WAVE MOTION 6 (1984) 407--418
NORTH-HOLLAND
407
REFLECTION OPERATOR METHODS FOR ELASTIC WAVES I - I R R E G U L A R
INTERFACES A N D REGIONS
B.L.N. K E N N E T T
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW,
England.
Received 28 July, 1983, Revised 19 October 1983
A wide range of reflection and transmission problems for elastic waves can be represented formally in terms of reflection
and transmission operators. The action of these operators on the incident field is to produce the corresponding reflected and
transmitted displacement fields. For interface problems, free surface reflections and reflections from a region, the operators
can be represented in terms of a surface integral of the product of the Green's tensor and a force system which may be
determined by the solution of a single integral equation.
Introduction
In a recent paper, De Santo [1] showed that the reflection of acoustic waves by an irregular interface
could be described in terms of a single integral equation. The aim of this paper is to generalize De Santo's
ideas to the case of elastic wave propagation, and to show that by suitable use of propagation invariants
for elastic waves we can develop a powerful formal method of solving elastic wave propagation problems.
The essence of the technique is to represent the reflected and transmitted fields from an interface, or a
region, in terms of surface integrals involving the Green's tensors for the materials on the two sides with
weighting functions dependent on the incident field. The reflected and transmitted fields can then be
regarded as derived from the action of a reflection or transmission operator on the incident fields. A
similar approach may be employed for free surface reflection problems which may also be described in
terms of a reflection operator, related to a Green's tensor integral over the free surface.
We consider media with horizontal irregularities superimposed on a predominantly vertical variation
in elastic properties, a situation which arises in many seismological applications but which is also appropriate
to non-destructive testing problems. For fiat interfaces and horizontal stratification the properties of the
reflection and transmission operators are convolutional in the horizontal coordinates. With a Fourier
transform over these coordinates we recover the conventional expressions for plane wave reflection and
transmission from the transforms of the operators.
The formal development represents the reflected or transmitted wave field at an interface by a set of
sources along the surface itself. The integral equation for these sources has a singular kernel which can
be removed by adopting a set of sources just above the interface for reflection, and just below in
transmission, following the approach suggested by Kupradze [11]. Where the material surrounding an
interface or region is heterogeneous, it will normally be necessary to work with approximations to the
reflection and transmission operators. Such approximations can be constructed in a systematic way by
employing the representations in terms of surface integrals.
0165-2125/84/$3.00 (~) 1984, Elsevier Science Publishers B.V. (North-Holland)
B.L.N. Kennett / Reflection operator methods I
408
If the region from which the reflection occurs is of finite extent, the representation of the reflected field
has a very close affinity to the transition matrix method [3-6]. Indeed this expansion of the incident and
reflected fields in terms of orthogonal basis functions may be regarded as providing a convenient algorithm
for the solution of the surface ihtegral equation for the reflected field. The T-matrix can therefore be
thought of as a specialization of a reflection operator.
The operator development becomes particularly effective when multiple reflections are important: as
in the combination of reflections from two adjacent regions, or for source problems. These two cases are
considered in detail in the accompanying paper [2]. Although we introduce the idea of reflection and
transmission operators in the context of elastic wave propagation, the concept is of much wider application
and can be used for other scalar and vector wave equations.
1. A propagation invariant
The use of integral representations of the elastic wavefield to solve elastodynamic problems was discussed
in detail by Kupradze [11], with the aid of a representation theorem for uniform isotropic media. In our
treatment of reflection and transmission we will make use of a similar class of results for heterogeneous
media. In particular we will need to be able to isolate portions of the elastodynamic field with specific
character. This can be achieved by exploiting certain invariance properties of displacement fields. In order
that the structure of these results may be clearly seen we present a direct derivation from the equations
of motion.
For simplicity, we will specialize to a time harmonic situation with time dependence exp ( - i t o t ) but
will not normally represent the time variation explicitly.
