SEISMIC RAY THEORYSeismic Ray Theory presents the most comprehensive treatment of the seismicray method available. This method plays an important role in seismology,seismic exploration, and the interpretation of seismic measurements.The book presents a consistent treatment of the seismic ray method, basedon the asymptotic high-frequency solution of the elastodynamic equation.At present, this is the most general and powerful approach to developing theseismic ray method. High-frequency seismic body waves, propagating in com-plex three-dimensional, laterally varying, isotropic or anisotropic, layered andblock structures are considered. Equations controlling the rays, travel times,amplitudes, Green functions, synthetic seismograms, and particle groundmotions are derived, and the relevant numerical algorithms are proposed anddiscussed. Many new concepts, which extend the possibilities and increasethe efciency of the seismic ray method, are included. The book has a tutorialcharacter: derivations begin with a relatively simple problem in which themain ideas are easier to explain and then advance to more complex problems.Most of the derived equations in the book are expressed in algorithmic formand may be used directly for computer programming. The equations and pro-posed numerical procedures nd broad applications in numerical modeling ofseismic waveelds in complex 3-Dstructures and in many important inversionmethods (tomography and migration among others).Seismic Ray Theory will prove to be an invaluable advanced textbook andreference volume in all academic institutions in which seismology is taughtor researched. It will also be an invaluable resource in the research and explo-ration departments of the petroleum industry and in geological surveys.Vlastislav Cerven y is a Professor of Geophysics at Charles University, Praha,Czech Republic. For 40 years he has researched and taught seismic raytheory worldwide. He is a member of the editorial boards of several journals,including the Journal of Seismic Exploration and the Journal of Seismology.In 1997 he received the Beno Gutenberg medal from the European Geophys-ical Society in recognition of his outstanding theoretical contribution in theeld of seismology and seismic prospecting. In 1999 he received the Con-rad Schlumberger Award from the European Association of Geoscientists andEngineers in recognition of his many contributions to asymptotic wave theoryand its applications to seismic modeling. He is an Honorary Member of theSociety of Exploration Geophysicists. He has published two previous books:Theory of Seismic Head Waves (with R. Ravindra, 1971) and Ray Method inSeismology (with I. A. Molotkov and I. P sen ck, 1977). He has also publishedmore than 200 research papers.SEISMIC RAYTHEORYV. CERVENYCharles UniversityCAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521366717 Cambridge University Press 2001 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2001 This digitally printed first paperback version 2005 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data erven, Vlastislav. Seismic ray theory / V. erven p. cm. Includes bibliographical references and index. ISBN 0-521-36671-2 1. Seismic waves Mathematical models. I. Title. QE538.5 .C465 2001 551.22015118 dc21 00-040355 ISBN-13 978-0-521-36671-7 hardback ISBN-10 0-521-36671-2 hardback ISBN-13 978-0-521-01822-7 paperback ISBN-10 0-521-01822-6 paperback ContentsPreface page vii1 INTRODUCTION 12 THE ELASTODYNAMIC EQUATION AND ITS SIMPLE SOLUTIONS 72.1. Linear Elastodynamics 82.2. Elastic Plane Waves 192.3. Elastic Plane Waves Across a Plane Interface 372.4. High-Frequency Elastic Waves in Smoothly Inhomogeneous Media 532.5. Point-Source Solutions. Green Functions 732.6. Application of Green Functions to the Constructionof More General Solutions 893 SEISMIC RAYS AND TRAVEL TIMES 993.1. Ray Tracing Systems in Inhomogeneous Isotropic Media 1023.2. Rays in Laterally Varying Layered Structures 1173.3. Ray Tracing 1243.4. Analytical Ray Tracing 1293.5. Ray Tracing in Curvilinear Coordinates 1373.6. Ray Tracing in Inhomogeneous Anisotropic Media 1483.7. Ray Tracing and Travel-Time Computations in 1-D Models 1603.8. Direct Computation of Travel Times and/or Wavefronts 1783.9. Perturbation Methods for Travel Times 1893.10. Ray Fields 1993.11. Boundary-Value Ray Tracing 2173.12. Surface-Wave Ray Tracing 2284 DYNAMIC RAY TRACING. PARAXIAL RAY METHODS 2344.1. Dynamic Ray Tracing in Ray-Centered Coordinates 2374.2. Hamiltonian Approach to Dynamic Ray Tracing 2594.3. Propagator Matrices of Dynamic Ray Tracing Systems 2784.4. Dynamic Ray Tracing in Isotropic Layered Media 2894.5. Initial Conditions for Dynamic Ray Tracing 3104.6. Paraxial Travel-Time Field and Its Derivatives 3224.7. Dynamic Ray Tracing in Cartesian Coordinates 3314.8. Special Cases. Analytical Dynamic Ray Tracing 341vvi CONTENTS4.9. Boundary-Value Ray Tracing for Paraxial Rays 3484.10. Geometrical Spreading in a Layered Medium 3564.11. Fresnel Volumes 3724.12. Phase Shift Due to Caustics. KMAH Index 3804.13. Dynamic Ray Tracing Along a Planar Ray. 2-D Models 3844.14. Dynamic Ray Tracing in Inhomogeneous Anisotropic Media 4005 RAY AMPLITUDES 4175.1. Acoustic Case 4195.2. Elastic Isotropic Structures 4495.3. Reection/Transmission Coefcients for Elastic Isotropic Media 4775.4. Elastic Anisotropic Structures 5045.5. Weakly Dissipative Media 5425.6. Ray Series Method. Acoustic Case 5495.7. Ray-Series Method. Elastic Case 5675.8. Paraxial Displacement Vector. Paraxial Gaussian Beams 5825.9. Validity Conditions and Extensions of the Ray Method 6076 RAY SYNTHETIC SEISMOGRAMS 6216.1. Elementary Ray Synthetic Seismograms 6226.2. Ray Synthetic Seismograms 6306.3. Ray Synthetic Seismograms in Weakly Dissipative Media 6396.4. Ray Synthetic Particle Ground Motions 643APPENDIX A FOURIER TRANSFORM, HILBERT TRANSFORM,AND ANALYTICAL SIGNALS 661A.1. Fourier Transform 661A.2. Hilbert Transform 663A.3. Analytical Signals 664References 667Index 697PrefaceThe book presents a consistent treatment of the seismic ray method, applicable to high-frequency seismic body waves propagating in complex 3-D laterally varying isotropic oranisotropic layered and block structures. The seismic ray method is based on the asymptotichigh-frequency solution of the elastodynamic equation. For nite frequencies, the raymethod is not exact, but it is only approximate. Its accuracy, however, is sufcient tosolve many 3-Dwave propagation problems of practical interest in seismology and seismicexploration, which can hardly be treated by any other means. Moreover, the computed raysmay be used as a framework for the application of various more sophisticated methods.In the seismic ray method, the high-frequency waveeld in a complex structure can beexpanded into contributions, which propagate along rays and are called elementary waves.Individual elementary waves correspond, for example, to direct P and S waves, reectedwaves, various multiply reected/transmitted waves, and converted waves. Abig advantageof the ray method is that the elementary waves may be handled independently. In the book,equations controlling the rays, travel times, amplitudes, Green functions, seismograms, andparticle ground motions of the elementary waves are derived, and the relevant numericalalgorithms are developed and discussed.In general, the theoretical treatment in the book starts with a relatively simple problemin which the main ideas of the solution are easier to explain. Only then are the more complexproblems dealt with. That is one of the reasons why pressure waves in uid models are alsodiscussed. All the derivations for pressure waves in uid media are simple, clear, and com-prehensible. These derivations help the reader to understand analogous derivations for elas-tic waves inisotropic andanisotropic solidstructures, whichare oftenmore advanced. Thereis, however, yet another reason for discussing the pressure waves in uid media: they haveoften been used in seismic exploration as a useful approximation for P waves in solid media.Throughout the book, considerable attention is devoted to the seismic ray theory ininhomogeneous anisotropic media. Most equations derived for isotropic media are alsoderived for anisotropic media. In addition, weakly anisotropic media are discussed in somedetail. Special attention is devoted to the qS wave coupling.A detailed derivation and discussion of paraxial ray methods, dynamic ray tracing,and ray propagator matrices are presented. These concepts extend the possibilities of theray method and the efciency of calculations. They can be used to compute the traveltimes and slowness vectors not only along the ray but also in its vicinity. They also offernumerous other important applications in seismology and seismic exploration such assolution of boundary-value ray tracing problems, computation of geometrical spreading,Fresnel volumes, Gaussian beams, and Maslov-Chapman integrals.viiviii PREFACEMost of the nal equations are expressed in algorithmic form and may be directly usedfor programming and applied in various interpretation programs, in numerical modelingof seismic wave elds, tomography, and migration.I am greatly indebted to my friends, colleagues, and students from Charles Universityand from many other universities and institutions for helpful suggestions and valuablediscussions. Many of them have read critically and commented on certain parts of themanuscript. I owe a special debt of thanks toPeter Hubral, BobNowack, and Colin Thomsonfor advice and constructive criticism. I am further particularly grateful to Ivan P sen ck andLud ekKlime s for everydaydiscussions andtoEva Drahotov a for careful typingof the wholemanuscript. I also wish to express my sincere thanks to the sponsors of the ConsortiumProject Seismic Waves in Complex 3-D Structures for support and to SchlumbergerCambridge Research for a Stichting Award, which was partially used in the preparationof the manuscript. Finally, I offer sincere thanks to my family for patience, support, andencouragement.CHAPTER ONEIntroductionThe propagation of seismic body waves in complex, laterally varying 3-D layeredstructures is a complicated process. Analytical solutions of the elastodynamicequations for such types of media are not known. The most common approaches tothe investigation of seismic waveelds in such complex structures are (a) methods based ondirect numerical solutions of the elastodynamic equation, such as the nite-difference andnite-element methods, and (b) approximate high-frequency asymptotic methods. Bothmethods are very useful for solving certain types of seismic problems, have their ownadvantages and disadvantages, and supplement each other suitably.We will concentrate here mainly on high-frequency asymptotic methods, such as the