GEOPHYSICS, VOL. 74, NO. 2 共MARCH-APRIL 2009兲; P. S33–S45, 15 FIGS., 2 TABLES. 10.1190/1.3077621
Designing stable extrapolators for explicit depth extrapolation of 2D and 3D wavefields using projections onto convex sets
Wail A. Mousa1, Mirko van der Baan2, Said Boussakta3, and Desmond C. McLernon4
seismic images where each designed operator output sample can be computed independently — even in parallel with other output samples. Parallel implementation is therefore straightforward 共Bhardwaj et al., 1999, 2000; Yilmaz, 2001; Phadke et al., 2002兲. Stable extrapolation of one-way wavefields through strongly laterally inhomogeneous media is possible. The main practical consideration is that we need short operator lengths to accurately handle strong lateral variations in velocities but long operators to correct for steep dips. Short operators are also desirable because computation times are proportional to the chosen operator lengths. In addition, the accuracy of wavefield extrapolation over large distances may deteriorate quickly because of operator instability 共Hale, 1991b; Thorbecke, 1997兲. Previous authors have described a variety of approaches for designing -x extrapolators 共Holberg, 1988; Hale, 1991b; Karam and McClellan, 1995, 1999; Soubaras, 1996; Thorbecke, 1997兲. However, there is still a need for improving the design of such filters 共FIR digital filters, as they are called within the signal processing community兲 used for this important geophysical data processing step. As a consequence, better subsurface images can be obtained, reducing exploration risks. Moreover, 2D extrapolation FIR filters 共extrapolators兲, used for 3D frequency-space 共 -x-y兲 migration, are circularly symmetric operators whose magnitude and phase wavenumber spectra are also circularly symmetric 共Hale, 1991a; Soubaras, 1996; Karam and McClellan, 1997; Yilmaz, 2001; Mousa et al., 2006兲. There exist few accurate 2D explicit-depth wavefield extrapolation design algorithms such as Laplacian synthesis 共Soubaras, 1996兲 and weighted least squares 共Thorbecke, 1997兲. Two important factors prevented the use of such accurate 2D operators. The first factor is computing/storage facilities, which have become less important as a result of recent advances in computer hardware. Second, most 1D wavefield extrapolation operator 共and operators or digital filters in general兲 design algorithms cannot be extended to 2D, e.g., the Remez algorithm 共Karam and McClellan, 1995; Çetin et al., 1997兲.
ABSTRACT We have developed a robust algorithm for designing explicit depth extrapolation operators using the projections-onto-convex-sets 共POCS兲 method. The operators are optimal in the sense that they satisfy all required extrapolation design characteristics. In addition, we propose a simple modification of the POCS algorithm 共modified POCS, or MPOCS兲 that further enhances the stability of extrapolated wavefields and reduces the number of iterations required to design such operators to approximately 2% of that required for the basic POCS design algorithm. Various synthetic tests show that 25coefficient 1D extrapolation operators, which have 13 unique coefficients, can accommodate dip angles up to 70°. We migrated the SEG/EAGE salt model data with the operators and compare our results with images obtained via extrapolators based on modified Taylor series and with wavefield extrapolation techniques such as phase shift plus interpolation 共PSPI兲 and split-step Fourier. The MPOCS algorithm provides practically stable depth extrapolators. The resulting migrated section is comparable in quality to an expensive PSPI result and visibly outperforms the other two techniques. Strong dips and subsalt structures are imaged clearly. Finally, we extended the 1D extrapolator design algorithm, using MPOCS for 2D extrapolation, to the 2D case to perform 3D extrapolation; the result is a perfect circularly symmetric migration impulse response.
INTRODUCTION The frequency-space 共 -x兲 explicit depth extrapolation method is one of the most attractive techniques for performing seismic migration 共Hale, 1991b兲. It uses spatial convolution to produce migrated
Manuscript received by the Editor 16 December 2007; revised manuscript received 16 August 2008; published online 4 March 2009. 1 Formerly The University of Leeds, School of Electronic & Electrical Engineering, Leeds, U.K; presently Schlumberger Dhahran Carbonate Research, Dhahran, Saudi Arabia. E-mail:
[email protected]. 2 The University of Leeds, School of Earth and Environment, Leeds, U.K. E-mail:
[email protected]. 3 University of Newcastle upon Tyne, School of Electrical, Electronic and Computer Engineering, Newcastle, U.K. E-mail:
[email protected]. 4 The University of Leeds, School of Electronic & Electrical Engineering, Leeds, U.K. E-mail:
[email protected]. © 2009 Society of Exploration Geophysicists. All rights reserved.
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For both reasons, researchers use approximations of 2D operators based on predesigned 1D operators. If we assume that the true 2D wavefield extrapolator is separable, then such operators can be designed, as is the case in Dudgeon and Mersereau 共1984兲. Another way is by using the McClellan transformation, which takes into account the circular symmetry of the operator’s wavenumber response 共Mecklenbräuker and Mersereau, 1976a; McClellan and Chan, 1977; Kayran and King, 1983; Dudgeon and Mersereau, 1984; Hale, 1991a; Lu and Antoniou, 1993; Karam and McClellan, 1997兲. Although such operators 共designed by approximations兲 are cheap to design and implement, these approximations come with a price: errors in extrapolated volumes 共Hale, 1991a兲. One mathematical technique whereby we can express our desired design characteristics easily is projections onto convex sets 共POCS兲. The POCS theory was developed by Bregman 共1965兲 and Gubin et al. 共1967兲. Because of its attractive properties, POCS has been used for designing other FIR filters 共Çetin et al., 1997; Haddad et al., 2000; Özbek et al., 2004兲. However, those filters were restricted to real-valued FIR filter coefficients. Generally speaking, the POCS design algorithm is very flexible because crucial filter design requirements can be incorporated very conveniently. It is an iterative technique that requires only two FFT computations per iteration. The resulting operators satisfy all predefined constraint sets as long as the imposed constraints are not contradictory and mutually exclusive. It is straightforward to extend the POCS design algorithm to obtain multidimensional operators, in contrast to many other conventional design techniques. In this paper, we first show how the POCS method can be used to design 1D and 2D stable wavefield extrapolation operators. We then
a)
c)
apply our 1D-designed operators to the SEG/EAGE salt model and compare their performance with other wavefield extrapolation methods. Finally, we present designs for the 2D wavefield extrapolators using our proposed POCS method and demonstrate their performance with impulse responses.
