Seismic waves in a stratified half space III. Piecewise smooth …rses.anu.edu.au/~brian/PDF-reprints/1981/gjiras-66-633.pdf · 2010-11-18 · Seismic waves in a stratified half space

Geophys. J. R. astr. SOC. (1981) 66,633-675

Seismic waves in a stratified half space - 111. Piecewise smooth models

B. L. N. Kennett and M. R. Illingworth Departmentof Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW

Received 1980 October 29; in original form 1980 July 7

Summary. A recursive procedure is introduced for the calculation of the properties of an earth model consisting of regions with smoothly varying properties separated by discontinuities in seismic parameters or parameter gradients. Within each region the solution is constructed from a Langer uniform approximation in terms of Airy functions supplemented by a series of terms representing multifold wave interaction with the parameter gradients. This procedure allows an efficient treatment of turning point problems and in general it is an adequate approximation to retain only the leading order term in the interaction series. At an interface between varying media the solution may be expressed in terms of generalized interface coefficients or alternatively recast into a form which separates the effects of gradients and the interface itself. In the latter case the plane wave reflection and transmission coefficients are to be used. The resulting calculation scheme for the reflection matrices is an extension of the recursive scheme for uniform layers. The simple phase delay transmission effect of a uniform region is replaced by the reflection and transmission from a gradient zone sandwiched between two uniform media.

This recursive scheme gives good results for crustal and upper mantle models, and only about a dozen subdivisions of the stratification are required down to 950 km. Checks on the accuracy of the computation from unitarity relations between reflection coefficients show that for periods less than 20 s, in the upper mantle, the error associated with neglect of the interaction series is less than 0.1 per cent except for a few cases. When a turning point is just at a structural boundary the error can increase to 2 per cent. For a low-velocity zone where close turning points occur for slownesses within the range of the inversion, rather larger errors may occur but these do not affect waves which turn well above or well below the zone. The leading order approximation allows no conversion of wave types except at a discontinuity in elastic properties and the error introduced by neglecting conversions at steep gradient zones of limited vertical extent (as at upper mantle discontinuities) may reach a few per cent.

634

1 Introduction

In the two previous papers in this series (Paper I - Kennett & Kerry 1979; Paper I1 - Kennett 1980) we have shown how the response of a stratified medium may be constructed from the reflection and transmission properties of portions of stratification. In this paper we turn our attention to the construction of reflection and transmission matrices for a medium consisting of regions with smoothly varying properties separated by discontinuities in elastic parameters or in parameter gradient.

As in Paper I we define these reflection and transmission matrices for a portion of the stratification (zA, zg) , in terms of relations between the upgoing and downgoing parts of the seismic wavefield at the limits of the region Z A - , z g t. The entries of the reflection matrices are the reflection coefficients for incident downward waves at zA, (RbB), and incident upward waves at zB (R6'). For P-SV waves we have a 2 x 2 matrix, e.g.

B. L. N. Kennett and M. R . Illingworth

with a similar form for downward transmission T t B . This matrix notation leads to an efficient scheme for handling the coupling between P- and SV-waves when we consider propagation in more than one region. For SH-waves we can use the same forms but now R fig is just the scalar reflection coefficient.

For models composed of uniform layers we have an exact decomposition into up- and downgoing waves in each uniform region. For general stratification we are able to achieve a unique and unambiguous definition of the reflection and transmission matrices R bB, T fiB, etc., by the following procedure. We 'imbed' the region (zA, z g ) between two uniform half spaces with continuity of properties at the limits of the region zA, zB. In each of these uniform half spaces we have then an exact decomposition of the seismic field into u p and downgoing waves and so we may construct Rbn, Tb", etc. These matrices will depend on the seismic parameter distribution within (zA, z B ) but we do not attempt to split up the field into up- and downgoing parts within an inhomogeneous region. Such an imbedding scheme is commonly used in radiation transport theory (see, e.g. Wing 1962).

For two superposed regions (zA, z B ) and (zg, zc) the overall reflection and transmission matrices can be constructed by using the addition rules for reflection matrices (Paper I - (4.21)), e.g.

A9 BC -1 A9 - R U R D T D

For piecewise smooth velocity profiles we may therefore build up the reflection response if we can determine the reflection properties of two basic elements. We need to find the reflection matrices for a gradient zone lying between two uniform half spaces and also for an interface separating two uniform half spaces. The latter reflection matrix is well known and is just the array of pkne wave reflection coefficients for a slowness p.

For the gradient zone the situation is more complicated since we have the possibility of the existence of turning points which for a given slowness p separate regions in which waves are propagating and evanescent. A downgoing wave will appear to be reflected from the turning point level. Following the general treatment of Wasow (1965), Chapman (1974b) showed how a suitable approximation to the behaviour in the neighbourhood of the turning point could be found in terms of Airy functions. Richards (1976) used this Langer uniform asymptotic representation but neglected any coupling between P- and S-waves. Woodhouse

Seismic waves in a stratified half space - III 635

(1978) has systematized the approximation, including coupling terms, and has shown how to derive the higher order coefficients in an asymptotic expansion in inverse powers of frequency. Cormier (1 980) has used Woodhouse's asymptotic results for the propagator matrix for a smoothly varying region to construct the response of a piecewise continuous model to point source excitation, and then has inverted the integral transforms to generate theoretical seismograms.

An alternative approach is proposed in this paper which makes use of the leading order term in the h g e r approach but in which corrections are determined by a convergent iterative scheme of Picard type. The successive correction terms in this scheme can be identified with higher order wave interactions with the medium. This technique parallels the development given by Chapman (1981) but is used here to find the reflection response of a gradient zone.

The present treatment differs from that adopted by Richards (1976) who chose to consider an interface separating two gradient zones as a single unit. He is therefore led to introduce frequency-dependent 'generalized' interface terms at fixed slowness p which include gradient effects at the interface. This procedure has the disadvantage that a wavefield decomposition is made within inhomogeneous regions.

With the procedure proposed in this paper we are able to achieve a flexible parameterization of the medium and accommodate gradient zones and uniform regions within the stratification. This enables us to construct the reflection response for crustal and upper mantle models with only a limited number of inhomogeneous 'layers'. For example, only 12 layers are needed to represent the T7 upper mantle model proposed by Burdick & Helmberger (1978) down to a depth of 950 km.

2 Stress-dependent vectors in vertically varying media

We consider a horizontally stratified half space with isotropic elastic properties (P-wave speed a, S-wave speed 8, density p ) depending only on the depth coordinate z (measured downwards). As in Papers I and I1 we take a cylindrical coordinate system (x, $, z) with corresponding unit vectors i, 4, i and express the elastic displacement W(x, $, z, t) and traction vector across a horizontal plane T(x, 4, z, t ) in terms of vector surface harmonics:

W(x, $, z , t ) = (2n)-'

T(x, $, z, t ) = (2n)-'

(URT t V S T t WTP),

(PRr t S S r + T T r ) ,

(2.1)

(2.2) m

where

RP = 2 Yr(x , $),

Sr = k-' Vl Yr(x , $),

T r = - i i ~ S r ,

with

Yr(x,$)=J,(kx)exp(im$), vl =ira,+ix-' a,. The summation over angular order m is restricted to I m I < 2 when we consider a point moment tensor source on the z-axis. On using the representation (2.1) and (2.2) in the equation of motion and constitutive relation the quantities U, V, W, P, S and T satisfy a

636 coupled set of first-order differential equations. In an isotropic medium this set decouples into P-SV and SH-wave parts; if we define the stress-displacement vectors

Bp = [U, V, w-'P, w-'SIT, for P-SV-waves, (2.4P)

BH = [ W, w-' 7'1 T, (2.4H)

then these vectors satisfy equations of the form

B. L. N. Kennett andM. R. IIIingworth

for SH- waves,

a,B(z) = wA(z) B(z) (2.5)

in a source-free region, where the forms of the matrix A are in Paper I. The stress-displacement vectors provide a very convenient means of calculating the

seismic displacements; but we are also interested in the reflection and transmission properties of the medium. We would therefore like to construct some framework which provides a closer relation to the propagation processes within the model.

In Paper I we have shown how we may relate the reflection and transmission matrices to the propagation matrix for some portion of the structure (equations 4.8, 4.1 1). Our object here is to find a suitable representation for a fundamental stress-displacement matrix B(z) in a smoothly varying medium from which we may find a propagator from P(z,zo)=

The most important characteristic of waves in a smoothly varying medium is the possibility of the existence of a turning level. For a medium in which the wave speeds increase with depth, above the turning level we can have travelling waves; beneath waves have an evanescent character. The result is that for waves incident on the gradient zone there appears to be reflection from the turning level. Our fundamental matrix forms must therefore be able to give an adequate representation of turning point phenomena.

B (z) B- (z0).

2.1 T H E D E F E C T S O F A N E I G E N V E C T O R DECOMPOSITION

We will consider the eigenvector decomposition used in Paper I for uniform media. We make a local transformation to the wave vector V,

B(z) = D(z)V(z) (2.6)

D-'(z) A(z) D(z) = iA(z). (2.7)

Ap = diag ( -qa, - q p , qa, q p } , for P-SV-waves (2.8)

where D(z) is the eigenvector matrix for A(z) so that

The diagonal matrix iA(z) has entries which are the eigenvalues of A(z)

A H = diag { -qP , q p } ,

with

for SH-waves

Im oq, 1 w4p 0,

in terms of the horizontal slowness p = k /o . In general stratification the wave vector V satisfies the equation

a,V = { iwA - D-'a,DlV. (2.9)


For a uniform medium, the second term on the right side is zero because A and so D are independent of z. In this case the elements of V may be identified with the amplitudes of up- and downgoing plane waves

V=[Yu, w T , (2.10)

while the columns of D may be identified as 'elementary' stress-displacement vectors corresponding to the different wave types bU, bD. For P-SV waves

4= IeabL, ~ p b ? ~ , EabL ~pbsD1, (2.1 1 P )

and for SH-waves

DH = [Epb;, ~ p b g ] . (2.1 1 H )

The full forms of D are given in Paper I and E,, E~ are scaling factors which are chosen to normalize the columns of D with respect to energy flux in the z-direction for propagating waves;

€a = (2pqa)-'/2, E p = (2pqp)-"*.

