pattern, an omni-directional source is assumed in this section from the beginning.12 12 In some situations it might be necessary to model the beam pattern of the sound source (e.g., of a dipole source such as a surface ship at low frequency—see Chapter 8). The eﬀect of doing so would be to increase PL at positions outside the main beam.

592 The sonar equations revisited

11.3.2

[Ch. 11

Source level (SL)

The broadband source level is the total source factor, integrated over the sonar bandwidth, in decibels. ð SL 10 log10 ðS0 Þf df dB re mPa 2 m 2 ; ð11:43Þ where, in terms of the spectral density of radiant intensity ðJf Þ, ð ðS0 Þf c Jf df :

11.3.3

ð11:44Þ

Broadband propagation loss (PL)

Broadband propagation loss is averaged over frequency. Speciﬁcally, if PL ¼ 10 log10 FBB ; then, assuming a white source spectrum (see Chapter 3)13 ð 1 fm þB=2 FBB ¼ F ð f Þ df ; B fm B=2 coh

ð11:45Þ

ð11:46Þ

where B is the sonar bandwidth; and fm is its center frequency. The integration of Equation (11.46) can be carried out more eﬃciently by ﬁrst simplifying the integrand. One approach is to smooth the integrand by replacing Fcoh with Finc , so that ð 1 FBB F ð f Þ df ; ð11:47Þ B inc as used in Chapter 3. Here the ‘‘inc’’ subscript is an abbreviation for ‘‘incoherent’’ in the ray sense of the word (which implies that the relative phase of multiple ray arrivals is neglected) and not the mode sense (which would refer to the relative phase of interfering modes being neglected). Thus, the step from Equation (11.46) to Equation (11.47) neglects coherent interference eﬀects such as cancellation between direct and surface-reﬂected paths. This coherent eﬀect is not negligible, even for incoherent broadband processing, if the distance between the sonar (or target) and the sea surface is a few wavelengths or less at the center frequency, in which case use of Equation (11.46) is required. If the bandwidth is small relative to the center frequency, the approximation of Harrison and Harrison (1995) may be used to avoid the cumbersome integration by replacing the average in frequency with one in range. As an example of broadband propagation loss, consider the Lloyd mirror problem from Chapter 3, with a linear variation of attenuation with frequency of 13

A more general expression, vaid for a colored source spectrum, is Ð Fcoh ð f ÞW Sf df Ð S FBB : W f df

Sec. 11.3]

11.3 Passive sonar with incoherent processing: energy detector

593

the form ð f Þ ¼ m þ f ð f fm Þ:

ð11:48Þ

In this situation, Equation (11.47) becomes FBB ¼

2 expð2 m rÞ sinhcð f BrÞ: r2

ð11:49Þ

For small B the broadband (BB) propagation factor FBB reduces to the usual expression for spherical spreading FBB

2e 2 m r : r2

ð11:50Þ

More generally (still for small bandwidth), FBB may be approximated by Finc ð fm Þ. Returning to Equation (11.49) and using the deﬁnition of sinhcðxÞ (see Appendix A) sinhcð f BrÞ ¼

expð f BrÞ expð f BrÞ ; 2 f Br

ð11:51Þ

it follows that (for a suﬃciently long-range or large bandwidth) sinhcð f BrÞ

expð f BrÞ : 2 f Br

ð11:52Þ

Therefore, in the same limit, FBB

11.3.4

exp½2rð m f B=2Þ

: f Br 3

ð11:53Þ

Broadband noise level (NL)

The broadband noise level is the noise spectral density, integrated across the sonar bandwidth, and expressed in decibels ð NL ¼ 10 log10 10 NLf =10 df : ð11:54Þ The previously introduced concepts of foreground and background noise, and of acoustic and non-acoustic noise (see Section 11.2.4), apply more or less unchanged here.

11.3.5

Processing gain (PG)

The array gain term (AG) from Chapter 3 is generalized here to processing gain (PG), which also includes ﬁlter gain (FG). The implied assumption of Chapter 3 is that the ﬁlter response is ﬂat (independent of frequency) within the sonar bandwidth. The eﬀect of a departure from this assumption is considered here.

594 The sonar equations revisited

[Ch. 11

Let RBB denote the SNR after all beamforming and (temporal) ﬁltering, while Rhp and Rarr are the SNRs at input and output of the beamformer, before ﬁltering.14 The respective gains due to beamforming and ﬁltering, respectively, are GA

Rarr Rhp

ð11:55Þ

GF

RBB : Rarr

ð11:56Þ

RBB ¼ GA GF : Rhp

ð11:57Þ

and

Total processing gain is GP In decibels, these become AG 10 log10 GA ¼ 10 log10

Rarr ; Rhp

ð11:58Þ

FG 10 log10 GF ¼ 10 log10

RBB ; Rarr

ð11:59Þ

and PG 10 log10 GP ¼ AG þ FG: 11.3.5.1

ð11:60Þ

Array gain (AG) and directivity index (DI)

For broadband sonar, array gain (AG) can be calculated using Equation (11.58). Because of the complexity involved in the full calculation, AG is sometimes approximated by DI.15 11.3.5.2

Filter gain (FG)

Filter gain can be calculated using Equation (11.59). Let Hð f Þ denote the combined transfer function of all ﬁlters.16 The post-ﬁlter SNR RBB is then given by ð Hð f ÞY Sf ð f Þ df RBB ¼ ð ; ð11:61Þ Hð f ÞY N ð f Þ df f 14

Or, strictly speaking, after a hypothetical ﬂat response ﬁlter. This statement begs the question of how to deﬁne DI for a broadband sonar. One possibility is as the (broadband) array gain, evaluated for a white plane wave signal in white isotropic noise. 16 The transfer function Hð f Þ incorporates the eﬀects of hydrophone sensitivity, analogue ﬁlters (such as an anti-alias ﬁlter), analogue-to-digital converter, and any digital ﬁlter. Recall that Rarr is a hypothetical SNR assuming a ﬂat response ﬁlter, whereas RBB is the true SNR, after all ﬁltering. 15

Sec. 11.3]

