GEOPHYSICS, VOL. 52, NO. 1 (JANUARY 1987); P. 11-25, 13 FIGS.

Diffraction tomography and multisource holographyapplied to seismic imaging

Ru-Shan Wu* and M. Nafi Toks6z*

ABSTRACT

Seismic tomography is emerging as an imagingmethod for determining subsurface structure. When theview-angle coverage is limited and the scale of themedium inhomogeneities is comparable with the wave­length, as is often true in geophysical applications, theperformance of ordinary ray tomography becomes poor.Other tomographic methods are needed to improve theimaging process. Here we study diffraction tomographyand multisource holography and evaluate their per­formances for surface reflection profiling (SRP), verticalseismic profiling (VSP), and cross-hole measurements.Theoretical formulations are derived for two­dimensional geometry in terms of line sources along asource line and line receivers along a receiver line. Thetheory for diffraction tomography is based on the Bornor Rytov approximation.

The performances of diffraction tomography andmultisource holography are evaluated by examining theinformation coverage in the spatial frequency domainand by numerical examples. Multisource holography,

INTRODUCTION

In this paper we study two approaches to seismic imaging:diffraction tomography and multisource holography. Tradi­tional holography, which uses a single source, is a method toreconstruct wavefronts from the phase and amplitude infor­mation of the scattered wave field. The reconstructed wave­front can give a three-dimensional (3-D) image of the object.(For the theory and application of optical holography, seeGoodman, 1968; Smith, 1969; and Francon, 1971.) Acousticholography followed the developments in optics, with appli­cations to nondestructive evaluation of materials, underwaterimaging, and medical diagnosis. There is vast literature onacoustic holography (e.g., Metherell et aI., 1969; 1970; Meth-

which is similar to Kirchhoff-type migration, often givesdistorted images of the object. This distortion causeslong tails of the image in the case of SRP and a strongnoise belt in the case of VSP and is due to incompleteand nonuniform coverage of the object spectrum. Thefiltering operation of diffraction tomography helps incorrecting the nonuniform coverage (including dupli­cation) of the object spectrum in the reconstruction pro­cess and therefore reduces the distortions. On the otherhand, multisource holography is better suited for imag­ing sharp boundaries with large acoustic impedancecontrasts since diffraction tomography is restricted, aspresently formulated, to weak inhomogeneities. In addi­tion, multisource holography has the flexibility to beused with an arbitrary number of sources (including asingle source). Its sampling interval is not restricted bythe Nyquist frequency.

Numerical examples show that combined data sets(such as surface reflection data combined with VSPdata, or cross-hole data combined with surface data,etc.) improve the image quality.

erell and Larmore, 1971; Wade, 1972; Green, 1973; Booth,1975; Kessler, 1976; Wang, 1979; Powers, 1981; Kaveh et aI.,1983). Hildebrand (1980) gives a short review on the advancesin this field. Several investigators have tried to use acousticholography for seismic exploration (Farr, 1968; Fontanel,1971; French et aI., 1973). The traditional single-source, single­frequency holography had poor longitudinal resolution (Fon­tanel, 1971; French et aI., 1973; Wu et aI., 1977). To overcomethis serious drawback, multifrequency holography and multi­source holography were introduced (Wu et aI., 1977; Wu andXu, 1979). Similar methods were introduced in seismic ex­ploration as Kirchhoff migration (French, 1975; Schneider,1978; Gazdag, 1978). In multisource holography, the imagereconstruction is a double-focusing process with both the

Manuscript received by the Editor October 18, 1985; revised manuscript received June 9, 1986.*Earth Resources Laboratory, Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA02139.© 1987 Society of Exploration Geophysicists. All rights reserved.

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12 Wu and Toksoz

where k = co/Co is the wavenumber of the field in the hostmedium. Let

where UO is the incident wave and U is the scattered wave.Substituting in equation (3),we have

THE BASIC SCATIERING EXPERIMENT AND THEFUNDAMENTAL RELATION OFDIFFRACTION TOMOGRAPHY

(1)

(4)

(3)

(2)

u(r) = uO(r) + U(r),

where u(r) is a scalar quantity of the field such as pressure, co isthe frequency, and V2 is the Laplacian operator.

Define the object function O(r) as

C2

O(r) = 1 - C2~r)'

The basic principle of diffraction tomography is simple, butit is often masked by the complexity of the algebra. To dem­onstrate the basic principle, we take an approach that sim­plifies the formulation.

Consider a plane wave incident on an object in a homoge­neous, infinite medium (Figure 1). Suppose the receiver is farfrom the object so that the scattered wave from the object canbe treated as a plane wave at the receiving point. We call thisa "basic scattering experiment." We derive the relation be­tween the scattered field and the object spectrum.

