GEOPHYSICS. VOL. 64, NO.5 (SEPTEMBER-OCTOBER 1999); P. 1524-1534,7 FIGS.
Extended local Born Fourier migration method
Lian-Jie Huang *, Michael C. Fehler*, and Ru-Shan Wu$
ABSTRACT
A migration approach based on a local application ofthe Born approximation within each extrapolation inter-val contains a singularity that can make direct applica-tion unstable. Previous authors have suggested addingan imaginary part to the vertical wavenumber to elimi-nate the singularity. However, their method requires thatthe reference slowness must be the maximum slownessof a given layer; consequently, the slowness perturba-tions are larger than those when the average slownessis selected as a reference slowness. Therefore, its appli-cability is limited. We develop an extended local BornFourier migration method that circumvents the singu-larity problem of the local Born solution and makes itpossible to choose the average slowness as a referenceslowness. It is computationally efficient because of theuse of a fast Fourier transform algorithm. It can handlewider angles (or steeper interfaces) and scattering effectsof heterogeneities more accurately than the split-stepFourier (SSF) method, which accounts for only the phasechange as a result of the slowness perturbations but notamplitude change. To handle large lateral slowness vari-ations, we introduce different reference slownesses indifferent regions of a medium to ensure the condition ofsmall perturbation. The migration result obtained usingthe extended local Born Fourier method with multiplereference slownesses demonstrates that the method canproduce high-quality images of complex structures withlarge lateral slowness variations.
INTRODUCTION
Prestack depth migration in three dimensions is widely usedfor imaging complex subsurface structures. Ray-tracing-basedKirchhoff migration methods, which are most commonly usedfor large 3-D imaging problems, have difficulty handling com-plex structures such as those containing salt or steep dips
(Hu and McMechan, 1986; Fei et al., 1996). Migration meth-ods based on a finite-difference solution of the full-wave equa-tion for a heterogeneous medium are quite accurate for com-plex structures with large lateral slowness variations (Changand McMechan, 1990), but it is difficult to apply such meth-ods to large 3-D problems because they are time consumingand require large computer memory. The phase-shift migra-tion method developed by Gazdag (1978) and the f -k migra-tion method developed by Stolt (1978) are implemented in thefrequency-wavenumber domain and have some attractive ad-vantages, such as the exact implementation of the transverseLaplacian operators in the wave equation, unconditional sta-bility and fast computation speed from the use of a fast Fouriertransform (FFT) algorithm. However, the methods use a con-stant velocity for each extrapolation depth interval, and theycannot handle lateral slowness variations.
The phase-shift plus interpolation (PSPI) method, imple-mented in the frequency-space and frequency-wavenumberdomains, was developed by Gazdag and Sguazzero (1984) asone way to handle lateral slowness variations. Several constantslownesses are used in the PSPI method to perform phase-shift migrations for each depth interval and the correspondingmigration results are interpolated to yield the final migratedimage. PSPI can handle large lateral slowness variations.
Another method implemented in the frequency-space andfrequency-wavenumber domains is the split-step Fourier (SSF)method introduced to seismology by Stoffa et al. (1990).Knepp (1983) and Martin and Flatt6 (1988) use the split-stepFourier approach to investigate wave propagation in randommedia and call it the phase screen method. In this method,wave propagation occurs across extrapolation intervals in twosteps. First, a phase shift is used to propagate through a ref-erence medium having a slowness equal to some average ofthe slowness of the real medium within the interval. Second,a correction is made to account for lateral slowness variationswithin the interval. The SSF method takes into account smalllateral slowness variations yet retains the advantages of thephase-shift method. It has been used for poststack migration(Stoffa et al., 1990), prestack migration (Huang and Wu, 1996b;
Popovici, 1996; Roberts et al., 1997; Tanis and Stoffa, 1997), andmodeling of forward and primary reflected wave propagation(Wu and Huang,1992; Wu et al., 1995).
