Geophysical Journal (1988) 92, 181-193
A ray-Kirchhoff method for body-wave calculations in
inhomogeneous media: theory
Tianfei Zhu
Tennessee Earthquake Information Center, Memphis State University, Memphis, Tennessee 38152, USA
Accepted 1987 June 5 . Received 1987 May 24; in original form 1987 February 26
SUMMARY
A ray-Kirchhoff method is developed for body-wave calculations which extends previous ray
methods to rapidly varying media. It is based on a newly derived integral solution to wave
equations which indicates that the wave field at a receiver point is given by a superposition of
ray solutions determined by the transport and extended eikonal equations. The latter is in
turn solved by an asymptotic series. In a slowly varying medium, only the leading term of this
series needs to be considered, and the extended eikonal equation reduces to the well-known
eikonal equation. Wave fields in this case can be calculated using asymptotic ray theory. For a
rapidly varying medium where velocity gradients are no longer small, the higher-order terms
of the series must not be disregarded. These frequency-dependent higher-order terms
represent the scattering effect of velocity gradients and provide a basis for avoiding caustics.
The new method also includes a procedure for estimating the errors introduced by truncating
higher-order terms from the asymptotic series. In particular, validity conditions for the
ray-Kirchhoff method in elastic media are formulated which indicate that the new method is
less restrictive than some previous ray methods such as the Gaussian-beam technique. For
implementation, a perturbation scheme is developed for solving the ray and transport
equations. In addition to computing the higher-order terms of the asymptotic series, this
scheme avoids most of the ray tracing required for computing wave fields in median with
weak lateral variations. Using this scheme, the ray-Kirchhoff method is extended to anelastic
media. Approaches for removing singularities on an integral surface used in the ray-Kirchhoff
method are proposed which not only prevent the infinite amplitude at a caustic, but also
predict the phase shift caused by this singularity.
Key words: Seismic body wave, ray-Kirchhoff method, perturbation, caustic
1 INTRODUCTION
Asymptotic ray theory (ART) provides a useful tool for
computing wave fields in laterally inhomogeneous media
(e.g. tervenf, Molotkov & PSenlSik 1977). This method has
the advantage of simplicity in that a single ray path is used
to determine the amplitude of a particular arrival. However,
ART cannot adequately model certain types of signals such
as diffractions and caustics that are vital to seismic
interpretation. Modified ray methods have recently been
developed to exploit the simplicity of the ray approach while
avoiding singularities of ART. Among these are the
Gaussian-beam technique (Cervenf, Popov & PSenZSik
1982), the Maslov method (Chapman & Drummond 1982),
and techniques based on extended Kirchhoff integrals (e.g.
Haddon & Buchen 1981; Frazer & Sinton 1984).
The Kirchhoff integral has long been applied to various
wave propagation phenomena in homogeneous media. For
example, it has been used in exploration seismology for both
* Now at Geophysics Division, Department of Physics, University
of Toronto, Toronto, Ontario M5S 1A7, Canada.
seismic modeling (e.g. Hilterman 1970, 1975; Trorey 1970,
1977) and migration (e.g. French 1974, 1975; Schneider
1978; Berkhout 1982). Scott & Helmberger (1983) have
used the Kirchhoff integral to model reflections from
mountain topography. More recently, the Kirchhoff integral
was extended to inhomogeneous media by Haddon &
Buchen (1981) and Frazer & Sinton (1984). In their
Kirchhoff techniques, rays are traced from both source and
receiver to an intermediate surface, and the wave field at the
receiver is given by an integral over the surface. The kernel
of this integral is constructed using ART. The choice of the
integral surface depends upon the problem considered. For
example, it can be an arbitrary surface between a source and
receivers (e.g. Haddon & Buchen 1981) or an interface of a
layered medium (Frazer & Sen 1985).
Similar to other ray methods, the previous Kirchhoff
techniques are valid only for slowly varying elastic media,
and they fail to model scattered waves such as those
partially reflected from a region of high velocity-gradient
(Chapman 1985). In addition, the use of ART to construct
the kernel of an extended Kirchhoff integral causes two
181
T. Zhu
182
major problems in the implementation of a Kirchhoff
method. One is that the computation of wave fields in
laterally inhomogeneous media by a Kirchhoff technique
usually requires rays to be traced from each receiver point,
and could be very timeconsuming when a large amount of
r w i v e r points is involved. Another problem occurs when
rays have caustics on the surface of integration. At these
points, the Kirchhoff techniques break down. Although a
number of approaches have been proposed to overcome this
problem, they seem to be unsatisfactory for various reasons
(Frazer & Sen 1985).
The primary purpose of this paper is to describe a new
Kirchhoff method which extends the previous Kirchhoff
techniques to rapidly varying media where the scattered
waves generated by the velocity gradients are no longer
negligible as in slowly varying media. New techniques are
also proposed to solve the problems encountered in the
implementation of this as well as previous Kirchhoff
techniques. Analytic examples are given in this paper to
illustrate the application of the new method. Numerical
examples will appear in a separate paper.
2 RAY-KIRCHHOFF FORMULA AND ITS
EVALUATION
The ray approach used here for deriving the integral
solution of scalar wave equations differs from previous ones
such as ART and the ray series method (Karal & Keller
1959). We start with an outline of this new approach.
Consider a scalar wave-equation in an inhomogeneous
medium with velocity V ( x )
(2.1)
is the wave field at a field point x. Its equivalent
where
form in the frequency domain is
PY
+ w 2 v - 2 Y = 0,
(2.2)
where Y is the Fourier transform of I#. Substituting a
harmonic solution Y = A(x, q,w ) exp [ i w t ( x ,x,, o)] into
(2.2) and equating the real and imaginary parts yields
(VZ)2=-
Figure 1. A ray from source point x, to field point x. The take-off
angles B0 and a,,arc measured from the z direction and x-z plane,
respectively. Also shown in the figure are the integration volume T
and surface S.
defined by (cerveng
et
al. 1977)
The solution of equation (2.5) is (e.g. Smirnov 1964)
where x’ is an arbitrary point between x, and x along the
ray. For a point source, wavefronts near the source are
small spheres. Let x’ be a point on such a sphere, then J(x‘)
in (2.7) becomes (r’)2sin Po with r’ = Ix’ -%I.
Substituting
this into (2.7) and letting r‘ approach zero gives
A = YAO,
(2.8)
where Ao=A0(Po, (yo) is the limit of r‘A(x’) as r‘
approaches zero; it represents the directivity of the source.