For further simplification, consider the case of the acousticwave equation with constant density. The object is describedby the velocity distribution C(r), where r is the position vector.The host medium has a velocity Co. The wave equation in thesource-free region is

migration, distorts the image of the object. Especially in thecase of vertical seismic profiling, there is a strong backgroundnoise belt. This noise belt is significantly reduced by the filter­ing operation of diffraction tomography.

and substitute in equation (1) to obtain

V2u(r) + k2u(r ) = k20(r)u(r),

receiver and source arrays (Wu and Xu, 1979). The methodcan be used to image both sharp boundaries (edges, corners,curved interfaces) and weak inhomogeneities.

Holography is not an exact inversion method; it often dis­torts the image. Recently, diffraction tomography, though notexact either, has been developed as a more accurate imagingmethod (for a review, see Kak, 1985; Devaney, 1985). Ordi­nary tomography (or ray tomography), such as X-ray tomog­raphy, refers to cross-sectional reconstruction of the mediumparameter distribution (velocity or absorption distribution)using transmission data and simple ray theory. When thecoverage of illuminating and viewing angles is limited, or thescale lengths of the medium inhomogeneities are comparablewith the wavelength, the performance of ray tomography be­comes poor. Diffraction tomography was introduced to im­prove the imaging process (Mueller et al., 1979; Mueller, 1980;Mueller et al., 1980; Devaney, 1982, 1983, 1984; Kaveh et al.,1983; Nahamoo et al., 1984). In diffraction tomography thescattered field (or diffraction field) is used to reconstruct theobject function. The basic principle is given in Wolf (1969), inwhich the scattered field based on the Born approximation isused to reconstruct the distribution of refractive index for aweak scattering object. Dandliker and Weiss (1970) discuss theinformation coverage in the spatial frequency domain for cer­tain experimental geometries. Iwata and Nagata (1975) gener­alize the method to the Rytov approximation.

For geophysical applications, Devaney (1984) derives theformulas for the special cases of borehole-to-borehole trans­mission and surface-to-borehole VSP measurements (Devaneyand Oristaglio, 1984). In Devaney's derivation the sources areassumed to be plane waves incident from all directions in ahemisphere from the upper half-space. Here we formulate theprocess in terms of point sources along a line. Such a sourcegeometry is more appropriate for the surface reflection, multi­offset VSP, and cross-hole measurements used in geophysicalexploration. We compare the results from diffraction tomogra­phy with those of multisource holography using syntheticdata. We show that diffraction tomography is equivalent tomultisource holography except that diffraction tomographyincludes an additional filtering operation. Multifrequency,multisource holography is equivalent to prestack migrationused in seismic exploration. We show by the informationcoverage in the spatial frequency domain and by numericalexamples that, without filtering, multisource holography, like

(5)

By using the free-space Green's function G(I r - r' I), we obtainU(r) from equation (5):

U(r) = -1 k20(r')u(r')G(1 r - r'l) dr', (6)

where the integration is taken over the volume of the object.Assuming the object is a weak inhomogeneity, the Born ap­proximation (u ~ UO) applies and equation (6)becomes

U(r) = - Lk20(r')G(1 r - r'l)uO(r') dr'. (7)

For our basic scattering experiment, the incident wave is aplane wave

r

uO(r') = exp (ikI• r'), (8)

FIG. 1. The geometry of the basic scattering experiment. where i is the unit vector in the incident direction. The free-

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Seismic Tomography and Holography 13

(11)

The integration in equation (11) is in the form of a 3-D Fou­rier transform. Define

exp (ikr) r [ ]U(r) = - r Jv k20(r') exp - ik(r - ~ • r' ds'.

Since the receiver is far from the object, we can use the Fraun­hofer approximation for the Green's function, that is,

GO r - r[) ;:::; exp [ik(r - r' . r)JI r, (10)

where r is the unit vector in the r direction. Substituting equa­tions (8) and (10) into equation (7),we obtain

(17)

(18)

(16)- i exp (iy d) _G(kg, r) = -2 g g exp (-ikg· r),

Yg

i exp (iys ds)G(ks' r) = - exp (-iks· r),2 Y.

Yg = Jk2 - k;,

where

G(r, r.), where r is the position vector of the object point and r,is that of the source point. From equation (7) we obtain

U(rg, rs) = - k2Iv O(r)G(r, rs)G(rg , r) dr, (14)

where U(rg, r.) is the scattered field measured at point rg whenthe point source is at rs, O(r) is the object function defined byequation (2), and G is the free-space Green's function.

