elastic waves snieder

528 General Theory of Elastic Wave Scattering

Chapter 1.7.1

GENERAL THEORY OF ELASTIC WAVESCATTERINGRoel SniederDepartment of Geophysics and Center for WavePhenomena,Colorado School of Mines,Golden, Colorado 80401,U.S.A.

PART 1 SCATTERING OF WAVES BYMACROSCOPIC TARGETS

Topic 1.7 Elastodynamic Wave (Elastic)Scattering: Theory

Contents

1 Introduction 5282 Principles of Elasticity 5283 The Anatomy of a Seismogram 5304 The Representation Theorem 5315 The LippmanSchwinger Equation for Elastic

Waves 5326 The Born Approximation of Elastic Waves 5337 The Greens Tensor in Dyadic Form 534

A Homogeneous Infinite Isotropic Medium 534The Ray-Geometric Greens Tensor for Body

Waves 535Other Examples of Dyadic Greens Tensors 536Physical Interpretation of Dyadic Greens

Tensor 537An Example of a Complicated Greens

Tensor 5378 Anisotropic Media 537

Plane-Wave Solutions 537An Isotropic Medium 539Distinction between Phase and Group

Velocity 539

1. IntroductionThe theory of elastodynamic wave scattering is basedon two foundations; continuum mechanics of elasticmedia and the general principles of scattering theory.In this chapter, the basic elements of the theory of elas-todynamic scattering theory are presented. In laterchapters this is applied to scattering of body waves(Chapter 1.7.2), scattering of surface waves (Chapter1.7.3), seismic imaging (Chapter 1.8.1) and nonde-structive testing (Chapter 1.8.2).

There is a wide body of literature on elastody-namic wave scattering. In their treatise of theoreticalphysics, Morse and Feshbach (1953a,b) include sev-eral aspects of continuum mechanics and elastic wavepropagation. The foundations of continuummechan-ics are described in great detail by Malvern (1969).

A comprehensive overview of scattering of acous-tic, electromagnetic and elastic waves is presented ina unified way by de Hoop (1995). The propaga-tion of elastic waves forms the foundation of seis-mology. The required theory and application toseismological problems is presented in great detailby Aki and Richards (1980) Hudson (1980) and byBen-Menahem and Singh (1981).

2. Principles of ElasticityIn this section an outline is given of the principles ofelasticity that are of relevance for elastic wave scat-tering. A more detailed description of the theoryof elastic waves needed for the description of elas-tic wave scattering can be found in Aki and Richards(1981). In general the derivations are given in thefrequency domain. The frequency-domain formula-tion and the time-domain formulation are related bythe Fourier transform. For the temporal and spatialFourier transforms the following convention is used:

f(t) =12pi

F()eitd, (1)

h(x) =12pi

H(k)eikxdk. (2)

For the inverse transform the exponents have theopposite sign and the factor 2pi is omitted.

A crucial element in the description of elastic wavesis the stress tensor . Consider an infinitesimal surfacewith surface area dS with normal vector n. The forceacting on this surface per unit surface area is calledthe traction; this quantity is given by

T = n . (3)The stress tensor is such a useful quantity because ex-pression (3) makes it possible to compute the forceacting on any surface once the stress tensor is known.The stress tensor is symmetric:

ij = ji. (4)

This property follows from the requirement thatangular momentum is conserved (Malvern, 1969;Goldstein, 1980).

Barbara MclenonThis chapter appears in Scattering and Inverse Scattering in Pure and Applied Science, Eds. Pike, R. and P. Sabatier, Academic Press, San Diego, 528-542, 2002.

General Theory of Elastic Wave Scattering 529

For a general medium the stress in the medium de-pends in a complicated way on the deformation ofthe medium. The deformation of the medium resultsfrom a displacement vector u(r) in the medium thatvaries with position, because a constant displacementvector u does not generate an internal deformation.For small elastic deformations the relation betweenthe deformation and the resulting stress can be lin-earised; this linearisation is expressed by Hookeslaw:

ij = cijklekl. (5)

Throughout this chapter the summation conventionis used where one sums over repeated indices. Inexpression (5) the infinitesimal strain tensor eij is de-fined by

eij =12(iuj +jui), (6)

where the notation i stands for the derivative withrespect to the xi coordinate: if f/xi. Thefourth-order tensor cijkl, called the elasticity tensor,generalises the concept of the spring constant of a sim-ple spring to three-dimensional elastic media. Oneshould keep in mind that Hookes law (5) is a lin-earisation that breaks down in situations when theresponse of the medium is nonlinear. This is for exam-ple the case at fractures or in the earthquake sourceregion, where nonlinear fracture behaviour governsthe response of the medium.

It follows from the definition (6) that the strain ten-sor is symmetric:

eij = eji. (7)

Because of this property, the elasticity tensor is sym-metric when the last indices are exchanged: cijkl = cijlk.Since according to (4) the stress tensor is also symmet-ric, the elasticity tensor is also symmetric in the firsttwo indices: cijkl = cjikl. Energy considerations (Akiand Richards, 1980) imply that the elasticity tensor isalso symmetric for exchange of the first and last pairof indices: cijkl = cklij. The elasticity tensor thereforehas the symmetry properties

cijkl = cjikl = cijlk = cklij. (8)

Inserting (6) in (5) and using the symmetry of the elas-ticity tensor in the first pair of indices, the relation be-tween the stress and the displacement can be writtenas

ij = cijklkul. (9)

In general a tensor of rank 4 in three dimensionshas 81 components. The symmetry relations (8) re-duce the number of independent components of theelasticity tensor to the number of independent ele-ments of a symmetric 6 6 matrix (Backus, 1970;Helbig, 1994), which means that the elasticity tensor

has in three dimensions 21 independent elements. Ofspecial importance is the case of an isotropic elasticmedium. In such a medium the elasticity tensor doesnot depend on some preferred direction, but only onthe Lame parameters and

cijkl = ijkl +(ikjl +iljk), (10)

where ij is the Kronecker delta defined by ij = 1wheni = j and ij = 0 when i 6= j. For the moment we considera general elasticity tensor.

