Geophys. J. R. astr. Soc. (1984) 79, 235-255

Guided wave propagation in laterally varying media - I. Theoretical development

B . L. N . Kennett Department of Applied Mathematics and Theoretical Physics, UniversitjJ of Cambridge, Silver Street, Cambridge CLij YEW

Accepted, in revised form, 1984 February 24

Summary. The propagation of surface waves in a laterally varying medium can be described by representing the wavetrain as a superposition of modal contributions for a reference structure. As the guided waves propagate through a heterogeneous zone the modal coefficients needed t o describe the wavetrain vary with position, leading to interconversions between modes and reflection into backward travelling modes. The evolution o f the modal terms may be described by a set of first-order differential equations which allow for coupling to both forward and backward travelling waves; the coefficients in these equations depend on the differences between the actual structure and the reference structure. This system is established using the orthogonality properties of the modal eigenfunctions and is valid for SH-waves, P-SV-waves and full anisotropy.

The reflected and transmitted wavefields for a region o f heterogeneity can be related to the incident wave by introducing reflection and transmission matrices which connect the modal coefficients in these fields t o those in the incident wavetrain. By considering a sequence of models with increasing width of heterogeneity we are able t o derive a set of Ricatti equations for the reflection and transmission matrices which may be solved by initial value techniques. This avoids an awkward two-point boundary value problem for a large number of coupled equations. The method is demonstrated for 1 Hz Lg- and Sn-waves in a multilayered model for which there are 19 coupled modes.

The method is applicable t o three-dimensional heterogeneity, and we are able to show that the interconversion between Love and Rayleigh waves, in the presence of gradients in seismic properties transverse t o the propagation path, leads t o a net rate o f increase of the transverse components of the seismogram at the expense of the other components.

1 Introduction

The outer 1000 km of the Earth is heterogeneous on both small and large scales and it is through this region that seismic surface waves principally propagate. The general behaviour

236 B. L. N. Kennett

o f such surface wavetrains can be reasonably well understood by working with radially stratified earth models i n which the main vertical heterogeneity is adequately represented. F o r such models the surface wavetrain arises as a superposition of modal contributions and with suitable source excitation many features o f observed seismograms can be explained in this way (see, e.g. Kennett 1983, chapter 1 I). In particular Knopoff, Schwab & Kausel (1973) were able to characterize the high-frequency Lg phase with group velocities in the range 3.2-3.7 km s-l as a superposition of surface wave modes (Rayleigh modes for the vertical and radial components. and Love modes for the transverse component). The faster Sn phase with gi-oup velocities around 4.5 km s- ’ can also be represented as superposition o f such higher niodes (Stephens & lsacks 1977).

In many areas interruptions in the propagation of these phases have been used to map the presence o f liorizontal heterogeneity in the crust or lithosphere, even though the mechanism of extinction is somewhat uncertain. Thus, lor example, Ruzakin et al. (1977) have used Lg t o map structural features surrounding the Tibetan plateau.

The extension of theoretical methods to deal with surface wave heterogeneity has been a topic o f interest for many years, and a wide variety of‘methods have been employed. For weak hetei-ogeneity, a number of authors have used considered perturbations from a strati- fied structure, and have used the first-order Born approximation t o calculate the field scattered from the heterogeneity, e.g. Knopoff & Hudson (1964) for stochastic media. Malin ( 1 978) tried to include the effects of large-scale heterogeneity by ‘renormalizing’ the field but his results are only valid for short times.

If the medium is slowly varying in the horimntal direction, then a solution may be sought as a superposition of ‘modes’ which themselves evolve slowly with position (Gjevik 1973; Woodhouse 1974). The local amplitudes are derived by a form of ray tracing and interconversions between modes are taken as negligible.

A number of problems have been tackled by using purely numerical solutions of the elastic wave equations using finite element methods ( e g . Lysmer & Drake 1972) o r finite difference methods (e.g. Boore 1972). Frequently the calculations have been performed for single frequencies. but recently Szelwis (1983) has introduced an approach in which an analytically defined Love wave pulse is propagated through a heterogeneous zone using a finite difference technique and the resultant waveform is then analysed into its constituent modal contributions.

The method presented here is based on the idea of retaining the representation of a surface wave as a multi-mode superposition but we allow the modal coefficients t o vary with position. If the model consists of a region of heterogeneity superimposed on an otherwise stratified medium, we work in terms of the modes of the reference stratification. When, however, the model is such that there is a slowly varying change in the large-scale structure, we work in terms of a superposition of ‘local modes’ (Woodhouse 1974). As the guided wavetrain progresses through the medium the relative proportions of the different modes change and reflected waves may be generated. The effects are described by an evolution equation l o r the modal coefficients.

Such methods were introduced in the study of fibre optics (Marcuse 1974) and have been employed in ocean acoustics (Kohler & Papinicolau 1977), and in these environments attention is concentrated on weak heterogeneity with long propagation paths and SO

reflected waves are neglected. Clarke (1982) was able to extend the appi-oach t o the elastic SH-wave problem, where variations in the shear modulus introduce complications which are not present in the acoustic case; he was also able to sketch the further extension to P-Sv- waves. Similar ideas were suggested by Herrera (1904) and applied t o linearized scattering of surface waves.

Propagation in laterally vacving media - I 237 In view of the observations of very poor transmission of surface waves through some

heterogeneous regions, we need to include the possibility of wave reflection in the modal evolution scheme. Such an extension is presented in this paper. The approach may be used for SH-waves, P-SV-waves or fully anisotropic propagation through 3-D heterogeneity superimposed on a stratified or slowly varying quasi-stratified medium. The heterogeneity is not required t o be small and we can consider waves in both reflection and transmission.

By working with a sequence of models with zones of heterogeneity of increasing width, we are able t o find a set of Ricatti equations for matrices of reflected and transmitted wave coefficients which can be integrated using initial value techniques and so considerably reduce the computational complexity of the problem. By this means interaction between 20 or more modes can readily be handled and so problems associated with the propagation of guided phases such as Lg in heterogeneous media at short periods can now be tackled.

2 Modal evolution equations

The equations of motion and stress-strain relations for an elastic medium can be recast in the form of a set of partial differential equations where one coordinate is singled out. Derivatives of displacement and traction with respect t o this variable are restricted t o the first order only, and can be separated from the remaining terms in which derivatives with respect t o the other spatial coordinates and time appear.

Up t o now, the choice of preferential coordinate has been the depth variable z . The resulting equations have been used for a laterally varying medium by Kennett ( 1 972), but normally the application has been t o a horizontally stratified medium. In this latter case Fourier transformation with respect to the horizontal coordinates and time is normally applied. After transformation we are left with the iamiliar set of coupled ordinary differen- tial equations of first-order, which for isotropic media separate into sets of P-SV- and SH-waves.