A fundamental stress-displacement matrix for a uniform medium is thus

B = D exp (iwAz). (2.12)

In a vertically varying medium the term D-' a,D introduces coupling between up- and downgoing waves. For P-SV-waves the coupling matrix has the form (cf. Chapman 1974a)

(2.13)

with (2.14)

638 B. L. N. Kennett and M. R. Illingworth

The zeroth order WKBJ approximation is simply to assume that the coupling matrixd is negligible and to use the matrix

(2.15)

to form an approximate fundamental stress-displacement matrix B = DV. However the coupling matrix A is not always negligible even for small parameter

gradients. A is singular whenever either of the vertical slowness radicals qa or 40 vanishes. This occurs at a turning point of a ray, which is just the region where the WKBJ approximation is known to fail. The singularity arises because the separation into up- and downgoing waves is not valid at a level where the wave is travelling close to horizontally. We therefore need an alternative decomposition which allows a uniform approximation near the turning point based on the Langer approximation (Wasow 1965; Chapman 1974b).

2.2 F U N D A M E N T A L M A T R I X D E V E L O P M E N T

The problems with the coupling matrix d arise from the presence of the radicals qa and q p in the eigenvector matrix D. We now seek a transformation matrix C free of the radicals.

Guided by the work of Chapman (1974b) and Woodhouse (1978) we introduce the decomposition

B = CU, (2.16)

where for P-SV-waves

(2.17P) P 2 P Z P Z . i l.ic P 2 P Z P 2 P ( 2 P 2 P 2 - 1 ) 0

0 P P 0 0 0 P

CP = ( P P Y p ( 2 p z p z -1) 0 0

As required C, does not contain the radicals qa and qp and so we may avoid the problems of the eigenvector decomposition. Cp has been chosen to have a block structure of the form (P IS), which is convenient for subsequent analysis. For SH-waves we take

CH = ( p p ) - q r P P P " The vector U satisfies the differential equation

a,U = {uC-'AC - C-' a,C)U,

where C-'AC is a block diagonal matrix H of the form

where

(2.1 7H)

(2.18)

(2.19)

with a similar definition for Hp. The block matrix form of (2.19) arises because P- and S-waves decouple at high frequencies (Richards 1974).

Seismic waves in a stratified half space - III The matrices of coupling terms I' = - C-I azC are given by

0 0 Yc\

and

639

(2.20P)

(2.20H)

where

YM = 31 az/-l/l,

Yc = 2P2P2 azcl/c(,

YA = - 31 azp/p -k yc,

YB = - azP/P 4- yc.

The coefficients yA, yM represent a rate of change of the P- and S-wave coefficients whilst ye, yc govern the cross-coupling between P- and SV-waves.

Following Woodhouse (1978) we attempt to match the term wC-'AC in (2.18) by a block diagonal matrix E whose entries depend on Airy functions

I - I

EP - = (; _ _ _ -+ ; - - _ _ _ ;j, h = E p , (2.21)

where, e.g.

sLy~1/61az@,l-112p1/2 Bi(-wy3@,) s , ~ ~ / ~ ) a ~ @ ~ l - ~ / ~ p ~ / ~ Ai (-urn@,)

w-l/61a @ 11/2p-1/2~i'(-w~3@,) w-1/61a @ (1/2p-1/2~'(-w2/3@,) 2 , z ,

E, =

with

$& = sgn (43 I % 7, I m,

J Z

We take z, to be some convenient reference level. When a turning point exists, i.e. 4, = 0 at z = Z , then 4, is regular and unique with z , = Z,. In the absence of a turning point in a region any choice of z, may be taken but in this case 4, is clearly non-unique. If, however, a turning point would occur just outside the region it is convenient to extrapolate the parameter distribution so that a turning point depth Z, is created, since this gives a good asymptotic form for E.

With a slight modification of a notation introduced by Kennett & Woodhouse (1978) we write (2.21) in the form

(2.22)

640

Now since the linearly independent Airy functions Ai and Bi in (2.21) are solutions of the equation

d’y/dx’-xy =0,

we find that E, satisfies the equation

B. L. N. Kennett and M. R. Illingworth

azE, ={OH, t az6,&;11 E,,

where a,& d,’ is a diagonal matrix with the form

(2.23)

We have a comparable structure for the S-wave matrix kp. The block diagonal matrix E will thus nearly match the dominant behaviour of (2.18) (at least at high frequencies) with a solution that is uniformly valid through the turning point region.

It is instructive to consider the asymptotic form of the matrix E,. Below a P-wave turning point (4: < 0) the asymptotic forms of the entries of E, are

Aj (wT,) - ‘/2 p”’ I 4, I-’’’ exp (- w I 7, I),

(2.24)

Bk (wT,) - p-”’ I 4 , I ”’ exp (w I T, I).

In this region E, gives a good description of the physically realisable evanescent fields. In the region above the turning point, however, Aj and Bk vary asymptotically as cos (WT, - n/4) and the functions Ak and Bj as sin (07, - n/4). In other words in this region they describe standing waves but we would prefer to consider travelling wave forms.

Since E, satisfies equation (2.23) each of its columns satisfies the same equation and so any linear combination of its columns wdl also be a solution. We choose

exp (- in/4) where now

E =-(

with

Ej = exp (in/4) (Aj - iBj),

Ek = exp (in/4) (Ak - iBk),

Far above a turning point we have the asymptotic forms

Ej (07,) - (4,/~)-”’ exp { i w ~ , L

Ek(w7,)- -i(q,/p)’” exp (iw‘T,),

Fj (w~, ) - (4 , /~ ) - ‘ / ’ exp {- i w ~ , } ,

Fk(w7,)- i(4,/~)~/’ exp { - i o T , ) .

i 1 s, Ej ( ~ 7 , ) s, Fj ( ~ 7 , )

d2 Ek(w7,) Fk 7, 1

Fj = exp (- in/4) (Aj t iBj),

Fk = exp (- Zn/4) (Ak + iBk).

(2.25)

(2.26)

(2.27)

Seismic waves in a stratified half space - 111 641

The z-coordinate increases downwards and so we see from (2.21) that r, above the turning point z = Z , is a decreasing function of z. When we take account of the exponential time factor exp(-iwt) in (2.1), (2.2) we may therefore recognize Ej and Ek as having the character of propagating upgoing waves. In a similar way we find that Fj and Fk have the character of downgoing waves. In passing we should note that the matrix E, is not useful below a turning point, since all the quantities Ej, Fj, etc., are exponentially increasing with increasing depth.

We will use E, below a turning point and E, above a turning point. They were both constructed to take advantage of the uniform approximation afforded by the Airy functions in the neighbourhood of an isolated turning point. When one has two close turning points the use of Airy functions gives non-uniform approximations and it is better to seek uniform approximations based on matrices containing parabolic cylinder functions, as sketched by Woodhouse (1978).

Since the P- and S-wave turning points occur at rather different levels in the medium it is cynvenient to introduce a third matrix for the P-SV case in addition to E and E. In the region below the P- wave turning point but above the S-wave turning point we use

(2.28)

since in this region the P- waves are evanescent although the S-waves are propagating. We require any fundamental matrix U for this present decomposition of the stress-

displacement field to satisfy equation (2.18). Now we have already shown that we can nearly match the dominant part of U at high frequencies with a block diagonal matrix (E, E or B) which is a solution of equation (2.23). We need therefore some means of constructing corrections to the block diagonal 'phase' matrix. In the following we shall consider the matrix E but the results will have the same formal structure in terms of E or E.

2.3 A N A S Y M P T O T I C E X P A N S I O N

One way of constructing a fundamental matrix U is to look for a matrix K which premulti- plies E, i.e. to take

U = KE. (2.29) Then K satisfies the equation

a,K = o [ H K - KH] - C1 a,CK - Ka,*Q-' (2.30)

so that K is independent of the choice of block diagonal matrix E and frequency dependence enters only through the commutator term. This approach is well-suited to an asymptotic expansion of K in inverse powers of frequency and the coefficients in the expansion are coupled by the commutator term:

K(z) = w-"k,(z), ko = I,

[Hk, +1- k, + I HI = - azkn + C-' a,Ck, + k, a,@*-'. (2.3 1)

These equations have been solved by Woodhouse (1978) who has presented detailed results for the coefficient k,(z). To this order an asymptotic fundamental stress-displacement matrix is:

0

n = O

(2.32)

642

where we have shown the explicit dependence of the various terms on frequency, slowness and depth. From a fundamental matrix we can easily construct the propagator matrix for a region (zA, zB)


p(zA, zB) = Bn(ZA) B,'(zB)

- C(ZA) [I kl (ZA)] E(ZA)E-' (ZB) [I - kl (ZB)] ~ - ' ( Z B ) . (2.33)

This asymptotic form is perfectly acceptable mathematically, but unfortunately the representation (2.33) has no clear representation in terms of internal reflections within the region (zA, zB) and so gives no insight into the actual interactions taking place between the wavefield and the parameter gradients as a body wave propagates. On the other hand the representation (2.33) has been used by Kennett & Woodhouse (1978) and Kennett & Nolet (1979) in shdies of the asymptotic behaviour of the normal mode frequencies of the Earth and it would appear that the asymptotic series approach is well-suited to such normal mode studies.

2.4 I N T E R A C T I O N S E R I E S

An alternative approach to constructing a fundamental matrix is to look for a matrix L which post-multiplies E, i.e. to take (cf. Chapman 1981)

U = E L . Then L satisfies the equation

(2.34)

a,L = E-'jEL (2.35)

and

j = - (c-l a,c t a,aa-l). Clearly here L is entirely controlled by the choice of E and depends on frequency through the phase matrices E-' and E. The equation (2.35) is best suited to an iterative solution in terms of an 'interaction series' in terms of the parameter gradients present in j through C-' a,C and a,+*-'. If these gradients are not too large we may hope to get a good approximation to L by the first few terms of an expansion

(2.36)

where E is a measure of the parameter gradients. Since all the elements of EJ = E-'jE are bounded (even at turning points) the series (2.36) will be convergent. The successive terms in the series are determined by the recursive relation

(2.37)

with lo = I . At each level in the recursion we introduce an additional J so that the coefficients in the series may be identified with successive interactions of the elastic waves with the parameter gradients. The lower limit in the integration g is chosen according to the situation. If we consider a region (zA, zB) in which there is a turning point then we would take q to lie at the turning point since we would expect to use different forms of phase matrix above and below the turning point. Otherwise, q can be chosen at any convenient reference level.