DESIGN OF 1D WAVEFIELD EXTRAPOLATORS USING PROJECTIONS ONTO CONVEX SETS The basic idea of POCS is as follows: We want to design a filter h 苸 H, where H is a given Hilbert space subject to certain constraints Ci 苸 H. We assume explicitly that the constraints Ci form closed convex sets in the Hilbert space. For m known properties, there are m closed convex sets C1,C2, . . . ,Cm. The design requirement for an unknown filter h is that h 苸 C0, where C0 denotes the intersection of all sets Ci, i.e., C0 ⳱ C1 艚 C2 艚 . . . 艚 Cm. Given the sets Ci, we derive the associated projection operators PCi for each i ⳱ 1,2, . . . ,m based on the nearest neighbor rule 共i.e., an orthogonal projection is used兲. Now the objective is to find a solution h within C0 that satisfies all constraint sets and therefore represents an acceptable solution. The solution is generally found iteratively using the recursion formula
hkⳭ1 ⳱ PCm PCmⳮ1 . . . PC1hk ,
共1兲
where we start with an arbitrary filter 共operator兲 h0. Iterations are stopped once the convergence rate drops below a predefined threshold ⑀ . The final solution h will converge to a point in C0, thereby satisfying all imposed filter design constraints as long as C0 is nonempty 共i.e., the constraints Ci are not mutually exclusive兲 共Haddad et al., 2000兲.
Problem formulation: Imposed constraints
b)
d)
The desired properties for obtaining stable explicit extrapolation operators, as described in Figure 1, are a stable and acceptably short length and an even, complex-valued operator that handles steep dips and strong lateral velocity variations. The proposed constraint sets that describe these characteristics are defined as follows. 1兲
2兲
Figure 1. The 1D explicit depth wavefield extrapolation operator requirements. 共a兲Anoncausal spatial operator of length N where the real 共R关n兴兲 and imaginary 共Ih关n兴兲 parts are of even symmetry. 共b兲 A short-length -x extrapolator for accuracy. Because of heterogeneous media within a layer, the velocity will also vary horizontally. At every lateral position, a new operator is used to perform migration on the data from one depth level to another. Within the operator length, the medium is assumed to be homogeneous. 共c兲 An accurate magnitude wavenumber response with a passband cutoff equal to kcp and an evanescent region cutoff kcs. 共d兲 An accurate passband phase response.
3兲
C1 — the set of all complex-valued vectors of length M with at most N odd nonzero members 共operator coefficients兲 that are noncausal and have even symmetry 共see Figure 1a兲 C2 — the set of all vectors that are complex valued and whose discrete-space Fourier transform 共DSFT兲 arguments are constrained to be equal to the exact extrapolation phase response 共see Figure 1d兲 C3 — the set of all complex-valued vectors whose DSFT magnitude spectrum is lower bounded by 1 ⳮ ␦ p in the passband 共see Figure 1c兲; ␦ p is the maximum allowable operator passband tolerance
POCS-based wavefield extrapolation 4兲
5兲
C4 — the set of all complex-valued vectors whose DSFT magnitude should not exceed the limit 1 Ⳮ ␦ p in the passband 共see Figure 1c兲 C5 — the set of all complex-valued vectors whose DSFT magnitude is limited to a maximum of ␦ s within the evanescent region
All of the constraint sets are closed and convex 共see Figure 1c兲. Note that ␦ s is the maximum allowable operator evanescent region tolerance. Now, based on a Lagrangian formulation for each constraint set, one can derive the associated projection operators of an arbitrary vector h 苸 CM , where h 苸 Ci for i ⳱ 1, . . . ,5 by the use of the nearest neighbor rule. Let kxs and kxp, respectively, be the evanescent and passband regions, and denote the exact phase response as 共kx兲 ⳱ 共⌬z/⌬x兲冑kc2pⳮkx2, where kcp is the passband cutoff, ⌬z is the depth sampling interval, and ⌬x is the spatial sampling interval. The associated operators with the above constraint sets can be derived to give
P C1h ⳱
冦
h关n兴, for 兩n兩 ⱕ 0,
Nⳮ1 2
otherwise,
冧
共2兲
P C2h ↔
再
兩H共kx兲兩cos共 hⳮ 共kx兲兲exp共i 共kx兲兲, if AC2 ⳮ 兩H共kx兲兩cos共 h ⳮ 共kx兲兲exp共i 共kx兲兲, if BC2 ,
冎
共3兲
where h ⳱ arg兵H共kx兲其, condition AC2 is cos共 h ⳮ 共kx兲兲 ⱖ 0, and condition BC2 is cos共 h ⳮ 共kx兲兲 ⬍ 0. The ↔ symbol denotes a Fourier transform pair, and i ⳱ 冑ⳮ1. While maintaining the phase response, PC3 limits the lower bound of any extrapolation operator within the passband 共Figure 1c兲 to 1 ⳮ ␦ p, as we see from
冦
H共kx兲,
if AC3
PC3h ↔ 共1ⳮ␦ p兲exp共i⬔H共kx兲兲, if BC3 H共kx兲,
otherwise,
where condition AC4 is 兩H共kx兲兩 ⬍ 共1 Ⳮ ␦ p兲 for kx 苸 kxp and condition BC4 is 兩H共kx兲兩 ⱖ 共1 Ⳮ ␦ p兲 for kx 苸 kxp 共see Figure 1c兲. Finally, we have
冦
冧
H共kx兲,
if AC5
PC5h ↔ ⳮ␦ s exp共i⬔H共kx兲兲, if BC5 H共kx兲,
otherwise,
冧
共6兲
where condition AC5 is 兩H共kx兲兩 ⬍ ␦ s for kx 苸 kxs and condition BC5 is 兩H共kx兲兩 ⱖ ␦ s for kx 苸 kxs. This projection ensures the evanescent region will be attenuated with a maximum magnitude of ␦ s as described in Figure 1c.