The force associated with the stress acting on a vol-ume is given by

T dS, where the surface integral is

over the surface that bounds the volume. By the the-orem of Gauss and expression (3), this force is givenby dV. This implies that the force generated by

the internal stress per unit volume is given by the di-vergence of the stress tensor (). In addition to thisforce, forces such as gravity or a seismic source thatare not resulting from the deformation may be opera-tive. These forces per unit volume are denoted by theforce vector f. Since the mass per unit volume is givenby the density , Newtons law can be written as

u = + f. (11)It should be noted that this is the linearised form ofNewtons law. Since we are dealing with a continu-ous medium the acceleration contains advective termsas well; hence, the left-hand side should be written asv/t+vv, with v being the velocity vector. How-ever, when thematerial velocity v is much smaller thanthe wave velocity, the advective terms can be ignored(Snieder, 2001). By inserting (9) in Newtons law,one obtains the wave equation for the displacementvector

ui = j(cijklkul

)+ fi. (12)

In the frequency domain this corresponds to

2ui +j(cijklkul) = fi. (13)

In a shorthand notation we will write this expressionalso as

Lu = f, (14)where the operator L is defined by

Lij = 2ij +kcikljl. (15)

The wave equation is a second-order differentialequation for the displacement vector u that needs tobe supplemented with boundary conditions at the sur-face of the medium. At a stress-free surface S, thetractions acting on the surface vanish:

T = n = 0 at S. (16)This boundary condition is appropriate when the sur-face S forms the outer boundary of the elastic bodythat is surrounded by empty space. Ignoring the stressimposed on an elastic body imposed by the gas or fluidin which the body is possibly embedded, the tractions


vanish at the surface. Using (3) and (9) this bound-ary condition can be expressed as a condition for thedisplacement at the free surface:

njcijklkul = 0 at S. (17)

For a number of applications it is useful to definethe power flux in an elastic medium. When a forceF acts on a particle moving with velocity v (given byu), the power delivered by the force is given by (Fv).Consider a surface element dS, according to (3) theforce acting on that surface element is given by TdS =ndS = dS. The power delivered at the surface istherefore given by v dS. The vector dS points out ofa volume; to obtain the power delivered to a volumewe should consider the energy flow into the volume byadding a minus sign. The power flux JP is thereforegiven by

JP= (u). (18)Expression (18) is the elastodynamic equivalent of thePointing vector in electromagnetism. The power thatflows through a surface is given by

P =

JPn dS. (19)

It can be shown (Ben-Menahem and Singh, 1981)that the energy density satisfies the conservation law

Et + ( JP) = (fu). (20)

The power flux and energy density depend quadrat-ically on the displacement because expression (18)contains the product of the displacement and thestress (which is a linear combination of partial deriva-tives of the displacement). One should therefore notsimply replace the real time-domain quantities by thecomplex counterpart in the frequency domain. Withthe Fourier convention (1) in the frequency domain,the power flux averaged over one period is given by

JP =

2Im(u), (21)

where the asterisk denotes the complex conjugate.

3. The Anatomy of a SeismogramAs an introduction to the different types of elasticwave propagation, we consider Fig. 1 in which thevertical component of the ground motion is shownafter an earthquake at Jan-Mayen Island in the NorthAtlantic. The ground motion was recorded by one ofthe stations of theNetwork of Autonomously Record-ing Seismographs (NARS) that was placed in Naroch,Belarus. The earthquake takes place at time t = 0. Inthe seismogram impulsive waves arrive around 300and 550 s. After this, a strong extended wave train ar-rives between 620 and 900 s. The impulsive arrivalsare waves that have propagated deep in the Earth;

Figure 1 Vertical component of the ground motion at a seismicstation in Naroch (Belarus) after an earthquake at Jan-Mayen Is-land. This station, part of the Network of Autonomously RecordingSeismographs (NARS), is operated by Utrecht University.

these waves are called body waves. In contrast tothese body waves, the wave train that arrives around700 s corresponds to waves that are guided along theEarths surface; for this reason such waves are calledsurface waves. The paths of propagation of thesetypes of waves are sketched in Fig. 2. The surfacewave propagates effectively in only two dimensionswhereas the body waves propagate in three dimen-sions. For this reason the surface wave suffers lessfrom geometrical spreading during the propagationand therefore has in general a larger amplitude thanthe body waves; this can be seen clearly in Fig. 1.

Elastodynamic waves are vector waves, with no re-striction on the polarisation. Because there are threespatial dimensions, elastodynamic waves can havethree different polarisations. In an isotropic medium,the body waves separate into a longitudinal wave and

Figure 2 Sketch of the paths of propagation and polarisation ofP-waves, S-waves and surface waves in the Earth.


transverse waves. The polarisation of these waves isindicated by the arrows in Fig. 2. Longitudinal wavespropagate at a higher velocity than do the transversewaves. The nomenclature P wave and S waveshistorically denotes the first arriving (primary) andlater arriving (secondary) body waves. The wave ar-riving around 300 s therefore is the P wave while thewave arriving around 550 s is the S wave. There aretwo directions of polarisation perpendicular to thepath of propagation; for this reason there are twoS waves in an isotropic medium that for reason ofsymmetry propagate at the same velocity. When themedium is anisotropic, there are still three polarisa-tions of the body waves, but they do not correspondto longitudinal and transverse directions of polarisa-tions. This is treated in more detail under Plane WaveSolutions.

The surface wave is guided along the Earths sur-face. For an isotropic Earth model where the elas-tic parameters and the density depend only on depththere are two types of surface wave: Rayleigh wavesand Love waves. The Love wave is linearly polarisedin the horizontal direction perpendicular to the prop-agation path, while Rayleigh waves have an ellip-soidal polarisation in the vertical plane through thepath of propagation (Aki and Richards, 1980). Loveand Rayleigh waves propagate in different modes, thenumber of modes depending in a nontrivial way onthe Earth model and on frequency.

The velocity of wave propagation changes withdepth in the Earth. Since the surface waves penetrateto different depths at different frequencies, the phasevelocity of each surface wavemode in general dependson frequency. This implies that surface waves are ingeneral dispersive. This can clearly be seen in the sur-face wave that arrives in Fig. 1; the early part of thewave train around 650 s has a frequency much lowerthan that of the later part of the wave train, whicharrives around 800 s.

4. The Representation TheoremGreens functions are very important in scatteringproblems. In elasticity theory the Greens functiongives the displacement generated by a point force in acertain direction. Since both the point force and thedisplacement are vectors with three components, theGreens function is a 33 tensor. The ith componentof a unit force in the n direction at location r is givenby in(r r ). The Greens tensor Gin(r, r ) is the dis-placement at location r in the i direction due to thispoint force

2Gin(r, r ) +j(cijklkGln(r, r )) = in(r r ), (22)

or, using the shorthand notation of Eq. (15),

LijGjn(r, r ) = in(r r ). (23)

In order to derive the representation theorem forelastic waves, let us consider a displacement u(1) thatis generated by an excitation f (1), and a displacementu(2) that is caused by an excitation f (2):

2u(1)i +j(cijklku(1)l

)= f (1)i ,

2u(2)i +j(cijklku(2)l

)= f (2)i . (24)

Multiply the first equation by u(2)i and the second by

u(1)i , subtract the two expressions and integrate overa volume V: {

u(2)i j(cijklku(1)l

)u(1)i j

(cijklku(2)l

)}dV

= {

f (1)i u(2)i f

(2)i u

(1)i

}dV. (25)

The volume integrals on the left-hand side can beconverted into surface integrals using the identi-ties

vij(cijklkul)dV =

j(vicijklkul)dV (jvi)cijkl(kul)dV =

njvicijklkul dS

(jvi)cijkl(kul)dV,

where Gausss theorem is used in the last equality.Applying this to both terms on the left-hand side of(25) gives {

nju(2)i cijklku

(1)l nju

(1)i cijklku

(2)l

}dS

{(

ju(2)i)cijkl

(ku(1)l

)(

ju(1)i)cijkl

(ku(2)l

)}dV

= {

f (1)i u(2)i f

(2)i u

(1)i

}dV. (26)

Because of the symmetry properties of the elasticitytensor as shown in the last identity of (8), the terms inthe second line vanish so that this expression reducesto {

nju(2)i cijklku

(1)l nju

(1)i cijklku

(2)l

}dS

= {

f (1)i u(2)i f

(2)i u

(1)i

}dV. (27)

This general expression holds for any volume V,which is not necessarily the complete extent of theelastic body through which the waves propagate.

