Marine Acoustics: Direct and Inverse Problems

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Author: James L. Buchanan | Robert P. Glbert | Armand Wirgin | Yongzhi S. Xu

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MARINE ACOUSTICS Direct and Inverse Problems

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MARINE ACOUSTICS Direct and Inverse Problems James L Buchanan United States Naval Academy Annapolis, Maryland

Robert P. Gilbert University of Delaware Newark, Delaware

Armand Wirgin Laboratoire de Mecanique et d'Acoustique Marseille, France

Yongzhi S. Xu University of Tennessee at Chattanooga Chattanooga, Tennessee

siam. Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2004 by the Society for Industrial and Applied Mathematics. 109876543 21 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Marine acoustics : direct and inverse problems/ James L. Buchanan ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 0-89871-547-4 (pbk.) 1. Underwater acoustics. I. Buchanan, James L. QC242.2.M37 2004 620.2'5—dc22

2003070359

This research was supported in part by the National Science Foundation through grants BES-9402539, INT-9726213, BES-9820813, the Office of Naval Research through grant N00014-001-0853, and the Centre National de la Recherche Scientifique through grant NSF/CNRS-5932.

siam

is a registered trademark.

Contents Preface Acknowledgments 1

xi xii

The Mechanics of Continua 1.1 Introduction 1.2 Survey of Previous Work 1.3 Underlying Principles of the Mechanics of Continua 1.3.1 Introduction 1.3.2 Lagrangian and Eulerian Coordinates, Deformation, Strain, Displacement, and Rotation 1.3.3 Deformation Gradients and Deformation Tensors 1.3.4 The Cauchy and Green Deformation Tensors 1.3.5 vStrain Tensors and Displacement Vectors 1.3.6 Infinitesimal Strains and Rotations 1.3.7 Lagrangian and Eulerian Strains in the Framework of Infinitesimal Deformations 1.3.8 Strain Invariants and Principal Directions 1.3.9 Area and Volume Changes Due to Infinitesimal Deformations 1.3.10 Kinematics 1.3.11 Material Derivatives of Line, Surface, and Volume Integrals over Regions Devoid of Discontinuities 1.3.12 Material Derivatives of Integrals over Regions Containing a Discontinuity Surface 1.3.13 Conservation of Mass Law for Uniform Bodies 1.3.14 Conservation of Momentum and Energy Laws 1.3.15 External and Internal Loads and Their Incorporation in the Conservation of Momentum Equation 1.3.16 Stress 1.3.17 Global and Local Forms of the Conservation of Momentum Law in Terms of Stress 1.3.18 Local Form of the Boundary Conditions on Discontinuity Surfaces 1.3.19 Thermodynamic Considerations V

1 1 5 9 9 10 11 12 13 15 16 17 18 19 21 23 24 25 25 26 27 28 29

vi

Contents

1.4

1.5

2

1.3.20 Constitutive Relations 33 Mechanics of Elastic Media and Elastodynamics 33 1.4.1 Definition of Elastic Media 33 1.4.2 Constitutive Equations 33 1.4.3 Linear Constitutive Equations (Linear Elasticity) 37 1.4.4 Symmetry Properties of the Elastic Moduli Tensor . . . . 41 1.4.5 The Wave Equation for Elastodynamics in Linear Elastic Media 42 1.4.6 Wave Equation for Elastodynamics in Compressible, Homogeneous Materials 43 1.4.7 Wave Equation for Elastodynamics in Heterogeneous, Isotropic Solids 43 1.4.8 Wave Equation for Elastodynamics in Homogeneous, Isotropic Solids 43 1.4.9 Obtaining the Wave Equation of Acoustics in Heterogeneous, Inviscid Fluids from Navier's Equation 45 1.4.10 Boundary Conditions between Two Linear, Isotropic, Homogeneous, Elastic Materials 46 Forward and Inverse Wavefield Problems 48 1.5.1 Introduction 48 1.5.2 The Frequency-Domain Equation for Propagation in an Unbounded, Heterogeneous, Inviscid Fluid Medium ... 49 1.5.3 The Frequency-Domain Radiation Condition at Infinity . 50 1.5.4 Governing Equations for the Frequency-Domain Formulation of Wave Propagation in an Unbounded, Heterogeneous, Inviscid Fluid Medium 51 1.5.5 Governing Equations for the Frequency-Domain Formulation of Wave Propagation in Two Contiguous, SemiInfinite, Heterogeneous, Inviscid Fluid Media 51 1.5.6 Governing Equations for the Frequency-Domain Formulation of Wave Propagation in an Unbounded, Heterogeneous, Isotropic, Elastic Solid 52 1.5.7 Governing Equations for the Frequency-Domain Formulation of Wave Propagation in Two Semi-Infinite, Heterogeneous, Isotropic, Elastic Solid Media in Welded Contact 52 1.5.8 Governing Equations for the Frequency-Domain Formulation of Wave Propagation in a Semi-Infinite Domain Occupied by a Heterogeneous, Inviscid Fluid Contiguous with a Semi-Infinite Domain Occupied by a Heterogeneous, Isotropic, Elastic Solid 54 1.5.9 Eigenmodes of a Linear, Homogeneous, Isotropic Solid Medium of Infinite Extent 55

Direct Scattering Problems in Ocean Environments 2.1 The Constant Depth, Homogeneous Ocean

57 57

Contents

VII

2.1.1

2.2

3

Point Source Response in a Constant Depth, Homogeneous Ocean 57 2.1.2 Propagating Solutions in an Ocean with Sound-Soft Obstacle 58 2. .3 The Representation of Propagating Solutions 59 2. .4 The Uniqueness Theorem for the Dirichlet Problem ... 61 2. .5 An Existence Theorem for the Dirichlet Problem 66 2. .6 Propagating Far-Field Patterns 69 2. .7 Density Properties of Far-Field Patterns 72 2. .8 Complete Sets in L2( ) 72 2. .9 Dense Sets in L2( ) 74 2. .10 The Projection Theorem in VN 76 2. .11 Injection Theorems for the Far-Field Pattern Operator . . 79 2. . 12 An Approximate Boundary Integral Method for Acoustic Scattering in Shallow Oceans 83 Scattered Waves in a Stratified Medium 92 2.2.1 Green's Function of a Stratified Medium and the Generalized Sommerfeld Radiation Condition 92 2.2.2 Scattering of Acoustic Waves by an Obstacle in a Stratified Space 96 2.2.3 Reciprocity Relations 98 2.2.4 Completeness of the Far-Field Patterns 101

Inverse Scattering Problems in Ocean Environments 107 3.1 Inverse Scattering Problems in Homogeneous Oceans 107 3.1.1 Inverse Problems and Their Approximate Solutions ... 108 3.1.2 Inverse Scattering Using Generalized Herglotz Functions 114 3.2 The Generalized Dual Space Indicator Method 123 3.2.1 Acoustic Wave in a Wave Guide with an Obstacle . . . . 123 3.3 Determination of an Inhomogeneity in a Two-Layered Wave Guide . . 129 3.3.1 Numerical Example 133 3.4 The Seamount Problem 133 3.4.1 Formulation 133 3.4.2 Uniqueness of the Seamount Problem 135 3.4.3 A Linearized Algorithm for the Reconstruction of a Seamount 139 3.5 Inverse Scattering for an Obstacle in a Stratified Medium 142 3.5.1 Formulation of the Inverse Problem 142 3.5.2 Uniqueness 144 3.5.3 An Example of Nonuniqueness 147 3.5.4 The Far-Field Approximation Method 148 3.6 The Intersecting Canonical Body Approximation 154 3.6.1 Forward and Inverse Scattering Problems for a Body in Free Space 154 3.6.2 A Method for the Reconstruction of the Shape of the Body Using the ICBA as the Estimator 156

viii

Contents 3.6.3

3.7

4

5

Use of the K Discrepancy Functional and a Perturbation Technique 3.6.4 More on the Ambiguity of Solutions of the Inverse Problem Arising from Use of the ICBA 3.6.5 Method for Reducing the Ambiguity of the Boundary Reconstruction The ICB A for Shallow Oceans: Objects of Revolution 3.7.1 Derivation of the Recurrences for Calculation of the Scattered Field 3.7.2 Numerical Simulation of Object Reconstruction Using ICBA 3.7.3 3D Objects in a Shallow Ocean

Oceans over Elastic Basements 4.1 A Uniform Ocean over an Elastic Seabed 4.1.1 The Boundary Integral Equation Method for the Direct Problem 4.1.2 Far-Field and Near-Field Estimates for the Green's Function 4.1.3 The Far-Field Approximation 4.1.4 Near-Field Approximations 4.1.5 Approximating the Propagation Solution 4.1.6 Computing the Scattered Solution 4.2 Undetermined Coefficient Problem for the Seabed 4.2.1 Numerical Determination of the Seabed Coefficients ... 4.3 The Nonhomogeneous Water Column, Elastic Basement System ... 4.4 An Inner Product for the Ocean-Seabed System 4.5 Numerical Verification of the Inner Product 4.6 Asymptotic Approximations of the Seabed 4.6.1 A Thin Plate Approximation for an Elastic Seabed . . . . 4.6.2 A Thick Plate Approximation for the Elastic Seabed . .

157 158 159 162 163 166 168 171 171 174 177 180 183 184 186 189 191 193 201 206 208 208 .214

Shallow Oceans over Poroelastic Seabeds 217 5.1 Introduction 217 5.2 Elastic Model of a Seabed 217 5.3 The Poroelastic Model of a Seabed 219 5.3.1 Constitutive Equations for an Isotropic Porous Medium . 219 5.3.2 Dynamical Equations for a Porous Medium 220 5.3.3 Calculation of the Coefficients in the Biot Model 222 5.3.4 Experimental Determination of the Biot-Stoll Inputs . . . 226 5.4 Solution of the Time-Harmonic Biot Equations 229 5.4.1 Simplification of the Equations 229 5.4.2 Speeds of Compressional and Shear Waves 232 5.4.3 Solution of the Differential Equations for a Poroelastic Layer 247 5.5 Representation of Acoustic Pressure 252

Contents

ix 5.5.1

5.6 6

Differential Equations for Pressure and Vertical Displacement in the Ocean 253 5.5.2 Interface Conditions 253 5.5.3 Green's Function Representation of Acoustic Pressure . .255 Sound Transmission over a Poroelastic Half-Space 257

Homogenization of the Seabed and Other Asymptotic Methods 267 6.1 Low Shear Asymptotics for Elastic Seabeds 267 6.1.1 The Wentzel-Kramers-Brillouin Expansion of the Displacements 269 6.1.2 The Regular Perturbation Expansion 270 6.1.3 A Singular Perturbation Problem for the Love Function . 271 6.2 Homogenization of the Seabed 273 6.2.1 Time-Variable Solutions in Rigid Porous Media 274 6.3 Time-Harmonic Solutions in a Periodic Poroelastic Medium 279 6.3.1 Inner Expansion and Homogenized System 281 6.3.2 Interface Matching and Boundary Layers 284 6.4 Rough Surfaces 290 6.5 A Numerical Example 296

Bibliography

299

Index

333


Preface This book is written with several audiences in mind. For those unacquainted with theoretical acoustics, the first chapter goes into some detail about the physics of vibrations, beginning with the Cauchy-Green deformation tensor, stress tensors, and symmetry properties of the elastic moduli tensor. This chapter concludes with a derivation of the wave equation for elastodynamics in heterogeneous isotropic solids. Finally, we discuss for propagation in an unbounded, heterogeneous, inviscid fluid; an isotropic, elastic solid; and a semi-infinite domain occupied by a heterogeneous, inviscid fluid contiguous with a semi-infinite domain occupied by a heterogeneous, isotropic, elastic solid. This first chapter contains all the physics necessary understanding for the book. The style of Chapters 2 and 3 is quite different. This material is written for mathematicians wishing to see a theorem-proof discussion of the direct and inverse problems of ocean acoustics, where the ocean is assumed to be a wave guide with a completely reflecting bottom. The surface is, as usual, considered to be a pressure release surface, i.e., the acoustic pressure vanishes there. The approach here is to show the existence and uniqueness of the acoustic scattering problem off of smooth inclusions and seamounts in the ocean. Based on these theorems the corresponding inverse problem is proposed. Namely, from acoustic far-field, the data discern the shape and position of an inclusion. Such problems are important in ecological challenges, such as determining the shape and size of methane cathrates on the seafloor. Natural methane hydrate exists in large quantities close to the earth's surface. The sudden release of this gas could significantly affect the global climate. There are numerous natural phenomena that continually alter the temperature and pressure profiles in seabottom sediments. This may result in occasional and potentially massive release of free methane into the atmosphere. Inverse ocean acoustic problems are important not only in locating hydrate-laden sediments on a particular ocean floor but in other ecological problems as well, such as locating sunken objects and pollutants. Chapters 4 and 5 treat more complicated ocean basements. Chapter 4 treats the case of an elastic seabed. It is important to consider this case, as much of the sound energy of the acoustic signal passes into the seabed. The inverse problems we investigate here are those of determining the elastic coefficients of the seabed. Knowing what type of basement one is dealing with is important for determining the spawning ground of different species of fish. The undetermined inclusion problem for an ocean over an elastic seabed is fraught with problems, one being that there is no existence and uniqueness result for the direct problem. This problem is currently out of reach of mathematics. Hence Chapter 4 is written more in xi

xii

Preface

the style of theoretical engineering, where we obtain representation formulas for the direct problem and algorithms for solving the associated inverse problem. In this chapter we take the point of view that the problem exists—What can we do with it computationally? After all, people watered their gardens even before Euler. Various simplified models of the seabed are suggested such as thin and thick plate approximations. Chapter 5 treats the case where the seabed is a poroelastic, Biot-type model. This chapter mainly focuses on the direct problem and the development of good three-dimensional codes for the propagating field. Finally, Chapter 6 returns once more to mathematics. Here we derive a mathematically rigorous treatment of poroelastic materials using the methods of homogenization theory. There appear to be several regimes, depending on the size of various physical parameters, that determine the macroscopic equations governing the propagation of sound in the medium. Only one of these regimes corresponds to a Biot-like material. Other regimes turn out to be more or less viscoelastic in nature. This suggests a new group of direct problems and consequently inverse problems. A final topic in this chapter concerns the treatment of rough seabed surfaces. Again, we use homogenization to obtain the correct macroscopic equations.

Acknowledgments Special thanks are due to the National Science Foundation, which supported our research through grants BES-9402539 and BES-9820813 from the Environmental Engineering Division and through grant NSFINT-9726213 from the International Program, and to the Office of Naval Research, who supported our research through grant N00014-001-0853. We also wish to thank Diane Klownowski, who debugged some of our James L. Buchanan Robert P. Gilbert Armand Wirgin Yongzhi S. Xu

Chapter 1

The Mechanics of Continua

1.1

Introduction

Bodies of water such as oceans, lakes, and rivers (in short, seas) cover more than twothirds of the surface of our planet. The climate of the earth is largely conditioned by exchanges of heat and mass between the seas and the atmosphere. Although much of human activity, i.e., shipping, fishing, extraction of natural resources (such as water itself, sedimentary solids, and petroleum), communication, traveling, washing, rejection of waste, warfare, etc., occurs on the surface and within these fluid masses. For humans, seas are, and remain, essentially a hostile, unknown, and unexplored medium (it being understood that the latter includes sedimentary layers located below the seafloor). For this reason ways have been sought of probing the sea at a distance. Doing this by optical means proved unsuccessful, except at rather small distances, because of (fluid) turbidity or (solid) opacity. Other electromagnetic waves are more or less absorbed due to the conductivity of seawater. On the other hand, elastic waves, i.e., longitudinal (acoustic) waves in fluids, or combined longitudinal-transverse waves in solids, propagate well over long distances (i.e., with little attenuation, this being less true in the sedimentary layers) in sea environments and thus constitute excellent vectors for gathering information (including that of a mechanical nature, of great importance in many applications) concerning what lies beneath the surface of the seas. The first way of probing the sea at a distance is achieved by detecting sounds (generated naturally within the sea or by some artificial source) by means of a set of hydrophones (i.e., microphones adapted for detecting sound in water) attached to, or suspended from, the hull of a ship (or other floating or submerged structure). Recordings of these sounds enable one to obtain a crude picture of sea activity and appearance (this may require repeating the operation at a series of locations near or below the sea surface). Recording sounds—a procedure called "data acquisition"—is usually not sufficient and must be followed by a procedure called "data processing" (i.e., unraveling the signals, which, as such, usually have no obvious meaning), so as to actually form the sought-after "image" of the sea (or a part thereof). The complete procedure, which can be termed "underwater acoustical imaging," falls

1

2

Chapter 1. The Mechanics of Continua

within the realm of inverse problems and, as such, can be divided into several categories, depending on what target (i.e., sources, boundary, wavespeed/material constant) one wants to identify. For example, sailors aboard submarines who have the task of identifying enemy vessels solve inverse problems. On the other hand, dolphins [ 13] and whales trying to locate and identify mates and prey also solve inverse source problems by processing (by means of acquired or hereditary expertise) sounds emitted by diverse sources on or below the sea surface. Electronic signal processing devices are often employed, together with human expertise [ 156], [ 113], in vertical echo sonar (for determining the depth of the water column, i.e., locating a boundary) and side-looking sonar (for obtaining an image of the seafloor and of objects lying thereon). This is accomplished with the help of a rather simple travel-time inversion formula appealing to geometrical acoustics (an interaction model that is valid due to the use of high frequencies in these sonars). Geometrical acoustics and/or mode theory are also employed to account for ducting phenomena in long-distance oceanic propagation and to recover the vertical distribution of wavespeed in the water (usually assuming a horizontal stratification of refractive index in the fluid medium). If all one seeks is a sort of qualitative picture of the target (i.e., source; medium; floating, buried (in the sediment), or submerged body), then these diagnostic tasks can be, and usually are, accomplished [136] without recourse to the rather elaborate machinery of what has come to be known, in the last 10 years [126], [371 ], [458], [341], as model-based inversion. Otherwise, model-based inversion (often termed matched field processing in the present context) must be implemented, but the question is, How? The literature on this issue is rare before 1990 and somewhat less scarce in the period 1990-1996. A chapter is devoted to this subject (more precisely, to the identification of underwater sources) in a well-known book [255]. More comprehensive treatments are given in the monograph [421], which is concerned with the use of matched field processing to identify sources and wavespeeds in rather small lateral stretches of range-independent water columns and sediment; in the proceedings [147], [449]; and in the monographs [28], [316], which deal with identification of ocean currents and other hydrodynamical features in rather large stretches of ocean. Little if anything has, until recently, appeared in books, conference proceedings, and articles concerning model-based inversions of the boundaries and material constants of finite-sized targets located in either the (especially shallow) water column or in the sediments. The reason for this may be that much of this work was financed and classified by military establishments. Since around 1997 the situation has changed, as is demonstrated by the contents of at least six conference proceedings.1 This is perhaps due to the decrease in tension between the East and the West, to the increased concern with environmental problems in general, and with removal of underwater ordnance (i.e., mines suspended in the water column, lying on the seabed, or buried in the sediments) in particular. Several features make underwater acoustical imaging a challenging inverse problem for applied mathematicians. The first is that reliable (i.e., reproducible, sufficient in quantity and nature) data (i.e., the input to the inversion procedure) is difficult to obtain, so that existence and uniqueness theorems, which usually concern complete sets of perfect data, need overhauling. Concerning real data obtained in the field (i.e., sea), the difficulties are mostly of a material nature: the sheer size of the zones to be covered and the cost of sending out ships loaded with sophisticated equipment and qualified personnel (this is not a mathe1

See [4], [417], [105], [373], [94], and [106].

1.1. Introduction

3

matical problem, but is mentioned to show that incomplete data is an irreducible feature of underwater acoustic inverse problems). The problem with data acquired in laboratory experiments (typically, where a tank replaces the sea) is that it is not always clear whether the experiment properly scales down and accounts for all the physical properties of the real sea configuration. The "reality" of data obtained by numerical simulation is also open to doubt, namely because of the difficulty of taking into account all the important aspects of the ocean medium and of its interaction with sound in the theoretical model (see why in the following description of the other features). The second feature is the complexity of the medium: the inhomogeneous nature (at various scales) of the water, the divided and anisotropic nature of the sediments, and the rough nature (at various scales) of the ocean surface and ocean bottom boundaries. The third feature is the nonstationary, often stochastic, nature of the medium and ocean surface due to tide, wind, and currents. The fourth is the nonstationary nature of the sources and detectors: the vessels or buoys that serve as emitters and/or receiving platforms move (notably due to gravitational wave motion), suspended hydrophones move (due to ocean currents and smaller scale convective movements), and targets such as plankton, crustaceans, fish, and submarines move, bubbles generated by living organisms or rotating boat propellers move and change size and shape. The fifth feature derives from the fact that the sea is a very noisy environment [264], [450] due to storms and rainfall, breaking surface waves, ship engines, drills on offshore platforms, whales, dolphins [13], snapping shrimp [14], earthquakes, etc.; considerable effort is required to filter out some of the contaminating components from the useful component of signals. The sixth feature is that the object to be characterized, whether a bounded body, layer, or interface, might be viewed (more appropriately "sounded") from the exterior or on the surface of the fluid medium rather than from the interior of the latter; moreover, the object is usually viewed via a limited rather than a full aperture (viewing an object from all sides is so-called full-aperture viewing). The following list gives an idea of current areas of practical interest in connection with underwater acoustical imaging: • Offshore petroleum and gas drilling platforms: characterization, prior to installation of the platform, of the sediment layer on/in which the infrastructure rests/is embedded • Offshore petroleum and gas drilling platforms: in situ nondestructive testing/monitoring of key elements of the infrastructure (steel/concrete columns or footings, etc.) • Bridges and harbor moorings: nondestructive testing of infrastructure in water or sediment • Localization and/or characterization of submerged/buried communication cables and petroleum/gas/waste transport pipelines • Imaging of the seabed for ecological survey and cleanup: localization, counting, and eventually identification of macro waste resting on the seabed (plastic or glass containers, tires, metallic containers (possibly with radioactive contents), toxic or nontoxic industrial garbage, building material debris (brick, plaster, cement, concrete, metallic armatures), scrap resulting from vessel dismantling, sunken ships, airplanes, and helicopters • Imaging objects of archaeological interest on and below the seafloor

4

Chapter 1. The Mechanics of Continua • Rheological characterization of the seabed for coastal ecological survey and harbor extension, evaluation of underwater landslide risks • Classification of, and communication with, submerged moving objects (e.g., submarines) [86], [297] • Localization and classification of dangerous submerged, moored, or idle objects (mines, navigational obstacles in harbors, etc.) • Localization and classification of dangerous objects embedded in the sediment (e.g., mines) • Localization and classification of mineral deposits in the sediment • Identification of the geometry of the water-sediment interface, i.e., mapping the seafloor [298]) • More precise characterization of underwater temperatures, currents, and sediment transport • Characterization of sea surface waves and ocean/atmosphere mass and heat exchanges • Identification of sea surface pollutants • Detection, localization, characterization, and monitoring of underwater earthquakes and the activity of underwater volcanoes [342] • Detection and characterization of fish and plankton swarms • Monitoring and characterization of the migratory patterns and populations of dolphins, whales, etc.

Many of these applications appeal to (or should appeal to) model-based inversions of acoustic data, and some require that this be done in near real-time. In all cases, the main problem is to find a realistic three-dimensional (3D) model of the interaction of sound with the chosen configuration of the sea (notably to solve the forward problem during the inversion). Often, the model should allow for a depth- and range-dependent fluid medium, a layered subbottom that is elastic-like or even poroelastic-like, and for bulk and boundary irregularities of the medium to account for reverberation and other troublesome effects, etc. To our knowledge such a (working) model, which should be able to account for the pressure field in the fluid medium (wherein the data is acquired) over a rather wide range of frequencies, does not yet exist. Thus, one either has to rely on 2D models or models that neglect many features of the medium and/or of its interaction with the acoustic wave [255]. This model mismatch (with respect to reality), plus the incomplete, fluctuating, and perhaps unrealistic (for the simulated or experimental variety) nature of the data, contributes to exacerbating the ill-posed nature of the inverse problem [90], [371]. Needless to say, much still has to be done before the above-mentioned practical problems are resolved. The purpose of this book is to indicate several current research trends in the field of underwater acoustic wave inverse problems. Essentially everything will be concerned with model-based inversion so that heavy emphasis is placed on the description and resolution of

1.2. Survey of Previous Work

5

the forward scattering problem. This is first done, once the material configuration is chosen and the related physics is defined, in a mathematically rigorous context. The rigorous forward scattering models are incorporated in inverse scattering schemes when the duration of the computations is not a problem; otherwise, approximate forward scattering schemes, some of which are described in detail, are employed to meet the fast imaging requirement.

1.2

Survey of Previous Work

The principal themes and techniques that have been studied in the realm of inverse scattering (which necessarily encompasses forward scattering) problems of marine acoustics are: (1) determining the velocity profile or other physical parameters of the underwater environment, generally modeled as a ID semi-infinite or thick layer (with acoustically rigid or penetrable bottom) stratified medium; for layers that are all fluid-like: [28], [30], [57], [54], [74], [94], [96], [102], [147], [203], [204], [188], [210], [212], [318], [335], [409], [415], [421], [427], [449]; for layers that are fluid- and elastic-like: [3], [80]; and for layers that are fluid- and poroelastic-like: [6], [65], [82], [84], [87]; (2) locating and characterizing a source situated in the water or on the sea surface: [85], [244], [238], [255], [333], [365], [397], [421], [422], [479], [481], [484]; (3) detecting the presence of an object (including the seafloor itself) located in the water column or on the water-seabed interface: [162], [168], [490]; (4) classifying an object (i.e., determining whether it is natural or manmade) located in the water column or on the water-seabed interface: [109], [114], [489], [490], [491]; (5) imaging an object located in the water column or on the water-seabed interface: [66], [67], [69], [71], [97], [366], [368], [466], [474], [477], [478]; (6) detecting and classifying an object located within the seabed (sediment): [223], [228], [247], [490].

[227],

Historically speaking, theme (1) was the first to become a topic of interest to researchers. As early as 1822, the Swiss physicist, J.-D. Colladon, attempted to compute the speed of sound in the waters (at a temperature near its surface of 8°C) of Lake Geneva [340]. To do this, he had an assistant, located at one end of the lake, strike an underwater bell and simultaneously flash a signal of light, while Colladon was on a rowboat at the other end of the lake, facing his assistant (to see the light signal and start his Swiss stopwatch at this moment), slightly inclined so that his ear was at the narrow end of a Swiss alpenhorn, the latter capturing at its other flared end (immersed in the water) the sounds emitted by the bell. By measuring the time (T) between the start of the light signal and the moment of hearing the bell, Colladon had known the soundspeed (C) in the water, knowing the distance (D) between him and his assistant, via the formula C = D/T, and so found C = 1435m/s. Obviously, if Colladon knew C and measured T, he could have computed, via the same formula, the distance of the source of sound (bell) from his boat, which is essentially the method (with various refinements) used for partially solving the inverse problems connected with theme (2).

6


The detection, classification, and imaging tasks associated with themes (3)-(6)) (see, e.g., [27]), are basically nonlinear problems (in terms of the to-be-reconstructed parameters) which must be solved by the aforementioned procedure called model-based inversion [458]. The latter involves the adoption of a model of the object, a model of its environment, a model of the interaction of sound with the object and/or environment, a model of the propagation of the wavefield to the locations of the detectors, and a systematic, automated algorithm for extracting the geometric and/or physical characteristics of the object by minimization of the discrepancy between the measured wavefield at the sensor locations and the predicted wavefield. Most of the published investigations on themes (3)-(6) (e.g., [458], [51], [21], [109], [115], [156], [176], [223], [226], [297], [426]) involve either very simple wavefield inversion schemes (such as that of Colladon) or no inversion at all, which is to say that they mimick classical optical imaging (i.e., the image is in more or less (because of noise and aberrations) one-to-one correspondence with the object) [458], [459]. The most widely used technique in connection with theme (3) is SONAR (i.e., SOund Navigation And Ranging), the sound equivalent of RADAR (RAdio Detection And Ranging). Sonar [297], [345], [396], is often employed to get either a picture of the immediate neighborhood of a vessel (such as for obstacle detection or avoidance, floating-mine hunting, or fishing), in which case it is catted forward-looking sonar, or a picture of the seafloor, in which case it is called downward-looking or bathymetric sonar [298]. When a highfrequency (~ 100kHz) acoustic wave encounters the obstacle or surface, which is generally uneven, at least a part of the energy associated with this wave is backscattered towards the emitter. In its simplest form, a sonar system sends out a single narrow beam and records the signal strength (information that is not always used) and travel time of the backscattered pulse. Assuming the medium is homogeneous between the emitter and the surface, and the velocity of sound therein is known, this time interval yields the distance of the target (a point on the surface) to the emitter. In side-looking sonar (SLR) [346], the object is to map large areas of the seafloor, and this is done by means of a narrow fan-shaped beam illuminating a swath parallel to, and off the side of, the emitter. The map (picture) of the seafloor is produced as the instrument travels along a line (shiptrack), sweeping its insonified swath along the surface underneath it. Objects or outcrops on the seafloor are recognized by the shadows (areas that are not insonified by the incident sound beam) produced in these pictures. Resolution in SLR is limited by the length of the emitting antenna. Synthetic aperture radar (SAR), and its sonar equivalent (SAS) [137], [154], [239], [426], overcome somewhat this limitation by employing a synthetic antenna of much larger length, thereby giving rise to a narrower beam and increased resolution. Techniques employing high-frequency sonar (as used for (3)-(5)) cannot be employed either for precise characterization of a closely packed composite object (e.g., pile of debris) lying on the water-seabed interface since only the insonified (upper) portions of the pile is insonified, nor for (6) due to the fact that high-frequency sound is highly scattered or absorbed in sediments and is therefore an unsuitable means of probing such a medium— except if the object is close to the seafloor surface (see, e.g., [291], [247], [447], [430], [429]). In studies of types (2) and (3) the detector is not necessarily close to the source and/or object, but in (4) and (5) the acquisition of data is carried out rather close to the object, just as in underwater optical imaging. In all but exceptional circumstances, the sources and detectors are in the water column so that it is usually impossible to employ the methods of (4) and (5) to the situation in (6). Most of the studies of types (4)-(5) are based on simple laws

1.2. Survey of Previous Work

7

of geometrical acoustics such as rays, well-defined shadows, one-to-one correspondence between a temporal echo and the presence of a target, distance of the target obtained by a time lag between emission and reception of an echo of the emitted pulse, so that the "inversion" reduces at best to a problem of signal processing [51] and at worst to one of image processing (e.g., to eliminate speckle or to distinguish features from background returns) [115]. Concerning (relatively) long-range detection and identification at (relatively) low frequencies, a large number of investigations assume point or line source approximations of the object, require a rather thorough knowledge of the sea environment which is assumed to be fluid-like and is often modeled by 1D or 2D geometries, and rely heavily on ray theory and much signal processing (see, e.g., [157], [136], [132], [55], [41], [448]). Another technique worthy of mention in connection with themes (2)-(4) can be termed "empirical" in that for the purpose of identification it relies not on any knowledge of how a sound is produced in the water column, but only on a comparison of this sound to a set of prerecorded sounds whose origins are known (e.g., those emitted by a whale or dolphin [13], swarms of shrimp [14], a motorized ship on the sea surface (constituting a potential threat to a submarine), a submarine (constituting a potential threat to a ship on the sea surface), etc.). This signature-recognition technique can be implemented either by trained operators aboard ships or submarines, or by neural networks [109], [489], [490]. Aquatic animals such as whales probably rely instinctively on such a technique to avoid prey and recognize offspring and mates. As mentioned earlier, papers on model-based inversion (MBI) (often termed wavefield inversion [458]) in the context of themes (3)-(6) were scarce before the end of the Cold War era. However, MBI was applied fairly often in the earlier period to problems concerning themes (1) and (2), notably for characterizing the deep-ocean wavespeed variation with depth [54], [53], [284], [286] and locating sources in such stretches of water [255]. More often than not, these studies employed ID multilayer [54] or continuum models [1 ], [451], [6] of the ocean (i.e., no lateral variation of wavespeed and density and all interfaces being horizontal planes), with each layer being occupied by a fluid-like medium. Various inversion schemes were either borrowed from the field of quantum mechanics [101], [371 ], [341] (e.g., the Agranovich-Marchenko scheme) and used frequency-domain data or were taken from the field of geophysical prospection (wave splitting, invariant imbedding and layer-stripping schemes [122]) and used time-domain data [130]. There is still much research being conducted to find the solutions to the inverse problem of determining the wavespeed distribution in the water column, notably by the so-called matched-field processing technique (a variant of MBI) [255], [420], [421 ], [422], [414] or by various other methods (e.g., plane-wave or Born approximations combined with genetic algorithms) [274], [275], [276], [272], [273]. The case of a source in a half-space layered medium such as a range-independent deep ocean constitutes a difficult problem, even in the forward-scattering context [255]. Research is still active on the corresponding inverse problem [333], [255], [420], [421], [422], [414], [237], [238]. MBI in general, and matched field processing in particular, have been (and continue to be) employed to determine the geoacoustic parameters of a shallow-water marine environment with horizontal interfaces [85] as well as one in which the bottom is sloped [149], [317] or uneven [481], notably to locate sources in such media [244], [355], [392] in such environments, which are increasingly considered to have 3D geometry, even the

8


forward problem [338] is a challenge, and a large variety of techniques have been employed to solve it: the parabolic equation method [412], [255], [283], [301], [335], [336], [337], [116], [117], [118], [119], [120], [282], [278], [158], [278], [270], [359]; the modal method [52], [325], [255]; the coupled mode method [160], [161], [370]; and the Green's function/transmutation method [152], [199], [201], [205], [203], [187], [206], [209], [212], [198], [292], [84], [89]. Time reversal constitutes a relatively new technique for locating sources in complex (with stochastic heterogeneity and/or multiphasic) media [172], [269], [32], [33], [50], [306], [431], [171]. The sea surface (with or without an ice cover), seafloor, and other interfaces in a real ocean are not generally flat and can affect underwater communication (through bottom loss) and imaging systems. To account for interface unevenness and reverberation introduced in the sound field, a large number of investigations, most of which are concerned with the forward problem, have been made [22], [25], [31], [42], [108], [138], [146], [155], [177], [180], [222], [196], [253], [256], [257], [281], [309], [310], [311], [324], [339], [353], [357], [378], [408], [411], [425], [434], [441], [438], [452], [453], [454], [455], [456], [457], [459], [466], [480], [485], [483]. Bubbles and small-scale heterogeneities in the water column produce volume reverberation which causes adverse effects similar to those produced by interface irregularities. They can also cause strong signatures (the case of bubbles produced by boat propellers) which are exploited for the detection of enemies. Research on these topics has been published in [100], [151], [174], [169], [229], [252], [324], [424], [433], [435], and [445]. Another topic that is receiving more and more attention is the propagation of acoustic waves within the sediment below the seafloor, this being of importance for the computation of bottom loss in underwater communication links, the geoacoustic characterization of the subbottom medium of the sea, and the detection and classification of objects buried in the sediment layer. When it is admitted that the wave penetrates the seafloor (otherwise, the latter is taken to be acoustically hard [135], [216], [413], [481], [485]), the sediment is more often than not modeled as a fluid-like medium (e.g., as in a Pekeris wave guide) [276], [272], [255], [227], [228], [67], [103], [46], [55], [54], [53], [149], [155], [158], [160], [161], [165], [166], [173], [194], [224], [237], [238], [247], [265], [268], [274], [275], [286], [287], [298], [307], [317], [338], [345], [361], [391], [416], [420], [421], [422], [423], [451], [466], [469], [489], or else as an elastic medium [79], [255], [116], [42], [44], [138], [184], [185], [186], [188], [222], [258], [301], [300], [303], [337], [390], [483]. In recent times it has also been modeled as a poroelastic (i.e., two-phase) medium, appealing either to the Biot or homogenization approaches [91], [88], [95], [102], [111], [113], [5], [26], [34], [36], [38], [40], [39], [76], [65], [133], [134], [143], [205], [206], [190], [193], [213], [231], [236], [242], [245], [288], [289], [291], [292], [314], [315], [322], [393], [399], [400], [402], [403], [404], [406], [407], [452], [482], [492]. The subject of forward and inverse scattering by a bounded object (e.g., a submarine or mine) in a marine environment, which is one of the principal themes of this book, has been investigated in several contexts and by a variety of methods such as domain integral, boundary integral, Rayleigh hypothesis, modified Rayleigh conjecture, extinction, discrete sources, partial waves, finite elements, finite differences, intersecting canonical body approximation (ICBA), boundary perturbation, Born tomography, level sets, etc. For an object in a homogeneous ocean with boundaries at infinity, some of the references are [ 12], [20], [29], [47], [48], [49], [56], [24], [62], [93], [104], [107], [110], [121], [124], [125],

1.3. Underlying Principles of the Mechanics of Continua

9

[126], [129], [131], [114], [139], [140], [144], [145], [226], [229], [230], [232], [250], [148], [266], [259], [263], [267], [271], [277], [293], [294], [321], [308], [319], [323], [327], [328], [329], [330], [344], [349], [351], [352], [348], [358], [360], [364], [369], [371], [377], [381], [382], [383], [384], [386], [387], [388], [398], [410], [419], [426], [436], [437], [444], [448], [457], [461], [462], [466], [478], [486], [487], [489], [491]. For an object in a deep ocean, with the sea surface taken into account, one can consult [43], 1150], and [477]. For an object (also a seamount) in a deep ocean, with the seafloor taken into account, the appropriate references are [42], [43], [109], [169], [170], [176], [179], [182], [363], [362], [217], [216], [258], [485]. For an object in a shallow ocean (the sea surface and seabottom are taken into account and are usually, but not always, considered impenetrable) the published works include [8], [9], [10], [11], [45], [66], [69], [70], [67], [71], [97], [131], [135], [141], [142], [162], [163], [167], [164], [184], [207], [208], [194], [215], [196], [197], [218], [225], [234], [235], [240], [241], [243], [251], [268], [285], [296], [304], [305], [365], [366], [368], [367], [372], [374], [375], [376], [379], [428], [442], [443], [23], [464], [465], [469], [471 ], [474], [480], [481], [459]. For an object (also a seamount) partially or fully imbedded in the sediment, the appropriate references are [89], [165], [168], [223], [227], [228], [247], [289], [290], [291], [378], [385], [447], [492]. Other references will be given further on in their specific contexts.