In the following, we consider only a symmetric point source,
and in this case A , becomes a constant determined by
source strength. The divergence coefficient y in (2.8) is
defined by
1 VzA
+V 02A
and
V2Z+--
/
2VA * VZ
- 0,
A
where x, is a source point, and A(x, q,, w ) and t ( x , q,, w )
are the amplitude and traveltime, respectively. Sometimes
the latter is also referred to as the phase function. Equation
(2.4), known as the transport equation, can be solved using
ray coordinates (s, Po, ao).Here Po and a. are the take-off
angles of a ray (Fig. l), and s is the arclength along the ray.
Using Smirnov’s lemma (Smirnov 1964, p. 442), (2.4) can be
rewritten in the ray coordinate system as
2 dA
_I _d ( J P ) -= 0,
JP ds
Ads
+
where P = 1 VrtJ is the phase slowness, and J is the Jacobian
where Po is the phase slowness at q,and E = (J/sin ,!30)1’2
has the dimension of distance and becomes r = Ix - x,,l in a
homogeneous medium. Substituting (2.8) into (2.3) yields
the extended eikonal equation
(Vr)”=
v-2+w-’(v*y/y).
(2.10)
Equations (2.9) and (2.10) determine the ray fields
generated by a unit symmetric point source, and they are
the basic equations for the ray approach used in this study.
This approach differs from previous ones in that equation
(2.10) contains a frequency-dependent term, w-’( V z y / y ) ,
which, as will be seen in the next section, represents the
scattering from velocity gradients.
We now derive the solution of wave equation (2.2) in an
unbounded medium using Green’s theorem. Let T be a
volume enclosed by a surface S with the outward normal n
(Fig. 1). Then, for any two functions G and Y with
Ray-Kirchhoff method in inhomogeneous media
continuous first and second derivatives within and on S it
follows from Green’s theorem that
jj/(YV2G
- G V 2Y)dT =
T
/I(
YVG - G G Y ) . n ds.
183
where E , is determined by the zero-order slowness po, and
the derivatives are evaluated at po. Substituting (2.16) and
(2.17) into (2.15), we obtain the following recursive
formulae for p2,,:
S
(2.
For our purpose, Y is chosen here to be the solution of (2.2)
while G ( x , q l , w ) is the Green’s function defined as the
wave field generated by a point source at xg. Based on the
above results, the Green’s function can be constructed as
(2.3 2)
G(x, %, w ) = y exp [ i w t ( x , xat w ) ] ,
where y and t are determined by (2.9) and (2.10).
Substituting (2.12) into (2.11) and after some intermediary
steps (see Appendix), we obtain the solution of (2.2):
Po = 1/v
P- = [3(@0)’ - 2Pov2P01/(8P:) i-L/(2Po)
P4= [-IOP;P;
f
(2. 18)
3vPo ’ vP2 - v2(PoP2)l/(4Pi)
+ ( N + P2LM)l(2Po)
...
where
and
(2.13)
where d r / d n = P cos 8 and 8 is the angle between n and the
ray. For an inhomogeneous wave equation, V2Y +
w ~ V - ~
= @,
Y an additional term
-
,(/j@y exp ( i w t ) dTl
4n
(2.14)
T
should be included in (2.13) to represent the wave field
resulting from the source potential @. Solution (2.13) is an
exact solution of wave equation (2.2) in unbounded
inhomogeneous media and has the same physical interpretation as the Kirchhoff integral in homogeneous media (Baker
& Copson 1939). On the other hand, it is clear from the
above discussion that (2.13) represents a superposition of
ray solutions determined by (2.9) and (2.10). For these
reasons, we refer to (2.13) as the ray-Kirchhoff formula, and
to the method based on this formula and its approximations
for computing scalar wave fields as the ray-Kirchhoff
method. The above procedure can be extended to obtain
integral solutions for more general wave equations such as
the generalized wave-equation considered by Smirnov
(1964, p. 449).
To evaluate integral (2.13), one must first determine y and t
from equations (2.9) and (2.10). In general, it is difficult to
solve (2.9) and (2.10) exactly and here we seek approximate
solutions. Inserting (2.9) into (2.10) and noting that P = I VtI
yields
p2 = v-2
f
o--2
+-
PE
In general, for an inhomogeneous medium there exists a
frequency wd below which series (2.16) diverges and the
asymptotic method described here becomes invalid. In some
cases, as shown by an example in section 7, this divergence
frequency is equivalent to the cutoff frequency defined by
Ben-Menahem & Beydoun (1985).
In slowly varying media where velocity gradients are small,
only the leading term in series (2.16) needs to be
considered, and (2.10) reduces to the well-known eikonal
equation
( vty = 1/v2
(2.19)
The corresponding divergence coefficient is given by
(2.20)
where V, is the velocity at x,. The ray theory based on
(2.19) and (2.20) is known as asymptotic ray theory (ART).
In a rapidly varying medium where velocity gradients are
no longer small, higher-order terms must be included in a
calculation. For example, when both the zero- and
second-order terms are retained, the extended eikonal
equation becomes
+ EV2(i)].
(2.15)
We solve (2.15) by assuming that the phase slowness P has
an expansion of the form
-
c w-2np2n,
n
P=
(2.16)
n =O
where coefficients pZnare functions of x that are to be
determined. The corresponding asymptotic expansion for E
is
E = E ( x , P) = E(x,po + rY2p2+ w-4p4 f . ‘
A)
dE
= E , + w-’p,-+
dP
In the case where E, equals zero, E , in (2.21) should be
replaced by the coefficient of the second term of series
(2.17), p,(dE/dP). The errors introduced by the approximations made in (2.19) and (2.21) can be estimated by
a procedure described in the next section.
Similar to ART, equations (2.9) and (2.21) can be solved
by ray tracing (e.g. Cerven-J et al. 1977). As pointed out
earlier, Eo in (2.21) is determined by equations (2.19) and
(2.20). In practice, it may be more efficient to approximate
Eo at each ray-tracing step with the one calculated using the
results from the last ray-tracing step. Such an approximation
184
T. Zhu
is valid as long as Eo is a piecewise smooth function and the
step size used in the ray tracing is sufficiently small. As will
be shown in the next section, for a far-field point, the
contribution of E,V2(1/Eo) in (2.21) to the wave field is
negligible, and (2.21) can be reduced to a more computable
form
The above procedure may, at least in principle, be extended
to the cases where more higher-order terms of (2.16) are
retained. The perturbation terms of series (2.17) such as
p2( d E / d P ) can be computed by a perturbation scheme
described in section 4.
