GEOPHYSICS,
VOL. 52, NO. 7 (JULY
On the imaging
Norman
1987); P. 931-942, 1 FIG.
of reflectors
in the earth
Bleistein”
ABSTRACT
The scale factor multiplying the singular function is
proportional to the geometrical-optics reflection coefficient. In addition to its dependence on the variations
in sound speed, this reflection coefficient depends on an
opening angle between rays from a source and receiver
pair to the reflector. I show how to determine this unknown angle. With the angle determined, the reflection
coefficient contains only the sound speed below the reflector as an unknown, and it can be determined.
A recursive application of the inversion formalism is
possible. That is, starting from the upper surface, each
time a major reflector is imaged, the background sound
speed is updated to account for the new information
and data are processed deeper into the section until a
new major reflector is imaged. Hence, the present inversion formalism lends itself to this type of recursive implementation.
The inversion proposed here takes then form of a
Kirchhoff migration of filtered data traces, with the
space-domain amplitude and frequency-domain filter
deduced from the inversion theory. Thus, one could
view this type of inversion and parameter estimation as
a Kirchhoff migration with careful attention to amplitude.
In this paper, I present a modification of the Beylkin
inversion operator. This modification accounts for the
band-limited nature of the data and makes the role of
discontinuities in the sound speed more precise. The
inversion presented here partially dispenses with the
small-parameter constraint of the Born approximation.
This is shown by applying the proposed inversion operator to upward scattered data represented by the Kirchhoff approximation, using the angularly dependent
geometrical-optics reflection coefficient. A fully nonlinear estimate of the jump in sound speed may be extracted from the output of this algorithm interpreted in the
context of these Kirchhoff-approximate data for the forward problem.
The inversion of these data involves integration over
the source-receiver surface, the reflecting surface, and
frequency. The spatial integmis are computzd by Thor
method of stationary phase. The output is asymptotically a scaled singulnrfunction
of the reflecting surface. The singular function of a surface is a Dirac delta
function whose support is on the surface. Thus, knowledge of the singular functions is equivalent to mathematical imaging of the reflector.
INTRODUCTION
array. Extension to other wave equations is also quite apparent from the presentation. Thus, the method provides a unified
inversion theory for all of the source-receiver configurations
used in practice, including vertical seismic profiles.
Despite the fundamental significance of Beylkin’s results, I
find that further analysis is necessary for implementation of
his method on reflection seismic data. For example, the results
leave imprecise the manner in which his method actually produces the discontinuities in the sound speed. The theory only
predicts that the result is a leading-order, high-frequency
asymptotic inversion operator. As presented, the theory does
not account for the band-limited nature of the input data and
does not relate the output to the information being sought.
Furthermore, because the inversion operator is based on a
This paper is motivated by the brilliant paper by Beylkin
(1985). Beylkin presented a theory for asymptotic inversion of
observations for the constant-density acoustic wave equation.
The method is based on the Born approximation for the forward problem. It allows for a completely general background
sound speed in the inverse problem, as well as an assortment
of possible source-receiver configurations broad enough to accommodate most of the cases of interest in seismic exploration
and other applications. For example, the method applies to
zero-offset data; common-source (or single-source), multireceiver array data (or the reverse); or fixed-offset data. The
inversion of the data is an integral over the source-receiver
Manuscriptreceivedby the Editor January31, 1986;revised manuscript received December 12, 1986.
*Center for Wave Phenomena, Department of Mathematics, Colorado School of Mines, Golden, CO 80401.
(~71987 Society of Exploration Geophysicists. All rights reserved.
931
Bleistein
932
LIST OF SYMBOLS
.4(x, x3 = Amplitude of ray-theoretic, or WKBJ,
Green’s function for background
sound speed with source at x, and
observation point x.
c(x) = Background sound speed above the
reflector.
c+(x) = Sound speed below the reflector.
D(& o) = Observed data us [x, (Q, x,(g), w]
[equation (3)].
6, C@(x,x’,
x, - x,)] = Band-limited Dirac delta function of 4,
(defined below) with x’, x,, and
x, evaluated at the stationary point,
defined by equation (A-l), as functions
of x [equations (18) and (19)].
6,(s) = Band-limited Dirac delta function of s,
the arc length normal to a surface on
which s = 0. Band-limited singular
function of ihe~ surfaces[equation (22)j.
F(w) = Filtered (smoothed and tapered) source
function in the Fourier domain.
@@, x’,
x,, x,) = Phase of the inversion operator applied
to Kirchhoff-approximate field data
[equation (9)].
[@,,I = Hessian matrix of the phase of the
inversion operator applied to Kirchhoffapproximate field data [equation (16)].
gj = First fundamental form of differential
geometry for the reflector S [equations
(11) and (12)].
y(x) = Singular function of the surface S.
representation of the upward propagating wave by the Born
approximation, one could only have confidence in the results
for small discontinuities in sound speed.
I propose here a modification of the Beylkin inversion operator that addresses each of these issues. I show that the smallparameter constraint of the Born approximation can be at
least partially eliminated by applying the inversion operator I
propose here to upward scattered data represented by the
Kirchhoff approximation, using the fully nonlinear and angularly dependent geometrical-optics reflection coefficient. This
operator differs from Beylkin’s in a power of w, the frequency
variable, and in a spatial factor as well. The output of the
inversion operator presented here is interpreted in terms of its
effect on these Kirchhoff data.
The model data are represented in the frequency domain as
an integral over the reflecting surface. The inversion of these
data requires filtering in the frequency domain, then integration over frequency and over the source-receiver surface. Thus,
the application of the inversion operator to these data is a
multifold integral over the source-receiver surface, the reflecting surface, and frequency. The spatial integrals are computed
ys(x) = Band-limited version of y(x).
h(x, 5) = Fundamental determinant of the Beylkin
inversion formalism [equation (5)].
ii = Upward normal vector on the reflector.
p(x, x,) = Vr(x, x,) [equation (C-2)].
R(x’, xs) = Ray-theoretic reflection coefficient
[equation (7)].
S = Reflector in Kirchhoff representation of
upward propagating wave.
S, = Observation surface.
S, = Domain of integration in 5 variables which
define the observation surface.
u = (a,, 02) = Parameters used to define the reflector.
r(x, x,) = Ray-theoretic traveltime between x and
x,.
8 = Angle between the normal to a surface at
the point x’ and the ray from x, or x, to
x’, under the stationarity conditions (A-l).
The openingsangle between these rayysand
normal is subject to Snell’s law of
reflection.
as cx, (5)?
x,(g) o] = Observed field data.
x = Point at which the output of the inversion
operator applied to O(& o) is to be
evaluated.
x’ = x’(a) = Point on reflecting surface.
x,, x, = Source and receiver coordinates,
respectively [equation (l)].