raymethod. The high-frequency asymptotic methods are based on an asymptotic solution ofthe elastodynamic equation. They can be applied to compute not only rays and travel timesbut also the ray-theory amplitudes, synthetic seismograms, and particle ground motions.These methods are well suited to the study of seismic waveelds in smoothly inhomoge-neous 3-D media composed of thick layers separated by smoothly curved interfaces. Thehigh-frequency asymptotic methods are very general; they are applicable both to isotropicand anisotropic structures, to arbitrary 3-D variations of elastic parameters and density,to curved interfaces arbitrarily situated in space, to an arbitrary source-receiver congu-ration, and to very general types of waves. High-frequency asymptotic methods are alsoappropriate to explain typical wave phenomena of seismic waves propagating in complex3-D isotropic and anisotropic structures. The amplitudes of seismic waves calculated byasymptotic methods are only approximate, but their accuracy is sufcient to solve many3-D problems of practical interest.Asymptotic high-frequency solutions of the elastodynamic equation can be sought inseveral alternative forms. In the ray method, they are usually sought in the form of theso-called ray series (see Babich 1956; Karal and Keller 1959). For this reason, the raymethod is also often called the ray-series method, or the asymptotic ray theory (ART).The seismic ray method can be divided into two parts: kinematic and dynamic. Thekinematic part consists of the computation of seismic rays, wavefronts, and travel times.The dynamic part consists of the evaluation of the vectorial complex-valued amplitudes ofthe displacement vector and the computation of synthetic seismograms and particle groundmotion diagrams.The most strict approach to the investigation of both kinematic and dynamic parts of theray method consists of applying asymptotic high-frequency methods to the elastodynamicequations. The kinematic part of the ray method, however, may also be attacked by somesimpler approaches, for example, by variational principles (Fermat principle). It is even12 I NTRODUCTI ONpossible todevelopthe whole kinematic part of the seismic raymethodusingthe well-knownSnells law. Such approaches have been used for a long time in seismology and have givena number of valuable results. There may be, however, certain methodological objections totheir application. In the application of Snells law, we must start from a model consistingof homogeneous layers with curved interfaces and pass from this model to a smoothlyvarying model by increasing the number of interfaces. Such a limiting process offers veryuseful seismological insights into the ray tracing equations and travel-time computations ininhomogeneous media, but it is more or less intuitive. The Fermat principle has been used inseismology as a rule independently for Pand Swaves propagating in inhomogeneous media.The elastic waveeld, however, can be separated into P and S waves only in homogeneousmedia (and perhaps in some other simple structures). In laterally varying media with curvedinterfaces, the waveeld is not generally separable into P and S waves; the seismic waveprocess is more complicated. Thus, we do not have any exact justication for applying theprinciple independently to P and S waves. In media with larger velocity gradients, the raymethod fails due to the strong coupling of P and S waves. Only the approach based on theasymptotic solution of the elastodynamic equation gives the correct answer: the separationof the seismic waveeld in inhomogeneous media into two independent wave processes (Pand S) is indeed possible, but it is only approximate, in that it is valid for high frequenciesand sufciently smooth media only.Similarly, certain properties of vectorial complex-valued amplitudes of seismic bodywaves can be derived using energy concepts, particularly using the expressions for theenergy ux. Such an approach is again very useful for intuitive physical understanding ofthe amplitude behavior, but it does not give the complete answer. The amplitudes of seismicbody waves have a vectorial complex-valued character. The waves may be elliptically polar-ized (S waves) and may include phase shifts. These phase shifts inuence the waveforms.The energy principles do not yield a complete answer in such situations. Consequently,they cannot be applied to the computation of synthetic seismograms and particle groundmotion diagrams.Recently, several new concepts and methods have been proposed to increase the pos-sibilities and efciency of the standard ray method; they include dynamic ray tracing,the ray propagator matrix, and paraxial ray approximations. In the standard ray method,the travel time and the displacement vector of seismic body waves are usually evaluatedalong rays. Thus, if we wish to evaluate the seismic waveeld at any point, we must ndthe ray that passes through this point (boundary value ray tracing). The search for suchrays sometimes makes the application of the standard ray method algorithmically veryinvolved, particularly in 3-D layered structures. The paraxial ray methods, however, allowone to compute the travel time and displacement vector not only along the ray but also inits paraxial vicinity. It is not necessary to evaluate the ray that passes exactly through thepoint. The knowledge of the ray propagator matrix makes it possible to solve analyticallymany complex wave propagation problems that must be solved numerically by iterations inthe standard ray method. This capability greatly increases the efciency of the ray method,particularly in 3-D complex structures.The nal ray solution of the elastodynamic equation is composed of elementary wavescorresponding to various rays connecting the source and receiver. Each of these elementarywaves (reected, refracted, multiply reected, converted, and the like) is described byits own ray series. In practical seismological applications, the higher terms of the rayseries have not yet been broadly used. In most cases, the numerical modeling of seismicwaveelds and the interpretation of seismic data by the ray method have been based on theI NTRODUCTI ON 3zeroth-order leading term of the ray series. In this book, mainly the zeroth-order termsof the ray series are considered. These zeroth-order terms, however, are treated here in agreat detail. Concise expressions for the zeroth-order ray-theory Green function for a pointsource and receiver situated at any place in a general 3-D, layered and blocked, structureare derived. For a brief treatment of the higher-order terms of the ray series for the scalar(acoustic) and vectorial (elastic) waves see Sections 5.6 and 5.7.As is well known, the ray method is only approximate, and its applications to certainseismological problems have some restrictions. Recently, several new extensions of the raymethod have been proposed; these extensions overcome, partially or fully, certain of theserestrictions. They include the method of summation of Gaussian beams, the method ofsummation of Gaussian wave packets, and the Maslov-Chapman method. These methodshave been found very useful in solving various seismological problems, even though certainaspects of these methods are still open for future research.The whole book may be roughly divided into ve parts.In the rst part, the main principles of the asymptotic high-frequency method as it isusedtosolve the elastodynamic equationina 3-Dlaterallyvaryingmediumare brieyexplained and discussed. Aparticularly simple approach is used to derive and discussthe most important equations and related wave phenomena from the seismologicalpoint of view. It is shown how the elastic waveeld is approximately separated intoindividual elementary waves. These individual waves propagate independently in asmoothly varying structure, their travel times are controlled by the eikonal equation,and their amplitudes are controlled by the transport equation. Various important phe-nomena of seismic waveelds connected with 3-D lateral variations and with curvedinterfaces are derived and explained, both for isotropic and anisotropic media. Greatattention is devoted to the differences between elastic waves propagating in isotropicand anisotropic structures. Exact and approximate expressions for acoustic and elas-todynamic Green functions in homogeneous media are also derived. See Chapter 2.The second part is devoted to ray tracing and travel-time computation in 3-Dstructures.The ray tracing and travel-time computation play an important role in many seis-mological applications, particularly in seismic inversion algorithms, even without astudy of ray amplitudes, polarization, and wavelet shape. In addition to individualrays, the ray elds are also introduced in this part. The singular regions of the rayelds and related wave phenomena are explained. Special attention is devoted tothe denition, computation, and physical meaning of the geometrical spreading. SeeChapter 3.The third part is devoted to dynamic ray tracing and paraxial ray methods. The paraxialray methods can be used to compute the travel time and other important quantities notonly along the ray but also in its vicinity. Concepts of dynamic ray tracing and of theray propagator matrix are explained. The dynamic ray tracing is introduced both inray-centered and Cartesian coordinates, for isotropic and anisotropic structures. Var-ious important applications of the paraxial ray method are explained. See Chapter 4.The fourth part of the book discusses the computation of ray amplitudes. Very generalexpressions for ray amplitudes of an arbitrary multiply reected/transmitted (possi-bly converted) seismic body wave propagating in acoustic, elastic isotropic, elasticanisotropic, laterally varying, layered, and block structures are derived. The mediummay also be weakly dissipative. Both the source and the receiver may be situatedeither in a smooth medium or at a structural interface or at the Earths surface. Final4 I NTRODUCTI ONequations for the amplitudes of the ray-theory elastodynamic Green function is lat-erally varying layered structures are derived. Great attention is also devoted to theray-series solutions, both in the frequency and the time domain. The seismologicalapplications of higher-order terms of the ray series are discussed. See Chapter 5.The fth part explains the computation of ray synthetic seismograms and ray syntheticparticle ground motions. Several possibilities for the computation of ray syntheticseismograms are proposed: in the frequency domain, in the time domain, and bythe summation of elementary seismograms. Advantages and disadvantages of indi-vidual approaches are discussed. Certain of these approaches may be used even fordissipative media. The basic properties of linear, elliptic, and