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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 configurationdirectly 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) acousticwave equa-
tion from boundaryvalue measurements.No a priori knowl-
edge of the subsurfacevelocity is assumedand 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 scatteringformalism.
INTRODUCTION
It is well known in seismic exploration (see Dobrin, 1976) that
ihe responseof the earth (recordedat or nearthe surfke) to a man-
made impulsive energy source (applied at or near the surface)
characterizesa set of reflection seismic data. This response is a
wave field and should, therefore, be characterizedby a wave equa-
tion or suitablepartial differential equation. The particular partial
differential equationwill naturallydependuponthephysical 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 associatedwith 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
someobject with known acousticproperties (e.g., velocity), then
solving for the scatteredwave field. The “inverse problem” or
inverse scattering problem is defined by specifying both the in-
cidentand scatteredwaves, then solving for the acousticproperties
(e.g., velocity) of the scatteringobject. 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 scatteringproceduresoriginally
developedin atomic scatteringtheory (Chadanand Sabatier, 1977)
and show how they may be applied to the reflection seismicprob-
lem. Specifically, we apply the mathematical techniquesused in
atomic scatteringfor obtainingthe 3-D potential in the SchrGdinger
equationto 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 mathematicsof atomic
scatteringto the physics of reflection seismology. For example,
in atomic scatteringtheory, the mathematical solutionsof S&r&
dinger’s equation are based on a physical experiment which
assumesthat the scatteredfield is recorded at a distance very far
from the scattering object. Although this far-field assumptionis
reasonablefor atomic scattering, it is not clear that this is a rea-
sonable assumptionfor reflection seismology. Thus, in applying
the atomic scatteringmathematicsto geophysicalseismicexplora-
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 suchthat our mathematics is consistentwith 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 necessarymodifications
are more involved.
The plan of this paper is asfollows. In the first sectionwe 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 measurementsto the far-field
assumption inherent in the inverse scattering mathematical
solution.
Manuscriptreceivedby theEditorJuly5, 1979;revisedmanuscriptreceivedSeptember8, 1980.
*Cities ServiceCo., EnergyResourcesGroup,P.0. Box 3908,Tulsa,OK 74102.
0016-8033/81/0801-1116$03.00.0 1981Societyof ExplorationGeophysicists.All rightsreserved.
1116
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3-D Velocity from Reflection Data
THREE-DIMENSIONAL INVERSE ACOUSTIC SCATTERING
WITH FAR-FIELD MEASUREMENTS
Our application of atomic inverse scattering to reflection seis-
mology is based on a result obtained by Moses (1956). Razavy
(1975) hasapplied the work of Moses to the problem of identifying
the velocity of a 1-D acoustic wave equation. Here we wish to
extend the work of Razavy to identifying the velocity c(r) in the
3-D acousticwave equation (1). In this development, we assume
that the 3-D scattered (reflected) field is recorded in a far-field
fashion.
Shot pant Earth’s surface
Observation
for Green’s
By Fourier transforming equation (1) with respect to time t,
we obtain the time-independent acousticwave equation
vzq + -g&-=0. (2)
where
_ _ 1 =
I$ = $(r, w, rs, t’) = -
i2rr _z
e’“‘+(r, t, r,, f’)dt
defines the temporal Fourier transform of the wave field +(r, r,
rs, t’) defined by equation (1). Let us now express the local
velocity c(r) as
FIG. 1. The coordinate system. The actual physical measurements
of the wave field are made at observation points which are
located on the earth’s surface. Here, the positive z-axis points
downward.
c2(r) =
co”
1 - a(r)
(3)
where cc is a constantreferencevelocity and(Y(r) = a isa spatially
varyingparameterwhich containsthedesiredvelocity information.
From equation (3), we see that
1
- = + [I - (u(r)1
c2(r)
(4)
so 0 < a < 1 means co”< c2(r) < =, --x. < cd% 0 means
0 < c2(r) 5 co”. Substitution of equation (4) into equation (2)
gives
026 + < (1 - a)& = 0. (5)
co
Let us now explain the reason for expressing the local velocity
c(r) by equations (3) and (4) in order to obtain equation (5). In
quantum scattering, the time-independent Schrodinger equation
can be written as
quency or wavenumber dependence, caution must be exercised
to apply the appropriate quantum inverse techniques to the
acousticproblem. Let us now proceed to solve equation (5) for
cx(r). We shall place no restriction on a(r) and, consequently, no
restriction on c(r). This is in contrast to both wave equation
migration (e.g., see Claerbout, 1971; Stolt, 1978; Schneider,
1978) and first Born approximation inverse scattering (e.g., see
Phinney and Frazer, 1978; Cohen and Bleistein, 1979). That is,
in the mathematical derivation of wave equation migration, the
local velocity is assumed to be constant. In first Born inverse
scattering,one assumesthat 1a 1+ 1, sofrom equation (3) we see
that the velocity c(r) is assumedto be close to the constantrefer-
ence velocity co.