In the present work we will concentrate attention on the coordinate along the line joining source and receiver, which we will designate x. This leads to a set of coupled partial differential equations in the components of displacement and horizontal traction. In order to allow for the inclusion of anelasticity we take a Fourier transform in time, and then treat the moduli as complex quantities which depend on frequency.

2.1 C O U P L E D E Q U A T I O N S F O R D I S P L A C E M E N T A N D T R A C T I O N

We work in a Cartesian coordinate system in which z is the depth variable, with displacement w = ( u , u, w ) and stress tensor T ~ , . We introduce the vector t = ( T x x . T , ~ , T ~ = ) whose components represent the traction on a plane x = constant. In terms of the w, t vectors, the propagation equations can be expressed in a form where all derivatives with respect to x appear on the left side of the equations:

where A,,, A,,, A , and A , are partial differential operators which depend on a/ay, a/&, frequency w and the local properties of the medium.

AS an illustration, we consider the generation of such equations for SH-waves in an isotropic medium, in the 2-D case for which the seismic properties do not depend o n t h e y

238 B. L. N. Koznett

coordinate (i.e. we assume no variation normal to the propagation path). The relevant equation o f niotion is

axTxy t = - p ~ 2 v (2 .2)

where, e.g. ax represents a/ax and p is the density. In terms of the shear modulus p ,

( 2 . 3 )

(2.4)

Tx,, = paxu, rYz = p a p ,

a x u = p- ' T x y .

and so we have two coupled partial differential equations

axrxy = - p o 2 u ~ a,(pa,V),

which we can rewrite in the form (2.1 ) as

For the corresponding P-SV case, the coupled equations are explicitly

a ax -

U

W

TXW

7x2

. (2.6)

where the composite modulus v = 4p( X + p ) / ( h + 2p) . In equation ( 2 . 6 ) the unclosed brackets indicate that the operator acts to its right. The corresponding expression for the equations (2.1) in a 3-D anisotropic medium are to be found in Appendix A.

2.2 M O D A L E Q U A T I O N S

The equations (2.1) nierely represent a reorganization of the basic seismic equations and d o not, by themselves, benefit us greatly unless we can make some suitable assumptions about the form of the displacement and traction field. We now concentrate attention on guided waves by using a representation as a superposition of modal contributions. In a horimntally stratified medium we can represent the part of the seismic wavetrain which propagates in the zone close to the free surface as a sum of normal residue terms combined with suitable phase factors (see, e.g. Kennett 1983, chapter 11). For a model in which the stratification is underlain by a uniform half-space with P-wavespeed aB and S-wavcspeed PB. such a representation arises directly from the poles in the response in the frequency- slowness representation for horizontal slownesses p > &'. lf the half-space is chosen to give continuity of properties a t the base. of the region of' interest (zB) the sum over the modal terms includes all S-wave phases with turning points above zB. The poles with largest slow- ness correspond to the main surface wave contribution and the remainder represent seismic phases built up by mutual interference of many waves such as Eg, Sa in different frequency bands. If the structure is extended in depth so that PB increases, more of the seismic wave- field can be represented in terms of modes. As pointed out by Harvey (1 981), almost all the P- and S-wavefields can be handled in this way if an artificial interface is created at a suffi- ciently deep zu below which a B , flu are made very large: so that the slowness 0;' can be brought below typical P-wave slownesses. An equivalent effect may be achieved by taking a

Propagatiorr it1 laterally varying media I 239 fluid layer beneath Z B (G. Nolet, private communication) since then we have poles all the way back to p = 0, and also on the imaginal-y p-axis. A closer analogy to the real spherical Earth, for which the entire wavel'ield is t-epresented by modes, is provided by extending the structure v ia a linear gi-adient in both wavespeeds which once again leads to an entirely modal representat ion.

We adapt the modal sum technique to a laterally varying medium by adding together modal terms with coefficients that have position dependence superimposed on the basic phase behaviour. We introduce the approach by considering a 2-D situation in which the structure is independent of the coordinate j'. 'The basic assumption we make is that to1 each x we can represent the portion of the wavcfield we wish to model as a sum of modes travelling in both the positive and negative x directions. The precise torm 01' the modal representation depends on the character of the lieterogencity i n the medium. If we have a region of' varying properties superimposed on an otherwise stratihed region. we can woi-k in terms of the modal eigenfunctions of the underlying stratification. Within the hetero- geneity the relative proportions of these modes needed to rcpt-esent the displacement and traction field will vary and we will seek to find equations for the modal coefficients. I n this cases the displacement and traction fields are taken as

w ( x , z ) = {c;(x) exp(ik,x) w:(k,, z ) + c;(x) exp(-ik,x) w:(-k,, z ) } . ( 2 . 7 ) r

t(x, z ) = {c,'(x) exp(ik,x) t:(k,. z ) + c,(x) exp( ik,.x) t,"( k,, 2 ) ) .

where the sum is taken over all the relevant modes a t 1.1-equency w . The displaccment eigen- functions w:(k,, z ) corresponding to t h e horizontal wavenumber k,. are determined by the requirement that they satisfy both the condition of vanishing traction a t the free surface and the boundai-y cunditioii imposed at the based of (lie stratit'ication (usually that Iw 1 + O a s z -+ m). The corresponding horizontal traction vector t:(k,. z ) is derived from the displace- ment w:(k,, z ) .

When the large-scale structure in the model is also varying, as for instance for a model of a continental margin, i t is no longer appropriate to work with fixed eigenfunctions w:(k,. z ) . Instead we use a representation in terms 01' 'local modes' and take the eigenfunctions f o r a stratified medium with the same structure as that a t x. This approach is discussed in more detail in Appendix B.

The representation o f the displacement and traction fields as in (1.7) is made possible by the orthogonality of the eigenfunctions at fixed frequency w

r

(1.8)

a result established. with diff'erent notation, by Herrera (1964) and McCarr & Alsop (1907). With the normalir.ation itnplied by (1.8) in a perfectly elastic medium, all eigenfunctions at frequency w carry thc same energy tlux across planes x = constant. Furthet-, we have

expressing orthogonality between modes travelling in opposite directions. For Love waves the eigenfunctions are also orthogonal to the continuous part of the spectrum. However, for P-SV-waves, although Kennett (1981) has proved similar properties at fixed wave- number k , we have as yet to conjecture that orthogonality to the continuous spectrum Persists for fixed W.

240 U. L. N. Kennett The representation (2.7) describes the modes which would exist in the reference stratified

medium, b y for example welding on the reference model t o the actual structure truncated a t the position x. In this way we can describe both the reflected and transmitted waves with- ou t having t o delve into the details of the wavefield in the heterogeneous medium.

We can use the orthogonality relation (2.8) t o derive a set of equations for the modal coefficients c,'(x), c;(x). We insert (2.7) into (2.1) t o obtain

where for brevity we have written

iV: = w:(- k,, z), i: = t;(-k,, z ) , c, = exp(ik,.u), C, = exp(-ik,x).