To handle large lateral slowness variations, Kessinger (1992)introduced the multiple reference slowness (MRS) logic of thePSPI method into the SSF method (hereafter abbreviated asSSF-MRS method). The physics base of wavefield extrapola-tion in the SSF method is de facto identical to the PSPI method.The differences between these two methods are in the ap-proaches for selecting reference slownesses and in the finalsteps used to obtain the wavefield at each depth level. Dif-ferent reference slownesses are selected in different regionsof a medium in the SSF-MRS method, while several refer-ence slownesses bounded by the minimum and maximum slow-nesses of a given layer are used in the PSPI method. The wave-field in each region is extrapolated using the correspondingreference slowness for that region in the SSF-MRS method;therefore, no interpolation is needed. Interpolation of wave-fields for different reference slownesses is used in the PSPImethod to obtain the extrapolated wavefield. A hybrid mi-gration method termed Fourier finite-difference migration isproposed by Ristow and Ruhl (1994) to obtain a better max-imum dip-angle behavior than the SSF method. This hybridmethod uses a finite-difference scheme in regions with largelateral slowness variations.
The local Born solution to the scalar wave equation canbe implemented in the frequency-wavenumber and frequency-space domains and used for migration and modeling followingan approach similar to the SSF method. It is valid for smalllateral slowness variations but its range of validity is generallygreater than that of the conventional SSF method. It has beenused for modeling primary reflected waves (Wu and Huang,1995) and prestack depth migration (Huang and Wu, 1996a).For migration, the method has been called the pseudoscreenmethod (Huang and Wu, 1996a). Adding an imaginary partto the vertical wavenumber in local Born Fourier solution hasbeen proposed by de Hoop et al. (1999) as a way to elimi-nate the singularity that exists in the local Born solution whenwave propagation in the reference medium is perpendicularto the main propagation direction. We call this method thecomplexified local Born Fourier method. To ensure a physi-cally acceptable wavefront, the method requires that the ref-erence slowness within each extrapolation interval be greaterthan or equal to the maximum slowness within that interval(de Hoop et al., 1999). Therefore, the slowness perturbationsare larger than those when the average slowness is chosen asthe reference slowness and the applicability of the method islimited.
We introduce an alternative approach to circumvent the sin-gularity of the local Born solution that makes it possible tochoose the average slowness as the reference slowness. Themethod is termed the extended local Born Fourier (ELBF)method since it uses the extended local Born solution and is im-plemented in the frequency-wavenumber and frequency-spacedomains. It can handle wider angle than the SSF method. Forlarge slowness variations, the local Born—based method is un-reliable or even unstable. Following Kessinger (1992), we in-troduce multiple reference slowness (MRS) logic of the PSPIinto our method. Different reference slownesses are chosenin different regions of a medium so that the slowness pertur-bations are small in all regions. The increased computation
time of the MRS method relative to the method with singlereference slowness (SRS) depends on the number of referenceslownesses selected. For example, if the average number of ref-erence slownesses per depth level is four, then the CPU timefor the MRS method is approximately three times more thanthe SRS method. As in the SSF-MRS method, no interpolationis needed in our method with MRS (hereafter abbreviated asELBF-MRS).
We present the formulations of the extended local BornFourier methods with single-reference and multiple-referenceslownesses. We numerically compare the accuracy of the ELBFmethod with the SSF method. In migration examples, we firstuse a simple model with a dipping interface to demonstratethat the ELBF method produces a more accurate image thanthe SSF method. A 2-D slice of the SEG/EAGE 3-D salt model(Aminzadeh et al., 1996) is then used to test the ELBF-MRSmethod. The result is compared with those obtained using SSFmigration, Kirchhoff migration (Fei et al., 1996), and FX mi-gration (Amoco, 1995).
EXTENDED LOCAL BORN FOURIER PROPAGATOR
Wavefield decomposition
By introducing a reference velocity vo(z), the scalar waveequation for a constant density medium
a 2a2 a2 Co2
ax + a + az + v x, z P(x , y, z; (0) = o (1 )z y z z z( y> )^
can be written in the form
a 2 a2 azaX2
+ ay, + Z2 +ko(z)p {x, y, z; cu}
_ —2k^s(x , y, z)P(x, y, z; w), (2)
with
irs(x, y, z) - v2 ^ (y) z) _1]. (3)
In the above equations, p(x, y, z; w) represents the pressurewavefield in the frequency domain, v(x, y, z) is the velocityof the medium, w is the circular frequency, and k o (z) is thereference wavenumber given by
ko(z) = v z) . (4)
Decomposing the pressure wavefield p into two parts, na-mely, the wavefield in the reference medium Po (i.e., incidentwavefield) that satisfies the homogeneous wave equation in thereference medium and the scattered field p,, i.e.,
p(x , y, z; a)) = po(x, y, z; w) + ps(x, y, z; w), (5)
we have
PS(x , y, z; w) = 2f dxt dyi dzi G(x, y, z; xl, yl, zi; cv)z
x k'(zt)E(x, , yt, zi)p(xl, yi, zl; w), (6)
where G(x, y, z; x i , y t , z 1 ; w) is the Green's function in the ref-erence medium, S2 is the integral volume, and (x i , yi , z 1 ) is a po-sition inside the volume Q. Fourier transforming equation (6)over x and y yields
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1526 Huang et al.