quasi-elliptic polariza-tion are described. The causes of quasi-elliptic polarization of S waves are brieysummarized. See Chapter 6.This book, although very extensive, is still not able to cover all aspects of the seismicray method. This would increase its length inadmissibly. To avoid this, the author has notdiscussedmanyimportant subjects regardingthe seismic raymethod(or relatedcloselytoit)or has discussed them only briey. Nevertheless, the reader should remember that the mainaimof this book is to present a detailed and complete description of the seismic ray methodwith a real-valued eikonal for 3-D, laterally varying, isotropic or anisotropic, layered, andblock structures. The author, however, does not and had no intention of including all theextensions and applications of the seismic ray method and all the problems related closelyto it. We shall now briey summarize several important topics that are related closely tothe seismic ray method but that will not be treated in this book or that will be treated morebriey than they would deserve.1. Although the seismic ray method developed in this book plays a fundamental rolein various inverse problems of seismology and of seismic exploration and in manyinterpretational procedures, the actual inversion and interpretational proceduresare not explicitly discussed here. These procedures include seismic tomography,seismic migration, and the location of earthquake sources, among others.2. The seismic ray method has found important applications in forward and inversescattering problems. With the exception of a brief introduction in Section 2.6.2; thescattering problems themselves, however, are not discussed here.3. The seismic ray method may be applied only to structural models that satisfy certainsmoothness criteria. The construction of 2-D and 3-D models that would satisfysuch criteria is a necessary prerequisite for the application of the seismic ray method,but is not discussed here at all. Mostly, it is assumed that the model is specied inCartesian rectangular coordinates. Less attention is devoted to models specied incurvilinear coordinate systems (including spherical); see Section 3.5.4. The seismic ray method developed here may be applied to high-frequency seismicbody waves propagating in deterministic, perfectly elastic, isotropic or anisotropicmedia. Other types of waves (such as surface waves) are only briey mentioned.Moreover, viscoelastic, poroelastic, and viscoporoelastic models are not considered.The exception is a weakly dissipative (and dispersive) model that does not requirecomplex-valued ray tracing; see Sections 5.5 and 6.3.5. In Sections 2.6.4 and 5.6.8,the space-time ray method and the ray method with a complex eikonal are brieydiscussed, even though they deserve considerably more attention. The computationof complex-valued rays in particular (for example, in dissipative media, in theI NTRODUCTI ON 5caustic shadow, and in some other singular regions) may be very important inapplications. Actually, the seismic ray method, without considering complex rays,is very incomplete.5. Various extensions of the seismic ray method have been proposed in the literature.These extensions include the asymptotic diffraction theory, the method of edgewaves, the method of the parabolic wave equation, the Maslov-Chapman method,and the method of summation of Gaussian beams or Gaussian wave packets, amongothers. Here we shall treat, in some detail, only the extensions based on the sum-mation of paraxial ray approximations and on the summation of paraxial Gaussianbeams; see Section 5.8. The method based on the summation of paraxial ray approx-imations yields integrals close or equal to those of the Maslov-Chapman method.The other extensions of the seismic ray method are discussed only very briey inSection 5.9, but the most important references are given there.6. No graphical examples of the computation of seismic rays, travel times, ray ampli-tudes, synthetic seismograms, and particle ground motions in 3-D complex modelsare presented for two reasons. First, most gures would have to be in color, as 3-Dmodels are considered. Second, the large variety of topics discussed in this bookwould require a large number of demonstration gures. This would increase thelength and price of the book considerably. The interested readers are referred to thereferences given in the text, and to the www pages of the Consortium ProjectSeismic Waves in 3-D Complex Structures; see http://seis.karlov.mff.cuni.cz/consort/main.htm for some examples.The whole book has a tutorial character. The equations presented are (in most cases)derived and discussed in detail. For this reason, the book is rather long. Owing to theextensive use of various matrix notations and to the applications of several coordinatesystems and transformation matrices, the resulting equations are very concise and simplyunderstandable from a seismological point of view. Although the equations are given in aconcise and compact form, the whole book is written in an algorithmic way: most of theexpressions are specied to the last detail and may be directly used for programming.To write the complicated equations of this book in the most concise form, we usemostly matrix notation. To distinguish between 2 2 and 3 3 matrices, we shall usethe circumex () above the letter for 3 3 matrices. If the same letter is used for both2 2 and 3 3 matrices; for example, M and M, matrix M denotes the 2 2 left uppersubmatrix of M:M=__M11 M12 M13M21 M22 M23M31 M32 M33__. M=_M11 M12M21 M22_.Similarly, we denote by q = (q1. q2. q3)Tthe 3 1 column matrix and by q = (q1. q2)Tthe 2 1 column matrix. The symbol T as a superscript denotes the matrix transpose.Similarly, the symbol 1T as a superscript denotes the transpose of the inverse, A1T=(A1)T. A1T= (A1)T.In several places, we also use 4 4 and 6 6 matrices. We denote them by boldfaceletters in the same way as the 2 2 matrices; this notation cannot cause any misunder-standing.In parallel with matrix notation, we also use component notation where suitable. Theindices always have the form of right-hand sufxes. The uppercase sufxes take the values6 I NTRODUCTI ON1 and 2, lowercase indices 1, 2 and 3, and greek lowercase indices 1, 2, 3 and 4. In this way,MIJ denote elements of M and Mi j elements of M. We also denote f (xi) = f (x1. x2. x3),f (xI) = f (x1. x2), [ f (xi)]xk=0 = f (0. 0. 0), [ f (xi)]xK=0 = f (0. 0. x3), [ f (xi)]x1=0 =f (0. x2. x3). The Einstein summation convention is used throughout the book. Thus,MI JqJ = MI 1q1 MI 2q2 (I = 1 or 2), Mi jqj = Mi 1q1 Mi 2q2 Mi 3q3 (i = 1. 2 or 3).Similarly, Mi J denotes the elements of the 3 2 submatrix of matrix M.We also use the commonly accepted notation for partial derivatives with respect toCartesian coordinates xi (for example, .i = ,xi, ui. j i = 2ui,xjxi, i j. j =i j,xj). In the case of velocities, we shall use a similar notation to denote the partialderivatives with respect to the ray-centered coordinates. For a more detailed explanation,see the individual chapters.In some equations, the classical vector notation is very useful. We use arrows aboveletters to denote the 3-D vectors. In this way, any 3-D vector may be denoted equivalentlyas a 3 1 column matrix or as a vectorial form.In complex-valued quantities, z = x iy, the asterisk is used as a superscript to de-note a complex-conjugate quantity, z = x iy. The asterisk between two time-dependentfunctions, f1(t ) f2(t ), denotes the time convolution of these two functions, f1(t ) f2(t ) =_ f1() f2(t )d.The book does not give a systematic bibliography on the seismic ray method. Formany other references, see the books and review papers on the seismic ray method and onsome related subjects (Cerven y and Ravindra 1971; Cerven y, Molotkov, and P sen ck 1977;Hubral and Krey 1980; Hanyga, Lenartowicz, and Pajchel 1984; Bullen and Bolt 1985;Cerven y 1985a, 1985b, 1987a, 1989a; Chapman 1985, in press; Virieux 1996; Dahlen andTromp 1998). The ray method has been also widely used in other branches of physics,mainly in electromagnetic theory (see, for example, Synge 1954; Kline and Kay 1965;Babich and Buldyrev 1972; Felsen and Marcuvitz 1973; Kravtsov and Orlov 1980).CHAPTER TWOThe Elastodynamic Equationand Its Simple SolutionsThe seismic ray method is based on asymptotic high-frequency solutions of theelastodynamic equation. We assume that the reader is acquainted with linear elas-todynamics and with the simple solutions of the elastodynamic equation in a ho-mogeneous medium. For the readers convenience, we shall briey discuss all these topicsin this chapter, particularly the plane-wave and point-source solutions of the elastodynamicequation. We shall introduce the terminology, notations, and all equations we shall need inthe following chapters. In certain cases, we shall only summarize the equations without de-riving them, mainly if such equations are known from generally available textbooks. Thisapplies, for example, to the basic concepts of linear elastodynamics. In other cases, weshall present the main ideas of the solution, or even the complete derivation. This applies,for example, to the Green functions for acoustic, elastic isotropic and elastic anisotropichomogeneous media.In addition to elastic waves in solid isotropic and anisotropic models, we shall alsostudy pressure waves in uid models. In this case, we shall speak of the acoustic case.There are two main reasons for studying the acoustic case. The rst reason is tutorial.All the derivations for the acoustic case are very simple, clear, and comprehensible. Inelastic media, the derivations are also simple in principle, but they are usually more cum-bersome. Consequently, we shall mostly start the derivations with the acoustic case, andonly then shall we discuss the elastic case. The second reason is more practical. Pres-sure waves in uid models are often used as a simple approximation of P elastic wavesin solid models. For example, this approximation is very common in seismic explorationfor oil.The knowledge of plane-wave solutions of the elastodynamic equation in homogeneousmedia is very useful in deriving approximate high-frequency solutions of elastodynamicequation in smoothly inhomogeneous media. Such approximate high-frequency solutionsin smoothly inhomogeneous media are derived in Section 2.4. In the terminology of the ray-series method, such solutions represent the zeroth-order approximation of the ray method.The approach we shall use in Section 2.4 is very simple and is quite sufcient to deriveall the basic equations of the zeroth-order approximation of the ray method for acoustic,elastic isotropic, and elastic anisotropic structures. In the acoustic case, the approach yieldsthe eikonal equation for travel times and the transport equation for scalar amplitudes. Inthe