We take dot products of the displacement and traction equations with 1:. iV: respectively t o give

1 - (c:e,w: . 1; + c;ert$ . i:j = f: . [Aww(c,?e,w;+ C;C,W;) +A,,(cJe,t; + c;err,O)]

and

1 - (c",ert: .w:+c;eri,O . ~ ; ) - ~ : . [ ~ , , ( c , ' e , w : + c ; e , ~ ; ) +A,(cJe,t,O +c;~,i ,O>l.

(2.1 1 )

a r ax (2.10)

a r ax

Now, on subtracting (2.11) from (2.10) and integrating over all z we are just left with the x derivative ofcg'e, on the left side, whilst the right side represents coupling between modes

a , ( c ; e , j = i c d z ~ { ~ ; I O . [ A , , ~ , O + A , , ~ , O I - ~ ; I O . [ A , ~ , O + A , ~ , O I } cfer r

+ i Lrn d z ~ ri: . [A, ,F,D + A ~ , I , O I - W;IO . [Anow: + A ~ ~ , o I ) c;?,. (2.12) r

We see that coupling to both forward and backward travelling waves is needed t o describe the evolution of the modal amplitudes as the position varies.

In the absence of heterogeneity, c; will be constant for the reference stratification and so we apply (2.12) t o the stratified case itself, denoting the corresponding operators by A&,,,, etc. we find

since

e i l axeq = ik, (2.13)

These new relations enable us t o restrict attention t o the effects of the heterogeneity alone, because when we subtract the zero terms in (2.13) from the corresponding elements in (2.12) we are left with differential operators of the type

A A w w = A,w-A,w 0

Propagation in laterally varying media - I 24 1 which just depend on the local departures from the reference stratification. These departures need not necessarily be isotropic and depend on both horizontal and vertical position.

In terms of these heterogeneity operators the set of coupled equations for the evolution of the modal coefficients may be written in the form

where the coupling coefficients K,, to similar forward travelling modes are given b y -00

(2.15)

and for coupling to backgoing modes, the coefficients L,, are

J o For the coefficients of modes travelling towards negative x, the corresponding coupled equations are a ax r r - (F, c; ) = ik 4 4 4 F c- -xiKqr2,c; -xiEqrerc: (2.18)

where Kqr, tqr are derived from K,,, L,, by the interchange of the roles of w i and Wi. The matrices of coefficients K, L describe the rate at which interconversions occur between the modes of the reference stratification as the wavetrain passes through the heterogeneous region. The amplitude of' a mode is modified and interconversions occur between modes so that if just a single mode enters the laterally varying zone, a multimode train will emerge from the region.

We may now extract the phase factors from the left side of (2, 18) to give equations for the modal amplitudes of the q th mode both forwards and backwards

a,c; = C i ~ , , e , e , c ~ +C~L,, e,erc;

a,c; = - C iLyr eq ere; - C iKqreq 2,c; . (2.19)

For a sum over n modes, (2.19) represents a set of 2 n coupled ordinary differential equa- tions which have to be solved subject to constraints on the form of the seismic wavefield at the limits of the heterogeneous region.

The set of coupled equations we have just constructed have the merit that they describe both reflection and transmission problems rather than transmission alone, as in previous treatments of this class of problem (see, e.g. Clarke 1982). The method also has the advant- age that we have been able for the first time to derive equations for arbitrary heterogeneity, which may be used for P-SV-waves or in the fully anisotropic case.

r r

r r

2.3 H E T E R O C EN E ITY C O E Fl- I C I E N T S

For SH-waves in the 2-D case the displacement has only a y component w = (O,y, 0) and the traction t = (0, pa,u, 0). The orthogonality relation for the displacement eigenfunctions has a particularly simple form because up(kp, z ) = up(-kp, z) and so (2.8) reduces to

(2.20)

242 B 1. N Keritiett

The heteiogeneity terms (2.16). (2.1 7) for isotopic heterogeneity now depend o n A(p- ' ) , Ap, Ap and have the explicit form

K , ~ =jam dz{iW2ap + p ; ~ ( ~ - l ) k , k ~ l u ; u: -apa,u~,a,U:)

Lql =

The corresponding terms for backward propagation are given by

dz { [w2Ap -pin( p-')kqklI u i u y - A p a , u ~ a , u ~ } . (2 .21)

Kql = s,/. Lq/ = L q / . (1.21)

For P-SV-waves, the displacement in the 2-D case is w = ( u , 0, w ) and the horiiontal traction is t = (T,,, 0, T . ~ ~ ) , which we will write as t = (I,, 0, I = ) . The sense of tlie horizontal component varies with the direction of travel o f a mode so that there is no simplification of the orthonormality relation (3.8). For an isotropic medium, the coupling terms introduced by the effect of the heterogeneity now depend on both the P and S wavespeeds through the Lame moduli h, p . The forward coupling term

(7 .23)

The last contribution will vanish at infinity, because of the decay of the modes. and the suilace term will vanish when we lake tlie sun1 over all forward travelling modes since

T,, = ua,w + h ( h + 3 p ) p Txz

and t:, will vanish via the traction conditions o n the eigenfunction. The expressions for L,, is obtained from (2.23) with w: replaced by $.

The structure of the P-SV- and SH-wave results are very similar. The correspondence can be made even closer when we are dealing with a wavetrain with slownesses such that P-waves are evanescent at the surf'ace ( p > 016'). In this case as shown by Kennett & Clarke (1983), the higher modes of Rayleigh waves can be represented almost entirely in terms of SV eigenfunctions whose vertical dependences are very close to those for the corresponding Love modes. At moderate frequencies and above, these higher Rayleigh modes are insensitive to the P wavespeed distribution (except just at tlie surface) and so mosr of the coupling between modes will depend on the shear modulus and the density variation. We may there- fore use the simpler SH-wave system as a guide to the behaviour of a general model train entering a heterogeneous zone.

3 Reflection and transmission problems

The first-order differential equations for the modal amplitudes can be written in a more compact form by introducing two new rz-dimensional vectors c+, c+ whose components are the amplitude coefficients c; , c; respectively. I n teiiiis oI'c+, c- we have

(3.1)

Propagation in laterally varying media - I 243

fransrn~ rreu

Figure 1. Configuration of heterogeneous zone and reflected and transmitted waves for an incidcnt wave from the left.

where the elements of the matrices B" etc are constructed from the heterogeneity terms introduced in (2.1 5), (2.18).