5 = (<,, 5,) = Parameters labeling source and/or
receiver points; i.e., x, = x,(g) and/or
x, = x,(g) [equation (l)].
by the method of stationary phase. I show that the output is
asymptotically a scaled singular function of the reflecting surface. The singular function of a surface is a Dirac delta function whose support is on the surface. Thus, knowledge of the
singular functions is equivalent to mathematical imaging of
the reflector. The effects of band limiting on the delta function
are readily determined and can be accounted for in the analysis of the output of the inversion operator.
The scale factor multiplying the singular function is proportional to the geometrical-optics reflection coefficient. In addition to its dependence on the variations in sound speed, this
reflection coefficient depends on an opening angle between
rays from a source and receiver pair to the reflector. These
two rays satisfy Snell’s law at the output point on the reflector.
The opening angle is unknown. I show how to determine this
unknown angle by using two inversion operators whose amplitudes differ only by a multiplicative factor. With this angle
determined, the reflection coefficient contains only the sound
speed below the reflector as an unknown, and it can be determined.
A recursive application of the inversion formalism is possi-
Imaging
ble. That is, starting from the upper surface, each time a major
reflector is imaged, the background sound speed is updated to
account for the new information and data are processed
deeper into the section until a new major reflector is imaged.
The inversion is produced pointwise, rather than globally as in
Fourier inversion methods. Hence, the present inversion formalism lends itself to this type of recursive implementation.
In these results, the upper surface is allowed to be curved.
Thus, the inversion eliminates two preprocessing steps usually
applied to seismic data. The first step is a static correction for
variable height of the source-receiver array. The second is
stacking to produce “equivalent” zero-offset (backscatter)
data.
Central to the derivation of these results is the method of
multidimensional stationary phase. The interpretation of the
output of the inversion algorithm in terms of a reflector image
and the estimation of the reflection coefficient arise in a natural way in this method. Furthermore, the method predicts that
such imaging will occur only at those points on the reflector
for which there is a specular pair of rays from the source and
the receiver to the surface point being imaged. This prediction
ties the inversion back to the required source-receiver array
necessary for imaging the reflectors in the subsurface.
The present method neglects multiple reflections. Furthermore, in order for a reflector to be located accurately, the
sound speed above the test reflector for the model data must
be close to the “true” sound speed. Finally, the estimate of
sound speed below the reflector is determined from eequations
involving the value of the sound speed above the reflector.
Thus, the background sound speed above the reflector of interest must be close to the true value for the value below that
reflector to be close to the “true” value. In this sense, I have
not completely dispensed with the small perturbations required for the Born approximation. However, having met
these criteria, the method allows the increment in sound speed
across the reflector to be arbitrary. By adjusting the background based on such an “accurately” determined estimate of
sound speed, one could hope that the necessary constraints
will apply at the next deeper reflector.
The verification of the two-and-one-half dimensional (2.5-D)
specialization of this modification of Beylkin’s result has already been presented in Bleistein et al. (1987). In that case, it is
assumed that the data are gathered only on a single line on
the surface and that all parameters of the subsurface are functions of the transverse variable along that line and that depth.
Significant simplifications occur in the analysis because fourfold integrals over two surfaces appearing in the analysis here
reduce to twofold integrals over two curves in the 2.5-D case.
Furthermore, a certain 3 x 3 matrix central to Beylkin’s approach appearing in the present analysis reduces to a 2 x 2
matrix in the 2.5-D case and can be analyzed much more
readily.
The modification of Beylkin’s approach that I use is a generalization of the method previously employed in Cohen and
Bleistein (1979), Bleistein and Cohen (19821, Bleistein and
Gray (1985), Cohen et al. (1986), and Bleistein et al. (1985,
1987). The essential feature of this modification is a fundamental result in Fourier analysis, namely, given the Fourier transform of a function with surfaces (3-D) or curves (2-D) of discontinuity, multiplication of the Fourier data by kik before
inversion produces the array of singular functions of the dis-
Reflectors
933
continuity surfaces or curves. In the application to Beylkin’s
result, I need only identify his inversion representation as an
inverse Fourier transform and then identify k for this representation. In fact, Beylkin (1985) provides the Fourier interpretation of his formalism. It then remains to verify the validity of the application of this theory to upward scattered data in
this general context, which I present here.
The main new results presented are as follows. First, I
modify Beylkin’s (1985) inversion operator to produce a reflector map. Second, I show how to estimate the new value of
sound speed below the reflector from the output. This estimate
is based on application of the inversion operator to bandlimited Kirchhoff data. Hence, it accounts for band limiting
and it is not subject to the small-perturbation constraint of the
Born approximation, even though Beylkin’s inversion operator was originally motivated by the Born representation of
the upward scattered data. This analysis is carried out for the
cases of common-source data, common-receiver data, and
common-offset data and a background propagation speed that
depends on all three spatial variables. The analysis is achieved
even though the geometrical-optics reflection coefficient depends on an a priori unknown opening angle between an
incident and reflected ray pair at the point in question.
Computer implementation of this algorithm proceeds along
the same lines as described in Bleistein et al. (1985) and Cohen
and Hagin (1985). First, each data trace is transformed to the
frequency domain, then it is filtered and inverse transformed
to provide a new data set of traces. For each output point, a
summation is then carried out over the traces in the sourcereceiver array. The time on each trace is taken to be the sum
of the traveltimes from the source and receiver points to the
output point. In this summation, one must honor causality
and antialiasing in the spatial domain. The computation of the
traveltimes and amplitude of the inversion operator becomes
progressively more intense as the complexity of the background is increased, from constant to depth-dependent to dependent on depth and one or both transverse variables.
The inversion algorithm [defined by equation (4)] takes the
form of a Kirchhoff migration (Schneider, 1978) of frequency
filtered and spatially weighted traces. Indeed, one might view
this type of inversion as a Kirchhoff migration with an integral kernel dictated by inversion theory. A major feature of
the frequency-domain filter is a factor of iw, which amounts to
a generalization of the 8/& which is part of the Kirchhoff
algorithm.
The zero-offset constant-background Born inversion (Cohen
and Bleistein, 1979; Bleistein et al., 1985) is a special case of
this general theory, as are the zero-offset and depth-dependent
background inversions (Bleistein and Gray, 1985; Cohen and
Hagin, 1985). The common-offset, constant-background inversion proposed by Sullivan and Cohen (1985) is also a special
case of this general theory. The Born inversion of Clayton and
Stolt (1981) is not a special case. There. all upper surface
points are required to act as both source and receiver locations of common-source or common-receiver experiments,
because the authors must take the Fourier transform with
respect to both source and receiver locations, Furthermore,
Fourier transformation over source-receiver locations requires
a flat upper surface; the present theory does not.
