Geophys. 1. Int. (1990) 103, 675-684
Wave propagation in laterally heterogeneous layered media
F. Wenzel, K.-J. Stenzel and U. Zimmermann
Geophysikalisches Institut, HertzstraBe 16, 7500 Karlsruhe, FRG
Accepted 1990 July 3. Received 1990 June 15; in original form 1990 January 5
SUMMARY
This paper derives a solution of the seismic response of an acoustic medium with
2-D geometry using Kirchhoff‘s representation of the wavefield. The medium can be
composed of several layers, separated by curved interfaces.
The difference between the approach presented here and other Kirchhoff-type
solutions of the multilayer case is that we take care of all diffractive effects during
wave propagation, including those during transmission from source to reflector and
from reflector to receiver.
A crucial point in this method is the replacement of the Kirchhoff-Helmholtz
formula, that requires both the knowledge of the wavefield and its directional
derivative at the medium’s interfaces, by the Rayleigh-Sommerfeld representation
where only the wavefield must be known o r approximated. The propagation of
acoustic or elastic waves through a stack of layers with curved interfaces can then be
described by a sequence of plane-wave compositions or plane-wave decompositions.
This point of view implies a reinterpretation of Kirchhoff’s formula, in terms of
plane waves emanating from a reflector, excited by an incident wavefield.
The outlined theory provides a convenient formulation for the transmission and
reflection of waves in a multilayered laterally inhomogeneous medium with good
accuracy as long as the radius of curvature of the boundaries clearly exceeds the
dominant wavelength and if the distance between interfaces is larger than a few
wavelengths.
Key words: Kirchhoff representation, seismic waves, synthetic seismograms.
INTRODUCTION
Kirchhoff s integral equation for a wavefield described by
the scalar wave equation has found several applications in
reflection seismology. Calculations of the seismic response
of a curved interface on the basis of Kirchhoff s theory were
performed by Hilterman (1970), Trorey (1970, 1977),
Berryhill (1977) and more recently by Carter & Frazer
(1983), Deregowski & Brown (1983), Frazer & Sen (1985),
Sen & Frazer (1985), and Kampfmann (1988). The theory
was also utilized for the inverse problem, the migration of
zero-offset data, in a classical paper by Schneider (1978).
Wapenaar & Berkhout (1989) give a comprehensive
overview on elastic wavefield extrapolation.
A paper by Hill 8t Wuenschel (1985) deals with refracted
waves travelling through the uppermost layers of the ocean
bottom and being disturbed by strong topography. It
describes a method to propagate waves through a layered
medium with curved interfaces that is based on Kirchhoff s
representation of the wavefield and is termed there
‘boundary hopping’. The referenced work stimulated a
systematic study of this approach, the result of which is
presented in this paper.
Two crucial points in the development of the method
must be stressed. First it is not attempted to use ray theory
to propagate the sourcefield towards the reflector and back
to the receiver. Instead plane-wave decompositions of these
wavefields are applied. The wavefield at the reflector is
evaluated with the standard Kirchhoff approximation. The
acoustic wavefield generated by a line source and reflected
from an interface can thus be expressed by a three-fold
integral: one integration is related to the plane-wave
decomposition of the source, another to the same operation
for Green’s function; the third variable of integration is the
path length along the reflector. At the first glance one is
tempted to approximate at least two of these integrals by the
method of stationary phase, a step that results in a single
integral along the reflector. This approach is employed by
most of the workers above. In case of several layers it can
be generalized: wave propagation from a source to the
reflector through a stack of possibly curved interfaces is
analysed by ray theory, as is the propagation from the
675
676
F. Wenzel, K.-J. Stenzel and U.Zimmermann
reflector back to the receiver at the surface. The Kirchhoff
representation is only used to connect both ray solutions at
the reflecting interface. Effects that are not covered by
simple ray theory (e.g. diffractions) are only taken into
account when produced by the reflecting interface. Non-ray
effects in the transmitted wavefield (upward and downward)
are neglected. This paper persues another path. The
Green’s function that has to be used in the Kirchhoff
formula is modified in a sense that the three-fold integral
can be broken in a sequence of single-fold integrals. The
Green’s function generally used for Kirchhoffs formula, e.g.
the Hankel function in the 2-D case, does not allow for the
required measure. Another Green’s function of RayleighSommerfeld type, see e.g. Goodman (1968) or Berkhout
(1984), must be used. The approximate construction of this
function is the second important point in the paper. The
downward propagation of a wavefield across interfaces, the
reflection at a curved boundary and the subsequent upward
transmission can be described as a sequence of plane-wave
decompositions and compositions with the help of which the
wavefield is propagated from boundary to boundary and
across boundaries.