(3.2)

We will suppose that the region of departure from stratification is isolated between the positions xL and xR (Fig. 1). If we now consider a surface wave incident from the left, this fixes cc(xL) and we will also have the requirement that for x > xR, c-(x) 0. This imposes a two-point boundary value problem for the 2n coupled equations (3.1), since the infor- mation we seek is the reflected field determined by C-(XL) and the transmitted field deter- mined by c + ( x ~ ) . For a large number of complex equations this poses an awkward numerical problem. One possible approach is to use initial-value methods and shooting techniques, adjusting C-(XL) until the boundary condition c-(xR) = 0 is met. Such a proce- dure is, however, rather cumbersome and computationally expensive if xL and xR are widely separated.

An alternative procedure is to concentrate directly on the reflection and transmission aspects of the problem. For a surface wavetrain incident from the left at xL, we define a transmission matr ixT++(xR, xL) connecting the C + fields at xR and xL so that

c+(xR) = T++(xR, xL)c+(xL). (3 .3)

We also introduce the reflection matrix for the region, R-+(xR,xL), connecting the left and rightward travelling wavefields a t xL

c-(xL) = R-+(XR, XL) C+(XL). (3.4)

Now, by adapting the invariant imbedding technique (Budden 1955) we are able to find a set of non-linear differential equations of Ricatti type for the two a x n matrices R-+, T++ as a function of XL with XR fixed. This corresponds to considering a sequence of problems with the region of heterogeneity progressively extended towards the left away from xR.

We derive the equations for R-', i"++ by differentiating (3.3) and (3.4) with respect to XL

244 B. L. N. Kennett

T h e derivatives of the coefficient vectors can be found from (3.1) in terms of c + ( x L ) alone, by substituting for c - ( x L ) from (3.4),

a - c + ( x L ) = B + + c + ( x L ) t B + - R - + c + ( x ~ ) ax L

a ~ c - ( x L ) = B - + c + ( x ~ ) t B - - R - + c + ( x ~ ) . axL

(3.6)

We back substitute from (3.6) back into (3.5) to give a set of equations in the derivatives of T++ and R-' multiplied by the factor c + ( x L ) . Since the incident wavefield is arbitrary we can extract it from the equations and obtain the coupled Ricatti equations for R-+, T". For the reflection matrix

(3.7)

We integrate these equations starting at X R and move t o the left with the initial conditions tha t , in the absence o f heterogeneity, there is complete transmission and no reflection, i.e.

where I is the n x n identity matrix. This set of two n x n matrix Ricatti equations replaces the original 2n coupled first-order

equations, and transforms a two-point boundary value problem into an initial value problem which is much easier t o solve numerically. An added advantage is that the zone of hetero- geneity can be extended beyond the region of existing calculation by using the current values of t h e reflection and transmission matrices as new initial values. The equation for the transmission matrix (3.8) gives us a means of assessing the circumstances in which we can neglect the reflected waves and concentrate instead on the interconversions between modes in transmission. If we can neglect the term T"'B+-R-' in (3.8) we have a relatively simple first-order equation for the matrix r'

(3.10)

which depends, as we expect, entirely on the coupling coefficients between forward travelling modes. Ki, in (2.15). This approximation will be successful when the cross- coupling matrices B-+, B + - are both small so that there is no feeding of energy into the reflected wave. For such weak heterogeneity (3.10) provides a rapid method of calculating the transmitted field.

From (3.2) we see that the relative sizes of the elements of the diagonal matrices (B-', B + - ) and off-diagonal matrices (B", B - - ) depend on the relation of the heterogeneity coefficients K, L . From the explicit forms in Section 2.3, we see that although K and L differ in the sign of some terms, their functional dependence is similar, and for density variations alone equal in magnitude. In most circumstances then, it would be dangerous to assume that transmission effects are dominant without a prior calculation including reflec- tion. if after a reasonable integration interval through the heterogeneous zone using the full

Propagation in laterally varying media -- I 245 equations (3 .7) , (3.8) we find that the reflected field is sufficiently small, then we can switch to the simpler approximate equation (3.10) for transmission, provided that the level of heterogeneity remains similar throughout.

A comparable set of Ricatti equations to (3 .7) , (3 .8) can be established for waves incident from the right at x R in terms of the reflection matrix R + - ( x R . XL) and the transmission matrix T - - ( x R , xL) but now in terms of differentials with respect t o xR . with xL fixed.

Although the dimensionality is greater, the properties of the reflection and transmission matrices R-+, R + - , "+', T-- are similar to those introduced by the author for coupled P-SV-wave problems in stratified media (Kennett 1974). In particular the symmetries of the original set of differential equations for displacement and horizontal traction (2.1) are such that we are able to extend the symmetry results estabished by Kennett, Kerry & Woodhouse (1978) t o these n x I Z matrices. With o u r choice of'eigenthnction normalization, we have that

where denotes matrix transposition, so that knowledge o f the ttansmission matrix for incident waves o n one side determines the transmission matrix for the other direction of incidence. Further, the reflection matrices are symmetric

which as we shall see provides a useful check on the accuracy of equations using the Kicatti equations.

4 A numerical example

As an illustration o f the method we have just introduced we consider a simple example of SH-wave propagation through a simple heterogeneous model (Fig. 2). We have taken as our reference stratification the crustal model with 2 km of superficial sediments used by Bouchon (1982) in his study of regional seismic phases. We have extended the model in depth to 70 km where we have introduced a jump in shear wavespeed t o 5 km s-', which has the result of allowing all the Lg and Sn behaviour to be modelled by modes. Into this model we have inserted rectangular blocks of uniform material 6 km in vertical extent. The first case (I) we discuss is where the inclusion lies between 2 and 8 km and has the same proper- ties as the sedimentary layer. The second case (11) has an inclusion at the base o f t h e crust

k m l s 0 1 2 3 L -w

case I 10

F X

FlguIe 2. Shear velocity model for the refercnce stratification down to 40 km, there is a jump to 5 kin S - '

70 km depth. Also illustration of the position of thc two cases of hcterogcneity considered in the text.

246 B. I,. N. Kerinett

km (bl

Figure 3. (a ) thc dispersion curvcs for Love modes o n the rcfcrcncc stratification as a function of wave slowness and trequency. (b) Normalized displacement eigenfunctions for the modes a t 1 HZ (marked on the dispersion curves above) plotted for the upper 60 km of the model.

between 24 and 30km with parameters intermediate between crust and mantle (0 = 4.3 km s-', p = 3.2 Mg m-3).

The dispersion curves for the Love modes on the reference structure as a function of frequency and slowness are shown in Fig. 3(a), there is a fairly complicated pattern of mode branches with a transition into sediment controlled modes at large downesses. We have chosen to perform the calculations at 1 Hz, since this gives a fairly large number of modes with slowness greater than 0.2 s km-', but is still computationally tractable. There are 19 modes for this frequency, modes 0-10 describe the Lg-waves trapped in the crust and modes 11-18 represent the Sn behaviour with energy travelling principally in the upper mantle. At 1 Hz the fundamental mode is almost entirely restricted to the sediments and SO

we have worked with the modes 1-18 which sample the crust and mantle. The normalised displacement eigenfunctions for these modes are shown in Fig. 3(b) with the zones of heterogeneity superimposed for reference.