In the P-SV-wave case we may partition j to separate the coupling within and between wave types

(2.38)

where

(2.39)

and jp has a similar definition to j,.

scheme takes the form The phase matrix has the block diagonal form (2.21) and so the kernel in the recursion

(2.40)

J is critically controlled by the choice of phase matrix. If we are in the purely propagating regime the phase matrix is indeed E and then, for instance, the Fwave part of the kernel is

9 (2.41) i Ej,Fk, t Ek, Fj, 2 Fj, Fk,

- 2 Ej, Ek, -(Ej,Fk, t Ek,Fj,)

i

2 (YA 'A azz@a/az@a) E-1' E = - - a Ja a

where, e.g. Ej, = Ej(o7,). In the asymptotic limit far above the turning point this reduces to

and

"/A + 112 azz~,/az@, - YP, where yp was defined in (2.13). The phase terms here are just those to be expected for P- waves reflected at the level 5. The matrix partition E j ' j p E p will similarly correspond to S- wave reflection at the level {. The cross-coupling terms are of the form

, (2.43) - {S,YB Ej, Fjp + S ~ Y C Ek, Fkpl

S,YB Ej,Ejp + S ~ Y C Ek, Ekp

- P,YB Fj, Fjp + s p ~ c Fk, FkpJ

S,YB Fj, Ejp + S ~ Y C Fk, Ekp

and in the asymptotic limit, taking s, = sp = 1 for simplicity

644 where YT, YR are defined in (2.13) and all phases are evaluated at (. The cross-coupling terms therefore include transmission losses from the diagonal terms and reflections with conversion of wave type from the off- diagonal coefficients.

With the recursion kernel (2.40) the structure of the interaction series L for P-SV-waves will be from (2.36)


(2.45)

L,, Lpp represent multiple interactions with a gradient without change of wave type. Lab, Lpo, allow for the coupling between P- and SV-waves that is not present in our choice of phase matrix E and which only appears after the first iteration with (2.37).

We can similarly evaluate the kernel J with the phase matrices appropriate to purely evanescent waves (E) or to evanescent P-waves and propagating S-waves (E). In the asymptotic limit each of these matrices (j and J) take on the character of reflection and transmission terms for their respective situations.

For SH-waves j has the simpler form

(2.46)

and expressions analogous to (2.41) are obtained in the propagating and evanescent wave cases.

From the form of the kernel J we see that the rth term in the series (2.36) may be identified with an r-fold interaction, including all reflection and transmission effects, with the parameter gradients. Thus, if these parameter gradients are small in some sense we would expect the higher order terms to be negligible. We will reconsider this point when we construct the reflection and transmission properties of a smoothly varying layer. In any case, we can find L(z; 7)) (where I) is the reference level in (2.37)) to any level of interaction with the parameter gradients and then a fundamental stress-displacement matrix can be approximated by

B I ( w P , z ) = C(p,z)E(w,p,z)L(w,p,z;q). (2.47)

Then, for a region (zA, zB) which is smoothly varying but which does not include a turning point, we may construct an approximation to the propagator matrix by

P(zA, ZB) % C(ZA)E(ZA)L(ZA; ~ ) L - ' ( Z B ; v)E-'(zB)C-~(ZB). (2.48)

To extend this expression to other cases we have to choose the form of phase matrix which is applicable to the layer and this in turn affects the form of L. If the layer (zA, zB) has a turning point at the level z,, say, then we choose to split the layering at that level. For instance for SH- waves we would write

p(zA, ZB) BI(ZA) B;'(zr) BI(zr) B;'(ZB), (2.49)

where BI(z) is the fundamental matrix calculated using the propagating form of the phase matrix (ED) and B&) is calculated with the evanescent phase matrix (k'p).

In the previous asymptotic treatment the solution obtained is most effective at high frequencies when only a limited number of inverse powers of frequency needs to be considered. The interaction series method is not, however, restricted in frequency. Although our starting point is a high-frequency approximation to the solution, this is compensated for by the presence of the same term in the kernel of the interaction series. The number of

Seismic waves in a stratified half space - III 645 terms required to get an adequate approximation to the wavefield characteristics depends on the size of the elements of the kernel J and thus is controlled both by the size of the parameter gradients and the frequency. At very low frequencies more terms are needed to describe the field.

In the leading order approximation we take the fundamental matrix

BO(W P , Z) = C ( P , ~ ) E ( w P , Z) (2.50)

and we neglect the departures of L from the unit matrix in (2.36). This choice gives no coupling of P- and SV-waves within a gradient zone. Such coupling is introduced at the first iteration with (2.37) and is a feature of all the subsequent corrections (2.45).

The approximation (2.50) means that except at a first-order interface we neglect the coupling between wavetypes. This leads to simple calculation schemes for the reflection and transmission coefficients as we shall see in Section 3.1.

2.5 R E L A T I O N T O E I G E N V E C T O R D E C O M P O S I T I O N

In a uniform medium we will use the eigenvector decomposition of the stress-displacement field into up- and downgoing waves (2.6), since it is then exact.

In a vertically varying medium where we have the possibility of turning points we shall adopt the alternative characteristic vector decomposition (2.16). As we have seen this latter approach enables us to deal conveniently with turning point problems and so it is interesting to compare the two decompositions.

Far away from any turning point the dominant term in the interaction (or asymptotic) fundamental matrix C(z) E(z) should essentially agree with the dominant term in the eigenvector decomposition which is just the zeroth order WKBJ approximation (2.15). That is their behaviour with z should agree but they may differ by constant amplitude or phase factors. To exploit this equivalence Richards (1976) chose to introduce generalized vertical wavenumbers in interface coefficients derived from the Langer approximation, whilst retaining the forms expected using the eigenvector approach.

As we shall see such a procedure is a natural consequence of a rearrangement of the elements of the dominant Langer term which has general validity and is not confined to interface problems.

2.5. I Purely propagating regime The dominant term in the fundamental matrix constructed from the ‘travelling’ wave phase matrix E(z) may alternatively be expressed as a product of a new transformation matrix D with a diagonal matrix of phase terms E :

C P ( Z ) EP(4 = DPEP (2.5 1)

(2.52)

646 We have here introduced the generalized vertical slownesses

ib , = - p Ek(or,)/Ej (or,), i b , d = p Fk(wT,)/Fj (wr,), (2.53)

with similar definitions for b p u and a@; these quantities depend on both frequency w through the phase terms and horizontal slowness p. By virtue of the dependence of r, on the properties of the medium they are sensitive to the velocity structure away from the point of definition. The forms (2.53) are equivalent to those given by Richards (1976) in terms of one-third order Hankel functions.


For SH-waves we may make a similar rearrangement of the dominant term

H H = [bu 9 bd IEH,

where

EH = (2pp)-'/' diag [Ej(wrp), Fj(wrp)],

and

c = [SpB1, --iP130pulT

(2.54)

= [ s p ~ ' , i ~ ~ j p d l ~ . (2.55)

On the other hand if we adopt the zeroth-order WKBJ approximation we have from (2.6), (2.1 1) and (2.15) an approximate fundamental matrix given by

(2.56)

where

and

yp =Iz: i w q p dz.

Far above any turning point we may use the asymptotic forms (2.27) for the Airy function terms, and then the b vectors pass over to the equivalent b forms since for q i , q i > 0;

b a u , bard - qor, ah, a p d - 40-

Also we would have, for example,

(2pp)-'l2 Ej(wr,)- ~ , e x p ( - J ~ i w q , dz), 2,

and so provided that we choose the same reference levels, i.e. z, = zo, the forms (2.5 1) and (2.56) are asymptotically equivalent.

Seismic waves in a stratified harfspace - III 647

2.5.2 Purely evanescent regime

As in the propagating case we may rearrange the dominant term in the stress-displacement fundamental matrix to give

Cp(z) Ep(Z) = Dp Ep

- P ^ P - s - s - = [ b ~ , bd, bu, bdlEp,

(2.57)

and the column vectors b differ from b by modified forms of the generalized slownesses

iB,, = - p Bk(w TJBj (w T,), i 6 , , = p Ak (w TJAj (w 7,). (2.58)

In this case the zeroth-order WKBJ approximation is similar to (2.56) with the o d y difference being that the radicals 4, and 40 are replaced by i I 4, 1 and i 140 1 respectively. Far below the turning points we have the asymptotic behaviour

Q,, d p d - i 14p I, Q,, b a d - i I4,L and so the b vectors pass over to the equivalent b forms. However the asymptotic form of E is not quite the same as the equivalent matrix in the WKBJ approximation since, for instance,

(PP)-’/’ Bj (0 7,) - ( P 1 4, 1 )-”’ exp (w I T, I = (2i)”’ E , exp (a 1 T, I),

(PP)-~” Aj 7,) - ‘h ( P I 4, I )-”’ exp (- I 7, I) = (i/2)”’ E , exp (-0 I T, I).

In other words the depth dependence is the same but the amplitudes and phases differ. The only effect of this is that the amplitudes associated with the various elementary stress- displacement vectors are different between the two leading order approximations.

2.5.3 Evanescent P and propagating S

Since this is just a hybrid of the previous two cases we see that we may make a comparable rearrangement of the leading order term

c p (z) Ep(z) = Dp Ep

-P - P s s - = [bu, bd, bu, bdl EP

with

Ep = (2pp)-’’’ diag[2’/’Bj(w~~)2~/’Aj(wT,), Ej(w~p), Fj(w7p)I.

(2.59)

648 3 Reflection matrices

In the previous section we have established an effective formal procedure for looking at wave propagation in media with smoothly varying properties. We now turn our attention to the construction of reflection and transmission matrices for a model composed of regions with properties smoothly varying with depth z, separated by discontinuities in elastic parameters or parameter gradients.

As we have seen in Section 2.2, far from any turning point we can identify the Ej and Fj fields as simply up- and downgoing waves in the propagating regime. However as we approach a turning point these fields no longer have such a simple, unambiguous, character. Indeed as can be seen from the iterative development of the entire wavefield by Chapman (1976), the Airy function terms include both up- and downgoing wave characteristics near the turning point. At the turning level itself, the distinction between up- and downgoing fields is removed but Ej and Fj remain independent solutions.

Since we seek an approach to the calculation of reflection matrices which is suitable for use over a very wide band of p values, it is advantageous to choose a representation that is readily adaptable to the levels of any turning points within the region of interest.

We can achieve this aim and also conform to previous definitions of reflection matrices (Paper I), if we characterize the stress-displacement field in terms of up- and downgoing waves only within a uniform medium. For uniformity this decomposition is unique and exact. If therefore we wish to consider the reflection response of a portion of the stratification we take the region to be imbedded between two uniform half spaces whose properties are chosen to give continuity at the limits of the region of interest. We may now connect up- and downgoing wavefields in the two half spaces and so define reflection and transmission matrices for the intervening region. This procedure is entirely consistent with that used in building up the reflection response for an extended region by the addition rule for reflection and transmission matrices for the subregions (Paper I, 4.21).