ps (kx , ky , z; co) position within the interval from z i to z i + Az. Equation (13)can be approximated by
= 2 f dxt dyt dz1G(kx, ky , z; xl, Yt, zt; w)2 ps(kx, ky , z i + Az; w)
x ko(zt)s(xl , Yi, zi)p(xi, Yt, zi; w). (7) ti 2Az f f dx dyG(kx , ky , zi + Az; x, y, z i ; w)Making use of equation (3), ko (zi )e(x, y, z i ) can be approxi-mated by x ko(z 1 )co0s(x, y, zi)P(x , y, zi; w), (14)
ko(zl)E(x , y, zi) w [s(x , y, zi) — so(zi)]
= coAs(x , Y, zl) , (g)
where s = 1/v and so =1/vo represent, respectively, the slow-nesses of the medium and the reference medium; As representsthe slowness perturbation. Substituting equation (8) into equa-tion (7) yields
ps(kx, ky, z; w) : 2 [ dx dy dzl G(kx , ky, z; x, y, zi; w)
x ko(zl)wAs(x, y, zt)p(x, y, z1; w) , (9)
where the integral variables x i and yl have been changed to xand y, respectively.
Wavefield extrapolation
To extrapolate the wavefield from z, to z i + Az where Azis the extrapolation interval, we must calculate the incidentand scattered wavefields at z i + Az using the known wavefieldp(x, y, z i ; w). The extrapolated wavefield at z i + Az is obtainedby
p(x , y, zi + Az; w) = po(x , y, zi + Az; w)
+ps(x, Y, zi + Az; (0), (10)
where the incident wavefield po (x, y, z i + Az; w) is given by
po(x , y, zi + Az; w)
_ -kxtky {e ikz(zi)4zJx,y{p(x , Y , zi; w)}} (11)
with
kz (zi) _ ^z —k 2 — ky. (12)v0 (zi )
In equation (11), ^x represents the Fourier transform overx and y, and Fkxlky represents the inverse Fourier transformover kx and ky . In equation (12), vo(z i ) represents the referencevelocity of the medium within the interval from z i to z i + Az.In the following, we focus on the calculation of the scatteredwavefield ps (x, y, z i + Az; w).
It follows from equation (9) that Ps (k5 , ky , z i + Az; w) is givenby
ps(kx , ky, zi + Az; to)
= 2f z +ozdz f f dx dyG(kx , ky , z i + Az; x, y, z; w)
z,
x ko(z)wAs(x, y, z)p(x, y, z; co), (13)
where the integral variable has been changed from z l to z.Hence, (x, y, z) in the integral of equation (13) represents the
where the down-going 3-D Green's function in the frequency-wavenumber domain G(kx , ky , z i + Az; x, y, z i ; £o) is given by(Clayton and Stolt, 1981)
G(kx ,ky ,zi+ Az; x,Y,zi;0))
l eikz(zi)Oze—i(kxx+kyy). (15)
2k (z1)
Substituting equation (15) into equation (14) yields
ps(kx, ky , zi + Az; C)) ti ko(zi) eikz(zi)4z
kz(zi)
x .Fx,y{[iwAs(x, y, zi)Az]p(x , y, zi; w)}. (16)
Inverse Fourier transforming equation (16) over kx and k),yields the scattered field in the frequency-space domain
(z zps(x, Y ,
zi + Oz; w) Fkxiky koi e (ik z i) oz
x .^',r , y {[iwAs(x, y, zi)Az] p(x, y, zi; w)}j. (17)
Equation (17) indicates that the scattered wavefield generatedby the heterogeneities within the interval from z i to z i + Az iscalculated by
1) multiplying the wavefield at z i in the frequency-space do-main with the term [iwAs(x, y, z i )Az] to take into ac-count the slowness variations within the interval,
2) Fourier transforming the result into the frequency-wavenumber domain,
3) freely propagating from z, to z i + Az,4) multiplying with a filtering term, and5) inverse Fourier transforming the scattered field from the
frequency-wavenumber domain to the frequency-spacedomain.
