GEOPHYSICS, VOL. 46, NO. 8 (AUGUST 1981): P. 1116-1120, 3 FIGS.

Obtaining three-dimensional velocity information directly from reflection seismic data: An inverse scattering formalism

A. B. Weglein*, W. E. Boyse*, and J. E. Anderson*

ABSTRACT

We present a formalism for obtaining the subsurface velocity configuration directly from reflection seismic data. Our approach is to apply the results obtained for inverse problems in quantum scattering theory to the reflection seismic problem. In particular, we extend the results of Moses (1956) for inverse quantum scattering and Razavy (1975) for the one-dimensional (1-D) identification of the acoustic wave equation to the problem of identifying the velocity in the three-dimensional (3-D) acoustic wave equa- tion from boundary value measurements. No a priori knowl- edge of the subsurface velocity is assumed and all refrac- tion, diffraction, and multiple reflection phenomena are taken into account. In addition, we explain how the idea of slant stack in processing seismic data is an important part of the proposed 3-D inverse scattering formalism.

INTRODUCTION

It is well known in seismic exploration (see Dobrin, 1976) that ihe response of the earth (recorded at or near the surfke) to a man- made impulsive energy source (applied at or near the surface) characterizes a set of reflection seismic data. This response is a wave field and should, therefore, be characterized by a wave equa- tion or suitable partial differential equation. The particular partial differential equation will naturally depend upon the physical model that is assumed for the subsurface (e.g., anelastic, elastic, or acoustic model). The acoustic model (an elastic model without shear waves) results in the ordinary three-dimensional (3-D) (time-dependent) wave equation

v2+ - 1 a2+

--co c2(r) at2 ’

(1)

where + = $ (r, t, rs, t’) is the wave field at observation point r and time t due to an energy source (shot) located at point r, and applied at time t’ (see Figure 1). Here c(r) is the local wave velocity or spatially varying parameter associated with the hyper- bolic system (1). The main objective of this report is to provide a formalism for identifying the 3-D local velocity or parameter c(r) directly from reflection seismic data.

By convention, the “forward problem” or forward scattering problem is defined by specifying a given wave incident upon some object with known acoustic properties (e.g., velocity), then solving for the scattered wave field. The “inverse problem” or inverse scattering problem is defined by specifying both the in- cident and scattered waves, then solving for the acoustic properties (e.g., velocity) of the scattering object. The problem in reflection seismology of estimating the subsurface velocity configuration from scattered seismic data is, in the above sense, an inverse problem. We exploit the inverse scattering procedures originally developed in atomic scattering theory (Chadan and Sabatier, 1977) and show how they may be applied to the reflection seismic prob- lem. Specifically, we apply the mathematical techniques used in atomic scattering for obtaining the 3-D potential in the SchrGdinger equation to the reflection seismic problem, where a 3-D velocity profile is of interest. In our application of the atomic scattering techniques to reflection seismology, we have (1) demonstrated how the mathematics of atomic scattering relates to the physics of atomic &cattering, and (2) applied the mathematics of atomic scattering to the physics of reflection seismology. For example, in atomic scattering theory, the mathematical solutions of S&r& dinger’s equation are based on a physical experiment which assumes that the scattered field is recorded at a distance very far from the scattering object. Although this far-field assumption is reasonable for atomic scattering, it is not clear that this is a rea- sonable assumption for reflection seismology. Thus, in applying the atomic scattering mathematics to geophysical seismic explora- tion, we are careful to preserve the under!ying physics of both atomic scattering and reflection seismology. As part of the pro- posed formalism, we explain how to modify reflection seismic data such that our mathematics is consistent with the physics. In a one-dimensional (1-D) acoustic formulation (Razavy, 1975), the reflection data can be easily modified. However, in extending Razavy’s result to three dimensions the necessary modifications are more involved.

The plan of this paper is as follows. In the first section we show how classic inverse scattering theory could be applied to a fur- field problem assuming an acoustic model of the subsurface [refer to equation (l)]. In the second section, we relate the near- field nature of reflection seismic measurements to the far-field assumption inherent in the inverse scattering mathematical solution.