elastic case, it yields an approximate high-frequency decomposition of the wave eldinto the separate waves (P and S waves in isotropic; qP, qS1, and qS2 in anisotropic media).Thereafter, it yields the eikonal equations for travel times, the transport equations foramplitudes, and the rules for the polarization of separate waves.78 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONSNote that Section 2.4 deals only with the zeroth-order approximation of the ray-seriesmethod. The higher-order terms of the ray series are not discussed here, but will beconsidered in Chapter 5 (Sections 5.6 for the acoustic case and Section 5.7 for the elasticcase). From the systematic and theoretical points of view, it would be more convenientto start the whole treatment directly with the ray-series method, and only then discussthe zeroth-order approximation, as a leading term of the ray-series method. The reasonwhy we have moved the ray-series treatment to Sections 5.6 and 5.7 is again tutorial. Thecomplete treatment of rays, ray-theory travel times, and paraxial methods in Chapters 3and 4 is based on eikonal equations only. Similarly, all the treatments of ray amplitudesin Sections 5.1 through 5.5 are based on transport equations only. Thus, we do not needto know the higher-order terms of the ray series in Chapters 3 and 4 and in Sections 5.1through 5.5; the results of Section 2.4 are sufcient there. Consequently, the whole ray-series treatment, which is more cumbersome than the derivation of Section 2.4, can bemoved to Sections 5.6 and 5.7. Most of the recent applications of the seismic ray methodare based on the zeroth-order approximation of the ray series. Consequently, most readerswill be interested in the relevant practical applications of the seismic ray method, such asray tracing, travel time, and ray amplitude computations. These readers need not botherwith the details of the ray-series method; the zeroth-order approximation, derived in Sec-tion 2.4, is sufcient for them. The readers who wish to know more about the ray-seriesmethod and higher order terms of the ray series can read Sections 5.6 and 5.7 immediatelyafter reading Section 2.4. Otherwise, no results of Sections 5.6 and 5.7 are needed in theprevious sections.Section 2.5 discusses the point-source solutions and appropriate Green functions forhomogeneous uid, elastic isotropic, and elastic anisotropic media. In all three cases, exactexpressions for the Green function are derived uniformly. For elastic anisotropic media,exact expressions are obtained only in an integral form. Suitable asymptotic high-frequencyexpressions are, however, given in all three cases. These expressions are used in Chapter 5to derive the asymptotic high-frequency expressions for the ray-theory Green functioncorresponding to an arbitrary elementary wave propagating in a 3-D laterally varyinglayered and blocked structure (uid, elastic isotropic, elastic anisotropic).The Green function corresponds to a point source, but it may be used in the repre-sentation theorem to construct considerably more complex solutions of the elastodynamicequation. If we are interested in high-frequency solutions, the ray-theory Green functionmay be used in the representation theorems. For this reason, representation theorems andthe ray-theory Green functions play an important role even in the seismic ray method. Therepresentation theorems are derived and briey discussed in Section 2.6. The same sec-tion also discusses the scattering integrals and the rst-order Born approximation. Theseintegrals contain the Green function. If we use the ray-theory Green function in these inte-grals, the resulting scattering integrals can be used broadly in the seismic ray method andin relevant applications. Such approaches have recently found widespread applications inseismology and seismic exploration.2.1 Linear ElastodynamicsThe basic concepts and equations of linear elastodynamics have been explained in manytextbooks and papers, including some seismological literature. We refer the reader toBullen (1965), Auld (1973), Pilant (1979), Aki and Richards (1980), Hudson (1980a),Ben-Menahem and Singh (1981), Mura (1982), Bullen and Bolt (1985), and Davis (1988),2. 1 LI NEAR ELASTODYNAMI CS 9where many other references can be found. For a more detailed treatment, see Love (1944),Landau and Lifschitz (1965), Fung (1965), and Achenbach (1975). Here we shall introduceonly the most useful terminology and certain important equations that we shall need later.We shall mostly follow and use the notations of Aki and Richards (1980).To write the equations of linear elastodynamics, some knowledge of tensor calculusis required. Because we wish to make the treatment as simple as possible, we shall useCartesian coordinates xi and Cartesian tensors only.We shall use the Lagrangian description of motion in an elastic continuum. In theLagrangian description, we study the motion of a particle specied by its original positionat some reference time. Assume that the particle is located at the position described byCartesian coordinates xi at the reference time. The vector distance of a particle at time tfrom position x at the reference time is called the displacement vector and is denoted by u. Obviously, u = u( x. t ).We denote the Cartesian components of the stress tensor by i j( x. t ) and the Cartesiancomponents of the strain tensor by ei j( x. t ). Both tensors are considered to be symmetric,i j = j i. ei j = ej i. (2.1.1)The strain tensor can be expressed in terms of the displacement vector as follows:ei j = 12(ui. j uj.i). (2.1.2)The stress tensor i j( x. t ) fully describes the stress conditions at any point x. It can be usedto compute traction T acting across a surface element of arbitrary orientation at x,Ti = i jnj. (2.1.3)where n is the unit normal to the surface element under consideration.The elastodynamic equation relates the spatial variations of the stress tensor with thetime variations of the displacement vector,i j. j fi = ui. i = 1. 2. 3. (2.1.4)Here fi denote the Cartesian components of body forces (force per volume), and is thedensity. The term with fi in elastodynamic equation (2.1.4) will also be referred to as thesource term. Quantities ui = 2ui,t2. i = 1. 2. 3. represent the second partial derivativesof ui with respect to time (that is, the Cartesian components of particle acceleration u). Ina similar way, we shall also denote the Cartesian components of particle velocity ui,tby vi or ui.The introduced quantities are measured in the following units: stress i j and tractionTi in pascals (Pa; that is, in kg m1s2), the components of body forces fi in newtonsper cubic meter (N/m3; that is, in kg m2s2), density in kilograms per cubic meter(kg m3), and displacement components ui in meters (m). Finally, strain components ei jare dimensionless.2.1.1 Stress-Strain RelationsIn a linear, anisotropic, perfectly elastic solid, the constitutive stress-strain relation is givenby the generalized Hookes law,i j = ci j klekl. (2.1.5)10 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONSHere ci j kl are components of the elastic tensor. The elastic tensor has, in general,3 3 3 3 = 81 components. These components, however, satisfy the following sym-metry relations:ci j kl = cj i kl = ci jlk = ckli j. (2.1.6)which reduce the number of independent components of the elastic tensor from 81 to 21.The components ci j kl of the elastic tensor are also often called elastic constants, elasticmoduli, elastic parameters, or stiffnesses. In this book, we shall mostly call them elasticmoduli. They are measured in the same units as the stress components (that is, in Pa =kg m1s2).If we express ekl in terms of the displacement vector components, see Equation (2.1.2),and take into account symmetry relations (2.1.6), we can also express Equation (2.1.5) inthe following form:i j = ci j kluk.l. (2.1.7)The components of elastic tensor ci j kl are also often expressed in an abbreviated Voigtform, with two indices instead of four. We shall denote these components by capital lettersCmn. Cmn is formed from ci j kl in the following way: m corresponds to the rst pair ofindices, i. j and n to the second pair, k. l. The correspondence m i. j and n k. l isas follows: 1 1. 1; 2 2. 2; 3 3. 3; 4 2. 3; 5 1. 3; 6 1. 2.Due to symmetry relations (2.1.6), the 6 6 matrix Cmn fully describes the elasticmoduli of an arbitrary anisotropic elastic medium. It is also symmetric, Cmn = Cnm and iscommonly expressed in the form of a table containing 21 independent elastic moduli:________C11 C12 C13 C14 C15 C16C22 C23 C24 C25 C26C33 C34 C35 C36C44 C45 C46C55 C56C66________. (2.1.8)The elastic moduli Cmn below the diagonal (m > n) are not shown because the table issymmetrical, Cmn = Cnm. The diagonal elements in the table are always positive for a solidmedium, but the off-diagonal elements may be arbitrary (positive, zero, negative). Notethat Cmn is not a tensor.A whole hierarchy of various anisotropic symmetry systems exist. They are describedand discussed in many books and papers; see, for example, Fedorov (1968), Musgrave(1970), Auld (1973), Crampin and Kirkwood (1981), Crampin (1989), and Helbig (1994).The most general is the triclinic symmetry, which may have up to 21 independent elasticmoduli. In simpler (higher symmetry) anisotropic systems, the elastic moduli are invariantto rotation about a specic axis by angle 2,n (n-fold axis of symmetry). We shall brieydiscuss only two such simpler systems which play an important role in recent seismologyand seismic exploration: orthorhombic and hexagonal.In the orthorhombic symmetry system, three mutually perpendicular twofold axes ofsymmetry exist. The number of signicant elastic moduli in the orthorhombic system isreduced to nine. If the Cartesian coordinate system being considered is such that its axes2. 