Consider two elastic displacement fields u~, u2 with associated stress-tensor fields rl, r2. In Cartesian
components, the equations of motion in the absence of sources are
3jTl/j = p O ) 2 U l i ,
Oj72/j = -- p W 2 U 2 i ,
(1.1)
where
(1.2)
T/J -~" C/jkl~ kUh
and the elastic modulus tensor has the symmetries
C/jkl : Cjikl = f/jlk = Ckl/j.
We form the scalar product of each equation of motion with the alternate displacement field and then
subtract to eliminate the frequency dependent term. Then, integrating over a volume V in which the
displacements and stresses are continuous
fv d V [U2iOj"l'l/j-
OjT"2ijUli ] "~- O.
(1.3)
Now, using the divergence theorem, we find that over the bounding surface S of V
f dS
n j [ u 2 i ' / ' l / j - - r2ijUli ] = O,
(1.4)
s
since the other terms cancel because of the linear constitutive equation (1.2). In terms of the traction
B.L.N. Kennett / Reflection operator methods I
409
vector t for the surface, we can rewrite (1.4) in the more compact form
fS dS[u2itli - t2iUli]
(1.5)
O,
where t~ = njr~j.
Although we have derived (1.5) under the assumption of continuous stress and displacement fields, we
may extend the result to include regions containing material discontinuities. With conditions of welded
contact both displacement and traction will be continuous across such a discontinuity. The additional
integrals introduced on the two sides of the discontinuity in the application of the divergence theorem
will then cancel.
Consider now a quasi-stratified medium in which the predominant variation in elastic parameters is in
the z direction (Fig. 1), with surfaces $1 and $2 spanning the whole x - y plane. Depending on the nature
of the model, a convenient choice for such surfaces would be to follow material discontintuities or contours
in elastic wave-speeds. If we choose the surface S in (1.5) to include both $1 and $2 with completion
surfaces S~o at infinity, we have
f , dS[u2itli - t2iUli] = f s dS[u2itli - t2iUli],
il
(1.6)
2
provided that ui, u2 decay sufficiently rapidly as S~ is moved to infinity. Such would be the case if they
were derived from sources lying outside the region S. Since the integrals in (1.6) will vanish identically
if Ul and u2 are the same field, we see that any part of u2 which is a multiple of//1 makes no contribution
to the invariant. With suitable choice of ui we can therefore isolate desired characteristics in the field u2.
Z
f
i
J
~X
J
.S2
Fig. 1. Integration surface in a quasi-stratified medium. The surfaces S~ and S 2 would be chosen to c o n f o r m to the nature of the model.
We will also need to make use of the elastodynamic representation theorem for heterogeneous media
in terms of a suitable Green's tensor (see e.g. Burridge & Knopoff [7]). For a volume V enclosed by a
surface S the displacement field u due to a set of forces//? can be represented in terms of the surface
values of displacement and traction as
O(X)l~k(X):Ivd3~Gkq(X,~)Eq(~)i-Isd2~[akq(X,~)tq(~)--Hkq(X,~)Uq(~)],
(1.7)
where
O(x)=l,
x e V,
O(x)=O,
x~V
and we have written H ~ for the traction components of the Green's tensor on S. The character of this
representation will depend on the boundary conditions imposed on the Green's tensor.
B.L.N. Kennett / Reflection operator methods I
410
2. A single interface
We introduce the idea of reflection and transmission operators by the example of a single non-planar
interface (Fig. 2).
N~
/
/
,4
.)¢
S
/
.X
R
!
I
J
O
J j
b
ffx, l
Fig. 2. The configuration for reflection from an irregular interface f(x~). The reflected field at XR is determined by the action of
a reflection operator R on the source field emerging from the source point x s.