The work reported by Beylkin and associates in the open
literature (Beylkin et al., 1985; Miller et al., 1986) focuses on
Bleislein
934
wide-band inversion and the imaging of reflectors as step functions characterizing the discontinuity in the medium parameters across the reflector. This is also true of the work of Esmersoy et al. (1985) and Esmersoy and Levy (1986).
ASYMPTOTIC INVERSION OPERATOR
reserved for the surface on which x, and x, are located. The
domain of integration in o is limited by the filter F(w). I take
this function to be symmetric and smoothly approaching zero
at the ends of its support. I think of F(w) as a smoothly
tapered version of the source wavelet. The functions T(X, x,)
and A(x, x,) [r(x, xr) and A(x, xr)] are the WKBJ (or raytheoretic) phase and amplitude of the Green’s function with
the source at x, [xr] and observation point x.
The function h(x, 5) is the essential element of Beylkin’s
result. It is the determinant
I consider a seismicexperiment carried out on the surface in
-which the source-receiver pairs x,_ and x,, respectively, are
identified by a parameter 5 = (c,, c2) as follows:
x, =x.~(&
x, = x,@.
(1)
For example, for the case of a common-source configuration,
x, would be a constant vector denoting that fixed position and
the function x1(c) would be a parametric representation of the
receiver surface; for the common-receiver case, the roles of x,
and x, would be reversed. For the common-offset case or
common-midpoint case, both x, and x, would vary with 5.
I assume that the sound speed c(x) is known down to a
reflecting surface S; below S, the sound speed takes on new
values c+(x) to be determined. along with the surface S itself.
It is assumed that u(x, x,, o) is the response to an impulsive
point source at x,. Above S. Msatisfiesthe wave equation
Vu + ;
u = -6(x - x,).
(2)
Below S, u satisfies this equation with the right side equal to
zero and c replaced by c+ Furthermore, u must satisfy two
continuity conditions on S, namely, that u and i?u/dnare continuous across S.
The total solution above S may be viewed as the sum of a
free-space Green’s function plus the upward-scattered response to the reflector. The first term satisfies equation (2)
everywhere, with c(x) analytically continued throughout space
as an infinitely smooth function. Since I only use the leadingorder asymptotic (WKB) Green’s function, the differences due
to different choices of c(x) will not affect the final result. The
second term I denote by us(x, x, , w). Thus, the data for the
seismic experiment are
I%> a) = us x1(5), %(5), 0
[
I
(3)
I introduce the following inversion operator to process these
data:
I h(x, 5 )I
4x7 x,)4x, x,) I Vr(x, x,) + Vr(x, x,) I
x
X
iwdw F(w)
exp
t(x, x8) + T(X, xJ
D(g, 0).
(4)
In this equation I have used the following notation. The
domain of integration Ss is the set of 5 values which are
required to cover the source-receiver array. The notation S, is
It is assumed throughout the discussion that follows that
h # 0. Interpretations of this result are provided in Beylkin
(1985) and in Beylkin et al. (1985). Beylkin includes in his
inversion operator a “neutralizer” function which is equal to
unity in a domain in which h is bounded away from zero from
below and vanishes infinitely smoothly in the neighborhood of
the zeroes of h. That is a technical detail which does not affect
the asymptotic analysis below. In practice, one would simply
carry out the numerical integration suggested by the above
formula over the domain where h is positive, perhaps tapering
the integrand near the zeroes of h.
The inversion formula (4) was deduced as follows. I started
from Beylkin’s (1985) equation (4.3) and expressed the timedomain data of that equation as an integral of the frequencydomain data. Asymptotic analysis indicates that for highfrequency band-limited data, this inversion formula will produce a band-limited step function wherever the sound speed
has a discontinuity. In a theory developed in Bleistein (1984),
it is shown that multiplication by an extra factor of iw will
produce a band-limited Dirac delta function which peaks on
the discontinuity surface; this delta function is the singular
function of the surface. The plot of this function is an image of
the reflector. The other factors in the amplitude provide the
desired scaling of the singular function, as is shown below.
The spatial integration in equation (4) is a Kirchhoff migration of the processed traces. The processing amounts to filtering and tapering in the frequency domain before inverse transforming to the time domain. The time at which the processed
traces are evaluated is the traveltime from source to output
point to receiver in the background sound speed. Any highfrequency model of the data can be expected to have the same
type of phase, except that the output point would be replaced
by a subsurface scattering point. Thus, the inversion may be
viewed as a matched, spatially varying filter, again with its
amplitude determined from the inversion theory. A major objective of the remainder of this paper is to verify that I have
picked this filter correctly.
The factor iw/l Vz(x, xS) + VT(X, xJI is a generalization of
the operator d/dz appearing in Kirchhoff migration (Schneider, 1978). In the frequency and wave-vector domain, i?/& is
equivalent to multiplication by ik, . It can be shown that for a
Imaging
reflector which is not horizontal, the latter filter will produce a
pulse, weighted (in part) by the cosine of the angle between the
normal to the reflector and the z-direction. The generalization
used here avoids that cosine factor. That is one of the main
results of the singular function theory (Cohen and Bleistein,
1979; Bleistein, 1984).
APPLICATION
TO KIRCHHOFF
935
Reflectors
where
In terms of these parameters,
dS’=\/;do,da,,
with g the first fundamental form of differential geometry for
S.
DATA
I now apply the inversion formula to the upward-scattered
response from a single reflector S. I use the Kirchhoff approximation to represent those data. This representation can be
found in many sources, including Bleistein [1986, equation
(4911.In the notation used here, the result is
D(&, 0) -
io
s
W,
xJA(x’, x$4(x’, xr)
s
xii.
1
V’r(x’, x,) + V’T(X’, x,)
x exp
1
II
dS’.
io ~(x’, x,) + r(x’, x,)
i[
(11)
(6)
In this equation, V’ denotes a gradient with respect to the x’
variables and R(x’, XJ is the geometrical-optics reflection coefficient
=/det[~.~~~,
k,m=l,2.
(12)
Here the bold times sign denotes the vector cross product and
the bold multiplication dot denotes the vector dot product.