3.1 A G R A D I E N T Z O N E

We consider a region (zA, zB) within which the elastic wavespeeds increase smoothly with depth so that s, = sp = 1. At the limits of the region we weld on uniform half spaces with continuity of properties with the gradient zone. At the bounding levels zA and zB we have then the likelihood of second-order discontinuities associated with the change in velocity gradient from its value within the gradient zone to zero in the uniform media.

As previously mentioned, the presence of the uniform media means that we may make an unambiguous decomposition of the seismic wavefield into up- and downgoing parts in z < zA and z > zB. We may therefore express the stress-displacement fields at z = zA and z = zB in terms of the corresponding wave vectors as

The two stress-displacement vectors are related by the propagator across the gradient zone

B@A) = P(zA, ZB) B(zB) (3.2)

and so the wave vectors are related by


We may recognize Q(zA -, zB t) as the wave propagator for this model and so we may relate its partitions to the reflection and transmission matrices for this gradient zone (Paper I, 4.8)

The form of the propagator matrix that we would wish to use in the construction of Q is determined by the position of the turning points within the structure for the slowness p under consideration. For each wave type we shall assume that there is at most one turning point, so that we can make use of the fundamental stress-displacement matrices with Airy function elements which we have constructed in Section 2.

3.1.1 Above all turning points

When the entire gradient zone lies in the propagating regime for both P- and S-waves we can use the interaction series form (2.47) for the stress-displacement matrix. With the representation introduced in Section 2.5 we may therefore express the wave propagator in the form

Q(zA-, Z B + ) = D-'(ZA-)BI(ZA)B;'(~B)D(ZB+)

= D-'(zA-) D(ZA t) E(ZA t) L(ZA;Q) L-l(Zg; Q) E-'(Zg-) o-'(z~-)D(Zg t). (3.5)

At the limits of the gradient zone (zA, zB) we will have second-order discontinuities associated with the abrupt change into a uniform medium. This is reflected by the presence of the terms D-l(zA) D(zA) and D-'(zB) D(zB). When we take account of the ordering of the columns of D into u p and downgoing parts and the arrangement of D by wave type (2.5 l), we find that

D-l(zA) D(zA) = XG(ZA). (3 -6)

For SH-waves X is just the unit matrix, but for P-SV-waves

0 O I, 0 0 1

0 1 0 \ o 0 0 1 )

to achieve the column reordering needed for Q. The G matrices have a relatively simple form. For SH- waves

(3 AH)

(3.8P)

650

The off-diagonal terms all arise from the departures of the generalized slownesses from the corresponding radical in the Uniform region, e.g. qp- a p d . We recall that a;)pd depends on the wavespeed within the gradient zone away from the interface itself, so that the reflections from the second order discontinuities should not be envisaged as occurring just at the level zA. A consequence of the partitioned form of (3.8) is that the interface matrices G do not couple P- and S- waves.

To get a good starting approximation for the phase matrix E it is convenient to extrapolate the gradient zone until turning points are reached and to reckon the phase delays 7, and T~ from these points. If the turning points lie well outside the gradient zone, the generalized vertical slownesses a,", tend asymptotically to the radical q,, and we have a similar effect for the S-wave terms. In this asymptotic limit G(zA) tends to a diagonal matrix and there will be no reflection from the second-order discontinuity. On the other hand when the turning points are sufficiently close to zA or zB, the generahzed slownesses depart significantly from the radicals and give rise to noticeable off-diagonal terms. In this case we get significant reflected waves, as in the work of Doornbos (198 1).


The wave propagator Q may be rewritten in terms of the interface matrices G as

Q(zA -, Z B +) = =G(zA) F(zA +, ZB -)G-'(ZB)=, (3.9)

where

F(ZA +, Z B -) = E(ZA +) L(zA, ZB; 77) E-'(zB -1, (3.10)

describes all the propagation characteristics within the gradient zone through the Airy phase terms in E(zA t), E-'(zB-) and the entire interaction sequence

L(ZA, ZB; 9) = L(zA; V ) L-'(zB; v), (3.1 1)

with Q as the reference level. L can, in principle, be found from the interaction series to any required order of interaction with the gradient zone Z A < z < Z B .

The structure of F may be obtained from (2.45) and (2.51): for P-SV-waves

)[:&a ________.; \ Lap )('%l(zB-) __--._---__ i I i I I + L @ 0 ~ Ei'(zB-)

(3.12P) where E,, Ep are diagonal matrices. For SH- waves

FH(ZA '3 ZB -) = Ep(zA '1 (1 ' Lhh)Ei'(zB-)- (3.12H)

To the leading order approximation discussed in Section 2.4, L will be the unit matrix. We see from (3.12P) that there is no wave coupling in this case, and so we have just the same structure for each of the P- and SV-waves as for SH-waves.

When the gradient effects leading to higher order interactions are taken into account, the presence of the Lc9, Lpo, terms means that P- and SV-waves are coupled. P to S conversions at the second-order discontinuity will be described via these terms.

Since E is its own inverse we may recast our expression (3.9) for the wave propagator into the form

Q(zA-, Z B +) == G(zA)E*Z F(zA +, zB-)E ' EG-'(zB)E,

= G'(zA) F ' ( z ~ t, Z B -1 G ' - ' ( z ~ ) , (3.13)

Seismic waves in a stratifed half space - III 65 1 where the factors G ', F ', G '-' are no longer organized into partitions by wave type. Thus

and in the asymptotic limit far from turning points, at least, these partitions connect up- and downgoing elements rather than a single wave type. For such a matrix Z we can express the partitions in a form reminiscent of the wave propagator (3.4)

(3.15)

provided each of the partitions of Z are non-singular. In a matrix product Z = XY the product of the partitions leads to the following matrix identies for the coefficients t f , t3, t i , etc.

rz = r i t t i r $ [ I - r t r 3 ] - ' t $ ,

(3.16)

where Z is the identity matrix. The structure of these relations is just that of the addition rule for reflection matrices (Paper I, 4.21) which we have reproduced in the introduction.

Hence by factoring the wave propagator Q(zA -, zB t) we can build up the reflection and transmission matrices R bB, T b B , etc., for the whole gradient zone from the f d , f d matrices for the factors. For the G' matrices these elements are easily calculated; for example for SH- waves we have from G ' (zA)

(3.17)

tuH = 2pep(apU mpd)4p/(qp apd),

and since P- and SV-waves are not coupled by G ' the rda, f d p sets have the same functional form.

The diagonal matrices E (zA t), E (zB -) governing the major phase dependence of F depend entirely on the Ej, Fj Auy function entries for the two wave types. It is convenient to introduce at z A , zR the parts which asymptotically have u p and downgoing wave characters ef, I$ and e:, e:. The main propagation term F' may then be written as

(: i 0 ) ( t L u u ~ Lud (.:I-' 0 (3.18) F'(zA+,zB-)= ---.-+ _ - - _ _ - _ - _ _ _ _ _ ______._ )(; _____. I _______)

edA Ldu ~ I t L d d (edB1-l

652

where we have reordered the interaction term L'. The diagonal nature of E means that the r d , t d terms derived from F'(zA t, zB -) are related to those for L' alone by


F F

(3.19)

The construction of the elements tk, r i , etc., is discussed in Appendix A.

matrix,rzo, rzo are zero and To the leading order approximation, when we neglect the departures of L from the unit

which just increment the phase terms across the layer. The existence of just the transmission terms simplifies considerably the construction of R fiB, TeB by application of (3.16) recursively. The approximate reflection and transmission matrices are thus given by

R A B - GA GA F G B t F GA F G B t F -1 GA

TfiE=t:BtdFg{f-ru LtuOrd do]} t d 9

DO - i d + t u [tugid d O l { l - r u [ c u o r d do]} t d , (3.21) GA F G B t F -1 GA

where, e.g. rzA, tdGA are the terms derived from G '(zA).

3.1.2 Below all turning points

When both P- and S-wavefields are evanescent we use the purely evanescent form B&) for the fundamental stress-displacement matrix when we construct the wave propagator

Q ( Z A - , Z B t ) = D - l ( z A - ) D ( z A ~ ) E ( z A ~ ) L ( z A ; ~ ) L - l ( z g ; ~ ) E-l(ZB-)D-l(ZB-)D(Zgt)

= z G ( Z A ) E ( Z A t ) i ( z A , ZB;q)E-'(zB-)G-'(ZB)z.

(3.22)

The G matrices are very similar to the G matrices introduced in (3.6), (3.8), the only differences being that %, is replaced by 6 and q,, q p are replaced by i l q , I and i lqpl respectively. As in Section 3.1.1 we may factor the wave propagator and build up the reflection matrix R fiB for the gradient zone from the i d , t d matrices for the factors. We do not expect significant reflections from the G matrices unless the turning points are close to the margins of the reflection zone.

The interaction matrix L(zA, zB; q) can be treated by methods similar to those for the purely propagating case (see Appendix).

3.1.3 Turning point problems

In the two preceding cases we have been able to use the same form for the fundamental matrix throughout the region occupied by the gradient zone. Since we have chosen to adopt different forms for the stress-displacement matrix in the propagating and evanescent regions we now split the calculation at a turning point. For simplicity we discuss the SH case first and then turn our attention to P-S V problems.


(a) A turning point for SH-waves We take the S turning point to lie at z, where zA<z, < z B . Then in Z A < Z < z , we have propagating waves and use the fundamental matrix B(z); whilst in z s < z < zu we have evanescent waves and so use B(z).

The propagator for the gradient zone may now be calculated using the chain rule (cf. 2.48)

(3.23)

= B(zA) NH(z,) B-'(zB).

The turning point contribution is similar to an interface term and explicity has the form, in terms of the full phase matrices,

NH(z,) = E-' (z, -) C-'(Z, -) C(Z, t) E(z, t). (3.24)

There are no interaction terms since the reference level will lie at the turning point. Now since the material properties are continuous across the turning level z,, C(z,-) = C(z,+) and

NH = E-' (z, -) E(z, t) = n, (3.25)

in terms of the conversion matrix from propagating to evanescent forms

so

) a

1 exp(in/4) exp (- in/4) n=- (

d 2 exp (- in/4) exp (in/4) (3.26)

The wave propagator for the gradient zone may now be written in the form

Q(zA-, ZB+)= zG(ZA)E(ZA+) L(ZA;Z~)NHL-'(ZB;Z,)E-'(Z~--)G-'(ZB)Z, (3.27)

and once again we may find the reflection and transmission matrices R kB, T k B in terms of the quantities f d , t d , etc., for the factors of Q.