The wavefield at z i is extrapolated to z i + Az using equa-tions (10), (11), and (17).
Singularity
There is a singularity in equation (17) when k, approacheszero which leads to an instability of the algorithm. Authorsde Hoop et al. (1999) present a general formulation of screenmethods for scattering of waves in inhomogeneous media usingpseudodifferential operators. They propose using
r_V 2k'(zi) = — (1 +i^) 2 (kz +ky) (18) )
to replace k5 in the denominator of the fraction ko /kz in equa-tion (17) to avoid the singularity. In equation (18), i is asmall real number. The corresponding propagator is hereaftertermed the complexified local Born Fourier propagator.
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Extended Local Born Fourier Migration
1527
The reliability of the local Born solution discussed heredegrades with increasing slowness perturbation because themethod was derived by assuming that the perturbation is small.This degradation is similar to that of the split-step Fouriermethod discussed by Huang and Fehler (1998). De Hoop et al.(1999) show that the reference slowness within each extrapo-lation interval must be greater than or equal to the maximumslowness within that interval (i.e., there are only negative slow-ness perturbations) to ensure a physically acceptable wavefrontwhen using the complexified local Born Fourier propagator.Consequently, the absolute values of slowness perturbationsare larger than what would result if the average slowness waschosen as a reference slowness; therefore, the applicability ofthe method is limited. In addition, in equation (18) must bethe same sign as that of the slowness perturbation. Since equa-tion (18) is used in the wavenumber domain, we cannot selecta single value of i for a given layer to ensure that the wave-field amplitude decreases when the average slowness withinthe layer is chosen as the reference slowness.
Extended local Born Fourier propagator
We introduce an alternative approach to circumvent the sin-gularity problem of the local Born solution. Let
( ko(zi) k
= 1— T (19)Q zi) = k(z) ko(zi)
where the transverse component of the wavenumber kT is givenby
kT = kX + ky. (20)
Since (kT / ko )2 <1 for one-way wave propagation, equa-tion (19) can be approximated by
or(z i ) 1 + A, (21)
where the term A can be obtained using different approxima-tions such as the Muir, Pade, and Taylor approximations. Forthe Taylor approximation,
2 4
A^0.5 kr + 0.375 ( k'
ko(z,) k0@) )6 8
+0.3125 ( k ) ) +O 2734375 ( ko(zi )
).(22)\\\ z,
Under approximation (21), equation (17) becomes
pe(x, Y , Zi + AZ; w) ti J7k,Iky {6(zi)e ikz(zi )4z
x FX,v{[iwOs(x, Y, zi)AZ] P(x , Y, zi; w)}}. (23)
There is no singularity in equation (23). Later, we numericallydemonstrate that the average slowness can be chosen as a ref-erence slowness to calculate scattered wavefields using equa-tion (23). This is an important advantage of using equation (23).
We note the following. First, equation (23) is derived fromequation (13), which is a nonlinear equation with respect tothe scattered wavefield Ps, but equation (23) is a linear equa-tion for p 5 . From this point of view, equation (23) involvesthe Born approximation within the interval from z i to z i + Az.Within each extrapolation interval, the incident wavefield is
the total wavefield from the preceding interval. This approx-imation is referred to as the local Born approximation. Sec-ond, equation (23) uses approximations (8) and (21). Third,the Fourier transform is a key tool in our wavefield extrapo-lation method. Therefore, the wavefield extrapolation methodusing equations (10), (11), and (23) is termed the extended lo-cal Born Fourier (ELBF) method. In this method, the Bornapproximation is applied within each interval using an inci-dent wavefield for that interval. The recursive application ofthe Born approximation from layer to layer means that multi-ple forward scattering is accounted for as the wave propagatesthrough a medium. Therefore, the use of the term Born ap-proximation does not necessarily mean that our method is asingle scattering approach.