Before leaving this section, it is worth noting that the
above results indicate that singularities of ART in the time
domain are caused by neglecting higher-order terms of
(2.16). It is well known that zeros of Jacobian exist at
caustics of rays which will cause (2.13) to become singular if
such zeros occur on the surface of integration. In the case
where rays are determined by (2.9) and (2.10), however, a
caustic is isolated at a single frequency component due to
the frequency-dependence of the phase velocity in (2.10) so
that the wave field from this point is integrable in the
frequency domain, i.e. regular in the time domain. This
becomes more obvious if one notes that the inverse Fourier
transform of (2.13) is an exact solution of (2.1) which should
be regular everywhere. On the contrary, in the case of
ART, the eikonal equation contains only the frequencyindependent zero-order term of series (2.16) so that a
caustic will persist through all frequency components,
resulting in a singularity in the time domain. Therefore,
near a caustic, the higher-order terms of (2.16), which
represent the scattering from velocity gradients, becomes
significant and should be included in a computation in order
to avoid the singularity. In the following, we will assume
that ray fields are regular everywhere. Techniques for
avoiding singularities are presented in section 6.
note in the Appendix), yields another solution of (2.2)
cos 6
V
dY
x exp ( i w t )dS,
dn
111
+ 45d
-
y V 2 y exp ( i w t ) dTo.
T
Solution (3.2) is the Fourier transform of Sobolev’s formula
which was originally obtained by Sobolev and Smirnov (see
Smirnov 1964, p. 441) using a somewhat more involved
approach. The surface integral in (3.2) has the same form as
(2.13) except that t and y are now determined by (2.19) and
(2.20) instead of (2.9) and (2.10). Thus this surface integral
is the approximate ray-Kirchhoff formula for slowly varying
media while the volume integral in (3.2) is the error term we
seek.
Physically, the volume integral in (3.2) represents the
scattered waves generated by the interaction between the
incident waves and the velocity gradients of the medium as
V 2 y depends on the derivatives of velocities [see (2.10) and
(2.15)]. In fact, if we replace the y in (2.10) with the one
determined by (2.20) and regard the second term of (2.10)
as a perturbation to the first term, then it can be shown by
the Born approximation approach (e.g. Bleistein 1984) that
the integral representation for the scattered waves due to
this perturbation has exactly the same form as the volume
integral in (3.2). Thus we see that the scattering from
velocity gradients can be modelled by two equivalent
approaches: either by evaluating the volume integral in (3.2)
using ART, or, as discussed in this study, by calculating the
surface integral (2.13) using the ray solutions determined by
(2.9) and (2.10). The former approach involves a costly
solution of an integral equation while the latter, in which the
scattered waves from inhomogeneities are absorbed by the
ray solutions, is more efficient and has the advantage of
physical clarity.
Based on order-of-magnitude considerations, it is possible
to obtain an explicit condition under which the error term in
(3.2) can be neglected. Consider the geometry shown in Fig.
2 where a point source (x,) is surrounded by two spherical
3 TRUNCATION ERROR A N D R A Y KIRCHHOFF FORMULAE I N SLOWLY
VARYING ELASTIC MEDIA
The errors in wave fields resulting from the truncation of
series (2.16) must be analysed. We first illustrate a
procedure for such analysis by considering slowly varying
media, and then generalize it to rapidly varying media. In
particular, we formulate the validity conditions for the
ray-Kirchhoff method in elastic media. These conditions
provide a basis for comparing the new method with some
other ray methods whose limitations have been examined by
Ben-Menahem & Beydoun (1985).
In a slowly varying medium, integral (2.13) is evaluated
by replacing t and y with those determined by (2.19) and
(2.20). To estimate the resultant error introduced by this
zero-order approximation, we use t and y defined by (2.19)
and (2.20) to construct another function G for (2.11)
G(x, xg) = Y exp [ i w t ( x , %)I,
(3.1)
which, after calculations similar to those for (2.13) (see the
S4
Figure 2. Integration volume, T, and surfaces, S, and S,, used for
error analysis. S, is a small sphere while S, extends to infinity. L
represents the portion of Slilluminated by a source at x.
Ray -Kirchhoff method in inhomogeneous media
surfaces, S1and S,. The latter extends to infinity so that the
contribution from this surface can be ignored. This
geometry is used here only for convenience, and the
conclusion drawn from the following analysis is independent
of this particular choice. The error term (Zl)can be written
as
Construct a smooth function M ( x ) 2 (PIwhich vanishes at
infinity and has much smaller spatial derivatives than y so
that (3.3) can be rewritten as
(3.4)
where MI is a typical value of M ( x ) on the small sphere S,.
In writing (3.4) we have assumed that V2y does not change
its sign on I”. In the case where this assumption is invalid,
the same result can be obtained by dividing T into
subregions, each maintaining a single sign for V’y, then
performing the above operation on each subregion.
Substitution of (2.20) in (3.4) yields
For a far-field point, the second term in the integral is
negligibly small compared to the first term so that
(3.5)
where A is the wavelength and 6V is the variation of V over
a wavelength. In ( 3 4 , we have replaced the integral area S,
with the area L (Fig. 2) because only the secondary sources
on this portion of S, contribute significantly to the wave field
at x. The magnitude of the second term in the surface
integral of (3.2) can be estimated as
-2nM, I I I y b c o s OdSol, (3.6)
L
where -n/2c: 01n/2
(Fig. 2) and Oscos 6 5 1 .
Comparing (3.5) and (3.6), we see immediately that the
condition for the error term, Z l , to be negligible in a far field
is
ISV/V(<< 1.
and 6p, meet the conditions:
In this case, q in (2.1) represents the displacement potential
for P- or S-waves and V(x) is the corresponding P- or
S-wave velocity. A similar mode decoupling condition was
also given by Ben-Menahem & Beydoun (1985). Clearly,
condition (3.7) can be derived from (3.8), indicating that the
approximate ray-Kirchhoff formula for slowly varying media
[i.e. the surface integral in (3.2)] can be used to compute the
decoupled P- and S-waves in an elastic medium where (3.8)
holds. We therefore conclude that the ray-Kirchhoff method
is valid in an elastic medium provided that (a) the mode
decoupling condition (3.8) is met by this medium, and (b)
the receiver point is in far field. A comparison of these
validity conditions with those given by Ben-Menahem &
Beydoun (1985) for other ray methods indicates that the
ray-Kirchhoff method is less restrictive than some of
previous ray methods such as the Gaussian-beam technique.