Taking the Fourier transform of equation (14) along boththe source line and geophone line, we have

U(kg, ks) = - k2Iv O(r)G(k., r)G(kg, r) ds, (15)

(9)G(I r - r'l) = exp (ikR)jR,

where

R=lr-r'J.

space Green's function can be written as

APPLICATION OF DIFFRACTION TOMOGRAPHYTO SEISMIC EXPLORATION

With the basic definition just given we can derive the for­mulation for the application of diffraction tomography to sur­face reflection, vertical seismic profiling, and cross-holemeasurements. For simplicity, we discuss only the two­dimensional (2-D) case, that is, O(r) = O(x, z). The gener­alization for the 3-D case is straightforward. Assume pointsources (i.e., line sources in the 3-D space) along the sourceline and point receivers along the receiver (geophone) line(Figure 2a). The perpendicular distances from the origin to thesource line and geophone line are d. and dg , respectively.Here, the incident field UO is equal to the point-source field

which is the plane-wave scattering response. From equation(11), we can obtain

Upl(U) = -k20[k(r-~l (13)

where 0 is the 3-D Fourier transform of the object functionO(r). Equation (13) is the fundamental equation for diffractiontomography or for the holographic imaging of a semitrans­parent object. It relates the plane-wave scattering responseUpI (f, r), where f is the unit vector for the incident directionand r is the scattering direction, to the 3-D spectral density ofthe object function O(K) at the 3-D spatial frequency K = k(r- f). Note that this simple relation is only valid when the

Born approximation holds. Equation (13) shows that theplane-wave scattering response of an object for a given scatter­ing angle is related to the 3-D spectrum of the object at onlyone point in the spatial frequency domain, namely, at K = k(r- f); while in the space domain it is related to all the points

of the object. Beyond the Born (or Rytov) approximation, thissimple relationship disappears.

The 3-D Fourier transform of an object is a representationof the object by a superposition of sinusoidal 1-D media withdifferent periods and in different directions. Therefore equa­tion (13) means that one basic scattering experiment candetect only one sinusoidal1-D component of the object.

Upl(r) = U(r)r exp (-ikr), (12)Ys = Jk2 - k; . (19)

U(kg, ks) denotes the double Fourier transform of the scat­tered field along both the geophone line and the source line, kgand k. are the wavenumbers along the geophone line andsource line, respectively, and r., and Y. are the correspondingperpendicular wavenumbers. G is the Fourier transform of G,and g and s are the unit vectors in the direction of propaga­tion of plane waves toward the geophone line and source line,respectively. Substituting equations (16)and (17) into equation(15)results in

k2

U(kg, ks) = 4-- exp (iYgdg+ iy.ds)Yg t,

x Iv O(r) exp [ -ik(g - ~ . rJ ds, (20)

where f = -s is the unit vector in the incident direction. Theintegration in equation (20) is in the form of a 2-D Fouriertransform. Therefore, equation (20)can be written as

4YgYsU(kg, k.) exp [-i(Ygdg+Y.d.)J=k20[k(g-~l (21)

where O(K) is the 2-D Fourier transform of O(r). Equation (21)is another form of the fundamental relation equation (13). Theleft-hand side is the plane-wave scattering response (or theangular spectrum of the scattered field) which is related to theobject spectrum in a simple way (right-hand side).

The generalization of equation (21) to the case of the Rytovapproximation is straightforward. Instead of equation (14), inthe case of the Rytov approximation we have (Chernov, 1960;Flatte et al., 1979; Devaney, 1984; Slaney et al., 1984)

cI>(rg , rs) = G(rg , r.)il'¥(rg , rs)

= - k2Iv O(r)G(r, r.)G(rg , r) dr, (22)

where G(rg , rs) is the primary field at the geophone when thepoint source is at r., and il'¥(rg , r.) is the complex phasedifference between the total field u(r g , rs) and the primary field

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14 Wu and Toksoz

uO(rgrs) = G(rg, rs), and is defined by

u(rg, rs)juO(rg, rs) = exp [ '¥(rg, rs)J!exp [ 'II o(rg, rs)J

= exp [.1'¥(r g, r s)} (23)

Therefore, instead of equation (21) the fundamental relationfor the Rytov approximation is

4YgYsc1l(kg, ks) exp [ -i(ygdg+ Ysds)J = k20[k(g - ~l

(24)

where c1I(kg , ks) is the double Fourier transform of <l>(rg , rs)'

which is defined as

<I>(rg , rs) = G(rg, rs).1'¥(rg, rs)

= G(rg, rs) lOge[u(rg, rs)/G(rg, r s)} (25)

It has the same form as equation (21) except the data U(kg , k.)are changed to c1I(kg , k.).