Manuscript received by the Editor July 5, 1979; revised manuscript received September 8, 1980. *Cities Service Co., Energy Resources Group, P. 0. Box 3908, Tulsa, OK 74102. 0016-8033/81/0801-1116$03.00. 0 1981 Society of Exploration Geophysicists. All rights reserved.

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3-D Velocity from Reflection Data

THREE-DIMENSIONAL INVERSE ACOUSTIC SCATTERING WITH FAR-FIELD MEASUREMENTS

Our application of atomic inverse scattering to reflection seis- mology is based on a result obtained by Moses ( 1956). Razavy

(1975) has applied the work of Moses to the problem of identifying the velocity of a 1-D acoustic wave equation. Here we wish to extend the work of Razavy to identifying the velocity c(r) in the 3-D acoustic wave equation (1). In this development, we assume that the 3-D scattered (reflected) field is recorded in a far-field fashion.

Shot pant Earth’s surface

Observation for Green’s

By Fourier transforming equation (1) with respect to time t, we obtain the time-independent acoustic wave equation

vzq + -g&-=0. (2)

where

_ _ 1 = I$ = $(r, w, rs, t’) = -

i 2rr _z e’“‘+(r, t, r,, f’)dt

defines the temporal Fourier transform of the wave field +(r, r, rs, t’) defined by equation (1). Let us now express the local velocity c(r) as

FIG. 1. The coordinate system. The actual physical measurements of the wave field are made at observation points which are located on the earth’s surface. Here, the positive z-axis points downward.

c2(r) = co”

1 - a(r) (3)

where cc is a constant reference velocity and (Y (r) = a is a spatially varying parameter which contains the desired velocity information. From equation (3), we see that

1 - = + [I - (u(r)1 c2 (r)

(4)

so 0 < a < 1 means co” < c2(r) < =, --x. < cd % 0 means 0 < c2(r) 5 co”. Substitution of equation (4) into equation (2) gives

026 + < (1 - a)& = 0. (5) co

Let us now explain the reason for expressing the local velocity c(r) by equations (3) and (4) in order to obtain equation (5). In quantum scattering, the time-independent Schrodinger equation can be written as

quency or wavenumber dependence, caution must be exercised to apply the appropriate quantum inverse techniques to the acoustic problem. Let us now proceed to solve equation (5) for cx (r). We shall place no restriction on a(r) and, consequently, no restriction on c(r). This is in contrast to both wave equation migration (e.g., see Claerbout, 1971; Stolt, 1978; Schneider, 1978) and first Born approximation inverse scattering (e.g., see Phinney and Frazer, 1978; Cohen and Bleistein, 1979). That is, in the mathematical derivation of wave equation migration, the local velocity is assumed to be constant. In first Born inverse scattering, one assumes that 1 a 1 + 1, so from equation (3) we see that the velocity c(r) is assumed to be close to the constant refer- ence velocity co.

Following Joachain (1975), we can rewrite equation (5) as

(V2 + /?)I$ = k%& (7)

where k = w/co. Consider k2cx$ as the inhomogeneous term in the differential equation (7). The general solution of equation (7) is then

V2$ + [k2 - V(r)]* = 0, (6)

where V(r) is a spatially variant potential (see Schiff, 1955). Hence, equation (5) is a Schrodinger-type equation where

&=60-J Go(r, r’, k) k2a(r’)&dr’, (8) ”

where $. is a solution of the homogeneous equation

“2 7 o(r) co

vzqo + kz&o = 0, (9)

and Go(r, r’, k) is a Green’s function (see Morse and Feshbach, 1953) (Figure 1)

plays the role of a spatially variant potential. Writing the acoustic equation in the form (5) allows us to apply certain results obtained in quantum inverse scattering theory, where the inverse problem, i.e., the solution for V(r), is well known. Thus, because of the mathematical duality between equations (5) and (6), we now have a solution for

(V2 + k2)Go(r, r’, k) = -Fi(r - r’). (10)

