Split-step complex Padé-Fourier depth migration

Transcription

1 Geophys. J. Int. (007) 7, doi: 0./j X x Split-step complex Padé-Fourier depth migration Linbin Zhang,,, James W. Rector III, 3 G. Michael Hoversten and Sergey Fomel 4 Department of Material Science and Engineering, University of California, Berkeley, CA 9470, USA. linbin.zhang@total.com One Cyclotron Road, Lawrence Berkeley National Laboratory, MS 90-6, Berkeley, CA 9470, USA 3 Department of Civil and Environmental Engineering, University of California, Berkeley, CA 9470, USA 4 Bureau of Economic Geology, The University of Texas at Austin, University Station, Box X, Austin, TX , USA Accepted 007 September 7. Received 007 August 7; in original form 006 November 8 SUMMARY We present a split-step complex Padé-Fourier migration method based on the one-way wave equation. The downward-continuation operator is split into two downward-continuation operators: one operator is a phase-shift operator and the other operator is a finite-difference operator. A complex treatment of the propagation operator is applied to mitigate inaccuracies and instabilities due to evanescent waves. It produces high-quality images of complex structures with fewer numerical artefacts than those obtained using a real approximation of a square-root operator in the one-way wave equation. Tests on zero-offset data from the SEG/EAGE salt data show that the method improves the image quality at the cost of an additional 0 per cent computational time compared to the conventional Fourier finite-difference method. Key words: depth migration, Fourier finite-difference method, Padé approximation. GJI Seismology INTRODUCTION Wave equation migration is a rapidly growing tool for complex structure imaging (Claerbout 985; Stoffa et al. 990; Lee et al. 99; Ristow & Ruhl 994; Wu & Jin 997; Bonomi & Cazzola 999). Wave equation migration methods can produce accurate images provided that the background velocity field is well defined. Most widely used wave equation methods are based on solving the one-way acoustic wave equation. There are many different methods to numerically solve the one-way equation. They can be classified into three main approaches: Fourier methods (Gazdag & Sguazzero 984; Stoffa et al. 990) solved in the wavenumber domain and space domain; finite-difference methods (FD) (Claerbout 985; Hale 99) and mixed methods that are a linear combination of spectral and FD methods (Ristow & Rulh 994; Biondi 00). The Fourier methods, such as Split-Step Fourier method (SSF) (Stoffa et al. 990) and Phase Shift Plus Interpolation (PSPI) (Gazdag & Sguazzero 984; Bagani et al. 995) can handle both wide-angle propagation and lateral velocity variations, but the imaging becomes less accurate when the lateral velocity variations are strong. The FD methods can easily handle strong lateral velocity variations. When implicit extrapolators are used in FD methods, the one-way wave equation is solved in the frequency-space domain, and the square root of the wave operator is approximated using continued fractions expansions (Claerbout 985). The essential shortcoming Now at: Total E&P USA Inc., 800 Gessner, Suite 700, Houston, TX 7704, USA. of this method is that it is not efficient for wide-angle propagation. High-order approximations to the square root operators are required for imaging of steep dips. When explicit extrapolators are used (Holberg 988; Hale 99), the exponential operator is approximated by finite length spatial filters. The disadvantage of explicit methods is that their stability is not guaranteed if the filter is not designed carefully. In 994, Ristow and Ruhl proposed a Fourier finite-difference method (FFD) which combined a phase-shift method in the frequency-wavenumber domain and an FD method in the frequencyspace domain. The FFD method combines the advantages of the FD methods and the phase shift methods and is more accurate than SSF, PSPI and FD methods. Huang & Fehler (00) proposed a SSF Padé migration method that allows for the use of a larger grid spacing than conventional FD migration methods. Huang and Fehler s method is equivalent to applying the Crank Nicholson scheme to the FD part in the FFD method of Ristow and Ruhl. In both FD and FFD migration methods, the real Padé approximation or a Taylor expansion is usually used to approximate the square-root operator. For example, the real [n/n] Padé approximation to the square-root operator + X is given by: + X + n j= a j X + b j X, where X = V (x,z), and a x j and b j are the real-valued coefficients. If X < (evanescent waves), the right-hand side of the above equation is real and the left-hand side is complex. The real approximation of the square-root operators is inconsistent and will cause accuracy and stability problems. It cannot handle evanescent ω 308 C 007 The Authors