If we can obtain O(K) for all the K from the scattering

measurements, we can, in principle, recover O(r) unambigu­ously. Because of limitations in experimental geometry forgeophysical applications, we generally obtain only partial in­formation about the object spectrum O(K), and thereforecannot recover O(r) precisely. In the following section, we dis­cuss the specific cases of surface reflection profiling (SRP) andvertical seismic profiling (VSP) and cross-hole measurements.

Surface reflection profiling (SRP)

In this configuration, we have both the source line andgeophone line along the surface. The scattering direction g isrelated to kgand Y9 by

kg = k cos e, Y9 = k sin e, (26)

where e is the angle between g and the geophone line. In thesame way

k; = k cos <1>, Ys = k sin <1>, (27)

where <I> is the angle between the incident direction : and thesource line. Therefore for one pair of tI, g), i.e., one element ofthe scattering matrix U(kg , ks)' we can obtain only the object

(b)

1\

ki

1\ 1\k(g- i)

Borehole

(a)

tc'"-.--_ X1!4"d"'-<:::------- d9-----

z

X Sources

o Geophones

(c)

FIG. 2.(a) The geometry of SRP and multioffset VSP. r: The position vector of the object point O(x, z). r,; rg : Theposition vectors of the source point and the geophone point respectively. d~, dg : The vertical distances from the originto the source line and geophone line, respectively. (b) In the case of SRP, lor one incident direction i the informationcoverage in the spectral domain from the data for all scattering directions g in the upper half-space is a semicircle.(c) In the case of VSP, the spectral coverage for one incident direction is a semicircle facing toward the borehole.

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Seismic Tomography arid Holography 15

~,..~ l' } '.\' / , II

... "" '" I\'--- '-~>

-- --".-

spectrum at one spatial frequency K = k(g - I) (see Figure 2b).For one incident angle I, the data from all the scatteringangles will cover the information on a semicircle in the spec­tral domain (Figure 2b). The semicircle is toward the gco­phone line. For all incident angles (0 s <p S n), the infor­mation coverage in the spectral domain is shown in Figure 3aby the shaded and cross-hatched area. Because the object isconsidered losslcss [i.e., O(r) is a real valued function], wehave

'.'. b . O*(K) = O( -K), (28)

"

FiG. 3. The spectral coverages of (a) the SRP data (the denselydotted area), (b) the VSP data (the sparsely dotted area),(c) the combination of both data sets, where ko is the wave­number of the wave field, K = (Kx ' K z) is the 2-D spatialfrequency vector of the object function.

where the asterisk stands for the complex conjugate. There­fore, the spectral coverage is conjugately symmetric. Thisproperty extends the coverage to the shaded region. FromFigure 3a it can be seen that the spectral coverage of thesingle-frequency surface reflection measurements is poor, es­pecially for the low-frequency components in the vertical di­rection.

Vertical seismic profiling (VSP)and the combination of VSP and SRP

Similarly, for one incident angle I (corresponding to one k,),the information in the data for all the scattering angles in theVSP case will cover a semicircle in the spectral domain, asshown in Figure 2c. For the whole set of data for all incidentangles, the spectral coverage is shown as the dotted region inFigure 3b. In some regions in the spectral domain the infor­mation has double coverage, but in other regions the infor­mation is missing. In general the VSP geometry providesbetter spectral coverage than the surface reflection data.

SourceBorehole

GeophoneBorehole

a) b)

FIG. 4.(a) The geometry of cross-borehole measurements. (h) The spectral coverage of cross-borehole transmissionmeasurements (shaded area).

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16 Wu and Toksoz

Figure 3c shows the spectral coverage of the combined dataset of VSP and SRP. The improvement in the informationcoverage is obvious.

Cross-hole measurements

b

e

c

az

k

-k

-k

In cross-hole transmission measurements, sources are in oneborehole (e.g., the left borehole), and geophones are in anotherborehole (e.g., the right borehole) as in Figure 4a. For oneincident angle i, the information coverage of all the scatteredwaves in the half-space in the right-hand side is a semicirciefacing the geophone borehole. The spectral coverage for thewhole transmission data set (including all the incident andscattering angles) is shown in Figure 4b (as derived by Deva­ney, 1984). The resolution in the z-direction is good, but reso­lution is generally poor in the x-direction. However, if theborehole reflection data can be combined with the cross-holetransmission data, the information coverage in the spectraldomain will be complete within the circle of radius 2k.