With the Ricatti equation approach we consider a sequence of models with increasing thickness of the heterogeneous zone and so we able to follow the evolution of the reflection and transmission matrices R-' and 7'" as the zone changes size. The interconversions between modes can be followed by studying the form of the matrices R - + , 7'" as a function of position. From (3.12) we expect the reflection matrix R-+ to be symmetric and because

Propagation in laterally varying media - I 247 Table 1. The reflection and transmission matrices at 1 H7. for a width of obstacle of 10 km comparcd for the two different cases of heterogeneity. Reflect>"" M t T 1 X w ~ 1 0 0 case

W e , 2 3 4 5 6 7 8 9 1 0 1 1

1 ? a 3 4 6 9 1 0 1 0 9 2 2 3 o 2 3 9 7 4 1 9 l l 5 I 1 2 4 4 1 3 b 1 9 3 1 2 8 1 9 1 7 2 2 b 8 2 4 9 1 1 28 2 9 22 I h b 2 1 b 2 5 10 5 1 9 2 7 1 9 1 6 8 5 1 2 1 6 10 1 1 2 1 7 1 6 I I 9 5 1 0 1 7 9 1 3 7 8 1 1 9 9 8 5 1 a z 2 2 z 5 9 9 1 0 1 1 8 2 9 2 3 b 3 1 5 8 1 0 1 1 1 7 3

1 0 2 4 8 h 3 1 5 8 1 1 1 2 4 1 1 0 1 2 7 1 0 1 2 3 4 1 17 1 2 4 4 7 1 1 3 5 6 2 , I 1 2 5 4 3 1 1 3 5 7 2 19 1 2 4 9 3 1 1 3 5 6 2 1.. 0 2 3 ? 2 1 1 1 4 5 7

17 0 2 4 4 3 2 1 2 3 5 2 18 0 2 5 5 4 3 7 2 4 6 2

16 0 2 3 3 3 2 1 2 1 5 1

1

12 I , 14 IS 16 I 7 1R

l l l 0 O O O 2 2 2 2 2 2 2 4 5 4 3 3 q 5 4 4 4 3 3 4 5 2 3 3 2 2 3 4 1 1 1 1 1 2 3 1 1 1 1 1 1 1 3 3 3 2 2 2 1 5 6 5 4 2 1 4 6 7 6 5 5 5 b 1 2 2 7 1 2 7 , 4 3 3 3 1 4

4 . + 4 3 3 4 * , 4 4 3 1 3 4

3 3 1 3 2 3 3 ? 3 1 2 2 3 + 3 4 3 3 3 4 4 4 4 4 , 4 4 5

TTanSmlSS,"" Clatr1x w ~ 1 0 0 case I

W e 1 2 2 9 5 6 7 8 9 10 1 1 12 13 1 4 15 16 17 18

1 2 0 3 4 7 7 7 5 3 5 1 8 3 5 6 5 5 5 b 8 2 3 4 7 5 1 9 1 " 5 3 4 4 3 0 0 1 1 1 1 2 2 3 2 7 1 9 6 9 2 7 1 n i i 7 8 7 5 2 3 4 4 9 5 6 a 4 7 1 0 L 7 7 1 2 1 1 8 9 6 3 3 1 1 2 1 1 1 2 2 5 7 5 1 8 Z 3 7 8 2 1 1 4 1 1 7 5 2 3 4 I 3 4 4 b 6 'i 3 17 19 2 1 72 23 21 17 1 3 5 n 10 10 9 9 12 15 7 3 9 7 9 1 4 2 3 78 22 21 17 6 10 1 2 1 1 1 0 1 0 1 3 l b B 5 4 8 6 I 1 21 22 74 2 5 2 7 7 12 1 4 1 3 11 11 1 4 1 7 9 7 3 7 3 7 17 21 2 5 7 2 28 9 1 4 16 14 1 2 1 2 13 15

1 0 8 0 5 3 9 1 3 1 7 22 L8 74 9 1 5 1 7 15 12 1 2 1 3 I 5 11 3 0 2 1 2 5 6 7 4 9 9 b 5 6 5 4 4 5 b 17 5 1 1 2 3 8 1 0 1 2 1 4 1 5 5 9 1 1 0 9 B 8 9 1 1 IJ 6 1 9 1 I 10 1 1 1 4 1 6 17 b 10 88 10 9 9 11 1 3 14 5 1 4 2 .) 1 0 1 1 1 3 1 4 1 5 5 9 1 0 89 8 9 1 0 1 3 15 5 1 4 1 3 9 1 0 1 1 1 2 1 2 + 8 9 8 9 2 8 9 1 2 16 5 7 5 2 4 9 1 0 1 1 1 2 1 2 4 8 9 9 8 9 2 1 0 1 2 17 6 2 6 2 5 1 2 1 3 1 4 1 3 1 3 5 9 11 10 9 1 0 88 16 18 8 3 8 3 b 1 5 16 17 1 5 1 5 6 11 1 3 1 3 12 12 1 6 8 3

1 2 3 1 7 1 6 1 4 1 2 1 2 1 0 1 0 9 9 2 4 4 * 3 7 3 9 2 1 7 1 2 1 2 1 0 8 9 7 7 6 6 2 3 2 1 ? 2 2 1 3 1 6 1 2 1 1 9 8 8 7 7 6 b 1 2 3 2 2 2 2 3 I 1 4 1 0 9 B 1 8 8 b h 6 1 2 3 2 2 2 2 2 5 12 8 8 7 1 0 1 3 1 7 8 7 1 2 0 I 2 2 3 2 4 4 6 12 9 8 8 1 1 1 , 2 5 1 1 b l i 2 4 5 5 9 4 4 1 7 1" 7 7 8 , 7 1 5 4 3 2 4 7 1 " 2 i 4 3 2 2 ? i ~~~

8 10 7 7 6 B l l 2 4 3 3 1 9 1 , 3 5 4 3 3 4 5 4

9 9 b b b 7 b 7 1 4 2 8 2 2 1 0 1 6 R 9 1 0 7

10 9 6 b 6 1 1 l i l O l l 2 2 2 h 0 2 4 b 7 7 b 2

11 2 2 1 1 0 2 2 3 l 0 1 l 1 i 0 0 0 1 12 4 1 2 2 1 1 ? 1 " 2 1 2 1 2 1 0 0 2 13 4 3 3 3 7 4 9 9 1 4 1 3 4 1 2 1 0 : 19 4 3 2 2 2 4 3 7 6 6 3 2 3 , 3 2 1 1

