Guided waves in three-dimensional structures

Geophys. J. Int. (1998) 133, 159^174

Guided waves in three-dimensional structures B. L. N. Kennett

Research School of Earth Sciences, Australian National University, Canberra ACT 0200, Australia. E-mail: [email protected]

Accepted 1997 October 18. Received 1997 October 18; in original form 1997 June 25

S U M M A RY A new formulation for the propagation of surface waves in three-dimensionally varying media is developed in terms of modal interactions. A variety of assumptions can be made about the nature of the modal ¢eld: a single set of reference modes, a set of local modes for the structure beneath a point, or a set of local modes for a laterally varying reference structure. Each modal contribution is represented locally as a spectrum of plane waves propagating in di¡erent directions in the horizontal plane. The in£uence of 3-D structure is included by allowing coupling between di¡erent modal branches and propagation directions. For anisotropic models, with allowance for attenuation, the treatment leads to a set of coupled 2-D partial di¡erential equations for the weight functions for di¡erent modal orders. The representation of the guided wave¢eld requires the inclusion of a full set of modes, so that, even for isotropic models, both Love and Rayleigh modes appear as di¡erent polarization states of the modal spectrum. The coupling equations describe the interaction between the di¡erent polarizations induced by the presence of the 3-D structure. The level of lateral variation within the 3-D model is not required to be small. Horizontal refraction or re£ection of the surface wave¢eld can be included by allowing for transfer between modes travelling in di¡erent directions. Approximate forms of the coupled equation system can be employed when the level of heterogeneity is small, for example the coupling between the fundamental mode and higher modes can often be neglected, or forward propagation can be emphasized by restricting the interaction to a limited band of plane waves covering the expected direction of propagation. Key words: 3-D structure, anisotropy, surface waves.

1

IN T ROD U C T I O N

Surface waves form a very prominent part of the recorded seismogram and in consequence it is desirable to be able to describe the behaviour of this part of the seismic wave¢eld as it interacts with 3-D structure. Most studies of surface-wave propagation in heterogeneous media have started from the properties of a strati¢ed medium. For slowly varying layered media, Woodhouse (1974) showed that the dispersion and trajectory of an individual mode could be described by using ray theory, where the local phase velocity and the depth dependence of the displacement match that of a strati¢ed medium with the structure beneath a point on the ray path. In this description there is no interaction between di¡erent modes, although the patterns of propagation can vary according to the nature of the modal dispersion. Woodhouse & Wong (1986) extended the ray approach to include longer-period normal modes with a partial allowance for amplitude e¡ects along the ray path. More complex structure can be included in a normal-mode description by allowing for coupling between modes. Such coupling between neighbouring multiplets along a mode branch has been considered by a number of authors (see e.g. Park 1987; Romanowicz 1987; Tsuboi & Geller 1989). The computational requirements for including coupling between normal-mode branches are more severe, but Yu & Park (1993) have included the e¡ects of anisotropy at long periods with limited cross-branch coupling. Tanimoto (1990) discussed the in£uence of signi¢cant lateral variations in phase velocity for fundamental-mode Love and Rayleigh waves propagating from the Whittier Narrows earthquake to stations in Northern California. His work was based on an approximate representation of the surface waves using a ¢nite di¡erence representation of the phase behaviour. ß 1998 RAS

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A more complete analysis requires the inclusion of coupling between mode branches. For 2-D structures Kennett (1984) introduced a description of the wave¢eld in terms of the modes of a ¢xed reference structure, and Maupin & Kennett (1987) analysed the in£uence of the truncation of modal expansions. Maupin (1988) introduced an alternative treatment for 2-D structures in which the expansion of the wave¢eld was made in terms of the local modes of the structure, with an allowance for coupling induced by the gradients in material properties. Maupin (1992) extended the coupled-mode treatment to handle ducted 3-D surface propagation along 2-D structures. A partial extension of Maupin's (1988) results to the 3-D case was made by Tromp (1994) who concentrated on mode conversions occurring along a given surface-wave trajectory. An approximate formulation for 3-D problems has been presented by Snieder (1986a,b) and Snieder & Nolet (1987) using a scattering development based on a ¢rst-order Born approximation using the modes of a ¢xed reference structure. This approach is able to account for changes in the direction of surface waves as well as intermode coupling induced by heterogeneity. The use of ¢rstorder scattering theory restricts the validity of the approximation to situations where the perturbation to the reference wave¢eld induced by the presence of heterogeneity is small. This approach is therefore suitable for localized moderate heterogeneity or for very weak heterogeneity extending over a broad area. An alternative approach for the 3-D case has been suggested by Bostock (1992) for the restricted situation where the pattern of heterogeneity can be represented in terms of radial variation about a source point modulated by an angular spectrum. The structure of the resulting coupled equations for the modal coe¤cients is very similar to that in Kennett's (1984) treatment of 2-D problems. Here we present a new formulation of the propagation of surface waves in three-dimensionally varying media, which leads to a set of coupled partial di¡erential equations for the weight functions for di¡erent modal contributions in an anisotropic model, including the e¡ects of attenuation. The basic equations can then be used with a variety of di¡erent assumptions about the nature of the modal ¢eld, for example the use of a single set of reference modes as in the 2-D work of Kennett (1984), local modes as employed by Maupin (1988, 1992), or a set of local modes for a laterally varying reference structure. Each modal contribution is represented locally as an angular spectrum of plane waves and the in£uence of the 3-D structure is included by allowing coupling between di¡erent modal branches and propagation directions. The representation of the guided wave¢eld requires the inclusion of the full set of modes, so that, even for isotropic models, both Love and Rayleigh modes appear as di¡erent polarization states of the modal spectrum. The coupling equations describe the interaction between the di¡erent polarizations induced by the presence of the 3-D structure. The level of lateral variation within the 3-D model is not restricted to be small, and horizontal refraction or re£ection of the surface wave¢eld is included by allowing for transfer between modes travelling in di¡erent directions. When the level of heterogeneity is small the complexity of the coupled equation system can be reduced by suitable approximations. For example, interactions between the fundamental modes and higher modes can be neglected, or the range of angular interaction can be restricted to emphasize forward propagation. For a 2-D model where the structure is independent of the x2 coordinate, the most important feature for a contribution to the surface wave¢eld is the sense of the propagation direction relative to the x1 axis. This applies for waves travelling in the x1 direction (Kennett 1984) or for plane waves at oblique incidence (Maupin 1988); the development of coupled-mode equations can thus be cast in terms of forward and backward propagation (relative to x1 ) for a set of mode branches. However, the situation is more complex for 3-D structures. The simplest assumption of great-circle propagation from source to receiver with no in£uence of transverse gradients in seismic structure has been used by Tromp (1994). In a high-frequency asymptotic viewpoint, the propagation for each mode branch follows a ray path dictated by phase velocity variation at each frequency (Woodhouse 1974). In a heterogeneous medium the paths can depart signi¢cantly from the great circle and the paths for di¡erent mode branches can have rather di¡erent character. Any method to handle 3-D structure must, therefore, be able to allow for deviations from the initial direction of propagation. In the treatment that follows, we are able to describe the full range of propagation phenomena by not only allowing transfer of energy between mode branches, but also including a spectrum of directions of propagation for each branch. A change in the direction of propagation of the wave¢eld can thus be described in terms of a modi¢cation of the angular spectrum associated with each mode branch.