During the reviewing process of this paper we learned
that similar ideas were presented by Sen & Frazer (1988) as
an extended SEG abstract. They call their approach the
‘Phase Space Path Integral’ method. The theory outlined in
the following pages considers 2-D propagation in a medium
controlled by the acoustic wave equation with layers of
constant velocity. These restrictions are introduced in order
to keep the theory simple, rather than because of intrinsic
deficiencies.
KIRCHHOFF-HELMHOLTZ INTEGRAL
REPRESENTATION
This section reviews basic properties of the representation of
a wavefield that satisfies the Helmholtz equation in 2-D,
known as the Kirchhoff-Helmholtz integral representation
(KHIR). Consider the case illustrated in Fig. 1: S is a closed
line, within and on which the arbitrary functions P and G
and their first two partial derivatives are continuous. Both
functions may depend on further parameters, e.g. frequency
or time. S encloses the area F, n is the outward normal unit
vector. Given these conditions Green’s second identity
holds:
If in addition both functions satisfy the homogeneous
equation on and inside S;
I
WL
V2P + -P = 0,
V2
W2
V2G +,G
21
=o,
(3)
the areal integral vanishes and Green’s second identity
degenerates to
(4)
Specification of G in a particular way may result in a
representation of P that enables physical insight in wave
propagation phenomena. One possibility is to define G as
the solution of the inhomogeneous Helmholtz equation valid
in the whole plane:
W2
V2G0+,Go=
V
-S(x-x,)6(z-zr).
It becomes unique with Sommerfeld’s radiation condition
where (xr, 2,) is a point inside the closed area and
r = [ ( x - x,)’ + ( z - z ~ ) ~ ] ’The
” . unique solution of (5,6) is
given by the zeroth-order Hankel function of the first kind,
Green’s function in 2-D:
i
Go(x,z , x,, z,, 0 ) = - H p
4
(7)
It should be mentioned that the actual form of equations
( 5 , 6 ) depends on the definition of the temporal Fourier
transform. The form used in this paper is
for the forward and
F p r e 1. Sketch of the geometry used in Kirchhoffs integral
representation of the wavefield P, equation (12). P must be a
solution of the homogeneous Helmholtz equation (2) inside the area
F and on its contour S. The unit vector n, perpendicular to the
contour line points outward. The Greeen’s function Go satisfies the
inhomogeneous Helmholtz equation with a singularity at (xr, zr),
equation ( 5 ) .
for the inverse transform. If one chooses the other sign
convention the sign in Sommerfeld’s radiation condition
must be changed and the Hankel function of the second
kind arises as the solution. For any definition of the Fourier
transform both Hankel functions satisfy (5); Sommerfeld’s
condition ensures that the ‘outgoing’ solution is taken. (6)
can also be viewed as a boundary condition at infinity that
guarantees both a unique solution of (5) and the
convergence of line integrals with integration paths tending
to infinity. The Hankel function (7) has an integral
Waves in heterogeneous media
meet infinitely far away from the origin. The counterclockwise integration path therefore consists of two parts. The
wavefield P is decomposed into plane waves:
representation
(10)
with
A ( p , w)eiO(px-qz)dP
= P,(x,
If p is understood as the ray parameter, (10) represents the
plane-wave decomposition of Go where a real q yields
homogeneous and an imaginary q yields inhomogeneous
plane waves. (10) is valid for the half-space above and below
the level z =I,. If only up- or downgoing waves are of
interest, the absolute value in the exponent of (10) can be
expressed as (z, - 2 ) or (z - z,.) with the z-axis pointing
downward.