From (3.2) and (3.4) we see that the last term in the
surface integral in (3.2) has the same order of magnitude as
the error term. Therefore, the approximate ray-Kirchhoff
formula for slowly varying elastic media can be simplified to
Here we have transformed the solution back to the time
domain because y and t in (3.2) are independent of
frequency. The square brackets in (3.9) indicate the
retarded value. In the case where the integral surface S is a
plane, (3.9) is further simplified to
(3.10)
Formulae (3.9) and (3.10) are similar but not identical to
those used by Haddon & Buchen (1981) and Frazer &
Sinton (1984) since the divergence coefficients in their
formulae were derived by the ray series method and involve
density explicitly.
It is not difficult to extend the above procedure to obtain
error terms for the higher-order approximation. For
example, in the case of (2.22), one can use T and y defined
by (2.9) and (2.22) to construct a function G similar to
(2.12). Substituting this into (2.11) leads to an integral
solution with a volume integral (i.e. the error term) of the
form
(3.7)
Thus we have obtained the far-field validity condition for
the ray-Kirchhoff method in a slowly varying medium.
As slowly varying elastic media are most frequently met in
seismic phenomena (Fu 1947), it is useful to relate condition
(3.7) to the mode decoupling condition for elastic media.
Zhu & Liu (1982) have shown that for elastic medium with
density p and Lam6 parameters A and p, the equation of
motion can be approximated by a set of decoupled scalar
wave equations of the same form as those in (2.1) and (2.2)
if the variations of A, p, and p over a wavelength, 61, 6p,
185
(3.11)
’ ( i w t ) . Similar to (3.4), it
where m = Y [ ( p o ( ~ ) / p 0 ( x ) ] ”exp
can be verified that the integral l ~ ~ m V Z ( l / EdT,
o ) in (3.11)
T
is equivalent to ll[O(E0’)] ds,. Thus, in the far field, the
first term in (3.11j is negligible, and the error term becomes
O(o-‘).In general, wave fields calculated by truncating the
terms higher than nth order from series (2.16) will involve,
in the far field, an error term O ( w P n ) .
186
T. Zhu
4 A PERTURBATION SCHEME FOR
COMPUTING THE R A Y A N D TRANSPORT
EQUATIONS
where q ( s ) is the unperturbed ray. The initial condition of
ray x(s) is
We present in this section a perturbation scheme for
computing the ray and transport equations. This scheme
provides a useful means for computing the higher-order
terms of series (2.17) and wave fields in various media. For
clarity, we develop this scheme here by considering the
propagation of seismic rays through a medium with weak
lateral variations. The slowness of this medium can be
expressed as
x(0) = 0,
where E is a small parameter and f ( x ) = O(1). While the
reference velocity c(z) varies only in one direction, the
perturbation term ~ f ( x ) is a 3-D function used to
accommodate the weak lateral velocity variations. We will
refer to the media with V(x) and c(z) as ‘perturbed’ and
‘unperturbed’, respectively. Perturbation methods based on
velocity model (4.1) have been used in inverse scattering
(e.g. Cohen & Hagin 1985) and seismic migration (Carter &
Frazer 1984). Their results show that these methods work
well even when the ‘small’ perturbation assumption is
violated.
Now we wish to solve the ray and transport equations
associated with velocity model (4.1). The basic approach
used here is similar to those of Keller (1963) and Moore
(1980) who solved the ray and transport equations for the
case of a constant reference velocity, and some of the
following formulae are essentially an extension of their
results to the cases of variable reference velocities.
The ray equation for the perturbed medium is (e.g. Aki &
Richards 1980, p. 92)
(q
G5
d V(x)h
=
+ EX&) + O ( E 2 ) ,
(4.4)
ds
Here we have assumed that rays x(s) and xg(s) start from
the same point (0 in Fig. 3) and have the same take-off
angles &, and a,. In equation (4.3) the pathlength s is taken
to be the independent variable, and I E X , ~ represents the
distance from the unperturbed ray terminus, x0(L), to the
perturbed ray terminus, x(L), where L is the total
path-length of the rays. Note that points x,,(L) and x ( L )
have the same ray coordinates (15,Po, a,).
First, setting E = 0 in equations (4.1) to (4.4) gives
(4.5)
This is the ray equation for the unperturbed medium. The
unit vectors of the corresponding ray coordinates are
defined by
&,/as
dx,/3/3,
= e,,
= h,e,,
dx,/aa, = h2e2.
(4.6)
where h , = ldxo/dPol and h, = Jdx,,/da,J are scale factors.
Next, differentiating equations (4.1) to (4.4) with respect
to E and setting E = 0 yields
1 VfK(x,,),
X,(O)
=o,
-(0)
dx,
=o,
ds
-
V[l/V(x)].
To find the perturbed ray x, we follow Keller (1963) and
Moore (1980) and express x as
x(s) = q ( s )
dx
-(0)= e,(O).
The operator Vf = V- eses V removes the component of 0
along the ray and is known as the transverse-gradient
operator. In slowly varying medium the second term on the
right side of (4.8)is much smaller than the first one and may
be disregarded so that
(4.3)
(4.9)
integrating (4.9) twice along ray x,, gives
X,(L) = r0c ( z ” )
I0” ~Y ( x o ) / c ( z o ~l
do.
(4.10)
Thus x 1 measures the cumulative effects of Vy(x,,) which in
turn represents the deflecting strength of the first-order
perturbations in the velocity field.
The time taken for energy to propagate from 0 to x(L)
along the perturbed ray is
I
L
tIx(L)l =
{P + N x ) l / c ( z ) )ds.
Taylor-expansion of f(x) and
unperturbed ray xo gives
Figure 3. A schematic diagram showing perturbed ray x,
unperturbed ray q,and integral surface S. Dashed line indicates
the perturbed ray between 0 and q ( L ) .
C(Z)
about their values on the
t[x(J5)1= tOl%(L)l+ -1[x,,(L)I + 0 ( E 2 ) >
where to= J-6&/c(zo) and
tl = ~~[f(x,J/c(z,)]ds
(4.11)
are the
Ray-Kirchhoff method in inhomogeneous media
traveltime for the unperturbed ray and first-order correction
due to the velocity perturbation, respectively.
Substituting (4.3) into (2.6) yields the expression for the
Jacobian at x(L):
(4.12)
where JO[%(L)]is the Jacobian at %(L) in the unperturbed
medium. The perturbation term J1[q(L)] is given by
(4.13)
as the first-order components perpendicular to e, contribute
only term O(c2) (Moore 1980). For a stratified reference
velocity, (4.13) can be simplified by using the orthogonal ray
coordinates defined in (4.6):
(4.14)
The above expressions for the Jacobian and traveltime are
associated with the perturbed ray from 0 to x(L). To
evaluate a ray-Kirchhoff integral, however, we must find the
expressions for the Jacobian, the traveltime and its
derivative d t / d n , and the takeoff angle /3; associated with
the perturbed ray from 0 to %(L) (dashed lines in Fig. 3)
because the end point of this ray is on the integral surface.