2

MODA L EVOLU T I O N E QUAT I O N S

The methods developed by Kennett (1984) and Maupin (1988, 1992) were based on writing the equations of motion and stress^strain relations for seismic waves in a form where a single horizontal coordinate is given a preferred status. A set of coupled partial di¡erential equations can be constructed for the components of displacement and the traction in the preferred direction. Derivatives with respect to the preferred coordinate are restricted to ¢rst order only and can be separated from the remaining terms in which derivatives with respect to the other spatial coordinates and time appear. This approach is well suited to models in which the structure is 2-D. Kennett (1984) has shown how some measure of 3-D e¡ects can be included under the assumption that the propagation path is not signi¢cantly perturbed from the direct path from source to receiver. However, this type of approach cannot readily be extended to general 3-D variation in structure. ß 1998 RAS, GJI 133, 159^174

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For surface waves in a three-dimensionally varying structure we have to be able to describe processes in which the direction of propagation of the wave¢eld varies spatially. We therefore need a representation of seismic wave propagation in which the action of the gradient in the horizontal plane is emphasized. A suitable set of equations can be found which depends only on ¢rst-order derivatives of displacement and traction with respect to the horizontal spatial coordinates, and for which only the vertical derivatives of the seismic parameters are required. There is some resemblance between these equations for the 3-D case and those previously derived for the 2-D situation by Kennett (1984), which can be used to guide a development in which the displacement and traction ¢elds for the guided waves are represented as a sum of modal contributions with spatially varying weighting coe¤cients. 2.1 Coupled equations for displacement and horizontal tractions We work in a Cartesian coordinate system with the x3 (z)-coordinate directed downwards and will emphasize propagation in the horizontal plane by introducing the horizontal coordinate vector x\ and horizontal gradient +\ with components ! ! x1 L/Lx1 x\ ~ , +\ ~ . (1) x2 L/Lx2 We will use roman subscripts i, j, k, l for unrestricted coordinates (e.g. i~1, 2, 3) and greek subscripts when attention is restricted to the horizontal coordinates, e.g. (x\ )a ~xa ,

(+\ )a ~La ,

a~1, 2 ,

(2)

where we have used the compressed notation L1 for L/Lx1. We employ the convention of summation over repeated su¤ces (both i, j, k, l and greek su¤ces). Consider a seismic wave¢eld with displacement u, stress tensor qij and elastic moduli cijkl ; we will concentrate on time-harmonic wave¢elds with angular frequency u, and can allow for attenuation by adopting complex moduli at each frequency. Following Woodhouse (1974) we introduce the elastic moduli matrices C ij such that (C ij )kl ~ckilj ~cikjl .

(3)

Then the stress^strain relation, qij ~cijkl Lk ul ,

(4)

can be written in terms of traction vectors ôi with components (ôi )j ~qij , in the form ôi ~C ij Lj u .

(5)

When we separate out the dependence of the tractions on the horizontal coordinates we obtain ôa ~C ab Lb uzC a3 L3 u , ô3 ~C 3b Lb uzC 33 L3 u .

(6)

The equation of motion, in the absence of sources, can be written as La ôa zL3 ô3 ~{ou2 u .

(7)

If we isolate horizontal derivatives on the left-hand side, L1 ô1 zL2 ô2 ~La ôa ~{ou2 u{L3 ô3 ,

(8)

and we can express ô3 in terms of displacement derivatives using (6) above. Our aim is to keep all derivatives with respect to the horizontal coordinates on the left-hand side of the equations so that we need to ¢nd expressions for L1 u, L2 u. From the de¢nitions of the tractions ô1 , ô2 we have C 11 L1 uzC 12 L2 u~ô1 {C 13 L3 u , C 21 L1 uzC 22 L2 u~ô2 {C 23 L3 u . We can write these two equations in a matrix form, ! ! ! L1 u ô1 {C 13 L3 u C 11 C 12 ~ , L2 u ô2 {C 23 L3 u C 21 C 22

(9)

(10)

and then it would appear that we could recover suitable expressions for L1 u, L2 u by inverting the matrix of moduli terms. Unfortunately, because of the symmetry of the stress tensor, q12 ~(ô1 )2 ~(ô2 )1 ~q21 , ß 1998 RAS, GJI 133, 159^174

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and the associated symmetry of the elastic moduli matrices, two of the six rows of the system (10) are identical and therefore it is singular. The problem can be overcome by working with the horizontal strains, which are symmetric in the displacement derivatives. This gives a condensed system of ¢ve equations which can be solved via the inversion of a non-singular 5|5 matrix whose entries are the elastic moduli. The horizontal derivatives of the displacement u can then be recovered from the strains and represented as L1 u~{[D11 C 13 zD12 C 23 ]L3 uzD11 ô1 zD12 ô2 ,

(12)

L2 u~{[D21 C 13 zD22 C 23 ]L3 uzD21 ô1 zD22 ô2 .

We can derive an evolution equation for the horizontal tractions from the equation of motion in the form (8) if we can represent the vertical traction ô3 in terms of u, ô1 and ô2 . From the de¢nition of the traction vector, ô3 ~C 31 L1 uzC 32 L2 uzC 33 L3 u .

(13)

On substituting for the horizontal derivatives of the displacement ¢eld from (12) we obtain the vertical traction in the form5 ô3 ~(C 33 {C 31 [D11 C 13 zD12 C 23 ]{C 32 [D21 C 13 zD22 C 23 ])L3 uz(C 31 D11 zC 32 D21 )ô1 z(C 31 D12 zC 32 D22 )ô2 .

(14)

The horizontal divergence of the traction ¢eld can therefore be expressed in a form in which only x3 derivatives act on the material properties: L1 ô1 zL2 ô2 ~{ou2 u{L3 [(C 33 {C 31 [D11 C 13 zD12 C 23 ]{C 32 [D21 C 13 zD22 C 23 ])L3 u]{L3 [(C 31 D11 zC 32 D21 )ô1 ] {L3 [(C 31 D12 zC 32 D22 )ô2 ] .