When a turning point occurs within the gradient zone the boundaries z A , zR are likelv to be not too far from the turning level and thus the off-diagonal elements of G, G may well be significant.

The main difference from the preceding case is in the nature of the propagation term

F(zA+, Z B - ) =E(zA t)L(ZA;Z,)NHL-'(zB;Z,) E-'(zB-).

If we neglect all gradient effects so that we take L, L to be unit matrices then

FO(ZA +, ZB -) = E(zA t) NHE-'(zB -),

(3.28)

(3.29)

and we no longer have the simple transmission effects as in (3.20). Indeed the reflection elements now correspond to the process of total internal reflection at the turning point. To this leading order approximation we now have, using the explicit form (3.26) for n,

(3.30)

654

The expression for r& displays the well-known phase shift of n/2 radians at total reflection. The other elements are numerically well behaved, even across this turning point, since they are damped through the evanescent region by 6; and (&:)-'I.

The reflection and transmission matrices for the gradient zone may now be obtained by using (3.16) to add in the effect of the interface terms G(zA), G-'(zB) in two stages.

Parameter gradient corrections must now be evaluated separately above and below the turning point (see Appendix). In order to include the full propagation effects in (3.28), the interaction elements t k , etc., must be combined with those for NH using (3.16) before the r i terms can be found.

(b) Turning points and P-SV-waves For the P-SV wave system additional complications are introduced by the different levels at which P- and S-waves turn, so that we must be able to handle problems in which one wave type is propagating and the other evanescent.

As an illustration of the procedure we consider the case when only a P turning point occurs at z p within the gradient zone (zA, zB). In the region zA < z < zp we have propagating P- and S-waves and so use the fundamental matrix B(z). In z P < z < z B we will have evanescent P- waves and then we employ the matrix

B(z) = bEL = CEL,

where the interaction term we split the calculation of the propagator at the turning level zp and so


(3.3 1)

is calculated with the fields appropriate to E. As for SH-waves,

(3.3 2)

(3.33)

and we choose the turning point as the reference level for the interaction terms. The turning level matrix is given by

N P = L- ' (zp- ;zp)E- ' ( zp- )~ (zpt )~ (zp+;zp) , (3.34)

using the continuity of material properties at zp. N, is most easily evaluated by using the original (PIS) block representation of the fundamental matrices B(zp-), B(zpt) and so we find

(3.3 5)

in terms of the conversion matrix n (3.26). The propagation within the gradient zone is thus represented by

F (zA t, ZB -) = E (Z A t) L(z,; Zp) N p L-' ( Z B ; Zp ) E-'(ZB -), (3.36)

and as the SH-wave case the presence of the conversion matrix n in N, (3.35) leads to internal reflection for the P-waves. The interaction terms have to be handled separately above and below the turning point (see Appendix).

Seismic waves in a stratified half space - III 655 Where there is only an S-wave turning point within the gradient zone we may make a

similar development to the above, but would then use B(z) in z A < z < z , and B(z) in z, < z < zB. The turning level matrix N, would then be found from

(3.37)

but the formal structure of the wave propagator would be otherwise unaltered, and so the calculation would proceed as before.

If both P- and S-wave turning points lie within the gradient zone we would wish to split the calculation at both the levels zp and z, so that

P(zA, zB) = B(zA) B-'(zp) B(zp) B-'(z,) B(z,) B-'(zB),

and

(3.38)

Q(zA-, ZB+) = Z G ( Z A ) E ( Z A ~ ) L ( Z A ; Z ~ ) N ~ L ( Z ~ ; Z , ) N , L - ' ( Z B ; ~ , ) E - ' ( Z B - ) G-'(zB)Z.

We now have to take account of the different reference levels appropriate to the various interaction terms in the presence of the two turning points.

In the leading order approximation we neglect the departures of the various L matrices from the unit matrix and thereby decouple P- and SV-wave propagation within the gradient zone. We may therefore calculate the PP and SS reflection coefficients separately by just the computational scheme which we have outlined for SH- waves. To this level of approximation the P-waves are insensitive to the presence of S turning points and vice versa. The non-zero elements of the approximate reflection and transmission matrices R bt, Tfi t can therefore be found by bringing together the results of scalar calculations.

Even when interaction terms are important, in all cases we can find the overall reflection and transmission matrices for the gradient zone by recursion over the rd, f d terms for the factors of the wave propagator.

If either of the P- or S-wave speeds is in fact smoothly decreasing with depth, so that either s, or sp is - 1, we may follow a similar approach to the one we have described but with the roles of up- and downgoing waves interchanged for that wave type.

3.2 P I E C E W I S E C O N T I N U O U S M O D E L S

As a simple prototype of models comprising smooth gradient zones separated by discontinuities in velocity or velocity gradient, we consider two gradient zones in (ZA, ZB) and (zB, zc) with an interface at zB (Fig. 1). To conform with our definition of the reflection and transmission matrices for this structure we append uniform half spaces with continuity of properties at zA, zc.

The wave propagator for the entire region (zA, zc) is then given by

Q(ZA -, zc +) = D-'(zA -)P(zA, zc)D(zc +I, (3.39)

in terms of the stress-displacement propagator P. It is convenient to break the calculation at the interface z = zB and so we write

p(zA, zC) = Ba(ZA) B,'(zB) Bc(ZB) B,'(zC)- (3.40)

656 B. L. N. Kennett and M. R. Illinworth

Figure 1. Division of a piecewise smooth model into gradient zones (a), (c) bordered by uniform media and an interface between uniform media (b).

The choice of fundamental matrices is dictated by the nature of the wavefields in ( zA, zB), ( Z B , zc). For illustrative purposes we will write

B(z) = D EL,

and then the wave propagator becomes

Q ( Z A -3 zC '> = =Ga(zA) E a ( Z A La(ZA, zB) E,'(zB -) D,'(zB -1 Dc(ZB E c ( Z B

X L C ( Z B , Zc) E,'(zc -1 G ~ ' ( z c ) ~ : , (3.41)

and it is this combination of terms which will determine the reflection and transmission matrices REc, TEc, etc., for the entire region (zA, zc). There are however a number of different ways in which this response can be determined with rather different physical interpretations.

3.2.1 Generalized reflection and transmission coefficients In the approach introduced by Richards (1976), and later elaborated by Cormier & Richards (1977) and Choy (1977), attention is concentrated on the effect of the interface and only the leading order terms considered within the gradient zones.

Thus if we write the wave propagator (3.41) in the form

Q(zA- , Zc +) = EG,(zA) Fa(ZA +, Z B -) D,'(ZB -) Dc(ZB+) F,(zB +, Zc -)GE'(Zc)E, (3.42)

the two gradient propagation terms would be approximated as, e.g.

Fa(ZA +, z B -1 Ea(ZA E,'(zB -). (3.43)

We then note that the interface term Di1(zB- ) Dc(zB t) is highly reminiscent of the interface term between two uniform media D;'(zB -) DC(zB +) but expressed in terms of generalized vertical slownesses a, rather than just the P- and S-wave radicals. One may therefore introduce generalized reflection and transmission coefficients at the interface z = Z B by a representation of the form (3.15)


However, except in the asymptotic regime well away from turning points, the r!' ti, etc., will not necessarily have the physical significance of reflection and transmission terms between purely up- and downgoing waves. In effect the stress-displacement field has been split up within vertically inhomogeneous regions into parts specified by Ej and Fj above the turning point and Aj, Bj below. As we have already noted these terms have rather mixed character, particularly close to the turning point.

The generalized coefficients defined by (3.44) are both frequency and slowness dependent, through the a terms. Once they have been evaluated we may use the factorization of the wave propagator implied by (3.42) to calculate the overall reflection matrices, e.g. Rbc, recursively using the addition scheme (3.16) for the propagation and interface elements.

At a discontinuity in velocity gradient the generalized interface matrix takes a rather simpler form, since the elastic parameters are continuous. Thus in this case for SH- waves

D,'(z~ -) D ~ ( Z t - (3.45) 1 P&j t a&

) - ( a k ' a j d ) (aj,-a&,) TjpU+aid

where we have written, e.g. a,, a + d for the generalized slownesses evaluated on the two sides of the interface. The elements rdH, f & are then i

(3.46)

Reflection from the second-order discontinuity is thus entirely controlled by the differences in the generalized slowness associated with the gradients on the two sides of the interface.

In this leading order approximation there will be no coupling of P- and SV-waves at a second-order discontinuity, and so the structure (3.46) will be valid for all wave types.

3.2.2 Phne wave interface coefficients

As an alternative to the procedure outlined in the previous section, we prefer to adopt a different, although equivalent, factorization of the wave propagator. We start from the matrix identity

Dal (ZB -> Dc(ZB = Da' (ZB -1 Da(ZB -1 Da' (zB -1 Dc (zB +> D,' (zB '1 Dc(ZB (3 .47) = G , ' ( z ~ -)ED;'(z~ -) Dc(zB + ) E G ~ ( z ~ t),

where we have introduced the unit matrix as the product of the eigenvector matrix D and its inverse. The G matrix is already familiar from our discussion of the gradient zone, and Dil(zB -) D,(zB t) is the plane wave interface matrix between two media with the elastic properties just at the two sides of the interface. With the substitution (3.47) the wave propagator for (zA, zc) can be written as

Q(ZA -, zc +) = Z G , ( z , ) Fa(z* +, Z B -) G,'(zB -)ZD;'(zB -)D,(zB +)

(3.48)

where we may recognize Q(zA -, zB -), Q ( z B t, zc +) as the wave propagators for the gradient zones in (zA , Z B -) and ( z B +, zc). Each of the terms in (3.48) have the interpretation of a wave propagator for either a gradient zone or an interface. We may therefore

658 calculate the overall reflection and transmission matrices R;', TAD", etc., by the addition rule for the reflection matrices for the regions (zA -, Z B -), (zB +, zc +) and the interface at ZB,

We may regard the representation (3.48) for the overall wave propagator as being equivalent to the parameterization of the medium indicated in Fig. 1. Basically we isolate the effect of the interface from the gradient zones by imbedding it between uniform media with the properties at the two sides of the interface. This also has the effect of leaving each gradient zone sandwiched between two uniform media, which is just the situation we have discussed in Section 3.1.

One advantage of this approach is that we separate the reflective properties of the interface from those due to the gradient zones. The reflection and transmission matrices for the interface are just the usual plane wave matrices and depend only on the slowness p . Thus since we have already shown in Section 3.1 how to find reflection matrices for the gradient zones, we have an effective procedure for calculating the reflection and transmission matrices for this model.

