Geophys. J. R . astr. Soc. (1981) 64,91-114

On coupled seismic waves

B. L. N. Kenne t t Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW

Received 1980 June 6 ; in original form 1980 March 27

Summary. The response of a stratified elastic medium can be conveniently characterized by the Green’s tensor for the medium. For coupled seismic wave propagation (P-SV or fully anisotropic), the Green’s tensor may be constructed directly from two matrices of linearly independent displacement solutions. Rather simple forms for the Green’s tensor can be found if each displacement matrix satisfies one of the boundary conditions on the seismic field. This approach relates directly to ‘reflection matrix’ representations of the seismic field.

For a stratified elastic half space the Green’s tensor is used to give a spectral representation for coupled seismic waves. By means of a contour integration a general completeness relation is obtained for the ‘body wave’ and ‘surface wave’ parts of the seismic field. This relation is appropriate for SH and P-SV waves in an isotropic medium and also for full anisotropy.

1 Introduction

The coupling of compressional and shear waves in the Earth is particularly important near major structural boundaries, in particular close to the free surface and the core-mantle boundary. In these zones rapid changes in the elastic parameters of the Earth with depth occur and lead to consequent interconversion of compressional and shear wave energy. In those regions where intracrustal discontinuities or the Moho are sharp, significant conversion between P and S waves can occur on reflection (Fuchs 1975). At greater depths the coupling through gradients in elastic parameters is generally modest at short periods (Jeffreys 1957; Richards 1974) but increases in the transition zone associated with the Upper Mantle discontinuities.

Recently most studies have concentrated on those situations where coupling between P and S waves is relatively unimportant as, for example, in the work of Dey-Sarkar & Chapman (1 978) in the construction of theoretical seismograms via modified ray theory.

However, in surface wave phenomena, and for seismic wave propagation to moderate ranges from a source, the coupling of the P and S wavefields cannot be neglected (Kennett & Kerry 1979; Kennett 1980a). The most common techniques which have been used to allow

92 B. L. N. Kennett for the coupling between compressional and shear waves are based on propagator matrix methods (Gilbert & Backus 1966) or the equivalent layer matrices for uniform media (Thornson 1950; Haskell 1953). These methods do not, however, lead to ready physical interpretation of the results.

In this paper we present a systematic approach to the construction of the Green’s tensor for a stratified elastic medium for arbitrary linear boundary conditions on the seismic wave- field at the limits of the stratification. Full account is taken of wave coupling in the stratification and the work generalizes results given by Lapwood & Hudson (1975) for the case of a stratified whole space. The general Green’s tensor is constructed from fundamental displacement matrix solutions of the elastodynamic equations and the role of the Wronskian in scalar Green’s functions is taken by a difference of products of displacement and stress matrices. This quantity is found to have a direct physical interpretation in terms of the propagation characteristics of the stratification.

The coupled wave formulation is not just restricted to the P-SV wave system and may be used for general anisotropic media where we have three coupled waves. For SH waves in an isotropic media for which there is no coupling, Green’s tensor expressions reduce to the usual scalar results.

We will consider a horizontally stratified media but the formal results for the Green’s tensor are directly applicable to spherical stratification. In a stratified elastic half space we may generate useful representations for the seismic radiation generated by a buried general point source in the half space. These forms for the seismic fields have a direct physical interpretation in terms of the reflection properties of the stratification and are shown to be equivalent to forms derived by rather different methods by Kennett & Kerry (1979) and Kennett (1 980b).

For a stratified region overlying a uniform half space we may use the Green’s tensor representation to generate a completeness relation for the vertical wave functions within the half space. This completeness relation includes an integral over a continuous spectrum constituting the ‘body wave’ part of the response for which waves are propagating in the underlying half space, and a discrete sum over ‘surface waves’ for which evanescent behaviour occurs in the underlying half space. This completeness relation extends the work of Herrera (1964) and Alsop (1968) to give a complete treatment of the orthogonality of body and surface waves for the P-SV system. For SH waves the completeness relation generalizes results due t o Kazi (1976) who considered a single layer over a half space.

2 Coupled equations for seismic waves

We consider a horizontally stratified elastic half space with isotropic elastic properties (P wave speed a, S wave speed /3 and density p ) which depend only on the depth coordinate z . Following Woodhouse (1978) and Kennett & Kerry (1979), we adopt a cylindrical set of coordinates (x, 9, z ) and represent the displacement w and traction across a horizontal plane t i n terms of a Fourier Bessel transform

w (x, 9, z , t ) = w,x -t W @ 3 t w z i m

(2.1) = [- d o exp (- iwt) k (URT t V S r -t WTr), 2 r r P r n m = - m

t (x , 9, z, t ) = rXz + rClz 3 t 7, i m

dk k 1 (PRT t SSr + T T r ) , (2 .2) m = - m

On coupled seismic waves 93

where RF, S T , TF are the vector surface harmonics

RF = i Y r (x, @),

with

ST = k-'V1 Y p (x, @), T F =- k-I i x V1 Y T (x, @), (2.3)

Y ~ ( x , $)=J , (kx)exp( im~) , v,=Xa, ++-la, . (2.4)

For a point source on the axis the summation in equations (2.1) and (2.2) is restricted to lml < 2.

The displacement and stress scalars appearing in (2.1) and (2.2) are related by

(2.5)

For this isotropic medium the SH part of the seismic field determined by W, T decouples from the P-SV system. It is customary to express the stress-strain relations and the elasto- dynamic equations of motion in terms of the stress and displacement scalars as sets of coupled first order ordinary differential equations (see, e.g. Kennett & Kerry 1979). However, there is an alternative representation in terms of coupled second order differential equations (Keilis-Borok, Neigauz & Shkadinskaya 1965; Lapwood & Hudson 1975).

For P and SV wave propagation we introduce the displacement and traction vectors

w = [U V ] T ,

t = [P, S I T ,

and then, in the absence of sources, w(z) must satisfy

a, [Aa,w + kBw] - kBTw + (pw21 - k2C)w = 0,

where I is the identity matrix and the 2 x 2 matrices A, B, C are given by

(2.6)

and T denotes a transpose. From equation (2.5) we may recognize the traction contribution to (2.7) as

t = Aa,w + kBw, (2.9)

and thus (2.7) may be written as

a,t + KW = 0, (2.10)

where

K =pw21 - k2C - kBT.