As the point ( x r , zr) is located inside the contour S,
requirement (3) is violated. The contour S must be modified
in a way that excludes the singularity of Go at ( x r , z,.). The
result of the procedure is the well-known KHIR:
P(x,, z,, w ) =
677
I,
(Go:-
2,
w)
+ P&,
2, w ) .
P, represents the upward travelling and Pd the downward
travelling wavefield. Evaluation of
with
dS, = -dx
and
d
-=
an
--a
dz
results in
2iwqA(p, w ) .
P"")dn dS.
This formula can be interpreted in terms of Huygens'
principle. Although the singularity of Go is located at ( x r , z,.)
the reciprocity property of Go suggests that the wavefield
measured at (x,., z,) is generated by line sources located
along the contour S and weighted by the value of the
wavefield and its derivative there.
If one wants to understand the physics of wave
propagation in terms of plane waves rather than with
Huygens' principle G is chosen as a downward propagating
plane wave;
(14)
(17)
The B term vanishes. If instead of (13) an upward travelling
plane wave is chosen, the term A vanishes. Therefore
with
a
dn
(19)
~ , ( x 2,
, w ) = e--iW(px-qz)
(13)
characterized by the ray parameter p . The integration path
is specified in Fig. 2. The full contour runs along the x-axis
(S,), and an arbitrary curved line in the lower half-space
(S,) described by a function d(x). S, and S, are assumed to
r
(--
\
\
I
I
I
\
I
Figure 2. Sketch of the geometry used for the derivation of the
plane-wave representation of Kirchhoffs integral formula (20). The
wavefield P is a solution of the homogeneous Helmholtz equation
inside the contour given by S, and S,; only the upward propagating
portion of P is considered. As Green's function G, is a downward
propagating plane wave, G, satisfies a homogeneous Helmholtz
equation and has no singularities inside the area bounded by S, and
$. S, is described by the function d(x).
Equation (18) can be expressed as
1 + P- d ' ( x )
4
P
A ( p , w ) = 1 ('.P
2 s2 iwq an [I + d"(x)]'"
x e-'w[PX-4d(x)l dF&
--I
(20)
According to (14) the contribution of a particular ray
parameter p to the upward travelling part of the wavefield
P, at an observational position (xr, z,.) is
Together with (20) the ray parameter p contributes
e - i o t ~ ( x , - * ) - q [ ~ , - d ~ x ) l ) ds
2'
(21)
The physical interpretation of (21) is quite obvious. The
678
F. Wenzel, K.-J. Stenzel and U. Zimmermann
I
X
In seismic applications the interesting quantity is the
portion of an incident wavefield that is backscattered from
an interface separating two regions of a medium with
different seismic impedance values. We assume that the
incident wavefield Do consists of downgoing waves only:
If the waves stem from a line source, located at (xo, 0), B ,
must, according to equation (lo), be expressed as
In the following, the behaviour of a particular incident plane
wave is studied in detail. For the moment the calculation
focuses on the reflected field in the immediate vicinity of the
reflector, because once it is known then the Kirchhoff
formula allows for its evaluation at any point above it. In
the vicinity of a specified point on the reflector [E, d(5)], see
Fig. 4, the reflected field may be easily approximated if the
curved interface is replaced by its tangent. If the incident
plane wave is
the reflection will be approximated by
Figure 3. Comparison of the two forms of Huygen's principle
according to equations (12) and (21). The upper panel sketches the
classical view. The wavefield at a particular point (q,z,) can be
composed out of contributions of individual line sources exploding
on a contour where the wavefield and its directional derivative are
known. The lower panel shows the plane-wave equivalent of
Huygen's principle. Plane waves with a ray parameter p emanate
from every point along the contour. Their sum results in the total
contribution of this ray parameter to the wavefield at (xr, 2,).
expression in curved brackets is a weight that depends on
the wavefield and its derivative at S,. The exponential term
represents a plane wave originating at a point on S, e.g.
[x, d ( x ) ] with ray parameter p . Therefore the contribution
to the upward travelling wavefield P, from ray parameter p
can be evaluated by shooting plane waves with p from each
location along S, weighted by a particular combination of P
and its derivative at S,. This interpretation is sketched in
Fig. 3 and can be viewed as a plane-wave equivalent to
Huygens' principle.
where P, 4 and p , q are related via Snell's law
or equivalently
with the derivative d'(E). The geometric meaning of the
quantities and equations used so far are sketched in Fig. 4.
