Beylkin Determinant

GEOPHYSICS, VOL. 52, NO. 7 (JULY 1987); P. 931-942, 1 FIG.

On the imaging of reflectors in the earth

Norman Bleistein

ABSTRACT

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 Kirch- hoff approximation, using the angularly dependent geometrical-optics reflection coefficient. A fully nonlin- ear estimate of the jump in sound speed may be extrac- ted from the output of this algorithm interpreted in the context of these Kirchhoff-approximate data for the for- ward 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, knowl- edge of the singular functions is equivalent to mathematical imaging of the reflector.

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.

INTRODUCTION

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 for- ward 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 ac- commodate 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), multi- receiver array data (or the reverse); or fixed-offset data. The inversion of the data is an integral over the source-receiver

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 Beylkins 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

Manuscript received by the Editor January 31, 1986; revised manuscript received December 12, 1986. *Center for Wave Phenomena, Department of Mathematics, Colorado School of Mines, Golden, CO 80401. (~7 1987 Society of Exploration Geophysicists. All rights reserved.

931

932 Bleistein

LIST OF SYMBOLS

.4(x, x3 = Amplitude of ray-theoretic, or WKBJ, Greens 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 Kirchhoff- approximate 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.

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 openings angle between these rayys and normal is subject to Snells 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 = (

Imaging Reflectors 933

ble. That is, starting from the upper surface, each time a major continuity surfaces or curves. In the application to Beylkins reflector is imaged, the background sound speed is updated to result, I need only identify his inversion representation as an account for the new information and data are processed inverse Fourier transform and then identify k for this repre- deeper into the section until a new major reflector is imaged. sentation. In fact, Beylkin (1985) provides the Fourier inter- The inversion is produced pointwise, rather than globally as in pretation of his formalism. It then remains to verify the validi- Fourier inversion methods. Hence, the present inversion for- ty of the application of this theory to upward scattered data in malism lends itself to this type of recursive implementation. this general context, which I present here.

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 natu- ral 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 main new results presented are as follows. First, I modify Beylkins (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 band- limited Kirchhoff data. Hence, it accounts for band limiting and it is not subject to the small-perturbation constraint of the Born approximation, even though Beylkins 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.

The present method neglects multiple reflections. Fur- thermore, 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.

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 source- receiver 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 verification of the two-and-one-half dimensional (2.5-D) specialization of this modification of Beylkins result has al- ready 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 four- fold 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 Beylkins 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 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 modification of Beylkins 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-

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

934 Bleislein

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 Es- mersoy et al. (1985) and Esmersoy and Levy (1986).

ASYMPTOTIC INVERSION OPERATOR

I consider a seismic experiment 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. M satisfies the 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/dn are con- tinuous across S.

The total solution above S may be viewed as the sum of a free-space Greens 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 leading- order asymptotic (WKB) Greens 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 x 4x7 x,)4x, x,) I Vr(x, x,) + Vr(x, x,) I X iw dw 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

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 ray- theoretic) phase and amplitude of the Greens function with the source at x, [xr] and observation point x.

The function h(x, 5) is the essential element of Beylkins result. It is the determinant

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 Beylkins (1985) equation (4.3) and expressed the time- domain data of that equation as an integral of the frequency- domain data. Asymptotic analysis indicates that for high- frequency 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 filter- ing and tapering in the frequency domain before inverse trans- forming 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 high- frequency 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 ob- jective 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 (Schnei- der, 1978). In the frequency and wave-vector domain, i?/& is equivalent to multiplication by ik, . It can be shown that for a


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 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 Vr(x, x,) + VT(X, x,) 1 x exp io

i[ ~(x, x,) + r(x, x,) II dS. (6)

In this equation, V denotes a gradient with respect to the x variables and R(x, XJ is the geometrical-optics reflection coefficient

R(x, XJ =

(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 x A(, x,)dx, X,) t VT(X, X,) + VT@, X,)1 d do F(o)

X s 4~~ WW, x,)A(x, x,) exp ~WD(X, XI, x,, x,) s xii- Vs(x, x,) + VT(X, x,) 1 dS (8)

In this equation

@(x, x, x,, XI) = T(X', Xs) + T(X', Xr)

- L T(X, XJ + 7(X, x,) 1 (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

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.

x = x(a), (10) The result of applying the method of stationary phase to

where

In terms of these parameters,

dS=\/;do,da,, (11) with g the first fundamental form of differential geometry for S.