1 LI NEAR ELASTODYNAMI CS 11coincide with the axes of symmetry, the table (2.1.8) of Ci j reads:________C11 C12 C13 0 0 0C22 C23 0 0 0C33 0 0 0C44 0 0C55 0C66________. (2.1.9)In the hexagonal symmetry system, one sixfold axis of symmetry exists. The numberof signicant elastic moduli in the hexagonal system is reduced to ve. If the Cartesiancoordinate system being considered is such that the x3-axis coincides with the sixfold axisof symmetry, the table of nonvanishing elastic moduli is again given by (2.1.9), but Ci jsatisfy the following four relations: C22 = C11. C55 = C44. C12 = C112C66. C23 = C12.If the sixfold axis of symmetry coincides with the x1-axis, the four relations are as follows:C22 = C33. C55 = C66. C23 = C222C44. C13 = C12.It is possible to show that the invariance of elastic moduli to rotation by ,3 in thehexagonal system (sixfold axis of symmetry) implies general invariance to rotation by anyangle. From this point of view, the hexagonal symmetry system is equivalent to a trans-versely isotropic mediumin which the elastic moduli do not change if the mediumis rotatedabout the axis of symmetry by any angle. Traditionally, the vertical axis of symmetry of thetransversely isotropic medium has been considered. At present, however, the transverselyisotropic mediumis considered more generally, with an arbitrarily oriented axis of symme-try (inclined, horizontal). If it is vertical, we speak of azimuthal isotropy (Crampin 1989).Inseismology, the most commonlyusedanisotropysystems correspondtothe hexagonalsymmetry. Systems more complicated than orthorhombic have been used only exception-ally. Moreover, the anisotropy is usually weak in the Earths interior. A suitable notationfor elastic moduli in weakly anisotropic media was proposed by Thomsen (1986).As we have seen, the abbreviated Voigt notation Cmn for elastic moduli is useful indiscussing various anisotropy symmetries, particularly if simpler symmetries are involved.If, however, we are using general equations and expressions, such as the constitutive rela-tions or the elastodynamic equation and its solutions, it is more suitable to use the elasticmoduli in the nonabbreviated form of ci j kl. Due to the Einstein summation convention, allgeneral expressions are obtained in a considerably more concise form. For this reason, weshall consistently use the notation ci j kl for elastic moduli rather than Cmn. As an exercise,the reader may write out the general expressions for some simpler anisotropy symmetries,using the notation Cmn for the elastic parameters.Elastic anisotropy is a very common phenomenon in the Earths interior (see Babu skaand Cara 1991). It is caused by different mechanisms. Let us briey describe three of thesemechanisms. Note, of course, that the individual mechanisms may be combined.1. Preferred orientation of crystals. Single crystals of rock-forming minerals areintrinsically anisotropic. If no preferred orientation of mineral grains exists, poly-crystalline aggregates of anisotropic material behave macroscopically isotropically.In case of preferred orientation of these grains, however, the material behavesmacroscopically anisotropically. Preferred orientation is probably one of the mostimportant factors in producing anisotropy of dense aggregates under high pressureand temperature conditions.12 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONS2. Anisotropy due to alligned inclusions. The presence of aligned inclusions (suchas cracks, pores, or impurities) can cause effective anisotropy of rocks, if observed atlong wavelengths. The most important two-phase systems with distinct anisotropicbehavior are a cracked solid and a poroelastic solid.3. Anisotropy due to regular sequences of thin layers. Regular sequences ofisotropic thin layers of different properties are very common in the Earths inte-rior, at least in the upper crust (foliation, bedding, and so on). If the prevailingwavelength of the wave under consideration is larger than the thickness of theindividual thin layers, the regular sequences of thin layers behave anisotropically.Let us give several typical examples of actual anisotropy symmetries important in seis-mology:a. Hexagonal symmetry. (i) Vertical axis of symmetry: periodic horizontal thin layer-ing. This symmetry is also known as VTI (vertical transverse isotropy) symmetry.(ii) Horizontal axis of symmetry: parallel vertical cracks. This symmetry is alsoknown as HTI (horizontal transverse isotropy) symmetry.b. Orthorhombic symmetry. (i) Olivine (preferred orientation of crystals) and (ii)Combination of periodic thin layering with cracks perpendicular to the layering.In the isotropic solid, the components of elastic tensor ci j kl can be expressed in termsof two independent elastic moduli and j as follows:ci j kl = i jkl j(i kjl ilj k); (2.1.10)see Jeffreys and Jeffreys (1966) and Aki and Richards (1980). Here i j is the Kroneckersymbol,i j = 1 for i = j. i j = 0 for i ,= j. (2.1.11)Elastic moduli . j are also known as Lam es elastic moduli; j is called the rigidity (orshear modulus).For an isotropic medium stress-strain relation, Equation (2.1.5) can be expressed in thefollowing form:i j = i j 2jei j. = ekk = uk.k = u. (2.1.12)Equation (2.1.12) represents the famous Hookes law. Quantity is called the cubicaldilatation. If we replace and ei j by the components of the displacement vector, we obtaini j = i juk.k j(ui. j uj.i). (2.1.13)Instead of Lam es elastic moduli and j, some other elastic parameters are also oftenused in isotropic solids: the bulk modulus (or incompressibility) k, the Young modulus E,the Poisson ratio , and compressibility . They are related to and j as follows:k = 23j. E = 3j_ 23j__( j). = k1=_ 23j_1. = 12,( j).(2.1.14)The physical meaning of these parameters is explained in many textbooks (Bullen and Bolt1985; Auld 1973; Pilant 1979).If j = 0, we speak of a uid medium(Bullen and Bolt 1985). In this case, =12, k = ,and Hookes law (2.1.12) reads i j = i j. Commonly, we then use pressure p instead2. 1 LI NEAR ELASTODYNAMI CS 13of i j,i j = pi j. p = 13i i. (2.1.15)Constitutive relation (2.1.12) then readsp = = k. (2.1.16)The uid medium is commonly used in seismic prospecting for oil as an approximation ofthe solid medium (the so-called acoustic case).Pressure p and elastic moduli , j, k, and E are measured in Pa = kg m1s2, thecompressibility is in kg1m s2, and is dimensionless.Constitutive relations (2.1.5), (2.1.12), and (2.1.16) are related to small deviations froma natural reference state, in which both stress and strain are zero. Such a natural referencestate, however, does not exist within the Earths interior due to the large lithological pres-sure caused by self-gravitation. It is then necessary to use some other reference state, forexample, the state of static equilibrium. By denition the strain is zero in this state, but thestress is nonzero. It is obvious that constitutive relations (2.1.5), (2.1.12), and (2.1.16), arenot valid in this case. If we study the deviations fromthe state of static equilibrium, we can,however, work with small incremental stresses instead of actual stresses. The incrementalstress is dened as the difference between the actual stress and the stress corresponding tothe static equilibrium state. This incremental stress is zero in the state of static equilibriumand satises constitutive relations (2.1.5), (2.1.12), and (2.1.16) if the deviations from thestate of static equilibrium are small. In the following text, we shall not emphasize the factthat i j actually represents the incremental stress, and simply call i j the stress. See thedetailed discussion in Aki and Richards (1980).All the elastic moduli are, in general, functions of position. If they are constant (inde-pendent of position) in some region, we call the medium homogeneous in that region.Even more complex linear constitutive relations than those given here can be consid-ered. In linear viscoelastic solids, the elastic moduli depend on time. In the time domain, theconstitutive relations are then expressed in convolutional forms. Alternatively, in the fre-quency domain, the elastic moduli are complex-valued and may depend on frequency. Suchconstitutive relations are commonly used to study wave propagation in dissipative media;see Kennett (1983). Unless stated otherwise, we shall consider only frequency- and time-independent elastic moduli here. For weakly dissipative media, see Sections 5.5 and 6.3.2.1.2 Elastodynamic Equation for Inhomogeneous Anisotropic MediaInserting relation (2.1.7) into elastodynamic equation (2.1.4), we obtain the elastodynamicequation for an unbounded anisotropic, inhomogeneous, perfectly elastic medium:(ci j kluk.l). j fi = ui. i = 1. 2. 3. (2.1.17)The elastodynamic equation (2.1.17) represents a system of three coupled partial dif-ferential equations of the second order for three Cartesian components ui(xj. t ) of thedisplacement vector u.Alternatively, the elastodynamic equation may be expressed in terms of 12 partialdifferential equations of the rst order in three Cartesian components vi(xj. t ) of the particlevelocity vector v = u, and 9 components i j of the stress tensor. We use Equation (2.1.4)14 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONSfor vi = ui and the time derivative of (2.1.7) for i j. We obtain vi = i j. j fi. i j = ci j klvk.l mi j. (2.1.18)Function mi j is nonvanishing in regions where the generalized Hookes law (2.1.7) is notvalid, for example, in the source region. It is closely related to the so-called stress-glut andto the moment tensor density. See Kennett (1983, p. 76). System (2.1.18) is alternative to(2.1.17), if we replace fi in (2.1.17) by fi mi j. j (equivalent force system).Inthis book, we shall use the elastodynamic equationinthe formof (2.1.17) consistently.If mi j. j ,= 0, it is understood that fi represents fi mi j. j. Equations (2.1.18) will be used inSection 5.4.7 only. It is, however, also possible to build the seismic ray method completelyon the elastodynamic equation in the form of (2.1.18). See, for example, Chapman andCoates (1994) and Chapman (in press).2.1.3 Elastodynamic Equation for Inhomogeneous Isotropic MediaAs inthe previous case, we insert (2.1.13) into(2.1.4) andobtainthe elastodynamic equationfor the unbounded isotropic, inhomogeneous, perfectly elastic medium,(uj. j).i [j(ui. j uj.i)]. j fi = ui. i = 1. 2. 3. (2.1.19)If we perform the derivatives, we obtain,( j)uj.i j jui. j j .iuj. j j. j(ui. j uj.i) fi = ui.i = 1. 2. 3. (2.1.20)This equation is often written in vectorial form:( j) u j2 u uj u 2(j ) u f = u. (2.1.21)As we can see, the elastodynamic equation (2.1.17) for the anisotropic inhomogeneousmediumis formally simpler than elastodynamic equations (2.1.19) or (2.1.20) for isotropicinhomogeneous media. This is, of course, due to the summation convention. Consequently,we shall often prefer to work with (2.1.17), and only then shall we specify the results forisotropic media.2.1.4 Acoustic Wave EquationThe elastodynamic equations remain valid even in a uid medium where j = 0. However,rather than working with displacement vector ui, it is then more usual to work with pressurep = 13i i and with particle velocity vi = ui. The acoustic wave equations for nonmovinguids are then usually expressed asp.i vi = fi. vi.i p = q. (2.1.22)Functions fi = fi( x. t ) and q = q( x. t ) represent source terms; fi has the same meaning asin (2.1.4) and q is the volume injection rate density; see Fokkema and van den Berg (1993).Brekhovskikh and Godin (1989) call the same term q the volume velocity source. For adetailed derivation of acoustic wave equations, even for moving media, see Brekhovskikhand Godin (1989). Equations (2.1.22) correspond to the elastodynamic equations (2.1.18)for j = 0, i j given by (2.1.15) and q = 13 mi i.2. 