This can be accomplished by Taylor expansion. The Taylor
expansion of 6; about its value on ray x is
Bb[%(L)I = B
a a ) - EXl(L)I + O(E2)
- E X , - V&[X(L)] + O(E2).
= &[X(L)]
As specified earlier, #?;t[x(L)]
= flO[q(L)] so that
Bb[%(L)I = /30[%(L)I+ EW30[%(L)l,
(4.15)
-
where DBO[q(L)]= -xl Vfi,[%(L)] is the first-order
correction.
Similarly, the traveltime for the perturbed ray 0 - % ( L )
has the expansion
Z[%(L)] = t.[x(L)] - EX1 * Vt[x(L)] + O(E2).
Substitution of (4.11) into this expression yields
T[%G)l= to[%(L)I + Etl[%(L)I - -1
It
- VTo[%(L)I + O(E2).
follows from Fermat’s principle that the term
x1 Vto[q,(L)] equals zero (e.g. Bishop et al. 1985) so that
-
t[%(L)l= to[%(L)I + ETI[XO(L)I + O(E2),
(4.16)
which is the same as equation (4.11). The derivative d t / d n
is computed by
a t / a n = at,/dn
+ E d t , / d n =-[1+1
c(4
~f(q,)]cos 8, (4.17)
where 8 is the angle between n and the unperturbed ray %.
In writing (4.17) we have used the relation d t , / d n =
(dzl/6’s)(ds/dn)= [f(~,,)/c(z,)] cos 8.
Finally, using Taylor expansion, the Jacobian at %(L) in
the perturbed medium can be written as
J[%(L)I = JO[%(L)I+ EDJO[%(L)I + O(E2),
(4.18)
187
where
DJo[Xo(L)I = JI[XO(L)I - x1 * VJO[%(L)I.
(4.19)
In (4.18), JO[q(L)] is the Jacobian associated with the
unperturbed medium while D J O [ q ( L ) ]represents the
first-order perturbation. As indicated by (4.19), this
perturbation term consists of two parts: the change caused
by the velocity perturbation at a fixed point in the ray
coordinate system, Jz[q(L)], and the change due to the
variations of the ray coordinates of that point,
x1 VJo[q(L)]. Inserting (4.1), (4.15), and (4.18) into (2.9)
yields the divergence coefficient at % ( L ) in the perturbed
medium (omitting the argument L )
-
where c ( 0 ) a n d f ( 0 ) are evaluated at point 0.
Thus, we have found the expressions for all quantities
needed for evaluating a ray-Kirchhoff integral in a medium
with weak lateral variations. To compute these quantities,
and hence the wave fields, only one set of rays needs to be
traced for the entire computation. It is interesting to note
that there is a class of 1-D velocity functions for which the
analytic solutions to the eikonal equation have long been
known (e.g. Fu 1947; Kaufman 1953). For these velocity
functions, as shown in section 7, analytic expressions for the
corresponding Jacobians may also be obtained. Thus, using
these velocity functions as the reference velocities in (4. l ) , it
is possible to evaluate formulae (4.10) to (4.20), and hence
wavefields in media with weak lateral variations, without
numerical ray tracing (Zhu 1987).
In a rapidly varying medium, the second term on the right
side of (4.8) is no longer negligible, and f(%) in (4.10),
(4.11), and (4.17) must be replaced by K(%). In this case,
similar to equation (2.21), integral (4.10) can be evaluated
by substituting x1 in K(%) with the one obtained in the last
integration step.
In the above derivation, we have assumed a stratified
reference velocity. The resulting formulae, however, are
equally applicable to the cases of 2-D and 3-D reference
velocities except that for a 3-D reference velocity, the ray
coordinates may not be orthogonal, and the J,[%(L)] in
(4.12) should be calculated by (4.13) instead of (4.14). Thus
the above formulae can be used to compute the
second-order term of asymptotic series (2.17),
~ - ~ p , d E / d P .In this case, the corresponding reference
velocity is l/po which may be 2-D or 3-D. In a similar
manner, one can also compute other perturbation terms of
(2.17) by including in velocity model (4.1) the corresponding
higher-order terms of series (2.16), although the computation will become increasingly complicated as more terms of
(2.16) are involved.
5 EXTENSION OF THE RAY-KIRCHHOFF
METHOD TO ANELASTIC M E D I A
Using the results of the last section, the ray-Kirchhoff
method can be extended to anelastic media. In such a
medium, the velocity becomes complex and may be formally
written as
V ( x ,w ) = V,(x) + V,(X,w )
+ iV,(X, w ) ,
(5.1)
188
T. Zhu
where V,, zll, and u2 are real (e.g. Kennett 1983). The
elastic velocity V, is frequency-independent while u 1 and u2
are not. Accordingly, the eikonal and transport equations
and the phase function also become complex. The phase
function is written as
t ( x , w ) =R(x, w )
+ iZ(x, w ) ,
(5.2)
where R ( x , w ) is the traveltime while Z(x, w ) represents the
cumulative energy-dissipation along the propagation path.
In principle, R ( x , w ) and Z(x, w ) can be determined by
simultaneously solving two equations obtained by separating
the real and imaginary parts of the complex eikonal
equation. In practice, however, the solutions are difficult to
obtain because of the coupling between the real and
imaginary parts of the phase function. It is even more
difficult to solve the complex transport equation as complex
ray coordinates are not involved. This perhaps explains the
lack of literature for rigorous treatment of seismic modelling
in inhomogeneous anelastic media with ray methods,
particularly for those with 3-D complex velocity functions.
Simplifying assumptions have been used by Frazer & Sen
(1985) to incorporate the attenuating effect of an anelastic
medium into their Kirchhoff-Helmholtz method.
Very often, however, the frequency-dependent part of the
complex velocity (5.1) is small compared to V, (e.g. Kennett
1983) so that we may solve the complex eikonal and
transport equations by the perturbation scheme described in
the last section. To do this, we set E = V,,/V,, where
V,, = max (V,) and V,, = max lvl iv2[. Then the eikonal
equation for velocity function (5.1) becomes
+
(5.3)
where f ( x , w ) = -Vom(v, + iv2)/(VoVlm). The slowness
inside the brace is identical to that in (4.1) except that the
perturbation term, @ ( x , w ) , is now complex. Using the
definitions of to, t,, J,, DJ,, Po, and DOo given in section
4, the complex phase function and divergence coefficient for
a slowly varying anelastic medium can be written as
t(%, w ) = To(%)
+ Etl(%,
w ) + O(E2),
(5.4)
and
V(%> w ) =
terms in (5.6) can be evaluated by integration along the rays
determined by the elastic velocity V,(x); hence, no extra ray
tracing is needed. From (5.6) we see that in an anelastic
medium, waveforms are affected not only by the complex
phase function (5.2) but also by the complex divergence
coefficient (5.5). The latter effect has often been neglected
or modelled using somewhat ad hoc assumptions in previous
ray treatments.