11.3 Passive sonar with incoherent processing: energy detector

and hence, from the deﬁnition of GF (Equation 11.56) it follows that ð ð S Hð f ÞY f ð f Þ df Y N f ð f Þ df ð ð GF ¼ : Hð f ÞY N Y Sf ð f Þ df f ð f Þ df

595

ð11:62Þ

As a ﬁrst approximation, it is common to assume GF 1 (i.e., FG 0 dB). This approximation is a good one if either the ﬁlter response or the SNR spectrum is ﬂat (i.e., either Hð f Þ or Y Sf ð f Þ=Y N f ð f Þ is approximately constant). However, if the narrowband signal-to-noise ratio varies with frequency, the broadband SNR, after all ﬁltering, depends on the frequency response of the true ﬁlter. In other words, if the receiver response is not ﬂat, it becomes necessary to account for departures from the hypothetical ﬂat response ﬁlter considered so far. An important special case is that of a pre-whitening ﬁlter,17 for which Hð f Þ ¼

1

and hence 1 GF ¼ B

ð

ð11:63Þ

YN f ðfÞ ð

Y Sf ð f Þ df YN f ðfÞ

ð

YN f ð f Þ df :

ð11:64Þ

Y Sf ð f Þ df

11.3.5.2.1 Filter gain for a white signal spectrum Consider a white signal spectrum and a power law frequency dependence for the noise of the form x YN ð11:65Þ f /f ; such that ð ð fþ 1 fþ x GF ¼ 2 f df f x df ; ð11:66Þ B f f where f ¼ fm B=2; ð11:67Þ B is the sonar bandwidth; and fm is the center frequency. If the noise is also white (i.e., x ¼ 0), the ﬁlter has no eﬀect and the ﬁlter gain is 0 dB.18 The special case x ¼ 1 (known as pink noise ) leads to GF ¼ 17

1 2þD loge ; D 2D

ð11:68Þ

A pre-whitening ﬁlter is one that is designed to achieve white noise (i.e., noise whose power spectral density is independent of frequency) at the ﬁlter output. 18 More generally, if the signal and noise have the same spectral gradient (in decibels per octave), the signal-to-noise ratio is independent of frequency and consequently there is no gain possible from further ﬁltering, irrespective of the functional form of Hð f Þ.

596 The sonar equations revisited

[Ch. 11

Table 11.4. Filter gain vs. bandwidth in octaves for a white signal and colored noise. No. of octaves

Pink noise (or blue) ðx ¼ 1Þ

Red noise (or violet) ðx ¼ 2Þ

Noct

D (Equation 11.69)

GF

FG

GF

FG

1

0.67

1.04

0.2 dB

1.17

0.7 dB

3

1.56

1.34

1.3 dB

3.04

4.8 dB

5

1.88

1.84

2.7 dB

11.0

10.4 dB

7

1.97

2.46

3.9 dB

43.0

16.3 dB

9

1.99

3.13

5.0 dB

171

22.3 dB

where D is the fractional bandwidth D B=fm :

ð11:69Þ

For other power laws, Equation (11.66) can be written GF ¼

1þx 1x 1x ð f 1þx þ f Þð f f þ Þ : 2 2 ðx 1ÞB

ð11:70Þ

An important special case is x ¼ 2 (‘‘red noise’’), as the high-frequency spectra for wind, shipping, and rain noise all approximate to this value (Chapter 8). For red noise, Equation (11.70) simpliﬁes to19 GF ¼

12 þ D 2 : 12 3D 2

ð11:71Þ

If jxj 2 and for a single octave, the eﬀects are less than 1 dB, increasing to up to 10 dB for ﬁve octaves, as indicated by the shaded entries in Table 11.4. 11.3.5.2.2 Filter gain for a colored signal spectrum Consider the noise spectrum of Equation (11.65) and the signal spectrum Y Sf / f y :

ð11:72Þ

For this combination, ﬁlter gain becomes ð fþ 1 GF ¼ B

ð fþ f

fy f df ð f þ fx f

19

Equation (11.71) holds also for x ¼ þ2.

f x df : y

f df

ð11:73Þ

Sec. 11.3]

11.3 Passive sonar with incoherent processing: energy detector

597

Provided that none of x, y, or y x is equal to 1, it follows that GF ¼

11.3.6 11.3.6.1

xþ1 yþ1 f yxþ1 f yxþ1 f xþ1 þ f þ : yþ1 ðx þ 1Þðy x þ 1Þ B f yþ1 þ f

ð11:74Þ

Broadband detection threshold (DT) Calculation of DT for given pfa

Apart from PG itself, the detection threshold (DT) is the sonar equation term for which the diﬀerence between coherent and incoherent processing most makes itself felt. The objective of incoherent processing is not to increase SNR, but to reduce the ﬂuctuations in both signal and noise in such a way as to facilitate detection at a lower SNR—in other words, to reduce DT. The net result is a completely diﬀerent form for DT, which varies with the sonar bandwidth B and incoherent integration time T in the manner described below. The deﬁnition of DT is in terms of the SNR at the output of coherent processing: DTð pfa Þ 10 log10 ðRBB Þ50 :

ð11:75Þ

A simple expression, valid for large BT and any false alarm probability, independent of signal ﬂuctuations for BT exceeding 20 or so, is20 pﬃﬃﬃﬃﬃﬃﬃ DTð pfa Þ 10 log10 ½erfc 1 ð2pfa Þ 10 log10 BT ; ð11:76Þ with an accuracy of ca. 1 dB. A good approximation for a non-ﬂuctuating signal (for pd ¼ 12 ), due to Albersheim, is (see Chapter 7) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:62 DTð pfa Þ 10x log10 loge 10 log10 2BT ; ð11:77Þ pfa where the factor x is 0:456 x ¼ xðBTÞ ¼ 0:62 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 2BT þ 0:44

ð11:78Þ

Equation (11.77) is valid for a stable (i.e., non-ﬂuctuating) signal in Gaussian noise. It holds for any BT across a wide range of pfa values. Figure 11.8 shows the quantity DT þ 5 log10 ð2BT Þ, evaluated using Equation (11.77), plotted vs. BT for pfa between 10 12 and 10 4 . From Chapter 7 it can be seen that, for pfa in the range 10 12 < pfa < 0:3, Equation (11.77) is accurate to within 0.5 dB. The main beneﬁt of Albersheim’s approximation is its validity for small BT, at least for a nonﬂuctuating signal. For large BT, Equation (11.76) (which is not restricted to a stable signal) is more accurate. If BT is large, it is not unusual for the broadband detection threshold to become negative (i.e., DT < 0 dB), which means that detection is possible even when noise 20 This and subsequent expressions for the detection threshold follow by substituting M ¼ 2BT and pd ¼ 12 into the relevant results from Chapter 7 for incoherent combination of multiple pings.