Following Joachain (1975), we can rewrite equation (5) as
(V2 + /?)I$ = k%& (7)
where k = w/co. Consider k2cx$ as the inhomogeneousterm 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 homogeneousequation
“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
equationin the form (5) allows usto apply certain resultsobtained
in quantuminverse scatteringtheory, where the inverse problem,
i.e., the solution for V(r), is well known. Thus, becauseof 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 correspondsto the wave field in the absenceof 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) hasa quadraticfre- $. may be any solution to the homogeneousequation (9) and can
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1118 Weglein et al
always be represented as a superpositionof 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
secondsection 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)
(seeJoachain, 1975). Thus, thegeneralsolutionof theSchrodinger-
type acoustic equation (5), which describes the total wave field
4 (r, k) resultingfrom 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 responseto 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 assumethat the measurementsare far field, that is,
the observationpoint r is very large in magnitude compared with
the object point r’, then
Ir - r’l
[
r - r’
‘=r I--
12 ’
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 assumedto be the plane wave (11). Let
us now discusstha 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 asthe 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
secondterm 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-
scattereddirection. 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 directionsabove the earthfrom reflection
seismic data. This will then be sufficient to determine a(r) and,
hence, the acousticvelocity configuration.
Razavy (1975) has given a procedure for identifying the 1-D
acoustic wave equation. Paralleling his development and using
his notation, the following equationsare 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 1dq, (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)]
requiredto realize the completevelocity determinationschemeout-
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 scatteringexperiment.
The slant-stacking technique of Schultz and Claerbout (1978)
is a very convenient form of seismic data arrangementin which
to discussour 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
scatteringexperiments. We now outlinetheessentialstepsinvolved
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
surfacedue to an incident plane wave $o(r, t, 2) with direction
denoted by the unit vector I (see Figure 2).
(2) Subtractthe 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 scatteredfield 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
hemisphereof directions above the earth. Given these b(k), the
procedureoutlined in the first section can now be realized.
DISCUSSION
As a formalism, thispaperservesan importantfunction, namely,
it demonstratesthe existenceof a complete solutionto the problem
of identifying the velocity in the acousticwave equation. The fact
that a solution exists encouragesthe search for practical imple-
mentations.In addition, it showshow seeminglyunrelatedresearch
in areasoutsideof geophysics, suchasquantuminversescattering,
can provide valuable insights into the problems encountered in
seismic exploration.
ACKNOWLEDGMENTS
The authorswould~!ikeTVthank W. G. Clement for his support-
and encouragement throughout this project. Cities Service Co,
is thankedfor permission to publish this work.
APPENDIX
In OUTformalism, as in all inverse scatteringprocedures, only
the waveform $o(r, t, r,, t’), which results from a shot in a
homogeneousmedium, is assumedto be known, The form F(e)
of the simulated incident plane wave $o(r, t, ii), formed by a
sequentialtime delay of these shots, is then also known; i.e., we
(a)
FIG. 2. (a) Illustration of a constantphasecomponentof the slant-
stackedsimulatedplane wave incident on the earth. (b) Side view
of (a).