Note that in Figures 3 and 4 the spectral coverages are forthe ideal theoretical case in which both the source line andreceiver line are infinitely long. In practice, because the sourceand receiver lines are finite, the spectral coverage is reduced.The primary effect is coarser sampling in the wavenumberdomain that affects the image quality. The "finiteness effect" isdifficult to demonstrate analytically. We later show the influ­ence of finite aperture on image quality using numerical exam­ples (Figures 6 and 7). These examples also show that imagequality varies with the position of the object. Generally speak­ing, the farther the object is from the center of the observationlines, the worse the spectral coverage.

Figures 3 and 4 also show that the maximum spatial fre­quency which can be recovered is 2ko , where ko = 21t/"- is thewavenumber of the wave field. I Iowever, if the spatial intervalof sources and receivers is greater than the half-wavelengths of

k

FIG. 6. The reconstructed images for a point scatterer in thecase of 16 sources on the surface and 16 geophones at theborehole along the right edge. (a) The object (a point scat­terer). (b) The image by multisource holography for the objectat the center. (c) The image by diffraction tomography for theobject at the center. (d), (e), (f), (g) The images by diffractiontomography when the point scatterer is at the upper left (d),upper right (e), lower left (f), and lower right (g) corners.

kg

FIG. 5. The window function w;,(kg, k,) and ~(kg, k,) in the(k" kg)domain. Cross-shaded areas are the rejected areas.

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Seismic Tomography and Holography 17

the incident waves, the spectral coverage will shrink accord­ingly. The reconstructed image will be a low-passed version ofthe full coverage and there may be aliasing.

COMPARISON OF DIFFRACTION TOMOGRAPHYAND MULTISOURCE HOLOGRAPHY

Multisource holography is a generalization of the tradition­al single-source holography. It is a numerical reconstructionprocedure based on the double focusing principle (for both thesource array and the receiver array) (Wu and Xu, 1979). Forany point (x, z) in the object space, the reconstructed imagecan be calculated by

M N

I(x, z) = I Bs(r" r) I Bg(r, rg)Umn, (29)m=l n=l

where

8.

c.

and

Bs(r" r) = exp [ -i ~o Rs(x, Z)}Umn is the scattered field measured by the mth receiver for thenth source. Bg is the back propagator from the geophone pointto the image point, B, is the back propagator from the imagepoint to the source point, Rg is the distance between the geo­phone and the image point, and R, is the distance between thesource and the image point. Equation (29) is an imaging pro­cess in that it focuses both the receiver array (inner summa­tion) and the source array (outer summation). This imagingprocess can be applied to two different classes of objects: theclass of discrete scatterers, such as sharp boundaries, inter­faces, edges, and corners; and the class of weak inhomoge­neities that satisfy the Born or Rytov approximations. Thelatter case has a close relationship to diffraction tomography.

To show the relationship to diffraction tomography, wereturn to equation (21) and calculate the object spectrum fromthe transformed scattered field (the angular spectrum). The2-D inverse Fourier transform of the object spectrum gives theband-limited reconstruction of the object function. However,reconstruction can also be done in the wavenumber domain of

b.

d.

FIG. 7. The case of 32 sources on the surface and 32 geophones along the borehole. (a) The object. (b) The image bymultisource holography.rc) The image by diffraction tomography. (d) The image by multi source holography with onlyone source but 32 more geophones on the surface.

a b c

FIG. 8. The reconstructed images for a square box composed of point scatterers, with 32 sources and 32 geophones, allon the surface (a) The object. (b) The image by multisource holography. (c) The image by diffraction tomography.

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18 Wu and Toksoz

The window functions are shown in Figure 5 in the (ks, kg)domain. Because O(K) is conjugately symmetric and band­limited (see Figure 3), we take the double of the real part ofequation (33) as the object image. The final reconstructionformula becomes

2 fkO(x, z) = - Re dk, exp (ik,» - iysZ)Ol(X, Z, ks) ,n -k

(35)

where

(36)

FIG. 9. The object for numerical experiments. The letters aremade up of point scatterers separated by 1 wavelength.

the scattered field (kg, k; domain) by filtered back propagation(Devaney, 1982, 1984). Starting from equation (21), we obtain

O(r) = (2~)2 II dK O(K) exp [iK .rJ

=~ ffdk s dkg J(K x, Kzlks, kg) 4Yks/

gU(kg, ks)

(2n)

x exp [ -i(Ysds + Ygdg)] exp [i(KxX + KzZ)J.