At a second-order discontinuity the interface term D,'(zB -) Dc(zB t) reduces to the unit matrix and in this case the overall wave propagator


Q(zA-, z c + ) = Q ( z ~ - , ~ g - ) Q ( ~ ~ + r Z c t ) . (3.49)

The reflective properties of the second-order discontinuity arise in this representation from the differences between Ga(zg -), G C ( z B +). In the expanded form of the wave propagator we now have a term

(3.50)

The rs, ti elements corresponding to GC(zg t) represent the transition from a uniform medium with the properties at z B into the lower gradient. The elements rd, ti from Gil(zB -) arise from the transition from the upper gradient zone into the uniform medium. The overall effect obtained by combining these elements via (3.16) may be thought of as building the actual gradient change from the two gradient steps on the two sides of the interface. In terms of Fig. 1, we just shrink the jump in properties at zB to zero.

For a more complex model with many interfaces we may extend the approach we have just described into a recursive calculation scheme, incorporating alternately the effects of gradient zones and interfaces. This scheme may be regarded as a generalization of the recursive approach for uniform layer models introduced by Kennett (1974) and discussed in Paper I (4.25), for there one alternately includes the phase delays for a uniform layer and the effect of the interfaces.

3.2.3 Approximations and limitations

In the previous section we have described a method for calculating the reflection and transmission properties of a piecewise continuous medium in terms of a recursive application of the addition rule for reflection matrices ( R t B , etc.) for each of the gradient zones and interfaces. Within each gradient zone we use a recursive development on the r d , t d matrices associated with portions of the response. With this system of inner and outer recursion it is useful to investigate whether there are any simplifying approximations which might reduce the computational labour of finding a solution. For example, under what circumstances can we neglect the interaction terms arising from the parameter gradients and just use the leading order representation for the response?

Seismic waves in a stratified half space - III 659 (1) Close turning points The derivation of a fundamental matrix in Section 2 in terms of Airy functions depends on making use of the uniform asymptotic approximation provided by the Airy functions across a single turning point for a particular wave speed. When there are two close turning points we no longer have a uniform approximation to the total response, even if we break the form of the medium representation to take account of the change of the slope of the parameter gradients. Two such circumstances are illustrated in Fig. 2. The range of slownesses for which such phenomena will occur is quite limited, although the nearness of turning points is determined by the rate of change of the arguments of the Airy functions and so depends on frequency. At low frequencies a wider range of slownesses are involved. We have tried to indicate by shading in Fig. 2 the region of the velocity models for which some error will be associated with the Airy function scheme. In general since our primary interest is in the pattern of response at the surface and receivers are not placed in the waveguide formed by the velocity inversion there will not be too much difficulty with the coalescing turning point problem. If however a velocity inversion is the dominant feature of the model as, for example, in the oceanic sound speed profile then more accurate approximations in terms of parabolic cylinder functions are needed (Ahawalia & Keller 1977).

In regions away from such complications the accuracy of the leading order terms in the Airy function approximation will depend on the choice of phase matrix.

( 2 ) Interaction terms As we have seen above, unless there is a turning point within a particular zone the phase terms h, @ p (2.21) are non-unique. Since we wish the character of the leading order terms to mirror, as closely as possible, the nature of the wave field, we choose to extrapolate the parameter distribution (usually with a simple linear gradient) to create a turning point lying outside the current gradient zone. In this way if for a slowness p the quantities 4:, 4; are not small, the E matrix we generate will have the character of up- and downgoing wavefields. Chapman (1981) has suggested that for reflected waves the reference level in phase calculation should be taken at the reflection level. Unless this reflection level is close to a turning point, the corresponding E matrix will not have the character of the actual field and so more interaction terms will be needed to correct for this choice.

How large then are the interaction series corrections to the reflection response if we have made a good choice of the leading order terms? In Appendix A we have made a perturbation expansion of the interaction coefficients in terms of a gradient measure E .

r k = er& + E2r: + . . . , t , L = I + E t : + E t : + ...

P- a I 7

p- l a I

I b Z

Figure 2. Two cases of the confluence of turning points: (a) a velocity inversion, (b) a change in the sign of the velocity gradient. The turning points are indicated by triangles and the region of the model for which the response is likely to be in error is indicated by shading.

660 The actual contribution of these terms to the reflection response of the gradient zone is modulated by propagation terms so that from (3.19) we are interested in, e.g.


(3.19')

in terms of the Airy function sets e t , d , t?:,d at the limits of the zone under consideration. In the presence of a turning point we deal with the parts above and below the turning level separately.

When we take account of the nature of the Airy functions it is possible, with perseverance, to place bounds on the interaction term contributions. Above the turning point we find that all the elements of the matrices r i , tb, etc., for a region (zA, zB) can be bounded by expressions of the form

(3.51)

where C is a constant of order unity, and e0 is an upper bound on the parameter gradient terms in (ZA, z g ) . Higher order terms behave like [ E ~ ( Z ~ - Z A ) ] ~ . In effect we can think of E~(zB -zA) as a measure of the change in parameter gradient terms across the zone. Now from the form of the coupling matrix j (2.38, 2.46) we see that the most significant contribution is likely to arise from terms like

2jH = azpip + azz$oiazGo. (3.52)

The logarithmic derivative of the shear modulus p will be common to all choices of the phase matrix, but the second term will vary. Above the turning point

(3.53)

and thus will be strongly dependent on the choice of reference level used to define the accumulated phase T~ We may therefore have reasonable confidence that the interaction correction terms are small if we choose to work with portions of the model such that the relative change in elastic parameters across a subdivision is small. For subsequent calculations we have arbitrarily chosen 10 per cent.

Although the quantities I Eri I may be small, they in fact constitute the entire reflection return from within a gradient zone. If therefore we decide that we are able to neglect them, it must be because we anticipate that the dominant features of the reflection response of the whole piecewise smooth model arise from the turning points and the first-order discontinuities and to a lesser extent from the second-order discontinuities. We shall see later that we have some computational checks on the accuracy of this approximation.

For regions below the turning point we have to take account of the character of the Airy functions Aj and Bj when assessing the significance of the interaction terms. Once we take account of the propagation effects (3.19) we find that once again the contribution of the gradient terms to the reflection and transmission response is 0 [ E & ~ -,??A)].

Thus if we neglect interaction corrections we are working to the same level of approximation throughout the model, and we shall endeavour to set up a piecewise continuous model in such a way that this is a good approximation. Within each portion of the stratification we will thus, in effect, treat each wave type independently and so significantly reduce the amount of computation required.

(3) Converted waves When we adopt the approximation of taking only the leading order terms we have restricted the places at which we allow interconversion of wave types to occur. Within any gradient

Seismic waves in a stratified half space - 111 66 1 zone we have made the assumption that P- and S-waves propagate independently. Since no coupling between P- and S-waves is introduced by the G matrices (3.8P) associated with second-order discontinuities in velocity, interconversion will occur only at interfaces of discontinuity in elastic parameters.

Richards & Frasier (1976) have shown that for strong parameter gradients P to S conversion can reach significant proportions and so should not be neglected.

If a fairly steep gradient zone is modelled with a sequence of second-order discontinuities, the use of the leading order approximation will not allow for interconversions between P- and S-waves. This problem could be overcome to some extent at least, by reparameterization of such a zone with a combination of interfaces with small jumps in properties. If the region over which the elastic properties change is relatively small compared with the dominant wavelengths of interest, the new model will have very nearly the same response as the one it replaces for P- or S-waves alone but an improved allowance will be made for the converted waves. A full solution awaits a convenient technique for handling strong gradient zones.

(4) Checks on the accuracy of the approximations For a perfectly elastic velocity model we may place certain checks on the accuracy of the approximation we have made in retaining only the leading order Airy function terms, by using reflection field identities (Kennet, Kerry & Woodhouse 1978).

From the unitarity relations for the seismic wavefield we have a number of requirements on any valid reflection matrix. The most useful cases to consider are when both P- and S-waves have turning points within the model and when P-waves are evanescent at the top but S-waves have a turning point. We will write the downward reflection matrix for the whole model, for P-SV-waves, as

and from general symmetry properties R Ls = R gp. Thus for turning points for both P- and S-waves we require

IRLp I = lRLs I, lRLp12 t1Rgp12 = 1REs12 t IRLs12=1.

(3.54)

3.55)

Whilst for evanescent P-waves at the top of the stratification but with a turning point for S- waves

lRgs I = 1,

arg ( R gp) = 7r/4 t lh arg ( R gs). For SH-waves, in the presence of a turning point

(3.56)

l R g H I = 1 . (3.57)

The size of departures from these identities in our calculations will provide a measure of the errors committed in our approximations.

4 The reflection response of piecewise smooth models

In Section 3 we have shown how we may obtain a good approximation to the reflection response of a piecewise continuous earth model by a recursive development using Airy

662

function representations in each segment. We here consider the construction of such a response for realistic earth models.


4.1 C O M P U T A T I O N A L ASPECTS

We choose to represent the model by specifying the elastic wave speeds and densities at a set of depth points with a dual entry on the two sides of any first- or second-order interfaces. Between interfaces we use a cubic spline fit to the parameter distributions except in the case of uniformity or linear gradients with depth. In order to satisfy our requirements on the relative change in parameters across any region, it sometimes proves necessary to break up a broad zone by the introduction of a very minor second-order discontinuity.

In order to optimize the efficiency of the reflection calculation we choose to fix the slowness p and then perform an inner loop over frequency w. In this way we can exploit the frequency independence of many of the quantities of interest. Thus we first calculate the plane-wave reflection and transmission coefficients at each interface and also the radicals 4,, 4p and accumulated phase terms T,, T~ at the top and bottom of each segment of the stratification. For linear gradient zones we find T,, TO directly from the analytic forms but in general for a smoothly varying medium perform a numerical integration of the vertical slownesses I 4, I or I 4 p I using a Richardson extrapolation technique, modified if necessary to allow for the derivative singularities at the turning points (Fox 1967).

For each frequency we start the recursive procedure for the calculation of the reflection and transmission matrices at the base of the model and work upwards a segment at a time. Within a region we calculate the Airy functions and the corresponding generalized slownesses a at each frequency. For the Airy terms we use a power series expansion for small arguments matched on to asymptotic expansions at larger arguments with up to fourth-order corrections. Once the generalized slownesses are known we can find the r d , f d terms for the G matrices and now carry the calculation up through a region adding in the phase terms with the recursive scheme. When we encounter a first-order interface we bring in the frequency independent interface terms with a further application of the addition rule for the reflection matrices (3.16). We then move up into the next region and continue the calculation until we reach the top of the stratification.