know the function F(e) where
+o(r, t, ii) = F(I * r - cot) (A-1)
and F(s) representsa wavelet with spectrumg(k). We now show
how to go from the total response$(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 usnow show how Green’s theorem is used to extrapolate
the scatteredwave 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 necessaryprocedure in order to evaluate the generalized
reflection coefficient b(k). Now in a homogeneousregion 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(seeMorse andFeshbach, 1953;Schneider,
1978) and choosing G(r, r’) such that it vanisheson 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 constructthe 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
scatteringtheory, 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-
geneousmedium 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., andClaerbout, J., 1978, Velocity estimationanddownward
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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Obtaining three-dimensional velocity information directly from reflection seismic data: An inverse scattering formalism- Prof. Arthur B. Weglein

  • 1. 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 configurationdirectly 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) acousticwave equa- tion from boundaryvalue measurements.No a priori knowl- edge of the subsurfacevelocity is assumedand 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 scatteringformalism. INTRODUCTION It is well known in seismic exploration (see Dobrin, 1976) that ihe responseof the earth (recordedat or nearthe surfke) to a man- made impulsive energy source (applied at or near the surface) characterizesa set of reflection seismic data. This response is a wave field and should, therefore, be characterizedby a wave equa- tion or suitablepartial differential equation. The particular partial differential equationwill naturallydependuponthephysical 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 associatedwith 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 someobject with known acousticproperties (e.g., velocity), then solving for the scatteredwave field. The “inverse problem” or inverse scattering problem is defined by specifying both the in- cidentand scatteredwaves, then solving for the acousticproperties (e.g., velocity) of the scatteringobject. 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 scatteringproceduresoriginally developedin atomic scatteringtheory (Chadanand Sabatier, 1977) and show how they may be applied to the reflection seismicprob- lem. Specifically, we apply the mathematical techniquesused in atomic scatteringfor obtainingthe 3-D potential in the SchrGdinger equationto 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 mathematicsof atomic scatteringto the physics of reflection seismology. For example, in atomic scatteringtheory, the mathematical solutionsof S&r& dinger’s equation are based on a physical experiment which assumesthat the scatteredfield is recorded at a distance very far from the scattering object. Although this far-field assumptionis reasonablefor atomic scattering, it is not clear that this is a rea- sonable assumptionfor reflection seismology. Thus, in applying the atomic scatteringmathematicsto geophysicalseismicexplora- 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 suchthat our mathematics is consistentwith 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 necessarymodifications are more involved. The plan of this paper is asfollows. In the first sectionwe 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 measurementsto the far-field assumption inherent in the inverse scattering mathematical solution. Manuscriptreceivedby theEditorJuly5, 1979;revisedmanuscriptreceivedSeptember8, 1980. *Cities ServiceCo., EnergyResourcesGroup,P.0. Box 3908,Tulsa,OK 74102. 0016-8033/81/0801-1116$03.00.0 1981Societyof ExplorationGeophysicists.All rightsreserved. 1116 Downloaded07/22/13to129.7.16.11.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/
  • 2. 3-D Velocity from Reflection Data THREE-DIMENSIONAL INVERSE ACOUSTIC SCATTERING WITH FAR-FIELD MEASUREMENTS Our application of atomic inverse scattering to reflection seis- mology is based on a result obtained by Moses (1956). Razavy (1975) hasapplied the work of Moses to the problem of identifying the velocity of a 1-D acoustic wave equation. Here we wish to extend the work of Razavy to identifying the velocity c(r) in the 3-D acousticwave equation (1). In this development, we assume that the 3-D scattered (reflected) field is recorded in a far-field fashion. Shot pant Earth’s surface Observation for Green’s By Fourier transforming equation (1) with respect to time t, we obtain the time-independent acousticwave equation vzq + -g&-=0. (2) where _ _ 1 = I$ = $(r, w, rs, t’) = - i2rr _z e’“‘+(r, t, r,, f’)dt defines the temporal Fourier transform of the wave field +(r, r, rs, t’) defined by equation (1). Let us now express the local velocity c(r) as FIG. 1. The coordinate system. The actual physical measurements of the wave field are made at observation points which are located on the earth’s surface. Here, the positive z-axis points downward. c2(r) = co” 1 - a(r) (3) where cc is a constantreferencevelocity and(Y(r) = a isa spatially varyingparameterwhich containsthedesiredvelocity information. From equation (3), we see that 1 - = + [I - (u(r)1 c2(r) (4) so 0 < a < 1 means co”< c2(r) < =, --x. < cd% 0 means 0 < c2(r) 5 co”. Substitution of equation (4) into equation (2) gives 026 + < (1 - a)& = 0. (5) co Let us now explain the reason for expressing the local velocity c(r) by equations (3) and (4) in order to obtain equation (5). In quantum scattering, the time-independent Schrodinger equation can be written as quency or wavenumber dependence, caution must be exercised to apply the appropriate quantum inverse techniques to the acousticproblem. Let us now proceed to solve equation (5) for cx(r). We shall place no restriction on a(r) and, consequently, no restriction on c(r). This is in contrast to both