(30)

where J(K x, K, I ks' kg) is the Jacobian of the coordinatetransformation. For the VSP case (see Figure 2a),

(31)

and

is the filtered angular spectrum of the scattered field. Here weassume ds = 0 and dg = X h (the horizontal distance from theorigin to the borehole). From equation (35), we can see that0 1(x, z, ks) is formed by back propagating D(kg , ks) from thegeophone line to the object space and O(x, z) is formed byback propagating 0l(X, z, ks) to the source line. This is adouble focusing process for D(kg , kJ Equation (35) is theholographic reconstruction process for D(kg, ks) using theangular spectrum method (Boyer, 1971). Diffraction tomogra­phy modifies the process into a velocity inversion method bywindowing and weighting the data in the spectral domain.Figure 3b shows that, in the VSP configuration, the infor­mation of the object spectrum is sampled non uniformly. Nearthe center in the upper right quadrant the sampling is muchdenser than elsewhere. Therefore, the holographic reconstruc­tion by back propagation without filtering will cause severedistortion of the image (this is shown later by numerical exam­ples). Diffraction tomography corrects this nonuniformity byweighting the data using the Jacobian of the coordinate trans­formation. The information duplication in some areas of thespectral domain also causes distortion of the image. Diffrac­tion tomography removes the duplication by windowing.

Similarly, for the case of SRP, we have

(37)

(39)

1 fkO(x, z) = - Re dk, exp tik,» - iysZ)Ol(X, Z, k.),n -k

(38)

where

K, = -(Yg + Ys),

d, = dg = O.

The reconstruction formula (33) becomes

where(34)

(32)

if kg ~ Y. for ks ~ 0

otherwise.

To eliminate duplicate information in the spectral domain, wedefine two windows to remove the high-frequency duplication(the upper right part in Figure 3b) and the low-frequencyduplication (the lower left part in Figure 3b):

if kg ~ -Y. for k, ~ 0otherwise

Kx=ks+Yg,

Therefore, we obtain

O(x, z) = :2 Idk, exp (iksx) exp (-iysz)

x {f dkgexp (ikgz) exp [ -iyg(dg- X)]

[O(k k)lkgks+ygy.IJ}. (33)

x g' s k2

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Sel8mlc Tomography and Holography 21

For the case of cross-hole transmission measurements, wehave

where

Kx=Yg-Ys ,

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mary purpose of this paper is to show the relative per­formances of diffraction tomography and multisource holo­graphy for different source-receiver configurations used in seis­mic imaging. We do not discuss the extrapolation and filteringtechnique, but simply use the minimum energy inversion onthe reconstruction.

For the reconstruction, we used the method of filtered backpropagation for diffraction tomography (Devaney, 1982,1984). The algorithms are based on equations (35), (38), (41)and the relative formulas. The reconstruction is accomplishedby back propagating the filtered scattered field. For example,in formulas (35) (for VSP), D(kg , k.) are the filtered data andexp [ - iy9 (x, - x)] and exp [ - iysz] are the back propagatorsin the wavenumber domain. The filtering is implemented by

1 fkO(x,z) =-Re dksexp(iksz -iYsx)Ol(X,z,ks)'11: - k

(41)SRP

NUMERICAL EXPERIMENTS

where

FIG. 12. The improvements in spectral coverage by usingmultiple frequencies for the case of (a) SRP and (b) VSP. Thedark areas are the extra information that can be obtained byusing multiple frequencies with bandwidth IV= (1/2)fmax.

(42)

In the case of the Rytov approximation, all the reconstruc­tion formulas are the same except that O(kg , ks) is replaced byif,(kg , ks)·

Devaney (1984) shows numerical simulation results for thecase of borehole-to-borehole transmission measurements andoffset VSP. In the simulations he uses the theoretical formulaof the 2-D spectrum of a circular or elliptical disc and gener­ates the angular spectrum of the scattered field O(kg , k.) fromthe object spectrum O(K), using a formula similar to equation(21). In such a simulation the distortion due to finite apertureand the influence of sampling interval cannot be tested. To bemore realistic and to compare the results with those of multi­source holography, we calculate the scattered field U(r g , r.) inthe space domain using equation (14) based on the Born ap­proximation. Since we can also generate the data set for theRytov approximation using the same formula [the right sidesof equations (22) and (14) have the same form], these numeri­cal tests are for both the Born and the Rytov approximations.These tests are only to demonstrate the influence of variousmeasurement geometries and the effects of filtering on theimaging process.