2 Split-step complex Padé-Fourier depth migration 309 waves properly (Milinazzo et al. 997; Yevick & Thomson 000). In FFD method, usually the minimum velocity is used as the reference velocity. In this case, most evanescent wave energy still exists in the wavefield. If the real complex Padé approximation is used, we will have inaccuracy and instability problem. To deal with this problem, a complex Padé approximant technique (Milinazzo et al. 997) can be employed. Before the Padé approximations, one can rewrite + X as: + X = e iθ/ + [(X + )e iθ ] = e iθ/ + X, where X = (X + )e iθ, and then apply the Padé approximation to the operator + X. This procedure maps the poles of the Padé approximant along the real axis for X < into the positive imaginary half-space. This method improved the stability significantly (Milinazzo et al. 997; Lingevitch & Collins 998; Yevick & Thomson 000). Zhang et al. (003) have applied this technique to the FD migration method. However, the FD method is not efficient for wide-angle propagation because using a [/] Padé approximant restricts the maximum migration angle to about 45. In this paper, we apply complex Padé approximant technique to FFD method to mitigate the inaccuracies and instabilities caused by evanescent waves. In the first part of the paper, we derive a split-step complex Padé-Fourier solution for the one-way wave equation using a Padé approximant. Then we present zero-offset migration results for the SEG/EAGE salt model. A numerical comparison of FFD, split-step complex Padé migration and our method highlights the advantages of the proposed method. METHOD The -D, one-way acoustic wave equation can be expressed in frequency domain as: P ω z = i V + x P = i ω + XP, () V where P = P(z, x, ω) is the wavefield, i =, and ω is the circular frequency. X = V ω x, () and V = V (z, x) is the velocity. The solution for eq. () is extrapolating in depth: P(z + z, x,ω) = e iδ +X P(z, x,ω), (3) where δ = ω z and z is the spacing in the z-direction. V Let δ 0 = ω V ref z and X 0 = V ref, where V ω x ref denotes the minimum velocity at the layer [z, z + z]. Multiplying and dividing by the approximate exponential operator e iδ 0 +X0, the exact exponential operator e iδ +X can be written as (Ristow & Ruhl 994; Huang & Fehler 00): e iδ +X = e iδ 0 +X0 +iδ 0 ( +X m + X m ), (4) where m = V V ref. Finally, we express the wave field depth extrapolation equation as (see Appendix A for the derivation) : P(z + z, x,ω) e iδ 0 +X0 e iδ 0( m ) ( ) a + b X P(z, x,ω). a + b X (5) The first two exponential terms are implemented using SSF algorithm (Stoffa et al. 990) and the third term is implemented by FD method. When θ = 0, eq. (5) is identical to FFD method with the Crank Nicholson scheme applied to solve the third term of eq. (5) (Huang & Felher 00). Therefore, the maximum migration angle of eq. (5) is the same as that of Huang and Felher s and FFD method. However, eq. (5) can properly handle the propagation of evanescent waves. To obtain more accurate results with FD method, we employ the Douglas operator to approximate the second derivative (Tsuchiya et al. 00): x x δ x x, (6) + /δx where δ x P(x) = P(x + x) P(x) + P(x x), x is the spacing in x-direction. One can also apply [/] Padé approximation to the exponential function E = e iδ 0 A to obtain: E + iδ 0 A + (iδ 0 A) iδ 0 A +. (7) (iδ 0 A) Using eq. (7), eq. (5) becomes: P(z + z, x,ω) e iδ 0 +X0 e iδ 0( m ) + iδ 0 A + (iδ 0 A) iδ 0 A + P(z, x,ω), (8) (iδ 0 A) Using the multistep method (Hadley 99) to solve the equation: P(z + z, x,ω) = + iδ 0 A + (iδ 0 A) iδ 0 A + P(z, x,ω), (9) (iδ 0 A) we get the following two-step algorithm: P(z + k z/, x,ω) = + a k(iδ 0 A) P(z + (k ) z/, x,ω), for k =, (0) a k (iδ 0 A) where a k = (3 + ( ) k 3i)/. The truncation error of [/] Padé approximant of eq. (5) is O( z ) while the overall truncation error of eq. (0) is O( z 4 ). So eq. (0) allows us to use a larger grid spacing z than that of eq. (5). NUMERICAL RESULTS We test our method with post-stack data for the SEG/EAGE salt model. The velocity model is shown in Fig.. The data set consists of 90 traces, each trace contains 66 time samples and a sampling interval of 8 ms (Fig. ). The velocity model grid is with x = z =.9 m. No additional filters were applied to the migration results. Figs 3(a) and (b) show the images using eq. (5) with θ = 0 (FFD) and θ = 5, respectively. Fig. 3(b) gives an image with fewer evanescent-energy artefacts than those produced by the FFD migration (Fig. 3a) while the accuracy of both migrations is similar to each other. The computational time of the Split-step complex Padé-Fourier migration increases 0 per cent compared to the FFD method and the memory usage is the same for both methods. Fig 3(c) shows the image using complex Padé migration method for the exploding reflector data set. Comparing Figs 3(b) and (c) we see that our method gives better images of the top and base interfaces of the salt body (Location A and B in those figures). Moreover, the image in Fig. 3(c) has fewer numerical artefacts than Fig. 3(a). Figs 4(a) (c) show the images with θ = 0,0 and 30, respectively. These images show that the artefacts decrease as θ increases, C 007 The Authors, GJI, 7,