598 The sonar equations revisited

[Ch. 11

Figure 11.8. Albersheim’s approximation for the detection threshold, in the form DT þ 5 log10 ð2BTÞ vs. BT for pd ¼ 0.5 and various pfa between 10 12 and 10 4 for a nonﬂuctuating signal in Rayleigh noise.

power exceeds that of the signal. This idea might sound counterintuitive, but what matters for detection of signal is the extent to which ‘‘signal plus noise’’ can be distinguished from ‘‘noise alone’’, which depends not only on the signal-to-noise ratio but also on the magnitude of any ﬂuctuations present. To illustrate this point, consider the extreme case for which noise is known to be precisely constant. Any departure from this constant, however small, can then be attributed to the presence of a signal. Incoherent processing has the eﬀect of smoothing out natural ﬂuctuations in both signal and noise, making it possible to detect a weak signal in the presence of a stronger noise background. For example, consider the case BT ¼ 500 and pfa ¼ 10 4 . From Figure 11.8, the value of the quantity DT þ 5 log10 ð2BTÞ can be read oﬀ as 6 dB, and hence DT ¼ 6 5 log10 ð1000Þ 9 dB: Thus, for this example, a 50 % chance of detection occurs when noise power is eight times greater than that of the signal.

11.3.6.2

Estimation of false alarm probability

The estimation of pfa can be simpliﬁed by assuming that the directivity gain GD is

Sec. 11.3]

11.3 Passive sonar with incoherent processing: energy detector

599

ﬁxed for all octaves (implying a nested array21 for broadband sonar). In this situation, Nbeams can be replaced by GD so that (from Chapter 3)22 pfa ¼

nfa Dt Nintervals Nbeams

ð11:79Þ

nfa Dt : Nintervals GD

ð11:80Þ

becomes pfa ¼

11.3.7

Worked example

The broadband passive sonar worked example of Chapter 3 (henceforth abbreviated BBp) is used here as a basis for a more realistic scenario. The main changes are — conversion to shallow water (30 m water depth, isothermal sound speed proﬁle. and a sand seabed23), with the target (communications transmitter) at a depth of 20 m; — reduction of integration time from 10 s to 0.1 s; — introduction of rough surface scattering; — introduction of shipping noise associated with a moderate shipping density of 500/Mm 2 ; — use of a vertical receiver array instead of a horizontal one. Detection performance is estimated below for this revised acoustic communications intercept problem. The disruptive eﬀect of rainfall is considered in Section 11.3.7.5. 11.3.7.1

Propagation loss

Propagation loss is plotted vs. range and depth in Figure 11.9 for the shallow-water sand (SWS) case, and compared also with the PL for the Chapter 3 BBp problem, which is plotted alongside for comparison. Losses are generally lower than for the BBp situation, due mainly to additional contributions to the signal resulting from multiple reﬂections from the seabed (see Chapter 9). 11.3.7.2

Signal-to-noise ratio

The source spectrum level SLf is equal to 100.9 dB re mPa 2 m 2 /Hz (unchanged from BBp). The resulting signal spectrum (SLf PLm ) and noise spectrum (NLf ) are 21

A nested array is a hydrophone array with non-uniform spacing between hydrophones, typically arranged with a high density (small spacing) close to the middle of the array and a low density (large spacing) at each end. The spacing is chosen in such a way that two or more subarrays can be made using selected hydrophones. For example, a short array of densely spaced hydrophones (the middle ones) is used at high frequency, whereas a long array with widely spaced hydrophones is used at low frequency. 22 If the footprint O is the same for all beams, the number of beams needed to cover 4

steradians of solid angle is 4 =O, which is equal to GD (Chapter 6). 23 The parameters correspond to a grain size of 1.5, or medium sand (see Chapter 4).

600 The sonar equations revisited

[Ch. 11

Figure 11.9. Propagation loss [dB re m 2 ] vs. range and depth for SWS (upper) and for the Chapter 3 BBp worked example (lower) (INSIGHT).

Sec. 11.3]

11.3 Passive sonar with incoherent processing: energy detector

601

Figure 11.10. Signal and noise spectra for SWS at range 10 km for array depth 10 m and target depth 20 m. ‘‘hp’’: values at the hydrophone, before beamforming; ‘‘in-beam’’: in-beam levels (IBSLf , IBNLf ) (INSIGHT).

plotted in Figure 11.10 as the thin solid and dashed curves, respectively (labeled ‘‘hp’’). The two curves labeled ‘‘in-beam’’ are the in-beam signal and noise (i.e., array response), calculated as IBSLf ¼ 10 log10 Y Sf

ð11:81Þ

IBNLf ¼ 10 log10 Y N f ;

ð11:82Þ

ð Y Sf ¼ Q Sf O BðOÞ dO

ð11:83Þ

and where

and YN f

ð

¼ QN f O BðOÞ dO:

ð11:84Þ

The in-beam signal and background spectrum levels (evaluated at the center frequency) are plotted vs. target distance in Figure 11.11 for the SWS case (thin lines) and for the original BBp problem (thick lines). As a result of reduced propagation loss, both signal and noise are higher than for BBp. At long range the signal increases by more than the noise, so the SNR increases there.

602 The sonar equations revisited

[Ch. 11

Figure 11.11. In-beam signal and noise levels vs. range for SWS (thin lines) and BBp (thick lines); the dip in IBSL at 0.1 km is caused by the beam pattern of the vertical line array, which results in negative signal gain at this distance (INSIGHT).