6 TECHNIQUES FOR AVOIDING
SINGULARITIES OF A R T ON A N
INTEGRAL SURFACE
In a 3-D inhomogeneous medium both caustics and foci
(also called the second-order caustics) can occur, and they
correspond to the first- and second-order zeros of the
Jacobian J ( x ) , respectively. Geometrically, the crosssectional area of a ray tube is reduced to a line at a caustic,
and reduced to a point at a focus (Chapman & Drummond
1982). In both cases, ART predicts infinite amplitudes, and,
in theory, the approximate ray-Kirchhoff formulae given in
section 3 break down when these singularities are located on
the surface of integration. Several approaches have been
proposed to overcome these problems. For example,
Haddon & Buchen (1981) suggested choosing an appropriate integral surface(s) between source and receiver on which
no singularities exist. This approach may be costly because it
may involve more than one integral surface which is chosen
empirically. A simple approach was used by Frazer & Sinton
(1984) who treated the singularities as integrable and
integrated over them. This approach, however, causes
spurious arrivals, and may lead to unreliable amplitudes.
Other approaches, as reviewed by Frazer & Sen (1985), are
also unsatisfactory in various aspects. New techniques are
presented in this section in an attempt to overcome some of
these difficulties. We will limit ourself to the caustics caused
by velocity gradients, and the following formulae may not
be applicable to a caustic associated with the reflection and
transmission from an interface.
As observed in section 2, singularities of ART in the time
domain can be avoided by including higher-order terms of
(2.16) in a computation. We thus consider the following
extended eikonal equation near a caustic
(Vt)’ = VO’
+ Hop,,
(6.1)
where V,(x) is the velocity of a slowly varying medium, and
In equations (5.4) and (5.5), to(%),
J0(%), and Po(%),
determined by V,, are real and frequency-independent ,
while the perturbation terms E Z ~ , EDJ,, and ED/^,(%),
determined by u1 iv,, are complex and frequencydependent. Inserting (5.4) and (5.5) into integral solution
(2.13) and neglecting the third term in that integral yields
+
x exp (ZwR - w l ) dS,,
+
(5.6)
where R = to(%)Re [ E T ~ ( % , w ) ] and Z = Im letl(%, w ) ] .
Here we have assumed x is a far-field point. Equation (5.6)
provides a simple yet rigorous formula for seismic modelling
in slowly varying anelastic media. The frequency-dependent
H=--
I‘
1 W,
V2V,
- +-2v0
4 v,
(
Wo-VEl
+E1V2(i).
V,E,
(6.2)
Equation (6.1) is equivalent to (2.21) except that E , in
(2.21) is now replaced by E l = E , + ~ - ~ p , ( d E / d in
P )order
to prevent the last two terms in (6.2) from becoming infinity
at a caustic. In a slowly varying medium the second term in
(6.1) is, in general, small so that we may compute the wave
fields using the perturbation scheme. We therefore rewrite
the phase velocity as
l l v ( x ) = 11+ E f ( X ) l / V , ( X ) ,
(6.3)
where f ( x ) = i ( V s ) and E = w-,. Assuming that the
integral surface is a plane, it is not difficult to see from
sections 3 and 4 that the far-field solution of (2.2) now takes
Ray-KirchhofS method in inhomogeneous media
the form
where
G = cos @[sin(Po + ~ D p ~ ) ] " ' [ v ( ~ ) v ( x ) ] - ~ / ~ .
When x, is regular everywhere on S, the first-order term in
the Jacobian, EDJ,, is small compared to the zero-order
term Jo(xo) and integral (6.4) produces essentially the same
results as (3.10). However, at a caustic 5 where Jo(6) = 0,
(3.10) breaks down while (6.4) still predicts a finite
amplitude at x
where S, is a small area around 5 and So represents the rest
of S. After replacing E with w-' the second integral
becomes
An additional multiplier i sgn w ( - w ) now appears in (6.6).
While the factor -iw corresponds to differentiation in the
time domain, i sgn w indicates a (n/2) sgn w phase shift.
The sign of the phase shift, (n/2) sgn w in this case, is
determined by the sign convention used in the Fourier
transform pair. Expression (6.6) was derived using
+"
q ( x , t)eio' dt
Y(x, w ) =
I_,
18'1
extended Green's function. Such extension is accomplished
by putting the source point in a complex space. If one
performs such extension to each secondary source point
involved in a ray-Kirchhoff integral, then this integral
becomes a summation of complex ray solutions. The
imaginary part of the starting position for each complex ray
will give a smoothing effect over the results. This smoothing
prevents the Jacobian from becoming zero, but it also causes
phase and amplitude distortions (Wu 1985). Alternati\rely,
the analytic extension of a Green's function can be done by
introducing a complex velocity as in section 5 . The
advantage of this approach is that the complex velocity can
be used to model the attenuating effect of a medium, and
hence does not introduce artificial phase and amplitude
distortions. Moreover, the complex velocity approach can
be readily combined with the previously described approach
to produce more accurate results. For example, we can
evaluate a ray-Kirchhoff integral in the same manner as that
described in section 5 when there are no singular points
involved on the surface of integration. On the other hand,
whenever a caustic associated with the elastic velocity
in
(5.1) occurs on the integral surface, we recompute the
perturbation terms in the Jacobian by including in the
complex eikonal equation a higher-order term such as that
in (6.1).
It is interesting to note that there is still another useful
closed-form solution of (2.1) which eliminates singularities
of wave fields. We derive this solution using
v)
G(x, xg) = exp [ i w t ( x ,x,)l/r
(6.7)
as the function G in (2.11). The t in (6.7) is determined by
the eikonal equaton (2.19) and r = (x - xo(. Similar to the
derivation of (3.2), substitution of (6.7) into (2.11) leads to
1
1dY
cos 6
w)=Gll(----iwy--rV
Y-an
L? r
~ ( x ,
and
1
S
r+"
Y(x, w)e-'O'dw.
q(x, t ) = 22n J_ m
If the complex conjugate of this pair had been used, the
phase shift would be -(n/2)sgnw. The accuracy of the
wavefield near a caustic can be estimated by the procedure
described in section 3. In the cases where equations (6.2)
and (6.4) are used, the error in the wavefield near a caustic
is 0 ( w p 2 ) .