The recursive use of the addition rule for reflection matrices has the merit of providing effective control of exponentially large terms occurring in evanescent regions. In the leading order approximation for a region where a wave type is purely evanescent we get no contribution from growing terms and of course there are no problems when a wave type is propagating. If a turning point occurs within a region, the character of the solution depends on how the coupling matrix N is introduced into the recursion but again exponentially large terms need not appear. The addition scheme has the merit that the same algorithm is used in all cases and all calculations for the coupled P-SV system are restricted to at most 2 x 2 complex matrix multiplications. On the other hand the approach due ' to Abo-Zena (1 979), which has been employed by Cormier (1 980), is based on a minor matrix development of the propagators and uses a variety of techniques in different parameter ranges with up to 4 x 4 complex matrix multiplications.

Hitherto most body wave calculations for complex models have used stacks of uniform layers to simulate the stratification. When a piecewise smooth representation is adopted, far fewer subdivisions are introduced into the velocity model since weak parameter gradients can be modelled directly. The recursive calculations with the Airy functions'can be carried out quite quickly. When we make use of the leading order approximation so that wave types are allowed to propagate independently with each region, a single layer recursion including


an interface takes about three times longer than the corresponding recursion for a uniform layer. For complex models the considerable reduction in the number of ‘layers’ in the present approach leads to significant overall savings in effort compared with uniform layer models. Indeed the accuracy of the present technique improves at higher frequencies (see Section 4.4) when finer sampling would be needed in terms of uniform layers.

4.2 A C R U S T A L M O D E L

As an illustration of the computational procedure which we have described, we consider the reflection response for the crustal model shown in Fig. 3. This model consisting of gradient zones with a significant velocity discontinuity at the Moho (28km) would need to be represented by a large number of uniform layers for an adequate approximation to the response.

We have here a linear gradient from the surface to a significant second-order discontinuity in velocity gradient at 2 km depth and a further linear gradient down to 12 km. Beneath this depth we have a non-uniform gradient zone down to 28 km and a further positive gradient below the interface. The response of this model is dominated by continuous refraction through the gradient zones and reflection from the Moho discontinuity.

The direct result of our calculations is to find a complex reflection matrixRD(p, w ) for the sets of slownesses p and frequencies w we have considered. In an attempt to display the phase behaviour of this matrix we have constructed a temporal reflection response kD(p, t ) by inverting the Fourier transform over frequency. The components of k D ( p , t ) represent the seismograms which would be obtained by illuminating the stratification from above with a plane wave of horizontal slowness p. Ideally we would have a delta function time input, but our calculations are necessarily band limited so that resolution is slightly reduced.

Figure 3. Crustal velocity structure employed in reflection calculations indicating the division of the stratification at the fist- and second-order interfaces, so that four inhomogeneous regions overlie the uniform half space (5).


=km3{ , 1, ,

0 p slkm 0 1 0 2

b

\, '. .. y - - -, - .-.- -a, \". \\ ..

, ,

\ ...._

2 - - - - - - - - - - - 0 0 0 1

Figure 4. Projective slowness display of the P-wave response of the crustal model in Fig. 3. (a) P-wave slowness (Y-' as a function of depth. (b) Intercept time map ~ ( p ) for P-wave propagation in the model, __ continuous refraction, - . - reflection from fiist-order discontinuity, - - - reflection from second-order discontinuity, . . . . . internal multiple (see text). The depth of reflection is marked for each phase. (c) The temporal reflection response K p p ( p , t ) for the frequency band from 0.01 to 4.0Hz.

Seismic waves in a stratified half space - 111 665

In Fig. 4 we show a projective plot of the PP reflection element ( R p p ( p , t ) ) for a frequency band from 0.01 to 4.0 Hz, together with an intercept time map and a representation of the velocity structure in the slowness domain, to aid the interpretation of the response. All three parts of Fig. 4 are on the same slowness and time-scales so that direct comparisons can be made.

The intercept time relation as a function of slowness ( ~ ( p ) ) is presented in Fig. 4(b) for the continuous refraction in the velocity gradients as a solid line, for the reflection from the Moho as a chain dotted line and for reflections at the second-order discontinuities by dashed lines. The depth of reflection is indicated in each case. For the Moho discontinuity at 28 km depth we have a critical point at a slowness of 0.125 s km-' which is marked with a star.

When we compare the ray times in Fig. 4(b) with the amplitude display in Fig. 4(c) we see that we get a prominent pre-critical reflection from the Moho with a rapid increase of amplitude as the critical point is approached. The large amplitude at the critical point is associated with the existence of head waves along the Moho interface, and we note the character of the response changes at this slowness. From 0.125 up to 0.145 skm-' we have post-critical reflection from the Moho with a progressive change of phase in the waveform and diminuition of amplitude. For the pre-critical reflection we get just a band passed delta function but in the post-critical range we have major positive and negative excursions in the waveform.

The onset of continuous refraction in the gradients is marked by a local increase in the amplitude. There is no significant reflection at the change in velocity gradient at 0.154s km-' and the transition is only marked by a slight increase in amplitude. A similar increase occurs at the rather stronger gradient change at 0.169skm-', but now there is a perceptible reflection at a T value of about 0.5 s for slownesses down to 0.08 s km-I. This second-order discontinuity also gives rise to internal multiples between the gradient change at 2 km and the Moho at 28 km. The first multiple is clearly seen through the large amplitudes associated with the refractions and emerges at a T of 11 s at 0.1 25 s km-' ; this phase is indicated on the intercept time map (Fig. 4b) as a dotted line. There is strong reflection from the sharp velocity gradient at the surface which diminishes rapidly once the P-waves become evanescent at 0.2 s km-I.

For this crustal structure we have thus been successful in determining the reflection response and the temporal-slowness display enables us to relate the features of the response directly to the velocity distribution. From this temporal reflection response Fryer (1 980) has shown, in a rather simpler case, how theoretical seismograms may be generated by integration (with interpolation) along linear trajectories through the map. Alternatively we may follow the more traditional route and use the frequency domain reflection matrix R D ( p , w ) with the appropriate phase terms for a given range, e.g. Jo(wpx) , and perform an integration over slowness p before that over frequency w, as in Paper 11.

In the approximate technique (WKBJ seismograms) introduced by Chapman (1978) and illustrated by Dey- Sarkar & Chapman (1 978) only continuous refraction and reflection from velocity discontinuities are considered (i.e. the solid and chain-dotted portions of Fig. 4b). However we note that a strong second-order velocity discontinuity gives rise to a small but significant amount of reflected energy and also internal multiples which would be neglected in Chapman's approach.

4.3 AN U P P E R M A N T L E M O D E L

The technique which we have discussed in this paper is particularly well-suited to investi- gating the reflection properties of the upper mantle of the Earth. Most seismic models of

666

this region consist of gradient zones separated by interfaces or zones of very rapid velocity change. As an example we consider the model T7 (Burdick & Helmberger 1978) derived from seismic observations in the western United States.

In order to find the reflection response of this structure we apply an approximate Earth flattening transformation designed to preserve travel times. We choose the density transformation to give the same reflection coefficients at normal incidence in both the spherical and flat models. Thus in the flattened model we take

z = a In (alr),


&f (z) = 0 (r) (air),

Bf(Z) = (air),

PfG) = P (r) (ria),

where a is the radius of the Earth. The flattened version of T7 is illustrated in Fig. 5, where we have also indicated the

subdivisions of the stratification used in the calculations. After flattening the model shows slight positive velocity gradients below Moho (35 km) underlain by a low-velocity zone for both P- and S-waves. At 170km depth we have a significant change in velocity gradient. Both the major upper mantle ‘discontinuities’ are modelled by narrow intervals of rapid velocity change which we represent in terms of closely spaced second-order velocity discontinuities around 400 and 7 0 0 h after flattening. The structural divisions we have employed were chosen with the object of satisfying our criterion for the neglect of all interaction terms. This leads to 12 active regions with bordering uniform half spaces, which should be compared with the 92 uniform layers used by Burdick & Helmberger (1978) in calculations with the Cagniard method.

We will first consider the P-wave response as represented by the App(p, t ) temporal behaviour for a frequency band from 0.03 to 1.0Hz. In Fig. 6 we show the time-slowness

subdivisions of


0 ' 1 0'15 p slkrn

Figure 6. PP response of model T7. (a) Intercept time map ~ ( p ) for P-wave propagation in the model. The same convention is used as in Fig. 4 ,md the approximate depth of reflection is marked for each phase. (b) The temporal reflection response R p p ( p , t ) for a frequency band from 0.03 to 1.0 Hz.

map of the reflection term and an intercept time display with the major features accompanied by their depth of origin. We use the same convention as in Fig. 4 so that solid lines correspond to continuous refraction, chain dotted lines to reflection from velocity discontinuities and dashed lines to reflections from changes in velocity gradient. The reflections from the 400 and 700 km 'discontinuities' are very clear, as also is the termina- tion effect of the second-order discontinuity at 925 km at the base of the model. The


Figure 7. S response of model T7 for frequency band 0.02-0.5 Hz. (a) SV response i i S S ( p , t ) , (b) SH response i i H H ( p , t ) .


low-velocity zone itself appears as a gap in the response and there is a small reflection from the velocity gradient change at 170 km. The presence of the low-velocity zone leads to noticeable multiples which, with the choice of time interval employed for the fast Fourier transform, are just aliased back into the early part of the response at small slownesses.

In Fig. 7 we compare the S-wave response in the SV and SH cases for the frequency band from 0.02 to 0.05 Hz. For an isotropic medium we use the same phase delays TO and Airy function terms for both SV- and SH-waves, so that there is a considerable saving in performing the reflection calculations for the P-SV system and SH-waves at the same time. To our leading order approximation, coupling between wave types is only introduced at first-order discontinuities and so within each region the SV and SH calculations have a common structure. As we would expect the temporal responses l?ss(p, t ) and l ? ~ ~ ( p , t ) are generally similar although they differ noticeably in some areas. In particular the reflection from the Moho shows a rather different character and for SV-waves we note a small excitation of converted P-waves below Moho for slownesses less than 0.125 skm-'. For the SV case there is more efficient trapping within the low-velocity zone and we can see two clear multiple trains as opposed to one for SH-waves. The reflections from second-order discontinuities are of a similar nature since, to the leadng order approximation, they are controlled by the same generalized interface coefficients. Differences arise, however, from the transmission through the overlying structure. For SV-waves these reflections from gradient changes diminish for smaller p values due to the effect of coupling between P and S in the shallow structure.