1.3 1.3.1

Underlying Principles of the Mechanics of Continua Introduction

Continuous media such as solids or fluids, or the solid and fluid components of poroelastic media, are deformed, rotated, and displaced when subjected to forces. The mathematical description of these mechanical phenomena is the subject of this and the following sections (1.3-1.5) and will draw heavily on Eringen's book Mechanics of Continua (including most of the notations) [159]. Since most of the acoustical phenomena in underwater environments occur in the water column, fluid acoustics will receive the largest emphasis herein. Insofar as acoustical waves propagating in water can encounter floating or immersed solid objects and penetrate the latter, the subject of the acoustics of solids must, and will, be treated in some depth. Since underwater sound also encounters and penetrates the sedimentary layers that lie underneath the seafloor, and the matter in these layers is neither solid nor fluid but a combination of both, the acoustics of poroelastic media will be treated at some length (Chapter 5). Continuum mechanics can be formulated by two mathematical descriptions. The lagrangian description emphasizes what happens to a particular particle of matter specified by its original position at a reference time. The eulerian description puts emphasis on what happens to a particle occupying a particular location. Often the eulerian description is employed for fluids and the lagrangian description for solids, but in the reference work of Eringen this separation is not so clear-cut, which is justified by the fact that fluids and (elastic) solids share many properties. Thus, the reader of the following material should not be surprised to often find side-by-side formulae applying to both descriptions.

10

1.3.2


Lagrangian and Eulerian Coordinates, Deformation, Strain, Displacement, and Rotation

The deformation, rotation, and displacement of matter, assumed to be confined to a finite region of space termed "body" B, can be related to the successive positions of material points. This suggests the introduction of two sets of cartesian coordinate systems, one for the undeformed body and the other for the deformed body [159, p. 7]. At time t = 0, B occupies the volume V whose boundary is S. The position of a material point P in B is denoted by the vector X extending from the origin O (of the cartesian system OXYZ) to P. At time T > 0 the body is displaced and deformed so as to occupy the region b having volume v and boundary s. Let p be the position of a material point of b and x the vector from the origin o (of a cartesian system oxyz attached to the body) to p. X, Y, Z are the lagrangian coordinates (also called material coordinates) and x, y, z the eulerian coordinates (also called spatial coordinates), which are related by

or, in shorthand notation,

Similarly,

The changes in the body due to the action of an external load are thus described by the fact that a material point P B at t = 0 is carried over to position point p b at time t > 0. Quantities associated with the undeformed body B are hereafter designated by uppercase letters and those with the deformed body by lowercase letters. Thus

wherein X1 = X, X2 = Y, X3 = Z, x1 = x, x2 = Y, X3 = z, ik Ik are unit (i.e., constant) base vectors in the two cartesian systems, and the Einstein summation formula for repeated indices (Roman letters for three dimensions, Greek letters for two dimensions and, later on, thermodynamic variables) is implicit. It follows that

with


11

the Kronecker delta symbols being equal to one when the two indices (both either lowercase or uppercase) are equal, and to zero when the two indices are unequal. By extension (but having different properties than the Kronecker symbols), the so-called "shifters" are

With w = wk(X)ik a vector in the euclidean frame xk, its components WK in the euclidean frame XK are

In a similar manner,

Putting the latter into the next-to-last relation gives wk — S Kk S ki iW i , but due to the fact that wk — S ki W 1 , one obtains

In the same manner,

1.3.3

Deformation Gradients and Deformation Tensors

With the notations

one can write

wherein Xk,k, XK,k are termed deformation gradients. Consider the product

This and a similar procedure show that

each of which represents a set of nine linear equations for the nine unknowns xk,K Using Cramer's rule one gets

or

XK,k

12


wherein J is the jacobian

and

klm

is the 3D permutation symbol defined by

where a single permutation of mkl is an interchange of any two of k, /, and m, and an even or odd permutation meaning an even or odd number of single permutations. Thus

Note that (1.16) and (1.17) imply

1.3.4

The Cauchy and Green Deformation Tensors

The insertion of (1.13) into (1.5) gives

with


13

One can express I and i in terms of c and C by employing (1.15):

Employing (1.29) and (1.31) in (1.6) leads to

wherein

is the Cauchy deformation tensor and

Green's deformation tensor. Both are symmetric and positive definite. 1.3.5

Strain Tensors and Displacement Vectors

The difference ds2 — dS2, for the same material points in B and b, is a measure of the change of length so that vanishing ds2 — dS2 denotes a situation in which deformation has not changed the distance between two neighboring points. If this is so for all points in the body, the body is subject only to a rigid displacement. This difference can be written in either of the two euclidian frames as

wherein

is the lagrangian strain tensor and

the eulerian strain tensor. When either of these strain tensors vanishes, one obtains, via (1.36) and (1.37), EKLdXKdXL — ekidxkdxt, so that

14


Recall that the original picture of a material point P in the undeformed body B referred to the euclidian frame XK with origin O, and a point p in the deformed body referred to the euclidian frame xk with origin o. Let b denote the vector from O to o (its length is constant with respect to XK and xk) and u the vector from P to p, while recalling that X is the vector from O to P and x the vector from o to p. Then the displacement vector u is given by

Since u — UlIl — ulil and b = BLIL, = blil

Employing (1.42)-( 1.43) and (1.31) enables the expression of the strain tensors in terms of the displacement:

A quantity of interest, CKL, in connection with the lagrangian strain, can, via (1.42) and (1 .7), be expressed as

which, when introduced into (1.38), gives

A similar procedure, starting with ckl,, leads to

Equations (1.44) and (1.45) entail dx = CKdXK = (S M K + U M , K ) l M d X K and d\ = ckdxk — (Smk — um,k)imdxk, and since dx — ikdxk and d\ — I k d X k , we have

Equations (1.47)-(1.48) show that the lagrangian and eulerian strain tensors are symmetric, i.e.,

In three dimensions, the lagrangian strain tensor thus has only six independent components, E11, E 22 , E33, E12, E 23 , E31, wherein E11, E 22 , E33 denote the normal strains and E12, E23, E31 the shear strains. The notions of strain (tensor) and displacement (vector) will be shown to be particularly important in connection with the description of phenomena in elastic media.

1.3. Underlying Principles of the Mechanics of Continue

1.3.6

15

Infinitesimal Strains and Rotations

A topic of particular relevance to wave propagation in materials is that of small deformations and linear elasticity. In this context, the nonlinear terms of the strain tensor are approximated or eliminated. The standard linear theory involves only the infinitesimal strain tensors EKL ekl and infinitesimal rotation tensors RKL, Rkl, which are approximations of (1.47) and (1.48):

With the notation

and

it is found that

Equations (1.53) and (1.54) indicate that EKL and ekl are symmetric tensors, each with six independent components:

whereas (1.55) and (1.56) indicate that RKL and rkl are skew-symmetric tensors with three off-diagonal nonzero components:

16


Similarly, one finds that U(k,l) is symmetric and V[k,l] antisymmetric:

Equation (1.58) substantiates the fact [220, p. 75] that second-order tensors can be expressed as sums of symmetric and skew-symmetric tensors. Since, in a 3D space, a vector generally has three independent components, one can define a so-called infinitesimal rotation vector r with components

and show that

Introducing (1.57) into (1.51) and (1.52) and using the properties of the Kronecker symbol entails

from which it is seen that in order for EKL

1.3.7

EKL, both EKL and RKL must be small.

Lagrangian and Eulerian Strains in the Framework of Infinitesimal Deformations

It is assumed that EKL ~ EKL and ekl ~ eu and that nonlinear terms in (1.49) can be neglected:

A consequence of (1.36) and (1.37) is

so that, with the aforementioned approximations, one finds

which gives

In the same manner, it is found that

Thus, within the framework of infinitesimal deformation theory, there is no distinction between eulerian and lagrangian strains.

1.3. Underlying Principles of the Mechanics ofContinua

1.3.8

17

Strain Invariants and Principal Directions

In [159, pp. 28-31 ], it was shown how the deformations in a body transform an infinitesimal sphere therein into an ellipsoid, called the strain ellipsoid. Deformation also rotates the principal directions of the ellipsoid. The strain components, referred to the principal axes of the strain ellipsoid, can be expressed in terms of those referred to any other axes once the cosine directors of the principal directions are found. An important task is to find the principal directions and primitive functions of strains that are invariant during such coordinate transformations. The so-called principal strains are solutions E of the equation

This leads to the cubic equation

wherein

The characteristic equation (1.72) possesses the three roots E1, E2, E3, which are termed principal strains. Then the coefficients IE, IIE, IIIE can be expressed in terms of these principal strains:

Thus, IE, I IE, 11 IE are invariant with respect to any coordinate transformation at P. This result substantiates the fact that in three dimensions, there exist no more than three independent invariants of a second-order tensor. Invariants may also be obtained via (1.38), (1.39) in terms of the strain tensors EKL, ekl ekl CKL, ckl. For instance,

with

18

1.3.9


Area and Volume Changes Due to Infinitesimal Deformations

An infinitesimal parallelepiped within the body, with edge vectors I 1 d X 1 , I 2 d X 2 , I 3 d X 3 , is transformed, subsequent to deformation, into a rectilinear parallelepiped with edge vectors C 1 d X 1 , C2dX2, CidX3, where (see (1.45))

One of the three area vectors is

which, on account of dA3 = d X 1 d X 2 and

becomes

With the definition (1.17), (1.19) of the jacobian,

one finds, with the help of (1.15),

which, upon introduction into (1.82), yields

More generally,

so that

Recalling (1.35), one finds, with the help of (1.17),

In the same way that (1.75) was obtained, one can show that


19

so that (1.88) gives

The infinitesimal volume of the rectilinear parallelepiped is

which, with the help of (1.7), (1.15), (1.79), (1.85), (1.90), and the fact that dA3dX3 = dV, becomes

With the help of (1.77)-(l .78) one can also write

and for inifinitesimal strains, so that

and

with the so-called dilatation, which, in infinitesimal strain theory, is a measure of the volume change per unit of initial volume. Note that (1.54) implies

1.3.10

Kinematics

Kinematics involves the time rate of change of various functions such as scalars, vectors, and tensors. The material time rate of change of a vector W is

which means that X is held constant in the differentiation. If W is a material function F, i.e.,

then

20


Otherwise, if W is a spatial function f (related to a position in the deformed state of the body), i.e.,

then

due to (1.2). The previous expression is written in more condensed notation as

wherein

is the material derivative and fk,l = fk/ xi. The material derivative of a material vector is just the ordinary partial derivative with respect to time of this vector, since

The velocity v is the material time rate of change of the position vector of a particle, which, on account of (1.4), is

If the identity of the particle is known (i.e., the particle is in the undeformed state of the body), then its velocity is

whence

The acceleration is the time rate of change of the velocity for a given particle, i.e.,

so that, by virtue of previous results concerning the material derivative,


21

whereas in lagrangian coordinates,

In the lagrangian representation, the particle with a given velocity or acceleration is known, as in the (classical) mechanics of the particle. In the eulerian representation, the velocity and acceleration at time t and spatial point x are known, but the particle occupying this point at this time is not known (it could be any of the material particles of the undeformed body). The material derivative of elements of arc, surface, and volume intervene in various balance and conservation laws. In [159, pp. 70-73], it is shown that (i)

with dxk = x k , l d X i ; (ii) the material derivative of a cartesian component of an element of area da is

and (iii) the material derivative of the volume element dv is

1.3.11

Material Derivatives of Line, Surface, and Volume Integrals over Regions Devoid of Discontinuities

The material derivative of a line integral of a scalar field 0 over a material line C is

A material line C can be described by the equation X = X(S), with S arc length so that no differentiation of C is necessary. This means that

which, on account of (1.110), yields

The time rate of change of a line integral over a spatially fixed line c is

22


with the difference from (1.115) being due to the motion of the particles of C. By employing (1.111) one shows that for a material surface S

and for a spatially fixed surface s

By choosing

as a component of a vector field 0, one obtains from (1.117)

and if it is assumed that S is an arbitrary material surface and the flux term constant, then

which constitutes a criterion for the flux of a vector 0 across any material surface to remain constant. A result similar to (1.119) is found for a spatial surface S(t) bounded by a closed curve c(t) moving with velocity v (s(t) + c(t) can be thought of as a set of fictitious material particles moving with velocity v):

which, with the help of Stokes's theorem [159, p. 427], yields

Finally, the material derivative of a scalar field over a material volume is, using (1.112),

wherein fk = f/ xk. If is assumed to be continuous throughout B, then the use of Gauss's (the divergence) theorem [159, p. 427] in the last expression gives


23

A remarkable result (due to the formal similarity with (1.124)) is that when V and S are replaced instantaneously with a fixed spatial volume v and boundary 5, then

which signifies that the rate of change of a scalar field 0 over a material volume V is equal to the sum of the rate of creation of 0 in a fixed volume v instantaneously coinciding with V and the flux Vk through the boundary surface s of v. Another remarkable result (again due to the formal similarity with (1.124), for an arbitrary spatial volume v ( t ) bounded by the closed surface s ( t ) and moving with the velocity v) is that

1.3.12

Material Derivatives of Integrals over Regions Containing a Discontinuity Surface

First consider the situation in which a discontinuity surface (t), moving with velocity v, cuts through the material volume V [159, p. 76]. This discontinuity surface divides V into two uniform portions V+ and V- bounded by S+ + + and S- + , respectively, so that applying (1.126) in each of these regions leads to

which, when added and after taking the limit

give

wherein [[A]] = A+ — A . Using the Gauss theorem, the last expression takes the form

A similar procedure can be applied [ 159, p. 77] to a material surface S bounded by the closed curve C on which a discontinuity line y(t) is moving with velocity v. At a given instant t, this line divides S into two portions S+ and S-. Assuming that the normal component of v to y is continuous, one finds, from (1.119) and with the help of the Stokes theorem,

24


wherein k is the unit vector tangent to y at the integration point in the last integral. Conservation laws are expressed by the following (for the moment, abstract) relations:

wherein q, h, and r are generally vectors and , and g are generally tensors. Of particular interest here is the case when region V contains a moving surface (t) or when a surface S contains a moving discontinuity line y(t). Applying (1.130) and the Stokes theorem to (1.131), and applying (1.129) and the divergence theorem to (1.132), gives

1.3.13

Conservation of Mass Law for Uniform Bodies

This law is actually one of the four fundamental axioms of mechanics and amounts to the statement that the total mass of a body is unchanged during its motion and deformation. This is a global conservation law. When, as is assumed herein, this axiom is thought to apply in an arbitrarily small neighborhood of all material points of the body, the mass is said to be conserved locally. Let designate the mass density in the spatial frame and the mass density in the material frame. For a system in which the mass does not evolve in time (i.e., there is no injection of matter into the system), the axiom is expressed by

wherein p is the mass volume density. When matter is entering the system, the previous relation is generalized to

wherein w designates the source density rate of injection of matter. Employing (1.123) gives


25

and since this global form of the law of conservation of mass holds for arbitrary V, one deduces from it the local form of the conservation of mass law

Although this expression is suitable for fluids, it turns out that another relation is more suitable for solids and is given by

or, on account of (1.92),

For this to be valid for every volume element, one must have (see also (1.92) and (1.77))

which is the local form of the conservation of mass law, equivalent to (1.138). 1.3.14

Conservation of Momentum and Energy Laws

These laws are actually the expression of three axioms. The first, deriving from Newton's second law, takes the form

wherein Fk is the kth cartesian component of the resultant force F acting on the body. A more detailed global form of this law is given later on from which will be derived a local form of the conservation of momentum law. Before doing this it is necessary to distinguish between external and internal forces (i.e., loads) and then introduce the notion of stress. Another equation translates the axiom of conservation of momentum, and yet another relation expresses the conservation of energy [159, pp. 84-86]. 1.3.15

External and Internal Loads and Their Incorporation in the Conservation of Momentum Equation

Bodies of matter are deformed and displaced under the influence of external forces. However, deformation is also caused by the action of internal forces. These forces have to be categorized in order to relate them to their effects. An almost trivial statement is that the resultant offerees acting on a body is

26


Insofar as the resultant of internal forces (due to the action of one particle on another) is nil (according to Newton's third law of action and reaction), F in the preceding equation can be considered an external, and therefore usually known, force. This force is due either to extrinsic body (or volume) loads such as gravity or to extrinsic surface (or contact) loads that arise from the action of one body on another across their boundaries. The extrinsic surface force per unit area is called the surface traction, an example of which is the hydrostatic pressure exerted by a fluid acting on a submerged body. Iff denotes the body force per unit mass and t(n) the surface traction per unit area acting on the surface of the body with exterior unit normal n, then

where the body force and surface force densities can eventually be considered distributions to account for concentrated forces acting at isolated points. Consequently, the conservation of momentum (Euler) equation (1.142) takes the general form

1.3.16

Stress

Consider a small closed subregion, of volume and boundary 5, fully contained in the body of volume V and boundary S. At a point p € s, V — 0 makes itself felt by (i) surface forces t(n) called stress vectors, (ii) surface couples termed couple stress vectors, (iii) body forces whose density is designated by pf, and (iv) body couples. The surface loads vary with the position p s and with the exterior unit normal vector at p. The law of conservation of momentum, expressed by (1.145), which was previously applied to the body V + S, holds also for the subbody o + s by replacing the domains of integration V and S by D and s, respectively. In [159, pp. 97-98], it is shown, by applying this conservation of momentum law and the conservation of mass law to a small tetrahedron-shaped subbody, that the stress vector t (n) at a point p 6 s is a known linear function of the stress vectors t* acting on the coordinate surfaces passing through this point, i.e.,

To render the notion of stress objective, i.e., independent of the coordinate frame, one introduces the stress tensor tki related to tk by

where, as before, the il are unit vectors along the coordinate axes. The first subscript in tkl indicates the coordinate surface xk = const, on which the stress vector t* acts, and the second indicates the direction of the component of tk so that t11, t22, t33 are normal stresses and the other components of tki are shear stresses. The units of both normal and shear stress are force per unit area. Note that the stress tensor is often designated by the symbol or


1.3.17

27

Global and Local Forms of the Conservation of Momentum Law in Terms of Stress

The point of departure is the conservation of momentum relation (1.142). Consider

which, by means of (1.123), becomes

From the local form of the conservation of mass relation (1.138) one obtains

so that (1.149) becomes

Equation (1.108) indicates that


or, for the subbody, in vector notation, and with the help of (1.145) and (1.146),

wherein a is the acceleration vector. Finally, by employing the Gauss theorem and (1.147), one obtains

which is the global form of the conservation of momentum law in terms of stress, incorporating the local form of the conservation of mass.

28

Chapter 1. The Mechanics of Continua For this law to be valid for an arbitrary volume , one must have

or, after use of (1.147),

which is the local form of the conservation of momentum law. By a similar procedure, based on an analysis of the conservation of the moment of momentum, it can be shown that, for a large class of (nonpolar) materials, the stress tensor is symmetric, i.e.,

The last two equations are Cauchy's first and second laws of motion.

1.3.18

Local Form of the Boundary Conditions on Discontinuity Surfaces

The starting point is (1.131)-(1.134). Taking = p therein gives, on account of (1.137) (i.e., this amounts to taking g = w and = 0),

Using (1.138) yields

which, for arbitrary a, implies

Considering the definition of v (it is the velocity of the moving surface a and therefore not discontinuous), the last expression can be written as

this being the local form of the boundary condition resulting from the continuity of mass relation. The choices 0 = pv l , rk = tk, and g = pft — wvl in (1.134) give


29

The first (volume) integral vanishes by virtue of (1.156) since V — a is uniform, so that

which, due to (1.147) and the fact that v is continuous, becomes

this being the local form of the boundary condition arising from the conservation of momentum and mass relations. Three cases arise in the application of these conditions depending on the physical problem at hand. The first is when a is a material interface, whence v = v, so that (1.162) is identically satisfied, whereas (1.165) yields

which expresses the fact that the surface traction is continuous across a material interface between two media. The second case corresponds to that of a body in a vacuum (medium with a + superscript), the discontinuity surface being the boundary of the body (medium with a — superscript). Then p+ = 0, v- = v, and again (1.162) is identically satisfied, whereas (1.165) yields

which, when tklnk is interpreted as an external (to the body) surface load, amounts to a boundary condition on surface traction. The third case corresponds to that of a fixed discontinuity surface so that v — 0 and (1.162) and (1.165) yield

and

1.3.19

Thermodynamic Considerations

The two main concepts in this subsection are energy and entropy, each governed by specific laws. The law of conservation of energy for a thermomechanical system such as those of interest in this book (in which only heat and mechanical energy are present) takes the form

and expresses the fact that the time rate of change of the kinetic energy and internal energy £ is equal to the rate of work W done on the body plus other energy rates (here, heat Q). The equality sign in this equation means that heat Q can produce changes in and/or , and changes in and/or £ can produce heat Q. Since in the study of continuous media one is

30


primarily interested in body deformation, the point of view is primarily one of thermostatics, this meaning that the change of kinetic energy is considered negligible compared to the otherthermodynamic quantities, i.e., 0, which, with = d , Wdt = W, Qdt = Q, enables one to obtain from (1.169)

which is an expression of the first law of thermodynamics. The units for Q and W in the above expressions are force times distance per unit time. In a uniform body V + S, heat can enter the body through S or be supplied from within V via distributed heat sources per unit mass h. Let q denote the heat vector per unit area acting at point x of S and directed outwards with respect to the body (whose outward unit normal vector is n at point x). The total heat input is

whereas, by definition,

and e is the internal energy density per unit mass of the body. The mechanical energy is the work done by surface and body forces per unit time:

Taking the time derivatives of the kinetic and internal energies and employing the divergence theorem in the expressions of Q and W yields


31

which, when inserted into (1.170), gives the following expression of global conservation of energy:

Henceforth, it is assumed that no matter is injected into the body, so that the second integral in the last expression vanishes. Also, on account of the conservation of momentum law, the third integral vanishes so that, for arbitrary V, one obtains the local form of the conservation of energy law:

Using the second Cauchy law of motion (1.158), and with

the deformation rate (symmetric) tensor, one can express the local form of conservation of energy as

The concepts of entropy and temperature are considered to have a self-evident status analogous to that of mass. Let designate the entropy density whose dimensions are energy per unit temperature per unit mass. Let 9 designate the temperature and recall that designates the internal energy density. The set of n + 1 independent variables , v = va, a = 1, 2 , . . . , n, on which depend e and 9 at a material point X corresponding to a position x at time t, defines the thermodynamic state of the body. The entries ( ) of v have the dimensions of mechanical entities, is independent of v so that

which are the so-called thermodynamic constitutive equations for internal energy and temperature. The choice of v is discussed in [159, p. 124]. The upshot is that v cannot be time, position, velocity, etc., and hence the internal energy and entropy density are not explicit functions of x and t, but they are dependent on the values of 77 and v at location x and time /. There is some evidence that for dilute gases and some fluids, one should consider V1 = 1 /p and all other entries of v to be irrelevant. The changes in e and due to changes in v and 9 are collectively termed a thermodynamic process.

32


Many of the working principles of thermodynamics, like those of continuum mechanics, derive from a series of axioms. The foremost of these is the Clausius-Duhem inequality

wherein F is the total entropy production, H the total entropy, S the entropy passing through S, and B the entropy resulting from sources in the body, which, for a uniform body, are given by

and b is the local entropy source per unit mass. The surface integral in (1.186) can be transformed into a volume integral via the Gauss theorem, enabling (for arbitrary V) the following relation, translating the local form of the production of entropy:

wherein y designates the local entropy production. The particular form of this equation depends on the type of process (thermal, diffusion, chemical, etc.). Whatever this type is, one can express S and b by

where is the entropy influx due to heat input, h/9 is the entropy source supplied by the energy source, and the remaining terms are due to other effects. Combining (1.191) and the local form of the conservation of energy relation results in

and introducing this result, as well as (1.190), into (1.189) gives

Henceforth, attention will be directed to simple thermomechanical processes (e.g., in which there are no chemical reactions) for which S1 = 0 and b\ — 0, so that (1.193) becomes

1.4. Mechanics of Elastic Media and Elastodynamics

33

this being the local form of the Clausius-Duhem inequality to which corresponds the global form inequality

A process (such as all those to be considered in the context of underwater acoustics) is said to be adiabatic if q = 0 and h = 0, from which it ensues that H > 0, signifying that the global entropy cannot decrease in an adiabatic process.

1.3.20

Constitutive Relations

The conservation of mass relation, the Cauchy first and second laws of motion, and the conservation of energy equation constitute a system of eight independent equations in the seventeen unknowns p, vk, k = 1, 2, 3; tkl, k, l = 1, 2, 3; qk = 1, 2, 3; if k = 1, 2, 3, and h are prescribed. Although the Clausius-Duhem inequality somewhat constrains the variation of the unknowns, it introduces two additional unknowns and , so that eleven additional relations must be found to make the problem solvable. Up to this point, abstraction has been made of the specific nature of the material of which the body B is composed. This information takes the form of constitutive relations which relate the above-mentioned unknowns to each other in specific functional forms that are constrained by axioms such as causality, material invariance, etc. [159, p. 145]. The materials considered herein fall into a rather wide class, termed thermomechanical materials, which are deformed and moved by heat or produce heat when they are deformed and moved. Actually, attention will be restricted to a subclass of this class, concerned with simple (thermomechanical) materials.

1.4 1.4.1

Mechanics of Elastic Media and Elastodynamics Definition of Elastic Media

Our focus henceforth is on so-called simple (thermomechanical) materials. Such materials are widespread and such that tkl, qk, , and at a given point of B are influenced only by what occurs in a small neighborhood of this point. A subclass of simple materials is that of elastic materials which are such that tkl, qk, , and at a given point P and instant t depend only on the deformation gradient xk and temperature at P, t, not on the whole past thermomechanical history of the material.

1.4.2

Constitutive Equations

The constitutive equations for elastic materials are

34


wherein t is the stress tensor and DK material descriptors indicative of the directional dependence of the constitutive functions t, q, , and . As stated above, the change in this material arises uniquely from a change of the configuration at time t. More precisely, if the reference configuration X is the undeformed, unstressed uniform-temperature initial configuration, the stress at time / is a result of the relative change of the configuration and temperature with respect to the initial state, regardless of intermediate changes. Equations (1.31) and (1.40) suggest that it can be useful to define the two functions

where FKL, Gkl , and are scalar invariants with respect to rigid motions of the spatial frame of reference (definition of objectivity). According to a theorem of Cauchy [159, p. 446], these functions must reduce to functions of the scalar products of the three vectors x,k and their determinant, i.e., (see (1.35)), of CKL and det(x k , k ) — po/p. This means that the constitutive equations take the form

wherein C is the Cauchy tensor. The argument p-1 can be discarded since, for given C, p-l is determined through the mass conservation relation. Also, since FKL and GK are referred to the material frame, it is not necessary to explicitly write their dependence on DK. Employing (1.197) and (1.15) gives


To these constitutive equations one must add the conservation of mass, conservation of momentum, conservation of energy, and Clausius-Duhem relations, which, for convenience, are rewritten as

On account of the symmetry of tkl, and with the help of a dummy index interchange, one finds that


35

so that the Clausius-Duhem inequality takes the form

Moreover, the constitutive relations, and the fact that X = 0, yield

so that (1.203) can be written as

The lagrangian and eulerian strain rates are

Equations (1.38) and (1.34), with the help of dummy index interchanges, entail

Consequently, (1.205) takes the form

This expression cannot be maintained for all d, these factors vanish, so that

unless the coefficients of each of

36


Again, use is made of the symmetry of the stress tensor and the fact [220, p. 75] that second-order tensors can be expressed as sums of symmetric and skew-symmetric tensors to obtain

A similar treatment of the bivector X k , K x i , L leads to

so that

The free energy is defined by

whence

and from (1.215) and (1.216):

On the other hand, (1.211) and the fact that

give


37

If

then (1.222) entails

whence

The use of (1.223) in (1.219)-( 1 -220) leads to

Materials characterized by (1.212) and (1.223)-( 1.226) are termed Green-elastic or hyperelastic materials.

1.4.3

Linear Constitutive Equations (Linear Elasticity)

Equation (1.38) entails

so that the principal constitutive equations of hyperelastic materials become, in terms of the lagrangian strain

It is assumed that the strains and rotations are small quantities, i.e.,

38


This authorizes a Taylor-series type of expansion of

(EKL)

wherein it is not unreasonable to assume

Consequently

and with the help of (1.141)

With

(1.229)-( 1.230) become

Use is now made of perturbation expansions in the small parameter , so that to zeroth order in 8

with (a consequence of (1.239))

To first order in 8 one obtains

with (a consequence of (1.239))

Equation (1.236) entails

and, in the same manner, one obtains, starting from (1.235),


39

Equations (1.58) and (1.237) imply

wherein use is made of (1.49) and (1.57), i.e.,

to find

whence

At this point, a digression is in order, the purpose of which is to evaluate This definition entails

which, on account of (1.12), results in

from which one must conclude that

The use of this result in (1.247)-( 1.248), and of various dummy index interchanges therein, entail

and the use of these results in (1.240)-(1.243), together with (1.234), gives

40


Combining the last two results into one leads, with the help of various dummy index interchanges and (1.234), to

Thus, the first-order perturbation approximation of the stress tensor is given by

It is henceforth assumed that the material is not prestressed in its natural state, i.e., , so that

The symmetry conditions (1.234) entail that the total number of elastic coefficients of is 21. The introduction of (1.70) into (1.259) yields the stress-strain relation for non-prestressed elastic materials

wherein

are the spatial elastic moduli subject to the symmetry relations

The relation of hypothesis:

to

is obtained via (1.233), with the help of the non-prestressed


1.4.4

41

Symmetry Properties of the Elastic Moduli Tensor

Due to (1.234) it is possible to regroup the nonzero elements of matrix

into the symmetric

Orthotropic materials are those that exhibit symmetry with respect to two orthogonal planes, say X3, = 0 and x1 = 0. Then it can be shown that

which indicates that the number of nonzero elastic moduli is reduced to 9. Isotropic materials, to which this book is henceforth devoted, possess no preferred direction with respect to their elastic properties. These materials form a subclass of orthotropic materials, characterized by

so that

wherein is the bulk modulus, UE the rigidity, and this pair is known as the Lame coefficients. The elements of the matrix in (1.267) can be expressed as

so that, by virtue of (1.261), (1.11), and three series of dummy index interchanges,

42


wherein and ue = uE Inserting this into (1.259), with a dummy index interchange and use of the symmetry properties of the strain tensor, gives the so-called Hooke-Cauchy law of non-prestressed isotropic elastic materials:

wherein

1.4.5

The Wave Equation for Elastodynamics in Linear Elastic Media

It was previously pointed out that hyperelastic (and therefore linear elastic) materials are in thermal equilibrium (i.e., q = 0). It can be shown [159, p. 193-194] that for such materials £ and n are not coupled, so there is no need to make further reference to these quantities. This means that all the information needed to solve problems of elastodynamics in nonprestressed media is contained in the conservation of momentum law, the relation of strain to displacement expressed by (1.271), and the Hooke-Cauchy stress-strain constitutive equation. It is convenient to rewrite the conservation of momentum law:

It is known that

so that

which, in the context of linear elasticity, is negligible because

. Consequently,

The introduction of (1.271) into (1.270) gives, after a dummy index interchange and use of the symmetry property of the elastic moduli tensor,

which, inserted into (1.271) together with (1.276) entails

which is the tensorial wave equation for linear elastodynamics. This set of partial differential equations can be recognized as linear in terms of the displacement u1.


1.4.6

43

Wave Equation for Elastodynamics in Compressible, Homogeneous Materials

This case corresponds to


1.4.7

Wave Equation for Elastodynamics in Heterogeneous, Isotropic Solids

Starting from (1.269) it can be shown, with the help of a few dummy index interchanges, that

so that

1.4.8

Wave Equation for Elastodynamics in Homogeneous, Isotropic Solids

The case of principal interest hereafter, that of waves in homogeneous (and isotropic) solids, is the one in which

The corresponding wave equation is

which is known as Navier's equation. Note that this equation is linear, like the wave equation for acoustics in homogeneous fluids, but unlike the nonlinear Navier-Stokes equation. By using (for any vector w)

one obtains the vectorial form of Navier's equation

44


It proves useful to invoke the so-called Helmholtz decomposition of u and f [2, p. 85]

Use is made of

and of (1.283) to obtain

which implies (for arbitrary u)

These two relations can be written as

wherein


45

Thus, the Helmholtz decomposition shows that the disturbances in a linear, homogeneous, isotropic elastic medium take the form of a combination of two types of waves, the first a scalar wave obeying the scalar wave equation (1.292), and the second a vectorial wave obeying the vector wave equation (1.293). The scalar wave is referred to as a bulk longitudinal (or compressional) wave and the vectorial wave as a bulk transverse (or shear) wave. The longitudinal wave travels with the wavespeed cp and arrives before the transverse wave, which travels with the wavespeed c, (the symbols p, s designate the order of arrival, i.e., p for primary, s for secondary). An interesting feature of these two wave equations is that they are uncoupled. As will be shown hereafter, this property is obtained at the expense of coupling of the boundary conditions, when the latter intervene in the problem at hand. Another important feature of the Helmholtz decomposition is that it relates three scalar components of the displacement (or force) to four scalars: the scalar potential 0 and the three components of the vector potential . This means that these potentials must be subjected to an additional constraint, called a gauge condition. Generally [2], the following choice is made:

1.4.9

Obtaining the Wave Equation of Acoustics in Heterogeneous, Inviscid Fluids from Navier's Equation

In inviscid, linear, Stokesian fluids, the shear strain vanishes and

so that (1.282) takes the form

If the association

is made, with the thermodynamic pressure and Ke the adiabatic compressibility, and both Ke and , as well as p, are assumed to be independent of the time variable t, then


46


which, after being submitted to the divergence operation, becomes

The reuse of (1.300)-(1.301) then gives

which is identical to the well-known acoustic wave equation, provided that the positiondependent wavespeed in the medium is defined as

When p is position-independent, (1.304) reduces to the standard wave equation of acoustics in homogeneous (in terms of bulk modulus and density) fluids.

1.4.10

Boundary Conditions between Two Linear, Isotropic, Homogeneous, Elastic Materials

Refer once again to the material in section 1.3.18. Equations (1.168)-(1.169) are the expression of the boundary conditions on the discontinuity surface a between two materials (fluid/fluid, solid/solid, solid/fluid). They reduce, after linearization, to

It can be shown that, for an interface between fluids, (1.307) takes the form

and (1.306) implies

from which it is found that

Designating the two contiguous fluids on the sides of an interface a by F1 and F2, these last two relations take the explicit forms


47

We recall (see sections 1.4.4 and 1.4.8) that for an arbitrary (solid or fluid), isotropic, elastic material

and at points of space on which there are no applied forces, and in the neighborhood of which the elastic material is homogeneous,

so that (1.307) and (1.309) take the forms

which, for the interface between two contiguous solid media S1 and 52, become

Combining these results, one obtains, on an interface between a fluid Fl and a solid 52,

The following limit cases, of considerable mathematical (if not physical) interest, are easily derivable from the preceding relations:

for a fluid/acoustically soft material interface, the pressure being negligible in the acoustically soft material;

48


for a fluid/acoustically hard material interface, the velocity being negligible in the acoustically soft material;

for a solid/rigid solid (in which the displacement is negligible) interface;

for a solid/vacuum (in which the stress is negligible) interface,

corresponding to the so-called impedance or Robin boundary condition (in which a and B are arbitrary real coefficients) of which the relations (1.323) and (1.324) are special cases (in all three cases, expressed by (1.323)-( 1.325), the stress and/or displacement are assumed to be negligible in the second medium). It is also possible to account for applied (and therefore assumed known) acceleration or stress on a portion of a:

for known applied acceleration on ones on and

(the other conditions being the previously evoked

for known applied stress on (the other conditions being the previously evoked ones on a — Formulas equivalent to (1.323)-( 1.327) in terms of velocity and pressure can be obtained for the case in which the adjoining media are both fluids.

1.5 1.5.1

Forward and Inverse Wavefield Problems Introduction

The fundamental ingredients of the forward and inverse wavefield problems considered in this book are (i) the system of partial differential equations (PDEs) for the wavefield, (ii) the boundary conditions (BCs), and (iii) the radiation conditions at infinity (RCIs) in case the spatial domain in which the solution sought is infinite. Since waves are spatiotemporal phenomena, they are treated, in the most natural way, in the space-time framework. The PDEs and BCs for this type of analysis were given in the preceding sections.