I5 3 7 2 2 3 4 2 3 8 7 0 1 2 3 , 2 2 1 16 3 I 2 2 3 4 1 4 9 7 0 0 1 2 2 3 3 2 17 3 7 ? 2 4 4 3 5 1 0 6 0 0 0 1 2 1 3 3 18 4 3 3 z 4 3 5 4 1 I 1 2 1 1 1 7 3 4

Tran5miss)on M L r l x 1 - 10 0 Case 1 1

w e 1 2 3 4 5 6 7 B 9 10 1 1 17 1 1 1 4 15 I h 17 1 8

1 ~ 1 1 5 1 4 1 7 1 0 1 0 a 8 7 a 2 3 ? 1 2 z 3 3 2 1 5 8 9 1 0 9 7 8 6 6 5 b l 2 1 2 2 2 2 2 3 1 4 1 0 9 0 8 7 7 7 5 5 b 1 i 2 2 1 1 2 2 + 12 9 6 9 1 5 9 3 h 6 1 1 3 3 3 ? , 2 2 5 10 7 7 5 8 8 8 1 6 0 6 1 0 1 1 1 0 0 I 7 4

b 1 0 8 7 9 0 8 3 1 6 1 3 7 b L 4 5 4 4 1 1 1

7 8 6 7 3 1 5 1 6 6 4 2 3 7 1 0 2 4 5 4 2 I 1 3

6 6 b 5 6 O l i 2 3 7 1 1 9 R 5 7 7 3 U + / l o 9 7 5 5 6 6 7 7 1 9 7 7 1 9 3 9 3 9 6 9 1 1 9

1 0 8 6 b 1 1 0 6 1 0 8 1 9 78 3 7 I I 1 3 12 I I a 1 11 2 1 1 1 1 2 Z 5 3 3 9 8 5 5 4 2 U 3 6 12 3 7 2 3 1 4 4 7 4 7 5 9 6 4 7 4 0 9 1 " 11 1 3 2 3 1 5 5 7 3 1 1 5 9 9 5 9 b 2 4 1 0 14 3 2 2 3 0 4 4 3 4 1 3 9 7 9 9 5 6 3 1 7

15 2 7 1 2 0 4 2 0 6 1 2 2 4 6 h 9 b 4 2 2 16 2 2 1 2 1 3 1 4 9 1 1 0 0 2 3 4 9 6 5 2 17 3 2 2 2 2 3 1 7 11 8 3 4 + 1 2 5 96 8

18 3 2 2 2 4 3 3 1 0 9 1 b 1 0 1 0 7 2 2 8 9 5

of the symmetry in the heterogeneous model T'+ will also be symmetric. We integrated the Ricatti equations using a Runge-Kutta scheme but did not build in the matrix symmetries. They were, however, maintained to high accuracy during the calculation.

The behaviour o f the reflection and transmission matrices is very different for the two locations of heterogeneity. In case I with a shallow inhomogeneity, the strongest inter- actions occur for the lower order modes (especially mode I ) which have their largest ampli- tudes in this zone. All these modes d e quite strongly reflected with the backward travelling energy distributed over a number of modes (see Table 1) . For the higher Lg type modes (6-10) the distribution of energy is more uniform throughout the crust and the coupling between modes is weaker. however. in transmission there is significant transfer to the Sn type modes (1 1-18). For case 11, with the inhomogeneity at the base of the crust. most of the coupling occurs between modes 6 and 10, but there is still weak interaction between the lower modes. The contrast between the two cases is illustrated in Figs 4 and 5 and Table 1.

In Fig. 4 we show the behaviour of the modal amplitudes as the width of the inhomo- geneity increases for the first four modes, when the incident wave is purely mode 2. For case I, the initial rate of change of the coefficients is high but becomes very slow for w > 8 km. In both reflection and transmission mode 2 is preferentially converted to mode 1 , and the reflected amplitude is greater than for mode 1 itself. For case 11. on the other hand, the build up of wave conversion is rather slower and is still increasing at \v = 10 km. In reflection the modes 1-4 are almost equally excited.

In Fig. 5 we consider mode 7 incident on the heterogeneous region, and the behaviour is very different. For case 11. with the deeper block, reflection back into mode 7 is most efficient, but there is a spread of encrgy across modes 5-9. In transmission, the relative amplitudes o f the modes change significantly with the width o f the obstacle. For case I ,

248 B. L. N . Keririett ."

Figure 4. The modulus of the modal reflection and transmission coefficients for modes 1 , 2 , 3 , 4 with an incident pure modc 2, as a function of the width of the obstacle for the two cases I . 11. bach curve is marked with thc corresponding mode number.

li Mode

I

t

0 5 w 10

0 0 5 w lo

0. w lo

0 5

10

T

0.5

---- 9

Figure 5. The modulus of the modal reflection and transmission coefficients for modes 5 , 6 , 7, 8 , 9 with an incident pure mode 7, as a function of the width of the obstacle for the two cases I , 11. Each curve is marked with the corresponding mode number.

however. although the behaviour of the reflection and transmission patterns is more complex than before, the relative amplitudes of the modes d o not vary greatly.

In Table 1 we show the full reflection and transmission matrices for both cases 1 and 11 for a width of inhomogeneity of 10 km, the limit of the range shown in Figs 4 and 5 . Since t h e elements of the matrices are complex we show only the absolute value of each element and use a discretized display with unity represented by 100. Each column of a reflection 01

transniission matrix shows the modulus of the modal coefficient corresponding to the inci- dence of a pure mode of that column number. The distribution of amplitudes down the

Propagation in laterally varying media - I 249

columns shows the partitioning of the energy between the different modes, the differences between cases I and I1 show up clearly in the patterns of the matrices.

These calculations show the power of our computational scheme when applied to large deviations from stratification which give rise t o substantial refections (reflection coefficients of more than 0.4 occur in each case). The behaviour we have described holds qualitatively over a range o f frequencies around 1 Hz but is affected by the different numbers ofmodes trapped in the waveguide at different frequencies.

In the companion paper (Kennett & Mykkeltveit 1984) we use this computation scheme to look at the effect of a graben structure with crustal thinning on the propagation of Lg-waves.

5 Three-dimensional variations

The approach we have described above can be extended to the case of 3-D heterogeneity superimposed on a stratified medium with the x-axis oriented along the line joining source and receiver. If there are systematic trends in structure transverse t o the path, then we need to make use of the representation in terms of local modes discussed in Appendix B.