Consider two different displacement fields in the stratified half space, w1 for frequency w I and horizontal wavenumber k , and w2 for frequency w2 and horizontal wavenumber k2, then from (2.7) and (2.10) we may show that

3, [WT(% kl)t2(% k2) - tT(% kl)W2(02> k2)l

=p(w ; - LI:)WTW~ + (k: - k;)wTCw, + ( k , - k2) { ~ , w T B w ~ + W T B ~ ~ , W , ) . (2.11)

94 B. L. N Kennett

The derivative terms on the right hand side of (2.1 1) will only vanish if the displacement fields wl, w2 have a common horizontal wavenumber k. With this restriction (2.1 1) becomes

a2 [WT(Ul, k)t2(U2, k ) - tT(w1, k)wz(w2, k)l = P<w; - wt>wTw2. (2.12)

If we specialize to the case where w1 and w2 are equal we see that, in the absence of sources, for displacement fields wl, w2 satisfying different boundary conditions

azh(wl, w,) = a, [w%J, k ) t 2 ( ~ , k) - tT(a, k)w2(o , k)i = 0, (2.13)

and so h(wl, w,) is independent of depth. The quantity h(wl, w2) plays a role similar to a Wronskian for P-SV wave propagation. The relation (2.13) may alternatively be derived from the properties of the coupled first-order equations (Kennett, Kerry & Woodhouse 1978, equation 23). Explicitly we have

h(w1, w?)= U1P2 + ViS, - Pi U2 - S1 Vz.

For SH waves the stress scalar

(2.14)

T = ppaz w, and the displacement W satisfies the second-order differential equation, in the absence of sources,

a,(ppZazw) + ( p a 2 - pp2k2)w = 0, (2.15)

which is similar in form to (2.10). If once again we consider two different displacement fields

(2.16)

In this case there are no derivative terms on the right hand side of (2.16) and so o and k play a more symmetric role. For a common horizontal wavenumber k we have

az [ W i ( ~ i , k)T2(~2, k ) - T i ( ~ i , k ) W z ( a 2 , k)I = P ( w ~ - ~ : ) w i W 2 , (2.17)

which we note is in an equivalent form to (2.12) for the P-SV wave case. For a common frequency o we have that

h(W,, W,) = w 1 7 2 - T1W2

will be independent of depth.

forms (2.12) and (2.13), with the interpretation, for SH waves

w=W, t = T . (2.19)

(2.1 8)

For wavenumber k we may combine the results for P-SV and SH waves by using the

3 Green’s tensor for coupled waves

A convenient way of representing the excitation of seismic waves by an arbitrary force system is in terms of the Green’s tensor for the coupled wave equations. Consider the inhomogeneous equation for the displacement vector w,

a,t +Kw = - f (3.1)

On coupled seismic waves 95

for some force system f(z), subject t o linear boundary conditions on the seismic field at the limits of the stratification z,, zB say. We look for a solution in the form

(3.2)

and will construct G (z, <) from linearly independent solutions of the homogeneous equation

a,t + KW = 0. (3.3) In the P-SV wave case, as noted by Lapwood & Hudson (1975), this means that we con- struct the 2 x 2 Green’s tensor from four independent displacement solutions. In the SH case the scalar Green’s function may be constructed from two independent solutions.

With respect to the variable z the columns of the Green’s tensor must satisfy the homogeneous equation (3.3), except at z = f , and also satisfy the boundary conditions at zA, zB. At the plane z = c, the Green’s tensor must satisfy continuity of displacement whilst the corresponding tractions have a unit discontinuity. Thus for P-SV waves we require

3.1 C O N S T R U C T I O N O F T H E G R E E N ’ S T E N S O R

For the P-SV wave system it is most convenient to use matrix methods to construct the Green’s tensor. We construct a 2 x 2 matrix W,(z) whose columns are two linearly indepen- dent displacement solutions of (3.3) satisfying the boundary condition at z = z,, and a similar matrix W2(z) whose linearly independent columns satisfy the other boundary condition at z = zB. For the displacement matrices W,, W, we define corresponding traction matrices

T1 = [Aa, + BIW,, T2= [Aa, t BIW,. (3.5)

Since G (z, [) has to satisfy the boundary conditions at zA, Z B with respect to z , we look for a solution in the form

and now the continuity conditions (3.4) across the plane z = [ require

W l ( 0 L - W2(!3M = 0, (3.7a)

Tl(0 L - T,(r)M = I. (3.7b)

Although we could solve (3.7) for L, M in terms of the inverses of W,, W,, etc., it is preferable to use a slightly more indirect approach. We premultiply (3.7a) by TT({) and premultiply (3.7b) by W: (5 )

(3.8a)

(3.8b)

96 B. L. A! Kennett We now define a composition of displacement matrices

W W l > W2) = W T ( W 2 ( 0 - T T ( W , ( t ) , (3.9)

and then the ijth entry of .% is the expression h(wIi, w2j) (2.14) constructed from the ith column of W, and the jth column of W,. From this definition we see that A? will be independent of the level C at which it is evaluated and further that

3 m v 2 , W , ) = - *(Wl> W2)T>

and

q w , , W,) = 0. (3.11)

Then, if we subtract (3.8b) from (3.8a), we find

m w 2 , W 1 ) L = w m (3.12)

and, with a simdar set of premultiplications on (3.7), that

=WWl, W,)M = - w m .

(3.10)

(3.13)

With these forms for L and M we may use (3.6) to write the Green’s tensor for P-SV waves at frequency w and horizontal wavenumber k as

(3.14)

where we have written 27, for .%(Wl, W,) since this is independent of depth. Although we have deduced (3.14) for the 2 x 2 P-SV wave system it is directly applicable

to the SH case if we replace matrices by scalars and adopt the convention (2.19). In a similar way if we consider the transformed equations for waves in a stratified anisotropic medium these may be written as three coupled second-order equations in the form (3.3). We may then form the Green’s tensor as in (3.14) but with now 3 x 3 displacement matrices W,, W,. Such an approach generalizes the Green’s tensor calculation given by Bedding & WiUis (1980) for an anisotropic layer in a half space.

3.2 A N A L T E R N A T I V E F O R M O F T H E G R E E N ’ S T E N S O R

In the construction of the expression (3.14) for G ( z , [) we have made use of displacement matrices W1, W, which satisfy the two sets of boundary conditions at z,, zB.

Consider a further displacement matrix W3, constructed from linearly independent solutions of (3.3) but which does not satisfy the boundary conditions at zA. We may still find the Green’s tensor in terms of W3 and W2 but will now have to look for a solution in the form

(3.15)

The relation between L and hl is determined by the requirement that the Green’s function must satisfy the boundary condition at z = zA. As an example we will suppose that traction must vanish at z = zA so that

T ~ ( Z A ) L + T,(ZA)M = 0. (3.16)

On coupled seismic waves 97 We now use the continuity conditions (3.4) at z = { to relate L and N, and thus we require

T3(0 L - T2(0 (N - M) = I.