1 LI NEAR ELASTODYNAMI CS 15If we eliminate the particle velocity vi from (2.1.22), we obtain the scalar acousticwave equation for pressure p( x. t ),(1p.i).i f p= p. where f p= q (1fi).i. (2.1.23)Alternatively, (2.1.23) reads (1p) f p= p. (2.1.24)If density is constant, Equation (2.1.24) yields2p f p= c2 p. c =_1, =_k,. (2.1.25)Equation (2.1.25) represents the standard mathematical formof the scalar wave equation, asknown frommathematical textbooks; see Morse and Feshbach (1953), Jeffreys and Jeffreys(1966), Bleistein (1984), and Berkhaut (1987), among others. Quantity c = c( x) is calledthe acoustic velocity. The acoustic velocity c( x) is a model parameter, c = ()1,2. Itdoes not depend at all on the properties of the waveeld under consideration, but only on and . Here we shall systematically use the acoustic wave equation with a variable density(2.1.23) or (2.1.24). We will usually consider fi = 0, so that f p= q.We can also eliminate the pressure from (2.1.22). We then obtain the vectorial acousticwave equation for the particle velocity vi( x. t )(kvj. j).i f vi = vi. where f vi = fi (kq).i.Here k is the bulk modulus. This wave equation is a special case of the time derivativeof the elastodynamic equation (2.1.19) for j = 0. For this reason, we shall not study itseparately. By the acoustic wave equation, we shall understand the scalar acoustic waveequation for pressure (2.1.23).2.1.5 Time-Harmonic EquationsIn this book, we shall mostly work directly in the time domain, with transient signals.Nevertheless, it is sometimes useful to discuss certain problems in the frequency domain.Let us consider the general elastodynamic equation (2.1.17) with a time-harmonic sourceterm f ( x. t ) = f ( x. ) exp[it ]. Displacement vector u( x. t ) is then also time-harmonic, u( x. t ) = u( x. ) exp[it ]. (2.1.26)Here = 2 f = 2,T is the circular frequency, where f is frequency and T is pe-riod. To keep the notation as simple as possible, we do not use new symbols for u( x. )and f ( x. ); we merely distinguish them from u( x. t ) and f ( x. t ) by using argument instead of t . Moreover, we shall only use arguments and t where confusion might arise.Instead of circular frequency , we shall also often use the frequency f = ,2.The elastodynamic equation (2.1.17) in the frequency domain reads(ci j kluk.l). j 2ui = fi. i = 1. 2. 3. (2.1.27)where ui = ui( x. ). fi = fi( x. ). Elastodynamic equations (2.1.19) through (2.1.21) canbe expressed in the frequency domain in the same way.For acoustic media, we introduce the time-harmonic quantities p( x. ), q( x. ), v( x. ),and f ( x. ) as in (2.1.26). The general acoustic equations (2.1.22) then readp.i ivi = fi. vi.i ip = q. (2.1.28)16 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONSThe acoustic wave equation for pressure (2.1.24) in the frequency domain reads as follows: (1p) 2p = f p. f p= iq (1fi).i. (2.1.29)If density is constant, wave equation (2.1.25) yields2p k2p = f p. k = ,c. (2.1.30)where k is the wave number. Equation (2.1.30) is known as the Helmholtz equation.Instead of the time factor exp[it ] in (2.1.26), it would also be possible to use factorexp[it ]. We will, however, use exp[it ] consistently, even in the Fourier transform.The Fourier transform pair for a transient signal x(t ) will be used here in the followingform:x( f ) =_ x(t ) exp[i2 f t ]dt. x(t ) =_ x( f ) exp[i2 f t ]d f.(2.1.31)Function x( f ) is the Fourier spectrumof x(t ). We againuse the same symbol for the transientsignal x(t ) (in the time domain) and for its Fourier spectrumx( f ) (in the frequency domain);we distinguish between the two by arguments t and f , if necessary. The sign conventionused in Fourier transform (2.1.31) is common in books and papers related to the seismicray method. For more details on the Fourier transform, see Appendix A.Note that the Fourier spectrum of the transient displacement vector satises elasto-dynamic equation (2.1.27), where = 2 f and f is the Fourier spectrum of the bodyforce. Similarly, the Fourier spectrum of pressure satises the acoustic wave equationin the frequency domain (2.1.29). The elastodynamic equation (2.1.27) in the frequencydomain remains valid even for viscoelastic media, where ci j kl are complex-valued andfrequency-dependent. See Kennett (1983) and Section 5.5. Similarly, (2.1.29) remainsvalid for complex-valued frequency-dependent .For a real-valued transient signal x(t ), the Fourier spectrum x( f ) of x(t ) satises therelation x(f ) = x( f ). For this reason, we shall mostly present the spectra x( f ) onlyfor f 0. Unless otherwise stated, we shall use the following convention: the presentedspectrum x( f ) corresponds to f 0 and can be determined from the relation x(f ) =x( f ) for negative frequencies.2.1.6 Energy ConsiderationsThe strictest and most straightforward approach to deriving the basic equations of theseismic ray theory is based on the asymptotic solution of the elastodynamic equation.Energy need not be considered in this procedure. Nevertheless, it does provide a veryuseful physical insight into all derived equations. We shall, therefore, also briey discusssome basic energy concepts. Some of themmay also nd direct applications in seismology,particularly in the investigation of the mechanism of the seismic source.In studying seismic wave propagation, the deformation processes are mostly consideredto be adiabatic. The density of strain energy W is then given by the relationW = 12i jei j; (2.1.32)see Aki and Richards (1980). Note that W is also called the strain energy function. In viewof linear constitutive relation (2.1.5),W = 12ci j klei jekl. (2.1.33)2. 1 LI NEAR ELASTODYNAMI CS 17In solids, the strain energy function satises the condition of strong stability for any non-vanishing strain tensor ei j (see Backus 1962),W = 12ci j klei jekl > 0. (2.1.34)In a uid medium (j = 0), the condition of weak stability, W 0, is satised.The density of kinetic energy K is given by the relation,K = 12 ui ui. (2.1.35)The sum of W and K is called the density of elastic energy and is denoted by E,E = W K = 12ci j klei jekl 12 ui ui. (2.1.36)All these equations can be found in any textbook on linear elastodynamics; see, for example,Auld (1973).For completeness, we shall also give the expression for the density of elastic energyux S, with Cartesian components Si,Si = i j uj = ci j klekl uj = ci j kluk.l uj. (2.1.37)See, for example, Kogan (1975), Burridge (1976), Petrashen (1980), and a very detaileddiscussion in Auld (1973). Elastic energy ux vector S is an analogue of the well-knownPoyinting vector in the theory of electromagnetic waves. It is not difcult to prove, usingelastodynamic equation (2.1.4), that the introduced energy quantities satisfy the energyequation, E,t S = f u. (2.1.38)We merely multiply (2.1.4) by ui and rearrange the terms.The energy quantities W. K. E. S introduced here depend on the spatial position and ontime t . Inhigh-frequencysignals and/or high-frequencyharmonic waves, all these quantitiesvary rapidly with time. It is, therefore, very useful to study certain time-averaged valuesof these quantities, not only instantaneous values. The time averaging can be performed inseveral ways. In a time-harmonic waveeld, we usually consider quantities time-averagedover one period. We denote the energy quantities averaged over one period by a bar abovethe letter, W, K, E and Si. For example,W(xi) = 1T_ T0W(xi. t )dt. (2.1.39)where T denotes the period of the time-harmonic wave under consideration. Amore generaldenition of the time-averaged energy values, applicable even to quasi-harmonic waves,was proposed by Born and Wolf (1959). The time-averaged value of W(xi. t ) is, in thiscase, given by the relationW(xi. t ) = 1t2t1_ t2t1W(xi. t )dt. (2.1.40)where time interval t2t1 is large (that is, considerably larger than the prevailing periodof the waveeld). Averaged quantities K(xi), E(xi), and Si(xi) can be dened in a similarway. If the waveeld is strictly harmonic, denition (2.1.40) yields (2.1.39) for t2t1 T.In seismology, however, harmonic and quasi-harmonic waveelds do not play the im-portant role that they play in other elds of physics; we usually consider signals of anite duration, including short signals. Quantity W(xi) would then depend on the choice18 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONSof t1 and t2. Assume that the signal under consideration is effectively concentrated in timeinterval tmin - t - tmax, where tmin represents the effective onset and tmax stands for theeffective end of the signal. Outside time interval (tmin. tmax), the signal effectively vanishes.It would then be natural to take t1 = tmin and t2 = tmax in (2.1.40). If, however, we taket1 - tmin and/or t2 > tmax, relation (2.1.40) would lead to a distorted result because the timeaveraging would also be performed over time intervals in which the signal is zero.In certain seismological applications, however, it may be useful to consider a slightlymodied denition (2.1.40) even in the case of shorter high-frequency signals. We modify itby removing constant factor (t2t1)1and take innite limits in the integral. As the signaleffectively vanishes outside time interval (t1 = tmin. t2 = tmax), we make no mistake if wetake (. ) instead of (t1. t2). However, we do not obtain time-averaged quantitiesbecause we have removed the averaging factor (t2t1)1. We shall call these modiedquantities time-integrated energy quantities and denote themby a circumex over the letter:W( x) =_ W( x. t )dt = 12ci j kl_ ei j( x. t )ekl( x. t )dt.K( x) =_ K( x. t )dt = 12_ ui( x. t ) ui( x. t )dt.E( x) =_ E( x. t )dt = W( x) K( x).Si( x) =_ Si( x. t )dt = ci j kl_ ekl( x. t ) uj( x. t )dt.(2.1.41)These time-integratedenergyquantities will be veryuseful inour treatment. Time-averagedquantitiesW. K. E. Si can be obtained from time-integrated quantities W. K. E. Si in avery simple way: we merely divide them by the effective length of the signal.For an acoustic medium (j = 0), the individual energy quantities W, K, E and Si caneasily be obtained from those given before:W =12p2. K =12vivi. E =W K. Si= pvi. (2.1.42)Here p is the pressure, vi are components of the particle velocity, and is the compress-ibility. See Section 2.1.4. The time-averaged and time-integrated energy quantities in theacoustic case can be constructed from (2.1.42) in the same way.Energy quantities W. K. E, W, K. E and W. K. E represent the energy density andare measured in J m3(joule per cubic meter) = Pa = kg m1s2. Time-integrated quan-tities W. K, and E are then measured in Pa s =kg m1s1. Finally, the densities of energyux Si. Si and Si are measured in J m2s1= Wm2(watt per square meter) = kg s3,and Si in J m2=Pa m=kg s2.In physics, the quantities with dimension of energy time are called the action; con-sequently, W. K. and E represent the density of action. We shall, however, not use thisterminology but shall refer to W. K. and E as the time-integrated energy quantities.Aphysical quantity that plays an important role in many wave propagation applicationsis the velocity vector of the time-averaged energy ux. We shall denote it UE, and its com-ponent UEi . The components are UEi are dened by the relationUEi = Si,E. (2.1.43)It is easy to see that an alternative denition of UEi isUEi = Si, E. (2.1.44)2. 