(15)

The set of equations (12) for the horizontal gradients of the displacement vector and (15) for the divergence of the horizontal traction constitute the coupled equations we are seeking for the horizontal evolution of the displacement and traction ¢elds. No horizontal derivatives of seismic parameters appear on the right-hand sides of these equations. We can rewrite this coupled set of equations in a form which displays the derivatives with respect to x3 explicitly: 12 L1 u~A1uu L3 uzA11 uq ô1 zAuq ô2 , 22 L2 u~A2uu L3 uzA21 uq ô1 zAuq ô2 ,

(16)

L1 ô1 zL2 ô2 ~{ou2 u{L3 [Aqu L3 u]{L3 [A1qq ô1 zA2qq ô2 ] , in terms of a set of operators A1uu ~{[D11 C 13 zD12 C 23 ] , A2uu ~{[D21 C 13 zD22 C 23 ] , A11 uq ~D11 , A12 uq ~D12 , A21 uq ~D21 ,

(17)

A22 uq ~D22 , Aqu ~C 33 {C 31 [D11 C 13 zD12 C 23 ]{C 32 [D21 C 13 zD22 C 23 ] , A1qq ~C 31 D11 zC 32 D21 , A2qq ~C 31 D12 zC 32 D22 . In this formulation, ô3 ~Aqu L3 uzA1qq q1 zA2qq q2 .

(18)

The presence of vertical derivatives means that in the presence of any material interfaces, delta functions and their vertical derivative will appear. The equations must therefore be viewed in a distributional sense, and the operators will always be tamed by application to a trial ¢eld. The matrices C ij , Dij reduce to relatively simple forms for materials with well-developed symmetry properties, for example isotropic media or media with hexagonal symmetry, which gives rise to transverse isotropy where the moduli in the plane perpendicular to the symmetry axis di¡er from those along the symmetry axis itself. We can make an explicit allowance for the presence of material discontinuities by introducing an interfacial force term. Consider the case of an inclined interface speci¢ed by the equation x3 ~hk (x\ ); across the boundary the normal component of ß 1998 RAS, GJI 133, 159^174

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traction must be continuous, so kz 1=2 [tk ]kz ~0 . k{ ~[ô3 {La hk ôa ]k{ /[1zLa hk La hk ]

(19)

The continuity condition is therefore equivalent to a jump in the traction ô3 , kz [ô3 ]kz k{ ~[La hk ôa ]k{ .

(20)

This traction discontinuity can alternatively be viewed as a localized volume force f situated at the material interface f~[La hk ôa ]kz k{ d(x3 {hk (x\ )) .

(21)

We can represent the coupled equations (16), together with the continuity conditions in the compact form La u~Aauu L3 uzAab uq ôb , La ôa ~{ou2 u{L3 [Aqu L3 u]{L3 [Aaqq ôa ]z

X k

[La hk ôa ]kz k{ d(x3 {hk (x\ )) ,

(22)

together with the continuity of displacement at any interface. The nature of the coupled equation system for displacement and horizontal traction (22) arises from the di¡erent rotation properties of displacement and traction. The displacement u and the vertical traction ô3 transform as vectors under rotation about a vertical axis. However, the horizontal tractions ô1 , ô2 undergo a tensorial transformation with more complex angular dependences, but their divergence La ôa has the same vector rotation property as the displacement. 2.2

Modal expansions

In principle, we can reduce the dependence of the coupled equations for displacement and horizontal traction, (22), from three dimensions to two by projecting the vertical (x3 ) dependence onto an orthogonal set of eigenfunctions. Thus we propose a representation of the displacement ¢eld as a sum of modal contributions with horizontally varying coe¤cients for each frequency u, for example XX cJh (x\ )ueJ (kJh , x3 )ikJh :x\ , (23) u~ J

h

where J is a mode-branch index and h represents an angular spectrum of local plane waves for each mode index. ueJ (kJh , x3 ) is the eigenfunction for the Jth mode with horizontal wavenumber kJh. Associated with the displacement (23) there would be a corresponding traction ¢eld XX cJh (x\ )ôe\J (kJh , x3 )ikJh :x\ . (24) ô\ ~ J

h

The angular spectrum employed in (23), (24) would be continuous in an analytic representation, but in any numerical implementation would be represented by a discrete sum. We have therefore chosen to start with a discrete formulation. The particular form of representation used in (23), (24) would be appropriate to a single set of reference modes for the entire region of interest and would be the 3-D analogue of the expansion used by Kennett (1984). Such a constant reference structure is likely to be helpful when the velocity model can be visualized as a set of perturbations (not necessarily small) from the reference structure. When the structural model includes systematic slow gradients in the seismic parameters, an alternative choice for the modal representation would be to use local modes, as in the work of Maupin (1988) and Tromp (1994), where the eigenfunctions are constructed for the 1-D model in the vertical column below each point. In such a coupled local-mode expansion the eigenfunctions are position-dependent and the phase terms include the cumulative phase history for each modal and angular component. A further possibility is to follow the procedure sketched in the Appendix to Kennett (1984) and work with the local modes of a reference medium which is a smoothed version of the actual structure. Such an approach combines some of the merits of the local-mode and reference-structure approaches and can be applied to models with velocity gradients and signi¢cant local variations. The representation of the guided wave¢eld requires the inclusion of a full set of modes, so that, even for isotropic models, both Love and Rayleigh modes must appear as di¡erent polarization states of the modal spectrum. The coupling equations we will derive below include the interaction between the di¡erent polarizations induced by the presence of the 3-D structure. In order to include in the subsequent analysis all these three types of modal representations, (1) ¢xed reference modes, (2) local modes and (3) local modes of a ¢xed reference structure, we will adopt a generic form for the modal representation of the displacement5 X cq (x\ )ueq (kq , x\ , x3 )eq (kq , x\ ) , (25) u~ q

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where we have used a single index q covering both the mode order and the angular spectrum, and eq represents the phase contribution. We will also use the index r in a similar way. The tractions corresponding to (25) will be written as X ô~ cq (x\ )ôeq (kq , x\ , x3 )eq (kq , x\ ) . (26) q

We will always represent summation over the mode index q, r explicitly but will continue to use the convention of implicit summation over repeated greek su¤ces a, b for the horizontal coordinates. On substituting the modal representations (24), (25) into the horizontal evolution equations (22), we ¢nd ! X X e e La cq uq eq ~ (Aauu L3 ueq zAab uq ôqb )cq eq , q

X

La

q

q

! cq ôeqa eq

~

X q

({ou2 ueq {L3 [Aqu L3 ueq ]{L3 [Aaqq ôeqa ])cq eq z

XX q

k

(27) [La hk ôeqa ]kz k{ d(x3 {hk (x\ ))cq eq

,

where we have allowed for the possible presence of sloping interfaces. Consider the terms on the left-hand side of (27), e.g. ! X X X e La cq uq eq ~ La (ueq )cq eq z ueq La (cq eq ) . q

q

(28)

q

The horizontal derivatives of the eigenfunctions ueq will appear for a local-mode treatment or where a smoothly varying reference medium is being employed, but will not be present in the case of a constant reference medium. e# We now introduce the dual modes ue# r , ôra associated with the wavenumber {kr which appear in the orthogonality condition between di¡erent modes derived in Appendix B. We de¢ne e ue# r ~ur ({kr , x\ , x3 , u) .