1.5. Forward and Inverse Wavefield Problems

49

It often proves useful to analyze the problem in the space-frequency framework— notably for treating wave phenomena in dissipative media. This is done by Fourier transformation of the space-time PDEs, BCs, and RCIs and will be described in subsequent sections. Since wave motion in the sea environment corresponds to many particular situations, the governing equations of only a few of these situations will be given in the remainder of this chapter. The others will be described in the applications part of this book (Chapters 2-5). Referring to the material in section 1.4.9, recall that f, , and u are generally functions of x and t, whereas c, , ue, and p are generally functions of position x = ( x 1 , x2, ..., x,,) (with n the dimension of the considered space), but not of t. The following symbol replacements will be made to conform with the usual notations:

1.5.2

The Frequency-Domain Equation for Propagation in an Unbounded, Heterogeneous, Inviscid Fluid Medium

Recall that the density is assumed not to depend on time t. The pressure and applied force are expressed by the Fourier integrals

and since p(x, t) and f1(x, /) are real functions, it is readily shown that

wherein the symbol * signifies the complex conjugate operator, so that

wherein R signifies "real part of." In spite of this latter result, it is preferable to employ (l.329)-(1.330), with the understanding that p(x, w) and f1(x, w) obey (1.331). The introduction of (1.329)-(l .330) into the time-domain wave equation gives (after interchange of integration and differentiation operators)

50


This relation can be true for all t > 0 only if [ . ] = 0, which means that

This is the sought-after frequency-domain wave equation (also termed the inhomogeneous Helmholtz equation) in heterogeneous fluids.

1.5.3

The Frequency-Domain Radiation Condition at Infinity

It is assumed that (i) space is divided into a bounded (inner) domain and an unbounded (outer) domain , (ii) the origin o of a euclidian reference system 0x 1 X 2 X 3 is entirely within , (iii) the support of the sources of the wavefield is entirely within , and (iv) the medium filling is generally heterogeneous, whereas the medium filling is homogeneous. One can show, as in the previous section, that the frequency-domain wave equation in , (which, it is recalled, is homogeneous, and wherein the wavespeed is the constant c) satisfied by the scalar potential and the three scalar components of the vector potential (in an elastic solid) is of the generic form

The case of a 3D space is treated first. A plausible guess is that the asymptotic solutions for large

are of the form

wherein

One finds that

which shows that the two functions written in (1.337) are indeed asymptotic solutions, for large , of the frequency-domain wave equation (1.336). One can write


51

The only way to ensure compatibility with the solution satisfying the time-domain radiation (i.e., outgoing wave) condition at infinity (TDRCI) is by choosing w + (x, w) instead of w - ( x , w). Thus, in three dimensions, the frequency-domain radiation condition at infinity (FDRCI) amounts to the choice

In the same manner one can show that the 2D version of the FDRCI corresponds to the choice

wherein x = x1i1 + x2i2.

1.5.4

Governing Equations for the Frequency-Domain Formulation of Wave Propagation in an Unbounded, Heterogeneous, Inviscid Fluid Medium

It was already shown that the PDE of this problem (see (1.335)) is

There are no initial conditions to cope with in the frequency domain since this is a permanent regime. The only remaining condition is the RCI, which takes the following form:

1.5.5

Governing Equations for the Frequency-Domain Formulation of Wave Propagation in Two Contiguous, Semi-Infinite, Heterogeneous, Inviscid Fluid Media

The media are the same as in section 1.5.2, so that the governing equations are (after use of (1.329))

52


When the domain occupied by the second fluid medium is bounded, the governing equations are the same as above, and the radiation condition (1.350) becomes superfluous.

1.5.6

Governing Equations for the Frequency-Domain Formulation of Wave Propagation in an Unbounded, Heterogeneous, Isotropic, Elastic Solid

The configuration is the same as in section 1.4.7, so that the governing equations are (after use of a Fourier transform such as (1.329))

1.5.7

Governing Equations for the Frequency-Domain Formulation of Wave Propagation in Two Semi-Infinite, Heterogeneous, Isotropic, Elastic Solid Media in Welded Contact

The media are the same as in section 1.5.6, so that the governing equations are (after use of (1.329))


53

Due to the facts that (i) the media were assumed to be homogeneous in the vicinity of the boundary a and (ii) the supports of the applied forces were assumed not to intersect a, and due to (1.353)-( 1.354), in which the terms in and u1 are dropped, it is preferable to replace (1.356) by

The remaining equations are

When the domain occupied by the second solid medium is bounded, the governing equations are the same as above, and the radiation condition (1.359) becomes superfluous.

54

1.5.8


Governing Equations for the Frequency-Domain Formulation of Wave Propagation in a Semi-Infinite Domain Occupied by a Heterogeneous, Inviscid Fluid Contiguous with a Semi-Infinite Domain Occupied by a Heterogeneous, Isotropic, Elastic Solid

The media are the same as in sections 1.5.2 and 1.5.6, so that the governing equations are (after the use of Fourier transforms such as (1.329))

Due to the facts that the media (i) were assumed to be homogeneous in the vicinity of the boundary a and (ii) the supports of the applied forces were assumed not to intersect , and due to (1.361), in which the terms in and u1 are dropped, it is preferable to replace (1.362) by

The remaining equations are


55

When the domain occupied by the solid medium is bounded, the governing equations are the same as above, and the radiation condition (1.366) becomes superfluous.

1.5.9

Eigenmodes of a Linear, Homogeneous, Isotropic Solid Medium of Infinite Extent

Since we are concerned here with eigenmodes, the governing equation is (1.351) in which the applied force term is taken equal to zero, so that

and owing to the fact that the medium is assumed to be fully homogeneous, position-independent, so that

u, and p are

Again, it appears to be plausible for the eigenfunctions of this equation to be of the form

wherein A is, in general, a constant and

with k a position-independent quantity. Equation (1.369) is the expression of a plane bulk wave. It is readily found that i so that (1.368) gives

which, in extended form, looks like

wherein which yields, after some algebra,

. Anontrivial solution is possible only if det ( ) = 0,

56


the (positive, since k > 0 for w > 0) solutions of which are

or, on account of previous results (1.294)-(l .295),

This result signifies that the wave equation in linear, homogeneous, isotropic, unbounded solid media admits two eigenvectors, corresponding to a bulk plane wave traveling with wavespeed cp and to another bulk plane wave traveling with wavespeed cs. As in the previous sections, dissipation can be accounted for by making the wavespeeds complex. This can be implemented by taking . and/or u complex.

Chapter 2

Direct Scattering Problems in Ocean Environments

2.1

The Constant Depth, Homogeneous Ocean

In Chapters 2 and 3 we consider in mathematical detail the direct and inverse scattering problems for an object in a wave guide.

2.1.1

Point Source Response in a Constant Depth, Homogeneous Ocean

In a homogeneous ocean of constant depth, the response to the point source time-harmonic acoustic wave (Green's function) satisfies the nonhomogeneous equation

Here, the source is located at ( X 0 , 0) in a cylindrical coordinate system, where A3 is the 3D Laplace operator. Assuming the ocean to be of constant depth h, the surface conditions are

z = 0 is referred to as a pressure-release boundary, and z = h is a totally reflecting boundary. Using the method of separation of variables with the boundary conditions (2.2), we may represent the Green's function as

57

58

Chapter 2. Direct Scattering Problems in Ocean Environments

where and are Hankel functions of order zero of the first and second kind, respectively. Since we restrict our attention to outgoing waves, the appropriate form of the radiation condition is

Here the coefficients an are the eigenvalues of the separated modal solutions

, i.e.,

Throughout this book, we shall refer to this condition (and some of its variations) as the outgoing radiation condition, and the corresponding Green's function shall be referred to as the outgoing Green's function. The outgoing Green's function has several equivalent representations [6], including the normal mode representation

and the integral representation

2.1.2

Propagating Solutions in an Ocean with Sound-Soft Obstacle

In this section we use the methods of integral equations to solve the direct scattering problem for the reduced wave equation.2 Integral equation methods are powerful tools for the investigation of the boundary value problems associated with the scattering of waves by bounded obstacles. For simplicity we concentrate on scattering surfaces that are smooth, say at least twice continuously differentiable. The reader may find further details concerning the regularity properties of surface potentials corresponding to nonsmooth surfaces in [ 125]. We suppose that the ocean occupies the space given by = {(x, z) R3; x = 2 (x1, x2) € R , 0 z h], h being the ocean depth. The submerged object occupies the space , abounded domain with a C2 boundary. The outward-pointing unit normal is denoted by v. Without loss of generality, it may be assumed that (0, Z0) € for some zo e [0, h]. 2

In this and subsequent chapters, unless otherwise stated we shall restrict our attention to the case of a finite, homogeneous ocean.

2.1. The Constant Depth, Homogeneous Ocean

59

In this and subsequent chapters we retain the following notation:

The direct scattering problem in a constant depth, homogeneous ocean with a pressurerelease surface and a rigid bottom may be formulated as seeking a function such that u satisfies the Helmholtz equation

with the pressure-release condition, the completely reflecting bottom, and the outgoing radiation condition. Moreover, on the submerged object the boundary condition

holds. It is well known that u has a normal mode representation

where un (x) is the nth Fourier coefficient with respect to the eigenfunction outgoing radiation condition

, which satisfies

uniformly for , where and The constant R is chosen so that We will refer to the problem described by (2.8), (2.2), (2.9), (2.10), (2.11) as Problem D.

2.1.3

The Representation of Propagating Solutions

If u is a solution of Problem D in on the boundary exists in the sense that the limit

, such that the normal derivative

60


exists uniformly on

, then it follows by Green's identity that

Here G is the Green's function defined in section 2.1.1, and

represents anE-neighborhood of (x, z); moreover, ,and e, R are positive numbers such that . It is clear that we can separate the polar singularity of into the form

where the regular part is given by

Moreover, that

and

is bounded and continuous at

It is easily seen


61

From (2.12) and using (2.2), (2.9), (2.10), we thereby obtain

Using the radiation condition (2.11) and the fact that it is easy to see that [467]

Therefore, we may conclude that if Problem D, then

Remark. Since satisfy the same condition.

2.1.4

is a solution to

satisfies the radiation condition, u(x, z) must also

The Uniqueness Theorem for the Dirichlet Problem

In order to show that our problem has a solution that uniquely depends on the boundary data, we need to show that the problem with homogeneous data has the trivial solution. In other words, we want to show that if is a solution of the homogeneous Problem D and for n — 0, 1, . . . , then in . We will prove this in several steps. Lemma 2.1. boundary data

, then for any

is a solution of Problem D with homogeneous such that

when u,, is the nth normal mode of the solution. Proof. By Green's identity, we have

62


Hence,

which follows from u vanishing on

. Now expanding u(x, z) in normal modes

we have

from which Lemma 2.1 follows. The approximation of the far field is provided by the next lemma. Lemma 2.2. Under the assumptions of Lemma 2.1, we may conclude that

Proof. In view of the radiation condition (2.11), for

But by Lemma 2.1,


and for

63

hence, the last term of (2.15),

becomes

Since the an are positive for n = 0, 1, . . . , N, (2.16) implies that as

Since for sufficient large R, any solution of Problem D can be written in the form

it follows from (2.17) that

Consequently we may bound the coefficients

However, since

it follows that

that is,

we have

64


Hence, we obtain the desired result, namely,

Now we may establish the following lemma, which is crucial for establishing our uniqueness theorem. The original idea of the proof is from [312]. Lemma 2.3. Let vx = (v 1 , v2, v3) be the outward-pointing normal vector of at (x, z) and vx = (v1 ,v2), x = (x 1 , x 2 ). If holds for any , and u satisfies the assumption of Lemma 2. /, then

Proof. Let and V be the outward-pointing normal of consider the function defined by the integral

. Then let us


65

where we have used the facts that v • X = r on Cr, v • x = 0 on Sr U Gr, and used the boundary data for u. On the other hand, since u — 0 on and on , from u == 0, 0 on Sr it follows that A(r) has the alternate reduction:

Hence,

Using Lemma 2.2 and letting

we obtain

66


where vx = — v . Therefore,

Finally putting together the above lemmas, we arrive at our goal, namely, the follow-

ing. be a bounded region with C2 boundary, such that Theorem 2.4. Let is a solution of Problem D with holds for homogeneous boundary data g — 0, then u = 0 in Proof. By Lemma 2.3, for all for some It implies that

2.1.5

Since u = 0 on in if in Therefore, by the real analyticity of

then

An Existence Theorem for the Dirichlet Problem

We have already seen that if there is a solution, it must be unique. Now we need to show that a solution actually exists. A useful trick for proving solvability for Problem D is to combine the double- and single-layer integral representations to obtain an integral equation corresponding to this problem. Consequently we seek a solution of the exterior Dirichlet Problem D in the form of a combined double- and single-layer potential:

where we use the subscript to denote the variables used to compute the normal derivative and perform the integration. The source point is at (x, z), and is an arbitrary real parameter. We recall that has a representation of the form (2.6). We define the operators S, K, K' from C( ) to C( by

3

From now on, in this and subsequent chapters we will assume that £2 satisfies the condition

on

.


67

Moreover, let S, K, K' be the operators defined by

where

Integration is with respect to the (

) variables here. We then have the decompositions

where Si, K1, K1 are the integral operators above with the continuous kernels

respectively. Based on well-known boundary properties for these operators S, K, and K' [125, Chap. 2], it is clear that S, K, K' satisfy the same jump condition as, respectively, theS, K, K'. The following two lemmas are useful in order to obtain existence. Lemma 2.5. The composite double- and single-layer potential u(x, z) defined by (2.18) is a solution of Problem D, provided that the density is a solution of the integral equation

Proof. The combined double- and single-layer potential u(x, z) obviously satisfies Helmholtz's equation (2.8) in , the radiation condition (2.3), and the surface and bottom

68


conditions (2.2). integral equation

So if

then by letting (

we obtain the

satisfies (2.19), then

Lemma 2.6. The integral equation (2.19) is uniquely solvable. Proof. Since K — in S is a compact operator, we need to prove that the only solution of the homogeneous form of (2.19) is trivial. Let be a solution to the homogeneous equation then u as defined by (2.18) solves Problem D with g = 0. Therefore, by the uniqueness theorem, in . Let , and define the interior limits as ( ,

From the jump relations we have

The first Green's identity implies that

Since k is real, it must follow that

and

on

Combining Lemmas 2.5 and 2.6, the conclusion then follows. Theorem 2.7. Problem D is uniquely solvable if

holds on

2.1. The Constant Depth, Homogeneous Ocean 2.1.6

69

Propagating Far-Field Patterns

We know that in a uniform wave guide there are only a finitely many propagating, modal solutions, whereas the other modes evanesce. Therefore, the far-field pattern in a uniform wave guide contains only the information emanating from the propagating modes; thereby much information is lost in the process. In this section, we present a representation of farfield patterns using double- and single-layer potentials, and discuss some of the properties of far-field pattern. We use the abbreviated notation R = |x — |, r = lxl r' — l lx = (r, ), £ = (r', 0'), R2 = r2 + r'2 - 2rr'cos( - ' . In addition, we make use of the Hankel function H (kR) expansion

where €0 = I, and €n = 2 for n > 1. In view of (2.20), we may expand the kernel as

where

70


Sinceforw > N the Fourier coefficients are imaginary, i.e., expansion

, we get the asymptotic

which implies that the kernel has the asymptotic expansion

Now from (2.18) (set n = 1),

where the

play the role of a far-field pattern and tion (2.19).

is the unique solution of integral equa-


71

We call the function

the propagating far-field pattern. From the representation of

Now let F be the set of all possible far-field patterns. In the case of R3 and R2, we know that there is a one-to-one correspondence between F and . In particular, if u is a solution to the Helmholtz equation in the exterior region satisfying the radiation condition, and if its far-field pattern vanishes identically, then in . Unfortunately, this is not true in a wave guide. The following is a typical example showing that it is, indeed, possible for the far-field pattern to be identically zero. In particular, if ,

for all n = 0, then by a simple application of the representation theorem, any solution to Problem D having has the asymptotic property

that is, its far-field pattern is identically zero.

72

2.1.7


Density Properties of Far-Field Patterns

In this section we discuss scattered waves and the corresponding far-field patterns for a given incident wave ui. The incident wave ul is scattered by ft, thereby producing a far-field pattern. We want to extract information about the far field in order to use it to investigate the object ft. A similar problem has already been investigated in R2 by Colton and Kirsch [123]. They introduced a certain dense subset of the far-field pattern. Colton and Monk [127], [128] were able to determine the shape of the object by introducing an extremal problem and solving it in projected subspaces. However, in the case of finite depth oceans, we have shown in Gilbert and Xu [211, 210] that the propagating far-field pattern can only carry the information from N +1 propagating modes, where N is the largest integer less than . This loss of information makes this nonlinear, improperly posed, inverse scattering problem very different from the case studied by Colton and Kirsch [ 123] and the others mentioned above. Let ul be the incident wave and us the corresponding scattered wave. Then, for a sound-soft object ft, the total field u = u' + us satisfies

In addition, the outgoing radiation condition must be satisfied:

where is the nth mode of us. In what follows we consider density properties of the propagating far field in a suitable subspace of , where C1 is the unit cylinder. The decomposition of the propagating far field into orthogonal components suggests a numerical algorithm for the express purpose of reconstructing the object ft.

2.1.8

Complete Sets in

In order to reconstruct an object it is useful to approximate the integral density by a complete set of functions on an arbitrary boundary . The relative complement of ft in is . We use J n ,(r) to denote Bessel's function of order n, and to denote Hankel's function of the first kind of order and an are defined as before by (2.4) and (2.5).

2.1. The Constant Depth, Homogeneous Ocean Theorem 2.8. Let for n = 0, 1, 2,

be a complex number such that 0 Then the sets of functions

73 Im

and

i/2h

are complete in L2 Proof. It suffices to show that if

if the following projections hold:

for m, n = 0, 1, . . . , , then g is identically zero on . Let (2.26), (2.27) be true for some and let be a solid cylinder containing in its interior, Then, when (x, z) , and ( ) we know that for , we can expand the Green's function as

Here we denote (x, z) in cylindrical coordinates by ( From (2.26) and (2.27), we can see that

) and denote

is identically zero for Since u, as defined by (2.29), is a solution of the Helmholtz equation in we can conclude by the analyticity of solutions to the Helmholtz equation [125] that u(x, z) is identically zero for ( Let (x, z) tend to . Then, in view of the ray representation for the Green's function (2.13) and from the properties of single- and double-layer potentials, we know (cf. [125], [262]) that

74

Chapter 2. Direct Scattering Problems in Ocean Environments Now let us denote by

the boundary values

and

Similar definitions are made for ( double-layer potentials,

and

(

From our knowledge of single- and

and

Since

, we have from (2.31) and (2.32) that

Hence u, as defined by (2.29), is a solution of the Helmholtz equation in assumes the boundary data (2.33) on It follows that u = 0 in . By the relation

we can conclude that g — O on

2.1.9

and continuously

This proves Theorem 2.8.

Dense Sets in

In this section we construct a dense set of functions in order to approximate boundary data on . We modify the notation of [123] to the case of namely, let N be the family of any finite subset of natural numbers containing 0, 1, . . . , N; C1 = [0, h] x [ ]; and let


75

Moreover, we set

and

We want to prove the following theorem. Theorem 2.9. Proof. Let

is dense in

.

such that

We need to show that (2.36) implies g — 0 on . If u is an arbitrary element of TD( then from the representation formula (2.14), we get

),

where Then u = 0, and (2.37) implies that

and

In view of the representation (2.13) for and the fact that it is symmetric with respect to the points (x, z) and ( ), it may be shown that K' is the adjoint operator to K, subject to the pairing

Moreover, it can be seen that I + K' + iS is invertible from section 2.1.5. Now from (2.38) and (2.39) it follows that

76


and hence

Substituting (2.41) into (2.36) yields

Since

, by using the Jacobi-Anger expansion,

we conclude that Hence, from Theorem 2.8, we get

and

) are elements of.

and

2.1.10 The Projection Theorem in VN We will now establish a condition for the far-field patterns to be dense in VN for arbitrary scattering regions which, moreover, satisfy the property on . Let N = , where [a] means the integer part of a, and let us introduce the product space

where the n = 0, 1 , . . . , N, are defined by (2.5). From section 2.1.6 we know that the propagating far-field patterns from the reduced wave equation, in a homogeneous wave guide, are contained in VN. We define the injections as follows:

and where


77

by is the propagating where far-field pattern of a solution of Helmholtz's equation in Let and We will prove the following decomposition for the space VN. Theorem 2.10.

where

I is the closure of F

Proof. By the representation formula (2.14), we have

and

We decompose the total field into the incident and scattered waves, u — u' + us, from which follows the integral identity

In view of the asymptotic behavior of Hankel's function and the representation

we obtain the asymptotic formula

The function the propagating far-field pattern.

is an alternate representation of

78


Let

i and

and (2.45), we have

Since

for any n — N + l , . . . ,

, the previous expression becomes

This proves the orthogonality

Now we prove that

In fact, if g e VN such that

then from (2.46)

then

. From (2.44)


79

where

is dense in Using Theorem 2.9 That is, and on Since VN is a Hilbert space, (2.47) implies the theorem.

so we can conclude that v = 0 It proves

Using the decomposition Theorem 2.10, we get the following density result. Corollary 2.11. A sufficient condition/or the far-field patterns of the problem (2.21)-(2.25) to be dense in VN is that ; i.e., the eigenfunctions of the Dirichlet problem are not elements of the set AN.

2.1.11

Injection Theorems for the Far-Field Pattern Operator

In view of the previous sections, we can represent the scattered wave us in the form of a combination of single- and double-layer potentials, namely,

where Im

and

We note that linear operator in For , function

satisfies

) is inverlible for any k > 0 and its inverse is a bounded , denoted by we recall the normal mode representation of the Green's

and in view of the asymptotic behavior of

has an asymptotic expansion

80


where Hence a natural way to define the far-field pattern operator

We know that

form a complete system in

) (Theorem 2.9). Now let

and i be the space orthogonal to ' product. Then where operator F. Hence, if then

in ) under the usual iinner i is the null space of the far-field pattern

i.e., the propagating far-field pattern of us is identical to zero. Next we wish to formalize a mapping from incoming waves to far-field patterns. At this stage, we think of the object as known and fixed. Let

for any

denote is invertible for any

, which is a continuous function on we can express

Combining (2.49) and (2.50), we define a mapping

. Since


81

Let

then we can see from (2.52) that Definition 2.1. Let

be two incoming waves. We say that , which is denoted by

is equivalent

Let [ul} be the equivalent class under this equivalent relation ~. Then for any given far-field pattern (i.e., in the range of Fan), there exists an equivalent class {u1} such that any element in the class is mapped onto /:

We refer to such {u1} as an equivalent class solution. Define

then we call

a minimal norm solution of integral equation (2.53) if

and

Theorem 2.12. //

Proof.

such that.

then

. We can represent

82


It follows that

and

Hence,

Consequently,

on Corollary 2.13. Let [ul} be an equivalent class solution of (2.53). Then there is a unique such that any element of{u'} can be written as

where Since

for any element of {u1}, from which we can conclude as follows. Theorem 2.14. Let {ui} be the equivalent class solution of (2.53), which has a unique decomposed expression

Then

is the minimal norm solution of (2.53).

Theorem 2.15. If such that the corresponding propagating far-field pattern then the corresponding scattered wave Proof. Let

such that

By Theorem 3.1, Hence us = —ui — 0 on . From the uniqueness theorem for the direct scattering problem, Theorem 2.15, it follows that


2.1.12

83

An Approximate Boundary Integral Method for Acoustic Scattering in Shallow Oceans

In this section we investigate an approximate boundary integral method of the scattering problem which describes the scattering of acoustic waves from a cylindrical object with a sound-soft boundary in a shallow ocean. This scattering problem is essentially a 2D problem and is modelled as a boundary value problem in a wave guide. Here, once more, the governing equation is the Helmholtz equation. Let ; be a region corresponding to the finite depth ocean, where d is the ocean depth. Consider an object imbedded in , which is a bounded, simply connected domain with a C2 boundary . An incoming wave u' is incident on and is scattered to produce a propagating wave u as well as its far-field pattern. If the object has a sound-soft boundary , this problem can be formulated as a Dirichlet boundary value problem for the scattering of time-harmonic acoustic waves in . Namely, one is to find a solution for the 2D Helmholtz equation (2.21)-(2.25), where the condition (2.24) is now replaced by the nonhomogeneous condition u = f on We call this new problem as Problem D2. The usual boundary integral equation (BIE) method uses the fundamental solution of (2.21) (2D) and reformulates the solution u as a layer potential. The BIE method reduces the problem to a problem in a lower dimensional space but leads to an integral equation on the boundary as well as the two unbounded boundaries. As a result, a complicated Wiener-Hopf integral equation system needs to be solved. To avoid the integral equation on the two unbounded boundaries, instead of using the fundamental solution, we may use the Green's function of the Helmholtz equation in :

which automatically satisfies the pressure release, the reflecting boundary conditions, and the radiation condition. Let vx = ( v 1 , v2) denote the outward normal vector (toward the interior of ) at the point x — (x1, x 2 ). By Green's identity we recall that

for any solution of Problem D2 For Problem D2, we know from section 2.1.4 that if holds for any ( ) , then Problem D2 has a unique solution. Moreover, introducing the double-layer potential

we know that the solution of Problem D2 is given by (2.55), where boundary integral equation

is the solution of the

84


Equation (2.56) has a unique solution when k is not an eigenvalue of the interior Neumann problem in £L. It is a challenge to develop an efficient numerical method for solving the boundary integral equation (2.56) where its kernel function is given in an infinite series, as there is no known method for the efficient evaluation of this series. An efficient method for evaluating the infinite series requires an appropriate truncation that preserves accuracy and also minimizes arithmetic operations. In this section, we shall focus on an efficient method for evaluation of the kernel of the boundary integral equation (2.56) and then present a quadrature method for solving this equation. The method is fully discrete and is estimated to have an O ( N - 3 ) rate of convergence, where N is the number of the quadrature knots distributed along the boundary. Our numerical experiments show that the method has good accuracy and involves a low CPU time. In the next section, we investigate in detail the approximation of the kernel of (2.56). Using these estimates, we discretize the integral equation (2.56) by a quadrature method. The quadrature method allows a procedure using the least number of arithmetic operations which provides an O ( N - 3 ) error estimate for the kernel, which in turn allows an O ( N - 3 ) truncation error. Some numerical results and convergence discussion are presented in [480]. Approximation of the Kernel The numerical solution of equation (2.56) requires the evaluation of the kernel Since G(x, y) is given only as a sum of the infinite series, this evaluation can only be done approximately. In this section we derive an approximation for the kernel The evaluation of the kernel is the most costly part in the numerical solution of equation (2.56), so it requires delicate estimates. We split G into G — GO + M, where GO is the Green's function for the Laplace equation satisfying the conditions (2.22)-(2.23). G 0 ( X , y) is singular at x = y,and M(x, y) is continuous, and hence GO is the dominant term in the splitting. It should be emphasized that the Green's function is given only for wavenumber . Otherwise one of the coefficients in the expression (2.54), , will become infinite. For simplicity of exposition, we assume that the depth and use the same notation as in (2.5) for , i.e.,

The function GO is defined by

and the function M by


85

has a simple expression (see, for example, 1.448.4

where c in [221])

where

Now we split the kernel into

Because GO has an analytic expression (2.57), can be evaluated analytically. Assuming a smooth boundary, a straightforward calculation leads to a well-known identity

and,for

we have the bound

where C is a constant. Now we focus on the approximation of

Let

Then

By a straightforward calculation we estimate the gradient

where

86


and

Direct analytical computations lead to the following results:

where G 0 ( X , y) is given by (2.58), and

where

2.1. The Constant Depth, Homogeneous Ocean (3) Now define the functions

and

Then

(4) If we define

and

then

(5) If we define

and

then

87

88

Chapter 2. Direct Scattering Problems in Ocean Environments (6) If we define

and

then,for

Using these results, we approximate |^ in the way that

is approximated by

where is evaluated analytically and is evaluated by the Gauss-Legendre quadrature rule. The choice of p relies on the size of , which will be discussed in detail in the next section. Numerical Solution of the Boudary Integral Equation By use of the splitting (2.59), equation (2.56) can be written in the form

In this section we discretize this equation by a quadrature method and replace the kernel by an ) approximation. We assume that the boundary is given by a -periodic parametric representation

with for all s. Furthermore, we assume that the kernel of the integral equation (2.60) by

and set

is a

function. If we denote


89

then (2.60) takes on the form

Recall from [125] that for a C2 boundary

, there is a constant C > 0 such that

Therefore, L 0 ( S , a) is continuous for The continuity of is obvious since, from results (l)-(6) above, we know that /j, j = 1 , . . . , 6, are L1 all a uniformly convergent senes. Moreover, letting

it can be shown that that

for j — 1, 2, 3, and

are continuous functions of .v and y, and

j — 2, 3, are continuous except at points where y1 = x1. In addition, for. = y(s)

and y = y( ), the function (I1 + /4)( , y) can be split into

where I 1 , 4 (s, a) is a smooth function of ( , ). Then

where

and

For this reason, we shall use the ordinary rectangular formula

90


the weighted quadrature formula

and the weighted quadrature formula

where tk = kh with h — the weights are given by

and N an even integer are the equidistant quadrature knots, and

and

Applying the quadrature formula (2.64), (2.65), and (2.66) to the integrals in (2.63), we replace the integral equation (2.56) by the linear system

for the approximate values Wj to w(tj), where gj — g(tj). This linear system has an O(h3) truncation, and so can produce an O(h3) rate of convergence for Wj to w(tj). The linear system (2.67) involves the calculation of functions a, b, L0, and L2 at the points (tj, tk), and the calculation of the weights. The evaluations of a (tj, tk) and b(tj, tk) are direct. L0(tj, tk) can be evaluated using the explicit formula (2.62). R 1 ( t j - k ) and R2(tj, tk) can be evaluated using the fast Fourier transform (FFT) with only O(N log N) arithmetic operations. It is difficult and expensive to evaluate L 2 (t j , tk), because J(x, y) is an infinite series function which has to be truncated properly. The following lemma provides an O( ) approximation to L2. Lemma 2.16. Let


91

and

where

are defined earlier, and p is chosen as

otherwise. Then for any

there is a constant C independent of N, s, and a such that

Proof. It is clear that we need only prove

For

we have from that

The proof for the other cases follows in a similar way. These estimates complete the proof of the lemma. The choice of the number p is made in order to save arithmetic operations when the truncation Jp is used to approximate . Using the approximation of, in integral equation (2.63), we arrive at a linear system

for the approximate values rate of convergence.

for w(t j ). From Lemma 2.16, the system (2.70) has an

92

2.2 2.2.1


Scattered Waves in a Stratified Medium Green's Function of a Stratified Medium and the Generalized Sommerfeld Radiation Condition

As previously mentioned, a point in R3 will be described simultaneously in Cartesian and spherical coordinates as

We use, furthermore, the notation

Let c(x, z) denote the sound speed at (x, z) € R3, C0 a reference sound speed, and w the frequency of the incident wave. We refer to k = w/co > 0 as the wavenumber and

as the index of refraction. The medium will be referred to as stratified if for some constant ao,

Moreover, for the simplicity of exposition, we assume that n some constants h1, h2. and positive constants n-,n+,

C1(R3)

such that for

The assumption that n o (Z) is constant outside of a slab is not essential and may be relaxed to conditions (A) and (B) in [470]. A function G( is the outgoing Green's function for the time-harmonic acoustic wave in a stratified medium if G(x, z satisfies

in the generalized function sense and satisfies the outgoing radiation condition. The outgoing radiation condition may be explained as follows. For the following equations, let pi(z, kd) be Jost functions:

2.2. Scattered Waves in a Stratified Medium

93

i.e.,

Using Fourier transforms [6], a Hankel transformed representation for may be obtained:

where W(ka) is theWronskianof p 1 ( z , ka) and p 2 ( z , ka}. Here the convergence of the integral is understood in the L2(R) sense. Let be the normalized eigenfunctions of (2.74), which have the same behavior as the Jost functions as z —> ±00, and ai (i = 1, 2 , . . . , N) the corresponding eigenvalues. Under the condition (2.72), we can show the following assertions (cf. [469], [451]). (1) There are only a finite number of eigenvalues, and all of them are simple. If constant, then N = 0. and there is such that

for 1 < i < N. Recall that

and W(ka) has simple zeros ai (i = 1, 2 , . . . , N), which are the eigenvalues of (2.74). We can rewrite the integral as a contour integral in the complex plane and obtain (cf. [470])

Here C is a contour consisting of the semicircle {z : Im > 0, \z\ = R}, a curve connecting — R to 0 slightly above the real axis, and a curve connecting 0 to R slightly below the real axis. Now we study the asymptotic behavior of as In view of the asymptotic behavior of Hankel's function

94


we have

where

It follows by the method of stationary phase (cf. [54]) that

The function called the free-wave far-field pattern from a point source in a stratified medium, has the representation

Furthermore, for given z, we have the asymptotic representation

It is natural to interpret the vector function

as the guided-wave far-field pattern vector from a point source in a stratified medium.

NUMERICAL ILLUSTRATIONS

Normwise

Comments on F Once more, one appreciates the different impact of normwise and componentwise perturbations. i) On the left-hand page, most of the right part of the spectrum is stable under componentwise perturbations. ii) On the right-hand page, it is, on the contrary, the right part of the spectrum that is most affected by normwise perturbations.

209

96


and where Liouville system

are, respectively, the nth eigenfunction and eigenvalue of the Sturm-

Hence, the free-wave far-field pattern is identical to zero, as we expected. The guided-wave far-field pattern is given by

Here N is the number of the propagating modes. From (2.77), (2.78), and (2.79) we have the following lemmas. Lemma 2.17. Forgiven is a solution to the corresponding homogeneous form of (2.73) with respect to For given each component of is a solution to the corresponding homogeneous form of (2.73) with respect to Lemma 2.18. For any given in for For any given and

are analytic4 is analytic in and for

In what follows we prove three reciprocity relations among the free-wave far-field patterns and the guided-wave far-field pattern vectors corresponding to incident distorted plane waves and normal mode waves. Then we prove conditions under which a set of far-field patterns is complete in a Hilbert space based on the reciprocity relation. These properties are important in investigating the inverse scattering problems.

2.2.2

Scattering of Acoustic Waves by an Obstacle in a Stratified Space

In a stratified medium, sound waves may be trapped by acoustic ducts and caused to propagate horizontally [470], [54], [451], [446]. In this case, the waves scattered by either a compact obstacle or a local inhomogeneity does not, in general, satisfy the Sommerfeld radiation condition, but rather a generalized Sommerfeld condition. Consequently, we refer to these scattering problems as generalized scattering problems. In this section, we investigate the scattering of time-harmonic, acoustic waves in a stratified medium with a local inhomogeneity. The uniqueness and existence of the generalized, direct scattering problem is established using integral equation methods. Relations between the far field of scattered acoustic waves and the sound profile of the inhomogeneity will be obtained. Using these relations, we shall prove three reciprocity relations among the free-wave far-field patterns and the guided-wave far-field pattern vectors corresponding 4

By analytic here we mean that the real and imaginary parts are real analytic functions.


97

to incident distorted plane waves and normal modes. Then, based on these reciprocity relations, we shall establish a condition under which a set of far-field patterns forms a complete subspace of a Hilbert space. These properties are important for investigating generalized, inverse scattering problems. Let be a bounded domain with C2 boundary. The obstacle, scattering problems for a time-harmonic wave can be described mathematically as follows: Given an incident satisfying wave,

find the scattered field

such that the total field, satisfies one of the following boundary conditions depending on the physical property of the obstacle. (1) Dirichlet condition (Problem D):

(2) Neumann condition (Problem N):

(3) Robin condition (Problem R):

where

and Moreover, the scattered wave us(x,z) satisfies the generalized outgoing radiation condition. That is, we define the scattered guided wave us and the scattered free wave usf as follows:

however,

and

satisfy

98


as

uniformly for

and

n = 1, 2, . . . , N,

as r ->•

uniformly for 0 e [0, 2 ],

where k — kn+ if e [0, /2) and k = kn- if 0 6 ( /2, ]. We refer to this set of conditions as the generalized outgoing radiation conditions. Using the generalized Sommerfeld condition, we established in [468] the uniqueness and existence of the solutions to the Dirichlet, Neumann, and Robin problems, respectively. Lemma 2.19. I f u s e C2(R3 \ ) C(R3 \ ) is the outgoing scattered field, i.e., i f u s satisfies (2.82) and the generalized outgoing radiation condition, then

and

2.2.3

Reciprocity Relations

We study far-field patterns for scattered waves in this section. Define S1 = {P e R3| \P\ — l},D 1 = {(x,z) R 3 | | x | = l } . We note the following equivalent notation for our presentation:5

We refer to

as a distorted plane wave with direction a and call

the jth normal mode wave with direction (a, 5

).

Here we are following the tradition for the labelling of functions in physics, rather than mathematics, in that we are using the same letter to describe a physical quantity even though the arguments are different.


