Applied Geophysics

, Volume 14, Issue 2, pp 258–269 | Cite as

Seismic wavefield modeling based on time-domain symplectic and Fourier finite-difference method

  • Gang Fang
  • Jing Ba
  • Xin-xin Liu
  • Kun Zhu
  • Guo-Chang Liu
Article
  • 65 Downloads

Abstract

Seismic wavefield modeling is important for improving seismic data processing and interpretation. Calculations of wavefield propagation are sometimes not stable when forward modeling of seismic wave uses large time steps for long times. Based on the Hamiltonian expression of the acoustic wave equation, we propose a structure-preserving method for seismic wavefield modeling by applying the symplectic finite-difference method on time grids and the Fourier finite-difference method on space grids to solve the acoustic wave equation. The proposed method is called the symplectic Fourier finite-difference (symplectic FFD) method, and offers high computational accuracy and improves the computational stability. Using acoustic approximation, we extend the method to anisotropic media. We discuss the calculations in the symplectic FFD method for seismic wavefield modeling of isotropic and anisotropic media, and use the BP salt model and BP TTI model to test the proposed method. The numerical examples suggest that the proposed method can be used in seismic modeling of strongly variable velocities, offering high computational accuracy and low numerical dispersion. The symplectic FFD method overcomes the residual qSV wave of seismic modeling in anisotropic media and maintains the stability of the wavefield propagation for large time steps.

Keywords

symplectic algorithm Fourier finite-difference Hamiltonian system seismic modeling anisotropic 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

We are grateful to Dr. Xiaolei Song from BP for vivid discussions and valuable suggestions. We also thank the developers of the Madagascar open-source software package (http://ahay.org) (Fomel et al., 2013a).