(29)

These dual modes were used implicitly by Kennett (1984) and explicitly by Maupin (1992) in a study of surface-wave guiding in a 2-D structure. We operate on the displacement equation (27) with the horizontal traction for the dual mode, X X X a e e# ab e La ueq :ôe# ueq :ôe# (ôe# (30) ra cq eq z ra La (cq eq )~ ra :Auu L3 uq zôra :Auq ôqb )cq eq , q

q

q

and on the traction equation (27) with the displacement in the dual mode, X X X e e# e e# a e La ôeqa :ue# ôeqa :ue# ({ou2 ue# r cq eq z r La (cq eq )~ r :uq {ur :L3 [Aqu L3 uq ]{ur :L3 [Aqq ôqa ])cq eq q

q

q

z

XX q

k

e kz La hk ue# r :[ôqa ]k{ d(x3 {hk (x\ ))cq eq ,

(31)

then subtract (31) from (30) and integrate over the full depth of the half-space. The resulting equations for the horizontal evolution of the modal contribution terms cq take the form X …? XX X …? e# e e kz dx3 (ueq :ôe# {u :ô )L (c e )z dx3 (re# :La ôeqa )cq eq { (La hk ,ue# a q q ra r qa r :[ôqa ]k{ )cq eq 0

q

~

q

X …? q

0

0

q

k

a e e# ab e 2 e# e e# e e# a e dx3 (ôe# ra :Auu L3 uq zôra :Auq ôqb zou ur :uq zur :L3 [Aqu L3 uq ]zur :L3 [Aqq ôqa ])cq eq :

(32)

The orthogonality property between di¡erent modes derived in Appendix B can be written in a comparable notation as …? e# e dx3 (ueq :ôe# i(kqa {kra ): ra {ur :ôqa )~0 ,

(33)

and the normalization when kq is equal to kr as …? e# e i dx3 (ueq :ôe# qa {uq :ôqa )~kqa /jkq j :

(34)

0

0

The equations (32) represent the most general set of coupled equations for the modal coe¤cients and can be simpli¢ed somewhat by making speci¢c assumptions about the form of the modal representation. ß 1998 RAS, GJI 133, 159^174

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Guided waves in 3-D structures The orthogonality condition (33) means that the vector integral …? e# e Lrq ~ dx3 (ueq :ôe# ra {ur :ôqa ) 0

165

(35)

will be non-vanishing for even the same mode travelling in di¡erent directions. When jkq j~jkr j, (33) requires Lrq to be oriented midway between the directions of propagation represent by kq, kr . If the angle between these directions is #, then the 1 and 2 components of Lrq have to be in the ratio 1z cos #5 sin #. For example, for a Love wave in a transversely isotropic medium with a vertical symmetry axis, …? e L1rq ~ikq (1z cos #)(2 cos #{1) dx3 Noe# q oq , L2rq ~ikq sin #(2 cos #{1)

…? 0

0

e dx3 Noe# q oq ,

and the integral is that expected in two dimensions, but modulated by the angular dependence (2 cos #{1). In the corresponding Rayleigh-wave case the 1 and 2 components are in the same ratio, but for #=0 a new term arises which is not present in the 2-D case. 3

EVOLU T I O N E QUAT I O N S FOR MODA L C O E F F IC I E N T S

3.1

Matrix equations

We can express the coupled equations (32) for the modal coe¤cients cq in the form of a set of N equations for N expansion coe¤cients {cq }: X X [Larq La (cq eq ){Jrq cq eq {Hrq cq eq ]~ Mrq cq eq . (36) q

q

The matrix elements on the left-hand side of (36) depend on the structural parameters for the medium purely through the functional form of the modal eigenfunctions5 …? e# e dx3 (ueq :ôe# Larq ~ ra {ur :ôqa ) , 0

Jrq ~{

…?

Hrq ~{

0

e e e# dx3 (ue# r :La ôqa {La uq :ôra ) ,

X k

(37)

e kz (La hk ,ue# r :[ôqa ]k{ ) ,

whereas …? a e e# ab e 2 e# e e# e e# a e Mrq ~ dx3 (ôe# ra :Auu L3 uq zôra :Auq ôqb zou ur :uq zur :L3 [Aqu L3 uq ]zur :L3 [Aqq ôqa ]) 0

(38)

depends on the eigenfunctions, the operators A, which depend explicitly on the seismic parameters (17), and their vertical derivatives. The orthogonality relation between di¡erent modes derived in Appendix B can be expressed in terms of the matrices Larq as i(kqa {kra )Larq ~0

for r=q ,

(39)

with the normalization iLaqq ~kqa =jkq j ,

no summation on q ,

(40)

for the diagonal elements. The full set of coupled equations can then be expressed in matrix form in terms of a vector cª whose entries are composed of modal coe¤cients and phase terms, cª q ~cq eq .

(41)

The phase term eq may be alternatively written as the action of a phase matrix E on a vector of modal coe¤cients so that (cª )q ~Ec ,

where

E~diag{eq } .

(42)

In terms of the vector cª the coupled equations for the evolution of the modal coe¤cients can be written in the compact form La La cª ~Jcª zHcª zM[A]cª , ß 1998 RAS, GJI 133, 159^174

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where we have indicated the explicit dependence of the matrix M on the structural operators A. We will ¢rst establish the equations appropriate to a strati¢ed medium and then use these to simplify the coupled equations in the case of 3-D heterogeneity.

3.2

A strati¢ed medium

Consider now a laterally homogeneous medium for which we will denote the di¡erential operators A0 . For such a strati¢ed medium, the modes must propagate independently. The eigenfunctions do not depend on horizontal position so Jrq ~0, and the interfaces are £at so that the interface term Hrq ~0. The result is signi¢cant simpli¢cation of the equations (43) for this strati¢ed case so that La La cª ~M[A0 ]cª .