11.3.7.3

Detection threshold and signal excess

The value of the detection threshold (DT) increases by about 10 dB due to the reduced incoherent integration time compared with BBp. If the permissible false alarm rate is held ﬁxed (at one per hour), the probability of false alarm must decrease to pfa ¼ 2:8 10 5 , which increases DT by a further 1.7 dB. The combined eﬀect gives DT ¼ 6.6 dB, calculated using Albersheim’s approximation (Equation 11.77). Given that processing gain is the sum of array and ﬁlter gains PG ¼ AG þ FG;

ð11:85Þ

the broadband sonar equation (Equation 11.42) can be written as SEBB ¼ IBSL IBNL DT þ FG;

ð11:86Þ

where IBSL is given by Equation (11.25) and IBNL ¼ NL þ NG:

ð11:87Þ

Equation (11.86) can be written in terms of the in-beam spectral density levels IBSLf and IBNLf SEBB ¼ IBSLf IBNLf DT þ FG þ fIBSL IBNL ðIBSLf IBNLf Þg; ð11:88Þ

Sec. 11.3]

11.3 Passive sonar with incoherent processing: energy detector

603

Figure 11.12. Signal excess [dB] vs. range and depth for SWS. White indicates areas of high detection probability ( pd > 0.5) and black means low detection probability ( pd < 0.5). In the gray region, detection probability is close to 50 % (signal excess is in the range 1 dB < SE < þ1 dB) (INSIGHT).

which, neglecting the term in curly brackets, approximates to24 SEBB IBSLf IBNLf DT þ FG:

ð11:89Þ

The right-hand side of Equation (11.89) is plotted vs. range and intercepting sonar depth in Figure 11.12, with spectral densities evaluated at the center frequency and FG ¼ 0 dB. The target (in this context the communications transmitter) is at a depth of 20 m. It is apparent that the achieved performance is sensitive to the chosen sonar depth, with the optimum value in the region of 15 m to 20 m, and poor performance close to the boundaries. A good choice of depth is therefore important, illustrating one of the uses of sonar performance modeling, namely as a decision aid for sonar deployment. A summary of relevant sonar equation terms is presented in Table 11.5. For a more detailed description of the scenario see Figure 11.13.

24 The introduction of the term in curly brackets in Equation (11.88), only to neglect it in the very next line, serves to make explicit the nature of the approximation leading to Equation (11.89).

604 The sonar equations revisited

[Ch. 11

Table 11.5. Sonar equation calculation for shallow-water sand (SWS). Description

Symbol

Value

Source spectrum level

SLf

100.9 dB re mPa 2 m 2 Hz1

Noise spectrum level (center frequency)

NLf

51.7 dB re mPa 2 Hz1

Array gain a

AG

14.0 dB re 1

Detection threshold

DT

6.6 dB re 1

a Array gain is approximated for this example by the directivity index, evaluated at the center frequency.

OPTIONS

Coherence Profile mode Signal Processing Passive Beam Pattern Wind Noise Shipping Noise Rain Noise Self Noise No. of bottom reflections No. of bottom refractions

Incoherent Value vs depth Broad band Vertical steered—Rectangular Wind noise model Shipping noise model Rain noise off Self noise off 25 5

PASSIVE PARAMETERS

Analysis frequency Array depth Array length Electronic steering angle Analysis bandwidth Detection threshold (broadband)

3. kHz 10. m 6. m 0. degrees 2.5e-004 kHz -6.6 dB

SOUND SPEED PROFILE

USER-DEFINED LAW: Water depth 30 m c (m/s) 1500 1500.16 1500.32 1500.48

z (m) 0 10 20 30

VOLUME LOSS

ALTERNATIVE LAW:

Thorp

SURFACE LOSS

ALTERNATIVE LAW: Wind speed

Marsh-Schulkin-Kneale 5. m/s

BOTTOM LOSS

GEO-ACOUSTIC PARAMETERS: Sediment depth cs/cw Sediment velocity gradient Maximum sediment velocity Sediment density 2.1 Sediment attenuation coefficient Sediment attenuation gradient

dc/dz (/s) 1.6e-2 1.6e-2 1.6e-2

H (m) 10 10 10

500. m 1.2 1.e-006 /s 1900. m/s 0.489 dB/m/kHz 1.e-006 dB/m/kHz/m

TARGET PARAMETERS

Target range Target depth Broad-band spectrum level at 1kHz Broad-band spectral slope Target bearing clockwise from North

* 20. m 100.9 dB/Hz 0. dB/octave 0. degrees

BACKGROUND PARAMETERS

Ship length Ship speed Shipping density Wind speed

100. m 5. m/s 5.e-004 /km2 5. m/s

Figure 11.13. Input parameters for shallow-water sand (SWS).

Sec. 11.3]

11.3.7.4

11.3 Passive sonar with incoherent processing: energy detector

605

Eﬀect of ﬁlter gain

The eﬀect of a non-zero ﬁlter gain is now considered. Applying Equation (11.74) to the spectral slopes measured from Figure 11.10, the gain from noise whitening is GF 1:12, which in decibels amounts to FG ¼ 0.5 dB. Applying this correction to Figure 11.12 results in an increase in the detection range of about 1 km.

11.3.7.5

Eﬀect of rainfall

The main eﬀect of rainfall is to increase the level of ambient noise by an amount that depends on frequency and rainfall rate. For a rain rate Rrain of 3 mm/h, the increase in noise level at 3 kHz is about 4 dB (see Figure 11.14). A second eﬀect is to ﬂatten the noise spectrum and hence reduce ﬁlter gain slightly. Figure 11.15 shows signal excess vs. range and rain rate (calculated assuming FG ¼ 0 dB, irrespective of rain rate). The detection range r50 varies from about 11 km with no rain to 5 km for Rrain ¼ 10 mm/h.

Figure 11.14. In-beam signal (solid curve) and noise (dashed curves) spectra [dB re mPa 2 /Hz] for SWS at a range of 10 km. The rainfall rate increases from 0 to 9 mm/h in steps of 3 mm/h (INSIGHT).