Applicability condition of the ELBF method
For a given extrapolation depth interval, the ELBF methodbecomes unreliable or even unstable when the lateral slownessvariations are strong and/or the frequency is high. The methodis accurate and applicable if and only if the Born approxima-tion is satisfied within each extrapolation interval during wave-field extrapolation. This requires that the scattered wavefieldp, (x, y, z i + Az; w) given by equation (23) be much smallerthan the incident wavefield po(x, y, z ; + Az; w) given by equa-tion (11). Equation (23) indicates that the scattered wavefieldps (x, y, z i + Az; w) is proportional to the term (wAs Az). There-fore, the applicability condition of the ELBF method is givenby
wmax{As)AzJ < P, (24)
where $ is a small real number and max{ As} represents themaximum slowness perturbation within the interval from z ;
to z, + Az. Numerical tests show that when condition (24) isviolated, the ELBF method may become unreliable or evenunstable (Huang et al., 1998). To stabilize the method, we canadjust the extrapolation interval Az such that condition (24)is always satisfied during wavefield downward continuation(Huang et al., 1998). In practice, the value of $ can be 0.1to 0.15.
Relation among the methods
When the propagation angle relative to the main propaga-tion direction (i.e., the positive direction of z-axis) is small, wehave
k0k ^1.
z
Equations (17) and (23) can be written as
Ps(x , Y, Zi + Az; w) ^ Fkxlk^ { e ikz (z')^z
X Fx , y {[iwAs(x, y, zi)Az]p(x, y, zi; w)}}. (25)
Substituting equation (25) into equation (10) and making useof equation (11) yields
P(x , y, z1 + Az; w) Fkx' {et)^z
x J{[1 + iwAs(x, y, z,)Az] p(x, y, z i ; w)}}. (26)
Making use of the approximation
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1528
Huang et al.
where is a small number, equation (26) can be written as
p(x, y, zi + Az; co).^kxtkY {e` kz (z i )oz
X .Ex,Y {p(x, y, zi; w)e iwAs(x,Y,zi)Az}} . (27)
Equation (27) indicates that the wavefield at z i is extrapolatedto z i + Az by
1) multiplying the wavefield at z i in the frequency-space do-main with a phase change term resulting from the hetero-geneities within the interval from z ; to z i + Az,
2) Fourier transforming the result to the frequency-wave-number domain,
3) freely propagating to the next depth level at z ; + Az, and4) inverse Fourier transforming into the frequency-space
domain.
Equation (27) is therefore termed the split-step Fourier(SSF) propagator (Stoffa et al., 1990; Huang and Wu, 1996b;Huang and Fehler, 1998). Since the heterogeneities are ac-counted for in equation (27) by a phase shift, this kind ofpropagator is alternatively called the phase-screen propaga-tor (Knepp, 1983; Martin and Flatt6, 1988). Unlike the ELBFmethod, the SSF method is inherently stable.
Huang and Fehler (1998) discuss different split-step Fouriermarching solutions for seismic modeling and migration. Forinstance, another split-step Fourier marching solution with al-most the same accuracy as equation (27) is given by
p(x , y, zi + Az; w) ti eim4s(x,Y,zi)4zFk t k I e ikz(zi)4z
x. y
x )c,Y{P(x , y, zi; w)}}. (28 )
In this extrapolation equation, the wavefield at z i freely propa-gates to z i + Az followed by a phase shift resulting from the het-erogeneities within the interval from z i to z ; + Az. Equation (28)is the form commonly used in seismic migration (Stoffa et al.,1990).
The term e 'wso (zi ) is independent of x and y; therefore,equation (27) can be rewritten as
1 k(zp(x, y ,
zi + Az; C^J) '';% .^'kx ky {B zzi)Az—cos0(i)Az]
X 1 x,Y 1p(x, y, zi; w)e ir&s(x,Y , zi)AzH, (29)
Equation (29) is the extrapolation equation used in the phase-shift plus interpolation (PSPI) method developed by Gazdagand Sguazzero (1984). In this method, a time shift is ap-plied to the pressure wavefield in the frequency-space do-main, the wavefield is Fourier transformed into the frequency-wavenumber domain followed by a phase-shift operation fora given reference slowness, then an inverse Fourier transformis applied to transform the wavefield to the frequency-spacedomain. Several different reference slownesses are selected toperform the same calculations, and the corresponding resultsare interpolated to obtain the wavefield at the next depth level.The reference slownesses are generally within the bounds of theminimum and maximum slownesses of each layer. The methodis capable of handling large lateral slowness variations.