POCS design algorithm The algorithm starts with an arbitrary complex-valued vector h0 of dimension M and uses the projection sequence in equation 1. For the kth iteration, we have the following process: 1兲 2兲 3兲 4兲
where h关n兴 is the extrapolator coefficient at the spatial index n. Note that PC1 truncates the impulse response to be of a length of N, i.e., it basically limits the operator to the desired short length N 共see Figure 1a and b兲. The second projection forces the phase to the required phase response 共Figure 1d兲 of the extrapolators while maintaining the magnitude of the wavenumber response:
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Take the FFT of hk. Impose the conditions C2–C5 共equations 3–6兲 in the Fourier domain. Take the inverse FFT of the output. Impose the space-domain constraint C1 共equation 2兲, which yields the output vector hkⳭ1.
If the mean square error between hk and hkⳭ1 is less than or equal to a predefined threshold ⑀ , stop the algorithm; otherwise, repeat steps 1–4. Figure 2 summarizes the design algorithm.
MPOCS algorithm for seismic wavefield extrapolation One disadvantage of the POCS design algorithm is that it requires many iterations to achieve convergence. A simple modification, however, significantly boosts the convergence rate. Recall that the projection PC1 given by equation 2 is a truncation applied to the output of the previous projections 共see Figure 2兲. This amounts to including a rectangular weighting window in the forward DSFT, inducing considerable artifacts. Therefore, we modify the
共4兲
where condition AC3 is 兩H共kx兲兩 ⬎ 共1ⳮ␦ p兲 for kx 苸 kxp, condition BC3 is 兩H共kx兲兩 ⱕ 共1ⳮ␦ p兲 for kx 苸 kxp, and ⬔ denotes the angle of H共kx兲. On the other hand, PC4 limits the upper bound of any extrapolation operator with a magnitude response greater than 1 Ⳮ ␦ p to be equal to 1 Ⳮ ␦ p within the wavenumber passband as
冦
H共kx兲,
if AC4
PC4h ↔ ⳮ 共1 Ⳮ ␦ p兲exp共i⬔H共kx兲兲, if BC4 H共kx兲,
otherwise,
冧
共5兲
Figure 2. The POCS design algorithm for designing wavefield extrapolators, where constraints C2–C5 are related to the wavenumber operator response and C1 is maintaining a short-length spatial operator.
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projection onto C1 by using the Kaiser window 共Oppenheim and Schafer, 1989兲 instead of the rectangular window to reduce the artifacts and flatten the magnitude response of our operators before applying the DSFT. This better satisfies constraint sets C3 and C4. In this case, our projected vector h at iteration k Ⳮ 1 approaches the solution set C0. As it turns out, the simple modification significantly boosts the convergence rate and leads to more robust wavefield operators. For example, designing a 25-coefficient operator using POCS with a wavenumber cutoff kcp ⳱ 0.25 and ⑀ ⳱ 10ⳮ15 required 33,819 iterations, but using the modified POCS 共MPOCS兲 for the same operator parameters took only 679 iterations. Thus, the new projection operator PˆC1 onto C1 will be
PˆC1h
冦
冋 冑 冉 冊册 2n Nⳮ1 I o关  兴
Io 
⳱ h关n兴
1ⳮ
2
Nⳮ1 , for 兩n兩 ⱕ 2
0, otherwise,
冧
g2,k ⳱ PC4g1,k
冦
G1,k共kx,ky兲, if AC4
↔ ⳮ 共1 Ⳮ ␦ p兲exp共i ⬔ G1,k共kx,ky兲兲, if BC4 G1,k共kx,ky兲, otherwise,
冧
共10兲
where AC4 is 兩G1,k共kx,ky兲兩 ⬍ 共1 Ⳮ ␦ p兲 and BC5 is 兩G1,k共kx,ky兲兩 ⱖ 共1 Ⳮ ␦ p兲, kx,ky 苸 k p. Third, we project g2,k onto C3 using
g3,k ⳱ PC3g2,k
冦
G2,k共kx,ky兲,
if AC3
↔ 共1 ⳮ ␦ p兲exp共i ⬔ G2,k共kx,ky兲兲, if BC3 G2,k共kx,ky兲,
otherwise,
冧
共7兲
共11兲
where Io关·兴 is the modified zeroth-order Bessel function of the first kind and  is the shape parameter that determines the trade-off between the main-lobe and peak side-lobe levels 共Oppenheim and Schafer, 1989; Jackson, 1996兲. The value of  can be found using
where AC3 is 兩G2,k共kx,ky兲兩 ⬎ 共1 ⳮ ␦ p兲 and BC3 is 兩G2,k共kx,ky兲兩 ⱕ 共1 ⳮ ␦ p兲, kx,ky 苸 k p. The fourth step is to project g3,k onto C2 using
⳱
冦
0.1102共Aⳮ8.7兲, for A ⬎ 50 0.5842共Aⳮ21兲0.4 Ⳮ 0.07886共Aⳮ21兲, for 21 ⱕ A ⱕ 50 0, for A ⬍ 21,
冧
g4,k ⳱ PC2g3,k 共8兲 ↔
where A ⳱ⳮ20 log10␦ s.