X ( W 2 , W3) L = w m , X ( W 3 , W2) (N - M) = - WT(S),

Eliminating N - M and L in turn from (3.1 7) we find

and so combining (3.16) and (3.18)

(3.17)

(3.18)

(3.19)

where X3,, = #'(W3, W2). We now substitute these forms for L, M and N into the representa- tion (3.15) for the Green's tensor to get

z < i - , T -1 T

w2 G(z, <)=- 1w3(z)- W2(Z)T;1(ZA)T3(ZA)} (**32> (3.20)

= - wZ(z) ((%2)-1wT(c) - T;'(zA)T3(ZA)(~)-1W:(5-)}, > r. l h s expression for the Green's tensor is entirely equivalent to our previous form (3.14) and although apparently rather cumbersome will be seen later to have the merit of emphasizing a different aspect of the excitation problem.

3.3 E X C I T A T I O N B Y A G E N E R A L POINT S O U R C E

For a force system f(z) the resulting displacement is given by

w(z) = G(z, C ) f ( D dt. (3.21)

If we adopt a general point source system at the plane z = z s consisting of an arbitrary body force and a moment tensor then f(z) may be represented as the sum of the contributions (Kennett I980b)

f(z) = f,6(z - zs) + f26'(z - z s ) , (3.22)

where f , arises from the body force and the Mxx, Mxy, MYX, My,, moment tensor components and fz from the Mxz, MYz, Mzx, Mzy, M,, moment tensor elements. The displacement arising from the point source will therefore take the form

ws(z) = G (z, zsY1 - dZSG (z, ZSY2. (3.23)

A more convenient expression for the displacement excitation may be obtained if we recognize that the effect of the general point source may be represented in terms of a discontinuity of both displacement and traction across the plane

1,:

(3.24)

98 B. L. N. Kennett

Explicit expressions for these discontinuities in terms of moment tensor components and body forces are given by Kennett & Kerry (1979). In our derivation for the Green’s tensor we have assumed only a traction discontinuity but the treatment is readily extended to include displacement discontinuity effects. The resultant displacement from the general point source is given by

z < zs,

z > zs. (3.25)

ws(z)= - wl@)(*:)-l [w:(zs>sT - m z s > s w 1 3

= - W , ( Z ) ( J y ; J l [WT(ZS)ST - TT(zs)swl,

This expression represents an extremely useful starting point for the discussion of the excitation of seismic waves in general stratification. Particular cases of (3.25) have been previously used, e.g. by Saito (1967), in a discussion of the excitation of surface waves.

An alternative form to (3.25) may be found in terms of the displacement matrices W3(z) and W,(z) by generalizing the algebra leading to (3.20). The resulting expression is most likely to be useful above a source; for example, when traction is to vanish at z = Z,

4 Green’s tensor and reflection matrices

We now introduce two sets of displacement matrices related to the propagation characteristics of the stratification. We use a Cartesian coordinate system in which z increases with depth, and follow the convention of Kennett & Kerry (1979) that downgoing waves travel in the direction of increasing z and upgoing waves in the opposite sense.

At a plane z = Z J within the stratification, consider the displacements which would be produced by unit amplitude upgoing P and S waves in a uniform medium with the same elastic properties as at z = ZJ

wuJ(zJ) = ML, (4.la)

where M & is a displacement submatrix of the wave eigenvector matrix D(zJ) for the uniform medium (Kennett & Kerry 1979, equation 2.17). The corresponding traction matrix is given by

TUJ(ZJ) = 4. (4.lb)

In a similar way we may construct displacement and traction matrices for unit amplitude downgoing waves

The forms of M:, M A and N:, N; are given in Appendix A. In a locally uniform region WUJ(z), W,J(Z) would represent the actual upgoing and downgoing wavefields, including the phase factors for propagation away from the plane z = Z J .

Away from ZJ we may construct the corresponding displacement fields by using the propagator matrix for the stratified medium (Gilbert & Backus 1966) at z = z K

On coupled seismic waves 99

Thus if Z K lies above ZJ we may use the representation of the propagator matrix in terms of the reflection and transmission properties of the region between ZJ and ZK (Kennett 1974) to get the explicit expressions for WUj(zK) , WDJ(ZK),

wUJ(ZK) = META~ - (ME + M E R A ~ ) (T;~)- 'R&~,

wDJ(zK)= (ME + M U R D ) (TD

Here RAK and T6K are the reflection and transmission matrices for downgoing incident waves at Z K . If Z K lies below Z J we use a slightly different representation for the propagator matrix so that,

wUJ(ZK)=(MUK+MDRU ) ( r u ,

wDJ(zK) = MDTD (ME i- MDRU ) ( ~ u R D ,

(4.4) ZK < Z J . K JK JK -1 ,

K JK JK -1

(4.5) ZK > ZJ. K J K - K JK J K -1 JK

The composition X ( W ~ J , WDJ) of the two displacement matrices W u ~ , W D ~ will be independent of depth and so may conveniently be evaluated at the level zJ itself and thus

with a normalization to unit energy flux for propagating waves (Appendix A). These two displacement fields will enable us to generate further displacement matrices

by linear superposition. Thus, for example, if we wish to have a displacement matrix with vanishing traction at z = 0 we may take

w1 (z ) = w U J ( z ) w D J (z ) R 6' > (4.7)

where the free surface reflection matrix REJ is to be chosen such that the surface traction vanishes, i.e.

T U J (0) + T D J (o)RgJ = 0 ,

and thus

4.1 A S T R A T I F I E D R E G I O N B E T W E E N T W O H A L F S P A C E S

As our first example for the Green's tensor we examine the problem studied by Lapwood & Hudson (1975). We consider a stratified region in ZA < z < Z B bordered by uniform elastic half spaces, so that our boundary conditions are that there should be only outgoing waves

A suitable choice for the displacement matrix W, is then the physical wavefield generated in z > z B , z < zA.

by the incidence of a unit amplitude upgoing wave at zB,

wl(z) = w U B ( z ) + WDB(Z)R$B, (4.9)

whch will consist of only upgoing waves in z < z,. Similarly for W2(z) we take the field generated by a unit amplitude incident downgoing wave at z,,

w,(z> = w D A ( z > i- w U A ( Z ) R ~ ~ . (4.10)

100 B. L. N Kennett

If we evaluate W1, W, at z = zA we have the transmitted field

W1(z>= M U T U A AB 7 (4.1 1)

from (4.4), and the incident and reflected displacement

w,(Z)= M e t M C R ~ ~ . (4.12)

The composition of W1 and W, is thus given by

(4.13)

using (4.6) and the symmetry properties of the transmission matrices (Kennett et al. 1978). We may now use the form (3.14) for the Green's tensor and obtain

AB -1 w AB T

AB -1 AB T

G (2, = - i [wUB(z) ' wD€3(z)R6B1 (TU [ D A ( o ' WUA(c)RD 1 9

[wUB(<) ' wDB(i?RU 1 3

< f , (4.14) > !?. = - i [ w D A ( z ) ' WUA(z)R6Bl (TD

This result when rewritten in component form agrees with that presented by Lapwood & Hudson (1975), although they did not recognize the present representation in terms of the reflection and transmission properties of the stratification.