R stands for the plane-wave reflection coefficient for the
incident ray parameter p . cpr represents an upward travelling
plane wave.
Approximation (24) is only valid if ( x , z) is very close to
the reflector at [E, d(E)]. At this point the value of the
reflected field is
The directional derivative (19) of (24) at
[ E, d(5)] is
KIRCHOFF'S APPROXIMATION
The KHIR, equation (12), and its plane-wave equivalent
(equation 20) do not contain any approximations so far. On
the other hand they cannot be termed solutions for wave
propagation problems, but rather as representations of the
wavefield which provide insight into the physics of wave
propagation, e.g. Huygens' principle or the plane-wave
interpretation discussed in the previous section. For the
solution of particular problems a further approximation,
usually known as Kirchhoff s approximation or Kirchhoff s
boundary condition, is convenient. It was introduced by
Kirchhoff (1883) in the study of light diffraction from a screen
with an aperture, and is also used in acoustic scattering (e.g.
Bleistein 1984).
R ( ~E,,
W)ei4~f+~d(E)J,
(25)
where Snell's law has been used to replace jj, (I by p , q.
With (24) and (25), approximate expressions for the
wavefield and its derivative along the reflector have been
found that can be inserted into the Kirchhoff formula (12).
Although the approximations for the wavefield and its
derivative are somewhat different from those originally
studied by Kirchhoff they are nevertheless referred to as the
'Kirchoff method or 'method of physical optics'; see
Beckmann & Spizzichino (1963, p. 6). The reflected
wavefield generated by a line source at (xo, 0) and recorded
Waves in heterogeneous media
I
679
x
>
F v 4. Geometry used for the derivation of the local plane-wave assumption, also called Kirchhoffs assumption for the case of a reflecting
interface, equations (23)-(25). If the curvature of the interface, defined by d(x) is small compared to the wavelength of an incident plane wave
an observer in the very vicinity of a point of the interface could describe the scattered field as a reflection from a tangent plane interface.
at ( x r , 0) can then be expressed as
For the derivation of (26) equations (14), (21) and (22) were
used. For completeness it should be mentioned that the
singularity of the incident line source has not been properly
treated so far. It requires an additional deformation of the
integration contour in (12) such that the singular source
point is excluded. The residual contribution from this step
will be
an inconsistency. For the optic screen problem PoincarC
pointed to this fact with arguments from potential theory
(Goodman 1968, 30 pp). Sommerfeld (1954) removed this
inconsistency. He introduced a Green’s function different
from (7) based on the idea that the KHIR holds for any
Green’s function that satisfies the Helmholtz equation (2)
inside the contour and has a singularity there of type (7). No
statements about its properties outside the contour are
required. Therefore the Hankel function (7) represents one
possible choice for Green’s function. It has only one
singularity in the whole plane. Any function with further
singularities located outside the contour which are
controlled by the Helmholtz equation can be inserted into
the KHIR (12). In the case of the optic screen problem an
appropriate choice is to add or subtract the same type of
function with a singularity that mirror-images the first one
with respect to the screen. At the screen, the total function
G2 itself or its derivative disappears.
In the second case the KHIR degenerates to
(27)
which physically represents the direct wave from the source.
If one is only concerned with the reflection response, this
can be ignored.
RAYLEIGH-SOMMERFELD INTEGRAL
REPRESENTATION
A disadvantage of the KHIR, equation (12), is the fact that
both the’ wavefield P and its derivative dPfdn must be
specified at the boundary in applications concerning the
optic screen problem and acoustic reflections. It is well
known, however, that Cauchy-type boundary conditions,
e.g. P and dPfan given at the boundary overdetermine the
solution of linear partial differential equations (Morse &
Feshbach 1953, p. 706). For an elliptic equation like the
Helmholtz equation (2), Dirichlet (P)or Neuman (dPldn)
boundary conditions yield unique and stable solutions.