When the heterogeneity is less organized, we can make a simple extension of the treat- ment for the 2-D case. As in (2.1) we can write

where the explicit forms of A,, etc. for an isotropic medium are given in (AS). We work with a modal development of the displacement and traction which allows for modes spread- ing in two horizontal coordinates. Provided that we are not too close t o the source we can work with the asymptotic form

w(x ,y , z )=x- l '* 1 (c:(x)e,w,O(k,,z)O, + C J X ) ~ , W , " ( - ~ , , Z ) ~ , 1, (5.2) r

where

e, = exp [ i (k ,x- n/4)] , 2,. = exp [ - i (k ,x - 77/4)j. (5.3)

and 0,. 8, are azimuthal factors depending on the source of t h e incident waves. The deri- vation o f the evolution equations for the modal coefficients is very close to the treatment in (2.12)-(2.18) but now the unperturbed solution changes slowly with distance. Nevertheless we can still extract the dependence of the modal coefficients c;, and cg on x in the same form as (2.19), thus

a,c; = 1 iK,,e, e,c: + 2 iL,, 2, Ere; r r

a x q c- = - 1 iLqre, ere: - C iKqreq e,c;

with the coupling coefficients K , L given by expressions similar t o (3.16). (2.1 7), e.g.

r r (5.4)

r m

L,, is found by exchanging the roles of w: and iV: and the coupling coefficients fot the backward travelling waves by interchanging w: and W:.

250 B. L. N. i i eme t t

We can think o f the Rayleigh and Love modes on an isotropic structure ns part of a 'super-niode' set with differing particle polarisations, i.e. for Kayleigh modes wHu = (u", 0. w") and for Love modes wLo = (0, u". 0). Such a viewpoint is natural for an anisotropic medium. At fixed slowness, the frequencies of Rayleigh and Love modes interleave (see, e.g. Kennet & Clarke 1983, fig. 4) so that it is possible to use a single index for the modes with even numbers corresponding to Rayleigh modes and odd to Love modes. With an index j corresponding to the mode number we can emphasize the difterence in the modal character by writing

1 (c:'',i"wl""Q:' +(;"c>i"wjL~!- I 1. i

Initially all Kayleigli modes will share the same a;.imuthal dependence. and the same will be true o f the Love modes. The effect of 3-D heterogeneity will be to introduce coupling between the Love and Rayleigli modes, via those terms in AA,, etc. which involve deriva- tives in y, transverse t o the propagation path.

The evolution equation for Rayleigh waves can now be written in the form

a c l i + =Ci,y4R,'< t;K e , K c, K+ + E i K F i?; ek c:' x 4

r 1

(5.6) R R - R - R R - R L - L - R L - + C iL,, eq e, c, CiL,l e4 el cl r 1

where we have used the index Y for coupling to Rayleigh modes and 1 for Love modes. The matrix elements K::, L:: are almost the same as i n the 2-D case (2.23) but there is an additional tei-m

dz iij; w: [O," I- ' ay [Apa,O; I

in KyR," and with w: replaced by $! i t1 1,;:. The major change is in the introduction of the cross-coupling terms K K L , LKL to Love waves arising from transverse gradients, for example

K:: = low dz {--A( A( h + 7 p ) - * J ?:, up 10: I - ' a,@: + W: [O:]-I a, [Ayu:d,Ob I

+ ic; [e," 1-1 a, laPofa, $1 , (5.7)

wliei-e y = 2 p h / ( h t 2p).

modal coefficients can be written as A similar situation prevails for the Love modes, for which the evolution equation for the

+ 2 i L q L : . ~ y L ~ ~ c ~ - + C i I , , L ' , y L ~ ~ c ~ - . (5.8) I r

The Ky"l", L:: cocf'ficien~s ar-e slightly modified from the 2-D case by the addition of an extra temi wiiicli tor K,!$ is

Propagation in laterally varying media - I 25 1 The cross-mode terms K L R , LLR again depend on linking via derivatives transverse to the path, for example

K;,? = Jam dz t ~ ~ , L ; l - ~ a ~ p q x ( x + mli t;repa,w,"g

(5.9)

We have seen, in Section 3, that 'line-of-sight' transmission of a surface wavetrain depends strongly on the diagonal matrix elements in the coupled equations and thus on the K coefficients. We may therefore get a measure of the relative rate of transfer between Love and Rayleigh modes by examining the relative sizes of the matrix elements KLR and K R L .

We assume that we are not close to the nodes of either Rayleigh or Love waves, so that the azimuthal dependence varies slowly and we can neglect t h e y derivatives of the azimuthal functions O R , O L relative to the transverse derivatives of the elastic parameters. In terms of the seismic wave speeds, a. p, the rate of conversion from Love to Rayleigh waves thus depends o n

(5.10)

Similarly the rate of conversion from Rayleigh to Love waves depends on

Although variations in the ratio of of S wavespeed t o P wavespeed (/3/cr) will occur, they are unlikely t o be much smaller than the variations in /3 itself. We can therefore make a further simplification in (5.1 1) so that

(5.12)

From the work o f Kennett & Clarke (1983) we already know that the depth dependence of the vertical displacement (wi ) in the q t h Rayleigh mode is very close to that o f the y component (ui) for the corresponding Love mode. The coefficient KkR will therefore be about twice the size of the element KZL. if the azimuthal terms are comparable.

In the mutual coupling between Rayleigh and Love wavetrains, the rate of transfer from Rayleigh to Love modes will therefore be about twice as large as the rate of conversion from Love t o Rayleigh modes. As propagation though the heterogeneous medium progresses, the presence of parameter gradients transverse to the path will lead t o a substantial net transfer to Love waves. The Love wavetrain will therefore build at the expense of the Rayleigh wave- train. The effect will be particularly marked for explosive sources where the initial field is purely Rayleigh modes and so all modes will have the same type of azimuthal dependence.

This result is in accord with the observations reported by Blandford (1981) where the transverse component of the Lg train at long range (- 900 km) from an explosion was of the m e order as the vertical component. Further examples are shown in the companion paper (Kennett & Mykkeltveit 1984).

This mechanism for Rayleigh to Love conversion involving mode coupling along the Propagation path leads to energy appearing on the transverse component at the same time as o n the vertical and radial components. We have not considered here the effects of surface Wavetrains arriving by alternative paths with distant scattering, although these can also have a significant effect o n the later part of the Lg train (Levshin & Berteussen 1979).

252 B. L. N. Kerirzeti

Acknowledgments This work was supported in part by a Senior Visiting Fellowship from NTNF held at NORSAR. I would like t o thank the staff at Norsar for the use of their facilities in terms of both records and computation. L am grateful to Drs E. S. Husebye, S. Mykkeltveit and D. Doornbos for many useful discussions on the character of Lg and other guided waves.

References Blandtord. I<. . 1981. Seismic discrimination problenis at regional distances. in Idenfifi'cation o.f Seismic

Earthquake or Explosion?, pp. 695 -740, eds Husebye, E, S. & Mykkeltveit, S.. Reidcl.

Boore, D.. 1972. 1,initc dit'fercncc methods for seismic wave propagation in heterogeneous materials.

Bouchon, M.. 1982. ?he coinpletc synthesis of seismic crustal phases a t regional distances, J. geophys.

Uudden, K. G. , 1955. The numerical solution of thc differential equations governing the rcflcction of long

Clarke. T. I . , 1982. Determining Earth structure from normal modes and guided seismic surface waves.