We now specialise momentarily for the casewherein the Greens function satisfies homogeneousboundary conditions on the bounding surface S. Thismeans that either the Greens function or its associ-ated traction vanish on the boundary: Gin(r, r ) = 0or njcijklk(Gln) = 0 when r is located on S. Whenpoint forces f (1)i (r) = in(r r1) and f

(2)i (r) = im(r r2)

are used to excite the wave fields u(1) and u(2), thesewave fields are by definition equal to the Greens ten-sors: u(1)i (r) = Gin(r, r1) and u

(2)i (r) = Gim(r, r2). For

this special excitation the following identity holds:f (1)i u

(2)i dV =

in(r r1)Gim(r, r2)dV = Gnm(r1, r2).Inserting these results in (27) then gives the reciprocitytheorem:

Gnm(r1, r2) =Gmn(r2, r1). (28)


This theorem states that n component of the displace-ment at r1 due to a point source excitation in r2 inthe m direction is identical to the m component ofthe displacement at r2 due to a point source excita-tion in r1 in the n direction. In other words, whenthe roles of source and receiver are exchanged, oneobtains exactly the same wave field.

Note that in deriving the representation theoremthe volume V can be any volume, and the wave-field solution can have any value at the surface S thatbounds this volume. Let us now leave out the super-scripts (1) in u(1) and f (1) and let the (2) solution bethe Greens function for a unit point force at r in then direction. This means that f (2)i (r) = in(r r ), andthe associated response is by definition given by theGreens tensor: u(2)i (r) = Gin(r, r

). Using this in ex-pression (27) yields

un(r ) =

Gin(r, r )fi(r)dV

+{Gin(r, r )njcijklkul(r)

ui(r)njcijklkGln(r, r )}dS. (29)

The reciprocity theorem (28) can be applied to theGreens tensors on the right-hand side. Exchangingthe coordinates r r in the resulting expression andinterchanging the indices i n then gives the repre-sentation theorem:

ui(r) =

Gin(r, r )fn(r )dV

+{Gin(r, r )njcnjkl kul(r )

un(r )njcnjkl kGil(r, r )}dS. (30)

Note that the integration and differentiation are nowover the primed coordinates. When the exciting forcefn(r ) is known within the volume and when the wavefield un(r ) and the associated traction njcijklkul(r )are known on the surface S, one can compute thewave field everywhere within the volume V. This ex-pression forms the basis of the elastic equivalent ofthe Kirchhoff integral.

5. The LippmanSchwinger Equa-tion for Elastic Waves

In order to describe scattering it is necessary to definea reference medium in which an unperturbed wavepropagates and a perturbation of the medium thatacts as a secondary source that generates scatteredwaves. For this reason we divide the elasticity ten-sor c into a reference tensor c(0) and a perturbationc(1), and do the same for the density:

c(r) = c(0)(r) + c(1)(r),

(r) = (0)(r) +(1)(r).(31)

With this decomposition one can also separate theoperator L defined in expression (15) into an oper-ator L(0) for the reference medium and an operatorL(1) associated with the perturbation of the medium.It should be noted that the reference medium is notnecessarily a homogeneous medium. For the Earth,for example, an Earth model in which the propertiesdepend only on depth is a natural reference model.Let u(0) be the solution for the reference medium, anddenote the Greens function for the reference mediumby G(0); this means that in operator notation

L(0)ij G(0)jn (r, r

) = in(r r ). (32)

Let us now assume that at the surface of the mediumthe tractions vanish. This boundary condition holdsboth for the reference medium and for the perturbedmedium.

The total wave field in the perturbed medium thatis excited by a force f(r) is given by(

L(0) +L(1))u(r) = f(r), (33)

or

L(0)u(r) = (f(r)+L(1)u(r)

). (34)

According to this expression the total wave field in theperturbed medium is identical to the wave field in theunperturbed medium that is generated by an effectiveforce given by f(r)+L(1)u(r). The representation theo-rem (30) for the reference medium can be applied tothis result. Using that the effective force is given byf(r)+L(1)u(r), the wave field is given by

ui(r) =

G(0)in (r, r)fn(r ) dV

+

G(0)in (r, r)L(1)nj (r

)uj(r ) dV

+ {

G(0)in (r, r)njc

(0)njkl

kul(r

)

un(r )njc(0)njkl kG(0)il (r, r

)}dS . (35)

Note that the Greens tensor for the reference mediumis used and that in the surface integral the elasticitytensor c(0) of the reference medium is used. The firstterm on the right-hand side is the unperturbed wave:

u(0)i (r) =

G(0)in (r, r)fn(r )dV . (36)

At this point we assume that the reference mediumhas zero tractions at the surface:

njc(0)njkl

kG

(0)il (r, r

) = 0. (37)

Because of this boundary condition the last term in(35) vanishes. The perturbed medium also has zerotractions at the surface S, but for this boundary con-dition we must use the perturbed elasticity tensor andthe total wave field:

nj(c(0)ijkl + c

(1)ijkl

)kul(r) = 0. (38)


From this it follows that at the surface njc(0)ijklkul(r) =

njc(1)ijklkul(r). This expression can be used in thethird term of the right-hand side of (35). Usingthese results and inserting also definition (15) for theoperator L(1) one obtains that

ui(r) = u(0)i (r) +

2

G(0)ij (r, r)(1)(r )uj(r ) dV

+

G(0)in (r, r) k

(c(1)nklj(r

) luj(r))

dV

G(0)in (r, r)njc

(1)njkl

kul(r

) dS . (39)

Note that the perturbed elasticity tensor c(1) is presentin the surface integral as well as in the volume integralin the second line. The theorem of Gauss can be usedto convert this volume integral in a surface integral:

G(0)in (r, r) k

(c(1)nklj(r

) luj(r))

dV

=

k(G(0)in (r, r

)c(1)nklj(r) luj(r

))

dV

k(G(0)in (r, r

))c(1)nklj(r

) luj(r ) dV

=

nkG(0)in (r, r

)c(1)nklj(r) luj(r

)dS

k(G(0)in (r, r

))c(1)nklj(r

) luj(r) dV . (40)

The surface integral in this expression is the same asthe surface integral in expression (39) but it has theopposite sign. When (40) is inserted in (39) theseterms cancel so that finally

ui(r) = u(0)i (r) +

2

G(0)ij (r, r)(1)(r )uj(r ) dV

k(G(0)in (r, r

))c(1)nklj(r

) luj(r) dV . (41)

This is the LippmanSchwinger equation for elasticwave scattering. In this expression the total wave fieldis decomposed in the unperturbed wave plus scatteredwaves that are generated by the perturbations of themedium. The strength of these scattered waves de-pends on the total wave field at the location of thescatterers as well as on the size of the perturbationsfrom the reference medium. Conveniently, expression(41) is simpler than the original expression (35) be-cause the surface integral has disappeared. In addi-tion, the Greens function and the wave field appearin a more symmetric way in the last term of (41) be-cause the gradient of both terms is taken.