We take an interface J : z = f(x±), where x± represents position in a horizontal plane i.e. x± = (x, y). In
order to give a clear separation between different classes of position vector, we will use G r e e k letters to
denote position on the interface so that e.g. ~:= (~±, f(~±)). Properties corresponding to the regions
z > f ( x ± ) and z < f ( x L ) will be denoted by the labels a, b. We work in terms of the G r e e n ' s tensors G",
G b for unbounded media with the properties of these regions a or b extended indefinitely. When the
regions are homogeneous this may be easily accomplished and we have analytic forms for the G r e e n ' s
tensor components. For heterogeneous media any convenient extrapolation of the material properties
outside the original zone may be made.
Consider then a general point source at the point Xs specified by a force system
E l ( X ) = F i ~ ( x - Xs) + o j [ m i j ~ ( x - Xs) ],
(2.1)
where we have combined a simple force F with a set of couples specified by the m o m e n t tensor M [8].
This allows a variety of sources to be simulated. For example, by taking M = d i a g ( 1 , 1, 1) we have an
isotropic (explosive) source; alternative combinations of couples can be used to describe shear sources
such as earthquakes.
In an unbounded medium the radiation due to such a source at a point XR takes the form
gk ( XR, XS ) = f f jGkj( XR, XS ) + Mjlc3tGkj( XR, XS ),
(2.2)
a linear combination of G r e e n ' s tensor components and their spatial derivatives.
When we have an interface present the total displacement field at a point above the interface must also
include a contribution reflected from the interface. We therefore seek the displacement field at an arbitrary
receiver point X R in z > f(x±) in the form
U(XR) = g(XR, XS) + Rig],
(2.3)
where R is a reflection operator acting on the field g which would exist at the position of the interface in
the absence of any material discontinuity.
In the lower region z < f(x±) all contributions to the displacement field must be transmitted through
the interface and so we seek the displacement through the action of a transmission operator T
U(XR)= T[g].
(2.4)
B.L.N. Kennett/ ReflectionoperatormethodsI
411
2.1. Reflection from the interface
Guided by the approach taken by De Santo [1] for scalar waves we will look for the total displacement
field in the region above the interface in the form
Uk(XR) = gk(XR, XS) + fj d2!dG~p(XR,~)Rp(~, Xs),
(2.5)
where the as yet unknown vector R determines the reflected field. This ansatz for the displacement field
can be regarded as a form of Huyghen's principle. The secondary sources along the interface J giving
rise to the reflections are represented as a force distribution R, so that the radiation may be represented
purely in terms of the Green's tensor components.
We now apply the representation theorem (1.7) to the region Va including both source (Xs) and receiver
(XR) points. The surface will consist of the interface J and a completion surface at infinity for which the
behaviour of the Green's tensor 13a will ensure that there is no residual contribution. Thus
Uk(XR) = gk(XR, XS) + fj d271[G~q(XR, ltl)tq(n)
-- H~q(XR,
I/) Uq(1J)].
(2.6)
A comparison of (2.5) and (2.6) shows that the force system R is controlled by the displacements and
tractions on the interface.
If, however, we apply the representation theorem to the region Vb both source and receiver points are
excluded and so
0 = fj d2~l[G~q(xR, n)tq(~l)--n~q(xR,
7/)uq(n)].
(2.7)
This equation will enable us to derive an integral equation for R, by using the continuity of the displacement
and traction across the interface J. We take our ansatz (2.5) for the total displacement field in the medium
a down onto the interface J and also use the equivalent form for the traction
tq(n) =
hq(rl, Xs) + f~ degUqp(~/, [j)Rp(g, Xs),
(2.8)
where we have written h for the traction corresponding to the source radiation g. Then, substituting for
the displacement and traction in (2.7) in terms of the force distribution R we have
0 = fj d2Tl[G~q(XR, TI)hq(n, Xs)--H~q(XR, ~l)gq(TI,Xs)]
+ ],d2. ; d2,[G~(XR, 71)Hqp(*h
~ ~) --Hkq(Xg,
b
r/) Gqp(~/, ~:)]Rp(~:, Xs).