The volume integral in equation (8) is for a volume v whose extent is defined by the support of a(r). In quantum or acoustic scattering, &o corresponds to the wave field in the absence of a potential V(r) or the variation in the index of refraction a(r),

respectively. In quantum scattering, &o is often chosen to be the plane wave

(11)

which is equivalent to knowledge of the local velocity c(r) via where k is the wave vector. We note that the incident wave field equation (3). Since our potential (w2/&) a(r) has a quadratic fre- $. may be any solution to the homogeneous equation (9) and can

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1118 Weglein et al

always be represented as a superposition of plane waves by the spatial Fourier transform

&a = j- eik”s(k)dk. (12)

For our purposes, we assume &a to be a plane wave. This assumption (see Moses, 1956) allows us to generalize the con- cept of a reflection coefficient from one dimension to three dimensions. Although the incident plane wave assumption appears to be somewhat restrictive, the slant-stack technique of Schultz and Claerbout (1978) allows us to apply the pro- posed formalism to reflection seismic data. More will be said about the application of slant stack to our formalism in the second section of this paper.

Now the Green’s function Gc(r, r’, k) is the solution to equa- tion (10) with “outgoing wave” boundary conditions, that is,

1 &klr-r’l Gc(r, r’, k) = - ~

4rr )r - r’l (13)

(see Joachain, 1975). Thus, the general solution of the Schrodinger- type acoustic equation (5), which describes the total wave field 4 (r, k) resulting from an incident plane wave (1 l), is the solution to the integral equation

eik.r

I eiklr-r’i

- - +(r, k) = (27F)3,2 v 47F,r _ r,, k’a(r’)+dr’, (14)

where v denotes the volume associated with an &homogeneity a(r’). Here we have changed our notation from +(r, o, r,, t’) to &(r, k) to emphasize the fact that the total wave field is now the result of an incident plane wave with wave vector k and not the response to a localized disturbance (or shot) at the point r,. More will be said about this in the next section. Equation (14) is called the Lippmann-Schwinger equation. This integral equation plays a central role in scattering theory.

If we now assume that the measurements are far field, that is, the observation point r is very large in magnitude compared with the object point r’, then

Ir - r’l [

r - r’ ‘=r I--

1 2 ’ r = Ir[ 9 Ir’l,

and

(r - r’l-’ = l/r.

Consequently, the Green’s function (13) may be written as

Gc(r, r’, k) z 2 eikre-ikF.r’, (15)

where P is a unit vector in the direction of r. Putting the far-field condition (15) into the Lippmann-Schwinger equation (14) gives

where k’ = kP and &, is assumed to be the plane wave (11). Let us now discuss tha concept of a 3-D reflection coefficient.

In a 1-D scattering experiment, an incident plane wave, say elkz, gives rise to a reflected or back-scattered plane wave b(k)e-““, which travels in the opposite direction and has ampli- tude b(k). The quantity b (k) is defined as the 1 -D reflection coeffi- cient. Now in a far-field 3-D scattering experiment, the incident plane wave (11) gives rise to the spherical wave defined by the second term in equation (16). From equation (16), the amplitude

of this spherical wave is

f(k’, k) = 2 1 emrk’.r ‘k2a(r’)$(r’, k)dr’, (17) ”

where k2 = w”/c,’ and k’ = k f. Equation (17) defines the scattering amplitude. The quantity f (k’, k) is the amplitude of the spherical wave in the direction of wave vector k’ resulting from an incident plane wave (11) with direction k. In order to generalize the 1-D concept of a reflection coefficient to three dimensions, we consider the scattering amplitude in the back- scattered direction. In terms of the quantity f (k’, k) in equation (17), we define the 3-D reflection coefficient by

b(k) =f(-k, k),

where

(18)

b(k) = - I, f$& [t]1’2 k2a(r’)$(r’, k)dr’. (19)

Notice that this generalized reflection coefficient is a function of both the magnitude and direction of the incident wave vector k. Recognizing the analytic properties of scattering amplitudes (see, e.g., Joachain, 1975), Moses (1956) and Razavy (1975) use the relationship

b*(k) = b(-k)

to show that b(k) for a hemisphere of incident directions will determine b(k) for all incident directions. Once b(k) is known for all k, a(r) is determined.