4.2 R E F L E C T I O N M A T R I C E S

We consider now a discontinuity in traction at the level zs and use this as the reference level for constructing displacement matrices which satisfy the boundary conditions at zA and zB. Thus we take

W ( Z > = W u d z ) + Wl>S(Z)Rl, (4.1 5)

where the reflection matrix R 1 is chosen to satisfy the upper boundary condition at zA. If we have an overlying half space we would take the upward reflection matrix

R , = Rhs , (4.16)

or if we have a free surface at zA we would take

R~ = R F ~ , (4.17)

defined as in (4.8) t o give vanishing traction at z = zA. For the other independent displace- ment matrix satisfying the lower boundary condition we take

bv2tz) = w D S ( z ) ' WUS(Z)RZ> (4.18)

for some appropriate reflection matrix R,. The composition of these matrices will be in- dependent of depth and is best evaluated at z = z s to give

(4.19)

On coupled seismic waves 101

However, X ( W D s , W,,), fl(W,,, Wus) vanish identically and from the general symmetries of reflection matrices (Kennett ef al. 1978) RT = R1. Thus using (4.6) we have

(4.20)

The inverse of qz is just the reverberation operator for the entire stratification, and singu- larities of (NlZ)-' will correspond to free modes on the structure (Kennett & Kerry 1979; Kennett 1980b).

In terms of these displacement fields W,, W z we may express the Green's tensor in the form

G(z, Z S ) = - i(WUS(Z) + wl,s(z)Rl)[I- R*Rll-'(WDs(zs) + Wus(zs>RdT3 z < zs, (4.21)

= - ~ ( W D S ( Z ) +WUS(Z)RZ)[I - RiRz]- '(Wus(Zs) + W ~ s ( z s ) R l ) ~ , z > 2s.

If we have a general point source excitation a t the plane we may represent this in terms of displacement and traction discontinuities across z s as in (3.24) with resulting displacement,

Here we have used the expressions (4.1), (4.2) for the displacements W D ~ , W U ~ and tractions TDs, Tus at the source plane z = zs. The two source level contributions in (4.22) may be represented as a linear combination of the two terms

(4.23)

With these source terms the displacement excited by a point source within the stratification may be written as

ws(zR)= [WUS(zR)+WDS(ZR)R1l [ I - R 2 R l I - ' [ Z l ( z S ) t R 2 Z Z ( z S > l , z R < z S ,

zR > z S , (4.24)

= [wDS(zR) -t WUS(ZR)RZI [ I - R 1 R 2 1 - I [ z Z ( z S ) R I Z l ( z S ) l ,

a compact and useful representation for the seismic wavefield.

4.2.1 A stratified half space I

If we specialize to a half space with a traction free boundary z = 0, these displacement expressions may be shown to be identical to those deduced by Kennett & Kerry (1979) by a rather different approach. We now take R 1 2 RgS the reflection matrix from the region above the source (including the free surface) and R z = RgL the reflection matrix for the region beneath the source. We will find the surface displacement at z = 0.

The reflection matrix

R ~ S = R ~ S + T ~ S R [ I - R D 0 s - R ] -1Tos , (4.25)

102 B. L. N Kennett

where 2 is the free surface reflection matrix, and so using (4.4) we find

wUs(zR) + wDS(zZR RE^ = (ME t MOD R)[I - ~ g ~ R 1 - l T,O'. (4.26)

The source terms Zl(zs), Z,(zs) may be equated directly to the discontinuities in wave vector introduced by Kennett & Kerry (1 979) to represent a general source,

(4.27)

since the partitioned inverse of D may be written in terms of its transposed partitions. The elements Z,(zs), Z,(zs) thus represent the upward and downward radiation from the source. From (4.24), (4.26) and (4.27) the surface displacement may be written

ws(0)= (M(: t ~ g k ) [ I - RgsI?]-'T$s[I- RgLRES]-'(RgLXg -CB), (4.28)

as in Kennett & Kerry (1979). The free surface reflections appear here in the reverberation through the entire half space [ I - RgLRES]-l and in the near receiver reverberation above the source level [ I - RgsR"]-'.

4.2.2 A stratified half space N

The choice of displacement matrices employed in deriving the expression (4.24) for the excitation was such that both the upper and lower boundary conditions on the seismic field were satisfied. As we have seen in Section 3.2 we may find alternative forms if we choose a displacement matrix such as

w3(z)= wUS(z) +wDS(z)R8s (4.29)

which does not match the free surface boundary condition on the half space at z = 0. From (3.26) the displacement excited by a general point source may be written as

SL -1 wS(z) = wUS(z) wDS(z)R$s - (wDS(z) WUS(z)RgL) (TDS(0) TUS(O)RD )

X (Tus(0) + TDs(0)R$s)} [ I - R ~ L R ~ s ] - l ( Z l ( ~ s ) + RgLZ2(zs)), (4.30)

in the region above the source z < zs The same source terms appear in (4.30) as (4.24) because we are still satisfying the lower boundary condition through Wz(z).

Consider once again the surface displacement at z = 0, the reflection matrix for the entire half space may be factored as

(4.3 1) 0 s SL -1Tos R g L = R g s t TgSRgLII - R u RD ] D , and so we find from (4.4), (4.31)

(4.32)

On coupled seismic waves 103

(4.33)

an expression derived by a different approach in Kennett (1980b). This form has the merit of separating the coupling between seismic wave types via free surface reflections in the first bracket, from coupling within the half space before waves are reflected from the surface.

5 Completeness relation for a stratified half space

In the previous sections we have shown how we may construct the Green's tensor for a stratified medium in terms of a pair of displacement matrices. We have demonstrated how this approach relates directly to the reflection matrix techniques introduced into seismology by Kennett (1974) and subsequently extended by Kennett & Kerry (1979) and Kennett (1 980b).

In this section we show how contour integration of the Green's tensor for coupled seismic waves leads to a spectral representation of the delta function in terms of a suite of displace- ment matrices. This representation may be viewed as an orthogonality relation between body and surface wavefields.