Therefore independent definition of P and dPfdn represents
and is referred to as the Rayleigh-Sommerfeld integral
representation (RSIR). Applications that approximate the
value of the wavefield along the boundary are no longer
inconsistent. (27) is well suited for the migration of seismic
data, e.g. wavefield continuation because seismologists
usually only measure one of the quantities P and dPldn,
never both. The RSIR in this context is discussed by
Schneider (1978) and Berkhout (1984). So far the utilization
of the RSIR is restricted to a straight line geometry as in
case of the optic screen or the continuation of seismic data
to depth from measurements along a straight line. An
extension of the approach to curved contour lines is not
obvious, and no exact solution is in sight. In the following,
we derive an approximate RSIR for this case from two
different viewpoints. First we extend the idea of
mirror-imaged singularities to a curved interface and thus
construct a Green’s function that approximately satisfies
(27). Second we use the plane-wave approach without
explicitly defining a Green’s function.
680
F. Wenrel, K.-J. Stenzel and U.Zimmermann
Construction of the Green's function
Suppose that the receiver coordinates (x,, 2), and another
point above the reflector are given according to Fig. 5. G, is
then constructed using the following procedure. Move from
(x, z ) vertically downward (in the positive z-direction) until
the reflector, defined by the function d ( x ) , is intersected at
the point [ x , d ( x ) ] ; construct the tangent to the reflector at
this point and define the mirror image (xi, z i ) of (x,, z,) with
respect to the tangent. The Green's function becomes
with
r, = [(x
- x,),
r, = [(x -xi)'
+ ( z - zr)2]112,
+ (z -
where (xi, zi) are functions of ( x , z ) and (x,, z,) according to
the geometry of Fig. 5. Algebraically the relation is
xi(x, x,, z,) = x, - 2 sin Cp{x, - x ) sin Cp + [d(x) - zr] cos @}
zi(x, x,, z,) = z,
+ 2 cos Cp{(x, - x ) sin Cp + [ d ( x ) - z,] cos Cp}
Application of the operator alan to C, and assuming that
W / V >> rl,, results after some algebra in
with
r, = { ( x
- x,),
+ [d(x) - z,]~})'".
Thus (28) establishes an approximate expression for a
Greens's function suitable for the RSIR (27). The
approximation is two-fold. First its definition according to
equation (28) might violate the condition that G2 has no
further singularities above the boundary than the one at
(x,, 2,). The image (xi, zi) might be above the boundary,
depending on the location of the point (x, z ) . The error for
the RSIR caused in this case cannot be estimated from G,
itself because it depends on the characteristics of the
incident wavefield P as well. Second, the derivative of G, at
the boundary, given by (32) requires a receiver position a
few wavelengths away from the boundary.
The reflected wavefield can be expressed if relation (32) is
inserted into (27):
(29)
with angle Cp as a function of x :
tg Cp = d ' ( x ) .
Note that the image point (xi,z i ) does not explicitly depend
on z . Evidently G, disappears at the boundary because there
r, equals r,:
G,[x, d ( x ) , x,, z,,
01 = 0.
In order to utilize equation (27), the value of the directional
derivative of G, at the boundary has to be evaluated. The
far-field or high-frequency approximation of C2 is
ei(w/u)n
ei(o/u)Q
(30)
and r, and r, are already defined in (28) and Fig. 5.
The plane-wave approach
Another approach for the derivation of a RSIR is given in
Hill & Wuenschel (1985). They do not construct an explicit
expression for Green's function but rather work with plane
waves. The geometry of the problem has already been given
in Fig. 2. The integration path consists of two parts, one
along the x-axis (S,), and one along the reflecting boundary
(S,) in counterclockwise direction. In order to study the
reflected field, the upward propagating pressure is composed
of upward travelling waves:
In a previous section of this paper a downward propagating
plane wave was inserted into Green's second identity (4) in
order to derive a KHIR. If a RSIR is requested another
function must be used:
G3(x, 2,
W ) = Gi(4 2, 0 )
+ G,(x, 2, 0 ) )
with the additional condition that C3 vanishes at S,
G3[x,d(x),
Figure 5. Geometry used for the construction of a Rayleigh-
Sommerfeld-type Green's function that vanishes at a curved
interface d ( x ) and has a singularity at (xr, 2,) above d(x), equations
(28) and (29).