If the PSPI algorithm is implemented without interpolationby choosing some average slownesses over given regions forextrapolation, the result is identical to that obtained using theSSF-MRS method where the slowness in each region is iden-tical to that used in the PSPI method. Thus the PSPI and SSF-
MRS methods can be regarded as the same method in spite oftheir derivations appearing to be quite different.
ELBF PROPAGATOR WITH MULTIPLEREFERENCE SLOWNESSES
The local Born approximation is valid only for small per-turbations in velocity or slowness. The ELBF propagator be-comes unreliable or even unstable when lateral slowness per-turbations are large and/or the frequency is high. FollowingKessinger (1992), we apply the MRS concept to the ELBFmethod as a way to limit the slowness perturbations. Equa-tion (23) then becomes
ps(x , y, zi + Az; w) 3(M(x, y, zi) — so(zi))
JX FkxIk,{ J(zi)eikzAzFx,YI[iwAsi(x,y,zi)Az]
x p(x, y, zi; w)}}, (30)
where M(x, y, z ; ) is a mapping function between spatial po-sition and reference slowness, j is the index of referenceslownesses at each depth level, the function S(.) meansthat 6(M(x, y, z,) — so(z 1 ))=1 for M(x, y, z ; )=so(z;) andS(M(x, y, z i ) — so (z i )) = 0 for M(x, y, z ; ) : s' (z ; ), and v' (z i ) isgiven by equation (21) for the reference slowness so (z ; ). Equa-tion (30) can also be written as
p .,(x, y, z i + Az; w) b(M(x, y , zi) — so(zi))
iX ^kx
1 kY {a j (zi)e iki0z^x,Y{TP(x,y,zi;w)}j, (31)
where
T = icw^S(M(x, y, z,) — so(zi))Asi(x, y, zi)Az. (32)
Equation (32) can be calculated as soon as reference slownesseshave been selected; therefore, it is more efficient to calculatescattered fields using equation (31) than using equation (30).For multiple reference slownesses, equation (11) can be writtenas
Po(x , y, zi + Az; w) 6 (M(x, y, zi) — so (zi ))j
iX Fkxiky Ie ikz4z
Y{p(x , y, zi; w)}1. (33)
In equations (31) and (33), the inner Fourier transforms aremade only once at each depth level. To reduce aliasing duringmigration, a Butterworth filter is applied in the wavenumberdomain and a Hanning taper is used near the lateral boundariesof a model.
In migration, the exponential terms and (iw) in all aboveequations must be changed to their complex conjugates forbackpropagating wavefields from receiver positions.
NUMERICAL EXAMPLES
We first compare the accuracy of the extended localBorn Fourier method and the split-step Fourier method forsimulation of wave propagation. Then we present migrationexamples for a simple model with a dipping interface and a
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Extended Local Born Fourier Migration
1529
complex model using the SSF and ELBF methods with sin-gle reference slowness (SRS) and multiple reference slowness(MRS).
Comparison between accuracy of propagators
We compare the accuracy of the ELBF and SSF propagatorsusing a 2-D homogeneous medium where we choose the ref-erence slowness to be different from the actual slowness. Thisallows a simple check to be made in the ability of the methodto correct phase for velocity perturbations (Huang and Fehler,1998). A 1024 x 100 grid with grid spacings along horizontaland vertical directions of 10 m was used. The velocity of themedium is 4000 m/s. We made two sets of numerical simula-tions using reference velocities of 3600 and 4400 m/s, respec-tively. Hence, the whole medium has a relative slowness per-turbation of —10% for the former case and 10% for the lattercase. A point source with a Ricker's time history and a dom-inant frequency of 20 Hz was introduced at grid site (512, 1).Seismograms were recorded at all grid sites from (512, 512) to(882, 512). The corresponding propagation angles relative tothe z-axis, which is the main propagation direction, range from00 to 75°. The frequency range used in the calculations is0.5-60 Hz. The SSF, complexified local Born Fourier (withrl = — 0.01), and ELBF calculations were made for 512 timesteps with a time sample interval of 0.004 s. No complexifiedlocal Born-Fourier calculations were made for the case of pos-itive slowness perturbation since the method is valid only fornegative slowness perturbations (de Hoop et al., 1999). For theextended local Born Fourier calculations, tests using the first 1,2, 3, and 4 terms on the right side of equation (22) were made.Traveltimes picked from the calculated seismograms were com-pared with those of seismograms calculated using an analyticalsolution.