I now apply the method of stationary phase to equation (8)
in the four variables (5, a). The phase Q is a function of these
variables through the dependence of x’ on o and the dependence of x, and x, on 5. Equation (9) is used to write the four
first derivatives of @ in terms of the derivatives of the traveltimes :
1dx,
1
.
d5,
+
R(x’, XJ =
v, tw, XI) -
7(x, XJ
and
(7)
The unit normal li points upward and a/an = ii - V’. This
result is substituted into equation (4) to obtain the following
multifold integral representation of the output B(x) when applied to these synthetic data:
I 4x, 5)I
xA(“, x,)dx, X,)tVT(X,X,) + VT@,X,)1
X
s
4~‘~ WW, x,)A(x’, x,) exp
d do F(o)
~WD(X, XI, x,,
x,)
s
xii-
V’s(x’, x,) + V’T(X’, x,)
1
dS’
(8)
In this equation
@(x, x’, x,, XI) =
T(X', Xs) + T(X', Xr)
-
L
1
T(X, XJ + 7(X, x,)
(9)
is the difference of traveltimes, i.e., the source point to the
input point to the receiver point, minus the source point to the
output point to the receiver point. The surface S is described
parametrically in terms of two parameters (0,. oZ) by an equation of the form
x’ = x’(a),
(10)
(13)
d@
= v’
do,
1$,’
T(X',Xs)+ T(X’, X,) .
m= 1, 2.
m
In this equation, V, is a gradient with respect to the variables
xs; similarly, V, is a gradient with respect to x,. The stationary
points- ins(5, 0) are determined by requiring that these first
derivatives all be equal to zero.
In Appendix A, I discuss the conditions under which @ is
stationary. The stationary phase conditions are stated as
equation (A-l). Also in that appendix I show that, for x on the
surface S, there is a unique stationary triple x’, x,~, and x,,
with x’ = x. This is shown for the following source-receiver
configurations of practical interest: common-source, commonreceiver, and common-offset. Although I only consider here
the fully three-dimensional problem. this analysis specializes
to the cases of 2.5-D inversion.
I focus attention on this stationary point when x is in the
neighborhood of S. That is, this is the stationary point which
has the limit x’ = x as x approaches S. If there were no
source-receiver pair in the seismic survey under consideration
which included the particular x, and x, needed to complete
the stationary triple, then the asymptotic contribution for that
point x would be of lower order in o and almost always of a
smaller numerical value after the w integration than the result
I obtain below. Thus, I proceed under the assumption that
such a stationary triple has been determined and that the
corresponding values of 6 and 5 are interior points of their
respective domains of integration.
The result of applying the method of stationary phase to
936
Bleistein
by equation (15) becomes
equation (8) is the following:
P(x)- - Rx’, x,)
4x’, x&w, x,)
4% x,).4x, x,)
I(x)=~ST(~)exp(iw4x,r’,x,,x,))dw.
I h(xv5)I
’
) det [$,I
Jtiz1VT(X, x,) + Vr(x, x,) I
x a . [V’r(x’,
XJ + Vt(x’,
x,)1&
I(K).
(14)
I assume that the original source was impulsive. Thus, from
the assumptions about F(w) in the previous section, it can be
seen that Z(x) is a band-limited Dirac delta function of the
argument G(x, x’, x,, x,). Therefore, I set
In this equation, CJis defined by equation (12) and
I(x) = 6,
I~x)=~SP(m)erpjio~x.x’,x,.x.)
+ i(sign w)(K/~) sig [@,:.I
do.
(15)
1
This integral, as well as the entire right side of equation (14),
are functions of x alone, because x’, x, , and x, are determined
as functions of x from the stationarity conditions (A-l). The
matrix [QJ is a 4 x 4 matrix
-
[
@(x, x’, x,, x,)
11
(19)
where I have used the subscript B to represent the band limiting.
The function Q is equal to zero on the surface S. Thus, the
support of this delta function includes S. In fact, this is the
only zero. To see why this is so, take the gradient of 0 with
respect to x, with x’, x,, and x, defined by the stationarity
conditions equation (A- 1):
i,m = 1, 2;
P&l =
(16)
det [0J denotes the determinant of this matrix and sig [Cp,,]
denotes the signature of the matrix, which is the number of
positive eigenvalues minus the number of negative eigenvalues
of the matrix.
Since it is expected that b(x) peaks for x on S, we are
interested in evaluating equation (14) for x near S. First consider the behavior of the matrix [@,,I in equation (16) when x
is on S. In this case, cs can be fixed b&we evaluating the
second derivatives with respect to & and 5,. In that limit,
Q, = 0; the entire 2 x 2 matrix in the upper left-hand corner of
[QJ is a matrix of zeroes. The determinant of [CJ,,] is just
the square of the determinant of the 2 x 2 matrix in the upper
right-hand corner:
det We,1=
(18)
!,et[e]]’
’
P[r(x’,
m
(20)
In this equation, each sum on k is zero, by the stationarity
conditions equation (A-l). Thus, the total derivative with respect to xj is just the partial derivative with respect to the
explicit xi in Q. One can now conclude using the following
argument that V@ is not zero. From equations (20) and (9),
V@ = -
(17)
with x’ evaluated at the stationary point x on S and x, and x,
evaluated so as to make the phase stationary.
From this result, the determinant is seen to be positive, so
that the eigenvalues of each sign must occur in pairs. Thus, the
only choices for sig (DC,) are -4 and 0 and the only effect that
the signature factor can have on the final result in equations
(14) and (15) is a multiplication by - 1 or + 1, respectively. In
Appendix B, I show that, in fact, the signature is zero and the
multiplier is + 1. I argue by continuity that if this signature is
equal to unity for x on S, then it must be equal to unity for x
in some neighborhood of S. I assume that this neighborhood
is at least a few wavelengths at the frequencies within the band
of the data. Then, the depiction of the output described below
will hold in a region around S sufficiently wide for the reflector to be detected. With sig [‘I+,,] = 0, the integral 1(x) defined
&(s)
$3(4
(21)
(22)
C-E
I V@ I
x,) + 5(x’, x,)]
I
By using Beylkin’s method [1985, equation (4.6)], which is an
expression for h(y, Q, one can show that h is, in fact, proportional to the magnitude of this vector. Thus, the assumption h # 0 assures V@ # 0. Consequently, the only zero of @
subject to the stationarity conditions [equation (A-l)] is the
surface S itself. By standard rules about delta functions, one
can now write I(x) in terms of a delta function of arc length
along a curve normal to S. Denoting that arc length by s,
_r(x)
%&,
Vr(x, XJ + V7(x, x,)
[
I wx,
x,1 + wx,
x,) I.
This delta function, with support on S, is the singular function of the surface S. Below, this function is denoted by y(x)
and its band-limited counterpart is denoted by yB(x). Determination of the singular function of a surface constitutes mathematical imaging of the surface. A plot of the band-limited
delta function ya(x) will, indeed, depict the surface. In fact,
standard seismic output depictsthe reflectors by plotting their
singular functions within a scale factor. By using the result
equation (22) in equation (14) with 6,(s) replaced by am, one
obtains
B(x)-- - R(x', XJ
4x’, x&qx’, x,)
4x, x,)A(x, x,)
I&, 5)I
xI detC@,,ll”2/ Vr(x, XJ + Vr(x, x,) 1’
x ii * [V’r(x’,
x,) + V’T(X’, x&&,(x).