Such spectral representations enable a general seismic field to be expressed in terms of body wave and surface wave contributions and have been employed for field matching in SH wave problems for discontinuous wave guides (Kazi 1978a, b).

5.1 H A L F SPACE G R E E N ' S T E N S O R

We now restrict attention to a half space consisting of a stratified region in 0 < z < zL overlying a uniform half space with uniform properties ( a ~ , PL, PL). The presence of this uniform region means that the boundary conditioil to be satisfied at z = Z L is a radiation condition. If we have a source within the stratification we require that we have either downgoing propagating waves or decaying evanescent waves in the lower half space. We thus take a displacement matrix W,(z) = WD,(z) which in z > ZL will consist only of outgoing waves. To exclude growing evanescent solutions, we require the vertical wavenumbers in the lower half space

VDL = (w2/@ - k y , V,L = (w2/cYi - k2)"2, (5.1)

Im (VDL, VaL) 0. (5.2)

to satisfy

In order to satisfy the free surface boundary condition at z = 0, we need to take a dis- placement matrix W,(z) for which the corresponding traction matrix vanishes at z = 0. We choose the 'regular' displacement matrix W,(z) such that

Wo(0) = I, T+(O) = 0. (5.3)

The composition of the displacement fields

1 04

where we have used the fact that X i s independent of depth and the properties of W@ (5.3). TDL(O) is the traction matrix at the free surface which is associated with unit ampli- tude outgoing waves in the lower half space.

B. L. N. Kennett

In terms of these displacement fields the half space Green’s tensor may be written as

for sources lying above z = zL. In (5.5) we must recall that G , W$, WDL, TDL are all functions of wavenumber k and frequency w.

5.2 T H E V - P L A N E

One of the commonest representations of the half space response is to consider fixed w and t o allow varying k. As we have seen in Section 2 a simpler structure is obtained in the P-SV wave case if we fix the wavenumber k and vary the frequency w.

We will therefore fii the wavenumber k in our subsequent treatment and note that we may write

where for brevity we have written v = vpL. From (2.1 2 ) and the definition of v = (a2//si - k2)”’, we see that at fixed wavenumber, we may use v as a measure of frequency w for our coupled wave equations. For frequencies w > PLk we have propagating S waves in the lower half space and v is real. For smaller frequencies w < PLk we have evanescent S waves in z > z L and v is positive imaginary from (5.2).

For surface wave modes to exist on the structure we require an evanescent P and S field in the lower half space (z > z,). Thus the surface wave poles in the Green’s tensor (5.5) regarded as a function of v for fixed k will lie on the imaginary axis on the Riemann sheet Im (v, v,,) 2 0 (Fig. 1). On the real v axis we will have branch points associated with the onset of evanescence for P waves at f Q, where

(5.7) Q = - &-2 112 01 L ) .

For Ivl > Q, both P and S waves are propagating in the lower half space but for I v I < Q, we have evanescent P waves. We may-find v,, from v via

v,, = (v2 - Q y ,

,Im v

Figure 1. The u plane showing the contour of integration employed for J(z). Surface wave poles occur along the imaginary u axis on the Riemann sheet Im u, u, 2 0. For real u such that lu I < 7,. P waves are evanescent in the lower half space.

On coupled seismic waves 105

and we choose the sign of the real part of vOL to match that of v. We have therefore in v a very suitable variable for handling the radiation boundary condition into the lower half space.

For a particular real value of v we may construct the ‘regular’ displacement matrix W@(v, z) from W D ~ ( V , z) and W D ~ ( - v , z)

w@(v, z) = w R ( v , z ) w R ( v , o)-’, where for real v, WR(V, z) is a non-singular matrix

W R ( ~ , Z) = WDL(- *, Z ) + W D L ( ~ , z)Ro(v),

with

Ro(v) = - TDZ (v, O)TD~(- v, 0).

(5.9)

This normalization satisfies the conditions (5.3), giving vanishing traction at z = 0. W, is a symmetric function of v

W@@, z) = Wq4- v, z), (5.10)

as may be verified by direct evaluation. Further, since all the elements of the elastodynamic propagator matrices are real W@ gives real displacements throughout the half space.

When both P and S waves are propagating in the lower half space / v J > va, then both v , vaL are real and

%I,- v, z) = WUL(V, z),

so that we have

w R ( v , Z ) = w u L ( v , Z) + wDL(v, z ) ~ E ~ ( v ) ,

where REL is the reflection matrix for the entire stratification above zL, including the free surface.

For general real v but not necessarily propagating P waves, consider the displacement field WR(v, z). If we take the composition of WDL and WR

x [ w D L ( v , z), wR(v, z)I =wTDL(v, z)[TDL(v, zPo + TDL(- v, z)I

(5.1 1)

- T Z L ( ~ , z)[wDL(v, z)~o + wDL(- v, 211, (5.12)

= %[WDL(V, z), WDL(- v, z>l.

Alternatively if we make use of the fact that Z [ W D L , W R ] is independent of depth and evaluate at z = 0

x [ w D L ( v , z), w R ( v , z)I = - TZL(~, 0) [wDL(v, o > ~ o + wDL(- V , 011. (5.13)

With our choice of downgoing wave field WDL(v, z) we may evaluate %[WDL(+v, z ) , WDL(- v, z)] a t z = zL to obtain

~ [ W D L ( ~ , z), WDL(- v, z ) l= - i$(v>,

where the projection operator waves in z > zL (4.13)

g ( v ) = diag (1, 11, Iv I > va, (5.15a)

(5.14)

diag Qa, j p ) is given by (a) for both propagating P and S

106 B. L. N. Kennett and (b) is only S waves propagating in z > zL

g(v) = diag (0, 11, l ~ l < vOr, (5.15b)

The action of J? is to exclude any evanescent waves which could possibly give rise to growing solutions. From the first two expressions (5.12), (5.13) for &[WDL, W,] and (5.14) we find that

w R ( v , 0) = wDL(v, ~ ) ~ o ( v ) -t wDL(- v,o> = i [ T ~ L ( v , o)I-' f < v > , (5.16)

the inverse traction matrix here is just that which appears in the half space Green's tensor (5.5). In a similar way we may show that

wR(- V , 0) = wDL(v, 0) -t wDL(- V, O)R& V ) = - i [T;~(- V , o)]-' $(.I. (5.17)

5.3 C O N T O U R I N T E G R A T I O N O F T H E G R E E N ' S T E N S O R

Guided by the approach used to determine a completeness relation for multi-channel quantum scattering from a central potential (Cox 1966) we consider taking a contour integral over the Green's tensor in the complex v plane. In our half space problem the free surface boundary condition has replaced regularity as r + 0 in the quantum problem and the radiation condition in the lower half space replaces the assumption of only radially outgoing fields as r -+ 00 in the quantum case.