606 The sonar equations revisited

[Ch. 11

Figure 11.15. Signal excess [dB] vs. range and rainfall rate for SWS (INSIGHT).

11.4

11.4.1

ACTIVE SONAR WITH COHERENT PROCESSING: MATCHED FILTER Sonar equation

The processing of both continuous wave (CW) and frequency-modulated (FM) pulses is considered in this section. A CW pulse can be thought of as a special case of an FM pulse, with a phase acceleration of zero (see Chapter 6). The active sonar equation, in its most general form, is SE ¼ ELE ðBLE PGÞ DT;

ð11:90Þ

where ELE is the echo energy level; and BLE is that of the total masking background. An equivalent form is SE ¼ EL ðBL PGÞ DT;

ð11:91Þ

where EL and BL are sound pressure levels of the echo and background, respectively, averaged over the echo duration; and PG is the gain in SNR due to all processing.25 The main qualitative diﬀerence in background level between active and passive sonar is the presence of reverberation in the former. Methods for the calculation of 25

Apart from initial ﬁltering into the processing bandwidth as described in Chapter 3.

Sec. 11.4]

11.4 Active sonar with coherent processing: matched ﬁlter 607

both echo and reverberation are described in Chapter 9. Discussions of the individual terms of Equation (11.91) follow.

11.4.2

Echo level (EL), target strength (TS), and equivalent target strength (TSeq )

Echo level (EL) depends on source level (SL), propagation loss (PL), and target strength (TS). In the presence of multipaths (multiple propagation paths), there is usually no simple equation enabling the calculation of EL from SL, TS, and PL separately. Instead, it is necessary, in general, to sum contributions to the mean square pressure (MSP) of the echo, over many possible multipaths. Speciﬁcally, if the sonar source factor is S0 and the diﬀerential scattering cross-section of the target is O ðin ; out Þ, the signal MSP can be written26 as a double integral over the incident and scattered grazing angles in and out ð Q S ¼ S0 bTx ðin ÞGTx ðin Þ O ðin ; out ÞGRx ðout Þ din dout ; ð11:92Þ where GTx and GRx are the diﬀerential propagation factors to and from the target (see Chapter 9); and bTx is the transmitter beam pattern. It is usual to characterize the properties of the target in terms of its target strength. This quantity was deﬁned in Chapter 3 for a point target in terms of ratios of MSP, and shown there to be equal to the diﬀerential scattering cross-section, evaluated in the backscattering direction and expressed in decibels. For a ﬁnite target, the relationship between TS and the backscattering cross-section is adopted here as a deﬁnition of target strength. Thus, TS 10 log10

back 4

dB re m 2 :

ð11:93Þ

The target strength of an object, as deﬁned by Equation (11.93), is a function only of the properties of the object itself and of sonar frequency. Unfortunately, there is no general way of converting Equation (11.92) into an equation containing only TS and the other terms in the active sonar equation. An alternative approach is to introduce the concept of an equivalent target strength, which is deﬁned here as TSeq EL SL þ PLTx þ PLRx ;

ð11:94Þ

such that without approximation EL ¼ SL PLTx þ TSeq PLRx :

ð11:95Þ

By comparison, the equivalent target strength (TSeq ), as deﬁned above, is a function 26

It is assumed here that the receiver is in the same vertical plane as the transmitter and target. This assumption makes it possible to restrict attention to propagation in that vertical plane. Equation (11.92) further assumes that the duration of the received pulse is equal to that of the transmitted one. In this situation the target strength based on energy (TSE ) and the one based on mean square pressure (TSMSP ) are equal, so no further distinction is made between them.

608 The sonar equations revisited

[Ch. 11

also of the propagation environment and sonar geometry. For this reason, TSeq is often approximated by TS in the same way as AG is often approximated by DI. For FM sonar (and for broadband sonar in general), it might be necessary to compute Equation (11.92) for all frequencies of interest and consider the resulting spectrum. This frequency dependence is ignored for the time being, focusing instead on the contributions from diﬀerent multipaths at a single frequency, assuming implicitly that beam pattern, propagation factor, and other frequency-dependent parameters do not change much over the bandwidth of interest. This weakness is addressed in the worked example of Section 11.4.6, where frequency dependence is considered for a broadband sonar. 11.4.2.1

Outward propagation loss (PLTx ) and sonar source level (SL)

To complete the deﬁnition of TSeq , it is necessary to deﬁne the quantities PLTx and PLRx for use in Equation (11.94). Propagation loss for the outward path is straightforward: PLTx SL Ltgt ; ð11:96Þ where SL is the sonar source level; and Ltgt is the sound pressure level that would be measured at the target position resulting from the transmitted sonar signal, if the target were not present. To deﬁne the sonar source level, let JTx ðÞ denote the far-ﬁeld radiant intensity of the transmitted sound as a function of elevation angle . Source level is then the value of this quantity evaluated on the sonar axis (i.e., its maximum, denoted here by the subscript zero), converted to MSP and expressed in decibels. In other words SL 10 log10 S0 ;

ð11:97Þ

S0 cðJTx Þ0 :

ð11:98Þ

where Outward propagation loss can be written in terms of the projector beam pattern bTx ðÞ and diﬀerential propagation factor GTx ðÞ as ð PLTx ¼ 10 log10 bTx ðÞGTx ðÞ d ; ð11:99Þ where the transmitter beam pattern is bTx ðÞ 11.4.2.2

JTx ðÞ : ðJTx Þ0

ð11:100Þ

Return propagation loss (PLRx )

By analogy with Equations (11.96) to (11.98), a natural deﬁnition for return propagation loss is PLRx 10 log10 ½cðJtgt Þ0 EL;

ð11:101Þ

where the notation ðJtgt Þ0 indicates the maximum value of the radiant intensity of the

Sec. 11.4]

11.4 Active sonar with coherent processing: matched ﬁlter 609

scattered ﬁeld as a function of the scattered elevation angle out . In other words

where

ðJtgt Þ0 max Jtgt ðout Þ;