DESIGN OF 2D WAVEFIELD EXTRAPOLATORS USING PROJECTIONS ONTO CONVEX SETS The 1D algorithm for designing 1D wavefield extrapolators is now extended to the 2D case. Without loss of generality, assume we are interested in designing 2D wavefield extrapolation operators for rectangular grids. Moreover, let ks and k p, respectively, be the stopband 共evanescent兲 and passband regions. We denote the exact phase response as 共kx,ky兲 ⳱ b冑kc2p ⳮ共kx2 Ⳮ k2y 兲, where kcp is the passband cutoff and kcs as the stop-band cutoff. Finally, we let ␦ p and ␦ s be, respectively, the maximum allowable passband and stop-band regions’tolerances. We start our 2D wavefield extrapolator design algorithm with an arbitrary complex-valued vector h0 of dimension M ⫻ M 共M is the number of FFT samples兲, which is much greater than the spatial operator length N ⫻ N. For the kth iteration, we project hk onto C5:
冦
Hk共kx,ky兲,
if AC5
g1,k ⳱ PC5hk ↔ ⳮ␦ s exp共i ⬔ Hk共kx,ky兲兲, if BC5 Hk共kx,ky兲,
otherwise,
where AC5 is 兩Hk共kx,ky兲兩 ⬍ ␦ s and BC5 is 兩Hk共kx,ky兲兩 ⱖ ␦ s, kx,ky 苸 ks. Next, we project g1,k onto C4 using
冧
共9兲
冦
兩G3,k共kx,ky兲兩 cos共 G3,k ⳮ 共kx,ky兲兲exp共i 共kx,ky,兲兲, if AC2 ⳮ兩G3,k共kx,ky兲兩 cos共 G3,k ⳮ 共kx,ky兲兲exp共i 共kx,ky兲兲,
if BC2 ,
冧
共12兲
where G3,k ⳱ ⬔ G3,k共kx,ky兲, AC2 is cos共 G3,k ⳮ 共kx,ky兲兲 ⱖ 0, and BC2 is cos共 G3,k ⳮ 共kx,ky兲兲 ⬍ 0. Finally, we project g4,k onto C1 by
hkⳭ1 ⳱ PC1g4,k ⳱
冦
g4,k关n1,n2兴, for 兩n1,n2兩 ⱕ 0,
otherwise.
Nⳮ1 2
冧
共13兲
If the mean square error between hk and hkⳭ1 is less than or equal to a predefined threshold ⑀ , we stop the algorithm. Otherwise, we repeat the above five steps.
MODIFIED PROJECTION DESIGN ALGORITHM FOR 2D MIGRATION COMPLEX-VALUED FIR FILTERS For the modified design algorithm, the 2D POCS design algorithm is used except that the projection of g4,k onto C1 is replaced by
POCS-based wavefield extrapolation hkⳭ1 ⳱ PC1g4,k
⳱
冦
冋 冑 冉 冊册 冋 冑 冉 冊册
Io  g4,k关n1,n2兴
1ⳮ
2n1
Nⳮ1
2
Io 
1ⳮ
2
2n2
Nⳮ1
I o关  兴 2
for 兩n1,n2兩 ⱕ
,
Nⳮ1 2
0, otherwise,
共14兲
where Io关·兴 is the modified zero-order Bessel function of the first kind and  is the shape parameter that determines the trade-off between the main-lobe and peak side-lobe levels 共see Lu and Antoniou 关1992兴 for details about the 2D Kaiser window兲.
1D WAVEFIELD EXTRAPOLATORS AND POSTSTACK MIGRATION OF THE 2D SEG/EAGE SALT MODEL Here we present simulation experiments evaluating our extrapolation operator design compared with existing techniques, leading to the application of our extrapolators to the problem of zero-offset migration of the 2D SEG/EAGE salt dataset. All of our experiments were performed using MATLAB software on a Pentium IV processor with 1 GB of RAM on a Linux-based operating system.