2 ELASTI C PLANE WAVES 192.2 Elastic Plane WavesIn this section, we shall discuss the properties of plane waves propagating in homogeneous,perfectly elastic, isotropic, or anisotropic media. Plane waves are the simplest solutionsof the elastodynamic equation. The procedure used here to derive the properties of planewaves will be used in Section 2.4 to study wave propagation in smoothly inhomogeneouselastic media. Before dealing with plane waves propagating in elastic media, we shall alsobriey discuss acoustic plane waves.2.2.1 Time-Harmonic Acoustic Plane WavesWe shall seek the plane-wave solutions of the time-harmonic acoustic wave equation(2.1.30), with c constant and real-valued and with vanishing source term f p. We shalldescribe the acoustic pressure plane wave by the equationp( x. t ) = P exp[i(t T( x))]. (2.2.1)where p is pressure, is the circular frequency ( = 2 f ), P is some scalar constantwhich may be complex-valued, and T( x) is a linear homogeneous function of Cartesiancoordinates xi,T( x) = pixi. (2.2.2)Here pi are real-valued or complex-valued constants. If pi are real-valued, plane wave(2.2.1) is called homogeneous (even for complex-valued P). For complex-valued pi, planewave (2.2.1) is called inhomogeneous. Here we shall consider only real-valued pi; inho-mogeneous waves will be discussed in Section 2.2.10. Coefcients pi are not arbitrary butmust satisfy the relations following from the acoustic wave equation. Inserting (2.2.1) into(2.1.30) with f p= 0, we obtain the conditionp21 p22 p23 = 1,c2. (2.2.3)where c = ()1,2. Equation (2.2.3) represents a necessary condition for the existenceof nontrivial plane waves (2.2.1) propagating in an acoustic homogeneous medium. Thiscondition is known by many names; here we shall call (2.2.3) the existence condition.Pressure p( x. t ), given by (2.2.1), is a solution of the acoustic wave equation and representsa plane wave only if constants p1, p2, and p3 satisfy existence condition (2.2.3). The scalarcomplex-valued constant P in (2.2.1) may be arbitrary.The pressure p( x. t ), given by (2.2.1) through (2.2.3), is constant along planesT( x) = pixi = const. (2.2.4)For t varying, equation t = T( x) represents a moving plane, here called the wavefront.This is the reason why wave (2.2.1) is called a plane wave. We shall denote the unit normalto the wavefront N. Ni = pi,pk pk. Hence, p = N,C. (2.2.5)where C represents the velocity of the propagation of the wavefront in the direction per-pendicular to it. This velocity is usually called the phase velocity. By inserting (2.2.5) intoexistence condition (2.2.3), we obtain C = c. Thus, the phase velocity of the pressure planewave propagating in a homogeneous uid mediumequals c = ()1,2. It does not dependon the frequency or on the direction of propagation N. Vector p with Cartesian components20 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONSpi, given by (2.2.5), is called the slowness vector; it is perpendicular to the wavefront, andits length equals slowness 1,c.In addition, it is also usual to introduce phase velocity vector C, which has the samedirection as slowness vector p but length C:C = C N = p,( p p). (2.2.6)The plane-wave solution (2.2.1) of the acoustic wave equation is complex-valued. Boththe real and imaginary parts of (2.2.1), and any linear combination thereof, are againsolutions of (2.1.30) with f p= 0.Plane-wave solution (2.2.1) can be expressed in many alternative forms; for example,p( x. t ) = P exp[iknxn] exp[it ].where kn = (,c)Nn are Cartesian components of wave vectork = p.The plane-wave solution of acoustic equation (2.1.28) with fi = q = 0 can be written asp( x. t ) = P exp[i(t T( x))]. v( x. t ) =NPc exp[i(t T( x))].(2.2.7)Here P is again an arbitrary constant; all other quantities have the same meaning as de-ned earlier. Note that the product of velocity and density c is also known as the waveimpedance.2.2.2 Transient Acoustic Plane WavesTransient pressure plane waves can be obtained from the time-harmonic plane waves usingFourier transform (2.1.31). We can, however, also work directly with transient signals. Forthis purpose, we shall use analytical signals F( ). We shall consider a general, complex-valued expression for the transient acoustic plane wavep( x. t ) = PF(t T( x)). (2.2.8)where P is again an arbitrary complex-valued constant and T( x) is a linear function of thecoordinates given by (2.2.2). For a given real-valued signal x(t ), the analytical signal F( )is dened by the relationsF( ) = x( ) ig( ). g( ) = 1P.V._ x() d. (2.2.9)where P.V. stands for the principal value. The two functions x( ) and g( ) form a Hilberttransform pair. More details on analytical signals can be found in Appendix A. A well-known example of the Hilbert transform pair are functionsx( ) = cos . g( ) = sin . (2.2.10)so that F( ) = exp(i ). Thus, time-harmonic plane wave (2.2.1) is a special case oftransient plane wave (2.2.8). Another important example of the Hilbert transform pair isx( ) = ( ). g( ) = 1,. (2.2.11)where ( ) is the Dirac delta function. The corresponding analytical signal, called theanalytic delta function, can be expressed asF( ) = (A)( ) = ( ) i,. (2.2.12)2. 2 ELASTI C PLANE WAVES 21As in the case of time-harmonic plane wave (2.2.1), transient plane wave (2.2.8) is a solutionof the acoustic wave equation only if constants pi in (2.2.2) satisfy existence condition(2.2.3). Constant P and analytical signal F( ), however, are arbitrary. For this reason, weprefer to work with the analytical signal solutions and not with the time-harmonic solutions.In the same way as in the case of the plane time-harmonic wave, we can again introducethe wavefront, the slowness vector, and the like.In realistic applications, of course, we can, generally, consider either the real or imagi-nary part of solution (2.2.8). For example,p( x. t ) = Re{PF(t T( x))]. (2.2.13)orp( x. t ) = 12{PF(t T( x)) PF(t T( x))]. (2.2.14)Here the asterisk denotes a complex-conjugate function. Moreover, any linear combinationof the real and imaginary parts of (2.2.8) is again a solution of the acoustic wave equation.2.2.3 Vectorial Transient Elastic Plane WavesIn a homogeneous elastic, isotropic, or anisotropic medium, the procedure for determiningplane waves and their characteristic parameters is more involved than in the acoustic case.The main complication is that we can obtain several types of plane waves, propagating inthe same direction with different velocities. The velocities can be determined by solvingeigenvalue problems for certain 3 3 matrices.We shall describe the vectorial transient plane elastic wave by the equation u( x. t ) = UF(t T(xi)). (2.2.15)where u is the displacement vector, U a complex-valued vectorial constant, and F, t , Thave the same meaning as before. Once again, we shall use F( ) for the analytical signal asin the expression for the transient pressure plane wave in the acoustic case; see (2.2.8). Wecan make this assignment because we shall treat the acoustic case independent of the elasticcase throughout the book. Actually, the analytical signals corresponding to displacement,particle velocity, and pressure are mutually related. If the analytical signal for displacementis F( ) (see (2.2.15)), then the analytical signals for the particle velocity and pressure areF( ).We wish to determine the number of plane waves with different velocities that canpropagate along a specied direction in the medium, as well as their velocities, slownessvectors, and polarizations. All this can be determined by inserting (2.2.15) into the elasto-dynamic equation. We shall consider elastodynamic equation (2.1.17) with elastic modulici j kl constant, density constant, and fi = 0. The elastodynamic equation then readsai j kluk.l j = ui. i = 1. 2. 3. (2.2.16)where we have used the notationai j kl = ci j kl,. (2.2.17)Thus, ai j kl are density-normalized elastic moduli of dimension m2s2(velocity squared).It seems surprising to start our treatment with anisotropic and not with isotropic media.Due to various symmetries and the Einstein summation convention, however, certain steps22 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONSof the mathematical treatment for the anisotropic medium are formally simpler than for theisotropic medium, which is, in some ways, a degenerate case of the anisotropic medium.Inserting ansatz plane-wave solution (2.2.15) into (2.2.16) and assuming F//,= 0, weobtain the following equations:ai j kl pj plUk Ui = 0. i = 1. 2. 3. (2.2.18)This is a system of three linear equations for U1. U2. and U3. If Ui and pi satisfy(2.2.18), expression (2.2.15) is then a solution of elastodynamic equation (2.2.16) andrepresents a transient elastic plane wave.2.2.4 Christoffel Matrix and Its PropertiesSystem of equations (2.2.18) can be simplied if we introduce a 3 3 matrix withreal-valued components Ii k given by the relationIi k = ai j kl pj pl. (2.2.19)Then (2.2.18) can be expressed in the following simple form:Ii kUk Ui = 0. i = 1. 2. 3. (2.2.20)Matrix , given by (2.2.19), will be referred to as the Christoffel matrix, and Equation(2.2.20) will be called the Christoffel equation. Traditionally, the term Christoffel matrixhas usually been connected with matrix ci j klNjNl; see Helbig (1994). The notation (2.2.19),which contains the components of slowness vector p, has been broadly used in the raymethod of seismic waves propagating in inhomogeneous anisotropic media, as it is verysuitable for the application of the Hamiltonian formalism. We hope no confusion will becaused if we use the term Christoffel matrix for given by (2.2.19).We shall now list four important properties of Christoffel matrix .1. Matrix is symmetric,Ii k = Iki.2. The elements of matrix , Ii k, are homogeneous functions of the second degreein pi. By homogeneous function f (xi) of the kth degree in xi, we understand afunction f (xi), which satises the relationf (axi) = akf (xi). (2.2.21)for any nonvanishing constant a. It is obvious from (2.2.19) that Ii k satises therelationIi k(apj) = a2Ii k( pj) (2.2.22)so that Ii k is a homogeneous function of the second degree in pi.3. Matrix satises the relationpjIi k,pj = 2Ii k. (2.2.23)This relation follows from Eulers theorem for homogeneous functions f (xi) of thekth degree, which readsxj f (xi),xj = k f (xi). (2.2.24)Eulers theoremimmediately yields (2.2.23). It is also not difcult to derive (2.2.23)directly from (2.2.19).2. 