6. The Born Approximation of ElasticWaves

Up to this point no approximations have been made.The simplicity of (41) is elusive because one needs toknow the total wave field at the scatterers in orderto compute the integrals on the right-hand side. For

many problems the Born approximation is a usefultool for computing the scattered waves. This approx-imation is obtained by replacing the total wave fieldon the right-hand side of (41) with the unperturbedwave. This means that in the Born approximationthe scattered waves uBi are given by

uBi (r) = 2

G(0)ij (r, r)(1)(r )u(0)j (r ) dV

k(G(0)in (r, r

))c(1)nklj(r

) lu(0)j (r

) dV . (42)

An alternative way to obtain the Born approximationis to iterate the integral equation (41). This result-ing series is nothing but the Neumann series solutionfor elastodynamic wave scattering where the wavefield is written as a sum of the unperturbed wave, thesingle-scattered waves, the double-scattered waves,etc. Truncating this series after the second term gives(42) for the scattered waves. This implies that theBorn approximation (42) retains the single-scatteredwaves in an elastic medium. Analytic estimates of thedomain of applicability for the Born approximationfor elastic waves are given by Hudson and Heritage(1982).

So far we have considered volumetric perturba-tions of the density and elasticity tensor. In many ap-plications, the perturbations of the medium are bestdescribed by perturbations in the position of inter-faces within the medium. Because the properties ofthe reference medium are discontinuous at interfaces,we consider perturbations of height h away from theirposition in the reference medium. For perturbationsof the boundary height small compared to the wave-length of elastic waves, the associated perturbation ofthe density is equivalent to a volumetric perturbation[(0)]over a thickness h across the interface (Hudson,

1977). The straight brackets denote the contrast inthe density across the interface. Similarly the associ-ated perturbation of the elasticity tensor is given a vol-umetric perturbation

[c(1)]over a thickness h across

the interface. Since h is assumed to be small com-pared to a wavelength, the resulting volume integralsin (42) reduce to surface integrals over the perturbedinterface multiplied with the height of the perturba-tion. This then generalises the previous expression toinclude perturbations of the interfaces in the referencemedium:

uBi (r) = 2

G(0)ij (r, r)(1)(r )u(0)j (r ) dV

k(G(0)in (r, r

))c(1)nklj(r

) lu(0)j (r

) dV

+2

G(0)ij (r, r)h(r )

[(0)(r )

]u(0)j (r

) dS

k(G(0)in (r, r

))h(r )

[c(0)nklj(r

)]

lu(0)j (r

) dS .

(43)

The summation is over the perturbed interfaces. Thedetails of this derivation are given by Hudson (1977).


One must be careful in the application of this ex-pression to the displacement of an interface that sep-arates a fluid from a solid (Woodhouse, 1976). Thereason is that at such a boundary the component ofthe displacement parallel to the interface is not contin-uous, and the equivalence of a displacement of suchan interface with a volumetric perturbation is notobvious. This is relevant for a variety of situationssuch as the scattering of elastic waves at the coremantle boundary in the Earth, the scattering of wavesby irregularities in the ocean floor and engineering ap-plications where an elastic body is surrounded by afluid.

7. The Greens Tensor in DyadicForm

In this section examples are given of the Greens ten-sor for different media. In all these examples theGreens tensor G(r, r ) can be written as a sum ofdyads of the form p(r)g(r, r )p(r ). (The dagger de-notes the Hermitian conjugate defined as the complexconjugate of the transpose: p =

(pT).) In this ex-

pression, the vectors p are polarisation vectors andthe scalar function g(r, r ) accounts for propagation.The fact that the polarisation vectors at the source rand the receiver r enter this expression in a symmet-ric way is due to the reciprocity property (28) of theGreens tensor.

A Homogeneous Infinite Isotropic Medium

As a first example, consider the Greens tensor fora homogeneous, infinite isotropic medium. ThisGreens tensor is given in expression (4.12) of Ben-Menahem and Singh (1981)

G(r) =ik

12pi(+2)(I h(1)0 (kr) + (I3rrT) h

(1)2 (kr)

) ik12pi

(2I h(1)0 (kr) + (I3rrT) h

(1)2 (kr)

), (44)

where r is the relative location of the observationpoint relative to the source. (In this expression Idenotes the identity matrix with elements Iij = ij.)The caret above a vector denotes the unit vector point-ing in the same direction, hence r r/r, where r is thedistance between the source and observation points.In expression (44) the wave numbers k and k aregiven by

k = / and k = /. (45)where is the wave velocity for compressive wavesand is the wave velocity for shear waves:

=

+2

and =

. (46)

These velocities are derived under An IsotropicMedium (8). In expression (44) the h(1)m are sphericalBessel functions of the first kind. It should be notedthat Ben-Menahem and Singh (1981) use a Fouriertransform with the opposite sign in the exponent tothe convention (1) that is used here. For this reason,the complex conjugate is taken, which implies thatthe spherical Bessel function of the second kind h(2)min their expression is replaced by the spherical Besselfunction of the first kind h(1)m . These functions aregiven by

h(1)0 (x) = ieix

x,

h(1)2 (x) =(

ix 3

x2 3i

x3

)eix.

(47)

Let us first analyse the Greens tensor in the farfield. The far field is defined as the region of spacethat is far away from the source compared to a wave-length; hence it is defined by the condition kr 1. Inthe far field only terms 1/kr are retained, but terms ofhigher powers in 1/kr are ignored. Using this in (47)and (44) gives the Greens tensor in the far field:

GFF(r) =1

4pi(+2)eikar

rrrT+

14pi

eikr

r

(I rrT

). (48)

Note that this expression is not yet in the form of asuperposition of dyads pg(r, r )pT. However, (48) canbe written in such form by using the closure relation

for the identity matrix: I = rrT+T+T, where r,

and are the unit vectors in the direction of increas-ing values of the variables r, and used in sphericalcoordinates. The directions of these unit vectors areshown in Fig. 3. Using this result and using the rela-tions +2 = 2 = 2/k2 and = 2 = 2/k2, thefar field Greens function can be written as

GFF(r) =k2

4pi2eikar

rrrT+

k24pi2

eikr

r

(T+T

). (49)

Let us consider the first term. For a general exci-tation f at the origin, the inner product of the forcewith the Greens tensor should be taken. This meansthat the first term gives a contribution proportional

to r(eikar/r

)(rf), and the displacement of this wave

is in the radial direction. This term therefore corre-sponds to a longitudinal wave. The wave propagateswith a wave number k through space, and the corre-sponding wave velocity is given by defined in (46).It follows from (46) and the fact that the Lame con-stants positive that is larger than . This means thatthe P wave arrives first in the time domain. Duringthe propagation the wave decays because of geomet-rical spreading with a factor 1/r. The excitation ofthis wave is given by the component (rf) of the forcein the direction of propagation. Since the unit vector


Figure 3 Definition of the unit vectors r, and that are usedin a system of spherical coordinates.

r gives the polarisation of this wave, it will be calledthe polarisation vector.