At a focus both Jo and DJ, vanish. To prevent the
integrand from becoming infinite, we must include in (6.4)
the second-order term E ~ D which
~ J will
~ cause an additional
multiplier w2. This indicates that at a focus the integrand of
a ray-Kirchhoff integral should be replaced by its second
derivative with a phase shift of n. Thus the new approach
has not only prevented an infinite amplitude at a singular
point, but also predicted the waveform change caused by
this singularity.
In the above discussion, we have replaced E , in (2.21)
with E l in order to prevent the zeros of E near a caustic.
This involves computing p,(dE/dP). Alternatively, zeros of
E, can be avoided by introducing a smoothing procedure.
Such a procedure is sometimes more desirable because it
also improves the stability of numerical computation
(Chapman 1985). Felsen (1984) and Wu (1985) have
suggested avoiding the singularities by using an analytically
iwmYexp(iwt)dT,,
X
(6.8)
+
where m = V. (Vr/r) V t . V(l/r). The volume integral in
(6.8), which represents the scattered waves, is now larger
than that in (3.2), and, in general, can no longer be
neglected even in a slowly varying medium. After some
manipulation, (6.8) becomes
where
Here we have transformed the solution back to the time
domain. Different from integral solutions (2.13) and ( 3 . 2 ) ,
solution (6.9) does not contain the Jacobian and hence
eliminates completely the difficulties caused by its zeros.
In general, evaluating (6.9) is more time-consuming than
a ray-Kirchhoff integral because it involves solving an
integral equation. This approach, however, does not require
190
T. Zhu
ray-amplitude computation and can be directly performed in
the time domain even in rapidly varying media. It provides
an alterantive to the finite-difference and finite-element
methods for direct numerical modeling, and could be useful
in the cases where the ray-Kirchhoff method fails, or the
Jacobian is difficult to find (e.g. in anelastic media).
In some cases, anlaytical solutions for extended eikonal
equation (7.5), and hence for the wave fields, are possible.
For example, in a medium with constant density and linear
velocity V(z) = Vo + Kz, it can be shown (Ben-Menahem &
Beydoun 1985) that the last two terms in (7.6) cancel each
other, and (7.5) becomes
7 C O M P U T A T I O N OF W A V E F I E L D S IN
RAPIDLY VARYING ACOUSTIC MEDIA
The phase velocity u is given by
As an example, we apply the new method to a rapidly
varying acoustic medium. In this case, p = 0, and the
equation of motion (Ewing et al. 1957) in the frequency
domain is
p ~ ~ u + i l v ( v . u )v+. u v a = o ,
(7-1)
where U is the Fourier transform of the displacement.
Substitution of U = p-’V(p”’Y) in equation (7.1) yields
(Gupta 1965)
dV-’Y
+ V2Y + h Y = 0,
(74
where h = [ V2p/p - (3/2)( Vp/p2]/2 and V = (A/p)’/’.
extended eikonal equation associated with (7.2) is
= V-’
(
+h 6 ’ -
o-’( V2y/y).
The
(7.3)
The coefficients p2,, in series (2.16) are now given by
Po = l / V
P z = [ 4 b g + ~(VPO)’
- 2P0v2P01/(8Pi)
~4
L/(2Po)
(7.4)
V P ~
- v 2 2 ( ~ 0 ~ 2 ) l / ( 4+~( iN) +P,LM)/(~P,)
= [ ~ ~ P O P Z- 1 0 ~ i b :+ 3
...
In the following we again retain only the zero- and
second-order terms; for simplicity, we also assume that il
and p depend only on depth. In such a case equation (7.3)
becomes
( V T )=
~ V-’
+ HIM-’,
(7.5)
where
u=
Vo+ Dz
= uo kz,
(1 - K 2 ~ - 2 / 4 ) 1 1 2
+
(7.9)
where
uo = Vo/(l - K2w-2/4)112
and
k = K/(1K2w-2/4)112. Equation (7.9) indicates that the gradient of
phase velocity u increases as the frequency decreases so that
low-frequency rays are more sharply bent. These sharplybent rays constitute the partial reflections from a
high-gradient region. From (7.9) we also see that the
divergence frequency in this case is wd = K/2 which is the
same as the cutoff frequency found by Ben-Menahem &
Beydoun (1985) for a linear velocity-distribution.
Since media with constant density have been used in
seismic migration, we derive here a wave extrapolation
formula for velocity distribution (7.9) assuming coincident
shot and receiver locations. This formula extends Kirchhoff
migration (French 1975; Schneider 1978) to inhomogeneous
media. The specific geometry of interest is illustrated in Fig.
4(a) where cylindrical coordinates (r, a,z ) are used. The
integration surface now consists of the recording surface
z = 0 and a hemisphere extending to infinity. Contribution
from this distant hemisphere is ignored. Mathematically,
wave extrapolation amounts to computing the wave field at
a depth point x1 from the boundary value at the surface
z = 0 (Schneider 1978; Carter & Frazer 1984).
The expression for the Jacobian associated with phase
velocity (7.9) can be obtained by considering a ray that
starts at x1 and emerges at x,, (Fig. 4(b)). Its ray parameter
is defined by
p = sin Pl/ul = sin /30/uo,
(7.10)
where /3 is the angle between the ray and z-axis, and the
subscripts ‘0’ and ‘1’ denote evaluation at q, and xl,
respectively. The traveltime between x1 and x,, is given by
(Kaufrnan 1953)
From (7.5) and (7.6) we see that in a rapidly varying
medium, phase velocity jumps are caused not only by the
discontinuities in elastic parameters but also by the
discontinuities in the derivatives of these parameters. As a
result, reflections may occur at the top and bottom of a
transition zone such as that considered by Gupta (1965).
As pointed out in section 2, one can solve (7.5) and (2.9)
by ray tracing, then proceed to compute wave fields by
(2.13). In general, determination of t and y at different
frequencies requires rays to be traced at each frequency
component. This could be time-consuming when a broad
frequency band is involved. In such a case, it is more
efficient to compute ray fields by the perturbation scheme
which, as discussed in section 6, avoids most of the required
ray tracing.
1 (1 + kz,/v,)[l + (1 -p2u$)1’2]
t = -kl n
2 2 112
1 + (1 -P u1)
(7.11)
The distance between q, and 0,ro, is determined by
ro = v,(cos
where
B1- cos P0)/(k sin PI),
(7.12)
Po and PI are related by
tan (P0/2) = exp ( - k z ) tan (P1/2).