ð11:102Þ

ð S0 Jtgt ðout Þ ¼ b ð ÞG ð Þ ð ; Þ din : c Tx in Tx in O in out

ð11:103Þ

With the above deﬁnitions of PLTx and PLRx , the equivalent target strength (Equation 11.94) becomes TSeq ¼ 10 log10 ½cðJtgt Þ0 Ltgt : 11.4.2.3

ð11:104Þ

Special case: separable target cross-section

A case of special interest is now considered to illustrate the calculation of propagation loss and equivalent target strength. The example chosen is for a simple target whose diﬀerential scattering cross-section is a separable function of incident and scattered angles: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ð11:105Þ

O ðin ; out Þ ¼

back ðin Þ back ðout Þ: 4

In this situation, echo MSP is ð ð qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ S Q S ¼ 0 bTx ðin ÞGTx ðin Þ back ðin Þ din GRx ðout Þ back ðout Þ dout : 4

ð11:106Þ The expression for PLTx (Equation 11.99) does not simplify for this case, but PLRx (Equation 11.101) becomes "ð sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ #

back ðout Þ GRx ðout Þ dout ; PLRx ¼ 10 log10 ð11:107Þ

back 0 where

back ¼ max back ðÞ: 0 Equivalent target strength is then 2qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð 6 TSeq ¼ 10 log10 6 4

back 0 4

3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ back bTx ðin ÞGTx ðin Þ ðin Þ din 7 7; ð 5 bTx ðin ÞGTx ðin Þ din

ð11:108Þ

ð11:109Þ

which simpliﬁes for the case of an omni-directional scatterer (i.e., one for which O is independent of angle) such that

back ðÞ ¼ 4

O ;

ð11:110Þ

610 The sonar equations revisited

[Ch. 11

to the true target strength TSeq ¼ 10 log10 O ¼ TS:

11.4.3

ð11:111Þ

Background level (BL)

Background level is a combination of noise and reverberation given by BL ¼ 10 log10 Q B ðÞ;

ð11:112Þ

Q B ðÞ ¼ Q N þ Q R ðÞ:

ð11:113Þ

where

The individual noise and reverberation terms are ð N Q ¼ QN f df and

ð Q R ðÞ ¼ Q R f ðÞ df ;

ð11:114Þ

ð11:115Þ

where the integration is over the processing bandwidth. In Equation (11.114), Q N f is the noise spectral density NLf =10 QN f ¼ 10

mPa 2 =Hz;

ð11:116Þ

where NLf is the noise spectrum level (see Section 11.2.4). Similarly, in Equation (11.115), Q R f ðÞ is the reverberation spectral density evaluated at the arrival time of the target echo.

11.4.4

Processing gain (PG)

Processing gain is the gain from all receiver processing except that from a nominal (ﬂat response) bandpass ﬁlter in the sonar processing bandwidth. That is, if Rout is the output SNR after all processing, then Rout ; Rin

ð11:117Þ

QS Q N þ Q R ðÞ

ð11:118Þ

PG ¼ 10 log10 where Rin is the input SNR Rin ¼

and the MSPs Q S , Q N , and Q R are all deﬁned in the processing bandwidth.

Sec. 11.4]

11.4 Active sonar with coherent processing: matched ﬁlter 611

It is conventional to distinguish between the gain from beamforming (AG) and from time domain processing such as a matched ﬁlter (MG), so that AG ¼ 10 log10

Rarr ; Rin

ð11:119Þ

MG ¼ 10 log10

Rout ; Rarr

ð11:120Þ

and therefore PG ¼ AG þ MG: 11.4.4.1

ð11:121Þ

Array gain (AG) and directivity index (DI)

The calculation of AG in the presence of reverberation is described below. If the signal can be approximated as a plane wave, AG can be written (see Chapter 3) AG ¼ 10 log10

Q N þ Q R ðÞ ; Q N GN þ Q R ðÞGR

ð11:122Þ

where Q N and Q R are the MSPs due to noise and reverberation, respectively, integrated over the processing bandwidth. Thus ðð QN ¼ QN ð11:123Þ f O df dO and R

Q ð Þ ¼

ðð

QR f O ðÞ df dO;

ð11:124Þ

R where Q N f O and Q f O are the respective diﬀerential spectral densities (i.e., the MSP per unit solid angle and unit bandwidth) of noise and reverberation. Noise gain is (see Chapter 6) ðð

QN f O bRx ðOÞ df dO

GN ¼

QN

;

ð11:125Þ

where bRx ðOÞ is the receiver beam pattern. Similarly, reverberation gain is ðð QR f O ðÞbRx ðOÞ df dO GR ¼ GR ðÞ ¼ : ð11:126Þ Q R ðÞ If the noise ﬁeld is isotropic, noise gain is equal to the reciprocal of the directivity factor GD ¼ 10 DI=10 , where DI is the directivity index. It is common to approximate GN by 1=GD , even when the noise ﬁeld is not perfectly isotropic, especially if noise directionality is unknown. By contrast, reverberation is usually of a strongly directional nature and so may not be approximated in this way. This point is addressed in the worked example of Section 11.4.6. The gain depends not only on the receiver beam pattern (through DI), but also on the source level, transmitter beam pattern, and on the details of propagation to and from the object responsible for the scattering.