(23)
937
Imaging Reflectors
Again, x’, x,, and x, are determined here as functions of x
by the stationarity conditions equation (A-l), so that the entire
result is a function of x. This result images the reflector
through the dependence of p(x) on the function ys(x). This
confirms part of the claim about the nature of the output of
the inversion operator equation (4) when applied to Kirchhoff
data. To complete the verification, it only remains to determine the peak amplitude of this result when x is on the reflector.
To determine this peak amplitude, I first introduce the
acute angle 8 between the upward normal to the surface and
the incident and reflected rays on the surface. Note that the
downward gradients V’r(x’, XJ and V’T(X’, x,) make angles of
(x - 0) with this normal and an angle of 20 with one another.
Therefore,
6 * V’T(X’, XJ + vqx’,
2 cos e
1
= - -
x,)
t3and c, [which is implicit in R(x, x,)] remain coupled in this
equation.
As a first step, I address the determination of 8. From equation (25), the first fraction in equation (28) can be recognized
as arising from the evaluation of IV$x, x,) + VT(X, x,)) at the
stationary point. This factor appears in the denominator in
the inversion operator defined by equation (4). By changing
the power of this factor in that inversion operator,
it is possible to change the power of the multiplicative factor
2 cos O/c(x) at the peak of the output of the inversion operator. Therefore, in addition to processing the data with the
inversion operator equation (4). I propose that the data be
processed with the operator
(24)
’
c(x’)
lk
x 4% 04x,
and
X
I Wx’,
x,) + V’r(x’,
x,) 12= - 2
c’(x’)
(1 + cos 20)
I
iw do
51
x,) I VW, x,) + wx, x,)lZ
F(o) exp
-iw
s(x, x3 + T(X, x,)
i
L
11
D(5, a),
(29)
2 cos 9 =
=
[ c(x’)
1
(25)
Finally, in Appendix C, I show that
I h(x>5)I
I det19eJII‘2
dii = I wx, %I + VT@,x,) I
=-
2 cos 0
x
c(x) ’
on
S.
(26)
By inserting the results equations (24)-(26) into equation
(23), one obtains the following result for p(x) at its peak; that
is, for x on S,
P(x) - R(x’> x,)yB(x),
x
on
S.
OF I3 AND c,
In order to determine the values of 8 and the velocity c+
below S, I use equations (22) and (25) to rewrite the result
equation (27) as
2 cos e
R(x, xs) &
P(x) * c(x)
F(o) dw,
x
on
Equaticn (28) shows that the-actual nmnericai value at the
peak depends on the area under the filter in the frequency
domain, the opening angle 0 between the normal and each of
the rays from x, to x on S and from x, to x on S, and the
reflection coefficient at that opening angle. We know the filter
and, hence, the area under the filter, but the separate elements
x
on
S,
(30)
and
P(x)
2 cos e
X
plo-
on
S.
(31)
’
c(x)
Consequently, when both inversion operators are applied to
the data, the locations of the peaks of either of them determine
the reflector and then the ratio of the peak values determines
cos 8. Thereafter, either peak amplitude provides a single
equation for the remaining unknown, c+ (x).
To see how thisworks out in detail, first rewrite the reflection coefficient in equation (7) in terms of 0 and x’ = x on S.
Note first that from the stationarity conditions,
&(x, XJ
an
S. (28)
s
F(o) do,
b,(x) - Rb, x,) ;
(27)
This confirms my original claim about the inversion operator
defined by equation (4). That is, when applied to Kirchhoff
approximate data and evaluated asymptotically, the operator
produces a band-limited singular function of the reflecting surface multiplied by a scale factor which, for x on S, is the
geometrical-optics reflection coefficient evaluated for some
particular choice of incident angle (through its dependence on
dr/c?n).
DETERMINATION
which differs from /3(x) in equation (4) by an extra factor of
(Vz(x, x,J + Vr(x, x,) ( in the denominator. Since it is necessary
to calculate IVT(X, x,) + v~(x, x,)1 anyway, simultaneous computation of this second inversion operator imposes no significant additional computer time
The asymptotic analysis of the output p,(x) applied to
Kirchhoff data is readily determined from the results for p(x).
This function also produces the band-limited singular function
yB(x) scaled by a different factor. At the peak, that scale factor
differs from the scale for p(x) by IVr(x, XJ + VT(X, x,)1-’
evaluated~at thenstationary pointyon S, given by equation (24)
with x’ = x. Thus,
cos 0
=- c(x)
With a slight abuse of notation, equation (7) can be rewritterr w fuilows :
cose r
1
--I-_-l
R(x, 0) =
c(x)
cos 8
Lc: (x)
r 1
4x1
1c:(x)
-+
---
sin’ Ol’/’
c2(x) 1
sin2 t31”’ ’
1
c?x)
(33)
938
Bleistein
Suppose that both operators p(x) and p,(x) have been computed for a data set. Furthermore, a particular point x has
been identified as being a peak of the band-limited singular
functions depicting the reflecting surface. Then cos 0 is determined from the ratio of the outputs. Furthermore, dividing the
peak value of p,(x) by the area under the filter in the frequency domain provides a value for the left side of equation
(33) at x. The solution of this equation for c+(x) is most easily
expressedin terms of the squared slowness.That is,
4R cos’ 0
(1 + R)’
1
(34)
This completes the determination of c, (x). This expression
is consistent with the angularly dependent geometrical-optics
reflection coefficient.
IMPLEMENTATION
As an integral operator, the inversion proposed here is a
linear operator on the data. Thus, given a standard data set
which is the response to many reflectors, the inversion will
treat the data from many reflectors (and, unfortunately, the
multiples) as described for a single reflector in the previous
sections. Reflector locations will be imaged in accordance with
traveltimes in the background sound speed.
The method can be used recursively. That is, suppose that
from the time section a first major reflector is identified in the
subsurface. Data are processed sufficiently deep to image that
reflector and estimate the sound speed below it. With this new
value of sound speed, the data are now processed through the
next major reflector, and so on, through the subsurface.
Computer implementation proceeds along the same lines as
in the papers cited is the Introduction. The original traces
must be transformed to the frequency domain, filtered, and
inverted to provide a modified data set in time Then, for each
output point, a weighted sum over the traces must be carried
out, with the weights as indicated in equation (4) or equation
(29). The time variable in the modified traces is taken to be the
sum of traveltimes from the source point to the output point
and from the output point to the receiver point.