In the same way, the last terms in (49) describethe propagation of S waves. These waves have awave number k and propagate away from the sourcewith a velocity defined in (46). For these waves,oscillation is in the direction u or w. As shown inFig. 3, both directions are orthogonal to the direc-tion of propagation r. This implies that the waves de-scribed by the last terms in (49) are transverse waves.The excitation of each of these waves depends on thecomponents of the exciting force (uf) or (wf) alongthe direction of polarisation. Note that there aretwo directions of oscillation for the S wave becausein three dimensions two vectors are orthogonal to agiven direction of propagation.

These results imply that both the P wave and thetwo S waves can be written in the same form. Foreach of these waves a polarisation vector can be de-fined; for the P wave the polarisation vector is givenby r, while for the S waves the polarisation vector isgiven by u and w respectively. The far field Greenstensor can be written as a sum over these polarisationvectors. Let the summation over the polarisation vec-tors be denoted by a Greek subscript, then the far fieldGreens tensor can be written as

GFF(r) = k2

4pi2p

eikr

rp , (50)

where k = k for the P wave and k = k for theS wave. Since the polarisation vectors are real, theasterisk has no effect here but it is used because,as shown in 1 of Chapter 1.7.3 the Greens tensorfor Rayleigh waves has polarisation vectors that arecomplex.

The terms in the first line of (44) depend only onthe P-wave velocity , while the terms in the sec-ond line depend only on the S-wave velocity . It istherefore tempting to identify the first line with theP-wave motion and the second line with the S-wavemotion. It follows from (47), however, that each ofthese terms has a 1/r3 singularity in the near field(kr < 1). A 1/r or 1/r2 singularity of the Greens ten-sor poses no problem because these singularities areintegrable in three dimensions. (They are absorbedby a contribution r2 from the Jacobian in sphericalcoordinates.) This means that the P-wave terms andthe S- wave terms separately have a nonintegrable sin-gularity at the origin. As shown by Wu and Ben-Menahen (1985), the sum of these terms is integrablein the near field:

GNF(r) = 18pi

(1

2

2

)1r(I3rrT). (51)

The lesson we learn from this is that the notion of Pand S waves is meaningful only in the far field; in thenear field the compressive and shear motions combinein an intricate way. Similarly, the distinction betweenbody waves and surface waves is physically meaning-less in the near field.

The Ray-Geometric Greens Tensor for Body Waves

As a second example of the Greens function we con-sider the ray-geometric propagation of body waves inelastic media. Because the algebraic complexity of raytheory for elastic media hides the physical simplicitya heuristic derivation is given here. Details are givenby Cerveny and Ravindra (1971), Cerveny and Hron(1980) and Aki and Richards (1980). Here we con-sider the wave field at location r due to a point forcef at source location rs.

When the variations of the properties of themedium and of the amplitude of the waves is smallover a wavelength, the equations of elastodynamicwave propagation are approximated well by theeikonal equation for the phase and the transportequation for the amplitude. Ray theory thus consti-tutes a high-frequency approximation. The eikonaland transport equations should be applied to the Pwave and the two S waves separately. Here we con-sider a mode of propagation that may be either theP wave or an S wave. The corresponding velocity isdenoted by c. The eikonal equation states that thegradient of the phase in the ray direction is given by/c. The arrival time of waves is determined by thephase in the frequency domain. Integrating along theray, the phase of mode is given by

= 1

cds =

kds. (52)

The transport equation ensures that energy is con-served by requiring that cJA2 is conserved, where A


Figure 4 Definition of the geometrical spreading factor J .

is the amplitude of the wave field and J is the geo-metrical spreading factor. As shown in Fig. 4, Jd isthe surface area spanned by a bundle of rays with asolid angle d at the source. This result implies thatthe amplitude of the wave is given by A = C/(

cJ),

where the constant C needs to be determined. Thepolarisation vector p is parallel to the ray directionfor the P wave and is orthogonal to the ray directionfor the S waves. Using these results one finds that thedisplacement for mode is given by

u(r, rs) =C

(r)c(r)J(r, rs)p(r)ei . (53)

The value of the constant C follows from the factthat close to the source, the medium can be treated asif it is homogeneous with the properties of the sourceregion. The corresponding wave field in the vicinity ofthe source is given by (50) contracted with the excita-tion f. (The contraction means that the quantities arecombined using the rules for matrix multiplication,i.e., (Gf)i stands for jGijfj.) For a homogeneousmedium, the rays are straight lines. It follows fromFig. 4 that, near the source, the geometrical spread-ing is given by J(r, rs) = |r rs|2. These results lead tothe following constraint on the constant C:

1

4pi(rs)c2(rs)

(p(rs) f

)|r rs|

=C

(rs)c(rs)|r rs|2. (54)

Solving this expression forC leads to the displacementof mode

u(r, rs) =1

4pi

(r)(rs)c(r)c3(rs)p(r)

ei

J(r, rs)

(p(rs) f

). (55)

Summing over the polarisations the complete ray-geometric Greens tensor can then be written as

GFF(r, rs) =

1

4pi

(r)(rs)c(r)c3(rs)p(r)

exp (ikds)

J(r, rs)p(rs). (56)

Note the similarity between this expression andEq. (50) for the Greens tensor in a homogeneousmedium. The polarisation vectors p(r) and p(rs)are defined by the ray direction at the point r andthe source point rs respectively. In (50) these vectorsare identical because the referencemedium is homoge-neous in that development. Since the rays in generalare curved, the polarisation vectors p(r) and p(rs)usually have a different direction. The integral of thephase is computed along the ray that joins the points rand rs. In the ray-geometric approximation the waveswith different polarisations are not coupled; this is re-flected by the fact that expression (56) contains a sin-gle sum over the different modes of polarisations.

The reader may be puzzled by the fact that theGreens tensor (56) appears to violate the principleof reciprocity (28) because the velocity at the sourceenters through a factor c3(rs), whereas the velocityat location r is present with a factor c(r). Sniederand Chapman (1998) show that for a point sourcein three dimensions the geometrical spreading factorsatisfies the reciprocal rule J(r, rs)c2(rs) = J(rs, r)c2(r).The velocity term in this expression ensures that theGreens tensor (56) indeed satisfies the reciprocityproperty (28).