(7.13)
From the geometry in Fig. 4(b) we see that the Jacobian at
x,, can be computed by
J = ( r o A ~ / A P 1=l Iro(Aro/co~e)/APII = lro dro/dp,/cos
pol.
Here we have used the relation 8 = JC - Po. It follows from
equations (7.12) and (7.13) that ar0/d@,= r, cos Polsin /3, so
Ray-Kirchhoff method in inhomogeneous media
191
(Berkhout 1982), such extension can be accomplished by
downward continuation of the wave field from the surface to
successive interfaces. When the wave field is downward
continued to the top of a layer, the coordinate system will
be rotated, if necessary, so that the z-axis is along the
direction of the velocity gradient of this layer and the wave
fields can be calculated using the above formulae. This
layered model can be further extended to accommodate
transverse velocity variations by the perturbation scheme.
8
XI
J
Z
Figure 4. (a) Integration surface for wave extrapolation in a 1-D
medium. Also shown in the figure is a ray from the depth point x1
to the surface point q:(b) A ray tube associated with the ray shown
in Fig. 4(a). 19 is the angle between the ray and the normal at q,
and Aq is the width of the ray tube at the surface.
that
J = rg/sin PI.
(7.14)
Combining equations (7.14), (2.9) and (2.13) and noting
that, for a planar integral surface, the first term in integral
(2.13) is equal to the sum of the second and third terms
(French 1975), we obtain a simple expression for the wave
field at x1 (assuming that x1 is a far-field point):
(7.15)
Here we have replaced t with -7 because wave fields are
extrapolated backward in time. Equation (7.15) is cast in
one-way traveltime so in practical computation, one must
either divide seismic section time scales by 2 or use a
velocity equal to one-half the true velocity. According to the
imaging principle described by Claerbout (1971) and
Berkhout (1982), the migrated depth section is given by
+(7.16)
v(xl, 0) =
Y(x,,
I
0)dw.
2n -m
I
Wave extrapolation formula (7.15) can be extended to a
medium consisting of dipping layers, each having a constant
velocity gradient. Similar to the case of homogeneous media
CONCLUSIONS A N D DISCUSSION
A new method has been developed for solving scalar wave
equations in inhomogeneous media. While retaining the
simplicity and physical clarity of ray approach, this method
extends the previous Kirchhoff techniques to rapidly varying
media, and can be used to model caustics and diffractions as
well as the scattering from inhomogeneities. Another useful
feature of the method is that it includes a perturbation
scheme for solving the ray and transport equations in
laterally varying media. This scheme avoids the extensive
ray tracing required for determining the Jacobian and
traveltime, and it provides a useful tool for evaluating a
ray-Kirchhoff integral in various inhomogeneous media. For
example, using this scheme, a simple yet rigorous integral
representation was obtained for the wave fields in an
anelastic medium. It has been demonstrated that zeros of
the Jacobian can be avoided by either introducing a complex
velocity or including higher-order terms in an extended
eikonal equation. The latter approach indicates that at a
caustic the kernel of a ray-Kirchhoff intergral should be
replaced by its phase-shifted derivative.
Most of the discussions in this study have been
concentrated on unbounded media with continuous
velocities. The resulting formulae and computation techniques, however, can be extended to media containing
interfaces where phase velocities are discontinuous. In
slowly varying media such interfaces are caused by
discontinuities in elastic parameters, while in rapidly varying
media they are caused by discontinuities in both elastic
parameters and their derivatives. Frazer & Sen (1985) have
developed expressions for boundary conditions for layered
media based on plane wave theory. Alternatively, an
interface can also be represented by a set of secondary
sources. The strengths of these secondary sources can be
determined by integral equations established using the
continuity conditions at the interface. This has been done
for acoustic media by DeSanto (1983) and extended to
elastic media by Kennett (1984). Coupled with these
expressions (or their modifications) for boundary conditions,
the ray-Kirchhoff method presented in this paper can be
used to model seismic responses from a layered inhomogeneous medium.
ACKNOWLEDGMENTS
This paper is based on part of my PhD thesis at Cornell
University. I am grateful to my advisors Drs J. E. Oliver
and L. D. Brown as well as Dr S. Kaufman for their support
and valuable advice during the course of this study. I thank
Dr R. A. W. Haddon for stimulating discussion, Drs B. F.
Chao and J. McBride for carefully reading the manuscript,
192
T. Zhu
and three reviews for their suggested revisions t o the
manuscript. This work was supported by National Science
Foundation grants EAR 82-12445 and EAR 83-13569, and
Memphis State University a n d t h e State of Tennessee
Center of Excellence program.
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APPENDIX: DERIVATION OF THE RAYKIRCHHOFF FORMULA
Consider the geometry shown in Fig. A1 where T is a
volume lying between surfaces S2 and S,. The latter is a
small sphere of radius R with its centre at a receiver point x.
is a source point. For the chosen functions Y and G,
Green’s theorem (2.11) is applicable on T , so that
///(YV2G
- GV’Y) dTo
T
=//+//(YVG-GVY).ndSo.
s
sz
From (2.12), we have
VG = exp (iot)(Vy + iuyVr),
Figure Al. Geometry for scalar integral representation.
(Al)
Ray-Kirchhoff method in inhomogeneous media
and
V2G = exp (ion)[V 2 y - yw2( Or)’]
+ io exp (iws)(2Vr. Vs + yV2z). (A3)
It follows from (2.4) and (2.8) that 2 V y . Vr + yV’r = 0,
which leads to
V2G = exp (iwz)[V2y - yw’( Vz)’].
(‘44)
Inserting (2.10) into (A4) yields
V2G = -exp ( i w t )yw’v-’.
(‘45)
193
Noting that for the small sphere S,, y = 1/R, and using the
mean-value theorem, we obtain
[
I”
B = 4n[exp ( i w t ) Y ] - 4nR exp ( i w s ) i w Y - - - ,
( d R R
where ‘-’
represents the mean value of the
corresponding functions on S,. Equation (A8) shows that B
approaches 4 x Y as R approaches zero. Substituting (A2),
(A6), and (A8) into (Al) and letting R approach zero yields
ray-Kirchhoff formula (2.13):
Substituting (A5) and (2.2) into the left side of (Al) gives
/ / / ( Y V Z G - GPY) d ~ =, o
(‘46)
T
Using (A2) and the fact that the outward normal to S, is
- R / R (Fig. Al), the first integral on the right side of (Al),
B, can be written as
Note: In deriving (3.2), (2.19) instead of (2.10) is used in
(A4), which leads to
V2G = exp (;or)[
V2y- ~ w ~ V - ~ ] .
It is the first term in the square bracket that produces the
volume integral in (3.2).
s1