Accuracy of 1D extrapolation operators First, we compare the accuracy of the 1D extrapolators designed using the POCS and MPOCS algorithms 共with ⑀ ⳱ 10ⳮ15兲 with the modified Taylor series method 共eight derivative terms兲 reported in Hale 共1991b兲.All operators were designed with an operator length of N ⳱ 25 and a passband wavenumber cutoff of kcp ⳱ 0.25. The magnitude response for the modified Taylor series method and the MPOCS method have a flatter amplitude within the passband. That is, the designed filters have less ripple than for the POCS result. This can be clearly seen in Figure 3a, where the passband magnitude response error for the MPOCS and the modified Taylor series methods
a)
0.05
The 2D impulse response For the 2D extrapolation impulse response, the synthetic seismic section used here is a zero section, with the zero-offset trace containing only three 25-Hz zero-phase Ricker wavelets centered at 0.3, 0.6, and 0.9 s. The depth interval ⌬z ⳱ 4 m and the lateral sampling interval ⌬x ⳱ 20 m is for a range of 1600 m. The time-sampling interval is ⌬t ⳱ 2ms and velocity c ⳱ 750 m/s; we have a maximum frequency of 45 Hz. Based on the above discussion, we design a set of 25-coefficient 1D operators using MPOCS and the modified Taylor series methods. Figure 4a shows the 2D migrated synthetic section using the MPOCS algorithm, with ␦ p ⳱ ␦ s ⳱ 10ⳮ3 and ⑀ ⳱ 10ⳮ15. For the same zero-offset experiment, Figure 4b shows the 2D migrated synthetic section using the modified Taylor series method. Clearly, the modified Taylor series method attenuates propagating waves having angles more than approximately 45°, unlike the MPOCS approach, which accommodates propagation angles up to approximately 70° for the same filter length. Also, unlike the impulse response of the modified Taylor series, the MPOCS impulse response has fewer numerical artifacts. Also, compared with the modified Taylor seismic impulse response, the impulse response of the MPOCS operators results in less dispersion noise. 0.15 POCS MPOCS Modified Taylor
0.1
0.03
Passband phase error
Passband magnitude error
are very close to zero. In other words, the MPOCS and modified Taylor series operators result in a more stable extrapolation than does the POCS method. In addition, the POCS and the MPOCS operators accommodate higher propagation angles than the modified Taylor series operators. This is seen in Figure 3a, where the POCS and the MPOCS magnitude response errors start to attenuate at wavenumber values higher than those of the modified Taylor operators. Figure 3b shows the passband phase response error for all of the designed filters. The passband phase response error indicates that the POCS, MPOCS, and modified Taylor series phase response errors are close to zero. Table 1 compares the CPU time for designing all the techniques. With MPOCS, we can save design time.
b)
POCS MPOCS Modified Taylor
0.04
冧
0.02 0.01 0 −0.01 −0.02 −0.03
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0.05
0
−0.05
−0.1
−0.04 −0.05
0
0.05
0.1
0.15
kx
0.2
0.25
0
0.05
0.1
0.15
0.2
0.25
kx
Figure 3. Accuracy of the designed 1D extrapolators using the POCS 共solid line兲, MPOCS 共dashed-dotted line兲, and modified Taylor series 共dotted line兲 algorithms, with N ⳱ 25 and kcp ⳱ 0.25, 共a兲 Magnitude response error and 共b兲 phase response error within the passband.
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The 2D dip accuracy and stability The same set of operators designed using the MPOCS algorithm and the modified Taylor series method for the impulse response experiments is used to migrate a synthetic time-space seismic section 共Figure 5a兲. This section contains dips with angles 0°, 30°, 50°, 60°, 70°, and 80°. It is constructed based on a Ricker wavelet input with a dominant frequency of 15 Hz and time duration of 0.2 s. Figure 5b shows the migrated section using the extrapolators designed with the MPOCS algorithm. Clearly, the operators accommodate dips up to 70° and a little of the 80° dip. Also, the MPOCS operators have shortened and moved the synthetic dip reflectors correctly updip. However, as seen in Figure 5c, the modified Taylor method attenuates dips above 50° and introduces background artifacts. The modified Taylor phase response error is much higher than the phase response error of the MPOCS algorithm 共Figure 3b兲, but its passband magnitude response propagates wavefields with much lower angles than the MPOCS operator 共Figures 3c and 4兲. Therefore, the MPOCS operators can handle large dips and are more stable over larger propagation distances than can modified Taylor filters. Note that we use nonlinear grayscale mapping to highlight the background artifacts for both migrated images.
Figure 8 shows migration results obtained by explicit depth extrapolation using operators designed with the MPOCS algorithm and the modified Taylor series method. For both results, we apply a 0.006 共1/m兲 cutoff high-pass-wavenumber depth filter per seismic trace to enhance the overall display of the results. The MPOCS result 共Figure 8a兲 shows a better overall image of the SEG/EAGE salt model than the result obtained using the modified Taylor filters 共Figure 8b兲. This is seen clearly in areas that are difficult to migrate, i.e., beneath the salt dome and at reflectors with steep dips. The result agrees with the synthetic results shown in Figures 4 and 5. For comparison, Figure 9 shows the migrated sections produced by other wavefield extrapolation techniques: the split-step Fourier method with 2048 FFT points 共Stoffa et al., 1990; Ferguson and Margrave, 2005兲 and the phase shift plus interpolation 共PSPI兲 technique with four reference velocities and 2048 FFT points 共Gazdag and Sguazzero, 1984兲. Although the PSPI method is expensive, it
a)
0 100 200 300 400 500 600 700 −800
−600
−400
−200
0
200
400
600
800
400
600
800
Lateral position (m)
b)
0 100 200
Depth (m)
Our main objective is to evaluate our MPOCS operators. Figure 6 represents the SEG/EAGE salt velocity model 共Aminzadeh et al., 1994, 1995; Aminzadeh et al., 1996; O’Brien and Gray, 1996兲. The salt body is embedded in sediments with smoothly varying velocities. The velocity model is composed of 1024 traces where each trace contains 1048 depth samples. A zero-offset section of the SEG/ EAGE salt velocity model 共Figure 7兲 is generated based on finite differences 共Ferguson and Margrave, 2005兲. It is composed of 1024 traces, and the time record length is 6 s 共3001 time samples per trace兲. Seven thousand frequency-velocity dependent operators were designed with the operator parameters given in the previous subsections and with a total operator length of 25 coefficients, but only 13 coefficients needed to be stored on account of symmetry conditions. We only display locations of 3000–17,000 m at a time interval of 0–5 s 共depth interval of 0–4000 m兲. A code was written to implement poststack explicit depth extrapolation as reported by Holberg 共1988兲, Hale 共1991b兲, and Yilmaz 共2001兲, where the operators are implemented as a spatially varying convolution to accommodate lateral velocity variations 共Holberg, 1988兲 共see Figure 1b for the concept of spatially varying extrapolation process兲.