Other Examples of Dyadic Greens Tensors

It is shown in 1 of Chapter 1.7.3 that the Greens ten-sor for surface waves in an isotropic half-space thatvaries only with depth can be written as

G(r, rs) = p(z,)

ei(kX+pi/4)pi2kX

p(zs,). (57)

In this expression the p are appropriate polarisa-tion vectors that describe the particle motion of Loveand Rayleigh waves, and X is the horizontal distancebetween r and rs. Again, the Greens tensor is givenby a sum over polarisation vectors that correspondto the (surface wave) modes of the system. Theterm exp (ikX)/

kX accounts for the phase shift

and geometrical spreading in the horizontal propaga-tion from the source at rs to the point r and denotesthe azimuth of the sourcereceiver direction.

The Greens tensor of the Earth can be expressed asa sum over normal modes of the Earth. These normalmodes also incorporate self-gravitation in the Earth.The theory of the Earths normal modes is describedin detail by Dahlen and Tromp (1998). Let the dis-placement of a normal mode be denoted by p(r)and let the mode have eigenfrequency . Accordingto Section 6.3.3.2 of Ben-Menahem and Singh (1981),the modes are orthogonal and can be normalised toform an orthonormal set with respect to the innerproduct

p|p= , (58)


where the inner product is defined by

u|v i

(r)ui (r)vi(r)dV =

(r)u(r) v(r)dV. (59)

Note the presence of the mass density in this in-ner product. As shown in Ben-Menahem and Singh(1981) the displacement for a point source f atlocation rs is given by

u(r) =

1

2 2p(r)p(rs)|f, (60)

where is the angular frequency of the excitation.This implies that the Greens tensor is given by

G(r, rs) = p(r)

1

2 2p(rs). (61)

Again the Greens tensor is written as a sum overdyads composed of the polarisation vector multipliedwith a function that measures the strength of theresponse. Note that the response is strongest whenthe excitation has a frequency close to a resonance( ). At a resonance ( = ) expression (61) forthe Greens tensor is infinite, but anelastic dampingprevents a singularity in the solution. In the presenceof anelastic damping, however, the eigenfunctions arenot orthogonal so one of the polarisation vectors ineach term of (61) must be replaced by the polarisa-tion vector of the dual eigenfunctions. The reader isreferred to Dahlen and Tromp (1998) for details.

Physical Interpretation of Dyadic Greens Tensor

In Eqs. (50), (56), (57) and (61) the far field Greensfunction is written as a sum of dyads:

G(r, rs) = p(r)R(r, rs)p

(rs). (62)

In this expression R(r, rs) is a response function thatmeasures the strength of the response. The displace-ment generated by a force f is given by

u(r) = p(r)R(r, rs)p(rs)|f. (63)

Reading this expression from right to left one can fol-low the life history of each mode . A mode can referto either a normal mode of the Earth or a surface wavemode, but it may also refer to the three polarisationsof body wave propagation. At the source the mode isexcited, the excitation given by the projection of theforce on the polarisation vector: p(rs)|f. The wavethen travels from the source to the receiver; this is ac-counted for by the response function R(r, rs). Finally,at the receiver location r the mode oscillates in thedirection given by the polarisation vector p(r). Thegeneral expression (62) therefore not only unifies thevarious Greens tensors from a mathematical point of

view, it also provides a physical description of the roleplayed by the Greens tensor in the life history of thewaves.

An Example of a Complicated Greens Tensor

The Greens tensors given in the previous sections arefor simple models of elastic media. These Greens ten-sors could be obtained because of the symmetries ofthe models considered and the simplifying assump-tions that have been made (such as smoothness of themodel). In practice the response of an arbitrary elasticmedium can be very complex. The Greens functiongives the impulse response of themedium. This meansthat for a localised earthquake, the recorded groundmotion gives a direct measurement of the Greensfunction in the Earth.

The top panel of Fig. 5 shows the vertical compo-nent of the ground motion recorded with the Park-field array in California (Fletcher et al., 1992) aftera local earthquake. The signals have a noisy appear-ance. At almost the same location a second earth-quake occurred on the same day; the resulting groundmotion measured by the same array is shown in thebottom panel of Fig. 5. The ground motion recordedafter these two earthquakes is virtually identical, eachtrace in the top panel matches the corresponding tracein the bottom panel wiggle by wiggle. In fact, thediscrepancies between the corresponding traces are onthe same order of magnitude as the fluctuations in thetraces before the first impulsive arrival.

This similarity implies that the noisy signals inFig. 5 are not noise but deterministic Earth response.At these frequencies the Earth is a strongly scatteringmedium, and the resulting ground motion is a com-plex interference pattern of multiply scattered andmode-converted elastic waves. The complexity of thewaves that propagate through the Earth is a strongfunction of frequency. At present it is not knownto what extent these complex waves can be used forimaging purposes. This issue and the possible re-lationship between classical chaos and wave chaosare discussed further in Scales and Snieder (1997),Snieder (1999) and Snieder and Scales (1998).

8. Anisotropic MediaPlane-Wave Solutions

When the elastic medium is anisotropic, interest-ing wave phenomena that have no counterpart inisotropic media occur. The phenomena are intro-duced here by considering a plane wave that prop-agates in a homogeneous elastic medium

u(r) = p eikr, (64)


Figure 5 Wave field recorded at the Parkfield array (Fletcheret al ., 1992) in California after two nearby earthquakes (top andbottom, respectively). Note the extreme resemblance of eachtrace in the top panel with the corresponding trace in the bottompanel. (Courtesy of Peggy Hellweg of the USGS).

where p is the polarisation vector and k the wave vec-tor. As illustrated in Fig. 6, the wave fronts of thisplane wave are perpendicular to a unit vector n. Withthe phase velocity denoted by v the wave vector canbe written as

k =

vn. (65)

Inserting the solution (64) into the equation of mo-tion (13) and using that and c do not depend on theposition one finds that

2pi cijklkjkkpl = 0. (66)With expression (65) for the wave vector this resultcan also be written as

Gp = v2p, (67)

with G defined by

ij = cikljnknl. (68)

Let us consider the direction of wave propagationn to be specified. This defines the matrix G throughexpression (68). Equation (67) then constitutes aneigenvalue problem; the eigenvectors of G define thepolarisation vectors p and the eigenvalues are given by

Figure 6 The phase velocity v = vn and the group velocity Uin relation to the orientation of the wave fronts in an anisotropicmedium. The transverse gradient 1 is defined in expression(76).

v2, so that the eigenvalues provide the phase velocityof the plane wave. Three important issues should benoted: (i) G is a real symmetric 3 3 matrix; hencethere are three real eigenvalues. (ii) Each eigenvaluegives a velocity v of the plane wave with polarisationvector p that corresponds to that mode of propaga-tion. The eigenvalues of G are equal to v2; theseeigenvalues are positive.1 This means that when v is asolution, then another solution is v; this reflects thereciprocity of waves in an elastic medium. (iii) Thepolarisation vectors and the phase velocity do not de-pend on frequency, this means that there is no disper-sion for the homogeneous medium.