Gjevik, B . , 1973. A variational method for Love waves in non-horizontally layered structures, Bull. seism.

Harvey,, D. J . , 1981. Seismogram synthesis using nornial mode superposition: the locked mode approsi-

Herrera, I.. 1964. Contribution to the linearised theory of surface wave transmission, J. geophys. Res.,

Kennett, B. L. N., 1972. Seismic waves in laterally varying media. Geophys. J. R. astr. Soc., 27, 310-325. Kennett, B. L. N., 1974. Reflections, rays and reverberations,Bull. seism. SOC. Am., 64, 1685-1696. Kennett. H . L. N.. 1981. On coupled seismic waves, Geophys. J. R. astr. Soc., 64, 91--1 14. Kennctt, B. L. N., 1983. Seismic Wave Propagation in Stratified Media, Cambridge University Press. Kennctt, B. L. N. & Clarke, T. J. . 1983. Seisniic waves in a stratificd halfspace 1V: P-SV wave decoupling

and surface wave dispersion, Geophys. J . R. astr. Soc.. 72, 633-646. Kennett, S. L. N., Kerry, N. J . & Woodhousc, J . l I . , 1978. Symmetries in the reflection and transmission

of elastic waves, Geophys. J. R. astr. SOC., 52, 215-229. Kennett, B. L. N. & Mykkeltveit, S., 1984. Guided wave propagation in laterally varying media - 11. Lg-

waves in north-western Europe, Geophys. J. R . astr. Soc., 79, 257 -267. Knopoff, L. & Hudson, J. A.. 1964. Scattering of seismic waves by small inhomogeneities, J. acoust.

Soc. Am., 36, 338-343. Knopoff, L., Schwab, 1,'. & Kauscl, 1:. G., 1973. Interpretation of Lg, Geophys. J. R. astr. Soc., 33,

389-404. Kohler, W. & Papinicolau. G. C., 1977. Wave propagation in a randomly inhomogcneous ocean. in

Wave Propagation and Underwater Acoustics, chapter IV eds Keller, J. B. & Papadakis, J . S. , Springer-Verlag, New York.

Levshin, A. L. & Bertcussen. K. A , , 1979. Anomalous propagation of surface waves in the Barents Sea as inferred froin NOKSAR recordings, Geophys. J. R. astr. Soc., 56, 97

Lysmer, J. & Drake, L. A,, 1972. A finite element method for seismology, Methods in Computations[ Ph,ivics, 11, 181 --216, ed. Bolt , B. A , , Academic Press, New York.

Malin, P. E., 1978. A first order scattering solution for modelling lunar and terrestrial seismic codas, PhD thesis, Princeton University.

McCarr, A. & Alsop, L. E., 1967. Transmission and reflection of Rayleigh waves at vertical boundaries, J. geophys. Res., 72, 2169--2180.

Marcuse. D., 1974. Theory of Optical Dielectric Waveguides, Academic Press, New York. Ruzaikin. A. I . , Nersesov, 1. L., Khalturin, V. 1. & Molnar, P., 1977. Propagation of Lg and lateral

Stephens, C. & Isacks, B. L., 1977. Towards an understanding of Sn: normal modes of Love waves in an

Szclwis. R., 1983. A hybrid approach to mode conversion, Geophys. J. R. astr. Soc., 74, 887-904. Woodhouse, J . H., 1974. Surface waves in a laterally varying layered structure, Geophys. J. H. astr. SOC.,

Sources Dordrecht.

Methods in Coniputarional Physics, 11, 1 - 38, ed. Bolt. B. A , , Academic Press. New York.

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Propagation in laterally varying media - I 253

Appendix A: general heterogeneity

We may establish the differential equations (2.1) for general 3-D heterogeneity by modifying results given by Woodhouse (1974). We define matrices Cij in terms o f the elastic modulus tensor ( c j j ) k l = Cijkl. and introduce the vectors w, t and t,

(W)i = wi, (t) i = T i l , (t ,)i = T,i, (5 = 2, 3.

in terms of the matrices

Q,,=c,,-C,iCi: C I U , ( A l )

the tractions

with summation over v.

derivatives with respect t o x(-xl) appear only on the left side We can write the equations of motion and the stress-strain relations in a form where

We can rewrite this in the form (2.1)

and identify the differential operators as

A,, = -c;; c1, A,, = C;:

For an isotropic material the elastic modulus tensor is

Cijkl = h6ij 6kI + p ( 6 i k 6jl + 6 2 6jk) ,

and the differential operators are then explicitly

-X(X + 2p)-'a,

0

0

( A + 2p)-' 0

254 B. L. N. Kennerl

i 0

A , = 0 - p a z - a, [ pa, -ay [vaY -pa2 - a, - aY [ pay , ra2 O - a y [ pa, a, [ Y a y -3, [ pay - ay [?a,

A, ,= - - a y [ h ( h + 2 p ) - * 0 -a, [ A( h + 2p)y1 o 0

whete v i" = 4p( h + p ) / ( h + 2p) , y = 2ph/( "ii), h + 2p). ('48)

Appendix B: varying background

When there is a large-scale but slow variation in the model with superimposed heterogeneity, we may still use a modal representation but work now in terms of 'local modes'. In the absence o f heterogeneity, wave propagation in the slowly varying structure can be described by

w = 1 c;+ ejwp(x, y , z )

where ej = exp[iOi(x, y , z ) ] . The phase term is such that VOi = ki, where ki is the wave- nunibet- vector for the j t l i mode. $ is the eigenvector for the vertical structure a t x, y with wavenumber I ki I. ( B l ) is just the first term in an asymptotic development (Woodhouse 1974). In this background medium, the modes are assumed to propagate independently with n o interconversions. For each mode the coefficients 3' may be determined by a set of ray- like equations.

With superimposed heterogeneity we will have intermode conversions and reflections superimposed on the basic propagation pattern (Bl) . For the 2-D case we represent the displacement field as

i

w(x, z> = 1 c7+eiwy(x, y , z ) (B2) i

where G: represents a backward travelling mode. As before we seek and equation for the rate of change of the coefficients c;, c; in terms of the departures from the reference struc- ture. We assume that the coefficients c?+ are already determined, the following a similar approach to that above we have

Propagation i i i laterally varyiiig media - I 255 In the reference structure we have n o cl-oss-coupling to other niod2s so the off-diagonal coupling terms vanish for the operators A t , , etc. NOW subtracting the expressions for the reference structure from (B3)

in terms o f the departui-2s from the slow varying I-eference structure. We therefore have a very biniilar situation to that foi- stratification.

In three dimensions we need to set up the modal system for the refer-ence structure i n two horizontal dimensions and then to work with the coupled partial dil'l'etential equations in a,c;, ayci t o find the local niodal field.