Depth (m)
Application to SEG/EAGE salt model
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Figure 4. The 2D impulse response sections, with ⌬z ⳱ 4 m, ⌬x ⳱ 20 m, ⌬t ⳱ 2 ms, and c ⳱ 750 m/s. The maximum frequency is 45 Hz using 共a兲 the MPOCS algorithm, where the operator parameters are ␦ p ⳱ ␦ s ⳱ 10ⳮ3 and ⑀ ⳱ 10ⳮ15 for N ⳱ 25 with an approximate resulting dip angle of 70°, and 共b兲 the modified Taylor series method, where N ⳱ 25 with an approximate resulting 45° dip angle.
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produces the best-quality image. This is clearly seen at areas beneath the salt dome and at steep dip reflectors. It thus serves as an overall quality benchmark. Table 2 shows the cost of all the compared migration schemes for a single angular frequency at a given depth slice. The most expensive is PSPI. All migrated results at the sediments above the salt dome as well as within the salt dome are imaged clearly. To highlight differences, Figures 10–12 enlarge structurally difficult areas 共respectively, different dips, steep dips, and subsalt structures兲. As anticipated, the PSPI technique produces the best migration result, although the explicit wavefield extrapolation using the MPOCS design algorithm yields a migrated section of comparable quality. Strong dips and subsalt structures are clear for images obtained using the MPOCS and PSPI methods. In other words, MPOCS and PSPI visibly outperform the other two techniques. Therefore, besides obtaining a much better-quality image than the
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Figure 5. 共a兲 A 2D synthetic time-space section containing dipping events with angles 0°, 30°, 50°, 60°, 70° and 80°. This section is constructed based on a Ricker wavelet input with a dominant frequency of 15 Hz and with a time duration of 0.2 s. Migration of the 2D synthetic time-space section contains dipping events with angles 0°, 30°, 50°, 60°, 70° and 80° using 共b兲 the MPOCS algorithm with N ⳱ 25 and 共c兲 the modified Taylor series method with N ⳱ 25.
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Figure 7. The time-space zero-offset section of the SEG/EAGE salt model, which is generated using finite differences. This data set serves to assess the structural migration capabilities of different poststack migration algorithms.
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split-step Fourier method and the modified Taylor filters, by using 25-coefficient explicit depth operators designed using MPOCS, a comparable overall image 共that contains insignificant salt-and-pepper noise兲 to the expensive PSPI method was obtained.
subsections are displayed with respect to their normalized wavenumbers. Finally, the simulations were performed in MATLAB installed on a Pentium IV processor with 1 GB of RAM on a Linux operating system.
EVALUATING 2D WAVEFIELD EXTRAPOLATORS DESIGNED USING 2D POCS ALGORITHM
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This section is divided into two main parts. The first part deals with designing 2D extrapolators using the 2D POCS and 2D MPOCS algorithms. These operators are compared with operators designed using the McClellan and improved McClellan 共McClellanHale兲 transformations 共Mecklenbräuker and Mersereau, 1976a, 1976b; McClellan and Chan, 1977; Hale, 1991a兲. The second part shows an application of the 2D MPOCS operator design algorithm to 3D synthetic seismic sections. The results obtained in all of our simulations can apply to all cases and are not restricted to the filters’chosen parameters. In addition, all of the filters shown in the following
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Figure 9. Migrated SEG/EAGE salt model using 共a兲 the split-step method and 共b兲 the PSPI method.
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Table 2. Comparison of computational cost (number of complex multiplications and additions) for each angular frequency at a depth slice for migrating the 2D zero-offset SEG/EAGE salt model using 25-coefficient 1D wavefield extrapolators, the split-step Fourier method, and the PSPI method (with four reference velocities and 2048 FFT points). The savings are calculated with respect to the PSPI method.
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Figure 8. Migrated SEG/EAGE salt model using 共a兲 the MPOCS method and 共b兲 the modified Taylor series method.
-x 共25 coefficients兲 Split-step Fourier 共2048 FFT points兲 PSPI 共4 reference velocities, 2048 FFT points兲
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2D extrapolators via the transformation are designed using our 1D MPOCS algorithm with N ⳱ 25, kcp ⳱ 0.25, and ␦ p ⳱ ␦ s ⳱ 10ⳮ3. Clearly, the magnitude response of the improved McClellan transformation in Figure 14c shows an improvement in the design when compared to the original McClellan transformation magnitude re-
The 2D wavefield extrapolator design using POCS and MPOCS
Simulation 2: Comparisons with McClellan transformations In Figure 14, the same 2D extrapolators are obtained with the McClellan transformation 共Figure 14a and b兲 and its improved version 共Figure 14c and d兲. The 1D extrapolators used to design these
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Figure 10. Detail of an area with different dips 共lateral position, 7500–9750 m; depth, 1200–1800 m兲 using 共a兲 MPOCS, 共b兲 modified Taylor series, 共c兲 split-step Fourier, and 共d兲 PSPI methods.