(44†

A single mode, u~ueq (x3 )eq (k0q ) ,

(45)

will satisfy the basic equations (44) without any need for coupling, La eq ~ik0qa eq ,

La cq ~0 ,

(46)

thus La (cq eq )~ik0qa (cq eq ) .

(47)

The entire contribution to La cª will come from the phase elements eq . Thus La cª ~ik0a cª , where

k0a

is a diagonal matrix whose entries are the a components of the horizontal wavevectors

ik0a ~diag{ik0qa } .

(48) k0q

for the strati¢ed medium, (49)

For isotropic media or transversely isotropic media with a vertical symmetry axis, the phase will be linear in x\ , so that we may write 0

eq ~ikq :x\ . However, for more general anisotropy there is the possibility of folded phase fronts even in a laterally homogeneous medium (Thomson 1997) and the phase requires a more complex representation. For this horizontally invariant case La ik0a cª ~M[A0 ]cª .

(50)

Since the modal weights are arbitrary, we can identify the contribution from the structural terms M(A0 ) with a weighted composition of the horizontal wavevectors, M[A0 ]~La ik0a .

3.3

(51)

A ¢xed reference structure

The ¢rst class of modal representation makes use of a ¢xed set of modal contributions across the whole region of interest and as a result there are no contributions from derivatives of the modal eigenfunctions, so that the J matrix vanishes and the main modal interaction comes from M. We consider an expansion (23) in terms of the modes of a ¢xed reference structure with propagation properties described by the di¡erential operators A0 . The same set of eigenmodes ue is used throughout, so there will be no contribution from the horizontal derivatives of the modal eigenfunctions (i.e. J = 0). The horizontal evolution equations for the modal coe¤cients are then La La cª ~[M[A(x\ )]zH(x\ )]cª ,

(52)

since we have to allow for the presence of sloping boundaries through the interface term H. The structural component M[A] is linear in the di¡erential operators A, so we can recast eq. (52) in terms of perturbations from the reference structure, through di¡erential operators of the form *Auq ~Auq {A0uq ,

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by using the representation (51) for M[A0 ]. We obtain La La cª ~[La ik0a {M[A0 ]zM[A(x\ )]zH(x\ )]cª ~[La ik0a zM[*A(x\ )]zH(x\ )]cª , where we have exploited the linearity of M with respect to A. The elements of the structural term have the explicit form …? a e e# ab e 2 e# e e# e e# a e Mrq [*A(x\ )]~ dx3 (ôe# ra :*Auu (x\ )L3 uq zôra :*Auq (x\ )ôqb zou ur :uq zur :L3 [*Aqu (x\ )L3 uq ]zur :L3 [*Aqq (x\ )ôqa ]) . 0

(54)

(55)

The derivation above parallels the approach used by Kennett (1984) for 2-D structures but is rendered more complex by the restricted orthogonality properties in the 3-D case. The resemblance of the representation of the coupling matrix Mrq to the 2-D forms is enhanced if we undertake an integration by parts with respect to x3 : …? a e e# ab e 2 e# e e# e e# a e dx3 (ôe# Mrq [*A(x\ )]~ ra :*Auu (x\ )L3 uq zôra :*Auq (x\ )ôqb :zou ur :uq {L3 ur :*Aqu (x\ )L3 uq {L3 ur :*Aqq (x\ )ôqa ) 0

z

X k

e e# a e kz [ue# r :*Aqu L3 uq zur :*Aqq ôqa ]k{ .

(56)

For modes travelling in the same direction, as in the 2-D case, the modal orthogonality property simpli¢es to iLarq ~drq kqa /jkq j . For this case of a ¢xed reference structure, we can demodulate the phase component from (54) to give a somewhat simpler form for the equations for the evolution of the modal coe¤cients. Following (42) we write cª ~E0 c ,

where La E0 ~ik0a E0 :

(57)

Then the horizontal derivative of cª is La cª ~La E0 czE0 La c~ik0a E0 czE0 La c ,

(58)

so La La cª ~La ik0a E0 czLa E0 La c ~[La ik0a zM[*A(x\ )]zH(x\ )]E0 c .

(59)

We recognize the appearance of La ik0a E0 c in each of the forms for La La cª , and we can therefore equate the remaining terms to produce a coupled set of equations for the modal weighting factors c themselves: La E0 La c~[M[*A(x\ )]zH(x\ )]E0 c ,

(60)

where M[*A(x\ )] includes the deviations of the seismic properties of the medium from their reference values and H(x\ ) introduces the e¡ect of any sloping interfaces. These equations (59) now need to be integrated from imposed initial conditions on the modal state of the wave¢eld. The complexity of the form of the equations arises from the nature of the orthogonality condition associated with the matrix L. We recall the indices q and r cover both the mode order J and an angular spectrum of plane waves h for each mode order. If we isolate the mode order, we can regard (59) as a coupled set of 2-D partial di¡erential equations representing the evolution of the angular spectrum for each mode: XX XX LaK#Jh e0Jh La cJh ~ [MK#Jh [*A(x\ )]zHK#Jh (x\ )]e0Jh cJh . (61) J

h

J

h

The original 3-D problem has been reduced to a set of coupled 2-D problems, and, where the portion of the wave¢eld of interest can be represented by a few modes, there is the potential for a signi¢cant reduction in the computational demands required. From the work of Maupin & Kennett (1987) for the 2-D case, we would expect that the use of this ¢xed-mode representation would be particularly useful where velocity perturbations dominate, and shifts in the positions of major interfaces are small. In particular, displacement of interfaces can lead to coupling across a signi¢cant number of mode branches. 3.4

Local modes

In contrast to the use of a ¢xed modal set, the use of a local-mode representation attempts to match the local behaviour as closely as possible and thereby minimize the number of modal branch interactions that need to be considered. ß 1998 RAS, GJI 133, 159^174

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In this case, the modal ¢eld used in the representation (23) is chosen to match the properties of the structure along a vertical pro¢le beneath each surface point. The eigenfunctions ueq (x\ , x3 ) will therefore vary with horizontal position, but their properties will be such as to annihilate any contribution from the integrals over the terms involving the di¡erential operators, so that the term in M vanishes. Because the modal information depends on local structure the derivatives of the modal eigenfunctions are of major importance and the matrix J represents the dominant modal interaction. From (43), using the local-mode eigenfunctions, La (x\ )La cª ~[M[A(x\ )]zJ(x\ )zH(x\ )]cª .

(62)

But, since the modes employed match the vertical structure at each point we have from (51) La (x\ )ika ~M[A(x\ )] .