To cope with the missing information in the object spec­trum (i.e., the" blind" areas in the spectral coverages for differ­ent source and receiver configurations , shown in Figures 3 and4), we adopt the criterion of "minimum energy inversion"(Devaney, 1984). That is, in the reconstruction the blind areasin the object spectrum are taken to be zero. If we do not haveany a priori knowledge about the object, this is probably thebest strategy for reconstruction. In practice, some a prioriknowledge of the imaged object may be available. In suchcases, there are ways of extrapolating in to the blind areas(Sezan and Stark, 1983). In order to reduce the side-lobe effectof sharp boundaries of the blind areas, some smoothing can beperformed using an appropriate filter (Menke, 1985). The pri-

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Seismic Tomography and Holography 23

equation (36) also in the wavenumber domain. Pan and Kak(1983) have compared this method with the method of inter­polation and Fourier transformation and have shown that forthe case of full angular coverage these two algorithms canhave the same results. Detailed comparison of different recon­struction algorithms is beyond the scope of this paper; thereader is referred to Pan and Kak (1983) and others (Orista­glio, 1983; Nahamoo et al., 1984). In the case of multisourceholography, the direct calculation of equation (29) is time­consuming. However, there is a fast algorithm using the fastFourier transform (Wu and Xu, 1978) for calculating equation(29) which makes the computing time of multisource holo­graphy of the same order as the filtered back propagationalgorithm for diffraction tomography.

Figures 6 through 13 show some examples of reconstructedimages from diffraction tomography and multisource holo­graphy using synthetic data calculated from equation (14). InFigure 6 there are 16 sources on the surface and 16 geophonesin the borehole. The object is a point scatterer (Figure 6a). Acomparison of images in Figures 6b and 6c shows that multi­source holography has a similar image quality (Figure 6b) tothat of diffraction tomography (Figure 6c). In these examplesthe object (point scatterer) was located at the intersection ofthe center lines of source and receiver arrays. When the objectwas moved away from the center, the image quality changed.Figures 6d, 6e, 6f, and 6g show four examples where the objectwas located sequentially at four corners of the square whosetwo edges are the source and receiver arrays. These images arereconstructed by diffraction tomography in the same way as inFigure 6c. The image quality is degraded toward the outeredges of the imaging plane. The poorest resolution is when theobject is at the farthest corner (Figure 6f). These indicate thateffective tomographic reconstruction is possible when sourceand receiver arrays bracket the object. When the numbers ofsources and geophones increase, diffraction tomography be­comes advantageous. The results for the same point scatterer,but with 32 sources on the surface and 32 geophones in theborehole, are shown in Figure 7. Diffraction tomographyachieves a slightly better image (Figure 7c) than does multi­source holography (Figure 7b). However, multisource holo­graphy has the advantage of flexibility. It can use an arbitrarynumber of sources depending upon their availability and therequirements of the image quality. Figure 7d shows an imageof the point scatterer obtained from only one source on thesurface, 32 geophones in the borehole, and 32 geophones onthe surface. The image quality is still acceptable.

Figure 8 gives the results for a square box composed ofpoint scatterers placed at half-wavelength intervals. There are32 sources and 32 geophones on the surface for both diffrac­tion tomography and holography. The image from diffractiontomography (Figure 8c) is defined better than the image ob­tained from holography (Figure 8b).

To test the quality of imaging for complex structures, wetook three letters, "ERL," as our object. The letters are madeup of point scatterers separated by one wavelength (Figure 9).Figures 10 through 13 show the results from multisource holo­graphy and diffraction tomography. Unless otherwise stated,we used 64 sources along the source line and 64 geophonesalong the geophone line. The sources and geophones areequally spaced at half-wavelength intervals. Figure 10 showsthe results from multisource holography and diffractiontomography for the case of surface reflection measurements.Note the long tails of the holographic image (Figure lOa), dueto the poor longitudinal resolution of single-frequency holo-

graphy and the decrease of this resolution with distance. Dif­fraction tomography reduces the long tails (Figure lOd) bycorrecting the nonuniformity of the spectral informationthrough weighting the transformed scattering data. FigureslOb and lOe show the results for the VSP case. The holo­graphic image has a strong belt-shaped noise band (FigurelOb) because of the nonuniform sampling of the object spec­trum and the lack of information about the object spectrum incertain directions as illustrated in Figure 3b. This noise belt isinherent in the holographic imaging process for this geometryand cannot be removed by increasing either the aperture orthe number of observation points. Figure l lb shows the holo­graphic image for the case of 128 geophones and 128 sources.The strong noise belt still exists. However, diffraction tomog­raphy reduces this noise belt significantly (Figure lOe) byweighting and windowing. Figures lOc and lOf give the com­parison between the results from holography and diffractiontomography for the case where both surface reflection dataand VSP data are used. Diffraction tomography gives betterresults.