=/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 :

1 . dx, d5, and

+ v, tw, XI) - 7(x, XJ 1 (13)

d@ - = v do,

T(X', Xs) + T(X, X,) 1 . $, 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, common- receiver, and common-offset. Although I only consider here the fully three-dimensional problem. this analysis specializes to the cases of 2.5-D inversion.

936 Bleistein

equation (8) is the following:

P(x) - - Rx, x,) 4x, x&w, x,) 4% x,).4x, x,) I h(xv 5) I

) det [$,I Jtiz 1 VT(X, x,) + Vr(x, x,) I x a . [Vr(x, XJ + Vt(x, x,)1& I(K). (14)

In this equation, CJ is defined by equation (12) and

I~x)=~SP(m)erpjio~x.x,x,.x.)

+ i(sign w)(K/~) sig [@,:.I do. 1

(15)

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

- -

P&l = i,m = 1, 2; (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 = !,et [e]] m

P[r(x, x,) + 5(x, x,)] %&,

(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

by equation (15) becomes

I(x)=~ST(~)exp(iw4x,r,x,,x,))dw. (18)

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

I(x) = 6, [ @(x, x, x,, x,) 1 1 (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) :

(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@ = - Vr(x, XJ + V7(x, x,) [ I (21)

By using Beylkins 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,

$3 (4 _r(x) C-E &(s)

I V@ I I wx, x,1 + wx, x,) I. (22)

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). Determi- nation 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 x I det C@,,l l2/ Vr(x, XJ + Vr(x, x,) 1 x ii * [Vr(x, x,) + VT(X, x&&,(x). (23)


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 Vr(x, XJ and VT(X, x,) make angles of (x - 0) with this normal and an angle of 20 with one another. Therefore,

and

t3 and 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

6 * VT(X, XJ + vqx, x,) 1 2 cos e = - - c(x) (24) lk 51 x 4% 04x, x,) I VW, x,) + wx, x,)lZ

I Wx, x,) + Vr(x, x,) 12 = - 2 (1 + cos 20) c(x)

2 cos 9

[ 1 = = c(x) (25) Finally, in Appendix C, I show that

I h(x> 5) I I det 19eJ I I2 dii = I wx, %I + VT@, x,) I

2 cos 0 =-

c(x) 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. (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 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 P(x) * c(x) R(x, xs) &

s F(o) dw, x on S. (28)

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 I iw do F(o) exp i L -iw s(x, x3 + T(X, x,) 11 D(5, a), (29)

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 I VT(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 then stationary pointy on S, given by equation (24) with x = x. Thus,

b,(x) - Rb, x,) ; F(o) do, x on S, (30) and

P(x) 2 cos e plo- c(x)

X on S. (31)

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 this works 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,

cos 0 &(x, XJ an =- c(x)

With a slight abuse of notation, equation (7) can be rewrit- terr w fuilows :

cos e r 1 sin Ol/ --I-_-l

R(x, 0) = c(x) Lc: (x) c2(x) 1 cos 8 r 1 sin2 t31 (33) -+ --- 4x1 1 c: (x) c?x) 1

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 expressed in terms of the squared slowness. That is,

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 Greens function amplitudes, as well as the determinant h(x, 5) defined by equa- ton (5). For a constant-background medium, all of these functions can be expressed explicitly as functions of the integration variables. For a variable background, the determination of these functions becomes progressively more computer inten- sive with increasing complexity of the background c(z), c(x, z), and c(x, y, z). 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. How- ever, 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

In current research at the Center for Wave Phenomena, we have had success with sparse computation of these functions and interpolation for intermediary values. These steps con- siderably 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 common- receiver configurations, h(x, 5) can be shown to be related to the same Jacobian as arises in determining the WKB amplitudes of the Greens 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 Greens 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 Reflectors 939

theless, as noted above, h(x, 5) is readily computed from infor- pany; Sun Exploration and Research; Texaco USA; Union mation on the ray direction. Oil Company of California; and Western Geophysical.