(2.9)
We may now change the order of integration in the double integral to derive an integral equation for R:
Lk(XR, XS) = fdJ d2~Kkp(XR, s~)Rp(~, Xs),
(2.10)
where
Lk(XR, XS) = fj
d2T/ [G~q(XR, 71)hq(TI,Xs)-- H~(XR,
71)ga(~l,Xs)]
B.L.N. Kennett/ ReflectionoperatormethodsI
412
and includes the dependence on the incoming radiation g. The kernel of the integral equation depends
on the dissimilarity between the Green's tensors for the materials on the two sides of the interface
Kkp(XR, ~)-'- fd 0ar/ [G~q(XR, r/)Hqp(r/, ~)--H~q(XR, r/)Gqp(r/, ~)].
(2.11)
If we take the observation point (XR) down on to the interface itself, the integral in (2.11) would be over
paths between different points in the interface with the Green's tensors corresponding to different notional
media along the path. From our discussion in Section 1 we see that the construction of L(XR, Xs) gives a
field in which all parts corresponding to radiation into the lower medium b have been removed.
This procedure will be effective even if parts of the interface lie in a geometrical shadow from the
source. In that case the action of the secondary sources will be to largely eliminate the incoming radiation
which we have supposed to be present in the region.
Since we are able to find a means of constructing the secondary source distribution R, we have also
achieved our goal of constructing a reflection operator for the interface. On comparison of (2.3) and (2.5)
we see that we can identify the action of the operator R on the source radiation with the secondary
radiation so that
Rk[g] = f j d2~ G~p(IR,
~)gp(~, Xs) ,
(2.12)
with Rp constructed from the integral equation (2.10). Although we have derived the reflection operator
(2.12) under the assumption that the incoming radiation was derived from a source at xs, the derivation
of the integral equation (2.10) for the source strength R does not depend on this assumption. It does
however require the specification of an incoming field g in (2.5) which determines the displacement at
the position of the interface in the absence of any jump in properties. We can stress this dependence on
the incoming field by rewriting (2.12) in the form
Rk[g]
= fj de~G~t,(Xg, ~)Rp(~; g).
(2.12')
The force representation R for the secondary radiation (2.5) does not provide any association of the
reflection process with particular wavetypes. In the case of uniform media on the two sides of the interface
we may split up the reflection by making an expansion of the Green's tensor. For arbitrary anisotropy a
convenient form is given by Willis [9] based on a plane wave decomposition of the delta function: the
Green's tensor components are
1 0 ~=1
~ ~,=1dS~ulr)(v)u}r)(v)[p(r~(v)]2.
Gij(x, ~:) -_ 8~r2
~(p.~)+~exp(i~op(r~lv.~l)
,
(2.13)
where ~ = x - ~:, and the u(r)(J') are the normalized eigenvectors of the tensor elastic equation corresponding
to wave slowness p(r)(p). From (2.2) a similar representation may be made for the field radiated from
the source. As a result we can work in terms of a matrix RrS(~, Xs) which describes the reflection and
interconversion of the different wavetypes as they impinge on the interface.
Once the interface is flat (z = constant) the integrals (2.10), (2.11) have convolution form and the
integral equation method is equivalent to the usual approach of working with a Fourier transform over
the horizontal coordinates to achieve a plane wave decomposition. In consequence we can recognize
B.L.N. Kennen / Reflection operator methods I
413
Rr~(~, xs) as being the convolution of the source radiation with the inverse Fourier transform of the
matrix of plane wave reflection coefficients.
2.2. Transmission through the interface
In transmission we look for a force system T which will give secondary radiation into the lower region,
so that for Zn > f(x~) the displacement field is taken as
Uk(XR)= Is
d2~: G~p(X' £) Tp(g, Xs).
(2.14)
Once again we make use of the elastodynamic representation theorem for the region Va including the
source point Xs but excluding the observation point XR, thus
0 = gk(XR, XS)+ fj
d2~: [G~q(XR,£)tq(~) --H~(XR, ~:) Uq(Se)],
(2.15)
where gk(XR, XS) is the field produced if radiation from the source has propagated throughout in medium
a. Now using the transmitted field representation (2.14) for the displacement and traction on the interface
we have
--gk(xn, Xs)= fjd2~l fsd2~[G~q(XR,,)Hbp(~l,~)-H~(XR, Ti)Gbp(ll,~)]Tp(~,Xs).