612 The sonar equations revisited

11.4.4.2

[Ch. 11

Matched ﬁlter gain (MG)

If the received pulse is a perfect scaled replica of the transmitted one, the gain of a matched ﬁlter for a pulse of bandwidth B and duration T is given by MG ¼ 10 log10 ðBT Þ:

ð11:127Þ

If the received pulse is a less-than-perfect replica of the transmitted one, the gain will be less than this, and the diﬀerence is known as coherence loss (CL). Thus, more generally (see Chapter 6) MG ¼ 10 log10 ðBTÞ CL: 11.4.5 11.4.5.1

ð11:128Þ

Detection threshold (DT) Calculation of DT from pfa

The active sonar detection threshold depends on signal statistics in the same way as for narrowband passive sonar, described in Section 11.2.7.1. An important qualitative diﬀerence can arise through background statistics. In both cases the background can be thought of as a summation of individual contributions from discrete events such as breaking waves at the sea surface. If the sonar averages over a large number of such events, a Rayleigh distribution is expected to result for the background amplitude. For modern high-resolution sonar, the number might not be so large, and in extreme cases discrete noise events (known as ‘‘clutter’’) would then be resolved. The eﬀect of clutter on background statistics is the subject of ongoing research (e.g., Abraham, 2003; Nielsen et al., 2008). 11.4.5.2

Estimation of pfa

A general expression for pfa for an active sonar with pulse repetition time Dt is n pfa fa Dt; ð11:129Þ Ntot where Ntot is the total number of detection opportunities per transmission, the value of which depends on the processing used. For a generic sonar capable of resolution in angle (with Nbeams sonar beams), range (Nranges range cells), and frequency (NDoppler Doppler cells), it is given by Ntot ¼ Nbeams NDoppler Nranges :

ð11:130Þ

For a CW sonar the number of range cells is Nranges ¼ or, equivalently (using the CW property

Dt ; t

ð11:131Þ

Df t 1) NDoppler

NDoppler Nranges Dt Df :

ð11:132Þ

Sec. 11.4]

11.4 Active sonar with coherent processing: matched ﬁlter 613

For FM the number of range and Doppler cells is Nranges ¼

Dt ¼ Df Dt 1=Df

ð11:133Þ

and NDoppler ¼ 1:

ð11:134Þ

Thus, Equation (11.132) holds for the FM case as well as for CW. Substituting Equations (11.130) and (11.132) into Equation (11.129) gives nfa pfa ; ð11:135Þ Nbeams Df where NDoppler f CW Df ¼ ð11:136Þ f FM. If the array directivity factor is known, the number of sonar beams can be approximated by GD ¼ 10 DI=10 . The bandwidth for use in Equation (11.135) can be estimated by knowledge of the task the sonar is required to perform. For CW it is related to the maximum Doppler speed of interest (vDoppler ) Df ¼

2vDoppler f; c

whereas, for FM it is determined by the required range resolution ðrÞ c Df ¼ : 2r Thus, 2 nfa c =ðvDoppler f Þ CW pfa ¼ 2cNbeams 4r FM.

ð11:137Þ

ð11:138Þ

ð11:139Þ

Consider a combined (CW plus FM) sonar system with common values of Nbeams ¼ 100, c ¼ 1500 m/s, and a maximum acceptable false alarm rate of one per hour. For an operating frequency of 1 kHz, if the maximum Doppler speed is 10 m/s and the FM range resolution is 1 m, then (from Equation 11.139) 2:1 10 7 CW pfa ¼ ð11:140Þ 3:7 10 9 FM. The point of this comparison is to demonstrate that, for a given false alarm rate nfa , the false alarm probability required of the FM system is two orders of magnitude smaller than for the CW sonar. This is a consequence of the large FM bandwidth, which, while delivering a high SNR on the one hand, results in a high false alarm rate on the other unless compensated for by a low false alarm probability. 11.4.6

Worked example

The example chosen here is inspired by the pioneering modeling work of Au et al. (2004), who consider the task faced by a killer whale (Orcinus orca) in hunting its

614 The sonar equations revisited

[Ch. 11

Figure 11.16. Geometry for worked example involving killer whale (orca) hunting salmon (adapted from Au et al., 2004, # American Institute of Physics, reprinted with permission).

prey, the chinook salmon (Oncorhynchus tshawytscha). The same basic problem is considered here, with the following changes: — the detailed ﬁsh model of Au et al. (2004) is replaced here by a point scatterer, with a representative TS value for salmon from Chapter 8; — reverberation from the sea surface, neglected by Au et al. (2004), is included in the sources of background. The orca is swimming close to the sea surface (shown at depth dO in Figure 11.16), looking ahead and down at the salmon. The ﬁsh, of length 80 cm, is at a depth of dF ¼ 25 m. For the present purpose the orca is assumed to use a baﬄed circular plate transducer, of diameter 10 cm, for both transmission and reception. The (measured) pulse shape and spectrum of an orca echolocation pulse are shown in Figure 11.17. The pulse has a duration of approximately 30 ms. The wide range of frequencies present in the orca’s transmitted pulse, which extends between 20 kHz and 80 kHz, means that the broadband nature of this problem, which has so far been ignored, needs to be considered. Questions to be addressed are: (1) What is the detection range in the absence of any masking noise or reverberation (i.e., if limited only by hearing sensitivity)? Assume that the spectral density of the transmitted pulse and the target strength of the ﬁsh are independent of frequency. (ii) What is the detection range in the presence of wind noise if wind speed (at 10 m height) is 2 m/s? (Surface reverberation is negligible at this wind speed). For the purpose of calculating DT, assume that the signal satisﬁes 1D þ R statistics in a Gaussian background and that the orca accepts a false alarm probability of 0.1 %. (iii) What is the sensitivity of the detection range to increasing wind speed if reverberation is neglected? (Consider wind speeds of 2, 6, and 10 m/s). (iv) What effect does reverberation have on detection range?

Sec. 11.4]

11.4 Active sonar with coherent processing: matched ﬁlter 615

Figure 11.17. Example measurements of orca pulse shapes and power spectra (from Au et al., 2004, # American Institute of Physics, reprinted with permission).