3 30 L. Zhang et al. Figure. SEG/EAGE -D velocity model. Figure. Exploding reflector modelled data (zero offset equivalent). but the events become weaker, for example at the location pointed to by the arrow. Thus it appears that there is a trade-off in image quality and θ. Experiments show that θ should be less than 5. Figs 5 and 6 show the images using eq. (0) with θ = 5 and vertical grid spacing.9 and m, respectively. Comparing Fig. 5 with Figs 3(a) and (b), the [/] Padé approximant yields improved image quality. The reflectors under the salt body are much clearer than those in Fig. 3. However it took double CUP times of that using eq. (8). With a coarse spacing, we can also obtain a good image (Fig. 6) but it is not as good as Fig. 5. This is because the true velocity model cannot be represented well by the coarse spacing of m. Therefore, we lacked necessary resolution in migration. CONCLUSIONS We have developed a split-step complex Padé-Fourier technique for seismic migration. Our method makes use of the advantages of both FD methods and spectral methods. Because the complex treatment of the propagation operator is used, it treats the evanescent waves with more accuracy, which improves the stability of the Figure 3. Comparison of migration images of an exploding-reflector data set for the -D model shown in Fig.. (a) Split-step complex Padé-Fourier migration with θ = 0 (FFD). (b) Split-step complex Padé-Fourier migration with θ = 5. (c) Split-step complex Padé migration with θ = 5. The images shown in (b) and (c) have less artefacts than that in (a). Both images in (a) and (b) give better images of the top and base interfaces of the salt body than that in (c) (Locations A and B in those figures). C 007 The Authors, GJI, 7,

4 Split-step complex Padé-Fourier depth migration 3 Figure 5. Split-step complex Padé-Fourier migration (eq. 8) with θ = 5 and vertical grid spacing.9 m. Figure 6. Split-step complex Padé-Fourier migration (eq. 8) with θ = 5 and vertical grid spacing m. numerical approximation. Synthetic data migration results show that the proposed method can produce higher quality images. ACKNOWLEDGMENT This work was supported by DOE. We are grateful to the editors and two reviewers for their useful comments and suggestions. REFERENCES Figure 4. Images with different θ: (a) θ = 0 ; (b) θ = 0 and (c) θ = 30. As θ increases the event that the arrow pointed to is weaker but the image contains less noise. Bagani, C., Bonomi, E. & Pieroni, E., 995. Data parallel implementation of 3D PSPI, in 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, pp Biondi, B., 00. Stable wide-angle Fourier finite-difference downward extrapolation of 3-D wavefields, Geophysics, 67, Bonomi, E. & Cazzola, L., 999. Prestack imaging of compressed seismic data: a Monte Carlo approach, in 69th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, pp Claerbout, J. F., 985. Imaging the Earth s interior, Blackwell Scientific Publications, Inc. C 007 The Authors, GJI, 7,