(63)

This result enables us to reduce (62) to the form La (x\ )La cª ~[La (x\ )ika (x\ )zJ(x\ )zH(x\ )]cª ,

(64)

where the term J(x\ ) depends on the horizontal gradients of the modal eigenfunctions. We can simplify (64) a little by demodulating the phase terms, as in our treatment of the ¢xed reference structure. Introduce the phase matrix E, such that La E(x\ )~ika (x\ ) , then (64) can be recast as an equation for the modal coe¤cients, La (x\ )E(x\ )La c~[J(x\ )zH(x\ )]E(x\ )c .

(65)

The gradients of the modal eigenfunctions appearing in the matrix elements J: …? e e e# Jrq ~{ dx3 (ue# r :La ôqa {La uq :ôra ) , 0

arise from the horizontal variation of the structural parameters o, C. Formally, using the chain rule, La ueq ~La oLo ueq zLa CLC ueq ,

(66)

La ôeq ~La oLo ôeq zLa CLC ôeq ,

depend on the horizontal gradients of the material properties. However, the formal expression (66) does not provide much insight into the character of the coupling term J, since the structural dependences are rather complex. The equations to be satis¢ed by the local modes at a point x\ are e ikeqp ueq ~Apuu L3 ueq zApl uq ôql ,

(67)

ikeqp ôeqp ~{ou2 ueq {L3 [Aqu L3 ueq zApqq ôeqp ] ,

subject to the boundary condition of vanishing traction ôeq3 at the surface and vanishing displacement at in¢nite depth. There are no interface terms because we are dealing with local strati¢cation. The equations (67) represent an eigensystem for the vector wavenumber keq. The dual local modes are described by a comparable set of equations: p pl e# e# {ikerp ue# r ~Auu L3 ur zAuq ôrl ,

(68)

p e# 2 e# e# {ikerp ôe# rp ~{ou ur {L3 [Aqu L3 ur zAqq ôrp ] .

Following the approach used by Maupin (1988) in the 2-D case, we take the horizontal gradient of (67) to yield p pl e e e La (ikeqp )ueq zikeqp La ueq ~La Apuu L3 ueq zLa Apl uq ôql zAuu La3 uq zAuq La ôql ,

La (ikeqp )ôeqp zikeqp La ôeqp ~{La (ou2 )ueq {ou2 La ueq {La3 [Aqu L3 ueq zApqq ôeqp ] : We now introduce …? ap e e e# Jrq ~{ dx3 (ue# r :La ôqp {La uq :ôrp ) , 0

(69)

(70)

as an integral over the gradient elements appearing in (69) and the required elements for the modal evolution equations can be found from aa ap ~Tr{Jrq }: Jrq ~Jrq

(71)

We also recall the de¢nition of Lprq through the orthogonality condition …? e# e i(keqp {kerp )Lprq ~i(keqp {kerp ) dx3 (ueq :ôe# rp {ur :ôqp )~0 . 0

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By combining the gradient of the local-mode equations (69) with the dual local-mode equations (68) and then integrating with ap respect to x3 we can obtain an equation for Jrq in terms of derivatives of the operators A, pa zLa (ikeqp )Lprq i(keqp {kerp )Jrq …? p e e# pl e e# 2 e e# e e# l e ~ dx3 (ôe# rp :La Auu L3 uq zôrp :La Auq ôql zur :La (ou )uq {L3 ur :La Aqu L3 uq {L3 ur :La Aqq ôql ) 0

z

X k

{

…? 0

kz e l e [ue# r :{La Aqu L3 uq zLa Aqq ôql }]k{ z

…? 0

p e e# pl e e# 2 e e# e l e dx3 (ôe# rp :Auu La3 uq zôrp :Auq La ôql zur :ou La uq zur :L3 [Aqu La3 uq zAqq La ôql ])

e pl e# e 2 e# e e# l e# dx3 (La ôeqp :Apuu L3 ue# r zLa ôqp :Auq ôrl zLa uq :ou ur zLa uq :L3 [Aqu L3 ur zAqq ôrl ]) ,

(73)

which can be simpli¢ed somewhat using the symmetry properties of the di¡erential operators (cf. Appendix C) to give …? ap p e e# pl e e# 2 e e# e e# l e i(keqp {kerp )Jrq zLa (ikeqp )Lprq ~ dx3 (ôe# rp :La Auu L3 uq zôrp :La Auq ôql zur :La (ou )uq {L3 ur :La Aqu L3 uq {L3 ur :La Aqq ôql ) 0

z

X k

e e e# kz La hk [ue# r :L3 ôq3 {L3 uq ôr3 ]k{ .

(74)

P e e e# kz In deriving (74) we have transformed the horizontal derivatives in the interface term k [ue# r :La ôq3 {La uq ôr3 ]k{ by introducing the horizontal derivatives of the shapes of the interfaces. ap . As in the 2-D case investigated The expression (74) is su¤cient for us to assess the functional dependence of the elements Jrq by Maupin (1988), the coupling terms depend on the inverse of the di¡erences in horizontal wavenumbers between the modal components. For this 3-D case the in£uence of the gradients of the horizontal wavenumber is more signi¢cant because of the more complex orthogonality condition and does not just a¡ect self-coupling as in the 2-D situation. Unfortunately, eqs (74) are not aa in the general 3-D case. su¤cient to enable the extraction of an analytic form for Jrq ~Jrq 3.5

Local modes for a smoothed reference structure

The direct use of the coupling treatment using a basis of local modes is most e¡ective when the heterogeneity in the medium is smooth and slowly varying so that the horizontal derivatives of the modal eigenfunctions remain small. However, when short-scale structure is present, it is preferable to adopt a representation of the wave¢eld in terms of the local modes of a smoothly varying reference medium which represents the long spatial wavelengths of the heterogeneity. The smaller-scale spatial variations of the seismic parameters can be included by an allowance for the departure from the reference model. This hybrid approach is very £exible and allows the representation of a wide range of structures. We start from the general equations (43) for the evolution of the modal coe¤cients, which we will write as š [A(x\ )]zJš (x\ )zHš (x\ )]cª , Lš a (x\ )La cª ~[M

(75)

where the overbar is to indicate the use of local modes for the reference structure. These modes will satisfy an analogue of (51), š [Ar (x\ )] , Lš a (x\ )ikra (x\ )~M

(76)

where kra is the diagonal matrix whose entries are the local horizontal wavenumbers for the reference medium and Ar are the di¡erential operators corresponding to the properties of the reference structure. Thus we have š [A(x\ )]{M š [Ar (x\ )]]cª Lš a (x\ )La cª ~[Lš a (x\ )ikra (x\ )zJš (x\ )zHš (x\ )zM š [*Ar (x\ )]]cª . ~[Lš a (x\ )ikra (x\ )zJš (x\ )zHš (x\ )zM

