Three-dimensional acoustic wave equation modeling based on the optimal finite-difference scheme
Cai Xiao-Hui1,2, Liu Yang1,2, Ren Zhi-Ming1,2, Wang Jian-Min3, Chen Zhi-De3, Chen Ke-Yang3, and Wang Cheng3
1. State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China.
2. CNPC Key Laboratory of Geophysical Prospecting, China University of Petroleum, Beijing 102249, China.
3. Exploration and Development Research Institute of Daqing Oilfield Company Limited, Daqing 163712, China.
Abstract Generally, FD coefficients can be obtained by using Taylor series expansion (TE) or optimization methods to minimize the dispersion error. However, the TE-based FD method only achieves high modeling precision over a limited range of wavenumbers, and produces large numerical dispersion beyond this range. The optimal FD scheme based on least squares (LS) can guarantee high precision over a larger range of wavenumbers and obtain the best optimization solution at small computational cost. We extend the LS-based optimal FD scheme from two-dimensional (2D) forward modeling to three-dimensional (3D) and develop a 3D acoustic optimal FD method with high efficiency, wide range of high accuracy and adaptability to parallel computing. Dispersion analysis and forward modeling demonstrate that the developed FD method suppresses numerical dispersion. Finally, we use the developed FD method to source wavefield extrapolation and receiver wavefield extrapolation in 3D RTM. To decrease the computation time and storage requirements, the 3D RTM is implemented by combining the efficient boundary storage with checkpointing strategies on GPU. 3D RTM imaging results suggest that the 3D optimal FD method has higher precision than conventional methods.
This research is supported by the National Natural Science Foundation of China (No. 41474110) and Shell Ph.D. Scholarship to support excellence in geophysical research.
Cite this article:
Cai Xiao-Hui,Liu Yang,Ren Zhi-Ming et al. Three-dimensional acoustic wave equation modeling based on the optimal finite-difference scheme[J]. APPLIED GEOPHYSICS, 2015, 12(3): 409-420.
[1]
Aoi, S., and Fujiwara, H., 1999, 3-D finite-difference method using discontinuous grids: Bulletin of the Seismological Society of America, 89, 918-930.
[2]
Baysal, E., Kosloff, D. D., and Sherwood, J. W. C., 1983, Reverse time migration: Geophysics, 48(11), 1514-1524.
[3]
Chang, S., and Liu, Y., 2013, A truncated implicit high-order finite-difference scheme combined with boundary conditions, Applied geophysics, 10(1), 53-62.
[4]
Chang, W., and McMechan, G. A., 1990, 3D acoustic prestack reverse-time migration: Geophysical Prospecting, 38(7), 737-756.
[5]
Chang, W., and McMechan, G. A., 1993, 3D prestack migration in anisotropic media: Geophysics, 58(1), 79-90.
[6]
Chang, W., and McMechan, G. A., 1994, 3D elastic prestack reverse-time depth migration: Geophysics, 59(4), 597-609.
[7]
Chen, J., 2012, An average-derivative optimal scheme for frequency-domain scalar wave equation: Geophysics, 77, T201-T210.
[8]
Claerbout, J. F., 1985, Imaging the Earth’s Interior: Blackwell Scientific Publications Inc., Palo Alto, CA.
[9]
Dablain, M. A., 1986, The application of high-order differencing to the scalar wave equation: Geophysics, 51, 54-66.
[10]
Di Bartolo, L., Dors, C. and Mansur, W. J., 2012, A new family of finite-difference schemes to solve the heterogeneous acoustic wave equation: Geophysics, 77, T187-T199.
[11]
Etgen, J. T., and O’Brien, M. J., 2007, Computational methods for large-scale 3D acoustic finite-difference modeling: a tutorial: Geophysics, 72, SM223-SM230.
[12]
Fei, T., and Liner, C. L., 2008, Hybrid Fourier finite difference 3D depth migration for anisotropic media: Geophysics, 73, S27-S34.
[13]
Graves, R. W., 1996, Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences: Bulletin of the Seismological Society of America, 86, 1091-1106.
[14]
Kelly, K. R., Ward, R., Treitel, W. S., and Alford, R. M., 1976, Synthetic seismograms: a finite-difference approach: Geophysics, 41, 2-27.
[15]
Larner, K., and Beasley, C., 1987, Cascaded migrations-improving the accuracy of finite-difference migration: Geophysics, 52, 618-643.
[16]
Li, Z., 1991, Compensating finite-difference errors in 3-D migration and modeling: Geophysics, 56, 1650-1660.