CONCLUSIONS The author also wishes to express his gratitude to Jack K.

Cohen for a critical reading of this paper and some helpful suggestions.

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. There- after, 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.

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: Aca- demic 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, 1253- 1265.

~ 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 seis- mology : 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

ACKNOWLEDGMENTS

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. _

The author gratefully acknowledges the support of the Office of Naval Research, Mathematics Division, through its Selected Research Opportunities Program, and the Consor- tium Project on Seismic Inverse Methods for Complex Struc- tures at the Center for Wave Phenomena, Colorado School of Mines. Consortium members are Amoco Production Com- pany : Conoco, Inc. ; Digicon, Inc. ; Geophysical Exploration Company of Norway A/S; Marathon Oil Company; Mobil Research and Development Corp.; Phillips Petroleum Com-

Esmersoy, C., Oristaglio, M. L., and Levy, B. C., 1985, Multidimen- sional 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 Phe- nomena. Colorado School of Mines; also Geophysics, 52, 745-754.

APPENDIX A

ANALYSIS OF THE STATIONARY PHASE CONDITIONS

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

magnitude l/c(x,) and is directed awny from the initial point x. Similarly, VT(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 in- traduced 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 p(x,,x)-+p(x,,x).-

dSm dSnl

=p(x,,X).~+p(x,,x).dx, dSm dS, m= 1,2.

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 (Snells 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 surfaces to the output point x. Con- sider 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 Snells 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 source- receiver 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

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.

with the normal. Use these as initial directions for rays from the point. Snells 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 source- receiver 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 sound speed

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 Snells 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. Simi- larly, 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 discuss how these are further constrained for particular source-receiver configurations and this flat-reflector, constant-background model.

Case 1: Zero offset

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 surfaces and variable background is fairly straightforward. Given x, find a normal ray from S which passes through 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


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.

Case 2: Common source

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-

background case, the essential features of this analysis still 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

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,.

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 B MATRIX SIGNATURE

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

and

X 1, = X -5 1, - 1,

xzs=x2,= 2, 5

x3, = x3, = 0,

x; = crl, (B-1)

x; =02,

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 = 1 x - x, I/c, (B-2)

and

7(x, x,) = 1 x - 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

(B-3)

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)

-

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, zero- offset, 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,

942 Bleistein

since det [QJ is nonnegative and by assumption, the signa- now add to that the assumption that our true model is not so ture changes. severely different from the flat-earth case to have caused h to

Appendix C shows that det [Q,,] is proportional to h(x, 5). pass through a zero on the way from one model to the other. It has been assumed that h is nonzero for the true model. I Thus. sig [Q,,] = 0 for the true model as well.

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

1 (C-1) In this equation, I have used the notation

p(x. x,) = Vt(x, x,), p(x. XJ = Vt(x, x,). K-2) To calculate this determinant, the matrix is multiplied by a matrix whose determinant is known:

where each vector represents a column of J$. Note that

(C-3)

(C-4)

with the second equality being equivalent to equation (12). Now, in multiplying 4 by the matrix in equation (C-l), the

first two elements of the first row are both zero by equation (A-l ), while the third element is given by

l p(x, x,) + p(x, x,) 1 2 cos 0 * ii = - 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

$ PW, XJ + p(x: x,) 1 dx & da, - a

= 2l - at, c;o,

5(x, XJ + T(X, x,) 1 ir+D =- ack do, k, m = 1, 2. (C-6) It now follows that if the matrix in equation (C-l) is multi-

plied by the matrix & before calculating the determinant, the following result is obtained:

2 cos 8 det h(x, 6),/i = -

c(x) det Q,, , [ 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 lrom equation (25).

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Beylkin Determinant