(77)

š and have set Here, once again, we have exploited the linearity of the operator M *Ar (x\ )~A(x\ ){Ar (x\ ) ,

(78)

which is the di¡erential operator for the di¡erences in structure between the actual medium and the smoothed reference structure. With the introduction of a reference phase ¢eld Er , such that La Er (x\ )~ikra (x\ ) , we can produce coupled evolution equations for the modal coe¤cients š [*Ar (x\ )]]cª : Lš a (x\ )Er (x\ )La c~[Jš (x\ )zHš (x\ )zM ß 1998 RAS, GJI 133, 159^174

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(79)

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Eq. (79) represents the 3-D generalization of the technique sketched for the 2-D case in the Appendix to Kennett (1984). The structure of Jš in terms of the smooth reference model will parallel the local-mode case (cf. 74), and the signi¢cance of the coupling š arising from the between modes will diminish as the inverse in the di¡erence in the horizontal wavenumbers. The elements M di¡erence between the actual structure and the smoothed reference model will take a comparable form to (56). 4 4.1

A PPROX I M AT E D EV E L O P M E N T S Restricting the number of modes

The coupled mode systems developed in Section 3 link the full system of modes representing the reference structure. Either by extending the structure in depth or by introducing a high-velocity re£ecting surface at depth (`the locked mode approximation', Harvey 1981) it is possible to include the full ¢eld of body waves within a modal sum. However, the number of modes required for such a full representation of the wave¢eld is large and grows rapidly with frequency. Since the system of coupled equations is itself quite complex, we would like to keep the number of mode sheets as small as possible. As illustrated in the schematic in Fig. 1, the modal decomposition procedure has transformed a 3-D problem into a coupled set of 2-D problems in which all modes are linked. This means that for the case of isotropic media both Love and Rayleigh modes must be considered at the same time. The coupling terms between Love and Rayleigh modes vanish if the modes are travelling in the same direction (as a result of the orthogonal polarizations), in agreement with the results of Snieder (1986a) using ¢rst-order scattering theory. However, as soon as the directions deviate somewhat the coupling is no longer zero and, since we need to consider an angular spectrum for each mode sheet, we have to retain Love^Rayleigh coupling. At low frequencies where the surface wave¢eld is dominated by fundamental-mode energy it is generally possible to restrict attention to just a few modes. The fundamental Rayleigh mode has a relatively rapid decay directly from the surface, in contrast to the higher modes, which are closer in character to Love modes but with a vertical polarization. In consequence, the fundamental Rayleigh mode does not have a strong coupling to the higher modes unless the shallower parts of the model show strong heterogeneity (as illustrated in the 2-D results of Kennett & Nolet 1990). The fundamental Love mode more closely resembles its higher modes, so there need to be links to the ¢rst two higher Love modes. Thus a basic system for considering lower-frequency propagation would link six mode sheets, including the fundamental and the ¢rst two higher modes of each wave type: 6 X X J~1

h

LaK#Jh e0Jh La cJh ~

6 X X J~1

h

GK#Jh (x\ )e0Jh cJh .

(80)

We have here used G to represent the various di¡erent forms of the coupling terms: (61) for a ¢xed reference structure, (65) for the use of local modes and (79) when we use the local modes of a smoothly varying reference structure. Similar approximations can be developed using other limited modal sums for a wide range of applications. In each case one has to be aware of the limitations imposed by such a ¢nite mode sum. There will be a limitation on the range of phase velocities which can be represented at each frequency so that as the frequency increases the range of angles from the vertical which are adequately represented by a given modal sum is diminished. Unless the P-wave velocities near the surface are very low, there will normally be little possibility of representing wave-type conversions. A fairly small number of modes can be used e¡ectively in a medium with no sharp horizontal changes in structure if attention is con¢ned to a single waveguide.

Figure 1. Schematic representation of the conversion of wave propagation in a 3-D model into a coupled set of 2-D systems for di¡erent mode sheets with horizontally varying angular spectra of plane waves. ß 1998 RAS, GJI 133, 159^174

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For a heterogeneous medium where di¡erent styles of guided wave propagation are possible, as for example the interchange between Lg waves in the crust and Sn waves in the mantle, the range of modes used needs to be su¤cient to describe the full range of possible wave interactions (cf. Kennett 1989). 4.2

Restricting the range of angular interaction

In the 2-D case, the set of coupled equations for the modal coe¤cients are to be solved subject to two-point boundary conditions at either edge of the region of heterogeneity being considered. Kennett (1984) has shown how the system of coupled linear di¡erential equations can be recast as a set of coupled Ricatti equations for re£ection and transmission matrices which describe the cumulative coupling between modes on passage through the heterogeneity. When the structure is relatively smooth, an approximation can be made to work with just transmission terms. Marquering & Snieder (1995) have further been able to show how the main contribution from the transmission terms can be expressed in terms of an approximation in which multiple mode coupling is represented through ¢rst-order coupling terms with suitable corrections. This `Scalar Exponential Approximation' is able to provide a signi¢cantly superior result compared with the neglect of mode coupling, without the computational burden of the full Ricatti system. In three dimensions the invariant embedding procedures used in the 2-D case can only be applied in highly specialized geometries, see e.g. Bostock (1992). In the formulation presented in this paper we represent the modal ¢elds through the use of an angular superposition of local plane waves. We are therefore able to impose various levels of approximation by suitable restrictions on the angular spectrum. Even when the full angular spectrum is employed we will need to take care with the number of wavenumbers employed in the ¢nite summation to ensure that the behaviour of the wave¢eld is adequately described. If the heterogeneity is rough then a wider angular spectrum is required. The process of refraction of modal contributions due to lateral gradients can be accounted for by progressive transfer between di¡erent angular directions; when structural changes are smooth and not varying too rapidly a fairly ¢ne discretization may be needed. A transmission approximation can be achieved by concentrating attention on a limited band of plane waves covering the expected direction of propagation. However, when there are structural gradients oblique to the direct path between source and receiver we can anticipate the possibility of conversion between modes of di¡erent polarization (Love and Rayleigh in the isotropic case) and the span of angular terms would need to be increased, but need not extend to cover backscattering in a smooth medium. 5