As noted in the Introduction, the structure of this implementation is exactly the same as in Kirchhoff migration. However, use of the specific frequency-domain filter and spatial
weighting provides an output for which the amplitude can be
interpreted in the context of the Kirchhoff approximation of
the upward-propagating wave and- the geometrical-optics reflection coefficient.
The factor io appearing in equation (4) or equation (29) is
the generalization of the a/o’= operator appearing in Kirchhoff
migration. That operator in the Fourier domain is of the form
ik,. Processing without one of these operators would produce
band-limited steps at each reflector. With one of these operators in place, the output at each reflector is a band-limited
pulse, essentially a scaled image of the inverse transform of the
filter F(o). It can be shown that for a/& or ik,, the scale
includes a factor of the cosine of the angle between the normal
to the reflector and the z direction. When iw is used instead,
this cosine no longer appears. Thus, when analyzing the amplitude, a postprocessing step of estimating the inclination of
the reflector is eliminated before estimating the change in
sound speed.
Given the background medium, one must compute the
WKB traveltimes, their gradients, and WKB Green’s function
amplitudes, as well as the determinant h(x, 5) defined by equaton (5). For a constant-background medium, all of these functions can be expressedexplicitly as functions of the integration
variables. For a variable background, the determination of
these functions becomes progressively more computer intensive with increasing complexity of the background c(z), c(x, z),
and c(x, y, z).
In current research at the Center for Wave Phenomena, we
have had successwith sparse computation of these functions
and interpolation for intermediary values. These steps considerably diminish computer time I believe that the method
works for two reasons. First, the background is approximate
anyway, and interpolation can be thought of as a small modification of the intended background to another one nearby.
Second, since the inversion is an integration process, it tends
to “smooth over” small errors.
For the cases of zero-offset, common-source, or commonreceiver configurations, h(x, 5) can be shown to be related to
the same Jacobian as arises in determining the WKB amplitudes of the Green’s function. Hence, computing h(x, 5) adds
no great computer burden. For the case of (nonzero) common
offset, the relationship between these Jacobians is not so
direct.
On the other hand, Beylkin has observed that h(x, 5) is the
Jacobian of a certain transformation between the unit sphere
and the surface S,. The points on the sphere are defined by
the direction of the vector V[Z(X, x3 + ‘c(x, x,)]. Each choice of
5 defines a value of this gradient, hence a direction, and thus
defines a point on the sphere. The assumption h(x, 4) # 0
assures that the correspondence goes the other way as well.
That is, each direction corresponds to at most one choice of 6,
and this functional relationship between & and directions is
differentiable.
Now consider a family of rays that might be generated as
part of the traveltime computation required for this algorithm.
Given a ray tube of differential cross-sectional area, the initial
directions of those rays map out a differential area element on
the unit sphere and their emergence on the upper surface
maps out a differential area on S,. Since h(x, 5) is the Jacobian
of the transformation between these variables, its value must
be the ratio of those differential areas on the unit sphere and
on S,. When the rays are determined, these differential area
elements can be determined, as well. Thus, computation
of h(x, 5) always requires only a minor increase in computer
time over the computation of the elements of the WKB
Green’s function.
For zero-offset, common-source, or common-receiver cases,
the unit sphere of directions for h(x, 5) is exactly the unit
sphere of directions for a family of rays from the output point
x to a neighborhood of either the source-receiver point, the
receiver point, or the source point, respectively. For the case of
common offset, the gradient direction depends upon a sum of
two ray directions associated with two different ray families,
one to the neighborhood of the source point, the other to the
neighborhood of the receiver point. That is why h(x, 5) is not
so directly related to a WKB ray Jacobian for this case. None-
imaging
theless, as noted above, h(x, 5) is readily computed from information on the ray direction.
CONCLUSIONS
Motivated by an inversion operator proposed by Beylkin
(1985) I have proposed two other inversion operators. Each of
those operators is shown by asymptotic analysis to produce a
reflector map when applied to Kirchhoff-approximate input
data. The peak value of the output of these operators is proportional to the geometrical-optics reflection coefficient. The
output also depends upon the opening angle between specular
rays from the source to the reflecting surface and from the
receiver to the reflecting surface. I show how to determine this
opening angle by comparison of the two inversions. Thereafter, the velocity below the reflector is determined as well.
These results are valid for three source-receiver configurations
of interest: common (or fixed) source, common-receiver, or
common-offset, with the last of these including the zero-offset,
or backscatter, case. The analysis allows for a curved datum
surface. The structure of the inversion operator is exactly the
same as the structure of a Kirchhoff migration operator for
the same source-receiver configuration and the same background sound speed. However, the amplitude of the inversion
operator, as dictated by the inversion theory, allows for interpretation of the amplitude of the output in terms of the
geometrical-optics reflection coefficient. Also, the filter io used
here, as opposed to the $6~ operator of Kirchhoff migration,
has certain inherent advantages when analyzing the amplitude.
ACKNOWLEDGMENTS
The author gratefully acknowledges the support of the
Office of Naval Research, Mathematics Division, through its
Selected Research Opportunities Program, and the Consortium Project on Seismic Inverse Methods for Complex Structures at the Center for Wave Phenomena, Colorado School of
Mines. Consortium members are Amoco Production Company : Conoco, Inc. ; Digicon, Inc. ; Geophysical Exploration
Company of Norway A/S; Marathon Oil Company; Mobil
Research and Development Corp.; Phillips Petroleum Com-
pany; Sun Exploration and Research; Texaco USA; Union
Oil Company of California; and Western Geophysical.
The author also wishes to express his gratitude to Jack K.
Cohen for a critical reading of this paper and some helpful
suggestions.
REFERENCES
Beylkin, G., 1985, Imaging of discontinuities in the inverse scattering
problem by inversion of a causal generalized Radon transform: J.
Math. Phys.. 26,999108.
Beylkin, G., Oristagho, M., L., and Miller, D., 1985, Spatial resolution
of migration algorithms, in, Berkhout, A. J., Ridder, J., and van der
Wal, L. F., Eds.: Acoustical imaging, 14, Plenum Press, 155-168.
Bleistein, N., 1984, Mathematical methods for wave phenomena: Academic Press Inc.
~
1986, Two-and-one-half in-plane wave propagation: Geophys.
Proso.. 34.686703.
Bleistein, N., and Cohen, J. K., 1982, The velocity inversion
problem- -Present status, new directions: Geophysics, 47, 14977
isle.
Bleistein, N., Cohen, J. K., and Hagin, F. G., 1985, Computational
and asymptotic aspects of velocity inversion: Geophysics, SO, 12531265.
~
1987, Two and one-half dimensional Born inversion with an
arbitrary reference: Geophysics, 52, 2636.
Bleistein, N., and Gray, S. H., 1985, An extension of the Born inversion procedure to depth dependent velocity profiles: Geophys.