To test the effect of multiple frequencies, we performed nu­merical simulation of multifrequency, multisource holography(see Wu et al., 1977)for the cases of surface reflection measure­ments and VSP. We used ten frequencies in the simulation.The bandwidth was about one-half of the highest frequencyfmax and the spacing of sources and geophones was equal tothe half-wavelength at fmax. For each frequency the datamatrix is 64 geophones and 64 sources. Figures l lc and l Idshow the results of reconstruction by multifrequency, multi­source holography. The image for the case of surface measure­ments (Figure l lc) has been improved relative to the single­frequency case. However, in the case of VSP (Figure lld) theimprovement is less significant and the noise belt still exists.We can explain this difference by the information coverage ofmultifrequency data in the spectral domain. Figure 12 showsthe improvements in spectral coverage by using multiple fre­quencies for the cases of surface profiling and offset VSP. Thedark areas are the extra information that can be obtained byusing multiple frequencies within the bandwidth t1f =

(1/2)fmax. In the case of surface profiling, the benefit of usingmultiple frequencies is much greater than in the case of VSP.In addition, the nonuniform sampling of the object spectrumis inherent to the VSP geometry. Therefore, use of multiplefrequencies cannot remove the noise belt.

Multifrequency, multisource holography is equivalent toprestack migration (Wu et al., 1977). Therefore, the problem ofa noise belt is also inherent in VSP migration. As demon­strated by Keho (1984), the image of a point scatterer by VSPmigration is a tilted and twisted "bow-tie" shape. This artifactis of the same type as the noise belt and can be reduced byfiltering the data based on the principle of diffraction tomog­raphy. Although the image quality of multisource holographyis not as good as that of diffraction tomography for weakinhomogeneities, it can be applied to the case of discretestrong scatterers in which diffraction tomography has littleuse. It also has the flexibility of using an arbitrary number ofsources or receivers.

The other flexibility of multisource holography is that thesampling interval is not restricted by the Nyquist frequency.Figure 13 shows the image of letters "ERL" composed ofpoint scatterers having an interval of five wavelengths (thesame as the sampling interval of the data) reconstructed bymultisource holography. Diffraction tomography cannot beused in this case. Apparently, the combination of borehole

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24 Wu and Toksoz

data and surface data provides much improved image quality(Figure 13c) over using either surface data (Figure 13a) orborehole data (Figure 13b) alone.

Figures 13d and 13e show the reconstructed images usingcross-borehole measurements. Figure 13d gives the result ofborehole-to-borehole transmission holography, with 32sources in one borehole and 32 receivers in the other borehole.Figure 13e shows the reconstructed image using both theborehole-to-borehole transmission data and surface reflectiondata. Both images have fairly good quality.

CONCLL'SIONS

We have derived the reconstruction formulas of diffractiontomography for the cases of surface reflection profiling (SRP),vertical seismic profiling (YSP), and cross-hole measurements.In the derivation we assumed line sources along a source lineand line receivers along a receiver line for a 2-D inhomoge­neous medium. We studied the relationship between diffrac­tion tomography and multisource holography through theo­retical analysis and numerical experiments.

The quality of the reconstructed image depends upon sev­eral factors:

(1) The object spectrum in the spatial frequencydomain has holes (i.e., information ..blind areas") foreach geometry (SRP, YSP, cross-hole). Thus none of thethree geometries alone can reconstruct the image per­fectly. Using a combination of two geometries (e.g., SRPand YSP, or SRP and cross-hole), the blind areas can bereduced and image quality can be improved.

(2) The object spectrum in the spatial frequencydomain has nonuniform information density and in­cludes duplication. The filtering operation used in thediffraction tomography reconstruction algorithm cor­rects the non uniformity of information density in theobject spectrum and therefore reduces the distortions ofthe image,

(3) Using multiple frequencies instead of a single fre­quency improves the information coverage and theimage quality. The improvement is more significant inthe case of SRP.

(4) Diffraction tomography as formulated here isbased on the Born or Rytov approximation and is re­stricted to weak inhomogeneities.

(5) Diffraction tomography is equivalent to multi­source holography with prefiltered data. Holography isnot an inversion method and often gives a distortedimage of the object. On the other hand, it permits theflexibility of using an arbitrary number of sources (in­cluding one source). It is suited for the imaging of sharpboundaries such as edges, corners, and interfaces.

ACKNOWLEDGMENTS

We are grateful to Dr. Shao-Hui Xu and Ticn-When Lo forthe helpful discussions and computational assistance. This re­search was supported in part by the Vertical Seismic Profiling­Reservoir Delineation Consortium at the Earth ResourcesLaboratory, Massachusetts Institute of Technology.

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