99

Using Lemma 2.17, functions and J = 1 , 2 , ..., N, satisfy and are legitimate incident waves. We will focus equation (2.81); i.e., on the boundary value problems with incident waves in the set U, where

and The scattered waves corresponding to are denoted by and respectively. From (2.80) and (2.86), we have the following two lemmas.

Lemma 2.20. The scattered wave corresponding to u1 (P; a) has the asymptotic expansion

and The jth component of where is denoted by (1) Ff(P, ) is thefree-wave far-field pattern corresponding to a generalized, incident plane wave with direction a, with the representation

is the jth normal mode of the guided-wave far-field pattern corre(2) sponding to an incident distorted plane wave with direction a, with the representation

Lemma 2.21. The scattered wave corresponding to sion

where by

and

The ith component of

has an asymptotic expan-

is denoted

100


(1) Ff(P a, ; j), the free-wave far-field pattern corresponding to the jth incident, normal mode with direction (a, ft), has the representation

(2) F' (x, z; a, ; j), the ith normal mode of the guided-wave far-field pattern correspending to the jth incident, normal mode with direction (a, ft), has the representation

There are reciprocity relations among far-field patterns F f ( P ; a ) , F'(x,z;a), i F f ( P ; a, ; j), and F g (x, z; a, ; j). We list the results here. The proofs are similar to that in a wave guide. For detailed proof, see [470], [473], [472]. Theorem 2.22. If us(P; a) is the scattered wave for either Problem D, Problem N, or Problem R, then its free-wave far-field pattern corresponding to the incident distorted plane wave u'(P;a) satisfies the reciprocity relation

for all P,a on the unit ball S1. Theorem 2.23. IfuSj(P;a, ) is the scattered wave for Problem D, Problem N, or Problem R, then its guided-wave far-field pattern corresponding to the incident normal mode u i j ( P ; a , ft) satisfies the reciprocity relation

for all (a, ft), (x, z) on the unit cylinder D\ and i, j = 1, 2 , . . . , N. Theorem 2.24. Let us (P; a) be the scattered wave for Problem D, Problem N, or Problem R corresponding to the incident distorted plane wave u'(P'; a), and let F , (x, z; a) be the jth mode of its guided-wave far-field pattern vector. Let us( P; a, ft) be the scattered wave for Problem D, Problem N, or Problem R corresponding to the incident jth normal mode u i j (P; a, ft) and let Ff(a; x, z; j) be its free-wave far-field pattern. Then there exists a reciprocity relation

for all a e S1, (x, z) € D1 and 0 < j < N.


2.2.4

101

Completeness of the Far-Field Patterns

We define the Hilbert space

with inner product

and norm ||f ||VN = (f,f) V N for f = ( f 1 , f2, ..., fN), g = (g 1 , g2, .. -, gN) VN A generalized, Herglotz, free-wave function has a representation of the form

where g e L 2 (S 1 ). A generalized, Herglotz, guided-wave function has a representation of the form

where h e VN.

A generalized, Herglotz, wave function has a representation or the form

where (g, h) L2( ) x VN. The functions (vector functions) g, h, and (g, g) are called the Herglotz kernel functions (vector functions) of Vf, V g S, and v, respectively. Clearly, Vf, vg, and v satisfy (2.81) in R3. The following results are useful in proving uniqueness of inverse scattering problems in a stratified medium. The proofs are similar to that in a wave guide. For details, see [470], [473], [472]. Lemma 2.25. Let vf, vg, and v be defined by (2.99), (2.100), and (2.101), respectively. (l)If v f ( P ) = O for all P R3, then g = 0 on S1. (2) If v g ( P ) = O for all P R3, then h = 0 on D1. (3) If v(P) = O for all P R3 then g = 0 on S1 and h = 0 on D1. We define the class

of free-wave far-field patterns by

102


Define the class Fg of guided-wave far-field pattern vectors by

Define the class F of far-field patterns as the direct sum of Ff and Fg

and Lemma 2.26. Let the scattered wave in Problem D corresponding to be denoted by and respectively. (1) For a given function g € L2 (S 1 ), the scattered wave solution to Problem D for the incident wave

is given by

which has a free-wave far-field pattern and a vector, guided-wave far-field pattern

and

(2) For a given vector function h the incident wave

Vn, the scattered wave solution to Problem D for

is given by

which has the free-wave far-field pattern and the vector, guided-wave far-field pattern


(3) For a given vector function (g, h) Problem Dfor the incident wave

103

L 2 (S 1 ) x VN the scattered wave solution to

is given by

which has the free-wave far-field pattern and the vector, guided-wave far-field pattern

and

Proof. (1) As discussed in [468], we can write

where

Then

in the form

satisfies

where

Multiplying (2.102) and (2.103) by g, integrating with respect to a over S1, and interchanging orders of integration, we obtain


104

where

and

satisfies

which implies the assertion in (1). We can prove (2) and (3) similarly. Clearly, we can prove parallel results for Problems N and R using the surface potential method as in the above proof. To avoid repetitive discussion, we will focus on Problem D in the following. Theorem 2.27. The set Ff is complete in L 2 (S 1 ) if and only if there does not exist a Dirichlet eigenfunction for that is a generalized Herglotz free-wave function. Proof. By the real analyticity of Ff, we need to prove that if there does not exist a Dirichlet eigenfunction for , then for g L 2 (S 1 ),

imply that g = 0 on S1. By Theorems 2.22 and 2.24, the above equations are equivalent to

This implies that

By Lemma 2.26 (1), the scattered wave us corresponding to incident wave


105

has vanishing far-field patterns given by (2.100) and (2.105), whence us is identical to zero By the Dirichlet boundary condition vf = vi = — us — 0 on in Hence Vf = 0 in R , for there does not exist any Dirichlet eigenfunction for From Lemma 2.25 (l) it follows that g = 0 on S1. Theorem 2.28. The set Fg is complete in VN if and only if there does not exist a Dirichlet eigenfunction for that is a generalized Herglotz guided-wave function. Proof. We want to prove that if there does not exist a Dirichlet eigenfunction for for h VN , we may conclude that h = 0 from

, then

By Theorem 2.23, equations (2.106) and (2.107) are equivalent to

By Lemma 2.26 (2), the scattered wave corresponding to incident wave

has vanished far-field patterns. Using the same argument as in the proof of Theorem 2.27, we know that vg = 0 in R3, hence h = 0 on D1 by Lemma 2.25 (2). Using similar proofs as in Theorems 2.27 and 2.28, we can show the following. Theorem 2.29. The set F is complete in L 2 (S 1 ) x VN if and only if there does not exist a Dirichlet eigenfunction for that is a generalized Herglotz function. Since the free-wave and guided-wave solutions are linearly independent, we have the following corollary. Corollary 2.30. If k is a simple Dirichlet eigenvalue in L2(S\) or Fg is complete in VN.

, then either Ff is complete in

Binary Trees, Left and Right Paths, WKB Expansions, and Painleve Transcendents* Charles Knessl*

Wojciech Szpankowski*

Abstract During the 10th Seminar on Analysis of Algorithms, MSRI, Berkeley, June 2004, Knuth posed the problem of analyzing the left and the right path length in a random binary trees. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right path lengths. In this paper, we mostly focus on the asymptotic properties of the distribution of the difference between the left and the right path lengths. Among other things, we show that the Laplace transform of the appropriately normalized moment generating function of the path difference satisfies the first Painleve transcendent. This is a nonlinear differential equation that has appeared in many modern applications, from nonlinear waves to random matrices. Surprisingly, we find out that the difference between path lengths is of the order n5/4 where n is the number of nodes in the binary tree. This was also recently observed by Marckert and Janson. We present precise asymptotics of the distribution's tails and moments. We shall also discuss the joint distribution of the left and right path lengths. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method.

1 Introduction Trees are the most important nonlinear structures that arise in computer science. Applications are in abundance; here we discuss binary unlabeled ordered trees (further called binary trees) and study their asymptotic properties when the number of nodes, n, becomes large. While various interesting questions concerning statistics of randomly generated binary trees were investigated since Euler and Cayley [8, 17, 18, 25, 27, 28], recently •The work was supported by NSF Grants CCR-0208709 and DMS 05-03745, NIH Grant R01 GM068959-01, and NSA Grant MDA 904-03-1-0036 ^Dept. Math., Stat. & Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045, U.S.A * Department of Computer Science, Purdue University, West Lafayette, IN 47907-2066 U.S.A.

novel applications have been surfacing. In 2003 Seroussi [22], when studying universal types for sequences and Lempel-Ziv'78 parsings, asked for the number of binary trees of given path length (sum of all paths from the root to all nodes). This was an open problem; partial solutions are reported in [15, 23]. During the 10th Seminar on Analysis of Algorithms, MSRI, Berkeley, June 2004, Knuth asked to analyze the joint distribution of the left and the right path lengths in random binary trees. This problem received a lot of attention in the community (cf. related papers [11, 20]) and leads to an interesting analysis, that encompasses several other problems studied recently [11, 15, 19, 20, 22, 23, 27]. Here, we mostly focus on the asymptotic properties of the distribution of the difference between the left and the right path lengths. However, we also obtain some results for the joint distribution of the left and the right path lengths in a random binary tree. In the standard model, that we adopt here, one selects uniformly a tree among all binary unlabeled ordered trees built on n nodes, Tn (where \Tn\ = (2^)^j- ^Catalan number). Many deep and interesting results concerning the behavior of binary trees in the standard model were uncovered. For example, Flajolet and Odlyzko [6] and Takacs [27] established the average and the limiting distribution for the height (longest path), while Louchard [19] and Takacs [26, 27, 28] derive the limiting distribution for the path length. As we indicate below, these limiting distributions are expressible in terms of the Airy's function (cf. [2, 3]). Recently, Seroussi [22, 23], and Knessl and Szpankowski [15] analyzed properties of random binary trees when selected uniformly from the set TI of all binary trees of given path length t. Among other results, they enumerated the number of trees in Tt and analyze the number of nodes in a randomly selected tree from Tt. We now summarize our main results and put them into a bigger perspective. Let Nn (p, q) be the number of binary trees built on n nodes with the right path length equal to p and the left path length equal to q. It is easy

198

Chapter 3

Inverse Scattering Problems in Ocean Environments

3.1

Inverse Scattering Problems in Homogeneous Oceans

The inverse scattering problem for acoustic waves consists of recovering the shape of a scatterer from the scattered field. Inverse problems have inspired a wide variety of techniques in the engineering sciences, such as remote sensing, nondestructive testing, and imaging etc., and for this reason have been the object of study by scientists in a number of diverse disciplines. Rapid progress in this field has been made since the early 1970s, and a survey of these results can be found in [126], [121 ], and the references cited there. However, most of the activity in this field has been directed to the cases of R2 and R3. It has been noticed that in some situations, for example, in a wave guide, remote sensing and imaging lead to more complicated problems. In a homogeneous, finite-depth ocean, Gilbert and Xu [211], [210] showed6 that the "propagating" far-field pattern can carry only the information from the N + 1 propagating modes, where N is the largest integer less than (2kh — )/2 and h is the depth of the ocean. This loss of information makes this problem different from inverse problems in R since the far-field pattern operator is no longer injective. A particular example of this occurs for 0 < k < /2h; then N — —\ and the far-field pattern is identically zero for any incoming waves. Even in the case of sufficiently large k, the vanished far-field pattern only implies that the N + 1 propagating modes are identically zero. Therefore, the far-field pattern operator F is not an injection on the Hilbert space L2( ). In Chapter 2, section 2.1.11 we established theoretical results for constructing an injective far-field operator. These results are essential for finding a generalized inverse for F. In section 3.1 we consider the following problem: Given the far-field pattern f(x, z, k) for one or several incoming (entire) waves ui, find the shape of the scattering object . Other inverse problems are considered in the following sections.

6

See also Chapter 2 of the present work.

107

108

3.1.1

Chapter 3. Inverse Scattering Problems in Ocean Environments

Inverse Problems and Their Approximate Solutions

Suppose ui is an incoming wave for which there exists a solution to (2.64), i.e.,

where T is defined by (2.63) and D is an auxiliary region contained in as define a far-field operator

. Then we can

For a given far-field pattern, this leads to an integral equation of the first kind, namely,

where From section 2.1.11, we know that F\ is an injection if k is not a Dirichlet eigenvalue of D and the domain of FI, D(F1), is UNHowever, in general, we cannot expect a solution to (3.3) to exist. One of the basic techniques for treating ill-posed integral equations of the first kind is the classical Tikhonov functional

After we have determined and its corresponding approximation usa for the scattered s wave u , we look for the unknown surface as the location of the zeros of usa + ui. As in the whole space case (cf. [260], [7]), one makes an a priori assumption about the unknown surface, namely, that if U is the set of all possible surfaces, the elements of U can be described by

Here B is the unit sphere, 0 < Zo < h is a known constant, and r(x) belongs to a compact subset

where C 1,B (B), 0 B 1, denotes the space of uniformly Holder continuously differentiable functions on the unit sphere with the usual norm. The functions r 1 (x) and r2(x) in the definition of V represent a priori information about the size of the object. If 3D is contained in the interior of the surface represented by r (x)x + (0, 0, zo), 7 we locate by minimizing

To simplify, we sometimes just say r(x).

3.1. Inverse Scattering Problems in Homogeneous Oceans

109

overall surfaces A in U; or, similar to [260], neglecting the Jacobian ofr(x), by minimizing

over all functions r V. Combining (3.4) and (3.5), we can formulate the inverse problem as an extremal problem: namely, minimize

here we use T to denote the single-layer acoustic potential

More precisely, we seek

and

such that

In the following, we establish existence of a solution to this nonlinear optimization problem and investigate its convergence properties as a —>• 0. Theorem 3.1. The optimization formulation of the inverse scattering problem has a solution. Proof. Let

be a minimizing sequence, i.e.,

Since V is compact, we may assume that In view of

we know that the sequence is bounded in the L2 norm. Hence, we may choose a subsequence that converges weakly to some as For simplicity also denote the convergent subsequence as Since F and T are compact operators, it follows that and

and

But then, from the above, we know that as


110

hence,

This, together with the weak convergence, implies that

and

since UN is a closed set. Hence,

This completes the proof. Theorem 3.2. Let that described by some

Proof. Let

and fo be the corresponding far-field pattern of a domain Then

be arbitrary. Then there exists

UN such that

due to Since the far-field pattern of the scattered wave depends continuously on the boundary data us, we can find a constant depending on such that

Since

on

we have


111

The theorem follows by letting Let ui be an incoming

be surfaces in Lemma 3.3. Let wave and {u 11 } and u* be scattered waves satisfying

and

then for any closed set G contained in the exterior of

*,

where A* is the boundary described by r* and

is the maximum norm over G.

Proof. As seen in Chapter 2, the exterior Dirichlet problem may be reformulated as a uniquely solvable integral equation of the second kind by representing the solution as a combined double- and single-layer potential

where An is the boundary corresponding to rn. Consequently, we reformulate our problem as seeking a solution to the integral equation

, where for the unknown density It is well known that K,, is a weakly singular kernel. Repeating the argument of Theorem 2.2 of Angell, Colton, and Kirsch [7] but replacing the Helmholtz fundamental solution by the wave guide Green's function

where GI is a continuous wave function, we know that the corresponding integral operators An : L2(B) L 2 (fi), defined by

satisfy the inequality

112

for


with some constant y. Consequently, we obtain the following error estimate:

where we have used the fact that (I + An)- l is uniformly bounded for n (see the proof of Theorem 2.2 in [7]). Substituting this into the combined double- and single-layer potential, it follows that for each closed set G contained in the exterior of A*,

This may be summarized as follows. and f Theorem 3.4. Let be an incoming wave such that be the corresponding far-field pattern of a domain and let be a solution to the minimization problem with regularization parameter a,,. Then there exists a convergent subsequence of the sequence {r n }. There is only a finite number of limit points, and every limit point represents a surface on which the total field us + ui vanishes.

Proof. From the compactness of V, there exists a convergent subsequence of {rn} which converges to, say, r*. For simplicity we also denote this subsequence as rn so that Let u* denote the unique solution to the direct scattering problem for the object as with boundary A* parameterized by r*. Then

The function un corresponds to the solution of an exterior Dirichlet problem with boundary values on a boundary , described by rn. From Lemma 3.3, we know the far-field patterns converge uniformly to the far-field pattern f* of u*, since


113

Moreover, by Theorem 3.1,

Therefore, we conclude that the far-field patterns coincide:

Recall that / is the far-field pattern with respect to an incoming wave u' e B(N, ) such that T0 = —ui admits a solution o € UN; therefore, we can represent the scattered wave as

Since / =

F1

it is implied from (3.15) that

as

It follows immediately that for any closed set G C R \ D,

Consequently,

due to (3.15) and (3.17), where G is any closed set in R3b \ D. In view of (3.17) and u* + ui = 0 on A* with A* c R D, we can conclude that

If there existed an infinite number of different limit points, then by the compactness of V we could find a cluster point of these limit points. Thus, it would follow that there was

114


an arbitrary small region for which us + u' is an eigenfunction for the Laplacian. This is a contraction to the generalized Faber-Krahn inequality

where is the principle eigenvalue of the Dirichlet problem for a 3D domain of volume V (see [178, p. 419]). Hence, the number of limit points is finite.

3.1.2

Inverse Scattering Using Generalized Herglotz Functions

Generalized Herglotz Functions In this section, we extend the method of Colton and Monk to seek an optimal solution of the inverse problem in the orthogonal complement of the closure of the set of far-field patterns [127], [128]. In this subsection we discuss the density and decomposition properties of far-field patterns for the Helmholtz equation in a finite depth ocean. As before, let u = ui + us, where ui is the incoming wave and us the scattered wave. for Then for any given incoming wave We assume that be the Green's function of the Helmholtz ui, u is determined uniquely. Let equation in R3b, which satisfies boundary conditions (2.2) and radiating condition (2.3). If the incoming wave u1 is

where

then the corresponding scattered wave

is given by Green's formula

The corresponding far-field pattern can be represented as


115

where The far-field pattern has the following property [214]. Theorem 3.5. For any

we have

Now we introduce a class of solutions to the Helmholtz equation defined in all of Definition 3.1. A solution v(x, z) of the Helmholtz equation in R satisfying boundary conditions (2.2), (2.3) and

where

is called a generalized Herglotz wave function.

Theorem 3.6. Any generalized Herglotz function may be represented in the form

for some g(x, z) € L 2 (C 1 ). Conversely, any solution in the form of (3.24) for some g L 2 (C 1 satisfies (2.2), (2.3), and (3.23). Remark. The function g(x, z)

L 2 (C 1 ) is called the generalized Herglotz kernel.

Proof. We first prove the direct part. Separation of variables in R leads to an expansion of the solution u(x, z) of Helmholtz equation

Since the Bessel function of the second kind of order m, Y m (r), is singular at r = 0, this implies dnm = 0, for n = 0, 1, 2 , . . . and m Hence, the solution takes on the form

The condition (3.23) now implies


116

Note that for any integer n > N, where N = [(2kh — imaginary number. Using the asymptotic expansion

we realize that since

, an defined by (2.6) is an

then cnm — 0 for any n > N and From this, (3.25) reduces to a propagating wave of the form

To show that u(x, z) may be written in the form (3.24), we rewrite (3.24) as

where r = |x| and expansion

are the cylindrical coordinates of (x, z). Using the Jacobi-Anger

we rewrite u(x, z) as

Now if g

is defined to be the function given by the series

then u(x, z)

v(x, z). Moreover, since

we may show that g

L 2 (C 1 ) if (3.28) is bounded. From (3.23) we have


117

Using the asymptotic expansion for the Bessel function of large argument, we have

for n — 0, 1, 2 , . . . and

This implies

To prove the converse part, we assume g

L 2 (C 1 ). From this we have

Consequently (3.23) is valid. The function v(x, z) was shown to satisfy the Helmholtz equation. Obviously it satisfies the boundary conditions. Define the function classes

and

where

and

has a limit point in

From Corollary 2.11, we have the following.

Theorem 3.7. If v defined by (3.24) is not an eigenfunction of the Dirichlet problem in then the set F is dense in VN. Now we consider the density of S in L2. If g

then

118


By analytic continuation of F(x, z; a, ) with respect to (a, a w1

), it follows that for every

where the cn are constants. Suppose now that not all the cn are equal to zero, and define

Then the far-field pattern of Us(x, z) is given by

Without loss of generality, we assume that cn — n (z 0 ), where (0, Z0) is in the interior of . It follows that for |x| > R, where R is a constant such that DR ,

where N contains no propagating modes. However, since g e S U s (x, z) can be expressed as

VN, for |x| > R,

Hence, we must have

The importance of the above result is that even though for a given far-field pattern we usually cannot determine a unique near field, we can find a function such as U s ( x , z) that is uniquely determined by the far-field pattern


119

Using (3.31), it follows that Us (x, z) is a solution to the Helmholtz equation in Due to the real analyticity of solutions to the Helmholtz equation, it follows that Us(x, z] can be uniquely continued up to the boundary A construction of U s ( x , z) for r R may be found in [214]. Similar to a theorem of Colton and Monk for the case of R2 [127], [128], we can prove the following theorem. (See [214] for details.) Theorem 3.8. Assume that k2 is not an eigenvalue of the interior Dirichlet problem and let v be the solution of the Dirichlet problem

such that

where U s ( x , z) is given by (3.31) and (r, , z) are the polar coordinates related to (x, z). Then (1) ifv is an entire Herglotz wave function with Herglotz kernel then (2) if v is not an entire Herglotz wave function, then The Inverse Scattering Problem

Based on the preceding analysis, we will reformulate the inverse scattering problem as a problem in constrained optimization. A similar formulation has been carried out by Colton and Monk for objects in R2 [127], [128]. We assume that is such that can be parameterized in the form where for and a, b are some positive constants. Let be the spherical coordinates with respect to (x, z). We define the sets U1 and U2 by

where M, C1, C2 are positive constants and denotes a Sobolev space with norm As in Colton and Monk [ 128], we know that U1 (M) and U2 are compact in C and , respectively. Hence, U(M) = U1(M) x U2 is compact in by Tikhonov's theorem. is the measured far-field pattern corresponding to the incident "plane" wave

120


then we define the optimization problem as

where

v is defined by (3.24). is said to be admissible if and only if Definition 3.2. A function the pair (g, p) U(M) minimizes (3.38) over U(M). It is clear that, from the compactness of U(M) and continuity of the integral in (3.38) with respect to p and g, there exists at least one admissible solution. In examining the relationship between admissible solutions and actual solutions of the inverse scattering problem, we can prove the following theorems in a similar way as the corresponding theorem in [127], [128]. Theorem 3.9. Let (F) be the set of admissible solutions corresponding to the far-field (F j ), then there exists a convergent subsequence of pattern F. If Fj F in VN, pj {Pj} whose every limit point lies in Proof. Since U2 is compact, without loss of generality we can assume that { P j } converges to p* U2 Let ( g j , PJ) be the corresponding pair such that the sequence (gj, PJ) (g*, p*) U(M). We need to show that

where v* is the generalized Herglotz wave function associated with g*, r* = p* sin z* — p* cos Now if p (F) has the corresponding pair (g, p), then

and


121

This completes the proof. Theorem 3.10. Assume that k2 is not the eigenvalue of the interior Dirichlet problem for , and is a bounded domain with C2 boundary : p = p( ) such that p U2Assume the solution of (3.36), (3.37) is a generalized entire Herglotz wave function with Herglotz kernel g If F is the far-field pattern corresponding to and the incident wave

then there exists a constant M0 < such that J(F, M, J, L) = 0 for each M Mo and integers J, L. For each J, L, let {p }, j = 1 , 2 , . . . , n j . l ., be the admissible functions. Then there exists a convergent subsequence of {p }, /, L = 0, 1, 2 , . . . , . The number of limit points of {pJj'L}, j — 1, 2 , . . . , nj,/,, J, L — 0, 1, 2 , . . . , , is finite. Proof. Since k2 is not an eigenvalue for the interior Dirichlet problem, the problem (3.36), (3.37) has a unique solution. Hence, g U1(M) is uniquely determined for M M0,

122


where Mo is a positive constant. From (3.30) and (3.37) we see that J (F, M, J, L) = 0 for each M M0 and every integer 7 and L. Now let {pJj'L} be as defined in the statement of the theorem. Then since U2 is compact, the sequence {p ' } has a subsequence converging to p* U2. Let U1(M) be a function associated with . Then { } has a subsequence converging to a limit point g* U 1 ( M ) . But it follows from the fact that J(F, M, J, L) = 0 for M M0 and for each J, L,

for i = 1,2, . . . , J , l = 1,2, . . . , L . Hence,

for i = 1, 2, ...,l = 1,2, In view of Theorem 3.7, F is dense in VN. Therefore, we now can conclude that g* = g. Now we prove that there is only a finite number of limit points p* lying in U2. Let {p-}. {gi} be the convergent subsequences defined above, then, since J(F, M, J, L) = 0 for M MO, we have that

and hence by passing to the limit

If there existed an infinite number of limit points p* in U2, then from the compactness of 1/2 the set of limit points p* would have an accumulation point. Hence, we could find a domain D* with an arbitrarily small area such that

would be an eigenfunction of D* with corresponding eigenvalue k2. But this is impossible due to the Faber-Krahn inequality (see [178, p. 419]). Hence, there is only a finite number of limit points p*. Numerical examples using this method may be found in [214].

3.2. The Generalized Dual Space Indicator Method

3.2 3.2.1

123

The Generalized Dual Space Indicator Method Acoustic Wave in a Wave Guide with an Obstacle

In this section we consider the acoustic imaging problem in a homogeneous, shallow-water wave guide. The shallow-water wave guide is denoted by , where h is the depth of the ocean. Let be a bounded obstacle imbedded in the wave guide. The total acoustic field u from a point source satisfies

u also satisfies the outgoing radiation condition

where is the location of the acoustic source, k > 0 is the wavenumber, and un is the nth normal propagation mode; i.e., if

then u has representation

On the boundary of , denoted by , u satisfies some unknown boundary condition of the Dirichlet, Neumann, and impedence types. We denote the boundary conditions as

We consider the total wave u as the combination of a prime field ui and a scattered field us, u — u' + us, where ui is the Green's function for the parallel wave guide without any obstacle:

124


where

From (3.44)-(3.47), we have

On

satisfies some unspecified boundary condition

where (3.55) may be data of the following types, namely Dirichlet:

Neumann:

Robin (impedence):

We assume that is regular enough to admit a solution and that k2 is not where an eigenvalue of the Dirichlet, Neumann, or Robin problem. Therefore, for each given xs, u i ( x ) = G(x, xs) is known and us(x) is determined uniquely for any given boundary condition B. Note. The uniqueness of the direct scattering problem in a shallow-water wave guide with an obstacle is still an open problem in general. For some recent discussion, see [475].


125

The Scattered Field on a Straight Line

Let

and

We assume that both and s are "above" the obstacle i.e., max (Note that here we denote the ocean surface by X2 = 0 and the ocean bottom by x2 = h > 0.) We consider the following problem: Given u(x, xs) for x and xs 5, construct the unknown obstacle without knowing which of the above three boundary conditions us satisfies on Here we may choose . Let D be a region containing For any y = (y1, yz) D, we consider the integral equation

where G(x, y) is the Green's function for the parallel wave guide without any obstacle. We have the following theorem [478]. Theorem 3.11. (l)For y (3.61) does not have a solution. (2) If (3.61) has a solution g(xs ; y) when xs and a given y , then the solution is unique, providing there does not exist an eigenfunctionfor with homogeneous boundary condition corresponding to any of the conditions (3.56), (3.57), and (3.58). (3) If (3.61) has a solution g for given y , then

From the above theorem, if we can solve (3.61) exactly for , then the norm blows up for , whereas the norm is finite when and (2.13) has a solution. Therefore, the shape of the obstacle may be revealed by plotting the norm as a function of y D. Unfortunately, (3.61) usually cannot be solved exactly. For the inverse scattering problem in a homogeneous space, a method was developed by Colton and Kirsch [124] to look for a solution that has the property

where is a small positive constant and u is the corresponding far-field pattern. Their method is based on the fact that the norm of the regularized solution ||g(; y)|| is unbounded near the unknown boundary. In [323] Norris considered a similar problem using the eigenvalue expansion of the far-field operator. His method is based on the observation that the

126


Figure 3.1. Circle: Local extrema of the norm.

Figure 3.2. Heart: Local extrema of the norm.

norm of the solution (in a series form) is divergent in the exterior of the unknown obstacle and convergent in the interior of the obstacle. However, in our underwater imaging problem, the scattered field is measured only at a finite number of points along a straight line. Our numerical experiments show that in many cases the norm changes gradually across the unknown boundary. Even when the measurements are taken at as many as 200 points, the norm of the solution is still not obviously larger when the source point is near the boundary. Without knowing the obstacle in advance, it is difficult to determine the boundary of the obstacle. Using numerical experiments, we notice that the norm of the solution of the integral equation has local extrema inside or near the boundary of the obstacle. This is particularly noticeable if the obstacle is convex with smooth boundary. See Figure 3.1 for examples. Figures 3.1 and 3.2 are the contours of the norm of the solution of the integral equation. The original objects are plotted for comparison. The setting of these two examples is the same as that in Example 3.1, except for the obstacles. Based on this observation we use the following inversion procedure. (1) Measure the scattered field at N + 1 points along a line (denoted by ) for each sound source on the same line. The measured data is saved in an (N + 1) x (N + 1) matrix. (2) Choose an exploration region that may contain the unknown obstacle (denoted by D). Compute the Green's function G(X, Y) for N + 1 points X and each Y D approximately (i.e., truncating at a suitable term). (3) Solve the regularized linear system for g(X; Y) for each Y D. Different regularization methods may be applied. (4) Compute the norm (L2-norm or other norms) for each Y D. Draw the contour of \\g(; Y)\\ as a function of Y on D. (5) Study the contour and set a filter to keep all (or most) of the local extrema in the picture. In this way we obtain a good image of the unknown obstacle.


Figure 3.3. No filtering.

127

Figure 3.4. Filtering at level 1.0.

Imaging Scheme and Numerical Examples We present some numerical examples here. The input data are obtained by solving the direct scattering problem using an approximate boundary integral equation method (see section 2.1.12). As in section 2, we plot the norm of the regularized solution of the integral equation as a function of the source point in the exploration region and choose a filter to keep all (or most) of the local extrema in the picture. For simplicity, let and v be the same line. The data u(xn; xs) are given at points {xn : n = 0, . . . , N} along a straight line for xs . In our numerical experiment, we choose xx {xn : n = 0, . . . , N}. Therefore, we have an N + 1 by N + 1 array of data. We approximate the integral by a trapezoidal rule. Let gj — g ( x j ) , fn = G(xn ; y), and S — ( s n j ) ( N + i ) x i N + i ) for n = 0 , . . . , N and j = 0 , . . . , N, where y D, the exploration area,and snj = hu(xn; Xj), whereh = (xN—xo)/N for O < n < Nandh = (xN—xo)/(2N) for n = 0 and n = N. Let / = ( f 0 , . . . , fN)T and g = (g0,..., gN)T. The integral equation (3.61) is approximated by

Note that / is a vector function of D. Hence g is also a vector function of y. The matrix S is an ill-conditioned matrix. In view of Theorem 3.11 (2), we use Tikhonov regularization; i.e., we solve instead the following regularized system

where S* denotes the conjugate of S. After solving g we compute its L 2 -norm for each y D and plot its level curves in D. Figures 3.3 and 3.4 give some numerical examples. Example 3.1. We use the following parameters for the shallow water wave guide. The wavenumber is taken to be k = 2.3, the depth of water is h = 10, and the obstacle lies

128




within the curve x = r cos( ), y = 6 + r sin( ), where

The Green's function G(x, y) is truncated at N = 20. The data are given on = {(—30 + 0.5n, 2) : n = 0, . . . , 120}. The exploration area is the square D = (-3, 3) x (3, 9). We use the regularization parameter = 10 -12 . We measure the scattered field along a line. The contour of (without filtering) is plotted in Figures 3.5 and 3.6, where the postprocessing filtering levels are set as indicated. The true boundary of the obstacle is plotted over the contour for comparison. From the contour we can see that all local extrema are located inside or near the boundary of the obstacle. Therefore, even though we do not know the shape of the object, we can set a filter such that all (or almost all) local extrema remain in the picture. In this way we can obtain a reasonably good image of the obstacle. In conclusion, the generalized dual space indicator method for the acoustic imaging of an obstacle in ocean environments is based on the observation that the combination (weighted integration) of the measured scattered field can approximate Green's function very well when the Green's function's source point is inside the obstacle, but not so well when the source is outside the obstacle. We set up an integral equation whose right-hand side is the Green's function with source point from an exploration region. From our numerical experiments, we notice that the norm of the solution of the integral equation has local extrema inside the obstacle. Plotting the norm as a function of the source point in the exploration region, and filtering out the region with no local extrema of the norm, we obtain a good image of the unknown obstacle. The advantage of this method is that we need not know the boundary conditions. As a tradeoff, this method does not reconstruct the exact shape of the obstacle. It is an open problem to show that (under some conditions) the local extrema of the norm of the regularized solution of the integral equation as a function of a source point location are in or near the obstacle.

3.3. Determination of an Inhomogeneity in a Two-Layered Wave Guide

3.3

129

Determination of an Inhomogeneity in a Two-Layered Wave Guide

Consider a 2D acoustic model of a two-layered wave guide:

Here d and h are constants, and h > d > 0. We assume that the inhomogeneity is contained in a bounded domain boundary having an outward pointing normal vector. The propagating solution

satisfies

with C2

130


and u satisfies the outgoing radiation condition (3.54). Assume K3 C the functions

and introduce

so that

Let G be the Green's function for the two-layered wave guide with an acoustic source at Assume that and are the limits of and respectively, as approaches from the interior. Similarly, we use approaches and to denote the limits of and as ! from the exterior. We have

where

satisfies (see [189] and [194])

We have the following theorem. (See [195] for the proof.) Theorem 3.12. If u satisfies the direct scattering problem (3.66)-(3.75), then (u, the integral equations (3.79) and (3.81), where is defined by (3.80).

) satisfies

3.3. Determination of an Inhomogeneity in a Two-Layered Wave Guide

131

is a solution of the integral equations (3.79) Conversely, if and (3.81), then u is a solution of the direct scattering problem.

and are small enough, then the system of integral Theorem 3.13. // equations (3.79) and (3.81) has a unique solution. In the special instance where p1 = p3, (3.81) becomes

and the system of integral equations reduces to a single integral equation

We can use an iterative algorithm for the integral equation (3.83) under the assumptions that the wave field can be determined in R by solving u(x, z) '.= u(x, z, xs, zs) in that

We construct the algorithm as follows: Start with

and then for n = 1, 2, 3, . . . , let

where This algorithm may be seen to converge if sufficiently small. The integral operator

has a singular kernel of the form

Now let us split

into two integrals:

is

132

For small

Chapter 3. Inverse Scattering Problems in Ocean Environments we estimate

Numerical examples of the above algorithm may be found in [195] and [194]. Now we consider the following inverse problem: Let be a subset of '. constant}, and be a subset of fixed}. Given for determine the inhomogeneity k 3 ( x , z). lie strictly above the inhomogeneity As before, assume that i.e., and max min In the 3D wave guide case, we can prove that the inverse scattering problem for given data on two planes has a unique solution. However, uniqueness is still an open problem in the 2D case, even if we take d = I and s = 2. Therefore, this section is merely a numerical investigation concerning the inverse problem. We now reformulate the inverse problem as an overdetermined linear system and use a nonlinear optimization scheme to solve the regularized nonlinear least squares problem. For simplicity we consider only the case where p1 — p3 moreover, it is assumed that k2 C(M1). Using (3.84) we can represent the acoustic field detected on with sources on as

where

satisfies (3.79) with

i.e.,

where

and

For given measured data u* = u* (x, z xs, zs), where (x, z', xs, zs) x , we reformulate finding the inhomogeneity as a minimization problem: namely, seek k2 — k (x, z),

133

3.4. The Seamount Problem

such that the functional the functional is defined by

Since max

i is minimized for some suitably chosen . Here

is small, we may approximate

which suggests solving the inverse problem by minimizing

The difference between the minimization of (3.93) and the minimization of (3.94) is that no integral equation needs to be solved in (3.94).

3.3.1

Numerical Example

We illustrate the method with a numerical example. The distributed inhomogeneity is contained in a rectangle {(x, z)|50 < x < 70, 75 < z < 90}. The measured data are from = {(-140 + 0.5m, 60)|m = 0, 1, 2 , . . . , 800}. The inhomogeneity is

The reconstruction is shown in Figure 3.7.

3.4 3.4.1

The Seamount Problem Formulation

In this section we continue to restrict our attention to constant depth oceans with completely reflecting seabottoms. However, in the present example we consider the case where there is a seamount on the ocean floor. We wish to reconstruct the seamount using far-field data. To this end we generate an acoustic field using a point source at a given location, say The acoustic pressure then satisfies

and the outgoing radiation condition. Here we assume that k2 is not an eigenvalue of the exterior boundary problem.

and

134


Figure 3.7. Determine inhomogeneity.