01= 0.
(36)
Physically (35) represents the sum of a downward travelling
plane wave and a reflected wavefield that consists
exclusively of upward travelling plane waves. The
amplitudes C are not specified but will be determined by
condition (36). A similar calculation as shown previously,
see equation (14) to (17), yields for the integral along S,:
-2iwqA(p, w). After the application of the acoustic soft
Waves in heterogeneous media
boundary condition at S,,
The sum of an upward travelling plane wave and the
downward scattered wavefield is inserted into Green's
second theorem :
G4(x,z, w ) = e-im(Px+qZz)
one can write
A(p, w ) =
681
I[
+
1+pd'(x)]
4
x ~ [ xd,( x ) , w]e-io[px-qd(x)l LLX.
(37)
I
qP,
jj, w)e-WPx-4zz)
dP?
-
where C is implicitly defined by the boundary condition
G4[x, d(x), w ] = 0.
The upward travelling wavefield results when (37) is inserted
into (34).
The same procedure as outlined previously results in
Comparison of results
B(p, w )=
Both results for the reflected wavefield, equations (33) and
(37) together with (34), are closely related. They can be
transformed into each other by using stationary phase
approximations.
If the integral expression of the plane-wave coefficient
(37) is inserted into (34) and the integration over the ray
parameter (p) performed with the method of stationary
phase one finds (33) as result with
sin y
p=-?
If conversely an approximate plane-wave decomposition for
the exponential term in (33) is used the solution derived by
the plane-wave approach is reproduced. Therefore both
methods for the derivation of an RSIR for the reflected
wavefield are essentially equivalent.
BOUNDARY HOPPING
Expressions for the reflected wave as derived in the previous
section can also be stated for the transmitted wave. We
assume the geometry of Fig. 6. The integration path again
consists of two parts. One coincides with the curved
boundary (SJ, defined by d ( x ) , and the other part S,
represents a level of constant depth several wavelengths
below S,. The (transmitted) wave propagates downward and
is therefore expressed as
(39)
I[ +
sz
d'(x)]
1
42
x ~ [ xd,( x ) , w]e-iW[pX+q2d(x)l
h.
(40)
Equations (34)-(37) and (38)-(40) form the basis for the
propagation of a wavefield through the medium and across
curved interfaces by a sequence of plane-wave compositions
(PWC) and plane-wave decompositions (PWD). Before we
write down the two-interface case that can be readily
generalized to an arbitrary stack of layers the formula for
the reflection by a single interface is presented in order to
emphasize the importance of the Rayleigh-Sommerfeld
integral representation (RSIR) for the boundary hopping
method.
The downward propagating part of a line source located
at ( x o , 0) has a PWD
The fraction of the wavefield U, that will be reflected at the
interface is calculated. Each plane-wave component B, must
be propagated towards the interface with the operator
exp [iwq, d , ( x ) ] , weighted by the plane-wave reflection
coefficient R,, which depends on the ray parameter p and
the position x where the wavefield is evaluated with a PWC:
I
u&, d , ( x ) , w ] = 2n:
IwI
~ , ( pw, ) ~ , ( px)eim[Px+qldl(x)ld
,
P.
(42)
Once U, is known along the interface the PWD of the
reflected wave can be calculated according to equation (37):
The velocity below S, is v2 and 4 , = (l/v; -p2)"'
e--i4p*--qi
di(x)l&
(43)
with the directivity function
I
"1
O,,(x, p ) = 1
P dI(x).
+41
I
\
,1
S
Figure 6. Geometry used for the derivation of the transmitted
wavefield propagating downward, equations (38)-(40).
The first suffix of 0 refers to the interface d,(x) and the
second specifies the layer into which the wavefield is
continued. At z = 0 the wavefield is given by a further
PWC:
Fig. 7 shows a scheme of the sequence of required steps.