wave equation migration (e.g., see Claerbout, 1971; Stolt, 1978; Schneider, 1978) and first Born approximation inverse scattering (e.g., see Phinney and Frazer, 1978; Cohen and Bleistein, 1979). That is, in the mathematical derivation of wave equation migration, the local velocity is assumed to be constant. In first Born inverse scattering,one assumesthat 1a 1+ 1, sofrom equation (3) we see that the velocity c(r) is assumedto be close to the constantrefer- ence velocity co. Following Joachain (1975), we can rewrite equation (5) as (V2 + /?)I$ = k%& (7) where k = w/co. Consider k2cx$ as the inhomogeneousterm 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 homogeneousequation “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 equationin the form (5) allows usto apply certain resultsobtained in quantuminverse scatteringtheory, where the inverse problem, i.e., the solution for V(r), is well known. Thus, becauseof 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 correspondsto the wave field in the absenceof 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) hasa quadraticfre- $. may be any solution to the homogeneousequation (9) and can Downloaded07/22/13to129.7.16.11.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/
  • 3. 1118 Weglein et al always be represented as a superpositionof 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 secondsection 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) (seeJoachain, 1975). Thus, thegeneralsolutionof theSchrodinger- type acoustic equation (5), which describes the total wave field 4 (r, k) resultingfrom 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 responseto 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 assumethat the measurementsare far field, that is, the observationpoint r is very large in magnitude compared with the object point r’, then Ir - r’l [ r - r’ ‘=r I-- 12 ’ 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 assumedto be the plane wave (11). Let us now discusstha 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 asthe 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 secondterm 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- scattereddirection. 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 directionsabove the earthfrom reflection seismic data. This will then be sufficient to determine a(r) and, hence, the acousticvelocity configuration. Razavy (1975) has given a procedure for identifying the 1-D acoustic wave equation. Paralleling his development and using his notation, the following equationsare 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 1dq, (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. Downloaded07/22/13to129.7.16.11.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/
  • 4. 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)] requiredto realize the completevelocity determinationschemeout- 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 scatteringexperiment. The slant-stacking technique of Schultz and Claerbout (1978) is a very convenient form of seismic data arrangementin which to discussour 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 scatteringexperiments. We now outlinetheessentialstepsinvolved 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 surfacedue to an incident plane wave $o(r, t, 2) with direction denoted by the unit vector I (see Figure 2). (2) Subtractthe 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 scatteredfield 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 hemisphereof directions above the earth. Given these b(k), the procedureoutlined in the first section can now be realized. DISCUSSION As a formalism, thispaperservesan importantfunction, namely, it demonstratesthe existenceof a complete solutionto the problem of identifying the velocity in the acousticwave equation. The fact that a solution exists encouragesthe search for practical imple- mentations.In addition, it showshow seeminglyunrelatedresearch in areasoutsideof geophysics, suchasquantuminversescattering, can provide valuable insights into the problems encountered in seismic exploration. ACKNOWLEDGMENTS The authorswould~!ikeTVthank W. G. Clement for his support- and encouragement throughout this project. Cities Service Co, is thankedfor permission to publish this work. APPENDIX In OUTformalism, as in all inverse scatteringprocedures, only the waveform $o(r, t, r,, t’), which results from a shot in a homogeneousmedium, is assumedto be known, The form F(e) of the simulated incident plane wave $o(r, t, ii), formed by a sequentialtime delay of these shots, is then also known; i.e., we (a) FIG. 2. (a) Illustration of a constantphasecomponentof the slant- stackedsimulatedplane wave incident on the earth. (b) Side view of (a). know the function F(e) where +o(r, t, ii) = F(I * r - cot) (A-1) and F(s) representsa wavelet with spectrumg(k). We now show how to go from the total response$(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 usnow show how Green’s theorem is used to extrapolate the scatteredwave 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 necessaryprocedure in order to evaluate the generalized reflection coefficient b(k). Now in a homogeneousregion 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(seeMorse andFeshbach, 1953;Schneider, 1978) and choosing G(r, r’) such that it vanisheson the surface of the earth, we obtain &(rP, o, 2) = - I +s(r’, o, II)‘~ dr’dy’, (A-4) az’ where Downloaded07/22/13to129.7.16.11.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/
  • 5. 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 constructthe 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 scatteringtheory, 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- geneousmedium 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., andClaerbout, J., 1978, Velocity estimationanddownward continuation by wavefront synthesis:Geophysics, v. 43, p. 691-714. Stolt, R. H., 1978, Migration by Fourier transform: Geophysics, v. 43, p. 23-48. Downloaded07/22/13to129.7.16.11.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/