Prosp., 33,999-1022.
Clayton R. W., and Stolt, R. H., 1981, A Born WKBJ inversion
method for acoustic reflection data: Geophysics, 46, 1559-1568.
Cerveny. V., Molotov, I. A., PSencik, I., 1977, Ray methods in seismology : Univ. Karlova.
Cohen, J. K., and Bleistein, N., 1979, Velocity inversion procedure for
acoustic waves: Geophysics, 44, 1077.-1085.
Cohen, J. K., Hagin, F. G. and Bleistein, N., 1986, Three-dimensional
Born inversion with an arbitrary reference velocity: Geophysics, 51,
1552-I 558.
Cohen. J. K.. and Haein. F. G.. 1985. velocity inversion using a
stratified reference: Giophysics, SO, 1689-l 700..
Esmersop C., and Levy, 9: C.. 1086, Multidimensional Born inversion
with a wide-band plane-wave source: Proc., Inst. Electr. Electron.
Eng., 74, 466475. _
Esmersoy, C., Oristaglio, M. L., and Levy, B. C., 1985, Multidimensional Born inversion: Single wide-band point source: J. Acoust.
Sot. Am., 78, 105221057.
Miller. D.. Oristaelio. M. L.. and Bevlkin. G.. 1986. A new slant on
seismic imagini:
Classical migration’ and integral geometry:
Schlumberger-Doll Research Note, Program GEO-002; also, 1987,
Geophysics, 52, this issue, 9433964.
Schneider, W. A., 1978, Integral formulation for migration in two and
three dimensions: Geophysics, 43, 49-76; Sot. Explor. Geophys.
reprint series.4. Migration of seismic data.
Sullivan, M. F.. and Cohen, J. K., 1985, Pre-stack Kirchhoff inversion
of common-offset data: Res. Rep. CWP-027, Center for Wave Phenomena. Colorado School of Mines; also Geophysics, 52, 745-754.
APPENDIX
A
ANALYSIS OF THE STATIONARY
In this appendix, I discuss the conditions of stationary
phase, that is, the conditions under which the four first partial
derivatives of @ in equation (13) are all equal to zero. Since
the traveltime r is symmetric in its initial and final coordinates, each of the gradients appearing in equation (13) is a
p vector directed tangentially to the ray. For example,
V, r(x’, x,) = p(x,, x’) is a p vector tangent to the ray from x’
to x, (from the second argument to the first argument) evaluated at x, (evaluated at the first argument). This p vector has
939
Reflectors
PHASE CONDITIONS
magnitude l/c(x,) and is directed awny from the initial point
x’. Similarly, V’T(X’, x,) = p(x’, xs) is evaluated at x’, has magnitude l/c(x’), and is directed away from x,.
The result equation (13) and the notation for gradients intraduced here are used to write the conditions that the phase
be stationary as follows:
fix’, X8) + p(x’, x,)
m=
1,2,
(A-J)
940
and
dx
dx
dSm
dSnl
p(x,,x’)-‘+p(x,,x’).-
=p(x,,X).~+p(x,,x).dx,
m= 1,2.
dS,’
dSm
It is assumed that a proper parameterization has been used for
which the two vectors in each case (m = 1, 2) are linearly
independent.
The first line in equation (A-l) has the interpretation that
the tangents to the rays from x, and x, to the surface point x’
have equal projections on two linearly independent tangents
in the reflecting surface. Consequently, the projections of these
two vectors onto S must be equal (Snell’s law for reflection).
The magnitudes of the p vectors must be equal [to l/c(x’)],
and hence the out-of-plane components must be equal in magnitude as well. Indeed, the normal components of these vectors
are of the same sign and must be equal.
The second line in equation (A-l) ties the points on the
reflector and observation surfacesto the output point x. Consider the rays from x to the upper surface points x, and x,.
Similarly, consider the rays from x’ to the upper surface points
x, and x, . For each pair of rays, take projections on tangents
at their respective emergence points. The sums of these projections for each pair of rays must be equal. This must be true for
two linearly independent tangents at each point.
At first glance, it may not seem apparent that such a condition can ever be satisfied. However, consider the case in
which x is on the reflecting surface S. Then, for x’ = x (and Q
chosen accordingly), the two pairs of rays overlie one another
and these stationarity conditions are automatically satisfied
for nny pair of surface points x, and x,. Thus, we would only
have to find such a pair for which Snell’s law is also satisfied.
Indeed, if there were no such pair in the seismic experiment
being modeled, then that subsurface point would not be one
for which the stationarity conditions are satisfied and that
point would not be ipaged.
On the other hand, there are many candidates for sourcereceiver pairs on the upper surface when x’ = x. To find them,
proceed as follows. At x’ = x, pass a plane through the normal
to S. In the plane, choose two directions making equal angles
with the normal. Use these as initial directions for rays from
the point. Snell’s law is satisfied for this pair of rays. Both of
the pair of emergence points at the upper surface are candidates for a source-receiver pair. Vary the opening angle of the
ray pair in the normal plane and rotate the plane, thereby
obtaining a two-dimensional continuum of candidate sourcereceiver pairs in the upper surface.
Suppose now that such a pair is available in a given seismic
survey when x is on S. Given that pair, it is argued by continuity that for x near S there must by points x’, x,, and x,
satisfying equation (A-l) and they must be near the solution
obtained in the limit when x is on S.
Constant background soundspeed
Further insight into the stationarity conditions is gained by
considering the case of constant background speed and flat
layers, as in Figure A-l. Given a point x, a perpendicular is
dropped to the surface S. This determines a point x’, Pass a
plane through the normal and draw the rays at equal angles
to the upper surface. This determines a pair of points as candidates for x, and x,. For this pair of points, the sum of projections on either side of the first line of equation (A-l) is equal
to zero. Thus, this triple of points satisfies both conditions of
stationarity.
The three points x’, x,, and x, must be in the same plane to
satisfy Snell’s law. If x were not in the same plane, then the
projections of its p vectors would no longer be collinear and
could not sum to zero. On the other hand, the sum of projections of the p vectors from x’ would remain zero. Thus, the
first condition in equation (A-l) could not be satisfied. Similarly, if x is in the normal plane but not on the normal line,
the first condition could not be satisfied. That is, the conditions of stationarity are satisfied by three points x’, xs, and
x, which, along with x, lie in a plane normal to the reflector
with x’ at the foot of the normal to S drawn from x. The only
freedom left in these conditions, then, is in the opening angle
of the rays at x’ and the orientation of the normal plane.
Below, I discusshow these are further constrained for particular source-receiver configurations and this flat-reflector,
constant-background model.