Figure 1 shows the relative errors of the traveltimes. Forthe medium having —10% relative slowness perturbation andallowing for a maximum of 5% relative error of traveltime,the SSF method can handle propagation angles of up to 52°relative to the main propagation direction. The ELBF method,using the first four terms on the right side of equation (22), canhandle propagation angles of up to about 75 0 . For a mediumhaving a 10% relative slowness perturbation and allowing fora maximum —5% relative traveltime error, the SSF methodcan simulate wave propagation at angles up to about 60°. Theangle for the ELBF method can be larger than 75°.
The ELBF method uses one more fast Fourier transformfor each depth step than the SSF method. However, the SSFmethod requires one more calculation of the exponent of com-plex numbers than the ELBF method. Our numerical simu-lations indicate that computation times for both methods arealmost the same.
Migration examples
We now present some examples showing the applicabilityof the extended local Born Fourier method to poststack mi-gration problems. We used finite-difference modeling to gen-erate an exploding-reflector dataset. For the ELBF method,we use the first four terms on the right side of equation (22).A model with a dipping interface as shown in Figure 2a wasused as the first example. The angle between the dipping in-terface and the horizontal direction (to the right) is 60°. Inthe exploding-reflector model, the plane wave radiated fromthe dipping interface propagates to the upper boundary of themodel along a propagation angle of 60° from the negative di-rection of the vertical axis (i.e., upward). A finite-differencescheme with second-order accuracy in time and fourth-orderaccuracy in space was used to solve the full acoustic wave equa-tion to generate the exploding-reflector data recorded on the
1:Split-step Fourier2: Extended local Born (1 terms)3: Extended local Born (2 terms)4: Extended local Born (3 terms)5: Extended local Born (4 terms)6: Complexified local Born
1:Split-step Fourier2: Extended local Born (1 terms)3: Extended local Born (2 terms)4: Extended local Born (3 terms)5: Extended local Born (4 terms)
Propagation Angle (degree)
Propagation Angle (degree)
a)10
FIG. 1. Relative errors of traveltime versus the propagation angle for the split-step Fourier method, the extended local Born Fouriermethod, and the complexified local Born Fourier method. The slowness perturbation in (a) is —10% and that in (b) is 10%.
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upper boundary of the model. Model densities were chosento eliminate impedance contrasts across model interfaces. ARicker's time history with a dominant frequency of 20 Hz wasused as the reflector sources. For migration, the model was de-fined on a 1024 x 500 grid with a grid spacing of 10 m. Thetime sample interval is 0.004 s, and the frequency range is 0.5-60 Hz. In the grid layers that contain the dipping interface, areference velocity of 3300.13 m/s was selected as the uniquereference velocity so the relative slowness perturbations onthe left and right sides of the dipping interface are exactly 10%and —10%, respectively. The exploding-reflector data were mi-grated using the SSF and ELBF methods with single referenceslowness.
Figures 2b, d show the migrated image with the model in-terfaces superimposed. The results indicate that the ELBF mi-gration gives a more accurate image of the dipping interfacethan the SSF migration. For comparison, the complexified localBorn Fourier method was also used to migrate the data. Forthis case, a reference velocity of 3000 m/s must be selected inthe grid layers containing the dipping interface. Consequently,the relative slowness perturbation on the right side of the dip-ping interface becomes —18.2%. The vertical grid spacing wasreduced from 10 to 2 m. The value of rl was chosen to be —0.01.Figure 2c shows the complexified local Born Fourier migrationimage along with the locations of the model interfaces. Theupper flat interface was imaged correctly because there is noheterogeneity in the region above that interface. Below that in-terface, the downward continued wavefield blows up becauseof the difficulty of calculating scattered wavefields in hetero-geneous media with large lateral slowness variations using thecomplexified local Born Fourier method.