(2.16)
With a change of order of integration we can recognize an integral equation for Tp with a similar kernel
to the reflection problem but with the roles of the media a, b interchanged. Now, however, it is just the
source radiation term which appears on the left hand side of the equation
-gk(XR, XS) = Is
d2r/Kpk(XR,
~) Tp(~, Xs),
(2.17)
involving the transpose of the reflection kernel
Kpk(XR,¢S) = fs
d2n
[O~q(XR,TI)nbp(n, ¢)--H~(XR, TI)Gbp(rl, ~:)].
It should not come as a surprise that the same matrix kernel appears in both the reflection and transmission
cases. It is entirely comparable to the presence of a common denominator in all the plane wave reflection
and transmission coefficients at a flat interface.
We can now identify the action of the transmission operator for the interface with the radiation integral
(2.13) so that
Tk[g] = dJ
f d% G~p(xR,#)T.(#, xs).
(2.18)
2.3. Approximations for reflection and transmission
We have already seen that the secondary force system along the interface for reflected waves is to be
sought from the solution of the integral equation (2.10)
Lk(XR, XS) = I.I j d2~Kkp(XR' ~)Rp(~, Xs),
(2.19)
414
B.L.N. Kennett / Reflection operator methods I
where both L and K depend on G r e e n ' s tensor terms. For transmission the integral equation (2.18) is of
similar nature. In each case the left hand side depends on the field radiated by the source, and the kernel
K on the surface displacements and tractions for the G r e e n ' s tensors G", G b. For uniform media on the
two sides of the interface we may calculate the G r e e n ' s tensors and the source radiation analytically and
the accuracy with which Rp can be determined is limited only by the numerical solution of the integral
equation (2.10).
With the sources R actually on the interface J, integrable singularities arise in the expressions for L,
K (2.10, 2.11) when the integration point 7/ coincides with the position vector ~: on J. This singular
behaviour can be avoided if the secondary sources for reflection R are assumed to lie on a surface J just above J and the observation point XR is also taken on J - . The analysis follows as before, but now
the integral over ~: in (2.19) is replaced by one over J - for the secondary sources, and the integral
equation can be reduced to a set of linear algebraic equations by using a numerical quadrature scheme.
A similar approach can be used in transmission with now the secondary sources T distributed along a
surface J + just below J. The resulting integral equation akin to (2.17) has the integral over ~ taken
along J + . This cure for the singular problems has the effect that the kernels for reflection and transmission
are no longer the same. Dravinski [13] discusses the conditions in which a representation by a discrete
set of sources on surfaces J - , J + will give a good representation of the scattered fields.
When, however, an interface separates two weakly heterogeneous regions we still have the same formal
integral equation (2.10) but it will not normally be possible to obtain exact expressions for G a, G b etc.
In this case we must make do with approximations to L and K and when these are used in (2.10) the
accuracy of the reflection operator estimate will depend on the quality of the approximations to the
G r e e n ' s tensors. If the heterogeneity is severe, the representation (2.5) may still be used but the physical
interpretation of the integral contribution as a reflected field will be lost.
One of the most convenient approximations is to use asymptotic ray theory estimates of the G r e e n ' s
tensor components. This will lead to an estimate of the reflection term R which will not include all the
possible wave propagation effects. It will however be based on an integral over the entire interface J, and
so include information on the slopes and curvature of the surface. Whereas, if ray theory is used throughout,
the reflection estimate is merely that for a plane wave impinging on a flat surface with the slope of the
tangent plane at the point a ray meets the interface.
We are therefore able to make systematic approximations to the reflection operators, including as much
as possible of the full wave propagation effects. Such approximations will normally be significantly more
accurate than alternative treatments of the reflection problem based on local approximations.