D represents the seamount, and M is the surface of the seamount, which has a parameterization

here a is some positive constant where we assume that the seabottom is flat for r > a. For a constant depth ocean without a seamount, the solution to (3.95)-(3.98) is the Helmholtz-Green function in , which has the form

where the g(z) are the point sources

The solution of problem (3.95)-(3.98) can be represented as

for

here

is the unique solution of the integral equation

135


and

The inverse problem is the following: Given constant} and seamount M.8

3.4.2

for all constant}, determine the

Uniqueness of the Seamount Problem

We assume that both (the receiving plane) and (the source location plane) are above the seamount. Our proof follows the uniqueness arguments for R3; in particular, see [261 ]. Theorem 3.14. Assume that D1 and D2 are two seamounts with rigid boundary M1 and M.2 such that the corresponding solutions of problem (3.95)-(3.98) coincide on for all where is the unbounded component of Then Dl = D2. Proof. Suppose that such that

Then without loss of generality we can assume there exists and We choose such that

is contained in where v is the unit normal vector. Consider the solution un,j to the problem (3.95)-(3.98) with Xo replaced by xn corresponding to the seamount Dj (j = 1,2). By assumption,

In view of the fact that un.l= un,2— 0 at z — h and that u,,j(j = 1,2) are outgoing, we know that un := Un,1 — un,2 0 in the region between and the surface z = h. Owing to the real analyticity of solutions of the Helmholtz equation, it follows that un 0 in . Hence, Un,1I = Un,2 in . Consider un = un,2 as the wave corresponding to seamount D2. We know that the

are uniformly bounded with respect to the maximum norm on M2 It follows from the continuous dependence on the boundary values for the exterior Neumann problem that the are uniformly bounded with respect to the maximum norm on closed subsets of In particular, we have the estimate

8

Here we assume that

and

lie strictly above the seamount, i.e.,

136


for all n and some positive constant C. On the other hand, consider un = un,1 as the wave corresponding to the seamount DI . From the boundary condition on M1,

where has continuous derivatives at which therefore implies that D1 = D2.

[197]. This contradicts (3.107),

In order to eliminate the requirement that incident waves must arrive from all directions, we need the following lemmas. Lemma 3.15. Let D be a bounded domain with C2 boundary and, moreover, let connected. D is located strictly below max Let G the Green's function with source at

Then H. is complete in L2( D). Proof. Assume

for all

satisfies

. Then the combined single- and double-layer potential

satisfies the Helmholtz equation in and

This implies that u 0 in Because of the polar singularity of

the outgoing radiation condition as

be be


where equation

is continuous at

137

by letting

we obtain the boundary integral

Here

The operator I + K — iS is invertible for and its inverse is a bounded linear operator in on 3D and the completeness of h is proved.

(see [197]) Hence, we have from (3.113)

Lemma 3.16. Let D be a bounded domain with C2 boundary connected and D is located strictly below Let Helmholtz equation. Then there exists a sequence vn in

such that is be a solution of the

such that

uniformly on compact subsets of D. Proof. By Lemma 2.1, there exists a sequence {vn} in V such that

Notice that condition

satisfies the Helmholtz equation in D and the impedance boundary

We represent wn as a sequence of single-layer potentials

where the I are density functions. Letting the boundary integral equations

approach a point on D, we obtain

138


where

and

For the integral operator (I + K' — iS) is invertible and has a bounded inverse (I + K' — iS) -1 ,

Hence, for

D, we have, applying the Schwarz inequality to (3.119),

Now L2 convergence of (3.125) follows from the uniform convergence of to u and

and

Theorem 3.17. Assume that D1 and D2 are two seamounts with rigid boundaries M1 and M2. such that the corresponding solutions of (3.95)-{3.98) coincide on for all. Then D1 = D2. Proof. We need to prove that, under the assumption of the theorem, the solutions corthe unbounded component of responding to DI and D2 coincide on for all First, for any we consider two exterior Neumann problems for outgoing solutions of the Helmholtz equation

with boundary conditions

We want to show that is connected, in span

in . We choose a bounded C2 domain D such that and Then by Lemma 2.2, there exists a sequence such that


139

uniformly on D1 U D2. In view of the fact that the v,, are linear combinations of point source waves from sources on PI, from the hypothesis it follows that the solutions v 1 and v 2 corresponding to the seamounts D1 and D2 coincide in PI. Using the same argument as that in the proof of Theorem 3.14 it follows that

Moreover,

As a consequence of the continuous dependence of the solution to the exterior Neumann problem on the boundary condition, along with the boundary condition (3.126) and the convergence (3.128), it follows that

uniformly on compact subsets of for j — 1, 2. Therefore, it must hold that p = p in . By Theorem 3.14, we conclude that D1 = Do.

3.4.3

A Linearized Algorithm for the Reconstruction of a Seamount

Let us consider the following linearized algorithm to find the shape of the seamount. Let

where fo(r, 0) is the initial guess for f(r, 9),

Substituting (3.132) into (3.95)-(3.98) and neglecting terms of O(82) and higher, we have

and

140


We can now use single-layer potentials to obtain a relation between Pn and fn in (3.140). Let us represent 8pn as

Then

(y) satisfies

and

Now we can establish an iterative algorithm for solving the inverse problem as follows: 1. 2. 3. 4.

Initial guess f0(r, 0); for n = 0, Solve for p n (x) from (3.137)-(3.140). Let 8pn(x) = p ( x ) - p,,(x), Solve (y) = n(y) for y E Mn from (3.143). For chosen En > 0, set

5. Let fn+1 =

f n +&/„.

Repeat steps 2-5 for (n = 1, 2 , . . . ) solving for pn, Spn, „, 8fn, respectively, until \ fn \ < e for some chosen €. Step 3 in the above algorithm solves an ill-posed integral equation, inherited from the original ill-posedness of the inverse problem. A proper regularization method must be adapted in order to solve (3.143). With this in mind we first discuss some properties of the integral operators T and TN defined by


141

We will need the following spaces that are weighted in x' = ( x 1 , x2) E R2'-

where we use the multi-index notation a — ( 1, 2), \ \ — \ 1\ + \ 2\, and L 2 denotes the space of square-integrable functions, and L 2 ( M ) , H 1 ( M ) , L 2 (M n ), and H l ( M n ) denote the usual Hilbert spaces and Sobolev spaces on the surfaces M and .M,,, respectively. Owing to the form of the normal mode expansion of the Helmholtz-Green function

where x — (r, 0, z), y = (r', 0', z') and an is purely imaginary except for a finite number of ns, E0 = 1, and €m = 2 for m 0. We know that G(x, y) is real analytic in x for any y E M. Moreover, we have for some constant C the estimates

holding uniformly for y E M as |x'| theorem. (See [216] for proofs.)

. From these facts we have the following

Theorem 3.18. (1) The operator T is compact from L2(M) into H l ' - s ( T 1 ) f o r s > 1/2. (2) The operatorTn is compact from L2(Mn} into H 1 - s ( r { ) f o r s > 1/2. Theorem 3.19. The operator T is injective and has dense range provided that the mixed boundary valued problem

has no nontrivial solution. Based on Theorems 3.18 and 3.19, we may apply the Tikhonov regularization to step 3; that is, we solve

142


with some regularization parameter a > 0 instead of (3.143). From the regularity of discrepancy principle for the Tikhonov regularization (see, for example, [123, Thm. 4.16, p. 99]), we have the following theorem. Theorem 3.20. If pn E T(L2(M)), then

approaches

3.5

Inverse Scattering for an Obstacle in a Stratified Medium

We consider an inverse acoustic scattering problem for an unknown object in a stratified medium. Suppose a compact obstacle, coinciding with the region , is imbedded in a stratified medium. We refer to the medium as being acoustically stratified when the refraction index varies with depth, which is frequently the case for oceans. In many cases this stratification occurs because warmer water is lighter and tends to move to the surface as heavier water sinks due to gravity. Other factors such as salinity also affect the refraction properties of the fluid column. In a stratified medium, sound waves may be trapped by acoustic ducts and caused to propagate horizontally. Therefore, the scattered energy flux does not spread spherically. Instead, there is a free-wave far field and guided-wave far field. Due to the nature of stratified media, some results that are valid for inverse scattering in a homogeneous medium may not hold for a stratified medium. For example, in R3 the shape of a scatterer may be determined uniquely from its far-field patterns. This may not be true for a stratified medium, unless the far field may be detected in a window large enough to contain both free waves and guided waves. In other words, two open sets are needed; one for the free-wave far field and one the for guided-wave far field.

3.5.1

Formulation of the Inverse Problem

To formulate the inverse problem, we use the same notations used in section 2.2, where the direct scattering for an obstacle in a stratified medium is discussed. We assume that the refraction index n(x, z) = n0(z) for (x, z) E R3. Moreover, we assume that n0 E C°(R), and for some constants h 1 ,h 2 and positive constants n_, n+

The scattering of acoustic, time-harmonic waves by a sound-soft obstacle in a stratified medium leads to the following exterior boundary value problem: Given ui E C 2 (R 3 ) satisfying

3.5. Inverse Scattering for an Obstacle in a Stratified Medium with positive wavenumber k, find the scattered field us E C 2 (R 3 \

143 ) satisfying

such that the total field u — u' + us satisfies boundary condition

Here is a bounded domain with C2 boundary . To ensure the uniqueness of the exterior problem, we require that the unknown scattered wave us satisfy the generalized Sommerfeld radiation condition. As discussed in [469], [473J, we are particularly interested in the scattering of the incident wave u' from the set

ui (f; a) is the distorted plane wave with direction a — (sin y cos 8, sin y sin 8, cos y) and given by

is the jth normal mode wave with direction

The inverse problem we consider in this section can be generally stated as follows: given the combination of the far-field patterns F f (-; a) 0 Fg(-', a) and F/(-; , ft; j) 0 Fg(-; , B; j) of the scattered waves u s ( - ; a) and u s .(•; a, B) for several incident waves u i '(-; a) and u i •(•; a, B) from with different incident directions a, ( , B) and different modes j E IN, determine the shape of the scatterer . We discuss the uniqueness of the inverse obstacle scattering in the next section and then consider the shape reconstruction of the unknown obstacle. An example is given to show a special feature different from that in inverse scattering in a homogeneous medium.

144

3.5.2


Uniqueness

We want to investigate the conditions under which an obstacle is uniquely determined by knowledge of the far-field patterns for incident waves. Our discussion here is inspired by that for obstacle scattering in a homogeneous medium. The uniqueness of inverse obstacle scattering in a homogeneous host medium may be seen in, for example, [280], [261 ], [126], and [350]. The surface of the obstacle, 3£!, may consist of several connected closed components. The given data for the inverse problem are the values of the incident distorted plane waves, the incident mode waves, and the corresponding far-field patterns u'(-; ) , u ' j ( - ; a , ft), F f ( P ; ) 0 Fg(x, z; a), and F f ( P ; , B} Fs(x, z; , B), where P, (x, z), a and ( , B) are from a subset in S1 x D1 x S1 x D1. This subset may be chosen in a number of ways. Case 1: Complete data of far-field patterns. Assume Ff(P; ) Fg,(x, z; ) and F f ( P ; , B- j) F g ( x , z; , ; j) are known for all P E S 1 , (x, z) E D 1 , j E I N , and a E S1 C S1, ( , B) 6 D1 C D1, where S1, D1 are subsets of S1 and D1. Theorem 3.21. Assume that and 2 are two sound-soft scatterers. If the far-field patterns F f ( P ; ) F g ( x , z; ) and F f ( P ; , B; j) F g (x, z; , ft; j) coincide for P E S 1 and (x, z) E D1 for incident waves ui (•; a) and u (•; B), a 6 S1, ( , B; j) E D1 x IN, then i = 2- Here S1 is a sequence of vectors , E S1 with a limit point 0 E S1; D1 = C1X R and C1 C C1 = {x E R2| |x| = 1} make up a sequence of vectors E C1 with a Proof. Let V and vs2 be the scattered waves with scatterers the function vs = v - v2 in BR = {P E R3\ \P\ < R} have coincident far-field patterns,

, U

respectively. Consider . Since v (7 = 1,2)

This implies that vs — 0 in R3 \ BR [469]. By the unique continuation property of elliptic equations (cf. [246]), vs = 0 in the unbounded component D of R3 \ ( U )- In particular, vs=Oon3D. Assume first that = (R3 \ D) \ 2 is not empty. Then v is defined in * and v2 = v + u' = 0 on * since v = —u1 on 2 and v = v — —u' on n D. Thus V2 solves the problem

That is,

with

2

is a solution of the eigenvalue problem

limit

p

3.5. Inverse Scattering for an Obstacle in a Stratified Medium

145

and

From the discussion in [446], we know that for such an eigenvalue problem, any eigenvalue k2 < has finite multiplicity. However, for distinct incident waves from U, the corresponding scattered waves are linearly independent in L2( *). In fact, if vj are scattered waves corresponding to u' (•; a) or ul •(•; a, B) from U, and

where cy are constants and m is an arbitrary integer, then from the unique continuation property of elliptic equations it follows that the above is valid in D. Rename Vj if necessary; we may assume that the first m1 of the {vj are scattered waves corresponding to incident free waves, and the others correspond to the incident guided waves. Thus,

where lj; E /#. The orthogonality of the eigenfunctions of ordinary differential equation (1.5) follows:

and

It implies that cj• — 0 for j — 1, . . . , m. This is a contradiction. If ft* = (R3 \ D) \ = 0, then ft* = (R3 \ D) \ . Using the same argument above in ft* for v1 — v + u', we conclude with a contradiction. Hence, there must be = .

Corollary 3.22. The shape of a scatterer is determined uniquely by F j ( P ; for all P E S1, (x, z) E DI, and a E S1.

) 0 F g (x, z; )

Corollary 3.23. The shape of a scatterer is determined uniquely by Ff(P; F g ( x , z; , B; j) for all P E S1, (x, z) E D1 and ( , B; j) E D1 x IN.

,B; j)

146


Case 2: Incomplete data of far-field patterns. 2(a). Assume that only the free-wave far-field patterns Ff(P; ), Ff(P; , ; j) are known for all P E S\ and a e 1 C S1, ( , ; j) E DI x IN c D1 x IN, where and 1 are defined as in Theorem 2.4. Theorem 3.24. Assume that and 2 are two sound-soft scatterers such that the free-wave far-field patterns coincide for incident free waves ui (•; a) for a E S1 and incident guided waves u'j(-; a, ft) for (a, ft; j) E D\ x IN. Then — Proof. By the reciprocity relation (cf. [469, Thm. 3.3]) we have

The condition in the theorem is equivalent to the free-wave far-field patterns Ff( ; ) coinciding on S1 for incident free waves u (•; a), a E 1, and the guided-wave far-field pattern vectors Fg( , B; P) coinciding on D1 = {( B) E D1 |( B) E D1} for incident free waves u i (•; P) for all P e S1. The real analyticity of with respect to 6 where a = ( ) follows that the guided-wave far-field patterns coincide on D\ for incident free wave for all , in particular for . Hence, by Corollary 3.23, Remark. Theorem 3.24 suggests that if, in a sound channel, the scattered data can only be obtained from readings above the scatterer, the shape of the scatterer may still be determined from sources in and above the sound channel. However, the fact that free-wave far-field patterns coincide for infinitely many distinct incident waves does not imply that two scatterers are identical. 2(b). Assume that only the guided-wave far-field pattern vectors Fg(x, z; a) and Fg(x, z; a, B; j) are known for all (x, z) e D1 and a e S1, (a, B; j) e D1 x IN. Theorem 3.25. Assume that and are two sound-soft scatterers such that the guidedwave far-field pattern vectors coincide for incident waves ui ( ) for a e S1 and -( a, B) for (a, B; j) e D1 x IN. Then . Proof. In view of the reciprocity relation (cf. [469, Thm. 3.3])

and the real analyticity of F f (a, x, z, j) with respect to and

for

, where

the condition in the theorem is equivalent to the far-fied patterns F f ( P ; a, B; j) F g ( x , z; a, B; j) coinciding for the incident guided wave .( a, b) with (a, b; j) e D1 x 1N. The theorem is followed by Corollary 3.23. Remark. One can obtain scattered information at a further location in a sound channel, where the scattered energy decays more slowly. Theorem 3.25 suggests that if one can


147

generate incident waves from sources, both in and out the sound channel, one can determine the scatterer using only guided waves. However, the fact that guided-wave far-field patterns coincide for infinitely many distinct incident waves does not imply that two scatterers are identical. 3.5.3

An Example of Nonuniqueness

In this section we construct an obstacle, where there exist infinitely many distinct, incidentwaves for which the corresponding scattered waves have vanishing free-wave far-field patterns. Let n 2 ( z ) be the refraction index such that there exist at least three guided modes. That is, equation (2.74) has at least three eigenvalues. The third eigenvalue and the third eigenfunction are denoted by a3 and (z), respectively. (z) has two zeros, denoted by Z1 and Z2 Let be the cylinder and Si the surface of the unit ball. Consider a sequence of incident waves defined by

where

for m > 0, and

and g n ,(P'), n = 1, 2, 3 , . . . , is a sequence of linearly independent functions on S1. The (P) are solutions of (3.153) and are linearly independent. The scattered waves corresponding to (P) satisfying (3.154), (3.155), and the radiation condition are

It is obvious that satisfy (3.154) and the radiation condition. To see that recall that for r > r',

satisfy (3.155),

148


Using (3.166), (P) for n = 1, 2, 3, ... have vanishing free-wave far-field patterns. This example shows a new feature of inverse scattering problems in a stratified medium; that is, a scatterer may not be determined by free-wave far-field patterns corresponding to infinitely many distinct incident waves.

3.5.4

The Far-Field Approximation Method

In this section we discuss an approximation method for the inverse obstacle scattering problem. Our results are inspired by the study of inverse obstacle scattering in a homogeneous medium, in particular the discussions in [7], [261], and [126]. We will restrict our discussion to the case of starlike domain . That is, we assume that is represented in the parametric form

with a positive function We define a Hilbert space

.

with inner product

and norm Let L2 (S 1 ) x VN be the product space of L2 (S 1 ) and VN , and let be a bounded domain with a C2 boundary having an outward normal vector. A far-field operator depending on is defined by

and

where and Ff-( a, b; j) F g ( a, B; j) are far-field patterns corresponding to incident waves u' ( or) and ( a, b), respectively.


149

Let B be the set of all starlike boundaries and W the set of all far-field operators. By the uniqueness theorem in section 3.5.2, the mapping is a one-to-one mapping from B to W. Define integral operators K, S by

It is easy to prove that for any

the operator

is compact.

Theorem 3.26. Let be a bounded domain with a C2 boundary having an outward normal vector. The far-field operator depending on has representation

for any

and

Proof. As discussed in [469], we can write us(P; a) in the form

where

Then

satisfies

Equation (3.175) has a unique solution for any u' e U, and the operator bounded inverse operator (/ + K + )-1 in C . Hence

and

Using the asymptotic representation of the Green's function (cf. [469], [473])

has

150


we have

The theorem is implied. Following Angell, Colton, and Kirsch [7], we consider the following continuous dependence on boundary. Let A,,, A be a sequence of starlike C2 surfaces having representation

By the convergence , we mean the convergence , , where is the Cl,u Holder norm on S1. A sequence of function fn from is L2 convergent to a function f in if

Theorem 3.27. Let be a sequence of starlike C2 surfaces that converges with respect to 1,u 2 the C norm to a C surface as , and let u,, and u be solutions to the Helmholtz equation in the exterior of and A, respectively. Assume that the continuous boundary values of un, on , are L2 convergent to the boundary values of u on . Then the sequence {«„}, together with all its derivatives, converges to u uniformly on compact subsets of the open exterior of A. Proof. Representing u in the form of the combined double- and single-layer potential (3.174), and using (3.175), we obtain

where


151

Writing

where

and

we have

where ci, c2, and 03 are C1 functions. Using the inequalities proved by Angell, Colton, and Kirsch (cf. [7] or [126, Lemmas 5.10 and 5.11]), we can prove that if R, Q e C l,u (S 1 ) and R(P), Q(P) > a for some positive constant a, then there exist constants M > 0 and 0 < u1, u2 < 1, such that

for all P, P' e S1. Proceeding along the line of the proof of [126, Thm. 5.9], we prove the theorem. Theorem 3.28. (1) The far-field operator F( (2) Let F1 := .F( ) o (/ F1( ) has a dense range in L 2 (Si) for (2.2) in , then the range of F(

) is injective. + K+ ): L2(Sl) x VN. x VN- Moreover, if k2 is not the Dirichlet eigenvalue ) consists of a complete set in L 2 (S 1 ) x VN.

Proof. (1) Let

and

Then

152


from which it follows that

Let P —> P0 E the jump relation of integral potential follows:

Hence is injective. (2) The adjoint operator

Let

: L 2 (S 1 ) x VN --> L2(

) of F1( ) is given by

Then the function

satisfies (3.153) in

such that

which implies that v = 0 in and by analytical continuation, v = 0 in R3. Lemma 2.25 follows that g = 0 on S1 and h = 0 on D1. Hence .F1*( ) is injective and its nullspace . Therefore, the completion of the range of

in

Then

We now show that if k2 is not an interior Dirichlet eigenvalue of (3.153) in , then is complete in . This will imply that is complete . Hence, is dense in L 2 (S 1 ) x VN. Let such that


153

This implies that the function defined by

has a vanishing far-field pattern. Hence v = 0 in R3 \ for k2 is not the eigenvalue.

, from which it follows that

=0

Now we construct an algorithm based on the above analysis which provides an approximation to the inversion. Given a set of incident fields and the corresponding measured far fields , find the optimal solution ( such that

where

for chosen 6 > 0. Here the surface A is taken to be starlike:

and R1 , R2 are functions chosen from a priori assumption. Similar to the discussion in [260], [261] (see also [126, Chap. 5.4]), we can prove the following theorems. Theorem 3.29. For each e > 0, there exists an optimal solution Theorem 3.30. Let belongs to B. Then we have

be the exact far-field pattern of a domain

. such that

Theorem 3.31. Let be a null sequence and let be a corresponding sequence of optimal surfaces for the regularization parameter €n. Then there exists a convergent subsequence of . Assume that , is the exact far-field pattern of a domain such that is contained in B. Then every limit point of represents a surface on which the total field vanishes. An alternate approximation method can be found in [476].

154

3.6


The Intersecting Canonical Body Approximation

The intersecting canonical body approximation (ICBA), for the cases R", n = 2, 3, was developed in the following series of papers: [384], [381], [382], [380], [460], [461], [386], [462], [463], [389], and [327]. The ICBA assumes that the amplitudes in the partial wave representation of the scattered field are nearly those of a canonical body, for example, a circular cylinder in the 2D problem. This is true locally for each observation angle, the canonical body having the same local radius at this angle as that of the real body. The reconstruction of the shape of the body, represented at a given angle by its local radius, then proceeds by minimizing the discrepancy between the measured or simulated data and the estimation thereof. This is done at each observation angle. In one of its forms, the procedure enables the reconstruction of the local radius of the body for a given polar angle by solving a single nonlinear equation [381], [382], [380], [460], [384], [386], [389]. Another variant consists of finding this local radius by minimizing the L2 cost functional of the aforementioned discrepancy. The remainder of this section is concerned with the use of the ICBA for solving boundary identification problems of a 2D body located in free space, and the following will be shown: (i) that the reconstruction of the boundary of the body is not unique for both (synthetic) simulated and (real) experimental data, and (ii) that it is possible to single out the correct solution by employing data at two frequencies. In the following section we show how to employ the ICBA for boundary identification of a 3D body located in an acoustic wave guide.

3.6.1

Forward and Inverse Scattering Problems for a Body in Free Space

Let u' (x; w) be an incident plane-wave monochromatic (pressure) wavefield (the exp(—iwt) time (/) factor, with w the angular frequency, is hereafter implicit) at point x of the x Oy plane (i.e., the field in the absence of the object); u(x; w) the total field in response to u i ( x ; w); (assumed to contain O) the subdomain of x O y occupied by the sound-hard cylindrical object in its cross section (x O y) plane (see Figure 3.8), and the trace in x O y of the boundary of the object, assumed to be representable by the parametric equation (p a continuous, single-valued function of , and r, z the cylindrical coordinates). u(x; w) and u' (x; w) satisfy

wherein c is the sound speed in the medium outside of the object, the angle of incidence of the plane-wave probe radiation, , and s = (p2 + p 2 ) 1/2 .

3.6. The Intersecting Canonical Body Approximation

155

Figure 3.8. Scattering configuration.

The forward scattering (measurement) problem is: Given w, c, u' (x; w), and , determinew(x; w) at all points on the circumscribing circle of radiusb > p = Max Of particular interest here is the Inverse scattering problem: Given to, c, u'(x, w), b, and the simulated or measured field on , determine the location, size, and shape of the object embodied by the so-called shape function , knowing a priori that (i) the origin O is somewhere within the object, and (ii)

wherein Po is assumed to be a known positive real constant (as is b). Note that in the inverse problem, M(X; w) is unknown everywhere except on ,.

156

3.6.2


A Method for the Reconstruction of the Shape of the Body Using the ICBA as the Estimator

The estimator appeals to the ICBA [461], [381], [382], [460], [380], the mathematical expression of which is

wherein

( . ) is the nth-order Hankel function of the first kind, Jn( . ) the nth-order Bessel function, , and is the trial boundary shape function (which, if the inverse problem is solved exactly, is identical to . The3ICBAfurnishes the exact solution for scattering from a sound-hard circular cylinder of radius a and center O (i.e., the case ) provided N -> . It also furnishes an approximate solution to the forward-scattering problem for cylindrical obstacles of more general shape [461]. Let designate synthetic (simulated) or real (experimental) data pertaining to the incident, scattered, and total field, respectively, for the (real) body whose (real) shape (defined by the function is unknown and to be determined. Let designate the estimated incident, scattered, and total field for a trial body with trial shape . To reconstruct the entire shape of the body requires finding for all values of . In principle, we can identify the reconstructed value of with that for which a discrepancy functional between the measured and estimated fields vanishes. Practically speaking, this is done at M measurement angles { m = 1 , 2 , . . . , M}, so that the discretized version of embodied in the set m = 1 , 2 , . . . , M}, is recovered from the set of M equations (i.e., the discretized form of the discrepancy functional):

wherein it is observed, with the help of (3.188)-(3.189), that the mth equation depends only on the mth trial boundary shape parameter . Although these equations are uncoupled in terms of = 1 , 2 , . . . , M}, each one is nonlinear because each member of the set is a nonlinear function of .

3.6. The Intersecting Canonical Body Approximation

3.6.3

157

Use of the K Discrepancy Functional and a Perturbation Technique

We assume that

where a, d, and e are positive real constants, , has the same functional properties as

has the same functional properties as and

To zeroth order in is just the field scattered by a circular cylinder of radius a and center at O, which, if it were simulated or measured in an exact manner, would be of the form

whereas the ICBA representation of the estimated scattered field takes the same form, with d replacing a. At this stage, rather than attempt to solve the system (3.190), we prefer to return to its continuous form, which we project onto the set of Fourier basis functions {exp( Z}. The inverse problem thus reduces to the recovery of d from the following set of equations:

which, after use of the identity

(with

, the Kronecker symbol), yields

from which we deduce, after using the relation [313] the Hankel (H), Bessel (J), and Neumann (Y) functions,

between

A consequence of this expression, from which we now try to determine d, is that if d is a solution of (3.197) for a finite subset of Z, then it must be a solution of (3.197) for the remainder of values of m in Z.

158


An obvious solution of (3.197) is d = a, which is the correct solution to order (ke)° for the shape function, since and to zeroth order in ks, but this does not necessarily mean that it is the only solution. Consider the subset of equations (3.197) for which Im(grad G(r, z, zo)) for the entire range. To perform the singular integrals, denote the positive poles by in increasing order, and compute

Then partition the interval into Hence,

4.1. A Uniform Ocean over an Elastic Seabed

185

Table 4.1. Spectral results for Example 4.1.

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14

I2.163998e-01 1.944158e-01 1.507746e-01 8.568 188e-02 1.312955e-02 -8.104280e-04 -1.138687e-01 -2.453701 e-01 -3.990689e-01 -5.744492e-01 -7.719074e-01 -9.912320e-01 -1.232466e+00 -1.495693e+00

an 1.419591e-04 1 .594572e-03 4.488249e-03 8.726347e-03 8.701583e-04 1 .447287e-02 2.203455e-02 3.094712e-02 4.110936e-02 5.281352e-02 6.600330e-02 8.058430e-02 9.669277e-02 1.142394e-01

n 15 16 17 18 19 20 21 22 23 24 25 26 27 28

an En -1.780802e+00 1.332268e-01 -2.087850e+00 1.537193e-01 -2.416858e+00 1 .756447e-01 -2.767765e+00 1.990276e-01 -3.140617e+00 2.239028e-01 -3.535413e+00 2.502126e-01 -3.952120e+00 2.779868e-01 -4.390771e+00 3.072452e-01 -4.851361e+00 3.379413e-01 -5.333867e+00 3.701047e-01 -5.838315e+00 4.0374716-01 -6.364700e+00 4.388300e-01 -6.913005e+00 4.753816e-01 -7.483251e+00 5.134086e-01

On intervals [£,- — a, f / + a], use the Cauchy principle value of the integral

By the assumption that Ej is a simple pole, function F(E) + F(2Ej — E) is finite on [Ej — a , E j + c c ] , where the value of the function at Ej, is defined by the limit. Hence, the integral becomes a regular one. Example 4.1. The following data are used to perform the numerical computation:

The zero En, and coefficients an, are in Table 4.1. Note that there are five positive En. To see the asymptotic behavior of En and an, they are plotted against n to obtain

which confirms the convergence of the far-field representation.

186

Chapter 4. Oceans over Elastic Basements

Figure 4.2. Comparison of far-field approximation and nearfield approximation.

Figure 4.4. Comparison of far-field approximation and nearfield approximation.

Figure 4.3. Comparison of far-field approximation and nearfieldapproximation.

G1

Figure 4.5. The computed (r, z, zo) -

To verify these formulae, G(r, z, z o ), (r, z, zo), and (r, z, z0) are computed for selected values of z and Z0 using both the far-field approximation and numerical integration, respectively. It is expected that the results of the two methods will agree in the middle field but disagree in the far field and near field. The comparisons are shown in Figures 4.2-4.4, and they confirm the dependence of the accuracy upon the range. This suggests that the number of modes used be range dependent in order to achieve the prescribed accuracy. The combined method can be used to compute G\ (r, z, zo) for all ranges (see Figure 4.5) [292].

4.1.6

Computing the Scattered Solution

We form the mesh on the surface of the scatterer using triangulation. For a scattering surface of arbitrary shape, generating an optimal surface mesh is a very complicated problem. One criteria for an optimal triangular mesh is that the ratio of the area to the square of the diameter should be as small as possible. Another is that for the prescribed number of triangles, the maximum diameter should be made as small as possible. Moreover, the maximum variation

4.1. A Uniform Ocean over an Elastic Seabed

187

of the normal on a triangle should be made as small as possible. Hence, this is a multiple objective optimization problem. See [185] for numerical data. Let be the approximation of the boundary , where each Tj is a planar triangle. Moreover, let (Xj, Zj) be the center of Tj, and nj the outward normal of 7). Approximate the unknown function fon Tj by the constant fj. Then the boundary integral equation is discretized as

where

with

To compute Aij, write

as

where . The last two terms are regular, and their contributions to AiJ can be approximated by using a routine quadrature formula for all i, j — 1, 2,..., N. For the first term, when i = j, since both (x, z) and (x j , zj) are on 7), {xi, — x, y1 —y, z1 —z} is perpendicular to ni, and hence

For i

j, since r1 > 0, a quadrature method is used to compute

but when r1 is very small, a more accurate approximation may be needed. Due to the complicated wave guide fundamental singularity, the construction of the stiff matrix A is a very time-consuming computation. This results from an essential difficulty of this problem, due to the interaction between the water column and the seabed. Compared with the scattering problems in the whole space, the computation of this step is slowed down roughly by a factor proportional to the average number of modes multiplied by the time for computing a Bessel function. Consequently, a parallel computation appears to be an efficient way to perform this calculation. Moreover, a number of parallelizations may be used; for example, let each processor in a parallel computer be responsible for a portion of elements and use the iteration algorithm. The following generic scheme seems to be efficient [292].

188


Figure 4.6. The mesh for discretizing the surface of the scatterer.

Figure 4.7. The contour Plot of , measuring uzo, at ocean surface.

1. For each processor PEk do in parallel: Solve L(E,h) = 0 for {Ej,} and compute the {a j } in kth subinterval E e (uk,

uk+1).

2. For each processor do in parallel: Refine the kth portion of a mesh. 3. For each processor PEk do in parallel: or i, j Ain the kth index subset. Compute iJ and 4. Solve the linear system in parallel. 5. For each processor do in parallel: Evaluate p(x, y, z) for (x, y, z) in the kth range. Depending upon the required accuracy and the computer, a more sophisticated parallel algorithm can be designed for this problem [254]. Finally, because of complicated data types involved in this problem, the program was written in C. The package [326] was employed to handle message passing during parallel computations. The subroutines for computing Bessel functions are taken from [279]. Example 4.2. The same values of the physical coefficients are used as in the previous example. Let the scatterer be a ball of radius 10 centered at (0, 0, h/2), and the time-harmonic point source be at (0, 15, /z/2). Using a mesh as shown in Figure 4.6, and implementing our algorithm on the T3D superparallel computer located in the Pittsburgh Supercomputing Center, we numerically solved the boundary integral equation. These results are shown in [292]. Results obtained by these computations are shown graphically in Figures 4.7-4.11.

4.2.

Undetermined Coefficient Problem for the Seabed

Figure 4.8. The contour plot of p(x, h), the pressure at sea

plot of

189

Figure 4.9. The contour p(x, 0)| and \p(x, h)\.

floor.

Figure 4.10. The effect of scattering: y dependent.

4.2

Figure 4.11. The effect of scattering: x dependent.

Undetermined Coefficient Problem for the Seabed

In this section we consider an inverse coefficient problem associated with an elastic seabed. The procedure is to use a point source to excite sound waves in the ocean, which are then used to determine the coefficients of the seabed. This method will use a time-harmonic point source; however, a similar procedure is possible with transient sources. In a shallow ocean the acoustic waves interact with the seabed, and hence the acoustic far field must contain information about the seabed. Our procedure will be based on finding a representation of the acoustic pressure in terms of the seabed constants, i.e., to find a representation of the direct problem in terms of these parameters. In the finite-depth, elastic seabed, we assume that it is a tightly packed and homoge-

190


neous. Moreover, as the water columns are shallow and of uniform depth, we assume it is also homogeneous. For our method to work, a representation for the acoustic pressure is necessary. We assume that b is the sediment thickness and is the frequency of the source. Our problem simplifies very much for a homogeneous seabed if we introduce the Helmholtz displacement potentials used earlier.12 For our finite-depth sediment model, we also assume that the sediment lies on a rigid rock foundation, for which we use the boundary conditions

Earlier we computed an analytical formula for the acoustic pressure, namely,

Using this representation of the acoustic pressure permits us to calculate the far field F, the coefficient of 1/r in the expansion of the acoustic pressure in descending powers of r, namely,

We pose our inverse problem as follows: (1) Let the measured acoustic far field in the ocean at the range r be given as Fmeas(r, Zj), j = 1 , , 2 , . . . , M

(2) Introduce the functional

where (3) Minimize G(U) over the parameter space, using say, Newton's method; i.e., we compute where grad G(U) can be easily obtained from (4.57) and (4.51) with the help of Macsyma. Numerical results concerning this method can be found in [77]. l2

This is the approach used in Gilbert and Lin [203], [204], and we present these ideas here.

4.2. Undetermined Coefficient Problem for the Seabed

4.2.1

191

Numerical Determination of the Seabed Coefficients

The discrete spectrum part of a Green's function representation gives a way to calculate far-field acoustic pressure in the water column arising from a point source at depth Z0:

where the eigenvalues Kn, are solutions to

(K) = 0.