The divergence and the curl of the displacement(64) are given by

(u) = iv

(np) eikr,

u = iv

(n p) eikr. (69)

For an anisotropic medium one cannot state that onesolution is polarised in the direction of propagation(p//n) and that two solutions are polarised in thetransverse directions (p n). It follows from (69)that the wave is curl-free when the polarisation islongitudinal (p//n) and that the wave is divergence-free when the wave is transversely polarised ( p n).Therefore, in a general anisotropic medium the planewave solutions are neither divergence-free nor curl-free.

Wave propagation in smoothly varying elastic me-dia can be described by a formulation of geometric raytheory that is suitable for such media (Vlaar, 1968;Cerveny, 1972). From the analysis of their work thelocal polarisation vectors and the local phase velocityagain follow from the eigenvalue problem (67), wherethe density and elasticity tensor are now evaluated atthe position of the ray for a given ray direction n.


An Isotropic Medium

Let us now for the moment consider the special caseof an isotropic elastic medium for which the elasticitytensor is given by (10). Using that njnj = (nn) = 1 onefinds that in this special case the matrix G from ex-pression (68) is given by ij = (+)ninj +ij, so thatin vector notation

G = (+)nnT +I. (70)

One eigenvector is given by p = n,

Gp = (+2)p. (71)

Hence the associated wave velocity is given by v2 =+2. This velocity is the P velocity, which is denotedby ,

=

+2

(46) again.

Note that since n = p, this means that the wavesare longitudinally polarised. For this type of waven p = 0; with (69) this implies that the P waves arecurl-free. The P waves thus involve only compressivemotion, but no shear motion.

Another eigenvector is such that the polarisationis orthogonal to the direction of propagation: p n,which means we now consider transverse waves. Itfollows from (70) that

Gp = p. (72)

According to (67) this corresponds to a wave velocityfor which v2 = . This is called the S velocity, whichis denoted by :

=

(46) again.

For these waves p n hence (pn) = 0. With (69) thisimplies that the S waves are divergence-free. S wavesdo not lead to compressive motion; they entail shearmotion in the medium.

Distinction between Phase and Group Velocity

Let us now return to a general anisotropic medium.For a given direction of propagation n, the polar-isation vectors follow from the eigenvalue problem(67). In general the polarisation vectors p are nolonger related in a simple way to the direction ofpropagation; they are not necessarily parallel or or-thogonal to the direction of propagation, but theyare mutually orthogonal. With (69) this implies thatthe waves in an anisotropic medium are neither curl-free nor divergence-free. In a general anisotropicmedium one cannot therefore speak of the P waveand of S waves. However, when the anisotropy isweak, one of the waves is polarised almost in thelongitudinal direction and two waves are polarised

almost orthogonally. This means that in a weaklyanisotropic medium one speaks of quasi P wavesand quasi S waves(Coates and Chapman, 1990).The quasi Pwaves are not quite curl-free and the quasiS waves are not completely divergence-free.

For surface waves the situation is similar; in ananisotropic medium one cannot speak of Rayleighwaves and Love waves, although for weakanisotropy one can distinguish quasi Rayleighwaves and quasi Love waves (Maupin, 1989).When the surface wave modes are nearly degener-ate (in the sense that their phase velocities are nearlyequal) these concepts must be used with care.

In an anisotropic medium the relation between thephase velocity and the group velocity is not trivial.This leads to observable phenomena. A plane wavehas in the time domain the form exp i(krt). Thephase slowness s (defined as the reciprocal of thephase velocity) is given by s = k/. The phase slow-ness is used rather than the phase velocity because thisis a tensor of rank 1, while the phase velocity does nottransform as a tensor. With (65) the phase slownesscan be related to the direction of propagation n andthe magnitude of the phase velocity v as

s =1vn. (73)

The phase slowness is shown in Fig. 7 for a numberof points in the (kx,kz) plane by the dashed arrows.The phase slowness is parallel to the wave vector k.

The group velocity of the plane wave is in contrastgiven by

U = =k. (74)

In this expression, =k is the gradient operator inwave-number space. The group velocity describes theenergy flow of the waves in the medium (Babuskaand Cara, 1991; Wolfe, 1998). Shown in Fig. 7 isthe contour = constant in the (kx,kz) plane. Sincethe gradient of (k) is orthogonal to the contour lines = constant, the group velocity is orthogonal to thesurface of constant values of . The direction ofgroup velocity vectors at points 1, 2, 3 and 4 in thewave-number plane is indicated by the solid arrows inFig. 7. It follows from the geometry of this figure thatthe group velocity and the phase slowness at any pointin an anisotropic medium in general have a differentdirection. This is possible because the directional-ity contained in the elasticity tensor of an anisotropicmedium breaks the symmetry of the system so thatthe direction propagation of the wave fronts and theenergy flow are not necessarily aligned.

The difference in direction of propagation betweenthe wave front and the wave group can be quanti-fied by decomposing the gradient =k into a compo-nent parallel to the wave vector and an orthogonalcomponent


Figure 7 Surface = const. in the wave-number plane. Thephase slowness s = k/ for the points 1, 2, 3 and 4 is indicatedby arrows. The direction of the group velocity U =k is given bythe solid arrows.

=k = n

k +1k

=1 , (75)

where =1 is the component of the gradient perpen-dicular to the direction of propagation. Using spheri-cal coordinates in wave-number space, the orthogonalcomponent can be written as

=1 = u

+w

sin

. (76)

Inserting = vk in (74) and using (75), the groupvelocity is given by

U(n) = v(n) +1v(n). (77)With (76) it then follows that

U(,) = v(,)n+ v u+1

sinv w. (78)

Thus, in general the phase velocity v = vn and groupvelocity U have different directions. The difference indirection is proportional to the change in the phasevelocity to the direction of propagation as describedby the derivatives v/ and v/.

Using that 1 n, one finds from (77) and fromv = vn that the components of the phase velocity andthe group velocity in the direction of propagation areidentical:

(nU) = (nv) = v. (79)This means that the group velocity differs from thephase velocity only because it also has a componentalong the wave fronts, as is illustrated in Fig. 6. Inanisotropic media the energy flow is therefore not

necessarily orthogonal to the wave fronts. It followsfrom the geometry of Fig. 6 and expression (77) thatthe angle between the group velocity and phase veloc-ity is given by

tan = |1v|v

. (80)

The nontrivial relationship between phase slow-ness and group velocity has interesting consequences.As an example, consider once again the group velocityat the points 1, 2, 3 and 4 in Fig. 7. The correspond-ing group velocities at these points are shown in thegroup velocity plane in Fig. 8. The group velocitiesindicated by the solid arrows have the same directionas the group velocity shown by the solid arrows inFig. 7. When we move from point 1 through point 2to point 3 in Fig. 7, the group velocity vector rotatesin the counterclockwise direction. However, at point3, the curve = constant has an inflection point, andthe group velocity vector at point 4 is rotated clock-wise compared to the group velocity vector at point3. This means that the group velocity vector tracesthe folded curve that is shown in Fig. 8. At point 3the group velocity is stationary for perturbations inthe direction of propagation. Therefore point 3 cor-responds to a caustic in the wave field. This exampleshows that caustics can form in a homogeneous elas-tic medium.