From our half space Green's tensor (5.5) we construct the contour integral

(5.18)

where the closed contour C consists of the real v axis and a large semicircle in the half plane of the v Riemann surface with Im(v, v,L) 2 0 (Fig. 1). H({) is a matrix which is well behaved as { + 00 but otherwise arbitrary. Since the form of G ( v , z, {) differs from z 2 { it is convenient t o split J into the two contributions

(5.19)

5.4 B O D Y W A V E A N D S U R F A C E W A V E C O N T R I B U T I O N S

We will consider the construction of J,(z) in detail. The Green's tensor G(v, z, {) is an analytic function of v on our chosen Riemann sheet except for simple poles on the imaginary v axis corresponding to surface wave modes when det TDL(v, 0) vanishes.

The residue contribution from these poles gives J (2 ) as

(5.20)

where the sum is taken over all the poles on the imaginary v axis at wavenumber k. In a surface wave mode the displacements must satisfy both the condition of vanishing traction

On coupled seismic waves 107 at the surface and the condition of evanescent P and S waves in the lowr half space. This means that one column of W,(vj, z) and WDL(vj, z ) must be simple multiples of the surface wave eigenvector wEj(z) for the jth mode, which satisfies both boundary conditions. The residue contribution to J , may be found in terms of the eigenvectors (Appendix B) and so we find

(5.2 1 a)

in terms of a regular displacement matrix W,(vi, z) at the pole. The mode weighting function

(5.21b)

and the summation is taken over all the N possible modes at fixed wavenumber k.

tion We now evaluate the integral Jl(z) directly. Over the large semicircle r we get a contribu-

When we take account of the asymptotic behaviour of the displacement fields as v + -with Im (v, vaL) > 0 (Appendix A) we find

(5.22)

in terms of the elastic properties of the underlying half space. We note then in (5.22) we have extracted a term H(z) appropriate to our receiver location.

There remains the 'body wave' contribution to the integral which comes from the real v axis portion of the contour

1 - _ - lim { jrdV i(2@,v)-' H (z) , IuI+m

= - %.$i2H(z),

JRI(Z) = - Irn dv 1' dc P ( r ) w D L ( v , Z)[TDL(v, O)l-'w,T(v, <)H(<), 0 - m

= - [ r v d v j 2d { P(o { w D L ( v , Z)[TDL(V, 011-1 - wDL(- v , Z)[TDL(-V, 0)l-l) 0

x W',<% OW), (5.23)

where we have combined the integrals for positive and negative real v and made use of the symmetry of W@ (5.10). To enable us to consider circumstances in which P waves are either propagating or evanescent in the lower half space, whilst S waves are propagating, we adopt the form (5.9) for W@. Now the bracketed term in (5.23) is given by

W D L ( ~ , Z)[TDL(V, 0)l-l -WDL(- v , Z)[TDL(- v, 0)I-l

= - ~ W D L ( V , z)Ro(v) + WDL(- v, 2) ) [TDL(- v, 0)l-l

= - WR(V, z)[TDL(- v, 0)l-l. (5.24)

108 B. L. N. Kennett

From the definition of W, (5.9) and the expression (5.1 6) for WR (v, 0) we find

WR(V,Z)= w@(v, z)wR(v, 0 )

= iW@P, Z)[TTDL(V, O) l - 'g (v) (5.25)

in terms of the projection operator 2. We now substitute for the bracket in (5.23) from (5.24), (5.25) and so obtain the body wave contribution in the form

JRI (z>= i Iomv dv I:d< P(c)w~(v , z)[T6L(v, o)l-'$(v)[T~>L(- v, o)l- 'W~(v, {)H(<).

(5.26)

We now bring together the two contributions (5.22) and (5.26) and equate them to the residue contribution (5.21) to obtain a representation for the matrix H(z)

H(z) = 2 /> ~ ( 0 (71-l JOrn P?v dv W&, z)ET~L(v, o) l - '$(v)[T~~(- v, O)l-'w:(v, 0 (5.27)

N

j = l + 1 K ; l a @ ( V j > z)Wi(vj , 0) ~ ( i - ) ,

where we have made use of the integrability of the Green's tensor terms to change the order of integration.

For the second integral J,(z) the calculation proceeds in a similar way and if H({) is sufficiently well behaved as < + m there will be no problem with the convergence of the integrals. From J,(z) we therefore have a second representation for H(z) in the same form as (5.27) but with the integral from 0 to z replaced by an integral from z to infinity. Adding these two representations for H(z) we obtain a further spectral representation of H(z) which is independent of the position of z:

(5.28)

Thus the expression in brackets in (5.28) plays the role of the delta function 6(z - c ) multiplied by the identity matrix.

We have therefore established our required completeness relation in terms of 'regular' solutions W+(v, z) for the half space

C r n

J O

N

j = 1 + 1 K;'w,(vj, Z ) W J ( V j , { ) p ( { ) = I6 (z ~ {).

(5.29)

In the derivation of (5.29) we have futed the wavenumber k and worked in terms of the vertical wavenumber u rather than frequency. We may conveniently recast this orthogonality relation for the regular solution in terms of a Stieltjes integral over frequency o, at fixed

On coupled seismic waves

N

j = 1 = 6 ( 0 - 0j)K$,

109

(5.30)

and the sum is to be taken over all the surface wave modes for the wavenumber k. Although we have used the P-SY wave system in an isotropic medium as our model in

constructing (5.30) the result may be easily extended to other cases. For SH waves we have only a single wave component and so the radiation condition is rather simpler. In consequence #(v) = 1, for all real v, and with this amendment (5.30) gives the completeness relation for SH body waves and Love waves.

In a fully anisotropic half space we have three coupled wave equations and so the require- ment of no growing evanescent waves in the lower half space must be met by choosing for v the vertical wavenumber of the wave which first becomes propagating as 0 increases for fixed k. The projection operator $(v) is here a 3 x 3 diagonal matrix with non-zero entries for wave types which are propagating in the lower half space. In terms of the 3 x 3 regular displacement matrices W, the completeness relation is again given by (5.30).

5.5 D I S C U S S I O N

We have succeeded in establishing a completeness relation for the seismic wavefield in terms of the regular displacement matrices W,(v, z ) , which have zero surface traction and unit matrix surface displacements. For fixed wavenumber k we have therefore established the orthogonality of the surface wave modes to each other and to the body waves, since from (5.29)

Jnffidt P( .~~ ; (v rn , t)fi,(vn, o 'Krn6mn2

(5.31)

dt P(c)w$(~, ~%p(p , , , , l) = 0, v real. J O

The first of these relations has been given in an equivalent form by Takeuchi & Saito (1972). We also have a mutual orthogonality condition for body wavefields

These results differ from most previous orthogonality relations (Alsop 1968; Herrera 1964) since they are at fixed wavenumber k.