Inverse Problem. Given N — 1 measurements of pressure Fj at evenly spaced depths zj at some distance r in the far field, find the density of the sediment ps , the compressional and shear wave speeds cp and cs, and the compressional and shear attenuation factors p and s. The Lame parameters formulas

and

are calculated from the data cp, cs,

t,,

and

s

by the

where n = ( We take the following approach. We seek a choice U = (ps, c p , c v , p, s) that minimizes

where the Fj is the measured value of pressure at depth z,. We use the Nelder-Mead simplex method to locate the minimum. To explore the feasibility of solving this inverse problem, we use estimates for these parameters for four sediments taken from Hughes et al. [249]. Sand: Input

Value

Density of sediment Compressional wave speed Shear wave speed Compressional attenuation Shear attenuation

ps — 2060kg/m3 cp = 1750m/s cs = 170m/s p = 0.46dB/wavelen s = 2.2ldB/wavelen

192


Glacial til: Input

Value


= 2100kg/m3 cp = 2000m/s cs = 800m/s p - 0.3dB/wavelen s — 1.4dB/wavelen

Input

Value


ps = 2200kg/m3 cp = 2400m/s cs = l000m/s p = 0.24dB/wavelen s = 1 .0dB/wavelen

Input

Value


ps = 2600kg/m3 cp = 5500m/s cs = 2400m/s p = 0.55dB/wavelen s = 0.14dB/wavelen

s

Chalk:

Granite:

Sand: Actual Data 2060 1750 170 0.46 2.21 Guess 2080 1770 190 0.36 2.0

: Density of sediment cp: Compressional speed in sediment cs: Shear speed in sediment p: Compressional attenuation s : Shear attenuation

s:

cp: cs: p: s:

Density of sediment Compressional speed Shear speed Compressional attenuation Shear attenuation

Initial numerical experimentation [83] indicated no strong correlation between number of data points or frequency. This suggested a stochastic approach. The following experiment was tried for the three soft sediments. Repeatedly choose a guess using a pseudorandom number generator from the intervals ps : [1700,2300],

4.3. The Nonhomogeneous Water Column, Elastic Basement System

193

cp : [1700, 2300], cs : [200, 800], p:

[0.2, 0.8],

s:

[0.1, 1.9],

and apply the simplex method. Take as the answer the result that gives the smallest value of/(£/). Sand: Range = 1km Frequency = 50Hz, N = 10 ps = 1835.25 cp = 1952.28, cs = 439.994

Guess (Try #8) p = 0.546962,

= 1.11647

Result

Guess (Try #1)

Result

A = 2073.37 cp = 1750.13,cs = 180.075 e = 0.463272, ft = 1.58999

ps = 1703.44 cp = 1910.37, cs = 246.503 p , =0.41 24, s = 0.103292 338 Iterations, f = 1.87466e-08

ps =2133.34 Cp = 1750.7 l,Cs= 218.928 p = 0.474387, s = 0.105031 122 Iterations, f — 2.38723e-06

Til: Range = 1 km Frequency = 50Hz, N = 10 ps =2109.79 cp = 1970.21, Cs =687.701

Guess (Try #5) ftp = 0.557647, s = 1.33528

Result

Guess (Try #4)

Result

, =2115.46 cp = 2003.55, cs = 799.492 p =0.353107, s = 1.27949 258 Iterations, f = 4.541 39e-08

p, = 2052.42 cp = 2010.93, cs =586.229 p = 0.40 2459, s = 1.48413

=2101.46 cp = 2000.66, Cs, = 799.631 P = 0.406849, s = 1.36628 94 Iterations, / = 2.1971e-06

Chalk: Range = 1km Frequency = 50Hz, N = 10 ps = 1945.52 Cp = 1931.87, cs = 779.098 Result ps = 2198.09 cp = 2399.85, Cs = 1000.33 p = 0.237793, s = 1.00884

4.3

Guess (Try #10) p = 0.692332, s = 1.11506

Guess (Try #5) p, = 2241.6 ps cp = 2012.12, Cs = 281.192 p = 0.692208, ft = 0.548457 265 Iterations, / = 2.57625e-08

Result = 2906.31 cp = 1770.6, c.y = 978.316 = 1.00186, s = 0.806622 p, 134 Iterations, f = 8.44138e-04

The Nonhomogeneous Water Column, Elastic Basement System

In this chapter we use the modal solution approach to treat the nonhomogeneous, elastic seabed. Knowledge of the modal solutions for an ocean-seabed system are useful, not only for constructing an ocean-seabed Green's function, but for also computing the transmission losses in the far field [65]. For analytical convenience we have the ocean surface at f, the ocean-seabed interface at fs, and the seabed bottom at s which are defined as

194


and

respectively. As before we consider both the water column and basement to be of constant depth. f and s, denote the regions occupied by the water column and the basement, respectively. The acoustic pressure, p(r, z), satisfies the Helmholtz equation in f and has a pressure-release boundary condition at z = h and a transmission condition at fs. The displacement field in the seabed will be seen to satisfy the Navier equations13 in S as well as boundary conditions on s and transmission conditions at s. As in the previous chapter we consider the boundary conditions

and the transmission conditions across the interface

fs

Alternatively, one might use the no-slip boundary condition ur = 0 at the bottom of the seabed rather than the slip condition rz = 0. As we wish to construct Green's functions and they are of the form G(|rexp — , it is convenient to introduce cylindrical coordinates. In terms of this coordinate system the displacement vector is denoted as u :— (ur, u , uz) and the axially symmetric strain tensor as

If the seabed is a nonhomogeneous, transiotropic, elastic material with the depth-dependent Lame coefficients , , , and , then from Hooke's law the stress tensor is given by

where the elastic coefficients are functions of the depth. Let us assume further that the system is in harmonic vibration with frequency ; then the equations of motion become

13

These equations are no longer homogeneous.


195

Substituting the strains (4.60) into the constitutive equations (4.61) and then these into the equations of motion (4.62), (4.63) yields the Navier equations for the displacements

As can easily be demonstrated using Hankel transforms [394], it is sufficient to assume that the displacements decompose as

where H0 and H1 are Hankel functions of order zero and one. Substituting these special solutions into the Navier equations (4.3), one obtains a coupled system of ordinary differential equations in and with k as a parameter:

The Hankel decomposition of the displacements leads the dilatation and stress components to take the form

If the system (4.68) is isotropic, then = and = For didactic purposes, in the future, we assume this is the case. The modal equations for the isotropic seabed are [233]

196


This seabed system can be uncoupled by setting

We notice that this is similar to the potential decomposition used in [2]. In the water column = 0 and (4.72) reduces to a self-adjoint differential equation with depth-dependent coefficients

To distinguish the values of Lame's coefficient in the two different media, we introduce

and

Therefore, in the sediment

S,

we have

which may be combined to obtain the uncoupled equations

where

are compressional wavenumber and shear wavenumber in the sediment, respectively. The boundary transmission conditions in terms of and may be written as

Using the last four conditions above, we can solve for 0 and express (0 - ) and (0 - ) as

in

5,

from which we may


197

By substituting the last two expressions into the second of the conditions (4.79), we obtain the acoustic boundary condition for the ocean floor [203], [204],

Then (4.74), (4.79), and (4.85) form a second-order, ordinary differential equation, eigenvalue problem. We stress that this eigenvalue problem is different from the normal Sturm-Liouville problem because one of the boundary conditions involves the eigenvalue itself. We can solve the water column equation using transmutation operators [98, 99].14 We first transform the differential equation (4.74) into canonical form by making the following transformations:

Then we have

Finally, it is computationally more convenient to rewrite the differential equation in the form

where

l4 Transmutation is a method which permits us to express the eigenfunctions of a complicated system in terms of the eigenfunctions of a simpler system by means of an integral representation in which the kernel is independent of the eigenvalues. This approach permits the derivation of an explicit characteristic equation for the eigenvalues.

198


The solution Z(z) can be written by means of a transmutation

where K(z, t) :— K(z, t, a ( . ) + K) and K(z, t, p(•)) is the solution to Gelfand-LevitanGoursat problem

Using this representation we obtain the characteristic equation for the eigenvalues of the modal solutions, namely,

The following three formulas are based on differentiation properties of Bessel functions:

Applying (4.94), (4.95) to the water-column equations, we obtain


199

Applying (4.94) to the dilatation and the stress components, we get

Letting H1 act on (4.62), H0 on (4.63), and applying (4.94) to them, we obtain

Inserting (4.99)-(4.101) into (4.102)-(4.103) yields

Similarly, the boundary transmission conditions are transformed into

The equations (4.96), (4.97), (4.104)-(4.110) form the Hankel-transformed ocean-seabed system. In the case of constant coefficients, we can solve it explicitly. If the coefficients are variable, we can construct a solution to this system and implement it numerically. For the case of depth-dependent coefficients, we associate the initial value problem with the Hankel-transformed modal equations for the water column,

The Green's function of (4.112) can be constructed using undetermined coefficients. As the Green's function, G(k2, z), satisfies (4.96)-(4.106), it must be expressed as

200


for some C1. Next, denote by (Y 1 (k 2 , z), Z 1 (k 2 , z)} the solution to the (seabed) initial value problem

[Y2(k2, z), Z2(k2, z)} is the solution to (4.114)-(4.116) and

Consequently, the solution [ur, uz} to (4.104), (4.105), (4.110) can be written as

Inserting (4.113), (4.119), and (4.120) into (4.107)-(4.109), we obtain a linear system for C1, C2, and C3:

where the matrix A and vector b are

where

Therefore, we can represent L( , c) as

Then we have

= |A| and

The substitution of C1 into (4.113) yields

4.4. An Inner Product for the Ocean-Seabed System

201

Now we can perform the same Hankel inversion procedure as in the previous section, but the asymptotic analysis is even more difficult. A rigorous proof, however, may be based on transmutation theory, as in [77]. We will establish the usual theorems concerning the existence of propagating solutions. Theorem 4.5. For frequencies

such that

the propagating solution G(r, z) exists, and

In particular, ifL(

, h) has no multiple zeros, then

The numerical implementation for constructing G(r, z) based on this analytical representation of the solution is clear. We solve numerically the two ordinary differential systems to construct the function L( , c). If we use the normal mode expansion method, it is clear that the characteristic equation is nothing more than L( , h) = |A| — 0. Therefore, the eigenvalues are just the poles of G( ,z).

4.4

An Inner Product for the Ocean-Seabed System

If we used the normal mode expansion only, then we would not know that the Fourier expansion had the coefficients

as indicated above. To construct the coefficients

from the eigenvalue problem, we first need to find an inner product under which the set of eigenfunctions are orthogonal. Let { be the set of eigenvalues and { ( g n , a11, B11)} be the corresponding eigenfunctions normalized by g' n (0) = 1. Then from (4.104)-(4.105), we have

202


By conjugating each of these equations, interchanging the indices, multiplying by (4.128) by gm, integration by parts over [0, h] and using (4.131) and (4.132), we obtain

This procedure may be repeated with other equations; for example, by multiplying (4.129) by am, integrating by parts over [h, b], and using (4.135) and (4.134), we have

By multiplying (4.130) by use of (4.133), we obtain

Using the notation

m

and performing the same procedure as the above, but making

4.4. An Inner Product for the Ocean-Seabed System

203

we notice that all terms except Enm and Hnm satisfy the symmetry relation

Consequently, the relations (4.136)-(4.140) become

By interchanging the indices n and m and conjugating each term, we obtain the relations

By combining (4.142)-(4.147) using Macsyma, we can obtain

Notice that

Therefore, if ( n, n) 0, then is real. But even though it is uncertain whether this condition is true for an arbitrary frequency, we are sure it is true for small frequencies, while for = 0 it is obvious. The answer is that ( n, n) 0 for all n and for all but a countable number of frequencies. The reason is that for each n, ( n, n) is an analytic function of w, and, moreover, ( n, n) 0 when w = 0 for all n. This means that since ( n, n) is a nontrivial analytic function of w, there are at most a countable number of w for which ( n, n;) is zero. A rigorous proof of this requires the theory of several complex variables because we are dealing a multiple-parameter spectral analysis problem. That is, the exceptional frequencies are constructed from the time domain spectrum, which are coupled to the spatial eigenvalues { n} [92], [377]. Let us suppose ( n,n n,) 0 for all n. Then ( m, n) can be used as an inner product, because according to (4.149), ( m, n) =0 for all m n. Now we are going to construct the Fourier coefficients { F n } assuming that G(r, z) has a normal mode expansion and ( n, n) 0 for all n. Using (4.149), we have

204


If we substitute (4.150) into the equation (4.23), using (4.128) and the differential equation for the Hankel functions

then we get

Since the source is located at (r = 0, z = Zo) in the ocean, it follows that

Using (4.151)-(4.152), by making use of the asymptotic behavior of Hankel functions

we obtain

These imply

Consequently,

By multiplying (4.154) by gm(z) and integrating over [0, h], then multiplying (4.156) by (z) and integrating over [h, b], then multiplying (4.155) by —

4.4. An Inner Product for the Ocean-Seabed System and integrating over [h, b], then multiplying (4.157) by [h, b], and using (4.149) and the orthogonality ( m, n) — ( m,

205 (z) and integrating over ) (m, n), we get m

Therefore,

Consequently,

We note that this Fourier coefficient expression has a quite different form from that in (4.127). The question that arises naturally is if the two types of expansions are equivalent. We are not able to prove the equivalence at this stage. We expect that numerical computations using this expression for the Fourier coefficients will return more accurate results than those in (4.127), since numerical integration is much more stable than numerical differentiation. In lieu of a proof, we should test them numerically. In fact, it is good to have two formulas to verify the correctness of our computations. Let us now use this method of Fourier to determine the acoustic field due to an axially symmetric source distribution inside a cylinder r < r0. We assume that the data ur(r0, z) are given on the truncated cylinder r = r0, 0 < z < h, and the data u r (r 0 , z), z u r (r 0 , z), r u r (r0, z) is given on r = r0, h < z < b. It is well known that an acoustic field is a linear combination of the modal waves

We can decompose this sum into M propagating modes, forming the far field, and the evanescent part. Since we can compute the propagating modes, the remaining problem is how to compute the Fourier coefficients an, n = 1, 2 , . . . , M. Now if there is a point source at z = Z0, 0 < Z0 < h, from the Helmholtz equation with the delta function source term we have

where p(r, z) = (z)e(r, z) is the pressure amplitude. In the seabed, the homogeneous Navier equations are satisfied. By substituting (4.160) into these equations, we can get

206


Then by combining the equalities above we have

If the water-column eigenfunctions are normalized by Z'(0) = 1, then the pressure in the ocean can be represented as

4.5

Numerical Verification of the Inner Product

A good way to verify the formulas and the computationally obtained eigenvalues is to evaluate the matrix }. This should be the identity matrix because of the orthogonality of the eigenvectors, if all computations were completely accurate. The numerical approximations will result in very small, nonzero, off-diagonal terms if there is no mistake in our formal derivation. We also compute the Fourier coefficients in (4.127), i.e., and those in (4.159), namely,

, which should agree.

Example 4.3. We use the following experimental data to check our solution:

This set of data suggests that the compressional wave speed is around 7000m/s and shear wave speed is around 4000m/s. We only search the positive eigenvalues that correspond to the propagating modes.

4.5. Numerical Verification of the Inner Product

207

Table 4.2. The comparison of Fourier coefficients.

n 1 2.719182e-02 2 9.176573e-02 3 1.353713e-01 4 1.569599e-01

in (4.127) -1.717591e-02 -1.203690e-02 7.586275e-03 2.312212e-03

Figure 4.12. Eigenfunctions g n ( z ) , n — 1,2,3,4, in the ocean.

The matrix

tions

in (4.159) -1.725774e-02 -1.205485e-02 7.610731e-03 2.319580e-03

Figure 4.13. Eigenfunc(z) in the seabed. n

is as follows:

Figure 4.15 shows the procedure for searching eigenvalues. We use the combination of a brute-force search and the bisection method. It turns out that computing is very expensive, so we approximate it by central differences. The eigenfunctions gn are plotted in Figure 4.12, n (z) in Figure 4.13, and Bn (z) in Figure 4.14. Finally, in order to see the effect of interaction of the seabed on the far field, we compare an elastic seabed with the totally reflecting seabed, using the following data as input: ocean as in Example 4.3; elastic seabed with constant c1 = 7000m/s, c, — 4000m/s, and b = 10m; frequency w = 600 1/s; depth of the source Z0 = 15m; range r = 2000m. Figure 4.16 shows the comparison between the case of a totally reflecting seabed and that of an elastic seabed. We can see from the far field that the effect of the seabed interaction is significant.

208


Figure 4.14. Eigenfunctions Bn (z) in the seabed.

Figure 4.15. The computed values of function L( ,h) during the brute-force searching and bisect searching.

Figure 4.16. A pressure comparison in the case total reflecting seabed with elastic seabed.

4.6 4.6.1

Asymptotic Approximations of the Seabed A Thin Plate Approximation for an Elastic Seabed

In this section we follow the ideas of Gilbert, Hackl, and Lin [202] to investigate asymptotic methods to approximate the ocean-seabed system. The purpose is to obtain a simple model that is amenable to posing the undetermined object problem. We shall, in this section, consider several models; however, each is based on the idea of replacing the seabed by either a thin or thick supported plate. We consider both the elastic and poroelastic case. Our work on the poroelastic seabed is based on a 2D asymptotic Kirchhof model of a poroelastic plate that was first developed by Hackl [233]. We shall assume now that the ocean is of uniform depth and occupies the region := R2 x [0, b], whereas the bottom is designated by R2 x {0} and the surface by b :•= R2 x {b}. If we describe the displacement

4.6. Asymptotic Approximations of the Seabed

209

vector u in the ocean in terms of a potential ,

the pressure is then given by

where

3

is the 3D Laplacian and (z) is the stratified bulk modulus, p then satisfies

At the ocean surface (z = b) we have the "pressure-release" condition p — 0, which implies = 0. The seabed is to be modeled as a plate; at the bottom of the ocean (z = 0) we assume that a Kirchhof plate (see [418]) exists whose displacement in the vertical direction obeys

where D is the plate stiffness and ps is the seabed density. 2 denotes the 2D Laplacian. Compare also the model of Bedanin and Belinskii [18], which differs slightly from ours. If we wish to construct the acoustic Green's function for this system, it is convenient to work with the pressure instead of the displacement potential. For the case where p() and A are constant we are led to consider

We Hankel-transform the pressure equation to obtain

where k denotes the modulus of the Hankel transform. For the source at zo solution to this equation has the form

(0, b) the

where :.= max[z, Z0], and z< := min[z, Z0]. We need to determine the coefficients A and B. One condition is given by the jump in the derivative of p(z) := p ( z , k ) at ZQ, namely,

which leads to

210


The other condition will come from the ocean-plate boundary conditions, which must be derived. The Hankel-transformed plate equation becomes

or

We obtain another condition from (4.168) and from the fact that for z Zo, p = — . Hence, at z = 0, p(0) = (0) = . This leads to

3

=

or

Using the above equations to solve for A, we obtain

and

The acoustic Green's function p(r, z) may then be found by using the Mittag-Leffler expansion of p(z, k) in the complex k-variable, the inverse Hankel transform, and the identity

Note that p(z, k) as given in expression (2.13) is an even function of is continuous across the branch cut. Setting = k2 we represent

so p(z, k)


211

where

and where

Notice that since

as | | —> with |Im | > € > 0, the integrand remains bounded except for the poles. As it may be shown that there exist no nonnegative, multiple poles, a representation for p(r, z, Z0) may be found in the form

where

Note that the aj do not depend on r, z, Z0, which thereby indicates that the Green's function p(r, z, Z0) is symmetric in z, Z0. In the remainder of this chapter we study other plate models of the seabed. The procedure for inverting these is the same as indicated here and in Gilbert and Lin [203]. Calculations of this type have been done symbolically for the case of finite and semiinfinite seabeds [204], [185], [186]. Having considered the case of a constant sound speed ocean, we treat next the stratified ocean where we employ again the method of transmutation [183], [209], [153], [83]. We attempt to represent the transformed acoustic pressure in the ocean by the transmutation

212


where H(z) is the Heaviside function. It turns out that the Gelfand-Levitan-type kernels [181] satisfy

where 0 < z < b, and we have suppressed the dependency of p on the transform variable k. This type of transmutation, in contrast to that used by Gelfand and Levitan, generates solutions of the nonhomogeneous acoustic equation. For the representation (4.180) to work, we need to determine p'(b), as was done in Gilbert and Lin [202] for a finite seabed. To this end we compute p(0) and p'(0) first; we have

and

As in the constant index ocean case we have the conditions and to solve for the "unknown" . We obtain


213

When we solved this case for a finite-depth seabed the expression (4.185) was quite long and is not presented here. The expression was stored in Macsyma and produced a Hankel inversion that compared excellently with the parabolic approximation [185]. Now we verify that the transmutation

generates solutions of

To do this we differentiate and then substitute into the differential equation. Differentiating the Heaviside function H ( Z 0 — z) leads to — (Z0 — z), and a second differentiation leads to a term with a (Z0 — z). We demand that the coefficients of (z — Z0) and (Z0 — z) independently vanish at z = Z0. The coefficient of (Z0 — z) is

which clearly vanishes at z — Z0. The coefficient of (Z0 — z) is

which also vanishes at z — Z0. The coefficient of H(zo — z),

214


vanishes for Z0 > z, providing

To make the remaining terms vanish it is sufficient to have

We recognize K(z, s) :— G(z + Z0 — b, s).

4.6.2

A Thick Plate Approximation for the Elastic Seabed

A Mindlin plate theory for an elastic isotropic plate of thickness h is given in [2, p. 256]. According to that theory the components are expressed in the form

where x and y are the local rotations in the x and y directions. For the case of harmonic vibrations we are led to the following system of equations:

where e := + . Here v is Poisson's ratio and K = 5/6 denotes the so-called shear-correction factor. By differentiating the first of these equations with respect to x and the second with respect to y and adding, we obtain the reduced equation

Hankel-transforming these equations leads to

We may solve this pair for e, w as


215

We choose for the constant index ocean the representation (4.180) for p ( z ) . We now proceed as in the thin plate approximation. Once again we can solve for the coefficients A and B. Substitution in (4.180) yields

with

For the stratified ocean we again use the same representation to determine p'(b). To this end we compute

with

Then we obtain

216

and we have


Chapter 5

Shallow Oceans over Poroelastic Seabeds

5.1

Introduction

In a shallow ocean, sound waves travelling distances of several kilometers will interact repeatedly with the underlying seabed. Consequently, in developing a mathematical means for predicting acoustic pressure in a shallow ocean the manner in which the seabed is modelled is important. In such computations the seabed typically is treated as a dense fluid, an elastic solid, or a poroelastic medium. As indicated in Vidmar [439], [440] the fluid model is appropriate for thick sediment layers, but thin sediment layers, where conversion of energy to shear waves is an important loss mechanism, require a model that supports shear effects. Examples of the poor predictions made by the fluid model for thin sediment layers can be found in Hughes et al. [249]. Since thin superficial sediment layers are common in shallow ocean environments, our concern will be a comparison of the solid elastic model with the poroelastic model developed by Biot in [36], [38], [40], [39]. This chapter is based on a series of papers by Buchanan and Gilbert [65], [75], [79], [81] and the paper [84] of Buchanan, Gilbert, and Xu.

5.2

Elastic Model of a Seabed

The elastic model of a seabed is widely used in ocean acoustics. Hence, before deriving Biot's model for a porous medium, let us summarize the equations for the elastic model. A sediment layer is treated as a viscoelastic slab depending upon the parameters , the aggregate density of the layer; . and , the compressional and shear Lame coefficients; and and , the compressional and shear attenuation coefficients. Let the vector u(x, y, z, t) = [ux(x, y, z, t), uy(x, y, z, t), uz(x, y, z, t)] track displacement of a material point in the seabed. The dilatation e — • u then measures the relative volumetric increment due to deformation. The constitutive equations for an isotropic elastic material are

217

218

Chapter 5. Shallow Oceans over Poroelastic Seabeds

where the strains are related to the displacements by

For constant parameters the equation of motion for an elastic solid is

By taking the divergence and curl of (5.3), separate equations for dilatational (compressional) and shear waves are obtained:

where = x u. It is of interest to know the speeds of time-harmonic waves of the two types in some direction, say the x direction. Substituting e — E exp(—ik p x + i w t ) and x = F exp(—ik s x + i w t ) into (5.4) gives

and thus the speeds for compressional and shear waves are

respectively. Attenuation is incorporated by making the Lame coefficients complex. Introducing the complex wavenumbers , where v — In 10/(40 ), and solving (5.5) with KptS replacing kptS for the Lame coefficients gives

where Cp and Cs are the complex wave speeds defined by

5.3. The Poroelastic Model of a Seabed

219

The normalization factor v is inserted so that the loss in decibels over one wavelength 2 /k is

The units for are said to be "decibels per wavelength." In ocean acoustic literature attenuation is often written in the form

where i is measured in dB/m, / is the frequency measured in kHz, and thus the constant K is in dB/m/kHz. In terms of the loss per meter is

which gives the formula for converting between the coefficients A and .

5.3

The Poroelastic Model of a Seabed

The most commonly used model for a poroelastic medium is that developed by Biot in [35], [36], [38], [40], [39]. In sections 5.3.1 and 5.3.2 we sketch the derivation of Biot's model. In section 5.3.3 we discuss the calculation of the parameters in the model from the set of input parameters introduced by Stoll [400]. In section 5.3.4 sets of Biot-Stoll parameters taken from the literature are given and the difficulties in determining these parameters are discussed.

5.3.1

Constitutive Equations for an Isotropic Porous Medium

The Biot model treats the medium as an elastic frame with interstitial pore fluid. Two displacement vectors u(x, y, z, t) = [ux(x, y, z, t), uy(x, y, z, t), uz(x, y,z, 01 and U(x, y, z, t) = [Ux(x, y, z, t), Uy(x, y, z, 0, Uz(x, y, z, t)] track the motion of the frame and fluid, respectively, while the divergences e • u and = • U give the frame and fluid dilatations. We shall treat only the case of an isotropic frame. In this case the frame has six components of stress, axx, cr vv , azz, axy, axz, ayz. The corresponding strains will be denoted by exx, eyy, ezz, exy, exz, eyz. The fluid stress in the pore space is given by (x, y, z, t) = — pf, where pf is the pressure of the pore fluid and the parameter is the fraction of fluid area per unit cross section. Biot makes the assumption of statistical isotropy, that is, that is the same for all cross sections. Thus ft is equal to the porosity of the medium (volume of the pore space per unit volume). In an isotropic medium the strain energy will be a function

where the Ij are the three elastic invariants (see Love [295])

220


For small amplitude vibrations we can neglect powers of the displacements above the first order and obtain linear constitutive equations. This corresponds to a strain energy function that is purely quadratic in the strains and hence it will be a linear combination of the four quadratic terms e2, I2, e , and €2:

The components of stress are related to the strain energy function by

This gives the constitutive equations

Since exx + eyy + ezz = e the equations can be put in the form

with = P — 2 . The symbols assigned to the parameters and are due to their formal analogy to the Lame coefficients in the constitutive equations (5.1) of an elastic solid. Indeed, the tangential stress equations (5.7)4 suggest that is the Lame coefficient of shear for the frame. However, as we shall see, A is not the frame compressional coefficient.

5.3.2

Dynamical Equations for a Porous Medium

In [36] Biot adopted the following form for the kinetic energy of the system


221

where ux = , Thus p11 and p22 are effective mass density parameters of the frame and fluid, respectively, and p12 is a mass coupling parameter for the frame-fluid interaction. A Rayleigh dissipation function

is introduced to account for energy loss due to the motion of the fluid relative to that of the frame. The lagrangian equations of motion are

Substituting the constitutive equations (5.7) and the strain-displacement relations (5.2) gives the equations of motion

222


In the case where all parameters are constant, we have, upon expressing the cross partial derivatives in terms of the derivatives of the dilatations, the considerably simpler vector equations

The form of the dissipation parameter b is complicated. In [38] Biot gave the formula

when the medium is undergoing time-harmonic oscillations of angular frequency w. Here pf is the density of the pore fluid, n is the viscosity of the fluid, k is the Darcy permeability of the medium, and a is referred to as the pore size parameter. The function F is given by

where T is defined in terms of Kelvin functions

Figure 5.1 shows the graphs of |F| and its real and imaginary parts. At low frequencies the flow of the fluid is of Poiseuille type; i.e., inertial effects are inconsequential relative to viscous effects. The purpose of the function F is to correct for the increased influence of inertial terms at higher frequencies. Biot estimates the range of validity of the Poiseuille regime to be 0 < < .

5.3.3

Calculation of the Coefficients in the Biot Model

The equations of motion (5.10) depend upon the parameters , R, Q, P11, p 12 , P22, and b, which need to be calculated from measured and estimated seabed parameters. We seek


line)ofF(

223

Figure 5.1. Real (dashed line), imaginary (dash-dotted line), and magnitude (solid ).

Table 5.1. Parameters in the Biot-Stoll model Symbol

Pf Pr

Kf Kr

n

k a a

Parameter Density of the pore fluid Density of sediment grains Complex frame bulk modulus Complex frame shear modulus Fluid bulk modulus Grain bulk modulus Porosity Viscosity of pore fluid Permeability Structure constant Pore size parameter

to determine these parameters in terms of the set of inputs used by Stoll [400], which are given in Table 5.1. Observe that the set contains , the Lame coefficient of the frame, and that the dissipation parameter b can be calculated in terms of a, pf, , n], and k using (5.11). Expressions for other parameters, , R, Q, p11, p12, and P22, remain to be calculated. To find , Q, and R we follow Stoll [400] and Biot and Willis [37] and consider two experiments. • The jacketed test: Consider a sediment sample contained in a flexible jacket. The sample is subjected to an external pressure p, and the fluid in the jacket is allowed

224


to drain out. Since the pore fluid pressure pf —> 0 we have from (5.7)5 that /e —> -Q/R and p'/e --->-Kb. • The unjacketed test: A sample of sediment is placed in a container and immersed in fluid. The fluid is placed under pressure p', whence pf — p'. In this situation

In both cases aggregate pressure in the three coordinate directions is


and thus from the constitutive equations (5.7),

The jacketed test then gives

The unjacketed test yields the system of equations

which are solved to obtain

Substituting these formulas into (5.12) and solving the system comprised of these two equations and (5.13) gives


225

where

The Lame coefficient of compression for the frame is = Kb — 2 /3. Hence the last term in (5.14)i introduces a correction for the presence of the pore space. Observe that it approaches 0 as B —> 0, Kb -> Kr. It remains to calculate the density parameters p11 pl2, and p22 In terms of the vector w = B(U — u), which tracks displacement of the fluid relative to the frame, the kinetic energy is

where p = (1 — B)p r + Bpf is the aggregate density of the medium and m is the effective mass for motion relative to the frame. Stoll wrote the latter parameter as

and referred to a as the structure constant. Stoll [400] stated that if the pores of the medium are oriented in the direction of the flow, then a — 1, while the value of — 3 at the other extreme corresponds to randomly oriented pores. Substituting U = w/B + u into (5.8) results in the kinetic energy function

Comparison with (5.16) yields the identities

226


Table 5.2. Grain and frame parameters for the Biot-Stoll model for five sediments. Sediment

FS CS CSFG SS SC

Pr

2670 2710 2680 2670 2680

k

Kr

4.0 5.6 4.0 3.8 3.5

x x x x x

10

10 1010 1010 1010 1010

0.43 0.38 0.30 0.65 0.68

-14

3.12 x 10 7.5 x 10-11 2.58 x 10-10 6.33 x 10 -15 5.2 x 10 -14

a

a

1.25 1.25 1.25 3.0 3.0

1.20x 6.28 x 1.31 x 4.25 x 1.24x

10 -6 10 -5 10 -4 10 -7 10 -6


In deriving the Biot model the frame was regarded as purely elastic. A realistic model must allow for attenuation due to intergranular friction. Biot [40], Stoll [400], and Stoll and Bryan [406] argued by appeal to viscoelastic models that this can be accomplished adequately by introducing frequency-independent imaginary components to the moduli Kb and . to obtain complex moduli that we will denote by and *. However, in [405] Stoll suggested that for fine, silty sediments a second loss mechanism, the "squeeze film" effect, which is frequency dependent, may be significant. Under what is termed the correspondence principle by Biot, the complex versions of the remaining moduli *, *, and Q* can be obtained by substituting and * into the relations (5.14).

5.3.4

Experimental Determination of the Biot-Stoll Inputs

In order to evaluate the Biot model we shall examine its predictions for the acoustic field for five different sediments encompassing the range of types that may be expected in a shallow ocean environment. The test suite, the parameters for which are shown in Table 5.2, consists of a coarse to medium sand (CS) sediment off of Daytona Beach, Florida and a silty sand (SS) sediment off of Corpus Christi, Texas, for which most of the parameters were taken from Beebe, McDaniel, and Rubano [26], and a coarse sand and fine gravel (CSFG) sediment, a fine sand (FS) sediment, and a silty clay (SC) sediment located at three different sites in the Gulf of La Spezia, Italy, with the parameters mostly taken from Holland and Brunson [245]. The parameters of Table 5.1 fall into three categories: pore fluid parameters pf, Kf, and n; grain parameters pr and Kr; and frame parameters Kb, , , k, a, and a. To illustrate the considerations involved in determining the Biot-Stoll parameters we shall examine the techniques used by the two sets of authors cited above. Further discussion and alternatives


227

Table 5.3. Fluid parameters to be used in the Biot model. All parameters are MKS (meters-kiloerams-seconds). Symbol Pf Kf V

Estimate 1000 2.4 x 109 1.01 x 10~3

can be found in Stoll [405], Turgut and Yamamoto [432], Chotiros [112], Cheng, Badiey, and Mu [299], and Hovem and Ingram [248]. The estimates were based upon core or grab samples taken at each site. From these the mineralogical nature of the grains, the porosity, and some measure of the statistical distribution of grain sizes were obtained. Pore Fluid Parameters

Fluid density, bulk modulus, and viscosity can be calculated from the temperature and salinity of the pore water. These apparently were not measured at the specific sites, but were based on generic estimates. Very similar values were obtained at all five sites. The values we shall use are given in Table 5.3. Grain Parameters

Grain density and bulk modulus were obtained from the literature once the mineralogical nature of the grains were determined from the sediment sample. For the silty clay sediment no estimates were found and the authors used generic soft sediment estimates. The values obtained by the two sets of authors as shown in Table 5.2 are similar for all five sediments. Chotiros [112] disputed using the handbook value of grain bulk modulus and argued for a significantly smaller value, Kr = 1 x 109Pa. We shall discuss this later. Frame Parameters

• Porosity was measured from sediment samples. • Permeability was inferred from the distribution of grain sizes obtained from the sediment sample. Holland and Brunson used the Kozeny-Carmen equation

where K = 5 is an empirical constant and S0 is the surface area per unit volume of the particles. The latter parameter was calculated as

228


from a discrete set (dn, wn) of grain sizes dn and proportions wn of total volume obtained by sorting the sample. Beebe et al. used a different empirical relation due to Krumbien and Monk, which depends upon the mean M and standard deviation of grain sizes in units

where d = 2-M

mm is the mean grain diameter.

• Shear and bulk moduli were calculated from generic parameters for the type of sediment. Holland and Brunson calculated the frame shear modulus from the empirical formula of Hardin and Richart for sands and clays,

in pounds per square inch. Here

is the voids ratio, and the mean effective stress due to over burden pressure is

where z is the depth in the seabed, K0 is the coefficient of earth pressure at rest, which is typically taken to be 0.5, and g is the acceleration due to gravity. The complex moduli were then calculated from

where and are the log decrements for compressional and shear vibrations and Rp is the Poisson ratio. Since Holland and Brunson were interested in wave velocity and attenuation near the sediment surface, they used a depth of z = 0.1m in (5.19). An alternative to (5.18) due to Bryan and Stoll [64], [405] is to assume a functional form for the shear modulus

where pa is the atmospheric pressure. Based on statistical regressions on laboratory results, Stoll arrived at the values a = 2526, b = -1.504, and n = 0.448. As field tests tend to lead to somewhat higher values for the shear modulus than laboratory results, Stoll suggested an empirical modification = FF • 1, where FF — 2. Beebe et al. used the formulas

5.4. Solution of the Time-Harmonic Biot Equations

229

of Stoll and Bryan [406]. Here VE and Vs are the uniaxial velocities for compressional and shear vibrations, respectively, and e and AS are the log decrements for the two types of vibrations. Thus both sets of authors use frequency-independent choices for the complex moduli, which means as mentioned earlier that they take into account intergranular friction, but not the squeeze film effect. • The pore size parameter is generally agreed to be proportional to . It was calculated by Holland and Brunson from an empirical relation of Hovem and Ingram [248]

Beebe et al., following Stoll, used the empirical relation a = • Both sets of authors follow Stoll in assigning a value of — 1.25 for "clean" sands and a — 3 for silty sediments. This parameter seems to be of little influence when conventional estimates of the grain bulk modulus Kr such as those in Table 5.2 are used, but is more significant if the lower value of this modulus suggested by Chotiros is employed. Table 5.2 gives estimates for some of the grain and frame parameters of the BiotStoll model for the five sediments. The values shown were taken from the articles cited above with the exception that the pore size parameter was calculated from (5.23) for all five sediments. The calculation of the moduli and * will be discussed in section 5.4.2.