Such caustics can be observed experimentally.Fig. 9 shows snapshots of waves that have propa-gated through a rectangular block of silicon at threedifferent times. (This figure was made available byJ.P. Wolfe (1998) who also describes the experimentin detail.) The waves are excited at one side of theblock. At t = 1.76 s the waves have propagated tothe opposing face where the wave field is measured,and at later times one sees the wave fronts spread overthe face of the crystal. Note that the wave fronts havethe same cusped structure as the group velocity shownin Fig. 8. This type of measurement provides informa-tion about the elasticity tensor of the crystal, which inturn gives information about the symmetry axes andelastic constants of the crystal.

It should be noted that in three dimensions theslowness and the group velocity as shown in Figs. 7and 8 are described by surfaces that can have avery complex shape, which gives rise to caustics inthe wave field with a complicated geometry (Helbig,1994; Wolfe, 1998). The only ingredients used in thissection were that the phase velocity is anisotropic andindependent of frequency, and that the group velocityfollows from expression (74). For this reason the the-ory of this section is not particular for elastic waves.In fact, the theory is equally applicable to other typesof nondispersive anisotropic media. The analogue ofthe theory of this section applied to electromagnetic


Figure 8 The group velocity for the points 1, 2, 3 and 4 thatare also shown in the previous figure. The solid arrows indicatethe group velocity. Note the cusps in the group velocity surface,which correspond to caustics in the wave field.

Figure 9 Snapshots of the wave field that has propagatedthrough a homogeneous silicon crystal recorded at three timesafter excitation on the other side of the crystal. Details of theexperiment are given by Wolfe (1998). (Courtesy of J.P. Wolfe.)

waves can for example be found in Chapter III andSection V.2 of Kline and Kay (1965).

Acknowledgements

I thank Ken Larner, Bob Nowack, Wolfgang Friede-rich, Bob Odom and Valerie Maupin for their criticaland constructive comments on this manuscript.

Note1. When the strain energy is positive, the quantity cikljvinkvlnj is

positive. It follows from this that ij is positive definitely sothat the eigenvalues v2 of G are positive.

References

Aki, K. and Richards, P. G. 1980,Quantitative Seismology.Freeman, San Francisco.

Babuska, V. and Cara, M. 1991, Seismic Anisotropy in theEarth. Kluwer, Dordrecht.

Backus, G. 1970, A geometrical picture of anisotropic elas-ticity tensors. Rev. Geoph. Space Phys., 8, 633671.

Ben-Menahem, A. and Singh, S. J. 1981, SeismicWaves andSources. Springer-Verlag, New York.

Cerveny, V. 1972, Seismic rays and ray intensities in inho-mogeneous anisotropic media. Geophys. J. R. Astron.Soc. 29, 113.

Cerveny, V. and Hron, F. 1980, The ray series methodand dynamical ray tracing system for three-dimensionalinhomogeneous media. Bull. Seismol. Soc. Am. 70,4777.

Cerveny, V. and Ravindra, R. 1971, Theory of SeismicHead Waves. Univ. of Toronto Press, Toronto.

Coates, R. T. and Chapman, C. H. 1990, Quasi-shear wavecoupling in weakly anisotropic media. Geophys. J. Int.103, 301320.

Dahlen, F. A. and Tromp, J. 1998, Theoretical Global Seis-mology. Princeton Univ. Press, Princeton, NJ.

de Hoop, A. T. 1995, Handbook of Radiation and Scatter-ing of Waves: Acoustic Waves in Fluids, Elastic Wavesin Solids, Electromagnetic Waves. Academic Press, SanDiego.

Fletcher, J. B., Baker, L. M., Spudich, P., Goldstein, P., Sims,J. D. and Hellweg, M. 1992, The USGS Parkfield, Cal-ifornia, Dense Seismograph Array: UPSAR. Bull. Seis-mol. Soc. Am. 82, 10411070.

Goldstein, H. 1980, Classical Mechanics, 2nd ed. Addison-Wesley, Reading MA.

Helbig, K. 1994, Foundations of anisotropy for explorationseismics, (K. Helbig and S. Treitel, Eds.). Pergamon, Ox-ford.

Hudson, J. A. 1977, Scattered waves in the coda of P. J.Geophys. 43, 359374.

Hudson, J. A. 1980, The Excitation and Propagation ofElastic Waves. Cambridge Univ. Press, Cambridge, UK.

Hudson, J. A. and Heritage, J. R. 1982, The use of the Bornapproximation in seismic scattering problems.Geophys.J. R. Astron. Soc. 66, 221240.

Kline, M. and Kay, I. W. 1965, Electromagnetic Theory andGeometrical Optics, Interscience, New York.

Malvern, L. E. 1969, Introduction to the Mechanics of aContinuous Medium. Prentice-Hall, Engelwood Cliffs,NJ.

Maupin, V. 1989, Surface waves in weakly anisotropicstructures: on the use of ordinary quasi-degenerate per-turbation theory. Geophys. J. Internat. 98, 553564.

Morse, P. and Feshbach, H. 1953a, Methods of TheoreticalPhysics, Part 1. McGraw-Hill, New York.


Morse, P. and Feshbach, H. 1953b, Methods of TheoreticalPhysics, Part 2. McGraw-Hill, New York.

Scales, J. and Snieder, R. 1997, Humility and nonlinearity.Geophysics 62, 13551358.

Snieder, R. 1999, Imaging and averaging in complex media.In Diffuse Waves in Complex Media (J. P. Fouque, Ed.),2000, pp. 405454. Kluwer, Dordrecht.

Snieder, R. 2001, AGuided Tour of Mathematical Methodsfor the Physical Sciences. Cambridge Univ. Press, Cam-bridge, UK.

Snieder, R. and Chapman, C. 1998, The reciprocity proper-ties of geometrical spreading. Geophys. J. Internat. 132,8995.

Snieder, R. and Scales, J. A. 1998, Time reversed imagingas a diagnostic of wave and particle chaos. Phys. Rev. E58, 56685675.

Vlaar, N. J. 1968, Ray theory for an anisotropicinhomogeneous elastic medium. Bull. Seismol. Soc. Am.58, 20532072.

Wolfe, J. P. 1998, Imaging Phonons, Acoustic Wave Prop-agation in Solids. Cambridge Univ. Press, Cambridge,UK.

Woodhouse, J. H. 1976, On Rayleighs principle. Geophys.J. R. Astron. Soc. 46, 1122.

Wu, R. S. and Ben-Menahem, A. 1985, The elastodynamicnear field. Geophys. J. R. Astron. Soc. 81, 609622.

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