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A 25⫻ 25 2D extrapolator is designed using the 2D POCS algorithm. The filter parameters are kcp ⳱ 0.25, kcs ⳱ 0.3841, and ␦ p ⳱ ␦ s ⳱ 10ⳮ3. Figure 13a shows the magnitude spectrum of such a filter that meets the filter magnitude spectrum constraints, i.e, C3, C4, and C5. Moreover, Figure 13b shows the phase spectrum in the passband for this designed filter where the phase in the passband meets the constraint C2 requirements, i.e., having circular symmetry. The design took 1286 iterations to converge with the parameter ⑀ ⳱ 10ⳮ14. Similarly, a 25⫻ 25 2D extrapolator is designed using the 2D MPOCS algorithm with the same parameters except for the stop-band cutoff wavenumber, which is in this case kcs ⳱ 0.401. The 2D MPOCS algorithm takes only 55 iterations to converge with the parameter ⑀ ⳱ 10ⳮ14. The obtained filter possesses a flatter magnitude response, as seen comparing Figure 13c and a. The ripple effect has been removed even in the phase response, as seen comparing Figure 13d and b, reducing phase error. Also, note that the 2D magnitude response of the MPOCS algorithm possesses a more circularly symmetric shape than the 2D extrapolator designed using the POCS algorithm. Finally, the MPOCS algorithm saves 共in this case兲 about 95% of the FFT computations when compared with the POCS algorithm. Thus, as in the case for the 1D MPOCS algorithm, the 2D MPOCS algorithm is a good choice for extrapolating 3D seismic volumes. This results from the reduced computations of the design, the circularly symmetric wavenumber response, and its magnitude response, which is expected to result in stable extrapolated volumes.
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In this section, we want to compare the accuracy of the 2D extrapolators designed with the two extended projection algorithms given earlier. Also, we want to compare these with the one designed by using the McClellan and ima) 1200 proved McClellan transformations.
Simulation 1: 2D wavefield extrapolator design using POCS and MPOCS comparisons
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Figure 11. Detail of an area with steep dips on the left flank of the salt model 共lateral position, 6000–8000 m; depth, 1800–2800 m兲 using 共a兲 MPOCS, 共b兲 modified Taylor series, 共c兲 split-step Fourier, and 共d兲 PSPI methods. The MPOCS technique provides stable results even in the presence of steep dips using only 25 operator coefficients.
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sponse 共Figure 14a兲.Also, because the predesigned 1D extrapolators satisfy the wavefield extrapolator characteristics, it is expected that these designs result in stable extrapolatoed volumes. However, both designs are approximations to the true 2D design and will not result in perfect circular symmetry of extrapolated sections when compared with those designed with the MPOCS algorithm 共see next section兲. Nevertheless, such transformations are much cheaper to design even when compared with the 2D MPOCS algorithm. Low-cost design 共implementation兲 techniques were proposed for extrapolating 3D volumes to speed up the migration process 共Yilmaz, 2001兲. That was back when computing facilities’ capabilities were more limited than today’s.
A set of 25⫻ 25 2D extrapolators was designed with the 2D MPOCS method, with M ⳱ 256 and for a depth step ⌬z ⳱ 2 m, inline and crossline ⌬x ⳱ ⌬y ⳱ 10 m, time sampling interval ⌬t ⳱ 0.004 s, ⳱ 50 rad/s, and c ⳱ 1000 m/s. We migrated a 3D zero-offset section with 1100 m for the inline and crossline apertures, up to a maximum frequency of 45 Hz. This time-space section contained one zero-phase Ricker wavelet centered at 0.512 s at x ⳱ y ⳱ 0. A 2D slice at depth z ⳱ 220 m of the extrapolated 3D volume is shown in Figure 15a using the proposed 2D MPOCS extrapolator design algorithm. In Figure 15b and c, respectively, we see 2D depth
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Figure 12. Detail of a structurally challenging subsalt area 共lateral position, 6500–8500 m; depth, 3400–4000 m兲 using 共a兲 MPOCS, 共b兲 modified Taylor series, 共c兲 split-step Fourier, and 共d兲 PSPI methods. The MPOCS technique again provides stable results.
The 3D seismic migration synthetic experiments
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Figure 13. 共a兲 Magnitude response of the POCSdesigned 2D extrapolator. 共b兲 Phase response of the POCS-designed 2D extrapolator, with N ⳱ 25, kcp ⳱ 0.25, kcs ⳱ 0.3841, ␦ p ⳱ ␦ s ⳱ 10ⳮ3, and ⑀ ⳱ 10ⳮ14. 共c兲 Magnitude response of the MPOCSdesigned 2D extraploator. 共d兲 Phase response of the MPOCS-designed 2D extrapolator, with N ⳱ 25, kcp ⳱ 0.25, kcs ⳱ 0.401, ␦ p ⳱ ␦ s ⳱ 10ⳮ3, and ⑀ ⳱ 10ⳮ14. Note that the 2D MPOCS algorithm resulted in better magnitude as well as phase wavenumber responses than the 2D POCS design algorithm result in 共a兲 and 共b兲.
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POCS-based wavefield extrapolation slices at z ⳱ 220 m of the same 3D seismic volume using the McClellan 共McClellan and Chan, 1977兲 and McClellan-Hale techniques 共Hale, 1991a兲. Both sets of operators designed with the McClellan transformations were obtained by the 1D MPOCS algorithm. Clearly, all three responses can accommodate dip angles of up to 65°, but Figure 15a results in a perfectly circular symmetrical impulse response with less dispersion. These results match our simulations for designing the 2D operators. As a consequence, we expect that extrapolated sections will improve significantly by using such true 2D operators designed using the 2D MPOCS algorithm when compared with the standard McClellan and McClellan-Hale techniques at the expense of higher migration costs.