References

  1. Alkhalifah, T., 1998, Acoustic approximations for processing in transversely isotropic media: Geophysics, 63(2), 623–631.Google Scholar
  2. Alkhalifah, T., 2000, An acoustic wave equation for anisotropic media: Geophysics, 65(4), 1239–1250.Google Scholar
  3. Alkhalifah, T., and Tsvankin, I., 1995, Velocity analysis for transversely isotropic media: Geophysics, 60(5), 1550–1566.Google Scholar
  4. Araujo, E. S., Pestana, R. C., and Santos, A. W. G., 2014, Symplectic scheme and the Poynting vector in reverse-time migration: Geophysics, 79(5), S163–S172.Google Scholar
  5. Ba, J., Cao, H., Yao, F. C., et al., 2008a, Double-porosity rock model and Squirt flow for laboratory frequency band: Applied Geophysics, 5(4), 261–276.Google Scholar
  6. Ba, J., Du, Q. Z., Carcione, J. M., et al., 2015, Seismic exploration of hydrocarbons in heterogeneous reservoirs: New theories, methods and applications: Elsevier, United Kingdom.Google Scholar
  7. Ba, J., Nie, J., Cao, H., et al., 2008b, Mesoscopic fluid flow simulation in double-porosity rocks: Geophysical. Research Letters, 35(4), L04303.Google Scholar
  8. Billette, F., and Brandsberg-Dahl, S., 2005, The 2004 BP velocity benchmark: 67th Annual International Conference and Exhibition, EAGE, Extended Abstracts, B035.Google Scholar
  9. Bonomi, E., Brieger, L., Nardone, C., et al., 1998, 3D spectral reverse time migration with no-wraparound absorbing conditions: 68th Annual International Meeting, SEG, Expanded Abstracts, 1925–1928.Google Scholar
  10. Du, Q.Z., Li, B., and Hou, B., 2009, Numerical modeling of seismic wavefields in transversely isotropic media with a compact staggered-grid finite difference scheme: Applied Geophysics, 6(1), 42–49.Google Scholar
  11. Du, X., Fowler, P. J., and Fletcher, R. P., 2014, Recursive integral time-extrapolation methods for waves: A comparative review: Geophysics, 79(1), T9–T26.Google Scholar
  12. Etgen, J. T., 1986, High-order finite-difference reverse time migration with the 2-way non-reflecting wave equation: Stanford Exploration Project, Report SEP-48 133–146.Google Scholar
  13. Etgen, J. T., and Brandsberg-Dahl, S., 2009, The pseudoanalytical method: application of pseudo-Laplacians to acoustic and acoustic anisotropic wave propagation: 71st Annual Meeting, SEG, Expanded Abstracts, 2552–2556.Google Scholar
  14. Fang, G., Fomel, S., Du, Q. Z., et al., 2014, Lowrank seismicwave extrapolation on a staggered grid: Geophysics, 79(3), T157–T168.Google Scholar
  15. Fang, G., Hu, J., and Fomel, S., 2015, Weighted least square based lowrank finite difference for seismic wave extrapolation: 85th Annual International Meeting, SEG, Expanded Abstracts, 3554–3559.Google Scholar
  16. Feng, K., and Qin, M., 2003, Symplectic geometric algorithms for Hmiltionian systems: Zhejiang Science and Technology Press, China.Google Scholar
  17. Finkelstein, B., and Kastner, R., 2007, Finite difference time domain dispersion reduction schemes: Journal of Computational Physics, 221(1), 422–438.Google Scholar
  18. Fomel, S., 2004, On anelliptic approximations for qP velocities in VTI media: Geophysical Prospecting, 52(3), 247–259.Google Scholar
  19. Fomel, S., Sava, P., Vlad, I., et al., 2013a, Madagascar: Opensource software project for multidimensional data analysis and reproducible computational experiments: Journal of Open Research Software, 1, e8.Google Scholar
  20. Fomel, S., Ying, L., and Song, X., 2013b, Seismic wave extrapolation using lowrank symbol approximation: Geophysical Prospecting, 61(3), 526–536.Google Scholar
  21. Fowler, P. J., Du, X., and Fletcher, R.P., 2010, Coupled equations forreverse time migration in transversely isotropic media: Geophysics, 75(1), S11–S22.Google Scholar
  22. Gazdag, J., and Sguazzero, P., 1984, Migration of seismic data by phase shift plus interpolation: Geophysics, 49(2), 124–131.Google Scholar
  23. Grechka, V., Zhang, L., and Rector, J. W., 2004. Shear waves in acoustic anisotropic media: Geophysics, 69(2), 576–582.Google Scholar
  24. Li, Y., Li, X., and Zhu, T., 2011, The seismic scalar wave field modeling by symplectic scheme and singular kernel convolutional differentiator: Chinese Journal Geophysics (in Chinese), 54(7), 1827–1834.Google Scholar
  25. Liu, Y., and Sen, M. K., 2009, A new time–space domain high-order finite-difference method for the acoustic wave equation: Journal of Computational Physics, 228(23), 8779–8806.Google Scholar
  26. Liu, J., Wei, X. C., Ji, Y. X., et al., 2015, Second-order seismic wave simulation in the presence of a free-surface by pseudospectral method: Journal of Applied Geophysics, 114 183–195.Google Scholar
  27. Lou, M., Liu, H., and Li, Y., 2001, Hamiltonian description and symplectic method of seismic wave propagation: Chinese Journal Geophysics (in Chinese), 44(1), 120–128.Google Scholar
  28. Ma, X., Yang, D., and Zhang, J., 2010, Symplectic partitioned Runge-Kutta method for solving the acoustic wave equation: Chinese Journal Geophysics (in Chinese), 53(8), 1993–2003.Google Scholar
  29. Pestana, R., Chu, C., and Stoffa, P. L., 2011, High-order pseudo-analytical method for acoustic wave modeling: Journal of Seismic Exploration, 20(3), 217–234.Google Scholar
  30. Pestana, R. C., and Stoffa, P. L., 2010, Time evolution of the wave equation using rapid expansion method: Geophysics, 75(4), T121–T131.Google Scholar
  31. Reshef, M., Kosloff, D., Edwards, M., et al., 1988, Threedimensional acoustic modeling by the Fourier method: Geophysics, 53(9), 1175–1183.Google Scholar
  32. Song, X., and Fomel, S., 2011, Fourier finite-difference wave propagation: Geophysics, 76(5), T123–T129.Google Scholar
  33. Song, X., Fomel, S., and Ying, L., 2013, Lowrank finitedifferences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation: Geophysical Journal International, 193(2), 960–969.Google Scholar
  34. Soubaras, R., and Zhang, Y., 2008, Two-step explicit marching method for reverse time migration: 70th Annual International Conference and Exhibition,EAGE, Extended Abstracts, F041.Google Scholar
  35. Stoffa, P. L., and Pestana, R. C., 2009, Numerical solution of the acoustic wave equation by the rapid expansion method (REM)-A one step time evolution algorithm: 71st Annual International Meeting, SEG, Expanded Abstracts, 2672–2676.Google Scholar
  36. Sun, G., 1997, Aclass of explicitly symplectic schemes for wave equation: Comput. Math. (in Chinese), 1 1–10.Google Scholar
  37. Sun, J., Fomel, S., and Ying, L., 2015, Low-rank one-step wave extrapolation for reverse time migration: Geophysics, 81(1), S39–S54.Google Scholar
  38. 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(1), 99–131.Google Scholar
  39. Tal-Ezer, H., Kosloff, D., and Koren, Z., 1987, An accurate scheme for seismic forward modeling: Geophysical Prospecting, 35(5), 479–490.Google Scholar
  40. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51(10), 1954–1966.Google Scholar
  41. Wu, W. J., Lines, L. R., and Lu, H. X., 1996, Analysis of higher-order, finite-difference schemes in 3-D reverse-time migration: Geophysics, 61(3), 845–856.Google Scholar
  42. Zhang, Y., and Zhang, G., 2009, One-step extrapolation method for reverse time migration: Geophysics, 74(4), A29–A33.Google Scholar

Copyright information

© Editorial Office of Applied Geophysics and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Gang Fang
    • 1
    • 2
  • Jing Ba
    • 3
  • Xin-xin Liu
    • 1
    • 2
  • Kun Zhu
    • 4
  • Guo-Chang Liu
    • 5
  1. 1.The Key Laboratory of Gas Hydrate, Ministry of Land and ResourcesQingdao Institute of Marine GeologyQingdaoChina
  2. 2.Laboratory for Marine Mineral ResourcesQingdao National Laboratory for Marine Science and TechnologyQingdaoChina
  3. 3.School of Earth Sciences and EngineeringHohai UniversityNanjingChina
  4. 4.School of GeosciencesChina University of Petroleum (East China)QingdaoChina
  5. 5.State Key Laboratory of Petroleum Resources and ProspectingChina University of Petroleum-BeijingBeijingChina

Personalised recommendations