32 Reflection and transmission from a region
In the preceding section we have seen how we can obtain a formal representation of the reflection and
transmission from an interface in terms of surface integrals. A similar approach may be used for a region,
provided that its internal propagation characteristics are well known.
Consider now a heterogeneous region bounded above by the surface A: z =fA(X.) and below by the
surface B: z =fB(x±). We will take the source to lie in the region a above the surface A and, as for a
single interface, try to represent the reflected and transmitted field via surface integrals of secondary force
terms. We will also denote the zone below the surface B as b.
B.L.N. Kennett/ ReflectionoperatormethodsI
415
For reflection with an observation point XR lying in ZR > fA(X±) we will look for the total displacement
field in the form
UK ( X R ) =
gk(XR, XS) 4- ffA d2~ G~p(XR, g)Rp(~, Xs),
(3.1)
Here g is the displacement radiated directly from the source (2.2) and as in our discussion of a single
interface G" is the Green's tensor for an unbounded medium with the properties of the region Va.
In transmission, where ZR <fB(x±), we will represent the displacement field as
Uk(XR) = f d2~ Gbp(xR, ~) Tp(~, Xs)
.1B
(3.2)
and again we take the G b to be the Green's tensor for an unbounded medium with the properties of the
region Vb.
In order to determine the force distribution R we start by applying the representation theorem to the
region Vb, which for reflection excludes both source and receiver points. Thus we have
0 = fR dE1?[G~q(xR' TI)tq(rl)--H~q(Xg' ~)Uq(~l)]
(3.2)
and we may now make use of the propagation invariant (1.6) to transfer this relation to the surface A.
Now we have
0 = fA d2~l[t~bq(xR' ~l)tq(rl) --fI~q(XR, ~/)Uq(~)],
(3.4)
where ~b represents the displacement field obtained by extrapolating the displacement and tractions
corresponding to the outgoing Green's tensor for the medium b from the surface B into the heterogeneous
region.
Once we have established the relation (3.4) for the surface A the calculation parallels that for the
single interface (equations 2.7-2.11). We use the ansatz (3.1) for the displacement fields on the surface
A in the integral (3.4), and are able to derive an integral equation for the force distribution R in terms
of invariant integrals of the Green's tensors and source radiation. The components of R are to be found
from the equation
Lk(XR, XS) = f d2!~Kkp(XR, ~)Rp(~, Xs),
JA
(3.5)
r
Lk(XR, XS) = JA d2~/[G~q(XR, ~?) hq( ~, Xs)-- fI~(XR, ~t)gq( ~l, XS)]
(3.6)
with
and kernel
Kkp(XR, ~) =
fa
d2"r/[GL(XR, ~)Hqp(~,~
Xs)--Hkq(XR,b
TI)Oqp(~l,XS)],
(3.7)
Provided that we exclude zones containing the source and receiver points, the two integrals (3.6) and
(3.7) are invariant and so may be evaluated over any convenient surface, with suitable extrapolation of
the fields occurring in the integrand.
416
B . L . N . K e n n e t t / Reflection operator m e t h o d s I
For the transmission problem we also m a k e use of the representation theorem, but now for the region
Va containing the source but excluding the observation point XR. Then
0 = gk(XR, XS) + fZ da~l[G'~q(XR' ~l)tq(ll)-
Hakq(XR,at/) Uq(It/)],
(3.8)
where gk(XR, XS) is the field which would be observed if both XR and Xs lay in material with the properties
a. As in the reflection case we may use the invariance properties of the integral to transfer it to the surface
B by extrapolating the G r e e n ' s tensor values G a. Thus
0 = gk(XR, XS)+
ffB
d2~ [Gkq(XR, ~ ) t q ( r / ) - / - t L ( x R , vl) Uq(r/)],
-a
(3.9)
in terms of the extrapolated field G~. With this representation at the surface B we can derive an integral
equation for the surface forces T in the form
-- gk(XR, XS) =
f d2rl I~kp(XR,~) Tp(~, Xs),
(3.10)
B
as in the case of a single interface. The kernel I( takes the form of an integral over the surface B,
Kkp(XR,~) =
"
SB
d2~7 [Gkq(XR, n)Hqp(rl,
b
-,
b
~)-- H~(XR,
"~)Gqp('O,
~)],
(3.11)
-°
but since it is an invariant can be transferred to other surfaces between A and B.