[17]
Liu, G., Liu, Y., Li, R., and Meng, X., 2013a, 3D seismic reverse time migration on GPGPU: Computers & Geosciences, 59, 17-23.
[18]
Liu, S., Wang H., Chen, S., and Kong X., 2013b, Implementation strategy of 3D reverse time migration on GPU/CPU clusters: Chinese Journal of Geophysics, 56(10), 3487-3496.
[19]
Liu, Y., and Sen, M. K., 2009, A new time-space domain high-order finite-different method for the acoustic wave equation: Journal of Computational Physics, 228, 8779-8806.
[20]
Liu, Y., and Sen, M. K., 2011a, Finite-difference modeling with adaptive variable-length spatial operators: Geophysics, 76, T79-T89.
[21]
Liu, Y., and Sen, M. K., 2011b, 3D acoustic wave modeling with time-space domain dispersion-relation-based finite-difference schemes and hybrid absorbing boundary conditions: Exploration Geophysics, 42, 176-189.
[22]
Liu，Y., 2013, Globally optimal finite-difference schemes based on least squares: Geophysics, 78, T113-T132.
[23]
Liu, Y., 2014, Optimal staggered-grid finite-difference schemes based on least-squares for wave equation modelling: Geophysical Journal International, 197, 1033-1047.
[24]
Moczo, P., Kristek, J., and Halada, L., 2000, 3D fourth-order staggered-grid finite-difference schemes: Stability and grid dispersion: Bulletin of the Seismological Society of America, 90, 587-603.
[25]
Pitarka, A., 1999, 3D elastic finite-difference modeling of seismic motion using staggered grids with nonuniform spacing: Bulletin of the Seismological Society of America, 89, 54-68.
[26]
Robertsson, J. O. A., Blanch J. O., Symes W. W., and Burrus C. S., 1994a, Galerkin-wavelet modeling of wave propagation: Optimal finite difference stencil design: Mathematical and Computer Modeling, 19, 31-38.
[27]
Robertsson, J. O. A., Blanch, J. O., and Symes, W. W., 1994b, Viscoelastic finite-difference modeling: Geophysics, 59, 1444-1456.
[28]
Shragge, J., 2014, Solving the 3D acoustic wave equation on generalized structured meshes: A finite-difference time-domain approach: Geophsics, 79, T363-T378.
[29]
Symes, W. W., 2007, Reverse time migration with optimal checkpointing: Geophsics, 72, SM213-SM221.
[30]
Takeuchi, N., and Geller, R. J., 2000, Optimally accurate second order time-domain finite difference scheme for computing synthetic seismograms in 2-D and 3-D media: Physics of the Earth and Planetary Interiors, 119, 99-131.
[31]
Tan, S., and Huang, L., 2014, An efficient finite-difference method with high-order accuracy in both time and space domains for modelling scalar-wave propagation: Geophysical Journal International, 197, 1250-1267.
[32]
Wang, B., Chen, J., and Chen W., 2012, Efficient boundary storage strategies for seismic reversetime migration: Chinese Journal of Geophysics (in Chinese), 55(7), 2412-2421.
[33]
Ristow, D., and Ruhl, T., 1997, 3-D implicit finite-difference migration by multiway splitting: Geophysics, 62, 554-567.
[34]
Wang, Y., Liang, W., Nashed, Z., Li, X., Liang, G., and Yang, C., 2014, Seismic modeling by optimizing regularized staggered-grid finite-difference operators using a time-space-domain dispersion-relationship-preserving method: Geophysics, 79, T277-T285.
[35]
Yomogida, K. J., and Etgen, T., 1993, 3-D wave propagation in the Los Angeles basin for the Whittier-Narrows earthquake: Bulletin of the Seismological Society of America, 83, 1325-1344.
[36]
Xu, S., Zhang, Y., and Tang, B., 2011, 3D angle gathers from reverse time migration: Geophysics, 76(2), S77-S92.
[37]
Zhao, Y., Liu Y., and Ren Z., 2014, Viscoacoustic prestack reverse time migration based on the optimal time-space domain high-order finite-difference method: Applied Geophysics, 11(1), 50-62.
[38]
Zhang, H., and Zhang Y., 2008, Reverse time migration in 3D heterogeneous TTI media: SEG Technical Program Expanded Abstrats, 2196-2200.
[39]
Zhang, Y., and Sun, J., 2009, Practical issues in reverse time migration: true amplitude gathers, noise removal and harmonic-source encoding: First Break, 26(1), 29-35.
[40]
Zhang, J., and Yao, Z., 2013, Optimized finite-difference operator for broadband seismic wave modeling: Geophysics, 78, A13-A18.
2015, Vol. 12 