If we examine the form of the original coupled wave equations (2.7) we see that, at fixed k , we have an eigenvalue problem for w2 when we take the boundary conditions into account, with a weighting function p(z). The factor a2 only arises from the original time derivatives in the equations of motion and so this structure holds in all cases. Since we have a

110 B. L. N . Kennett

simple weighting function we were able to consider a weighted contour integral of the Green’s tensor

J(z>= j C V d V l o k P(C)G(V, z , OH(C)

and by equating different expressions for J establish the completeness relation (5.30). When we fix the frequency o, the situation is rather different. In the SH wave case we

still have an eigenvalue problem in kZ with weighting function p ( { ) P 2 ( { ) and this enabled Kazi (1976) to find a completeness relation at fixed frequency for SHwaves in a layer over a half space. For a general stratified half space a generalization of Kazi’s result may be obtained for SH waves by replacing p ( { ) in (5.30) by p(C)/3’({) and modifying the definition of the normalization constant Ki in the same way.

For the P-SV wave system the behaviour is more complex, from (2.7) we see that k appears in the coupled wave equations on its own as well as with the k2 term comparable to the SH wave case, We cannot therefore work with an eigenvalue problem just for kZ. Although we could set up a weighted Green’s tensor integral similar to (5.33) it is not clear which weighting function should be chosen. Residue contributions are readily obtained (Appendix B) and may be simplified using Rayleigh’s principle, Jeffreys (1961). The corresponding normalizations of the modes (B16-B17) (cf. Herrera 1964) do not suggest any direct attack.

References

Alsop, L. E., 1968. An orthonormality relation for elastic body waves, Bull. seism. Soc. Am., 58 , 1949-

Bedding, R. J. & Willis, J. R., 1980. The elastodynamic Green’s tensor for a half-space with an embedded

Cox, J . R., 1966. Many-channel Gelfand-Levitan equations, Ann. Phys., 39, 216-236. Dey-Sarkar, S. K. & Chapman, C. H., 1978. A simple method for the computation of body wave seismo-

Fuchs, K., 1975. Synthetic seismograms of PS-reflections from transition zones computed with the

Gilbert,F. & Backus, G. E., 1966. Propagator matrices in elastic wave and vibration problems, Geophysics,

IlaskeU, N. A., 1953. The dispersion of surface waves on multilayered media, Bull. seism. Soc. Am., 43,

Herrera, I . , 1964. On a method to obtain a Green’s function for a multilayered half space, Bull. seism.

Jeffreys, H . , 1934. The surface waves of earthquakes, Mon, Not. R. astr. Soc., Geophys. Suppl., 3, 253-

Jeffreys, H., 1957. Elastic waves in a continuously stratified medium, Mon. Not. R. astr. Soc., Geophys.

Jeffreys, H., 1961. Small corrections in the theory of surface waves, Geophys. J. R. astr. Soc., 6, 115-

1954.

anisotropic layer, Wave Motion, 2, in press.

grams, Bull. seism. Soc. Am., 68, 1577-1593.

reflectivity method, J. Geophys., 41, 445-462.

31, 326-332.

17-34.

SOC. Am., 54, 1087-1096.

261.

SUPPI., 7, 332-337.

1 1 / .

Kazi, M. H., 1976. Spectral representation of the Love wave operator, Geophys. J. R. astr. Soc., 47,

Kazi, M. H., 1978a. The Love wave scattering matrix for a continental margin (theoretical), Geophys. J.

Kazi, M. H., 1978b. The Love wave scattering matrix for a continental margin (numerical), Geophys. J .

Keilis-Borok, V. I., Neigauz, M. G. & Shkadinskaya, G. V., 1965. Application of the theory of eigen-

Kennett, B. L. N., 1974. Reflections, rays and reverberations, Bull. seism. Soc. Am., 64, 1685-1696.

225-249.

R. astr. Soc., 52,45-64.

R. astr. Soc., 53, 227-244.

functions to the calculation of surface wave velocities, Rev. Geophys., 3, 105-109.

On coupled seismic waves 111 Kennett, B. L. N., 1980a. Seismic waves in a stratified half space - 11. Theoretical seismograms,

Kennett, B. L. N., 1980b. Elastic waves in stratified media, in Advances in Applied Mechanics, vol. 21,

Kennett, B. L. N. & Kerry, N. J., 1979. Seismic waves in a stratified half space, Geophys. J. R. astr. SOC.,

Kennett, B. L. N., Kerry, N. I. & Woodhouse, J . H., 1978. Symmetries in the reflection and transmission

Lapwood, E. R. & Hudson, J . A , , 1975. The passage of elastic waves through an anomalous region - 111.

Richards, P. G., 1974. Weakly coupled potentials for high frequency elastic waves in continuously

Saito, M., 1967. Excitation of free oscillations and surface waves by a point source in a vertically hetero-

Takeuchi, H. & Saito, M . , 1971. Seismic surface waves, in Methods of Computational Physics, vol. 1 1 ,

Thomson, W. T., 1950. Transmission of elastic waves through a stratified solid medium, J. appl. Phys.,

Woodhouse, J . H. , 1978. Asymptotic results for elastodynamic propagator matrices in plane stratified and

Geophys. J. R. astr. SOC., 61, 1-10.

ed. Yih, C.-S., Academic Press, New York.

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of elastic waves, Geophys. J. R. astr. Soc., 52, 215-229.

Transmission of obliquely incident body waves, Geophys. J. R. astr. SOC.. 40, 255-268.

stratified media,Bull. seisrn. Soc. Am., 64, 1575-1588.

geneous Earth, J. geophys. Res., 12, 3689-3699.

ed. Bolt, B. A,, Academic Press, New York.

21,89-93.

spherically stratified earth models, Geophys. J. R. ustr. Soc., 54, 263-280.

Appendix A: uniform media solutions

For an isotropic elastic medium with properties a, p , p at frequency w and horizontal wave- number k , the vertical wavenumbers for P and S waves are

V , = (w’/a’ - k2)”’,

vp = (0’//3’ - k2)1’2,

and we choose Im (v,, up) > 0. The displacement partitions of the eigenvector matrix for the medium are given by

MD, U =

where the normalization is with respect to energy flux for propagating waves (Kennett et al. 1978)

E , = (2/pw3v,)”Z, €0 = (2/pw3vp)’/2. (‘43)

The corresponding traction partitions are

where

r = 2k2 - w2/p2 = k2 - v$ = w2//32 - 2 4 .