11.4.6.1

Part (i) maximum audibility range (no background)

If the background term is neglected, there is no need for the sonar equation in the usual sense. Instead, the requirement for audibility is that the echo level must exceed the animal’s hearing threshold. The echo level and hearing threshold are considered below. 11.4.6.1.1

Echo level (EL)

Echo level is given by EL ¼ 10 log10 Q S ;

ð11:141Þ

where, generalizing Equation (11.106) to the broadband case and further assuming spherical spreading (neglecting surface reﬂection) and an omni-directional scatterer ( back ðÞ independent of angle), gives for the signal at the center of the beam (using bTx ¼ 1) ð fmax expð2 sÞ 2 back S Q ¼ ðS0 Þf df : ð11:142Þ 4

s2 fmin where ðS0 Þf is the source factor spectral density, such that SLf ¼ 10 log10 ðS0 Þf :

ð11:143Þ

The integral over frequency in Equation (11.142) is necessary because of the broadband nature of the animal’s sonar. The lower and upper limits of integration

616 The sonar equations revisited

[Ch. 11

are fmin ¼ fm B=2

ð11:144Þ

fmax ¼ fm þ B=2:

ð11:145Þ

and Numerical values of fm and B are 50 kHz and 60 kHz, respectively. If target and source spectra are both white, the only frequency dependence arises through the absorption term in the propagation factor, and Equation (11.142) can therefore be written

back F ; Q S ¼ ðS0 Þf BFTx ð11:146Þ 4 Rx where (see Equation 11.49) 1 expð2 m sÞ sinhcð f BsÞ s2 ð 1 1 fm þB=2 ¼ expð4 sÞ df : FTx Bs 4 fm B=2

FTx ¼ and FRx

ð11:147Þ

ð11:148Þ

Substituting Equation (11.146) into Equation (11.141) gives EL ¼ SL ðPLTx þ PLRx Þ þ TS;

ð11:149Þ

where SL ¼ 10 log10 ½ðS0 Þf B ;

ð11:150Þ

PLTx ¼ 10 log10 FTx ;

ð11:151Þ

PLRx ¼ 10 log10 FRx ;

ð11:152Þ

back : 4

ð11:153Þ

and TS ¼ 10 log10

The terms on the right-hand side of Equation (11.149) are discussed below. Source level (SL). The value quoted for the peak-to-peak source level of an orca in Chapter 10 is SLp-p ¼ 218 dB re mPa 2 m 2 , which is close to the highest values measured (see Figure 11.18). These measurements show signiﬁcant variability, with a systematic decrease in level with decreasing distance to the target. For a representative value, consider the black curve in Figure 11.18, which follows the relationship due to Au et al. (2004) (see also Chapter 10) SLp-p ¼ 181:4 þ 20 log10 s:

ð11:154Þ

Putting s ¼ 25 m (the largest distance for which a source level measurement is available) gives a value of 209.4 dB re mPa 2 m 2 for SLp-p . Before it can be used in Equation (11.149), this value needs to be converted to an RMS level. The peakequivalent RMS value (corresponding to a hypothetical average over a full cycle at peak amplitude) is obtained by subtracting 9 dB (see Chapter 10), and the true RMS level is less than this because on average the amplitude is lower than its peak value.

Sec. 11.4]

11.4 Active sonar with coherent processing: matched ﬁlter 617

Figure 11.18. Variation in orca source level (SLp-p ) with distance from target (reprinted with permission from Au et al., 2004, # American Institute of Physics).

The diﬀerence between RMS and peRMS levels depends on the pulse shape and the choice of averaging time, with realistic values, from the last column of Table 10.19, between 1.0 dB and 3.4 dB. Taking the average of these two values (2.2 dB) as representative for the orca pulse, the resulting (RMS) source level is 11.2 dB below the peak–peak value: SL ¼ 209:4 11:2 ¼ 198:2

dB re mPa 2 m 2 :

ð11:155Þ

Propagation loss (PLTx þ PLRx ). Equation (11.148) is similar to the corresponding integral for the broadband passive example (Equation 11.47), and can be evaluated in the same way. Assuming a linear variation of attenuation with frequency as in Equation (11.48), the result is FTx FRx ¼

1 expð4 m sÞ sinhcð2 f BsÞ: s4

ð11:156Þ

Figure 11.19 (upper graph) shows PL (calculated using spherical spreading plus absorption) plotted at three nominal frequencies of 20 kHz, 50 kHz, and 80 kHz. For use in the broadband sonar equation, broadband PL values are necessary and these are shown in the lower graph, calculated using Equations (11.147) and (11.156). This second graph shows that, for a target at a distance of 500 m, the broadband propagation loss for the outward path is 2 dB lower than PLð fm Þ (i.e., PLTx ¼ 60.5 dB re m 2 at this distance). For the return path, the loss is 2 dB lower still, so that PLRx ¼ 58.5 dB re m 2 , making the sum of these two terms (still for a distance of 500 m) PLTx þ PLRx ¼ 119 dB re m 4 :

ð11:157Þ

618 The sonar equations revisited

[Ch. 11

Figure 11.19. Upper: propagation loss PLð f Þ vs. distance for f ¼ 20, 50, 80 kHz; lower: broadband correction PLð fm Þ PLBB , where PLð fm Þ is the value for f ¼ 50 kHz from the upper graph and PLBB is one of PLTx and PLRx as indicated by the legend. The distance s is the slant range, equal to ðx 2 þ ðdF dO Þ 2 Þ 1=2 as illustrated by Figure 11.16.

Sec. 11.4]

11.4 Active sonar with coherent processing: matched ﬁlter 619

The reference unit used here (m 4 ) is explained as follows: — before dividing by reference units, both PL terms have dimensions of area (see Chapter 3); — addition of logarithms implies multiplication of propagation factors and hence also of their reference units; — the product of two areas is a squared area, with reference unit m 2 m 2 ¼ m 4 . Target strength (TS). Salmon is a physoclist, which means that its bladder is closed (see Appendix C). The target strength for physoclists can be estimated using the formula (see Chapter 8) TS 10 log10 L 2 ¼ 27:1 1:7

dB:

ð11:158Þ

Using L ¼ 0:8 m for the salmon length gives TS ¼ 29.0 dB re m 2 . 11.4.6.1.2

Hearing threshold (HT)

Based on measurements by Hall and Johnson (1972) and Szymanski et al. (1999), Wensveen and Van Roij (2007) propose a hearing threshold (in dB re mPa 2 ) for the orca of the form 8 0:05401 > 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 <0 HðxÞ 1=2 x ¼ 0 : 1 x > 0, the sign function 8 < 1 x < 0 sgnðxÞ 0 x¼0 : þ1 x > 0 and the rectangle function

8 jxj < 1=2 > <1 PðxÞ 1=2 jxj ¼ 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 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)

707

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)

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