In the second part of the paper, we show how we can find b(k) for a hemisphere of directions above the earth from reflection seismic data. This will then be sufficient to determine a(r) and, hence, the acoustic velocity configuration.

Razavy (1975) has given a procedure for identifying the 1-D acoustic wave equation. Paralleling his development and using his notation, the following equations are an extension of Razavy’s work to three dimensions:

T(p, k) = &(p, k) - k2 I

dqG(p> q)T(q, k)

q2 - k2 - ie (20)

2 1’2 b(k) W(k) = - ;

[ 1 7 + k2 T*(k, q)T(-k, q) . I

I H(k) + H(-k)

q2 - k2 - iE q2 - k2 + ie 1 dq, (21)

(22)

where

1 6(k’, k) = 3 dr e- ik’.r,(r)eik.r,

T(k’, k) = 1 f$ .

. V(r)$(r, k)dr

(23)

and

1, k>O

H(k) =

0, k < 0.

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3-D Velocity from Reflection Data

Hence, given b(k) the three coupled integral equations (20)-(22) are solved for W(k), h(k, k’), and T(p, k). From W(k) we can

solve for a(r) using equation (23). The acoustic velocity profile c(r) is then obtained via equation (3).

RELATING THE NEAR-FIELD REFLECTION DATA TO THE FAR-FIELD INVERSE SCATTERING SOLUTION

In this section, we show how to go from seismic reflection data to the kind of information [i.e., 3-D reflection coefficients b(k)] required to realize the complete velocity determination scheme out- lined in the first part of this paper. That is, we now outline a pro- cedure that relates the realities of near-field seismic exploration to an effective far-field scattering experiment.

The slant-stacking technique of Schultz and Claerbout (1978) is a very convenient form of seismic data arrangement in which to discuss our velocity determination scheme. Since slant-stacked seismic data simulate the response of the earth to an incident plane wave, it is a natural way to relate the seismic and quantum scattering experiments. We now outline the essential steps involved in our inverse scattering formalism. For specific details, see Appendix A. In going from field data recorded at the surface to 3-D reflection coefficient information b(k), there are four steps:

(1) Slant stack the reflection data $ (r, t, rs, t’) to obtain $(r, t, &), the total time-dependent wave field on the earth’s surface due to an incident plane wave $o(r, t, 2) with direction denoted by the unit vector I (see Figure 2).

(2) Subtract the incident wave field +o(r, t, ii) from the total wave field 4 (r, t, ii) to obtain the time-dependent scattered~ wave field +s(r, t, ii), namely,

&(r, t, ti) = +(r, t, ti) - $o(r,t, N.

(3) Fourier decompose +s(r, t, 5) to obtain &s(r, w, a), the single frequency scattered field on the earth’s surface.

(4) Use Green’s theorem (see Appendix A) to extrapolate &,(r, o, ii) in the direction -I, far from the earth’s surface, to evaluate the generalized reflection coefficient b(k), that is,

b(k) = b(M) = rem’kr&s(r, w, si),

where k = w/co. By evaluating b(kl) for all temporal fre- quencies o and directions %, we obtain b(k) for all k on a hemisphere of directions above the earth. Given these b(k), the procedure outlined in the first section can now be realized.

DISCUSSION

As a formalism, this paper serves an important function, namely, it demonstrates the existence of a complete solution to the problem of identifying the velocity in the acoustic wave equation. The fact that a solution exists encourages the search for practical imple- mentations. In addition, it shows how seemingly unrelated research in areas outside of geophysics, such as quantum inverse scattering, can provide valuable insights into the problems encountered in seismic exploration.

ACKNOWLEDGMENTS

The authors would~!ike TV thank W. G. Clement for his support- and encouragement throughout this project. Cities Service Co, is thanked for permission to publish this work.