Case 1: Zero offset
FIG. A-l. Triple x’, x,, x, satisfying the stationary phase conditions of equation (B-l) for a horizontal observation surface,
horizontal reflector, and constant background sound speed.
When the source and receiver are coincident, the opening
angle of the rays at x’ must be zero for both the source and
the receiver; both rays from x’ to x, and xr must be the
normal ray to the surface, passing through x. The stationary
point on the upper surface and the point x’ must have the
same transverse coordinates as x itself. The stationarity conditions are completely satisfied by these three points. Because
of the degenerate nature of this case, a specific normal plane is
not ‘determined. However, that is secondary to determining
the actual triple of points.
The generalization of this result to curved surfacesand variable background is fairly straightforward. Given x, find a
normal ray from S which passesthrough x. The initial point of
that ray on S is the point x’. The point where the ray emerges
on the upper surface S, is the source-receiver point which
completes the triple of points satisfying equation (A-l). For x
941
Imaging Reflectors
background case, the essential features of this analysis still
on S, there is clearly only one stationary triple. On the other
hand, for x on the evolute of S (the envelope of normals to S),
there will be more than one triple. In order for the asymptotic
methods used here to be valid, it is necessary to assume that
this evolute is a few wavelengths (at least three) away from S.
Thus, it is assumed that the reflector is not severely curved;
that is, the principal radii of curvature of the reflector must be
a few wavelengths long.
apply.
Case 3: Common receiver
One need only interchange the subscripts s and r in the
discussion of case 2 (common source) to obtain a completely
analogous conclusion for a common-receiver configuration.
Case 4: Common offset
Case 2: Common source
It is assumed that all of the offset pairs lie on lines that are
parallel. I rotate the normal plane containing x and x’ until it
is parallel to this set of lines. Indeed, the intersection of the
normal plane and the upper surface contains one of those
lines. Choose the opening angle of the rays from x’ so that the
rays emerge at the upper surface at a separation distance
equal to the common-offset distance. The emergence points
are the pair x, and x,.
Suppose now that the source point is fixed. Given x, drop
the normal to S and thereby determine x’. Pass a plane
through x,, x, and x’. This plane is normal to S. Draw the ray
from x, to x’. Draw the reflected ray in the given normal
plane, The emergence point on S, is the point x,. If x is on S,
set x’ = x and use the normal at that point and the fixed point
x,~to determine the normal plane. Then proceed to determine
x, as before, with x not on S.
In a theoretical model, receivers are spread over the entire
upper surface. In practice, the spread is finite. Thus, the spread
need not extend to the determined x,. In that case, the determined point x’ will not be part of a triple satisfying equation
(A-l) and will not be imaged. In the text I proceeded as if such
candidate points are indeed stationary.
Again I argue by continuity that for curved surfaces and
variable c(x) that does not differ greatly from the constant-
Case 5: Common midpoint
There will only be a solution to equation (A-l) in this case if
the common midpoint and x lie along a common normal to S.
Furthermore, in that case, all source-receiver pairs are stationary points. The method of stationary phase breaks down since
the stationary points are no longer isolated. This is a case
which requires further investigation.
APPENDIX
MATRIX
The purpose of this appendix is to show that the signature
of the matrix [@,,,I defined by equation (16) is equal to zero. I
consider first the special case in which the background sound
speed c in the region between the upper surface and the reflecting surface is constant, the layers are flat, and there is zero
offset between sources and receivers. In this case, the upper
surface and the reflecting surface are defined, respectively, by
X 1, =
X 1, --5
xzs=x2,=
B
SIGNATURE
and
7(x’, x,) = 1x’ - x, I/c
These results are used to simplify a, as defined by equation
(9), and then to compute the determinant in equation (16). The
analysis is further simplified by setting x’ = x. The result is
1,
(B-3)
5 2,
x3,
=
x3,
x;
=
crl,
x;
=02,
=
0,
(B-1)
For this matrix it is fairly straightforward to calculate the
characteristic equation. The result is
det~~V-~I]=~(l-L)+l]lIIHc~=O.
(B-4)
and
-
x; = H.
Furthermore, the traveltimes are just the distances between
the initial point and the final point, divided by c:
7(x, x3 = Ix - x, I/c,
7(x, x,) = I x - x, I/c,
T(X’, XJ = 1x’ - x, I/c,
(B-2)
This equation has two double roots, h = (1 f $)/2.
Since
two of the roots are positive and two are negative,
sig [@,,I = 0.
Now consider deforming this constant-background, zerooffset, flat-layered model into the true model. If the signature
is to change as the model is deformed, then at some point in
the deformation at least one eigenvalue must be zero. In fact,
exactly two eigenvalues would have to be zero at this point,
Bleistein
942
now add to that the assumption that our true model is not so
severely different from the flat-earth case to have caused h to
pass through a zero on the way from one model to the other.
Thus. sig [Q,,] = 0 for the true model as well.
since det [QJ is nonnegative and by assumption, the signature changes.
Appendix C shows that det [Q,,] is proportional to h(x, 5).
It has been assumed that h is nonzero for the true model. I
APPENDIX
C
RELATION BETWEEN Ir(x, 5) AND
det [@cJ AT THE STATIONARY POINT
In this appendix, equation (26) will be verified. !t is necessary to evaluate ( h(x, k) 1 as defined by equation (5) subject to
the stationarity conditions equations (A-1) and the additional
condition that x = x’. As a first step, x is replaced by x’. The
result is
first two elements of the first row are both zero by equation
(A-l ), while the third element is given by
2 cos 0
p(x’, x,) + p(x’, x,) * ii = -
l
1
CM ’
(C-5)
which follows from equation (24). Thus, in expanding the determinant of the product by the first row, it is only necessary
to consider the lower left 2 x 2 matrix after multiplication.
Now consider a typical term
1 (C-1)
$
dx’
PW, XJ + p(x: x,)
&
In this equation, I have used the notation
p(x’. x,) = V’t(x’,
x,),
p(x’. XJ = V’t(x’,
a
= 2l at, c;o,
x,).
K-2)
To calculate this determinant, the matrix is multiplied by a
matrix whose determinant is known:
(C-3)
1
k, m = 1, 2.
(C-6)
It now follows that if the matrix in equation (C-l) is multiplied by the matrix & before calculating the determinant, the
following result is obtained:
det h(x’, 6),/i
with the second equality being equivalent to equation (12).
Now, in multiplying 4 by the matrix in equation (C-l), the
d
‘ a,
5(x’, XJ + T(X’, x,)
ir+D
=ack
do,’
where each vector representsa column of J$.Note that
(C-4)
1
2 cos 8
= det Q,, ,
c(x’)
[
1
(C-7)
for x’ = x on S. The outer equality in equation (26) follows
from this result. The right equality in equation (26) follows
l”romequation (25).