5.4 5.4.1

Solution of the Time-Harmonic Biot Equations Simplification of the Equations

For a medium that is assumed to oscillate harmonically in time, we have u(x, y, z, t) = u(x, y, z ) e l w t , U(x, y, z, t) = U(x, y, z ) e ' w t , . . . . Substituting these representations into the constant parameter equations (5.10) gives

where

Taking the divergence and curl of both equations in (5.24) yields the system

230


where := x u and := x U. In the first two equations of (5.26) we make the change of dependent variables

the inverse transformation for which is

where

with

Note that for the transformation given by (5.27) to be nonsingular it must be the case that d 0, . From (5.14)

where D is given by (5.15). Since Kr > Kb, Kf implies that D > Kr > Kb, it follows that 0 0. The sign of Re kp± may be either positive or negative. If we write

where v = In 10/(40 ) and the attenuation coefficients are measured in decibels per wavelength, then the compressional wave speeds and attenuation coefficients for the two waves are given by

A similar analysis of equation (5.32) gives


233

Table 5.4. Bulk moduli of five sediments. Sediment

Stoll-Bryan

Bryan-Stoll (z = 1m)

FS CS

4.8 x 107 + 6.7 x 105i

CSFG

5.9 x 107 + 8.2 x 105i 2.9 x 107 + 1.3 x 106i 2.6 x 107 + 1.2 x 106i

3.9 5.1 7.1 6.0

ss sc

5.3 x 10 7 +7.4 x 105i

x x x x

107+ 107 + 107 + 106 +

1.9 x 2.4 x 3.4 x 6.6 x

106i 106i 106i 105i

Chotiros 4.0 x 10 9 +1.9x 108i 7.9 x 109 + 2.5 x 108i 7.6 x 109 + 3.6 x 108i

3.9 x 106 + 6.2 x l05i

Table 5.5. Shear moduli of five sediments. Sediment

FS CS CSFG SS SC

Stoll-Bryan 6.7 x 7.4 x 8.3 x 2.9 x 3.4 x

7

10 +4.3 x 10 7 +4.7 x 10 7 + 5.3 x 107 + 1.3 x 107 + 7.2x

Bryan-Stoll (z = 1m) 6

10 i 106i 106i 106i 106i

2.4 3.1 4.3 3.6 2.3

x x x x x

107 + 107 + 107 + 106 + 106 +

1.1 x 1.8x 2.0x 5.7 x 3.7 x

106i 106i 106i 105i 105i

for the speed and attenuation of shear waves through the frame. The compressional waves corresponding to the wavenumbers kp+ and kp- are sometimes referred to as Type I and Type II compressional waves. As we shall see, Type I waves correspond in magnitude to the compressional waves of the elastic model. Type II waves, which are slower and more strongly attenuated, do not occur in the elastic model. In section 5.3.4 two approaches to estimating the complex moduli and * were described. One possibility is to use the Stoll-Bryan formulas (5.22). When using these formulas we follow [26] in taking the rod velocity to be VE — 300m/s, the shear velocity to be Vs — 210m/s, and the log decrements to be — 0.45 and AS = 0.6 for the two silty sediments and = 0.15 and AS = 0.2 for the other three sand sediments. The precise values of VE and Vs are not important as the values of the moduli are insensitive to them in the range around the stated values. Another way to calculate the moduli is to use formula (5.18) or (5.21) in conjunction with (5.20). In formula (5.20) we will use a Poisson ratio of Rp — 0.25, and following Stoll and Kan [407] we will use log decrements of = =0.15 for the three sand sediments and = = 0.5 for the two silty sediments. For uncon soli dated sediments the moduli are not very sensitive to the precise choice of Rp unless it is very near the theoretical upper limit of 0.5. It should be noted, however, that there is not a consensus on the values of the log decrement, with some authors such as Turgut and Yamamoto [432] and, later, Stoll in [405] using lower values of , and in the 0.05-0.1 range. Tables 5.4 and 5.5 give the computed complex bulk and shear moduli computed according to the Stoll-Bryan formula (5.22) and the Bryan-Stoll formula (5.21) at depth of 1 m into the sediment for each of the sediments of Table 5.2. Also shown in Table 5.4 are the substantially higher values of the bulk modulus when computed

234


Figure 5.2. Speeds of Type I compressional waves for five sediments. Coarse sand and fine gravel (CSFG), coarse sand (CS), fine sand (FS), silty sand (SS), and silty clay (SC). The Stoll-Bryan formulas were used to calculate the moduli and

according to the scheme of Chotiros. This will be discussed later. In the elastic model, compressional and shear waves speeds are independent of frequency. As Figures 5.2, 5.3, and 5.4 show, this is not the case for the Biot model, which predicts an increase in wave speed with increasing frequency for all three types of waves. This is most pronounced for coarse sediments; for finer sediments the transition to higher wave speeds occurs above the frequency range used in most ocean acoustic experiments. For Figures 5.2-5.4 the complex moduli and were calculated using the Stoll-Bryan formulas (5.22) with the parameter values given above. In the elastic model, attenuation is proportional to the first power of frequency when measured in dB/m and independent of frequency when measured in dB/wavelength. Figure 5.5 shows that for the coarse sand and fine gravel sediment of Table 5.2 the dependence of attenuation of Type I compressional waves on frequency predicted by the Biot model is more complicated, increasing more rapidly than fl at low frequencies, but less rapidly than fl at high frequencies. This is typical of coarse sediments. Figures 5.6 and 5.7 show that for finer sediments the Biot model predicts attenuation proportional to f1 in the l0Hz-l0kHz range, which is typically used for in situ measurements. The dashed line in these figures indicates the attenuation predicted by the elastic model using the attenuation predicted by the Biot model at 400kHz, a frequency that is typical for laboratory measurements, for the coarse sand and fine gravel, fine sand, and silty clay sediments of Table 5.2. As can be seen, the losses predicted by the two models at lower frequencies would be substantially different. Figures 5.8-5.10 show the attenuations of all three types of waves for the five sediments of Table 5.2 when rendered in terms of dB/wavelength. Observe that the peak attenuations occur in the frequency interval in which the velocity curve is inflecting. Table 5.6 summarizes some of the data on measured wave speeds and attenuations given in [26] and [245]. The compressional wave speeds are reasonably close to or fall within the range of values predicted by the Biot model, but the superficial values of shear


235

Figure 5.3. Speeds of Type II compressional waves for five sediments. Coarse sand and fine gravel (CSFG), coarse sand (CS), fine sand (FS), silty sand (SS), and silty clay (SC). The Stoll-Bryan formulas were used to calculate the moduli and

Figure 5.4. Speeds of shear waves for five sediments. Coarse sand and fine gravel (CSFG), coarse sand (CS), fine sand (FS), silty sand (SS), and silty clay (SC). The Stoll-Bryan formulas were used to calculate the moduli and

wave speed of Akal cited in [245] are much lower than those predicted by the Biot model when the moduli are computed from the Stoll-Bryan formulas (5.22). The compressional attenuations for the fine sand and silty clay sediments are close to the measured values at 400kHz, the frequency used for the laboratory measurements of core samples from the two sites. However, as Figures 5.6 and 5.7 indicate, the predictions based on the data at 400kHz would disagree substantially at lower frequencies even though both predict attenuation proportional to f1 at lower frequencies.

236


Figure 5.5. Biot model's prediction for compressional wave attenuation for coarse sand and fine gravel using the Stoll-Bryan formulas to calculate the moduli Kb and .

Figure 5.6. Biot model's prediction for compressional wave attenuation for fine sand using the Stoll-Bryan formulas to calculate the moduli Kb and

We now consider the predictions for wave speed and attenuation when formulas (5.20) and (5.21) are used. We start with the quasi-static case in which the derivative terms in (5.34) and (5.35) are neglected so that formula (5.36) remains applicable. As mentioned in section 5.3.4 Stoll advocated computing the real part of the shear modulus as = FF • , where is computed from (5.21) and the field factor is FF = 2. Figures 5.11 and 5.12 show the predictions of the quasi-static equations for the fine sand and silty clay sediments of Table 5.2 in the first two meters below the sediment surface when FF = 1 and FF = 2. Also shown are some measured shear speeds taken at these sites, which were extracted from Richardson et al. [356, Fig. 3]. The range of frequencies used in these measurements was


237

Figure 5.1. Biot model's prediction for compressional wave attenuation for silty clay using the Stoll-Bryan formulas to calculate the moduli Kb and .

Figure 5.8. Biot model's prediction for compressional wave attenuation for coarse sand and fine gravel using the Stoll-Bryan formulas to calculate the moduli and

200-2000Hz, but the Biot model predictions for shear speed vary only slightly over this range. As can be seen, the choice FF = 1 better fits the measurements. Hence we shall simply use equation (5.21) to compute the real part of the shear modulus and disregard the field factor. Figures 5.13 and 5.14 show the predictions at frequency 320Hz for the speed of Type I compressional and shear waves for the five sediments of Table 5.2 as functions of depth when the real part of the shear modulus was computed using (5.20) and (5.21). Figure 5.15 shows that the Stoll-Bryan formula (5.22) and the Bryan-Stoll formula (5.21) at a depth of 1 m make very similar predictions for the speed of Type I compressional

238


Figure 5.9. Biot model's prediction for Type II wave attenuation for coarse sand and fine gravel using the Stoll-Bryan formulas to calculate the moduli and

Figure 5.10. Biot model's prediction for shear wave attenuation for coarse sand and fine gravel using the Stoll-Bryan formulas to calculate the moduli and

waves, but Figure 5.16 indicates that there is a discrepancy in the predicted shear wave speeds. However, as can be seen in Table 5.7, this difference is not great for the three sand sediments of Table 5.2. The Stoll-Bryan formula's prediction for shear wave speeds is about that of the Bryan-Stoll formula at a depth of ten meters or less, and thus within the confines of a constant parameter model, either set of parameters can be regarded as representative of the sediment layer unless it is very thin. On the other hand, the predictions for compressional and shear wave speeds of the Stoll-Bryan formula for the two silty sediments is that of the Bryan-Stoll formula at a depth of over a hundred meters, which generally would be greater than the width of the layer. Thus the disparity here may be more significant.


239

Table 5.6. Measured Type I compressional and shear wave speeds and attenuation of compressional waves. (a) range of majority of core measurements at 400kHz, from Holland and Brunson; (b) from Beebe, McDaniel, and Rubano; (c) superficial value due to Akal, cited in Holland and Brunson; (d) Biot model prediction at a depth of 0.1 m and a frequency of 400kHz; (e) Biot model prediction at a depth of 0.1m and a frequency of l0Hz. Sediment Fine sand Coarse sand Coarse sand, fine gravel Silty sand Silty clay

cp 1650-1700a 1720b 1900C 1490b 1480-1520a

Attenuation (dB/m) 125-200a ,209d

Cs

70c , 66e

90C, 83e 30-100a, 123d

30C,23e

Figure 5.11. Predictions of shear speed of the Biot model at 320Hz for fine sand along with measured data extracted from Richardson et al. The solid and dashed lines are for field factors of F F — 1 and F F = 2, respectively.

Table 5.7. Compressional and shear waves predicted by the Stoll-Bryan formula at 320Hz, and the depth in meters at which the Bryan-Stoll formula predicts the same speed. Sediment Coarse sand, fine gravel Coarse sand Fine sand Silty sand Silty clay

S-B cp 1890 1742 1647 1518 1507

B-S depth | S-Bcs 1 202 2 193 3 186 110 155 159 240

B-S depth 4 7 10 180 520

240


Figure 5.12. Predictions of shear speed of the Biot model at 320Hz for silty clay along with measured data extracted from Richardson et al. The solid and dashed lines are for field factors of F F = 1 and F F = 2, respectively.

Figure 5.13. Speeds of Type I (top) and shear waves (bottom) as a function of depth as predicted by the quasi-static Blot model for fine sand (FS), coarse sand (CS), and coarse sand and fine gravel (CSFG).

Figure 5.17 shows little difference between the two predictions for the attenuation of Type I waves for the three sand sediments, but Figure 5.18 indicates a more substantial difference for the two silty sediments. The data in Hamilton [236, Fig. 18] indicates a compressional coefficient Ap (see (5.6)) in the range 0.05 —> 0.3 for sediments with a porosity of around 0.65. This corresponds to a range for p of 0.07 —» 0.45 for a wave speed around 1500 m/s and the Stoll-Bryan formula's prediction falls in the lower end of this range, while formulas (5.20) predict a value p = 0.02 well below it. Thus we have


241

Figure 5.14. Speeds of Type I (top) and shear waves (bottom) as a function of depth at 320Hz as predicted by the quasi-static Biot model for silty sand (SS) and silty clay (SC).

Figure 5.15. Type I compressional wave speeds as predicted by the Stoll-Bryan formula (solid line) and the Bryan-Stoll formula (dashed line) at a depth of 1 m below the sediment surface for coarse sand and fine gravel (CSFG), coarse sand (CS), fine sand (FS), silty sand (SS), and silty clay (SC).

the dilemma that the Bryan-Stoll formula makes better predictions for compressional and shear wave speeds for silty sediments, whereas the Stoll-Bryan formula's predictions for compressional wave attenuation seem more in accord with the literature. There is other evidence to support the Biot model's predictions. For Type I compressional waves a frequency dependence conforming reasonably with the Biot model predictions has been confirmed both in laboratory measurement for a porous structure composed

242


Figure 5.16. Shear wave speeds as predicted by the Stoll-Bryan formula (solid line) and the Bryan-Stoll formula (dashed line) at a depth of 1 m below the sediment surface for coarse sand and fine gravel (CSFG), coarse sand (CS), fine sand (FS), silty sand (SS), and silty clay (SC).

Figure 5.17. Type I compresional wave attenuation as predicted by the Stoll-Bryan formula (solid line) and the Bryan-Stoll formula (dash-dotted line) at a depth of 1 m below the sediment surface for coarse sand and fine gravel (CSFG), coarse sand (CS), and fine sand (FS).

of glass beads (Hovem and Ingram [248]) and in situ (Turgut and Yamamoto [432]). The latter authors also adduce evidence for dispersion of compressional attenuation with respect to frequency. The laboratory measurements of attenuation of shear waves given in Brunson and Johnson [63] are also in accord with the Biot model's predictions. On the other hand, in his 1980 survey article [236] Hamilton found that the evidence from in situ studies at


243

Figure 5.18. Type I compresional wave attenuation as predicted by the Stoll-Bryan formula (solid line) and Bryan-Stoll formula (dash-dotted line) at a depth of 1 m below the sediment surface for silty sand (SS) and silty clay (SC).

Figure 5.19. Comparison of predicted Type I wave speed for coarse sand and fine gravel as a function of depth at 10Hz between the variable coefficient equations (solid line) and the quasi-static equations (dash-dotted line).

high frequencies confirmed the elastic model's prediction of a variation of loss, measured in dB/m, that depends linearly on frequency (see formula (5.6)). As indicated earlier, for coarse sediments the Biot model predicts a dependence on frequency that is greater than f1 at low frequencies and less than f1 at high frequencies. However, Hamilton's doubts were less about the accuracy of the Biot model than about whether most naturally occurring sediments are permeable enough for the two models to yield significantly different predictions.

244


Figure 5.20. Comparison of predicted Type I wave speed for fine sand as a function of depth at 10Hz between the variable coefficient equations (solid line) and the quasi-static equations (dash-dotted line).

Figure 5.21. Comparison of predicted Type I wave speed for silty clay as a function of depth at 10Hz between the variable coefficient equations (solid line) and the quasi-static equations (dash-dotted line).

As mentioned above, Figures 5.13 and 5.14 were generated using the quasi-static version of the coefficient formulas (5.34) and (5.35). When the derivative terms in these formulas are not neglected, seeking solutions of the form (x) = c 1 e - i k x , (x) = c 2 e - i k x , and uz(x) = c3e-ikx to equation (5.33) results in a cubic equation in k2. This equation may be solved, but associating the three roots with the three types of waves is problematic since Type II and shear waves have similar velocities. The quasi-static case suggests that at low frequencies Type II waves have the lowest velocities of the three types, while Type I waves


245

Figure 5.22. Comparison of predicted shear wave speed for coarse sand and fine gravel as a function of depth at 10Hz between the variable coefficient equations (solid line) and the quasi-static equations (dash-dotted line).

have the highest speeds. Based upon this ansatz, Figures 5.19-5.21 compare the predictions for Type I waves as a function of depth at a frequency of 10Hz for the coarse sand and fine gravel, fine sand, and silty clay sediments. As can be seen, the variable coefficient formulas predict very low speeds for Type I waves near the sediment surface with the disparity being greater for coarser sediments. On the other hand, Figure 5.22 shows that the predictions for shear speed as a function of depth differs little between the quasi-static and the variable coefficient cases for coarse sand and fine gravel. This is also the case for the other two sediments. Figures 5.23, 5.24 and 5.25 show that at a depth of 1m below the sediment surface, the predictions for the speed of Type I waves are in disagreement below 200Hz, while the predictions for Type II and shear wave speeds are in agreement down to about l0Hz. We know of no experimental data that confirms the low velocities of Type I waves predicted by the variable coefficient model, however, most measurements encountered in the literature that are associated with a specific frequency are at a frequency of a kilohertz or higher. Moreover, Burridge and Keller [91], using the mathematical technique of twoscale homogenization, were able to derive the Biot equations from the microstructure in the case of constant seabed parameters. However, they found that the variable coefficient equations differed from the Biot equations. Thus there are also theoretical grounds to doubt the predictions of the variable coefficient model. Observe from Figures 5.26 and 5.27 that Type II waves are highly attenuated with respect to range and thus cannot be detected at distances far from the source. Stoll [401] estimated the energy loss due to conversion to Type II waves at the ocean-sediment interface to be around 1 % at frequencies in the kilohertz range, but smaller at lower frequencies. The high attenuation of Type II waves makes them inherently difficult to detect, however Plona [343] cites possible evidence for their existence. Also, Chotiros has attempted to categorize waves that have been detected experimentally near the sediment surface at short source-toreceiver distances and shallow grazing angles with speeds in the 1000-1300m/s range as

246


Figure 5.23. Comparison of the variable coefficient (solid line) and quasi-static (dash-dotted line) predictions for Type I wave speeds at a depth of 1 m into the sediment for coarse sand and fine gravel (CSFG), coarse sand (CS), fine sand (FS), and silty clay (SC).

Figure 5.24. Comparison of the variable coefficient (solid line) and quasi-static (dash-dotted line) predictions for Type II wave speeds at a depth of 1 m into the sediment for coarse sand and fine gravel (CSFG), coarse sand (CS), fine sand (FS), and silty clay (SC).

Type II waves. In order to get the Biot model to predict Type II waves of this velocity, he suggests a value, Kr = 7 x 109Pa, for the grain bulk modulus that is significantly lower than the handbook values used in Table 5.2. Use of this value in turn makes the Biot model far more sensitive to the value of the structure constant a (for the values given in Table 5.2, this parameter has little influence). Chotiros [112] advocated a value for the structure constant in the range 1.7-1.9. The use of the lower value of Kr substantially reduces the predicted


247

Figure 5.25. Comparison of the variable coefficient (solid line) and quasi-static (dash-dotted line) predictions for shear wave speeds at a depth of 1 m into the sediment for coarse sand and fine gravel (CSFG), coarse sand (CS), fine sand (FS), and silty clay (SC).

Type I wave speeds. To deal with this, Chotiros simply adjusted the real part of the bulk modulus K to produce some target value. Figures 5.28-5.31 show the result of modifying the Biot parameters of the three sand sediments of Table 5.2 as just described. The target values for Type I wave speed were those of Table 5.6 at a frequency of 100Hz. The structure constant was taken to be a = 1.8, and log decrements of Kb = 0.15 were used in each case. The shear moduli were calculated from the Stoll-Bryan formulas. As can be seen in Figure 5.28 this procedure does produce Type II wave speeds in the desired range at the frequencies, 5-60kHz, used in the experiments. Note also from Figures 5.28 and 5.30 that the dispersion of Type I speed and attenuation with respect to frequency is less than that predicted with the conventional estimates, especially for the coarse sand sediment. As Figure 5.31 indicates, Type II waves remain highly attenuated at low frequencies.

5.4.3

Solution of the Differential Equations for a Poroelastic Layer

For computing acoustic pressure in a wave guide, it is appropriate to work in cylindrical coordinates and suppress the dependence upon the angular variable whence the displacement vectors are now denoted as u(r, z) = (ur(r, z), u z ( r ,z ) ) , U(r, z) — (Ur(r, z), Uz(r, z)). In this situation the relevant constitutive equations (5.7) and strain-displacement relations (5.2) are

and

248


Figure 5.26. Attenuation of Type II waves as predicted by the Stoll-Bryan formulas for coarse sand and fine gravel (top), coarse sand (middle), and fine sand (bottom).

Figure 5.27. Attenuation of Type II waves as predicted by the Stoll-Bryan formulas for silty sand (top) and silty clay (bottom).


249

Figure 5.28. Type I wave speeds as a function of frequency for three sand sediments when subject to the modifications suggested by Chotiros. CSFG: coarse sand and fine gravel; CS: coarse sand; FS: fine sand.

Figure 5.29. Type II wave speeds as a function of frequency for three sand sediments when subject to the modifications suggested by Chotiros. CSFG: coarse sand and fine gravel; CS: coarse sand; FS: fine sand.

respectively. In (5.26) the angularly independent Laplacian is

We introduce a reference wavenumber k0 = w / C 0 , where C0 is a representative sound speed in the ocean. Since the dependence on r of separated solutions is well known, we

250


Figure 5.30. Attenuation of Type I waves as a Junction of frequency for three sand sediments when subject to the modifications suggested by Chotiros. CSFG: coarse sand and fine gravel; CS: coarse sand; FS: fine sand.

Figure 5.31. Attenuation of Type II waves as a function of frequency for three sand sediments when subject to the modifications suggested by Chotiros. CSFG: coarse sand and fine gravel; CS: coarse sand; FS: fine sand.

seek solutions of the form

where H , j = 0, 1, are outgoing Hankel functions and Im (k)

0 is required for solutions


251

to approach zero as r —> . The presence of the Hankel functions requires the use of the time-harmonic factor e-iwt rather than the factor eiwt used in connection with (5.24). This can be accomplished by using the complex conjugates of the various coefficients used in section 5.4. Substituting the assumed forms into the system (5.24) gives

where

The general solution to this system is

where zd is the depth of the layer surface and

with the branch cut for the square root function is chosen so that Im(m±) 0. The frame and fluid dilatations now can be computed from (5.28). From (5.31) and (5.40) the depth factor for vertical displacement of the sediment frame now can be obtained by solving the differential equation

where

The solution is

252


From (5.30) vertical displacement of the pore fluid is given by

Finally, the definitions of the dilatations

and (5.40) yield solutions for the depth factors for radial displacement

Similar solutions can be obtained for the elastic model. In the elastic case, from (5.3) and (5.4)1, the formulas corresponding to (5.41) are

where

The dilatation and vertical displacement are then

5.5

Representation of Acoustic Pressure

In this section we present appropriate interface conditions and then derive a representation for pressure in an ocean over a poroelastic sediment.

5.5. Representation of Acoustic Pressure

5.5.1

253

Differential Equations for Pressure and Vertical Displacement in the Ocean

In the case of an ocean in which the water density p0 is constant and the sound speed profile c(z) depends only upon depth z, the differential equations for acoustic pressure P0(r, z) and vertical displacement UZ()(r, z) arising from a time-harmonic point source of angular frequency w located at a depth z = Z0 are given by

In the first equation k0 = w/C0 is the reference wavenumber corresponding to some representative sound speed c0. The refractive index n(z) is given by C 0 / C ( Z ) .

5.5.2

Interface Conditions

We assume that the ocean is of constant depth z = Zd and lies over a multilayer seabed and seek the interface conditions between two successive layers. In the upper layer we denote the dilatation, vertical displacement and so forth by eu(r, z), u z u (r, z),..., while in the lower layer the corresponding quantities are denoted e l (r, z), u z l (r, z), At the surface of the ocean a pressure-release condition is imposed p 0 (r, 0) = 0.

The conditions used at the ocean-sediment boundary are particularizations of those between two sediment layers, hence we treat this case first. At an interface z = Zb between two poroelastic layers, continuity is required for vertical displacement ws, aggregate normal stress zz + , pore fluid pressure / , specific flux

tangential stress

rz,

and radial displacement ur. This gives

Condition (5.45)4 was obtained from the continuity of vertical frame displacement and flux conditions. At the interface between the ocean and a poroelastic sediment we use (5.45)4

254


with = 1 on the ocean side, equate both sediment side normal stress and pore stress to P0, and set tangential stress to zero on the ocean side. This gives

For a poroelastic layer over an elastic layer, continuity is required for both skeletal and fluid vertical displacement, aggregate normal stress, tangential stress, and radial displacement. This gives

For an interface between two elastic layers we require continuity of vertical displacement u z ,normal stress zz, tangential stress rz, and radial displacement ur. At the boundary z = zb between the two elastic layers this gives

At the interface z — zd between the ocean and an elastic sediment, the radial displacement condition is dropped and tangential stress is set to zero on the ocean side. This gives

The bottom-most, substrate layer will be treated as a half-space. We require as asymptotic conditions that all dilatations and displacements vanish as z —> . Thus, if the top of the substrate layer is a distance zb from the ocean surface and it is poroelastic, the solutions (5.42) have the form

5.5. Representation of Acoustic Pressure

255

Similarly, for an elastic substrate we have from (5.44)

5.5.3

Green's Function Representation of Acoustic Pressure

We now construct a modal representation for acoustic pressure in the ocean, following the formulation presented in Boyles [52]. The process is similar irrespective of which model, elastic or poroelastic, is used for the sediment layers. We illustrate with the case of an ocean over a poroelastic half-space. More detail can be found in Buchanan, Gilbert, and Xu [84]. A Green's function representation

is sought. The contour C0 must enclose all singularities of G2 and exclude those of H - We choose it to be the slit cut enclosing the positive real axis, oriented counterclockwise. The depth Green's function G2 then satisfies the differential equation

where a 0 ( z ,

K)

:=

k0

, with the interface conditions

at the ocean surface and

at the source depth. Let (z, K) and (2, K) be solutions to

satisfying the conditions

256

The functions and sound speed C0,


are entire functions of K.. In the case of an isovelocity ocean with

with a ( ) (k) = k0 • When the sound speed varies with depth, the functions and will in general have to calculated by numerical solution of the initial value problem (5.52), (5.53). The general solution that satisfies the surface and source interface conditions is

where H is the unit step function, and the Wronskian of the solutions

is nonvanishing and entire in K. If the sediment is modelled as a single half-space layer, then denoting the three constants occurring in the solutions (5.47) as C2, C3, C4, we have upon substituting G2, , ,uz, and Uz into the ocean-sediment interface conditions a matrix equation

for the determination of the constants C1, C2, C3, C4. Since and are entire functions of K, the singularities of G2 are those of the constant C1, which from (5.55) has the representation

where 0 := det(M) and I is the numerator determinant for C\ in Cramer's rule. The singularities are thus the zeros of 0 (the eigenvalues of the problem) and any nontrivial branch cuts. These are the m± and au branch cuts. To compute the contour integral in (5.48) we introduce counterclockwise slits Cm± and Cau about the three branch cuts. The depth Green's function is now analytic outside of the contour Cm+ + Cm_ + Cau except at the eigenvalues {K n ,}. Computing the residues at the eigenvalues gives the representation for pressure:

5.6. Sound Transmission over a Poroelastic Half-Space

257

When the sediment has multiple layers, then in all layers except the bottom one the solutions (5.42) can be written as

and so forth, and thus the solutions are entire in K. Therefore, the integrals in (5.56) need only be computed for the bottom, half-space, layer. It should be noted that while the solutions , , and uz can in theory be calculated numerically, thereby permitting consideration of sediments with depth-varying coefficients, this is not practical because the two solutions corresponding to the two m+ terms in (5.47) undergo enormous variations in magnitude over even a small change in depth, making accurate determination of their values difficult. A more feasible approach for the case of depth-varying parameters is the numerical technique parabolic approximation. This method is discussed in Buchanan and Gilbert [78], but we shall not pursue it here.

5.6

Sound Transmission over a Poroelastic Half-Space

In section 5.4.2 we considered the effect of the parameters on the speeds and attenuations of the different types of waves predicted by the Biot model. Investigating such effects has been the primary concern of many of the articles cited. Perhaps more important but less well studied is the ability of the model to predict acoustic pressure. Of the articles cited, only [26] considers the Biot model's predictions of acoustic pressure, and the model used there was not an implementation of the full set of Biot equations. As seen in the previous section, it is possible to completely solve these equations in the case of constant seabed parameters. The predictions made by these solutions will be the concern of this and the next section. Let us first consider the predictions of (5.56) for an isovelocity ocean over a poroelastic half-space. We shall use transmission loss as a measure of the strength of the acoustic field P 0 (r, z, Z 0 ). Transmission loss, normalized to be zero decibels 1m from a point source, is

From (5.56) the total acoustic field consists of a discrete and a continuous spectrum term. The eigenvalues { K n } in the discrete term are found numerically by minimizing | O(k)|, since this is found to be more reliable than solving 0(k) = 0 numerically. The derivatives (kn) in the discrete spectrum term and the integrals along the branch cuts in the continuous spectrum term are computed numerically. At very low frequencies there are no eigenvalues, and hence all the energy is in the continuous spectrum. The frequency at which the first eigenvalue emerges is the modal cutoff. As the frequency increases, more and more of the energy in the field shifts to the discrete spectrum. The mathematical mechanism for this shift is the emergence of eigenvalues from the branch cuts at certain frequencies. Having emerged from the branch cut the eigenvalues migrate toward the point K = 1 (Figure 5.32 (top)). The clustering of the eigenvalues near K = 1 means that locating

258


Figure 5.32. Eigenvalue map (top) and transmission loss as a function of range (bottom) at 200Hz for coarse sand and fine gravel. The ocean depth was 50m, the source and receiver depth were 25m.

Table 5.8. Location of the three branch cuts at l00Hz. Sediment Fine sand Coarse sand Coarse sand, fine gravel Silty sand Silty clay

m_ 0.83 0.76 0.65 0.98 0.99

+ 0.00098i + 0.01 7i + 0.030i + 0.003 \i + 0.0030i

m+ -39086 + 864952i 68.4 + 33 li 89.2 + 96.2i -944368 + 6839930i - 125238 + 90691 9i

au 65.2 + 4. 1i 61.9 + 5.3* 56.8 + 5.6i 83.5 + 15.9i 88.3+16.9i

all of the eigenvalues in this vicinity will become increasingly difficult with increasing frequency. However, examination of the spacing between eigenvalues leads us to believe that determination of all eigenvalues appears feasible for frequencies up to at least 1kHz. Table 5.8 shows the locations of the tips of the branch cuts at l00Hz for the five sediments whose parameters are given in Table 5.2. The branch cuts run leftward from these points. Observe that only the m_ branch cut lies near the interval 0 K . The presence of the Hankel functions in (5.56) causes the contribution of eigenvalues and branch cuts distant from the positive real k-axis to be quite small. Hence the contribution of eigenvalues emerging from the m+ and au branch cuts as well as the line integrals about them can be neglected for most purposes. Observe also that the tip of the m_ branch cut lies farther from K = 1 for the coarser sediments. Eigenvalues are observed to emerge at lower frequencies for the coarser sediments. Figures 5.33-5.37 show transmission loss at l00z from a distance of 1m from the source outward to 500m for the five sediments of Table 5.8. In these computations only the contribution of the integral along the m_ branch cut is taken into account (cf. formula (5.56)). The integrands along this branch cut are highly oscillatory and become more so


259

Figure 5.33. Total loss as a function of range for coarse sand and fine gravel. The dashed line is the loss predicted by using only the discrete spectrum term. The frequency is l00Hz, the ocean depth is 50m, the source and receiver depth are both 25m.

Figure 5.34. Total loss as a function of range for coarse sand. The dashed line is the loss predicted by using only the discrete spectrum term. The frequency is 100Hz, the ocean depth is 50m, the source and receiver depth are both 25m.

with increasing frequency and range. Numerical experimentation indicated that accurate computation of the contribution of the integrals along the other two branch cuts, which are distant from the real k-axis, was probably not possible. However, it was found in all cases that the estimated total loss at a distance 1m from the source was near the expected value of zero, leading us to believe that these integrals are of little practical consequence. Also shown in these figures is transmission loss if only the discrete spectrum is used. As can be seen, the contribution to the loss of the continuous spectrum is smallest for the

260


Figure 5.35. Total loss as a function of range for fine sand. The dashed line is the loss predicted by using only the discrete spectrum term. The frequency is l00Hz, the ocean depth is 50m, the source and receiver depth are both 25m.

Figure 5.36. Total loss as a function of range for silty sand. The dashed line is the loss predicted by using only the discrete spectrum term. The frequency is l00Hz, the ocean depth is 50m, the source and receiver depth are both 25m.

coarsest of the sediments, fine sand and coarse gravel, and greatest for high porosity, low permeability sediments, silty sand and silty clay. It is of interest to compare the predictions of the Biot model with those of the elastic model for the same sediment. For the elastic case in the formula (5.56), there are two rather than three branch cuts in the integral term, but the representation is otherwise the same. Figure 5.38 shows the eigenvalue map and transmission loss curve for the coarse sand and fine gravel sediment with the elastic parameters of Table 5.9 (discussed below). In the


261

Figure 5.37. Total loss as a function of range for silty clay. The dashed line is the loss predicted by using only the discrete spectrum term. The frequency is l00Hz, the ocean depth is 50m, the source and receiver depth are both 25m.

Figure 5.38. Eigenvalue map (top) and transmission loss as a function of range (bottom) at 200Hz for coarse sand and fine gravel modelled as an elastic sediment. The ocean depth was 50m, the source and receiver depth were 25m.

elastic model the a branch cut typically lies above and near the real k-axis in a position similar to that of the m_ branch cut in the poroelastic model. Figures 5.32 and 5.38 indicate that transmission loss is often a complicated function of range. For most purposes a detailed picture of the fluctuations is not needed. In such cases, following [249], we will use the smoother curves obtained by making a least square

262


Table 5.9. (a) measured value from Holland and Brunson; (b) measured value from Beebe, McDaniel, and Rubano; (c) Estimate using the Biot model at a frequency of 320Hz; (d)from Hamilton; (e) from Beebe and McDaniel. Sediment Fine sand Coarse sand Coarse sand, fine gravel Silty sand Silty clay

P

Cs

Cp a

1850 2060b 2180a 1500b 1540a

a

1675 1720b 1900a 1490b 1500a

c

185 195C 200C 165c 160C

P

0.4,1.0" 0.4, 1.0d 0.4, 1.0d 0.05,0.3d 0.05,0.3d

s 2.4d 2.5d 2.6d 2.8" 2.7"

fit of the detailed data to a curve of the form

The result of such a smoothing is shown by the dashed line in Figures 5.32 and 5.38 (bottom). Table 5.9 shows the sets of elastic parameters that we shall use for the seabeds of Table 5.2. Comparison of the two models is problematic, since as indicated earlier the Biot model is dispersive in frequency with respect to both wave speed and attenuation. The articles [245] and [26] cited above contain measured Type I compressional wave speeds, which we shall use as the compressional wave speed for the elastic model. Sediment densities are either given in these articles or inferable from porosity and grain and water density. For consistency we shall use the shear speeds predicted by the Stoll-Bryan formulas (5.22) for the Biot model as the shear wave speeds in the elastic model for all five sediments. However, as noted in Table 5.6, the measurements of superficial shear wave speed of Akal cited in [245] are somewhat lower, with the predictions of the Bryan-Stoll formulas (5.21) being in better accord. For the attenuation of compressional waves we will use the two values shown in Table 5.9. These are based on Hamilton [236, Fig. 18], which plots attenuation versus porosity, and are chosen to encompass the range of attenuations of most of the plotted samples at the porosity of the particular sediment. For the attenuation of shear waves we used a value of As — 13.2dB/m/kHz for the three sediments and As = 17dB/m/kHz for the two silty sediments (cf. equation (5.6)). Figures 5.39-5.41 compare the predictions for transmission loss of the elastic and poroelastic half-space models for the three nonsilty sediments of Table 5.9 for a mid-depth source and both mid-depth and bottom-mounted receivers. The geoacoustical parameters used were those of a Scotian shelf site discussed in [249] (see also Table 5.10). The source was at a depth of 18.3m and the source-to-receiver range was 12.7km. In this and all subsequent simulations the loss at each one-third octave of frequency was calculated by computing the loss at the indicated frequency over a range from 0.75 to 1.25 times the sourceto-receiver distance, fitting a curve of the form (5.58) to the data and then using the source-toreceiver distance in the resulting curve. For the elastic model, transmission loss is plotted for both values of compressional attenuation shown in Table 5.9. As can be seen in the figures, compressional attenuation influences the predictions of the elastic model more substantially at lower frequencies. At higher frequencies the elastic and poroelastic model were in better agreement for a mid-depth receiver than for a bottom-mounted receiver. For the fine sand


263

Figure 5.39. Transmission loss at one-third octave intervals for fine sand as predicted for (V) the elastic model with p — 0.4dB/ ; (A) the elastic model with p — 1 .0dB/ ; (o) the poroelastic model. The receiver was at 35m (top) and 69m (bottom).

Figure 5.40. Transmission loss at one-third octave intervals for coarse sand as predicted for (V) the elastic model with yp — 0.4dB/ ; (A) the elastic model with P — l.0dB/ .; (o) the poroelastic model. The receiver was at 35m (top) and 69m (bottom).

sediment the poroelastic model predicted less loss, especially at low frequencies, whereas for the coarsest of the sediments, coarse sand and fine gravel, it predicted more. Perhaps the most interesting case is the intermediate one, coarse sand, where the poroelastic model predicted less loss at low frequencies, but more at high frequencies. The article of Hughes et al. [249] contains experimental measurements of transmission loss at a Scotian shelf site with a superficial layer of sand thick enough that a half-space model

264


Figure 5.41. Transmission loss at one-third octave intervals for coarse sand and fine gravel as predicted for (V) the elastic model with p = 0.4dB/ .; (A) the elastic model with p = 1.0dB/ ; (o) the poroelastic model. The receiver was at 35m (top) and 69m (bottom).

Table 5.10. Geoacoustic and elastic parameters for a Scotian shelf site. Depth is measured from the surface of the water. Layer Water Sediment

Depth

70

P 1000 2060

CP

1460 1750

P

\ cs |

0.46

170

s

2.2

Figure 5.42. Comparison of the predictions of transmission loss of the elastic model ( ) and the Biot model for coarse sand ( ) with experimental measurement ( ). The source depth was 18.3m, the receiver depth 69m, and the source-to-receiver range 12.7km.


265

Figure 5.43. Predictions of the Biot model for coarse sand for different permeabilities: ( ): k = 1.0 x 10-11; ( ): k = 2.5 x 10-11; ( ): k = 5.0 x 10-11; ( ): k = 7.5 x 10-11m2. The source depth was 18.3m, the receiver depth 69m, and the sourceto-receiver range 12.7km.