We see therefore that with suitable knowledge of the wave propagation in the region AB, we are able
to construct the secondary force distributions R and T from equations (3.5) and (3.10). This enables us
to construct the reflection (R AB) and transmission (TAB) operators for the regions A B in the form
R~BCg]=I d2gG~p(xR,~)Rp(~,Xs),
A
T~Btg]=I
d2~Gbp(xn,~)Tp(~,Xs).
(3.12)
B
With uniform media on the two sides of the region A B we can introduce representations of G r e e n ' s
tensors G a, 13b in terms of the different wave types as in (2.13). The reflection and transmission between
different wave types can then be represented by matrix force systems. For horizontal stratification the
results we have just obtained reduce, under Fourier transformation over the horizontal coordinates x~,
to those presented by Kennett [10] who also worked in terms of propagation invariants.
Although we originally introduced the expressions (3.1) and (3.2) to describe the reflected and
transmitted fields outside the region AB, once we have found the force systems R and T we can extend
the range of these representations into the region AB. This requires that we use the extrapolation of the
G r e e n ' s tensors G a and 13b and the source radiation g into the heterogeneous region.
Although we have obtained a formal representation of the reflection and transmission operators for a
region in terms of surface integrals (3.12), this will not normally prove to be a suitable numerical approach.
Frequently, it is possible to build up the response of a region from simpler elements of the structure and
this procedure is described in detail in the accompanying paper [2].
B.L.N. Kennett / Reflection operatormethods I
417
4. Free suriace reflections
In the previous section we have treated a case where elastic waves can pass through a region. If, however,
the further boundary of the zone of interest is a free surface, we can still construct a suitable reflection
operator but with different integral expressions.
Suppose therefore that the surface B is replaced by the free surface F: z = F(x±) and look for the
displacement field in reflection at a point XR in Va as
Uk(XR)
=
gk(XR, XS) q - f A d2~ G~p(XR, ~)RIp(g, Xs).
(4.1)
The boundary condition at the free surface is that of vanishing traction, and we may match this condition
by extending the definition (4.1) to the free surface by extrapolating the source radiation and Green's
tensor into the region AF. We then require
0 ~- hk(XF, XS) q-
fA
d2~/-t~p(XF'~:)glP(~:' Xs),
(4.2)
where XF is a point on the surface, and hk(xf, XS) is the traction which would be produced by the source
at the surface F in the absence of the free surface and ~ a is the traction obtained by extrapolating the
Green's tensor G a from the surface A to the free surface F. We have thus generated an integral equation
for R I and so may determine the corresponding reflection operator.
The analogue of the transmission problem in this case is the generation of the surface displacement
from a given source field. At the free surface F
Uk(XF) = ~k(XF, XS) + fA d2~ G'~p(XF,~)Rfp(g, Xs)
(4.3)
and we may deduce an integral equation for U(XF) by eliminating R I between equations (4.2) and (4.3).
We can do this by making use of the vanishing of the invariant integral (1.5) so that
f F d2~ Uk(~, XS)I'TI'~,(~,XA) = f F d2~ [gk (~, Xs)ffI~,(!;, XA)-- hk(~, Xs)t~,(~, XA)]
(4.4)
for points XA lying in the surface A, and the kernel of the integral equation is once again the traction
corresponding to the Green's tensor Ga at the surface F.
The solution of the integral equation (4.4) can be represented in terms of the transfer operator W IA
which connects the surface displacement U(XF) to the source radiation
U(XF) = wrA[g].
(4.5)
This operator plays an important role in source problems [2].