APPENDIX

In OUT formalism, as in all inverse scattering procedures, only the waveform $o(r, t, r,, t’), which results from a shot in a homogeneous medium, is assumed to be known, The form F(e) of the simulated incident plane wave $o(r, t, ii), formed by a sequential time delay of these shots, is then also known; i.e., we

(a)

FIG. 2. (a) Illustration of a constant phase component of the slant- stacked simulated plane wave incident on the earth. (b) Side view of (a).

know the function F(e) where

+o(r, t, ii) = F(I * r - cot) (A-1)

and F(s) represents a wavelet with spectrum g(k). We now show how to go from the total response $(r, t, ii), due to the general time-dependent plane wave source (A-l), to the scattered field I$, (r, w, A) resulting from a specific single frequency plane wave source. The pertinent equation is

1 &(r, w, i) = -

2Tg (k) I eicUt [+(r, t, a) - $O(r, t, A)]dt,

W-2)

where k = o/co.

Let us now show how Green’s theorem is used to extrapolate the scattered wave field 6,(r, o, A) on the earth’s surface to a far-field point above the earth in the direction -A. Recall that this is a necessary procedure in order to evaluate the generalized reflection coefficient b(k). Now in a homogeneous region above the earth, the following equations apply

(V2 + k2)qs(r, w, 3) = 0,

and

(V2 + k2)G(r, r’) = -6(r - r’). (A-3)

Using Green’s theorem (see Morse and Feshbach, 1953; Schneider, 1978) and choosing G(r, r’) such that it vanishes on the surface of the earth, we obtain

&(rP, o, 2) = - I

+s(r’, o, II)‘~ dr’dy’, (A-4) az’

where

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1120 Weglein et al

I_ r plrp-r’l

G@P, r’) = 4, 1 (r eiklr”-r’l 1

P- rl, - ,f _ &, J

and r” is chosen to be the image point of rp (see Figure A-l). We can use equation (A-4) to construct the effective asymptotic measurements +~~(r~. o, A). Thus, b(k) is given by

b(k) = e -ikrP rP+s(-rP&, w, A).

REFERENCES

Chadan, K., and Sabatier, P. C., 1977, Inverse problems in quantum scattering theory, Springer, New York.

Claerbout, 1. F., 1971, Toward a unified theory of reflector mapping: Geophysics, v. 36, p. 467-481.

Cohen, J. K., and Bleistein, N., 1979, Velocity inversion procedure for acoustic waves: Geophysics, v. 44, p. 1077-1087.

Dobrin, M., 1976, Introduction to geophysical prospecting, 3rd Ed., New York, McGraw-Hill Book Company, Inc.

Joachain, C. J., 1975, Quantum collision theory: Amsterdam, North Holland Publ. Co.

Morse, P. M., and Feshbach, H., 1953, Methods of theoretical physics. Moses, H. E., 1956, Calculation of the scattering potential from re-

flection coefficients: Phys. Rev., v. 102, p. 559-567. Phinney, R. A., and Frazer, L. N., 1978, On the theory of imaging by

Fourier transfonn: submitted to Geophysics. Razavy, M., 1975, Determination of the wave velocity in an inhomo-

geneous medium from the reflection coefficient: J. Acoust. Sot. Am., v. 58, p. 956-963.

Schiff, L. I., 1955, Quantum mechanics: New York, McGraw-Hill Book Company, Inc.

Schneider, W. A., 1978, Integral formulation for migration in two and three dimensions: Geophysics, v. 43, p. 49-76.

Point at which b(6) IS evaluated

Surface boundary used - I” Green’s theorem.

J

7, = Image point of rp

FIG. A-l. Shaded area is the support of ~1, i.e., the region of interest. Gre_en’s theorem is used to extrapolate the scattered wave field +,(r, 9, k) from the earth’s surface to the distant point rp.

Schultz. P. S., and Claerbout, J., 1978, Velocity estimation and downward continuation by wavefront synthesis: Geophysics, v. 43, p. 691-714.

Stolt, R. H., 1978, Migration by Fourier transform: Geophysics, v. 43, p. 23-48.

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