A 2-D slice of the complex SEG/EAGE 3-D salt model de-scribed by Aminzadeh et al. (1996) was used in the next mi-
gration example (see Figure 3). The finite-difference schemementioned above was used to generate the exploding-reflectordata for the model. The reflector source function is a Ricker'stime history with a dominant frequency of 20 Hz. The abso-lute values of the reflectivity were used as the amplitudes ofthe exploding reflector sources. For migration, the model wasdefined on a 1024 x 320 grid with a grid spacing of 12.192 m.The time sample interval is 0.004 s and the frequency rangeis 0.5-60 Hz. For migrations with single reference slowness,the average slowness within each depth level was used as thereference slowness for that level, For migrations with multiplereference slownesses, the reference slownesses were chosenin different regions to ensure that the absolute values of theslowness perturbations were <10%. Figure 4 shows migrationimages obtained using the split-step Fourier methods with SRSand MRS. The SSF-MRS method gives an image (Figure 4b)much better than the SSF-SRS method (Figure 4a), particu-larly in regions A, B, and C where the structures are complex,the lateral slowness variations are strong, and the dip angles ofthe interfaces are large. The ELBF migration image with SRSis shown in Figure 5a. There are a lot of artifacts because ofthe difficulty in calculating scattered fields for large slownesscontrasts between the salt body and the surrounding media.Figure 5b is the migration image obtained using the ELBF-MRS method, which clearly images the lower part of the saltbody interface and regions A, B, and C. No artifacts associatedwith the use of multiple reference slownesses were observedfor our numerical examples.
For comparison, the corresponding images cut from 3-Dexploding-reflector migration images obtained using Kirchhoffmigration with first-arrival traveltimes calculated by a finite-difference scheme (Fei et al., 1996) and using FX-migration(Amoco, 1995) are given in Figures 6a, b, respectively.
FIG. 2. Migration images of an exploding reflector dataset for the model shown in (a) using (b) the split-step Fourier method, (c)the complexified local Born Fourier method, and (d) the extended local Born Fourier method using the first four terms on the righthand side of equation (22). Locations of correct model interfaces are shown in (b), (c), and (d).
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A detailed comparison of expanded images within the rect-angular areas around location A in Figures 4b and 5b is dis-played in Figure 7. It shows that the ELBF-MRS migrationgives a slightly better image around location F on the rightside interface of the V-shaped interface than does the SSF-MRS migration.
CONCLUSIONS
We have developed and tested the extended local BornFourier migration method. Like the split-step Fourier method,the extended local Born Fourier method is based on theassumption of small slowness perturbation. The extended
FIG. 3. A 2-D slice of the SEG/EAGE 3-D salt model. Dark black region is salt.
FIG. 4. Migration images obtained using the split-step Fourier method with single referenceslowness (a) and multiple reference slownesses (b).
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FIG. 5. Migration images obtained by the extended local Born Fourier method using the firstfour terms on the right side of equation (22) with single reference slowness (a) and multiplereference slownesses (b).
FIG. 6. Kirchhoff migration image (a) (Fei et al., 1996) and FX-migration image (b) (Amoco,1995) of 3-D exploding reflector data.
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local Born Fourier method is more accurate than the split-stepFourier method. To handle large lateral slowness perturbations,we have developed the extended local Born-Fourier migrationmethod with multiple reference slownesses. Different refer-ence slownesses are selected in different regions of a mediumso that slowness perturbations are small in all regions. Theextended local Born Fourier migration method with multiplereference slownesses can be used to image complex structureswith strong lateral slowness variations. The increased compu-tation time for the method with multiple reference slownessesrelative to the method with single reference slowness dependson the average number of reference slownesses.
ACKNOWLEDGMENTS
We greatly appreciate the valuable comments and sugges-tions from reviewer Biondo Biondi, two anonymous reviewers,and the associate editor Alexander Mihai Popovici. We thankAmoco Production Company for providing the FX-migrationimage for the SEG/EAGE 3-D salt model, and Tong Fei andthe Gulf of Mexico Subsalt Imaging Project for providing theirKirchhoff migration image. The work of Huang and Fehlerwas funded by the US Department of Energy (DOE) Officeof Basic Energy Sciences as a part of the Advanced Computa-tional Technology Initiative (ACTI) through contract W-7405-ENG-36 to the Los Alamos National Laboratory. The work ofWu was supported by the ACTI project of the University ofCalifornia at Santa Cruz, granted from the DOE and admin-istered by the Los Alamos National Laboratory. Calculationswere done at the Los Alamos National Laboratory AdvancedComputing Laboratory.
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