For each of the wave types we see from the temporal-slowness maps Figs 6 and 7 that we would expect the approximate technique of Dey-Sarkar & Chapman (1978) to give a good account of the main refracted arrivals but this approach will miss much of the reflections from the sharp gradient zones.

0.7 p slkm 0.2

Figure 8. Converted PS temporal response of model T7.

670 Although our approximations have restricted the possible levels at whch conversion of

wave types can occur within the model, the converted response has interesting features ( R p s ( p , t ) - Fig. 8). Since the 400 and 700 km 'discontinuities' are represented as gradient zones, no conversion arises at these regions; the size to the error thus introduced is discussed in the next section. The converted energy appears in two major portions which are principally associated with conversion at the Moho. For slownesses less than 0.125 skm-' and intercept times greater than 20 s we see energy which has propagated as P-waves below Moho. For slownesses greater than 0.135 s km-' we see waves which propagate as S beneath the Moho. This latter group shows a rather interesting illustration of wave tunnelling, since at 0.155 s km-' P-waves start to became evanescent in the crustal region, even though S-waves may still propagate. This phase may be seen to lose amplitude and to increase its period as p increases and so P-waves become more evanescent. However, for a range of slowness the decay associated with a single evanescent leg in the crust is not sufficient to prevent long- period propagation.


4.4 ACCURACY O F APPROXIMATIONS

In Section 3.2.3 we showed that we have some effective checks on the accuracy of the approximations we have employed from the unitarity relations (3.55-3.57). We now see how far we can justify the use of just the leading order approximation within each subregion, recalling that we have attempted to divide the layering to satisfy our relative change of properties criterion within each region.

We look first at the situation when P-waves are evanescent and S-waves have a turning point. In this region we should have unit modulus for the Rss and RHH reflection coefficients for the slownesses in this range. In Fig. 9 we show the amplitude JRss (p , w ) 1 for the model T7, the temporal response for which appears in Fig. 7. For slownesses greater

Y I I I I 0.1 0.2

p slkrn

Figure 9. The amplitude of the reflection coefficient for SV-waves IRss(p , w)l as a function of frequency and slowness. The effects associated with second-order discontinuities are indicated by the full triangles and those with the low-velocity zone by an open triangle. The region in which the moduli I Rpp I and I R S S I should be equal is indicated by shading.

Seismic waves in a stratified half space - III 67 1 than 0.166skm-' the exact result is a level unit amplitude over the whole range. We note that this is indeed achieved for most of the response, but there are some departures from the expected behaviour. A very similar pattern is also seen for the SH-wave case IRHH(p, w ) I.

The most significant features in the large slowness range which lead to departures from unity of up to 25 per cent for slowness between 0.216 and 0.221 skm-' are associated with the low-velocity zone for S-waves. The errors here, which occur over a small range of slownesses arise from the problem of coalescing turning points discussed in Section 3.2.3. As the frequency increases the effective separation of the functional behaviour in the neighbourhood of the two turning points is increased, and so the errors introduced by our non- uniform approximation in the low-velocity zone are diminished.

We see that there are three slight departures of IRss(p, w ) I from unit modulus at slownesses around 0.177, 0.187 and 0.207skm-' which are the slownesses associated with the second-order discontinuities in the S-wave velocity distribution at 413, 391 and 170km depth. For each of these second-order discontinuities the amplitude of the departures from unity decreases with increasing frequency. The most significant effects occur at the velocity gradient change at the top of the '400 km discontinuity'. The low-frequency departures at 50s period can reach about 5 per cent but at 2.0 s period are reduced to 0.2 per cent.

These errors occur when a turning point lies just at the level of the discontinuity and except for the lowest frequencies are confined to almost a single slowness. The size of the effect is accentuated in Fig. 9 by the interpolation used to produce the plot. Errors arise at a slowness p such that the turning point in one medium lies at virtually the same depth as the 'fictitious' turning point in the extrapolation of the structure on the other side of the discontinuity. When this occurs the actual situation is closer to a two turning point case than the single turning point Airy forms assumed in the leading order approximation in each medium. For this narrow band of slownesses the interaction of internal reflections from the gradients with the second-order discontinuity become important and these are given by the higher order corrections.

A similar effect occurs as a turning point just touches an interface with a discontinuity in properties. There can be local departures of lRss I from unity of a few per cent, which in this case decay more slowly with frequency. Once again the interaction terms represented by the higher order corrections ought not to be neglected for these slownesses.

We construct theoretical seismograms by p integration in either the time-slowness domain or via a spectral response by integration at futed frequency. In each case the overall error introduced by the localized errors for slownesses at structural boundaries will be very small at moderate frequencies (> 0.05 Hz) and should amount to at most a few tenths of a per cent error in the form of the seismogram. The errors in the low-velocity zone range are much more significant but will have almost no effect on the response for waves which turn either above or below the velocity inversion.

In the leading order approximation conversion between wave types can occur only at first-order interfaces, and a sensitive test of the accuracy of this part of the approximation is provided by the relation (3.55). For the slowness range from 0.135 to 0.166 s km-' both P- and S-waves have their turning points within the structure and so the amplitude of their reflection coefficients (Rpp I, lRss I should be equal. Their phases will be very different since the P-waves are turning in the crust and S-waves below 500 km depth. We also require

over this slowness interval. Both of these checks on the coupling behaviour show that, in general, neglecting conversions at velocity gradients is a reasonable approximation. For periods less than 30 s the error in the relations (3.55) is rather less than 2 per cent for most

672 frequency/slowness points. More significant fluctuations of up to 4 per cent arise at low frequencies and for slownesses such that the S-wave turns at a second-order discontinuity whilst the P-wave is reflected from, say, the Moho. The conversion errors could be reduced by using a combination of interfaces with small jumps in parameters and gradients in the representation of the upper mantle ‘discontinuities’.

For upper mantle calculations, at least, we have shown that the use of the leading order Airy function approximations in each subregion of the velocity model gives very good results for the reflection response, when the division of the stratification is chosen to give only a limited relative change in parameters across any region. For periods less than 20 s the errors in the response may be well controlled even for conversion of wave type.

B. L. N, Kennett and M. R. Illingworth

Acknowledgment During the course of this work M. R. Illingworth was supported by a research studenthip from the Natural Environment Research Council.

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Appendix: reflection terms for the interaction matrices

A.l A B O V E A L L T U R N I N G POINTS

When both P- and S-waves are propagating within the gradient zone we have shown in (3.19) how the reflection and transmission terms for the full propagation function F'(zA t, zB -) are related to those for L'(zA, zB; Q).

In this case it makes little difference whether we calculate the rk, r i , etc., from the interaction series expansions or calculate them directly. The latter procedure will be introduced here, since in the evanescent regime it has considerable advantages. The numerical problems associated with exponentially growing terms can be avoided by direct calculation.

The interaction matrix L' is governed by the set of coupled first-order equations

a, LYZ, r;; V) = EJYZ, V) LYZ, r;, 71, (Al)

where E is a measure of the parameter gradients. The coupling terms J were defined in (2.40), but we now reorder them by up- and downgoing parts of the wavefield so that

The matrices ck, r i can then be shown to satisfy a coupled set of matrix Ricatti equations which may be used for a direct numerical solution

The leading order term in L' is the identity matrix and so for small parameter gradients we anticipate that we will have an expansion of the form

r; = erh t E2ri + . . . ) c = I + et: + e2 c: t . . . ) r i = er; t E2rL t . . . , c k = I t E c k + eZc{ t ... .

23

674 On substituting the expansions (A4) into the coupled equations (A3) we find that the first-order corrections satisfy


a,r& = J 1 2 ,

a,r: = - J z l ,

azti = - JZ2,

a,t: = J l l ,

and so depend directly on the coupling gradients. The second-order corrections depend only on the first-order set and so require a further integration

a,r i = J l l r b - r b J z 2 ,

a , ti = - JZ1r& - tb Jz2 ,

a,t: = J l l t : - r h J z l ,

a r'' = - J t' z u 21 " - t b J z l .

Higher order corrections obtained by this approach would require a further integration over the zone for each order of correction. In general therefore we would expect to solve (A3) numerically once we have found the phase matrix and constructed the coupling matrix J. The perturbation expansion does, however, have the merit of allowing us to estimate the size of the gradient correction terms.

A.2 B E L O W A L L T U R N I N G P O I N T S

The interaction matrix L(zA, zB; 77) introduced in (3.22), evaluated with Aj, Bj phase terms in an interaction series expansion, can be treated as above and its effects are most easily calculated from the differential equations (A3) for rd , t d with the appropriate form of the coupling coefficients J. The Airy phase terms appearing in j-are now Aj, Ak and Bj, Bk and the latter two terms increase exponentially with increasing depth. The differential equations will be stable if integrated in the direction of decreasing z, i.e. towards the turning point. We may alternatively make an expansion in terms of the parameter gradients via (A4), (A5), and the only large coefficient will be j2 , which depends on the products BjBk. However the contribution made to the overall propagation terms from this source is small because the propagation contributions from (3.19) more than compensate for the size of the coefficient.

A.3 T U R N I N G P O I N T P R O B L E M S

In the presence of a turning point we are interested in interaction contributions from both the propagating region above the turning point and the evanescent region below. These effects are linked by a coupling matrix N to give, for example

I = L(ZA; z,) N L - ' ( z ~ , z,) (A71

as in (3.28). Above the turning level z, we may construct either L(zA; z,) or the corresponding

coefficients tk, etc., from their respective differential equations (Al), (A3). Below this level it is advantageous to work directly with L- l (zB , z,). The inverse interaction matrix satisfies the ordinary differential equation

Seismic waves in a stratified half space - III 675 and the rb , f terms for this inverse satisfy a further set of coupled matrix Ricatti equations

a,t$ = E ( j Z Z - r t j l Z ) t $ ,

a , r t = € j z l - E r t j l l + ~ j ~ ~ r t - r t j ~ ~ r t .

We note that compared with (A3) the roles of ru and rd have essentially been reversed. The set of equations (A9) will be well-behaved numerically if we here integrate in the direction of increasing z, i.e. away from the turning point. Once again we may make a small gradient expansion in terms of E with first-order corrections given by

a,r: = j Z l r a,t: = - j l l ,

a,r& = -J ,2 , a , tb =&.

The rd , f d terms for the composite I may now be found by successive applications of (3.16) to include the main effect of the turning point and the parameter gradient corrections.

For the P-SV system we will usually have a turning point for only one wave type in the gradient zone, and then the form of the J matrix needed to construct the L matrix must be chosen to fit the context. Here again we will make a split at the turning level.

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