2 ELASTI C PLANE WAVES 234. Matrix is positive denite. This means that Ii k satises the inequalityIi kaiak > 0. (2.2.25)where aj are the components of any nonvanishing real-valued vector a. Property(2.2.25) follows from the condition of strong stability (2.1.34). As ei j and ekl maybe quite arbitrary, we can take ei j = ai pj, ekl = ak pl. Then (2.1.34) yields (2.2.25).We shall now discuss the eigenvalues and eigenvectors of the matrix . We denote theeigenvalues by the letter G. Eigenvalues G are dened as the roots of the characteristicequationdet(Ii k Gi k) = 0. (2.2.26)that isdet__I11 G I12 I13I12 I22 G I23I13 I23 I33 G__= 0. (2.2.27)This equation represents the cubic algebraic equationG3 PG2 QG R = 0. (2.2.28)where P, Q, and R are invariants of matrix :P =tr . R = det .Q =det_I11 I12I12 I22_det_I22 I23I23 I33_det_I11 I13I13 I33_.(2.2.29)It is obvious from (2.2.28) that matrix has three eigenvalues. We denote them Gm,m = 1. 2. 3. Because matrix is symmetric and positive denite, all the three eigenvaluesG1, G2, and G3 are real-valued and positive.It is not difcult toconclude from(2.2.26) that eigenvalues G1, G2, G3are homogeneousfunctions of the second degree in pj, similar to Ii k. Thus,Gm(apj) = a2Gm( pj). (2.2.30)andpjGm,pj = 2Gm. (2.2.31)The second relation follows from Eulers theorem (2.2.24).Now we shall discuss the eigenvectors of . We shall denote the eigenvector corre-sponding to eigenvalue Gm by g(m), m = 1. 2. 3. It is dened by the relations(Ii k Gmi k)g(m)k = 0. i = 1. 2. 3. (2.2.32)with the normalizing condition,g(m)k g(m)k = 1 (2.2.33)(no summation over m). As we can see from (2.2.33), we take the eigenvectors as unitvectors. Thus, matrix has three eigenvalues Gm (m = 1. 2. 3), and three correspondingeigenvectors g(m)(m = 1. 2. 3). Unit vectors g(m)are also mutually perpendicular.24 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONSUsing (2.2.32) and (2.2.33), we can nd an expression for Gm in terms of Ii k and g(m)i ,which will be useful in the following. Multiplying (2.2.32) by g(m)i and taking into account(2.2.33), we obtainGm = Ii kg(m)i g(m)k = ai j kl pj plg(m)i g(m)k . (2.2.34)In the degenerate case of two identical eigenvalues, the direction of the two correspondingeigenvectors cannot be determined from(2.2.32) and (2.2.33); only the plane in which theyare situated can be determined. This plane is perpendicular to the remaining eigenvector.2.2.5 Elastic Plane Waves in an Anisotropic MediumConditions (2.2.20) followingfromthe elastodynamic equationcanbe expressedas follows:(Ii k i k)Uk = 0. i = 1. 2. 3. (2.2.35)Equation (2.2.35) represents a system of three homogeneous linear algebraic equations forU1, U2, and U3. The system has a nontrivial solution only if the determinant of the systemvanishes,det(Ii k i k) = det__I111 I12 I13I21 I221 I23I31 I32 I331__= 0. (2.2.36)Here again, Ii k = ai j kl pj pl; see (2.2.19). This equation plays a very important role ininvestigating the propagation of plane waves in a homogeneous elastic medium. Planewave (2.2.15) is a solution of the elastodynamic equation if, and only if, its slowness vector p (with Cartesian components pi) satises Equation (2.2.36). As in the acoustic case, weshall refer to (2.2.36) as the existence condition for plane waves in homogeneous anisotropicmedia. It represents a polynomial equation of the sixth degree in the components of theslowness vector pi. In the phase space with coordinates p1. p2. and p3, Equation (2.2.36)represents a surface, known as the slowness surface. For more details on the slownesssurface, refer to Section 2.2.8. The slowness surface has three sheets, corresponding to threeeigenvalues, which will be discussed later. Existence condition (2.2.36) also represents acubic equation for the squares of phase velocities C(m), m = 1. 2. 3, of the plane waves,which can propagate in the medium described by density-normalized elastic moduli ai j kl,in any selected direction specied by unit vector N, normal to the wavefront. Since pi =Ni,C(m), Ii k = ai j klNjNl,(C(m))2.Here, we shall discuss Equation (2.2.35) not only from the point of view of the exis-tence condition, but also from a more general point of view. We also wish to determine thedirection of displacement vector amplitude U. In this discussion, we shall apply the eigen-value formalism. This approach, based on the solution of the eigenvalue problemfor matrixIi j, will be applied in Section 2.4 to a more complicated problem of wave propagation insmoothly inhomogeneous media.Equation (2.2.35) constitutes a typical eigenvalue problem; see (2.2.32). Comparing(2.2.35) with (2.2.32), we can conclude that (2.2.35) is satised if, and only if, one of thethree eigenvalues of matrix equals unity,Gm( pi) = 1. m = 1 or 2 or 3. (2.2.37)In the phase space with coordinates p1. p2, and p3, these equations represent three branchesof the slowness surface (2.2.36). The corresponding eigenvector g(m)then determines the2. 2 ELASTI C PLANE WAVES 25direction of U so thatU = A g(m). (2.2.38)where A is a complex-valued constant scalar amplitude quantity.As matrix has three eigenvalues Gm, Equations (2.2.35) are satised in three cases.These three cases specify the three plane waves, which can generally propagate in a ho-mogeneous anisotropic medium in the direction of N.Eigenvectors g(m)determine the polarization of the individual waves so that they canbe referred to as polarization vectors. If G1 ,= G2 ,= G3, they can be strictly determinedfrom (2.2.32) and (2.2.33). If two eigenvalues are equal, the relevant polarization vectorscannot be determined uniquely from (2.2.32) and (2.2.33); some other conditions must betaken into account. We shall speak of the degenerate case of two equal eigenvalues. Themost important degenerate case is the case of isotropic media (see Section 2.2.6); the otherdegenerate case is the case of the so-called shear wave singularities in anisotropic media.Let us now determine the phase velocities C(1), C(2), and C(3)of the three plane wavespropagatinginananisotropic homogeneous mediuminthe directionspeciedby Ni. Puttingpi = Ni,C(m)(2.2.39)for the mth plane wave (m = 1 or 2 or 3) and inserting it into (2.2.37) yieldsC(m)2= Gm(Ni). (2.2.40)due to (2.2.30). Here Gm(Ni) is obtained fromstandard eigenvalue Gm( pi), if pi is replacedby Ni. Thus, Gm(Ni) is the eigenvalue of matrix Ii k = ai j klNjNl. Matrix Ii k has the sameproperties as Ii k; only pi is replaced by Ni in all relations. It is obvious that Gm(Ni) isalways positive. Both matrices Ii k and Ii k have the same eigenvectors g(m), m = 1. 2. 3.Equation (2.2.40) yields the square of the phase velocity, not the phase velocity itself.We shall only consider the positive values of C(m):C(m)=_Gm(Ni). (2.2.41)Thus, the phase velocities of plane waves in a homogeneous anisotropic medium dependon the direction of propagation of wavefront Ni.From a practical point of view, the three phase velocities C(1), C(2), and C(3)for a xeddirection Ni are determined by the solution of cubic equation (2.2.28), where P, Q, and Rare given by (2.2.29), with pi replaced by Ni (that is, Ii k is replaced by Ii k = ai j klNjNl).The square roots of the three solutions yield the phase velocities of the three waves.The plane wave withthe highest phase velocityis usuallycalledthe quasi-compressionalwave (or quasi-P or qP wave). The remaining two plane waves are called quasi-shear waves(quasi-S1 and quasi-S2 waves, or qS1 and qS2 waves).If we combine Equations (2.2.15), (2.2.2), (2.2.38), and(2.2.39), we obtainthe followingnal expression for any of the three plane waves propagating in a homogeneous anisotropicmedium in a xed direction Ni: u(xi. t ) = A g(m)F_t Nixi,C(m)_. (2.2.42)where A is an arbitrary complex-valued constant and C(m)is given by (2.2.41). The planewave is linearly polarized.26 ELASTODYNAMI C EQUATI ON AND I TS SI MPLE SOLUTI ONS2.2.6 Elastic Plane Waves in an Isotropic MediumIn isotropic media, the phase velocities can be determined analytically. The elements ofmatrix are simply obtained by inserting (2.1.10) into (2.2.19):Ii k = jpi pk ji k pl pl. (2.2.43)Thus, eigenvalues G can be determined from relation (2.2.27), which takes the followingform:det_____j p21 j pi pi G j p1p2j p1p3j p1p2j p22 j pi pi G j p2p3j p1p3j p2p3j p23 j pi pi G_____= 0.This determinant can be expanded in powers of (1jpi pi G). It is easy to see that theterms with (1jpi pi G)nvanish for n = 1 and 0 so thatdet(. . .) =_jpi pi G_3_jpi pi G_2 jpk pk.which leads todet(. . .) =_jpi pi G_2_ 2jpi pi G_.Putting =_ 2j_1,2=_k 43j_1,2. =_j_1,2(2.2.44)nally yieldsdet(. . .) = (2pi pi G)(2pi pi G)2.Thus, in an isotropic homogeneous medium, matrix has the following three eigenvalues:G1( pi) = G2( pi) = 2pi pi. G3( pi) = 2pi pi. (2.2.45)Two of these eigenvalues, G1 and G2, are identical. If we replace pi by Ni and take intoaccount NiNi = 1, we obtainG1(Ni) = G2(Ni) = 2. G3(Ni) = 2. (2.2.46)Equations (2.2.41) then yield the following three phase velocities C(m):C(1)= C(2)= . C(3)= . (2.2.47)Thus, the isotropic mediumrepresents a degenerate case of ananisotropic medium, withtwoidentical phase velocities. Only two plane waves can propagate in an isotropic medium in axeddirection Ni. The faster, propagatingwithphase velocity, is calledthe compressionalwave (P wave). The slowness vector components pi corresponding to this wave must satisfythe existence condition of P waves everywhere:p21 p22 p23 = 1,2; (2.2.48)2. 2 ELASTI C PLANE WAVES 27see (2.2.45) for G3 = 1. The slower plane wave propagates with velocity and is calledthe shear wave (S wave). The slowness vector components pi corresponding to this wavemust satisfy the existence condition of S waves everywhere:p21 p22 p23 = 1,2. (2.2.49)Velocities and do not depend on the position or the direction of propagation.As we can see from (2.2.48) and (2.2.49), the slowness surfaces in the phase space p1,p2, and p3 of both P and S waves are spheres. The radius of the sphere is 1, for P waves(the slowness of P waves) and 1, for S waves (the slowness of S waves).We will now determine the eigenvectors of Ii k in isotropic media. As we know, wecan determine strictly only the eigenvector g(3), but not eigenvectors g(1)and g(2), sinceG1 = G2. Eigenvector g(3)satises relation (2.2.32), with Ii k given by (2.2.43) and withG3 = 1,_ jpi pk ji k pl pl i k_g(3)k = 0.If we insert pi = Ni, and multiply the equation by g(3)i , we obtain (Nig(3)i )2= 1. Thisresult yields g(3)= N. (2.2.50)Thus, the displacement vector of the plane P wave is perpendicular to the wavefront.Whereas the eigenvector g(3)corresponding to the P wave is uniquely determined,eigenvectors g(1)and g(2)corresponding to the plane shear wave cannot be strict