Asymptotically as lvpl --z - with k fixed

M D , U - ivpepI,

with

N D , U - - P ~ ~ V $ E ~ I

112 B. L. N. Kennett Consider the entire displacement field in a stratified half space expressed in terms of

v = upL appropriate to a uniform underlying half space. If we construct the integral equations for the Green's tensor G(u, z, {) and a displacement matrix in terms of auxiliary solutions for a uniform half space, as in Lapwood & Hudson (1975), then we may show that as I v I - f m

where GL, W, are the Green's tensor and corresponding displacement matrix for a uniform half space with the properties a,, PL, p L . From (A4) we therefore have the asymptotic relations

- - %i(pP2v)-' {exp [iv(z +{)I + exp [iv(z - {)]}I.

Appendix B: residue contributions for coupled wave problems

Consider the Green's tensor for the P-SV wave case

G ( v , z, t) = W&, Z)[T;fL@, o)I - lwDL(v, {I. (B1)

The displacement matrix WG is constructed from two linearly independent column vectors wl, w2 and WDL from the column vectors w3, w4 and so

1 %-'(u, 0 ) = [T;fL(v, 0)I-l =

where h31= h ( w , wl) is the composition of vectors introduced in (2.13). We are interested in the residue contributions from (BI) associated with simple poles

on the imaginary v axis and therefore wish to construct the matrix a.W/av. We start by constructing an expression for det 3Z Now

a (det W/az = 3(h42 h31- h32 h 4 1 ) / a ~ (B3)

= h42g31 + h31g42 - h32841 - h41g32,

where azh31 =g3]; for fixed wavenumber k , for instance g3, = p i p ( v : - v:)w;w, (2.12), if w3, w1 are evaluated for vertical wavenumbers v3, v 1 respectively. Integrating over the entire interval in z

det x ( 0 ) = /:d{ {h42({)g31({) +h316)g42({) - h32({)g41({) - h41({)g32({)}.

Now at the exact surface wave pole we have a displacement eigenvector WE(Z) which satisfies both sets of boundary conditions, i.e. vanishing tractions at z = 0 and the radiation condition into the lower half space. At the pole we can therefore choose w1 = wE and also w3 = ywE for some constant y, but w2 and w4 will be distinct and will correspond to displacements which satisfy only one of the boundary conditions. Since w1 and w2 satisfy the same free surface boundary conditions hI2 = 0, and also since both w3 and w4 satisfy the radiation

On coupled seismic waves

conditions h34 = 0. Thus at the pole

k32 = yk12 = 0 ,

k41 = y-'h4, = 0,

and since w1 and w3 are multiples

h31 = 0,

113

(B5)

there is therefore only one non-zero entry in .w from k42.

9 t Av and then as Av -+ 0, w3 -+ ywl. We construct

(Av)-' det .%?= (2vj + Av)

and recall that at the pole itself det Z'= 0. Thus in the limit as Av -+ 0

For fixed wavenumber k , we consider wl, w2 evaluated at the pole vj and w3, w4 at

d< (h42g31 +h,ig42 - h32g4i - k4ig32) j0= m a

- av det 21 = 2vjPZ j0 dS ~ ~ ~ ( < ) P ( ~ - ) w T ( < ) w I ( < ) (B7)

but at the pole h42 is independent of depth and w3 = ywl. We will work in terms of the mode eigenvector w E ~ ( z ) and then

-1 - a [ . % ? - l ( ~ , O ) ] / ~ = ( ~ d e t . l k " l ~ ) (:42

a v alJ

= (2vjljp:y)-'

and we have used (B5)-(B7). The full residue contribution from the Green's tensor is obtained when we premultiply (B8) by W,(vj, z) and post multiply by W;,(q, z), the action of the single non-zero element is to extract just the contribution from the column vectors wl, w3. The entire residue contribution for the j th mode may therefore be written in terms of the eigenvectors wEj(z) as

Res {G(vj,z, <)I Ik = ( 2 v j 8 2 l j ) - l ~ ~ j ( z ) ~ ~ j ( < ) .

At the surface wave pole we may form a regular displacement matrix by taking multiples of the eigenvector as columns

m@(vj, Z) = w E ~ ( z ) c ~ (B11)

in terms of a constant vector c. We require W@(v,, 0) = l a n d so c = ~ ~ j ( ~ ) [ ~ ~ j ( ~ ) ~ ~ j ( ~ ) ] ~ l . Thus we find

114 B. L. N Kennett

The Green's tensor residue contribution (B10) may thus be rewritten in terms of the regular matrix Go(?, z )

Res {G(vj, z , C)) lk = (2vjP2Kj)-'W~(vj, z)W;(vj, 0313)

with

Kj = Ij [ w E ~ ( O ) W E ~ ( O ) ] - ~ .

We note that the matrix w,(y, z ) does nor have linearly independent columns. For the fully anisotropic case, where we have to consider three independent solutions for

each boundary condition, the analysis parallels the above and again we recover (B13). In the scalar SH wave case the algebra is slightly simpler since we have to deal with only a single solution for each boundary condition, and with the convention (2.14) we obtain (B9) as before.

If we work at fixed frequency w rather than at fixed wavenumber k, the formal analysis above is still appropriate provided that we use the correct expression for e.g. a,h12. For P-SV waves we see from (2.1 1) that at fixed frequency dzh12 is rather more complex than at fixed wavenumber and cannot be written in terms of k2 alone. If we introduce the quantity

R = 4 p p 2 k ( l - P 2 / a 2 ) V + ( 1 -2p2/a2)P 0314)

a ~ h ~ ~ = ( k ~ - k ~ ) { V ~ R , + U ~ S , + V,R1+U2S1) (B15)

then at frequency w

and the expression in brackets is equivalent to that introduced by Herrera (1964), Alsop (1 968). At the pole in this case

where

Lj =k[' d< [vEj(c)REj(c) uEj(<)SEj(<)]. 10-

Now we may relate Lj to the previous normalization factor by making use of Rayleigh's principle (Jeffreys 1934,1961) and so following Takeuchi & Saito (1972) we find

k1'l /:d{ { V j j R ~ j + U ~ j s E j } = c;Uj d < p ( { ) w E j ~ ~ j 1:- where t+ is the phase velocity and Uj the group velocity for the j th surface wave mode. Thus

and this result is in fact independent of the types of waves being considered, although the details of the calculation differ. At fixed frequency w we have thus a Green's tensor residue contribution at the j th mode

Res { G ( v j , Z, r)} I w = (2vjqUjKj)-'\iii,(vj, z)N$(vj, C)

for SH and P-SV waves in an isotropic medium and in the fully anisotropic case.