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width calculation of a low-pass operator, which can be computed using 共Çetin et al., 1997; Madisetti and Williams, 1998兲
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DISCUSSION There are many practical issues to be discussed in designing 1D and 2D wavefield extrapolation operators using the POCS or MPOCS methods. One of these is initialization of the algorithm. A good choice is to start with an approximation of the ideal filter response to initialize the algorithm 共Sezan, 1992; Çetin et al., 1997; Stark and Yang, 1998兲. This is obtained by inverse Fourier transforming the ideal filter wavenumber response into the space domain, saving a number of the early iterations 共Sezan, 1992; Stark and Yang, 1998兲. Furthermore, the design of our extrapolators requires implementation in the wavenumber domain of the constraint sets Ci, where i ⳱ 2, . . . ,5. Ideally, they should be implemented 共realized兲 on a discrete wavenumber grid via the discrete space Fourier transform 共DSFT兲. An M-length DSFT is implemented using the FFT algorithm for which M is much greater than the operator spatial length N. The choice of M ⱖ 10N 共M is a power of two for the FFT algorithm兲 results in satisfactory designs, so we have M discrete wavenumber values over the normalized wavenumber interval. In addition, for the 1D or 2D POCS design algorithms, the stopband cutoff wavenumbers are chosen based on the transition band-
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We use the 2D Kaiser window given in equation 14. This window is composed mainly of the product of two 1D Kaiser windows where they result in satisfactory filter wavenumber responses 共Lu and Antoniou, 1992兲. Another way to obtain a satisfactory Kaiser window is to transform the 1D Kaiser window into an equivalent 2D window by replacing the spatial index n with 冑共n1兲2 Ⳮ 共n2兲2, as originally demonstrated by Huang 共1972兲. In addition, although we design 2D operators for equally spaced sampled data, i.e., ⌬x ⳱ ⌬y, we are not prevented from adapting the 2D POCS and MPOCS design algorithms for data with different inline and crossline sampling intervals 共⌬x ⫽ ⌬y兲. This can be achieved easily by using an FFT with ⌬x ⫽ ⌬y. Alternatively, one can follow the work of Levin 共2004兲, where he extends the McClellan-Hale method for the design of 2D operators with ⌬x ⫽ ⌬y. Our focus is to compare both McClellan transformation designs with our 2D design algorithms based on POCS and MPOCS. In practice, the 1D operators are designed and the Chebychev structure 共Lu and Antoniou, 1992兲 is used along with these predesigned 1D filters to realize 共implement兲 the 2D operators 共Mecklenbräuker and Mersereau, 1976b; Lu and Antoniou, 1992; Yilmaz, 2001兲. Finally, Figure 14. 共a兲 Magnitude and 共b兲 phase response of the McClellan-designed 2D extrapolator. 共c兲 Magnitude and 共d兲 phase response of the improved McClellan-Hale designed 2D extrapolator. All were designed based on a 1D MPOCS extrapolator with N⫽25, kcp ⫽ 0.25, kcs ⳱ 0.401, ␦ p ⳱ ␦ s ⳱ 10ⳮ3, and ⑀ ⳱ 10ⳮ14. The result in 共c兲 shows a better approximation of the 2D extrapolator than the result in 共a兲. However, we still see an error along line kx ⬇ ky.
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Robust explicit depth extrapolation operators can be designed using the POCS method. The operators are acceptable in the sense that they satisfy all required extrapolation design characteristics. Through a simple modification of the POCS algorithm into the MPOCS, we achieved faster convergence of the operator design and enhanced stability of the extrapolated sections. In addition, 2D poststack depth migration of the SEG/EAGE salt model using explicit depth wavefield extrapolators designed with the MPOCS scheme compared favorably with the more computationally expensive PSPI method; our approach saved 80% of the total computational cost when compared to the PSPI technique. Also, it outperformed several other wavefield extrapolation techniques such as the modified Taylor method at the same computational cost as well as the split-step Fourier method, where savings of 40% were realized. Thus, practically stable and short-length explicit depth extrapolation can be achieved in structurally challenging areas involving steep dips or underneath salt using these MPOCS operators. It is straightforward to extend the POCS/MPOCS design algorithms to design 2D extrapolators for 3D explicit depth extrapolation. Circularly symmetric migration impulse responses are obtained easily. This should lead ultimately to more accurate 3D migrated images.
ACKNOWLEDGMENTS
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The authors thank Schlumberger Dhahran Center for Carbonate Research for sponsoring this work; C. Bagaini and A. Özbek, Schlumberger Cambridge Research, for their helpful suggestions; and R. Ferguson, The University of Texas at Austin, for providing us with the SEG/EAGE synthetic data set. Finally, we thank the associate editor and reviewers for all their valuable comments and suggestions.
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REFERENCES
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Figure 15. For N ⳱ 25, the 3D seismic migration impulse response 2D slice at depth z ⳱ 220 m with ⌬z ⳱ 2 m, and ⌬x ⳱ ⌬y ⳱ 10 m for a range of 1100 m for the inline and crossline sections. Time sampling interval is ⌬t ⳱ 4 ms, c ⳱ 1000 m/s, and maximum frequency is 45 Hz using 共a兲 true 2D filters designed with MPOCS and an approximate resulting dip angle of 65°, 共b兲 the McClellan transformation method, and 共c兲 the improved McClellan-Hale transformation method. The result in 共a兲 has less migration noise and better circular symmetry than 共b兲 and 共c兲.
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