5. A finite region
We have so far considered problems involving reflections from a region which is indefinitely continued
in the horizontal coordinates x~. A similar analysis may be applied to a finite region Vb bounded by a
surface J. We take G ~, G b to correspond to the Green's tensors for unbounded media with the exterior
and interior properties respectively.
B.L.N. Kennett / Reflection operator methods I
418
The results for the scattered field outside J are entirely equivalent to the previous results for a single
interface, so that the displacement field including reflected waves takes the form
Uk(XR) = gk(XR, XS) + fl
d2g
G~kp(XR'~)Rp(~, Xs),
(5.1)
where R p ( ~ , Xs) is the solution of the integral equation (2.10).
The integral equation approach represents an alternative to the transition matrix method for elastic
waves [3-6]. In this scheme the source field g is expanded in terms of a set of regular spherical harmonic
functions, and the scattered contribution (the integral in (5.1)) in terms of outgoing harmonics. The
interior field is also expanded in terms of regular harmonics. The transition matrix T is then introduced
to relate the coefficients in the outgoing wave field to those in the source field, so that it has essentially
the role of a reflection operator. The matrix itself is determined in terms of surface integrals of the type
we have already encountered, but with the displacement and tractions for the various harmonics replacing
those for G a, G b.
For a finite region bounded above by a free surface, Dravinski [12, 13] has adopted a representation
of the field similar to (2.5) but has worked with the Green's tensor for the half space. He uses a set of
secondary sources on surfaces inside and outside the boundary, following Kupradze [11 ], and then matches
the representations on the region boundary itself to find an integral equation for the source terms. This
procedure requires that both interior and exterior sources be found at once, whereas use of the representation theorem allows the separation of the problem into two separate equations (2.10), (2.17) with the
same matrix kernel.
References
[1] J.A. De Santo, "Scattering of scalar waves from a rough interface using a single integral equation", Wave Motion 5, 125-135
(1983).
[2] B.L.N. Kennett, "Reflection operator methods for elastic waves II - Composite regions and source problems", Wave Motion
6, 419-429 (1984).
[3] P.C. Waterman, "New formulation for acoustic scattering", J. Acoust. Soc. A m . 45, 1417-1429 (1969).
[4] P.C. Waterman, "Matrix theory of elastic wave scattering", J. Acoust. Soc. Am. 60, 567-580 (1976).
[5] Y.-H. Pao and V. Varatharajulu, "Huygben's principle, radiation conditions and integral formulas for the scattering of elastic
waves", J. Acoust. Soc. A m . 59, 1361-1371 (1976).
[6] Y.-H. Pao, "The transition matrix for the scattering of acoustic waves and for elastic waves", in Modern Problems in Elastic
Wave Propagation. ed. J. Miklowitz & J. Achenbach. Wiley, New York, 123-144 (1978).
[7] R. Burridge and L. Knopoff, "Body force equivalents for seismic dislocations", Bull Seism Soc. A m . 54, 1875-1888 (1964).
[8] F. Gilbert, "The excitation of the normal modes of the earth by earthquake sources", Geophys. J. R. Astr. Soc. 22, 223-226
(1971).
[9] J.R. Willis, " A polarization approach to the scattering of elastic waves I - Scattering by a single inclusion", J. Mech. Phys.
Solids 28, 287-305 (1981).
[10] B.L.N. Kennett, Seismic Wave Propagation in Stratified Media. Cambridge University Press, (1983) section 5.2.
[11] V.D. Kupradze, "Dynamical problems in elasticity", in Progress in Solid Mechanics, 3, (1983), ed. 1.N. Sneddon & R. Hill,
North-Holland, Amsterdam.
[12] M. Dravinski, "Influence of interface depth upon strong ground motion", Bull. Seism. Soc. A m . 72, 597-614 (1982).
[13] M. Dravinski, "Scattering of plane harmonic SH waves by dipping layers of arbitrary shape", Bull. Seism. Soc. A m . 73,
1303-1319 (1983).