D I SC US S I O N

The methods developed in this paper provide a means for including the e¡ect of 3-D structure on guided wave propagation which can naturally be analysed in terms of surface-wave modes. We have developed a ¢rst-order system of equations for the displacement and the horizontal tractions in which horizontal propagation is emphasized by isolating all horizontal derivative terms on the left-hand sides of the equations. With the aid of this representation, and the introduction of dual modes with a reversed horizontal direction of propagation, we have been able to produce a coupled system of equations for the horizontal evolution of the weighting functions for the various modal contributions representing di¡erent mode branches and directions of propagation. The coupling terms depend on the deviations between the reference model and the actual model and/or the horizontal derivatives of the modal eigenfunctions. The functional forms of the coupling terms are closely related to an orthogonality property between modal eigenfunctions which has been established for an attenuative, anisotropic medium. In the horizontal plane, the di¡erence between the horizontal wavenumbers of two modal terms is orthogonal to a depth integral of displacement^traction products. This restricted orthogonality property makes the coupled equation system signi¢cantly more complex than the corresponding forms for 2-D heterogeneity developed by Kennett (1984) and Maupin (1988). The new formulation goes beyond that proposed by Tromp (1994) because it is not restricted to considering the in£uence of heterogeneity along the path between source and receiver, but can allow for the in£uence of lateral gradients in seismic properties via the transfer of energy within the angular spectrum for each mode. The implementation of the methods described in this paper for a 3-D structure requires a computational e¡ort of a similar order to a direct numerical solution. A signi¢cant advantage of the mode-coupling approach is that it is possible to provide physical insight into the nature of the propagation processes which cannot be readily extracted from the numerical solutions. The coupling terms themselves provide information on the extent to which the approximations conventionally employed in the analysis of surface wave trains can be justi¢ed, as for example the assumption of independent mode propagation along great-circle paths. AC K NOW L E D GM E N T S I would like to acknowledge very useful discussions with Prof. C. Thomson on the properties of surface waves in anisotropic media, and with Dr K. Koketsu on the in£uence of sedimentary structures. This study was completed at the Earthquake Research Institute, University of Tokyo, with support from the cooperative research program (1997-VI-02). ß 1998 RAS, GJI 133, 159^174

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REF E RE NC E S Bostock, M.G., 1992. Re£ection andembedding approach, transmission of surface waves in laterally varying media, Geophys. J. Int., 109, 411^436. Harvey, D.J., 1981. Seismogram synthesis using normal mode superposition: the locked mode approximation, Geophys. J. R. astr. Soc., 66, 37^70. Kennett, B.L.N., 1984. Guided wave propagation in varying mediaöI. Theoretical development, Geophys. J. R. astr. Soc., 79, 235^255. Kennett, B.L.N., 1989. Lg-wave propagation in heterogeneous media, Bull. seism. Soc. Am., 79, 860^872. Kennett, B.L.N. & Nolet, G., 1990. The interaction of the S wave¢eld with upper mantle heterogeneity, Geophys. J. Int., 101, 751^762. Kennett, B.L.N., Koketsu, K. & Haines, A.J., 1990. Propagation invariants, re£ection and transmission in anisotropic laterally heterogeneous media, Geophys. J. Int., 103, 95^101. Koketsu, K., Kennett, B.L.N. & Takenaka, H., 1991. 2-D re£ectivity method and synthetic seismograms for irregularly layered structuresöII. Invariant embedding approach, Geophys. J. Int., 105, 119^130. Marquering, H. & Snieder, R., 1995. Surface-wave mode coupling for e¤cient forward modelling and inversion of body wave phases, Geophys. J. Int., 120, 186^208. Maupin, V., 1988. Surface waves across 2-D structures: a method based on coupled local modes, Geophys. J. Int., 93, 173^185. Maupin, V., 1992. Modelling of laterally trapped surface waves with application to Rayleigh waves in the Hawaiian swell, Geophys. J. Int., 110, 553^570.

A PPE N D I X A :

Maupin, V. & Kennett, B.L.N., 1987. On the use of truncated modal expansions in laterally varying media, Geophys. J. R. astr. Soc., 91, 837^851. Park, J., 1987. Asymptotic coupled-mode expressions for multiplet amplitude anomalies and frequency shifts on an aspherical earth, Geophys. J. R. astr. Soc., 90, 129^169. Romanowicz, R., 1987. Multiplet-multiplet coupling due to lateral heterogeneity: asymptotic e¡ects on the amplitude and frequency of the Earth's normal modes, Geophys. J. R. astr. Soc., 90, 75^100. Snieder, R., 1986a. 3D scattering of surface waves and a formalism for surface wave holography, Geophys. J. R. astr. Soc., 84, 581^605. Snieder, R., 1986b. The in£uence of topography on the propagation and scattering of surface waves, Phys. Earth planet. Inter., 44, 226^241. Snieder, R. & Nolet, G., 1987 Linearised scattering of surface waves on a spherical earth, J. Geophys., 61, 55^63. Tanimoto, T., 1990. Modelling curved surface wave paths: membrane surface wave synthetics, Geophys. J. Int., 102, 89^100. Thomson, C., 1997. Modelling surface waves in anisotropic structures I. Theory, Phys. Earth. planet. Inter., 103, 195^206. Tromp, J., 1994. A coupled local-mode analysis of surface-wave propagation in a laterally heterogeneous waveguide, Geophys. J. Int., 117, 153^161. Tsuboi, S. & Geller, R.J., 1989. Coupling between the multiplets of laterally heterogeneous Earth models, Geophys. J. Int., 96, 371^380. Woodhouse, J.H., 1974. Surface waves in a laterally varying layered structure. Geophys. J. R. astr. Soc., 37, 461^490. Woodhouse, J.H. & Wong, Y.K., 1986. Amplitude, phase and path anomalies of mantle waves, Geophys. J. R. astr. Soc., 87, 753^774. Yu, Y. & Park, J., 1993. Upper mantle anisotropy and coupled-mode long-period surface waves, Geophys. J. Int., 114, 473^489.

A PRO PAGAT I O N I N VA R I A N T FOR S U R FAC E WAV E S

Tromp (1994, Appendix B) has demonstrated an important property of trapped wave¢elds in a 3-D model using a dyadic notation. Tromp's analysis was restricted to perfectly elastic models and was interpreted as a consequence of conservation of energy. We will show that this result can be extended to attenuative anisotropic structures. Consider a half-space x3 §0 with two displacement ¢elds u, v with associated stress tensors qij , pij in a source-free zone. For the ¢rst ¢eld we express the equation of motion in a form which isolates the dependence on horizontal coordinates, La qaj zL3 q3j ~{ou2 uj ,

(A1)

where, as in the body of the text,we have used the summation convention with greek subscripts con¢ned to the values 1, 2. We have an equivalent equation for the second ¢eld5 La paj zL3 p3j ~{ou2 oj :

(A2)

We now contract (A1) with v and (A2) with u and subtract to obtain oj La qaj {uj La paj zoj L3 q3j {uj L3 p3j ~0 :

(A3)

A consequence of the ij