F. Wenzel, K.-J. Stenzel and U.Zimmermann
682
I
I
'x*
-
,
X
u,
/
Combining (41) to (44) one finds
(45)
t l
uo
I '
Figure 7. Scheme of boundary hopping procedure in calculating the
reflection response of an interface, source at (xo, 0), receiver at
( x , O ) . The source field is decomposed in downward propagating
plane waves (PWD) with amplitudes B , ( p , o),equation (41). A
summation over the individual plane-wave contributions at the
interface d,(x) (PWC) results in the reflected,wavefield (42). It is
again decomposed into upward propagating plane waves with
amplitudes A,, equation (43). The plane waves are propagated to
the receiver level (z = 0) where the wavefield results from a PWC,
equation (44).
X
xo
z.1
"3
Figure 8. Scheme of boundary hopping procedure in calculating the
reflection response of an interface d2(x) underneath an interface
d,(x), source at (xo, 0), receiver at ( x , 0). The source field is
decomposed in downward travelling plane waves (PWD) with
amplitudes B , ( p , o),equations (41). A summation weighted with
the transmission coefficient yields the transmitted wavefield, (PWC)
at d,(x), equation (46).A further PWD in downward propagating
waves according to (47) allows for the calculation of the reflected
wavefield at the second interface d,(x) with a PWC, equation (48).
In order to propagate it to the surface, a PWD in upward travelling
waves (49) with amplitudes A, i s required. These waves are
weighted with the transmission coefficient at d , ( x ) and combined
with a PWC to the wavefield at d,(x) (50). A further PWD with
amplitudes A, and PWC at z = O results in the wavefield at the
receiver level.
= P ( x - x o ) + 41 d*(x),
t 2 =B(xr
- x ) + q1 d l ( x ) .
A comparison of this formula, derived from an RSIR and
equation (26) based on the KHIR, reveals that the only
difference is the directivity factor. In the case of the KHIR it
depends on all three variables of integration whereas for the
RSIR it only depends on two of them. This property allows
us to break the three-fold integral into a sequence of PWC's
and PWD's. A similar procedure is not feasible with (26).
The boundary hopping procedure follows for the
two-layer case where the reflection from the second
interface is to be calculated. Geometry and notation become
obvious by looking at Fig. 8. As in the one-layer example
one starts with a PWD of the downward travelling part of
the source field, B l ( p , w ) of equation (41). The fraction of
the wavefield at the first interface that will propagate further
down is expressed via a PWC weighted with the
transmission coefficient :
(46)
Note that the only difference compared to (42) is the use of
the transmission instead of the reflection coefficient. The
PWD of the wavefield propagating towards the second
interface is given according to equation (40):
Once the PWD above the second interface is known the
value of the reflected wavefield at the interface is similar to
(42):
U2[x,d,(x), w ] =
I
14 B,(p, w)R,(p, x)eiw~Px+qzdz(x)l
dP.
(48)
A PWD yields the amplitude of the upward travelling
waves, according to (43):
022(x, p ) = 1
P d;(x).
+q2
In order to get the fraction of the wavefield at d,(x) that
propagates towards the surface, a PWC weighted with the
upward transmission coefficient is required:
(50)
The next two steps correspond exactly to (43) and (44) in
the single-layer case.
The extension of the scheme of PWD's and PWC's for an
arbitrary stack of layers and arbitrary propagation paths
including multiples is evident.
SYNTHETIC SEISMOGRAMS
We present two examples of synthetic seismograms
calculated with the formulae outlined in the preceeding
Waves in heterogeneous media
683
(a)
v, = 5 KMlSEC
f1 = 2.4GICM’
a
x
b
$= 2.8WCM3
Y
v2 = 6 KNlSEC
$2 = 2.8 GICM’
10..
\5 = 7 KMlSEC
f3
= 3.2 WCM’
S
E
C
0
10
20
KM
30
Figure 9. Acoustic model (a) and corresponding synthetic
reflections calculated with formulae (42)-(44). The step in the
model causes a time delay in the seismograms and diffractions. As
the critical distance is about 20 km, the total reflection beyond that
point results in increasing amplitudes. The last two traces exhibit
the evolving refraction as a forerunner of the reflection.
section. First we show the acoustic reflection response of a
step-like interface (Fig. 9a). Fig. 9(b) displays the synthetic
seismograms in an offset range between 0 and 30 km to the
right of the shotpoint, with a 15 Hz Kiipper source wavelet.
