1887
Volume 69 Number 1
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

We propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time‐steps than traditional finite‐difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the first‐order linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post‐stack migration results are also shown using the proposed method.

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/content/journals/10.1111/1365-2478.13033
2020-12-12
2024-04-19
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References

  1. Araujo, E.S. and Pestana, R.C. (2019) Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method. Geophysical Prospecting, 68, 572–590. https://doi.org/10.1111/1365-2478.12868.
    [Google Scholar]
  2. Araujo, E.S., Pestana, R.C. and Santos, A.W.G. (2014) Symplectic scheme and the Poynting vector in reverse‐time migration. Geophysics, 79, S163–S172.
    [Google Scholar]
  3. Barucq, H., Boillot, L., Calandra, H. and Diaz, J. (2014) Absorbing boundary conditions for 2D tilted transverse isotropic elastic media: Esaim. Proceedings and Surveys, 45, 400–409.
    [Google Scholar]
  4. Bonomi, E., Brieger, L., Nardone, C. and Pieroni, E. (1998) 3D spectral reverse time migration with no‐wraparound absorbing conditions. Geophysics Area CRS4.
    [Google Scholar]
  5. Chen, J. (2009) Lax‐wendroff and nystrom methods for seismic modeling. Geophysical Prospecting, 57, 931–941.
    [Google Scholar]
  6. Chen, K.Y. (2014) Finite‐difference simulation of elastic wave with separation in pure‐ and s‐modes. Journal of Computational Methods in Physics, 43, 108713. http://doi.org/10.1155/2014/108713.
    [Google Scholar]
  7. Cherifi, C., Cherifi, H., Karsai, M. and Musolesi, M. (Eds.). (2017) Complex Networks and Their Applications VI. Proceedings of Complex Networks 2017 The Sixth International Conference on Complex Networks and Their Applications. Berlin: Springer.
    [Google Scholar]
  8. Collino, F. and Tsogka, C. (2001) Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics, 66, 294–307.
    [Google Scholar]
  9. Gao, Y., Song, H., Zhang, J. and Yao, Z. (2015) Comparison of artificial absorbing boundaries for acoustic wave equation modelling. Exploration Geophysics, 48, 76–93.
    [Google Scholar]
  10. Guo, D.L., Tian, M.Z. and Zhong, C.J. (2000) Stability of the staggered‐grid high‐order difference method for first‐order elastic wave equation. Chinese Journal of Geophysics, 43, 904–913.
    [Google Scholar]
  11. Hall, B.C. (2015) Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Berlin: Springer.
    [Google Scholar]
  12. Han, B., Seol, S.J. and Byun, J. (2012) Elastic modelling in tilted transversely isotropic media with convolution perfectly matched layer boundary conditions. Exploration Geophysics, 43, 77–86.
    [Google Scholar]
  13. Hu, F.Q. (1996) On absorbing boundary conditions of linearized euler equations by a perfectly matched layer. Journal of Computational Physics, 129, 201.
    [Google Scholar]
  14. Igel, H., Mora, P. and Riollet, B. (1995) Anisotropic wave propagation through finite‐difference grids. Geophysics, 60, 1203–1216.
    [Google Scholar]
  15. Jin, Q., Shiguo, W. and Ruofei, C. (2013) Accuracy of the staggered‐grid fi nite‐difference method of the acoustic wave equation for marine seismic reflection modeling. Chinese Journal of Oceanology and Limnology, 31, 169–177.
    [Google Scholar]
  16. Lin, P.Z. and Xu, W.S. (2005) A staggered‐grid high‐order finite‐difference modeling for elastic wave field in arbitrary tilt anisotropic media. Acta Seismologica Sinica, 18(4), 471–482.
    [Google Scholar]
  17. Liu, H., Fox, A., Sliz, K., Zhang, H. and Zhao, Y. (2018) A stable constraint for pseudo‐elastic anisotropic reverse time migration. SEG International Exposition and 88th Annual Meeting, 4347–4350.
    [Google Scholar]
  18. Long, G., Zhao, Y. and Zou, J. (2013) A temporal fourth‐order scheme for the first‐order acoustic wave equations. Geophysical Journal International, 194, 1473–1485.
    [Google Scholar]
  19. Pei, Z., Fu, L., Sun, W., Jiang, T. and Zhou, B. (2012) Anisotropic finite‐difference algorithm for modeling elastic wave propagation in fractured coalbeds. Geophysics, 77, C13–C26.
    [Google Scholar]
  20. Stickel, E.U. (1994) A splitting method for the calculation of the matrix exponential. Analysis, 14, 103–112.
    [Google Scholar]
  21. Tal‐Ezer, H. (1986) Spectral methods in time for hyperbolic problems. society of industrial and applied mathematics. Journal on Numerical Analysis, 23, 11–20.
    [Google Scholar]
  22. Tal‐Ezer, H., Kosloff, D. and KOren, Z. (1987) An accurate scheme for seismic forward modelling. Geophysical Prospecting, 35, 479–490.
    [Google Scholar]
  23. Thomsen, L. (1986) Weak elastic anisotropy. Geophysics, 51, 1954–1966.
    [Google Scholar]
  24. Ting, C. and Bing‐Shou, H. (2014) A normalized wavefield separation cross‐correlation. Applied Geophysics, 11, 158–166.
    [Google Scholar]
  25. Virieux, J. (1986) P‐SV wave propagation in heterogeneous media: velocity‐stress finite‐difference method. Geophysics, 4, 889–901.
    [Google Scholar]
  26. Xu, Z., Cai, L. and Cong, X. (2015) An imaging condition for reverse time migration based. Global Geology, 18(2), 122–126.
    [Google Scholar]
  27. Yao, G., Silva, N.V. and Wu, D. (2018) An effective absorbing layer for the boundary condition in acoustic seismic wave simulation. Journal of Geophysics and Engineering, 15, 495–511.
    [Google Scholar]
  28. Yao, Z. and Margrave, G.F. (2000) Elastic wavefield modeling in 3D by fourth‐order staggered‐grid finite difference technique. CREWES Research Report, 12.
    [Google Scholar]
  29. Yongming, L. and Qiancheng, L. (2018) Non‐overlapped P‐ and S‐wave Poynting vectors and their solution by the grid method. Journal of Geophysics and Engineering, 15, 788–799.
    [Google Scholar]
  30. Yoon, K. and Marfurt, K.J. (2006) Reverse‐time migration using the Poynting vector. Exploration Geophysics, 37, 102–107.
    [Google Scholar]
  31. You, J., Liu, X. and Wu, R. (2017) First‐order acoustic wave equation reverse time migration based on the dual‐sensor. Pure Applied Geophysics, 174, 1345–1360.
    [Google Scholar]
  32. Zhou, M. (2001) A well‐posed pml absorbing boundary condition for 2D acoustic wave equation. Journal of Computational Physics, 173, 455–480.
    [Google Scholar]
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  • Article Type: Research Article
Keyword(s): 2D FD modelling; Acoustic; Anisotropy

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