Figure 5.44. Comparison of the predictions of transmission loss of the elastic model (o) and the Biot model for coarse sand with a permeability of k = 1.5 x 10 - 1 1 m 2 (A) with experimental measurement ( ). The source depth was 18.3m, the receiver depth 69m, and the source-to-receiver range 12.7km.

might be applicable. Table 5.1 Ogives the geoacoustical and elastic seabed parameters used in [249]. While no Biot model parameters were given, the elastic parameters suggest a sediment similar to the coarse to medium sand (CS) sediment of Table 5.2 (cf. Table 5.9). Figure 5.42 shows the measured data, extracted from [249, Fig. 1 ], compared to the predictions for the elastic model for the parameters of Table 5.10 and the Biot model for coarse sand when

266


the Stoll-Bryan formulas (5.22) were used to compute the bulk and shear moduli. In the experiment the source was at a depth of 18.3m and the receiver was bottom-mounted. The source-to-receiver distance was 12.7km. The experimental measurements differ from the simulations in that they are averages in frequency over one-third octave intervals at a fixed distance, rather than the averages over a range of distances at a single frequency which we use in our simulations. As can be seen in Figure 5.42, the elastic model predicts approximately constant transmission loss above 20Hz, while the experimental measurements indicate a rise in loss of about l0dB between the middle- and high-frequency range. The Biot model predictions follow the experimental data qualitatively, but overestimate the loss by about 5-10dB. It turns out that this rise is strongly influenced by the permeability. Figure 5.43 shows the predictions of the Biot model for coarse sand when the permeability is changed from k = 7.5 x 10-11m2 to k = 5.0 x 10 - 1 1 ,2.5x 10 -11 ,or l . 0 x l0 - 1 1 , and the pore size parameter is changed according to formula (5.23). Best agreement with the data occurs for a permeability in the range of k = 1.0 - 2.5 x 10 -11 m 2 . Figure 5.44 shows that a permeability of k = 1.5 x 10 -11 m 2 gives good agreement with the experimental data. In computing the predictions of the elastic and Biot models used in Figures 5.42-5.44, only the discrete spectrum part of formula (5.56) was used. The modal cutoff for both the elastic and Biot models was about l0Hz, so there were no predictions below this frequency. However, the measured values in the 2.5-10Hz range were likely affected by layers below the superficial one, and thus the predictions would not be expected to be accurate in any event.

Chapter 6

Homogenization of the Seabed and Other Asymptotic Methods

6.1

Low Shear Asymptotics for Elastic Seabeds

In this section on small shear asymptotics, we find it convenient to have the ocean surface at z = —h, the bottom at z = 0, and the seabed lying in the region z > 0. For simplicity we treat only the case of a 2D ocean-seabed system. In this section we report on the work of Gilbert and Makrakis [188] on the asymptotic behavior of 2D time-harmonic acoustic waves in an ocean lying on an elastic seabed, where the modulus of the shear modulus is small compared with its bulk modulus. We assume that the ocean D = {(x, z)\x R, —h z 0} is filled by an ideal acoustic fluid with constant density pf, while the bottom is filled by a homogeneous and isotropic elastic material, with Lame constants , u and density p. The compressional and shear wave velocities are cp = and cs = . Under the above assumptions, the pressure in the ocean is governed by the Helmholtz equation and we assume, as customary, that the surface is pressure released. Let u, v be the displacements of the elastic bottom B = {(x, z)\x R, z 0} in the directions x, z, respectively. For an isotropic medium the stresses xx, zz, xz are given in terms of the displacements by the relations

Recalling Chapter 4, the equations of motion in terms of the stresses are given by

while in terms of the displacements u — (u, v), they satisfy

The usual continuity conditions are satisfied on the interface between the ocean and 267

268

Chapter 6. Homogenization of the Seabed and Other Asymptotic Methods

the seabed, namely,

It is convenient to formulate the problem in terms of the Lame displacement potentials 0 and . These potentials are related to the displacements by

In terms of these potentials the equations of motion are equivalent to the Helmholtz equations

and kp = w/cp, ks = w/c s are the compressional and shear wavenumbers, respectively. By introducing the Love function L, which is related to the displacements by15

the equations of motion are equivalent to the metaharmonic equation

where

In terms of the Love function, the stresses are given by

We observe that as u -> 0, the Navier system (6.1) leads formally to a singular perturbation problem, while from (6.9) it follows that the zero shear condition is trivially satisfied. In the Lame potential formulation, we observe that only the equation for the potential becomes a singular perturbation, while the zero shear boundary condition is again trivially satisfied. In the formulation with the Love function, the metaharmonic equation (6.9) leads to a singular perturbation, as it reduces to the Helmholtz equation for L as —> 0, and the conditions (6.7), (6.8), and (6.9) imply that xz = 0, whereas xx, ,zz remain finite on the boundary v = 0 only if xx = 0 = zz on z = 0, which is a trivial case. In the formal limit — 0, the compressional velocity cp tends to c0 = , and we put Following physical considerations, we anticipate that the limit under investigation must correspond to a fluid seabed, which suggests that the shear boundary condition should become trivial. In fact, it turns out that one of the elastodynamic equations plays the role of a kinematic constraint. In the next subsection we investigate the singular perturbation nature starting with the displacement formulation, which is the most natural one. 15

See [347], [354], [395].

6.1. Low Shear Asymptotics for Elastic Seabeds

6.1.1

269

The Wentzel-Kramers-Brillouin Expansion of the Displacements

In terms of the small parameter

= u/( + 2u), the system (6.1) takes the form

and the boundary conditions the form

where f (x) = zz(x, 0) is the interfacial normal stress, which obviously depends on the small parameter 6. In the case of a homogeneous ocean, where kf is constant, using the Dirichlet to Neumann map on the interface [302], we can explicitly construct the Fourier transform Of f ,

is the Rayleigh function. In the formal limit = 0, f0 coincides with that obtained directly assuming the bottom to be fluid. However, after inverting f it appears quite complicated to recover f 0 (x) from the asymptotic expansion of the inverse Fourier integral for f (x). Therefore, it is more convenient to apply the Wentzel-Kramers-Brillouin (WKB) technique directly for the system (6.1), (6.2). Assuming a WKB expansions is valid,

with

In order to match the phases in the equations (6.11), we choose the order, we obtain for

,z

0. By separating

270


and

Obviously (6.15) can have nontrivial solutions since the determinant of the system for U0, V0 is zero, and the boundary conditions (6.16) can be satisfied if ( )2 — ( )2 — 0 for z = 0. On the other hand, all these conditions are trivially satisfied for 9 — 0, which should be the case when the source term in the nonhomogeneous Helmholtz equation, which describes the signal in the ocean, does not contain an oscillatory term like exp( ). Therefore, the problem at hand is a regular perturbation problem.

6.1.2

The Regular Perturbation Expansion

We now assume that the displacements and the normal boundary traction have the following regular perturbation expansion:

where = cs/cp. Substituting these expansions into the equations of motion (6.1), and separating the various orders, we obtain the following hierarchy of equations in B:

where k = par/X.. Moreover, substituting the expansions (6.17) into the boundary conditions (6.7), (6.9), (6.8) we construct the corresponding boundary conditions on z — 0, namely,

from the normal stress, and

from the zero shear stress. First we consider the zeroth-order system (6.18), (6.21), and (6.24). Moreover, let

be the dilation and the rotation corresponding to the displacements (uv, vv). By differentiating the equations (6.18) with respect to x, z, respectively, we find that

6.1. Low Shear Asymptotics for Elastic Seabeds

271

which is to say that the displacement field (u0, v0) is rotation free. This corresponds to the fluid limit, as —» 0. Using now the condition W0 = 0 as a kinematic constraint in the form zu0 = , we rewrite (6.18) as a decoupled system:

which satisfies the coupled boundary conditions (6.21) and (6.23). Similarly, we can show that the (uv, vv), v 1 satisfy the uncoupled equations

The above systems for uv, vv, can be solved by using Fourier transform methods and the correct radiation conditions as z —> . For example, for the zeroth-order displacements we find

where f0( ) = f f0(x), given by

exp(ix ) f 0 ( x ) d x is the Fourier transform of the boundary traction

The analysis of this section and the expressions (1.7) for the displacements in terms of the Lame potentials suggest that a similar regular perturbation expansion can be constructed using the potential formulation. The constructed regular perturbation for the elastodynamic field in the bottom leads to a regular perturbation expansion for the pressure field p = p(x, z), the zeroth-order term of which coincides with that corresponding to the limiting fluid bottom.

6.1.3

A Singular Perturbation Problem for the Love Function

Introducing the new unknown / by L = €l and putting u — ./(l — 2 ), we see that / again satisfies the metaharmonic equation (6.6), and we rewrite the displacements (6.5) and the stresses (6.7)-(6.9) in the form

272


and

Then, in the formal limit € — 0, the shear stress problem

xz

vanishes, and / satisfies the Neumann

which is easily solvable using the Fourier transform, and with zz(x) = f0(x), it leads again to the solution (6.30)-(6.32). This suggests the regular expansion l(x, z) = (X, z), which, in fact, leads to the same results as the regular expansion for the displacements constructed in section 6.1.2. We close this section with some remarks concerning the singular perturbation problem for the Love function. We introduce the scale function ( ) = 1/2. So we put y = , and we rewrite the Laplacian as = . Then, the metaharmonic operator is written in the form

The characteristic equation, which is the most singular term in (4.1),

is 4 + k2 2 = 0, which has a double real root = 0 and two imaginary roots = ±ik. The existence of imaginary roots for the characteristic equation violates Eckhaus's fundamental assumption for treating elliptic degenerations. Second, applying the Fourier transform with respect to x to (6.6), we obtain

where l(y, ) = exp(i x )l(x, z)dx, with transformed boundary conditions (6.2) are

= 6(1 — 2 2 ) - l , y — kz, and

= k . The

We observe that the principal symbol of the ordinary differential operator in (6.40) coincides with the symbol of

6.2. Homogenization of the Seabed

273

This operator has been investigated by Frank [175] in the framework of coercive singular perturbations, for the case of e-independent boundary conditions that appear when ID problems for elastic rods are investigated. In our case, the symbols of the boundary operators in (6.41) and (6.42) are e-dependent, and moreover the highest derivatives degenerate as € goes to zero, a fact that does not allow existing results on coercive perturbations to be applied.

6.2

Homogenization of the Seabed

We model the seabed as a porous medium, arranged as a periodic packing of the pores into cells. The vibrational motion is assumed to be stimulated acoustically by a signal whose wavelength is X. For an averaging procedure to work, we need the wavelength to be large compared to a typical cell size I. Assuming in addition that A. is comparable to the characteristic macroscopic size L of the problem and the fluid phase is incompressible, we follow the classification of models after Auriault [16] who, depending on the magnitude of various physical parameters, found four different types of possible macroscopic behavior. • Model I. The acoustics of a fluid in a rigid porous matrix regime. This case corresponds to a seabed consisting of hard gravel. • Model II. Diphasic macroscopic behavior of the fluid and solid matrix. This case is dominated by the relative velocity between the fluid and solid phases. • Model III. Monophasic elastic macroscopic behavior, where the ensemble acts as a single elastic body. • Model IV. Monophasic viscoelastic macroscopic behavior. Denote the ratio between the cell size, , and a macroscopic length, L, by , L = . The geometrical structure inside the unit cell Q = ]0, 1[3 has a solid part, Qs, which is a closed subset of Q and a fluid part, Qf = Q\QS • Now we assume Qs is periodically repeated over R" and set Qsk = Qs + k, k e Z". Obviously the closed set Xs = is a closed subset of R3 and X* = R3\XS is an open set in R". We make the following assumptions concerning Qf and Xf: (i) QJ is an open connected set of strictly positive measure with a smooth boundary, and Qs has strictly positive measure in Q as well. (ii) Xf and the interior of Xs are open sets with the boundary of class C , which are locally situated on one side of their boundary. Moreover, Xf is connected. Now we see that X — ]0, L[3 is covered with a regular mesh of size e, each cell being a cube Q], with 1 < i < N(e) = |X|(e)- 3 [l + 0(1)]. Each cube Q is homeomorphic to Q, by linear homeomorphism , being composed of a translation and a homothety of ratio .

We define and

274


For sufficiently small

> 0 we consider the sets

and define

Obviously, = X S . The domains X£s and X f represent the solid and fluid parts of a porous medium X, respectively. For simplicity we suppose L/ N. Then K — 0.

6.2.1

Time-Variable Solutions in Rigid Porous Media

A seabed consisting of gravel or sandstone might be modeled as a rigid porous media. Consider the acoustic equation

where a is a positive constant and is an integer, f C ((—L, L)3 x[0, + the boundary value problem for (6.43) with conditions

)). Consider

The problem (6.43)-(6.47) could be extended on the layer {x1 (-L, L)} without X f by means of an even extension of the right-hand-side functions f, g and the solution u . So we consider further the problem (6.43), (6.44), (6.47), and (6.45) for x1 — —L with the conditions of 4T-periodicity in x2 and X3,. We build the asymptotic expansion of the solution u as e —> 0. We shall apply the method of boundary layers in homogenization to obtain the asymptotic expansion of the solution of the problem posed in the partially perforated layer. We seek the asymptotic expansion in the form

6.2. Homogenization of the Seabed

275

where

where

and

Nq,i( ) are 1-periodic of R3, N , N , and M°q, i(£) are the boundary layer functions, 1-periodic in and exponentially decaying as | | —> +00,

Substituting (6.44) into (6.43), (6.45), (6.46), and the interface conditions on {x1 = 0},

we obtain

276

In (6.54),


T

,

T

,

T

,T

have the form

We obtain the following sequence of problems [200]:

6.2. Homogenization of the Seabed For

Nq,i

277

1-periodic in

For N

1 -periodic in

For M

1-periodic in

where constants C and C are chosen in such a way as to provide the exponential decay as | | —>• +00 of N and M ( ), respectively. For N 1 -periodic in

For the function v we obtain the problem

278


Here C — 0 and C = 0. This problem could be reduced by substitution of (6.51) to a sequence of problems for the Vj:

where

The existence and exponential decaying of the boundary value problems could be proved in an analogous manner to that in the works [334], [331], [219]. For the truncation series

where

one can obtain the following estimates. Theorem 6.1.

where 8 > 0 and

> 0 are independent of .

6.3. Time-Harmonic Solutions in a Periodic Poroelastic Medium

279

The last estimate justifies the homogenized model of the second order of accuracy if v (1) is a 4L-periodic in x2 and x3 asymptotic solution of the problem

Then

Particularly in the asymptotic expansion when f = f (x1, t), g = g(t), then v V (X1, t) is an asymptotic solution of the problem

6.3

=

Time-Harmonic Solutions in a Periodic Poroelastic Medium

If the matrix is assumed to be elastic instead of rigid, we have a so-called poroelastic material. Sediments such as glacial til or fine sand might fall into this category. Consider an infinite -periodic medium composed of an elastic solid and a compressible viscous fluid. Let Xs and Xf denote the domains occupied by the solid and fluid, respectively. Their common boundary S is assumed to be a smooth manifold of codimension 1. The displacement vector u16 satisfies the system of equations (written componentwise):

l6

For convenience of notation we use u for u in the following discussion.

280


in Xs, and

in Xf. Moreover, on the interface S, the following transmission conditions hold:

where vj denote components of the unit normal to S pointing inside of Xs. In the solid part, components of the stress tensor s satisfy Hooke's law:

with coefficients a

C (XS) satisfying conditions of symmetry and positivity:

In the fluid part Xf, the stress tensor satisfies the Navier-Stokes law:

where viscosity coefficients u and

satisfy

If the displacement is small, we can linearize the equation near the reference state characterized by the known reference densities ps and pf. The linearized equation of state relates pressure P to the perturbation of density p:

where c is the speed of sound. Moreover, linearizing the conservation of mass equation permits the pressure to be eliminated from (6.83):

where

Denote by aijkl the components of symmetric fourth-order tensor equal to a

in Xs.


6.3.1

281

Inner Expansion and Homogenized System

Assuming that u is a time-harmonic vector with angular frequency a), then the amplitude u (x, w) satisfies

where u(x) is an n-component vector function of x and Akj( ,w) are periodic n x n matrices with components A given by

in Xs and by

in Xf. On 5, the following transmission conditions are to be satisfied:

where vj are components of the unit normal to S. We look for a solution asymptotic in :

Substituting (6.90) into the system (6.87), we obtain

Then it may be seen that (6.91) transforms into

where Hp,i depend on N P , i , Akj, w, and p. Since the left-hand side is of order

, we obtain

282


where / denotes the unit matrix. Also, we assume that Np,i = 0 if at least one of p, \i\ is negative. Then, collecting terms in (6.92), we have the following expressions for H p , i :

if p >2,

and for p

0, |i| > 1,

If we require that Hp,i be constant, then (6.93), (6.94) can be used to determine Np,i recursively. All equations above are of the form

where

Note that Tp,i depends on the previously obtained Np',i' with p'+ \i' \ < p + \i\. Wespecify the constants Hp,i to be ( T p , i ) , and write

This choice of HP,i guarantees that each cell problem is uniquely solvable up to a constant matrix. To show this, consider the variational formulation of cell problems (6.95). The variational formulation of a cell problem now reads: find u V1 such that

where


283

for some f V° and all v V1. Of course, the actual cell problem (6.95) is a matrix one, so we have to solve several vector problems and determine columns of an unknown matrix one by one. Theorem 6.2. There exists a unique solution of the problem (6.96). Proof. See Gilbert and Panchenko [ 192]. The asymptotic series for u now takes the form

Representing v as an asymptotic series

we obtain a chain of averaged problems for successive determination of the vq:

where

and

and, generally,

284


The first equation in the chain (6.97) is the homogenized system:

The matrices N0,i2 above are obtained as solutions of the cell problem:

satisfying the periodic boundary conditions and the transmission conditions:

and

on the interface hypersurface S.

6.3.2

Interface Matching and Boundary Layers

In the previous section we did not consider boundary conditions, so the construction above applies only locally in R3. To investigate the nature of the changes needed to incorporate boundary effects, consider the following model problem. Suppose that the plane interface {x : X3, = 0} separates two different periodic media. We assume that the equations (6.87) together with the constitutive relations (6.88) with possibly different 1 -periodic matrices A are valid in the half-spaces K+ = {x : x3 > 0} and K~ = {x : x3 < 0}, respectively. A particular case of this is acoustics in a two-layer media of the type with homogeneous fluid above, fluid-saturated sediment below. At this point we prescribe no conditions on u as \X3\ —> . Our primary interest is to investigate how the presence of the interface affects homogenization. For any z = ( z 1 , • • •, z3) R3, let z denote the vector ( z 1 , Z2, 0). In what follows, we use the following notation:

and

with

modified accordingly. We also denote


285

and

Denote by H l ( w ( a , b)) the space of locally H1-functions 1-periodic in x. We recall that L denotes the differential operator in (6.87). A function u H l ( w ( a , b)) is a solution of the problem that is weak periodic in x: Lu = f in w(a, b) if for any v

H l ( w ( a , b)) such that v = 0 on

the following holds:

Below, the quantities with sub- or superscript + are defined in K+, and similarly — refers to a quantity defined in K-. In K+, we look for asymptotic expansion of the form

where v± are asymptotic series formed by (so far) arbitrary solutions of the chain of homogenized problems (6.97) in K±. In K~ we look for a similar expansion with all plusses replaced by minuses and vice versa. Matrices Np,i are as above, and M and S are matrices 1-periodic in . Substituting (6.100) into the original equations (6.87) and repeating the calculations of section 6.3.1, we obtain the identical equations for M and S , written explicitly only for M :

in K+, where MM

are of the form

where

To start the chain, we set M = M = S = S = 0. On the interface xn = 0 we impose the following transmission conditions:

286


where

These conditions arise due to requirements of continuity of displacements and stresses. Differentiating u± and shifting indices in the sums in the same fashion as in section 6.3.1, we obtain

and a similar expression for a (u- )n. Substituting into the second equation in (6.102), collecting terms, and combining with the equations (6.101) we obtain the following transmission problems for determination of the pair of matrices M , S :

in K+, in K- , with the interface conditions

where k , are constant matrices. We look for the solution of this problem in a class of 1 -periodic-in- matrices that decay exponentially as | | —> . Similarly, the pair M , S should be a solution to the problem

in K-,

in K+,

and

6.3. Time-Harmonic Solutions in a Periodic Poroelastic Medium at the interface. Let us define the operator L to be L+ in K+ and L~ in K-. problems above can be written in common form:

287 Then the

In order to formulate the solvability theorem, we first introduce some definitions. Definition 6.1. Let u(x, xn) L be a vector function 1-periodic in x. We say that u has one-sided exponential decay if the estimate

holds either for s Z+ or s Z- with positive constants C independent of 5. If (6.108) holds for all s Z, we will say that u has two-sided exponential decay. To describe behavior at infinity we will use the following definition. Definition 6.2. A vector function u will be called one-sided exponentially stabilizing if there exists a constant vector w such that the function u — w satisfies the estimate (6.108) either for s Z+ or s Z~. If there is a pair of constant vectors w+, w~ such that one-sided estimates (6.108) hold for both respective differences, we will call u two-sided exponentially stabilizing to w+, w-. Theorem 6.3. Suppose that f in (6.107) has two-sided exponential decay. Then there exist constant vectors t, w+, and w- such that the problem (6.107) has a 1-periodic-insolution u such that e(u) has two-sided exponential decay and u is two-sided exponentially stabilizing to w+ and w-. Moreover,

Proof. See [ 191 ] for details.

Theorem 6.4. Let s > h > 0 be integers and let u be a periodic-in-x solution of

in w(s — h, s + h + 1). Suppose that P(s — 1, u) = 0. Then

288


where A is a positive constant independent ofs,h. Proof. See the paper [ 191 ] for details. Next, we need to generalize this to the case when / and P(s — h, u) are nonzero. The prototype of the main estimate is given by the following lemma. Lemma 6.5. Let N be a positive integer. The system

with the boundary conditions

satisfying the compatibility condition

has a unique solution U satisfying the estimate

where

Proof. See [ 191 ] for details. Using the above lemma, we prove the following theorem. Theorem 6.6. Let u be a periodic in x solution of

in co(ti, ^2), where t^ > t\ + 2, tj are integers, t^ > 0, t\ < 0. Then for any integer s, h > 0 such that s — h > t\, s + 1 + < t\, the estimate


289

holds with C independent of s ,h. A is a constant from Theorem 6.4. Proof. See [ 191 ] for further details. As a consequence, we derive the following theorem. Theorem 6.7. Let fi be a vector function satisfying the inequality

where c, a are positive constants independent of s. Let u be a periodic solution of the system

s = 1 , 2 , . . . , where c is a constant independent of s, A is a constant from Theorem 6.4, and 8 is a constant such that 0 < A. Then there exist constants C\,C2,a 1 ,a2 independent of s and a constant vector w such that

Proof. See [191] for further details. The next result is an existence theorem of the type needed for construction of the boundary layer. Consider the problem

290

in

Chapter 6. Homogenization of the Seabed and Other Asymptotic Methods We assume that fi

for all t2 > 0, t1 < 0, and periodic in

Theorem 6.8. Suppose

where M, & are constants from the previous theorem, 0 Then for any constant vector q there exists a solution of the problem (6.120) such that P(0, u) = q and the following estimate holds:

for k = 1 , 2 , . . . . Here C is independent of k, Proof. See [191] for further explanations.

6.4

Rough Surfaces

Next we turn our attention to the interface between two materials. Essentially we are concerned with a poroelastic seabed that has a rough interface with the water column lying above. For example, see Figure 6.1 for an illustration. The pore fluid is the same as in the water column and is assumed to be Newtonian and incompressible. Moreover, we restrict our interest to time-harmonic oscillations. We adopt Auriault's [15], [17], [16] notation for the homogenized system of equations, namely

In these equations, I is the unit tensor, the bulk stress tensor, C the elasticity tensor of the skeleton, p the pressure, a) the acoustic frequency, the displacement of the solid part, / the displacement of the fluid part, e( ) the strain tensor, / the porosity, and K the generalized Darcy permeability tensor. These terms come about using standard homogenization. It is well known that the system (6.123)-(6.126) may be reformulated in terms of the solid displacement and the fluid pressure, p, leading to (6.127) and (6.128) below. We assume that there is a periodic roughness to the surface; namely, we cover R2 by 6 squares whose sides are , such that in each square the interface between the two different poroelastic materials are described by X3, = h ( x 1 , x2). In each of the two half-spaces, separated by the rough surface, the physical parameters are assumed to be constant. We shall use the summation convention where repeated indices mean summation; moreover, Latin subscripts run from 1-3 and Greek subscripts run from 1-2:

6.4. Rough Surfaces

291

Figure 6.1. Schematic representation of the rough interface.

where

and [•] denotes the jump across the interface X3, = h(x 1 , x2), i-e.,

We introduce stretched variables

In terms of the

(6.127) and (6.128) can be written as

1, 2 to obtain the jump conditions

292


Figure 6.2. Schematic representation of the x1x2 profile of the periodicity cell Y.

Assuming that

Letting

and q have asymptotic expansions for small ,

denote the real part of

Then it is seen that

must satisfy

By scalar multiplying (6.139) by (0), then integrating the product over the periodicity cell Y (see Figure 6.2) and applying the F-periodic condition and the jump conditions, we obtain

from which we may conclude [190] that Using a similar argument on the imaginary part of Therefore, does not depend on

, we may also conclude that

6.4. Rough Surfaces

293

A similar argument [190] leads to

i.e., p(0) does not depend on Next, we consider the

equations, which are

To solve (PI), we introduce a new unknown

from which we can see that the

by writing p ( l } in the form

will solve (PI) if

satisfies

As we also require to be y-periodic, continuous in Y, and have zero average over the Y cell, the are uniquely determined [320]. A similar analysis for the displacement leads to the system

To solve (P2), we write

in the form

which implies

Therefore,

solves

satisfies

We also require Xkmn to be continuous in Y, Y periodic, and have zero average over the Y cell. These conditions uniquely determine Xkmn [320]. Next, we consider the equations. We integrate (6.147) with respect to over Y and divide it by the area of Y, which is denoted by A. The first and the third terms can be converted to integrals around F and Y by the divergence theorem. The integrals along dY vanish because of the assumed F-periodicity of and We thus obtain

294


where {•) is the "averaging operator" defined as

By using (6.98), the first integral in (6.147) can be written as

where A\ is the area of Y+ [190]. By direct calculation and using the facts that Y+ and Yare functions of x3 only and doesn't depend on y, the second integral in (6.147) can be converted to

The fourth integral in (6.147) is equal to because is not a function of and Cjjki are constants in Y+ and Y- . Realizing that (C ijkl ) depends only on X3, and

we may further write

Finally, we substitute (6.148), (6.149), and (6.150) into (6.147), then replace pl and by respectively. Note that the last term in (6.148) cancels with the last term in (6.150). This yields a new equation:

6.4. Rough Surfaces

Introduce the "effective" parameters

295

and

defined by

Then (6.151) can be rewritten as

which is the homogenized equation of (6.127). Similarly, we first apply (•) to (6.97). Using the divergence theorem on the first and fifth terms, we convert the integrals to be boundary integrals of jumps around F. Second, we apply (6.99) to rewrite the boundary integral of the jump. Third, we replace pl and u\ respectively. Finally, (6.97) becomes

Introducing another set of "effective" parameters

296


(6.156) can be written as

This is the homogenized equation of (6.128). We summarize our discussion as follows. Theorem 6.9. Let and p satisfy the dynamic equation (6.127) and the continuity equation (6.128) on both sides of the periodic surface x3 = h (x 1 , x2) with constant solid density p, fluid density pi, Darcy permeability K, porosity f, bulk compressibility Kb, solid compressibility Ks, and fluid compressibility Kf on each side. Suppose u(x,t,€) and p also satisfy the continuity conditions (6.129)-(6.132) across the interface x3 = h(xi,x2). If and p have the asymptotic forms (6.137) and (6.138), respectively, then and p satisfy the system of homogenized equations

Here a is defined in Figure 6.1, (•} the average operator, and the effective parameters are defined in (6.152)-(6.154) and (6.157)-(6.159). Remark 6.1. By the same way the other asymptotic expansion could be considered when the set is connected (see Figure 6.2). This case models the wave propagation in a porous media bounded above by a fluid.

6.5

A Numerical Example

In this section, we will construct the effective parameters of a special case of the previous discussion. For simplicity, the interface is assumed to consist of very narrow truncated uniform cylinders with radius equal to 0.25e (see Figure 6.3). In this case, h = and the unit normal vectors on the interface are proportional to rather than Accordingly, the effective parameters and are modified to be

whereas the other effective parameters remain unchanged. Suppose the upper layer is of fine sand and the lower layer is of silty clay. Using the experimental data in [245] and the formulas (58)-(71) in [79], [81] we list the Lame

6.5. A Numerical Example

297

Figure 6.3. Single cell of the periodic interface.

Table 6.1. Physical parameters of poroelastic media.

X

Kij

Fine sand 7.12 x 10 6 -2.3 x l05I 1.68 x 109- 1.04263 x l05I (3.09 x 10-11 - 5.55 x 10-16I)

Silty clay 7.86 x 10 6 -2.5 x 104I 3.5 x 10 8 -9.6x 104I (5.15 x l 0 - 1 1 - 2.34 x 10-15I)

coefficients and Darcy tensor of each layer in Table 6.1. The frequency co is chosen to be 200Hz. The value of effective parameters may be found in [190].


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Index 1 -periodic matrices, 284

comparison of Biot with elastic model, 260 complete sets of functions, 72 complete system, 80 completeness of far-field patterns, 101 complex Lame coefficient, 218 compressional attenuation coefficients, 217 compressional wave, 177 compressional wave speed, 206 conservation of energy, 31, 32 conservation of momentum, 42 constant depth ocean, 57, 58, 171 constitutive equations, 34, 195, 220 constitutive laws, 31 constitutive relations, 33, 172 contour integral, 256 contour integral representation, 93 correspondence principle, 226 cost functionals, 158 cylindrical coordinates, 154, 178, 247

acoustically hard, 48 acoustically soft, 47 adjoint operator, 75, 152 admissible solution, 120 analytical continuation, 152 angle of incidence, 154 approximation of the kernel, 84 asymptotic behavior of Hankel's function, 77 asymptotic expansion, 269-271,274,279, 285 asymptotic representation of Green's function, 94 attenuation coefficients, 232 Bessel function, 72, 156, 187 Bessel function of large argument, 117 Biot model, 217, 241 boundary conditions, 46 boundary integral equation, 176 boundary integral method, 83 boundary perturbation, 163, 168 boundary transition conditions, 179 branch cut, 180,232 bulk modulus, 41, 267

Darcy permeability, 222 Darcy permeability tensor, 290 decomposition on the far field, 72 decomposition theorem, 78 dense sets of functions, 74 dense subsets of far-field pattern, 72 dilatation, 217 diphasic behavior, 273 direct scattering problem, 59 Dirichlet eigenfunction, 104 Dirichlet problem, 66 Dirichlet to Neumann map, 173, 269 double-layer potential, 83 dual space indicator method, 123

Cauchy deformation tensors, 13 Cauchy law of motion, 28 Cauchy principle value, 185 Cauchy stress tensor, 172 Cauchy tensor, 34 cell, 273 cell problem, 282 Clausius-Duhem, 32, 34 coercive singular perturbation, 273

eigenfunctions, 55, 117 333

Index

334

eigenmodes, 55 eigenvalue problem, 197 eigenvalues, 58, 84, 133, 256 Einstein convention, 10 elastic invariants, 219 elastic model, 217 elastic seabed, 171 elastic transmission conditions, 173 entire Herglotz wave function, 119 equation of state, 280 equivalence class, 81 eulerian coordinates, 9 eulerian strain tensor, 13 evanescent modes, 69 exceptional frequencies, 182 exploration region, 128 extremal problem, 109 Faber-Krahn inequality, 114, 122 far-field approximation, 62, 177, 180 far-field operator, 151 far-field pattern, 69, 70, 110 far-field pattern operator, 80 fluid model, 217 flux conditions, 253 Fourier coefficients, 59, 169, 205 frame displacement, 253 free energy, 36 free-wave far-field pattern, 94 frequency domain, 49 fundamental solution, 111 Gauss theorem, 32 Gauss-Legendre quadrature rule, 88 Gelfand-Levitan-Goursat problem, 198 generalized function, 92 generalized Herglotz functions, 104, 114 generalized Herglotz wave functions, 115 generalized scattering problem, 96 generalized Sommerfeld condition, 96 Green's deformation tensors, 13 Green's function, 57, 84, 199 Green's function integral representation, 58 Green's identity, 60, 61, 83 guided-wave far-field pattern, 94

Holder norm, 150 Hankel function, 58, 72, 156 Hankel function expansion, 69 Hankel inversion, 201 Hankel inversion theorem, 180 Hankel transform, 93, 178, 195 Heaviside function, 179 Helmholtz decomposition, 44, 45, 177 Helmholtz equation, 59, 172 Herglotz kernel, 119 homogeneous oceans, 57, 107 homogenization, 274 Hooke-Cauchy law, 42 Huygens's principle, 174 hyperelastic materials, 37 ICBA, 154, 167 ill-posedness, 140 imaging problems, 107 impedance boundary condition, 48 improperly posed inverse problem, 72 inhomogeneity, 132 injection theorem, 79 injections, 76 inner expansion, 281 inner product, 201 integral equation, 68 interface conditions, 253 interface matching, 284 interior Neumann problem, 84 intersecting canonical body approximation, 154 inverse problem, 142, 157 inverse scattering problems, 107, 119 inversion procedure, 126 invertibility of transformed Green's function, 182 isotropic, 41 isovelocity ocean, 256 iterative algorithm, 131, 140 Jacobi-Anger, 116 lost functions, 92 jump relation, 152 Kelvin functions, 222

335

Index kinetic energy, 220 kinetic energy function, 225 Kirchhof model, 208 Kozeny-Carmen equation, 227 lagrangian coordinates, 9 lagrangian equations, 221 lagrangian strain tensor, 13 Lame coefficients, 41, 196, 217 Lame displacement potentials, 268 Laplacian in cylindrical coordinates, 57 layer potentials, 66, 111 linearized algorithm, 139 linear elasticity, 15, 172 linearized equation, 280 local entropy, 32 local extrema of the norm, 128 locally Hl -functions, 285 Love function, 268, 272 macroscopic size, 273 measured data, 133 measured far-field pattern, 119 meromorphic function, 180 metaharmonic, 268 Mindlin plate, 214 minimization, 133 Mittag-Leffier, 181 modal representation, 255 modal solutions, 58 monophasic elastic behavior, 273 monophasic viscoelastic behavior, 273 MVB, 168 Navier equations, 194 near field, 118 near-field approximation, 177, 180, 183 Neumann function, 158 nonlinear optimization, 132 nonlocal condition, 173 normal mode, 61 normal mode representation, 58, 59 nullspace, 152 numerical example, 127, 185, 191, 206, 296 numerical methods, 84, 88

object reconstruction, 166 objects of revolution, 162 optimal solution, 153 optimal surfaces, 153 optimization problem, 109 orthotropic, 41 outgoing condition, 51, 59 outgoing Green's function, 58 outgoing Hankel functions, 250 outgoing radiation condition, 58 parallel computer, 187 perfectly reflecting seabed, 183 periodic roughness, 290 Poiseuille fluid, 222 Poisson ratio, 214 pore size parameter, 222 poroelastic, 279 poroelastic coefficients, 297 poroelastic plate, 208 poroelastic seabeds, 217 pressure-release boundary, 57 principal symbol, 272 projection theorem, 76 propagating far-field pattern, 71, 80 propagating solution, 129 radiation condition, 48, 58, 61, 62 random number generator, 192 ray representation of Green's function, 73 Rayleigh dissipation function, 221 real analyticity, 119 reciprocity relations, 98, 100 reconstruction, 133 recurrence relations, 163 reduced wave equation, 57 reflecting seabottom, 133 regular perturbation, 270 regularization parameter, 112 representation for pressure, 256 residues, 256 rigid matrix, 274 rigidity, 41 rough surfaces, 290 sand, chalk, til, granite, 191

Index

336

scattered wave, 72, 99, 100, 104 seamount problem, 133 separation of variables, 57 set of far-field patterns, 71 shape function, 155 shear attenuation coefficient, 217 shear boundary condition, 268 shear modulus, 267 shear wave, 177 shear wave speed, 206 simple materials, 33 single-layer potential, 109, 174 singular perturbation, 268 singularity of Green's function, 60 Sobolev space, 141 Stoll-Bryan formulas, 233, 238 strain energy, 219 strain energy function, 220 strain rates, 35 strains, 218 stratified medium, 92, 142 stress tensor, 36 stress-strain relation, 40 Sturm-Liouville problem, 197 thin plate approximation, 208 Tikhonov functional, 108

Tikhonov regularization, 141 time domain, 49 totally reflecting boundary, 57 transisotropic, 194 transmission conditions, 178, 281 transmutation, 212, 213 transmutation operator, 197 triangular mesh, 186 two-dimensional model, 129 two-layered wave guide, 129 two-sided exponential decay, 287 undetermined coefficient problem, 189 uniquely determined, 144 uniqueness, 135 uniqueness of direct scattering problem, 124 uniqueness theorem, 61, 66 variational formulation, 282 wave equation, 43 wave guide, 83 wave speeds, 232 weak convergence, 109 weighted integration, 128 Wiener-Hopf integral equation, 83 WKB expansion, 269

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