These are calculated with formulae (42) to (44) with the ray
parameters replaced by angles of propagation and with
appropriate angular+quations
(42) and (44)-and
spatial-equation (43bwindows. The step in the interface
causes a delay in the traveltime curve beyond 6km. In
addition to simple ray theory, diffractions at the termination
of both branches of the seismic onsets are generated. With
increasing distance the amplitude of the reflections grows
despite the increasing spherical spreading. This is caused by
the total‘reflection of waves in the acoustic case. On the last
two traces the refraction starts to evolve. A refracted wave
would not be visible if seismograms were calculated with
classical Kirchhoff methods (e.g. Kampfmann 1988). O n the
contrary, an artificial diffraction would be associated with
the critical distance.
The second example is a two-layer acoustic case with two
interfaces (Fig. 10a). The first one at 4 k m depth is
deformed by a synclinal structure; the second one at 10 km
depth is plain. The seismogram section of Fig. 10(b)
0
10
20
30
KM
Figure 10. Acoustic two-layer model (a) with a bump in the first
interface. The synthetic section (b) shows the reflection from the
second interface calculated with formulae (42)-(SO). The
irregularity of the first interface results in a focusing of energy at
8 km, in diffractive onsets around 8 km and in distortions of the
wavefront beyond 18 km.
represents the response from the second plain interface. It is
calculated with formulae (42)-(50), again with the ray
parameters replaced by angles and associated with
appropriate spatial and angular windows. Without the first
interface a simple subcritical reflection would be expected
because the maximal observational distance of 30 km is
smaller than the critical distance. All complexities of the
wavefield are therefore caused by the irregularities of the
first interface. These irregularities consist of the effect of
energy focusing at 8 km and the diffractive features around
this distance. A more surprising point is the complex
waveshape at distances beyond 18 km. It is actually a result
684
F. Wenzel, K.-J. Stenzel and U.Zimmermann
of diffractive effects during downward and upward
propagation of the wavefield through the first interface.
coworkers. This is a contribution in the context of the
German Deep Continental Reflection Seismic Program
(DEKORP), Geophysical Institute Contribution No. 416.
CONCLUSIONS
A new method for calculating synthetic seismograms with an
approach based on Kirchhoff-type representations of the
wavefield governed by the acoustic wave equation is
presented. It can handle the transmission and reflection of
acoustic waves in laterally inhomogeneous layered media.
For the case of a reflection from an interface overlain by
homogeneous material the solution can be transformed to
the classical Kirchhoff method (e.g. Kampfmann 1988) by
application of stationary phase approximations.
If the reflection response from a second interface is to be
calculated, the available solutions, e.g. Deregowski &
Brown (1983), d o not take diffraction effects into account
that arise when the wavefield travels through the overlying
curved interfaces. The boundary hopping solution, however,
reproduces these effects in the frame of the local-plane-wave
assumption. The derivation of the method implies a new
look on Huygens’ principle, e.g. its reformulation in terms
of plane waves rather than cylindrical or spherical waves.
An important step is the replacement of the KirchhoffHelmholtz integral representation (KHIR), which involves
knowledge of the wavefield and its directional derivative at
every boundary, by a Rayleigh-Sommerfeld integral
representation (RSIR) which contains only the wavefield.
The construction of the RSIR requires the design of a new
Green’s function which can be viewed as an extension to the
non-planar case of the one given by Sommerfeld for the
optic screen problem. It allows us to write the propagation
of waves through a stack of layers with curved interfaces as
a sequence of plane-wave decompositions (PWD) and
plane-wave compositions (PWC). With a KHIR this would
not be possible. Numerical examples indicate the superiority
of the new method in the single layer case because
refractions are handled appropriately in contrast to the
standard Kirchhoff approach. A two-layer case shows strong
diffractive effects of interfaces in the wavefront.
ACKNOWLEDGMENTS
Financial support for this work was granted by Federal
Ministry of Research and Technology, project R G 78070.
We owe our thanks to G. Miiller (Frankfurt) and C. P. A.
Wapenaar (Delft) for useful critical comments on the
manuscript. From N. Frazer (Hawaii) we learned that a
similar approach has been worked out by himself and
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