5 3 L. Zhang et al. Gazdag, J. & Sguazzero, P., 984. Migration of seismic data by phase shift plus interpolation, Geophysics, 49, 4 3. Hale, D., 99. Stable explicit depth extrapolation of seismic wavefields, Geophysics, 56, Hadley, G.R., 99. Multistep method for wide angle beam prpagation, Opt. Lett., 7, Holberg, O., 988. Towards optimum one-way wave propagation, Geophys. Prospect., 36, Huang, L. & Fehler C.F., 00. Split-step Fourier Padé migration, in 7th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, pp Lee, M., Mason, L.M. & Jackson, G.M., 99. Split-step Fourier shotrecord migration with deconvolution imaging, Geophysics, 56, Lingevith, J.F. & Collins, M.D., 998. Wave propagation in range-dependent poro-acoustic waveguides, J. Acoust. Soc. Am., 04, Milinazzo, F.A., Zala, C.A. & Brooke, G.H., 997. Rational square-root approximations for parabolic equation algorithms, J. Acoust. Soc. Am., 0, Ristow, D. & Ruhl, T., 994. Fourier finite-difference migration, Geophysics, 59, Stoffa, P.L., Fokkema, J.T, Freir, R.M. & Kessinger, W.P., 990. Split-step Fourier migration, Geophysics, 55, Tsuchiya, T., Anada, T., Endoh, N., Nakamura, T., Aoki, T. & Kaihou, I., 00. An efficient method combined the Douglas operator scheme tosplitstep Pade approximation of higher-order parabolic equation, in Ultrasonics Symposium, 00 IEEE, pp Wu, R. & Jin, S., 997. Windowed GSP(Generalized Screen Propagators) migration applied to SEG-EAGE salt model data, in 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, pp Yevick, D. & Thomson, 000. Complex Padé approximants for wide-angle acoustic propagators, J. Acoust. Soc. Am., 08, Zhang, L., Rector, J.W. & Hoversten, G.M, 003. Split-step complex Padé migration, J. Seismic Explor.,, APPENDIX A To handle evanescent waves, we first rotate the square-root operators in eq. (4) by θ (Milinazzo et al. 997), + X = e iθ/ [( + X)e iθ ] + == e iθ/ + X, (A) [( ) ] + Xm = eiθ/ + Xm e iθ + = e iθ/ + X, (A) where X = ( + X)e iθ and X = ( + X m )e iθ. Eq. (4) becomes: e iδ +X = e iδ 0 +X0 +iδ 0( m )+iδ 0[ m ( +X ) ( +X )], (A3) ω V ref ze iθ/. Let s set E = e iδ 0 [ m ( +X ) ( +X )] = e iδ 0 A, where A = m ( + X ) ( + X ). Then we apply where δ 0 = Taylor expansion to A, A ( ) X m X 8 ( ) X m X + = ( ) 8 (e iθ ) m (5 e iθ ) + ( ) e iθ (3 e iθ )X 4m m e iθ 8m ( m 3 ) X +, using the approximation: ax ax( bx +...). + bx We obtain: A 8 (e iθ ) ( ) m (5 e iθ ) + ( 4m Applying Padé approximation to e iδ 0 A,wehave: E + iδ 0 A iδ 0 A = a + b X a + b X, m ) e iθ (3 e iθ )X + (m +m+) m e iθ X. (A4) (3 e iθ ) C 007 The Authors, GJI, 7,

6 Split-step complex Padé-Fourier depth migration 33 where ( ) a = + iδ 0 8 (e iθ ) m (5 e iθ ), b = (m + m + ) m { e iθ ( m 3 (3 e iθ ) + ) (e iθ )(5 e iθ )e iθ iδ 0 6m 3 (3 e iθ ) ( ) a = iδ 0 8 (e iθ ) m (5 e iθ ), b = (m + m + ) m { e iθ ( m 3 (3 e iθ ) ) (e iθ )(5 e iθ )e iθ iδ 0 6m 3 (3 e iθ ) + m } 4m e iθ (3 e iθ ), + m } 4m e iθ (3 e iθ ). (A5) C 007 The Authors, GJI, 7,