Cite this article
2
Download
44
Views
Journal Browser
Volume | Year
Issue
Search
News and Announcements
View All
ARTICLE

High-order pseudo-analytical method for acoustic wave modeling

REYNAM PESTANA1 CHUNLEI CHU2 PAUL L. STOFFA3
Show Less
1 Federal University of Bahia (UFBA), Center for Research in Geophysics and Geology (CPGG), Rua Barão de Geremoabo, Salvador, Bahia, Brazil.,
2 ConocoPhillips, 600 N. Dairy Ashford Rd., Houston, TX 77079, U.S.A.,
3 Institute for Geophysics, The University of Texas at Austin, 10100 Burnet Rd., Bldg. 196, Austin, TX 78758, U.S.A.,
JSE 2011, 20(3), 217–234;
Submitted: 19 December 2010 | Accepted: 23 May 2011 | Published: 1 September 2011
© 2011 by the Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Abstract

For the time evolution of acoustic wavefields we present an alternative derivation of the pseudo-analytical method, which enables us to generalize the method to high-order formulations. Within the same derivation framework, we compare the second-order pseudo-analytical method, the Fourier finite difference method, and the fourth-order Lax-Wendroff time integration method. We demonstrate that the pseudo-analytical method can be regarded as a modified Lax-Wendroff method. Different from the fourth-order time stepping method, both the second-order pseudo-analytical method and the Fourier finite difference method use pseudo-Laplacians to compensate for time stepping errors. The pseudo-Laplacians need to be solved in the wavenumber domain with constant compensation velocities for computational simplicity and efficiency. Low-order pseudo-Laplacians are more sensitive to the choice of compensation velocities than high-order ones. As a result, we need to use the combination of several pseudo-Laplacians to achieve the required accuracy for low-order pseudo-analytical methods. When using the pseudospectral method to evaluate all spatial derivatives, the computation cost for the second-order pseudo-analytical method, the Fourier finite difference method, and the fourth-order Lax-Wendroff time integration method is approximately the same. Both the second-order pseudo-analytical method and the Fourier finite difference method have less restrictive stability conditions than the fourth-order time stepping method. We demonstrate with numerical examples that the second-order pseudo-analytical method, greatly improves the original pseudo-analytical method and as a modified version of the Lax-Wendroff method, is well suited for imaging seismic data in subsalt areas where reverse-time migration plays a crucial role.

Keywords
acoustic wave equation
seismic modeling
pseudo-method
pseudo-analytical method
pseudo-Lapliacan operator
Fourier finite-difference method
Lax-Wendroff method
Fourier pseudo-method
References
  1. Chen, J.-B., 2006. Modeling the scalar wave equation with Nystrém methods. Geophysics, 71:151-158.
  2. Chen, J.-B., 2007. High-order time discretizations in seismic modeling. Geophysics, 72: 115-122.
  3. Chu, C., 2009. Seismic modeling and imaging with the Fourier method: Numerical analyses andparallel implementation strategies. Ph.D. thesis, The University of Texas at Austin.
  4. Chu, C., Stoffa, P.L. and Seif, R., 2009. 3D elastic wave modeling using modified high-order timestepping schemes with improved stability conditions. Expanded Abstr., 79th Ann. Internat.SEG Mtg., Houston, 28: 2662-2666.
  5. Crase, E., 1990. High-order (space and time) finite-difference modeling of the elastic wave equation.
  6. Expanded Abstr., 60th Ann. Internat. SEG Mtg., San Francisco, 9: 987-991.
  7. Dablain, M.A., 1986. The application of high-order differencing to the scalar wave equation.Geophysics, 51: 54-66.
  8. Etgen, J.T., 1989. Accurate wave-equation modeling. Technical Report, Stanford ExplorationProject, 60: 131-147.
  9. Etgen, J.T. and S. Brandsberg-Dahl, 2009. The pseudo-analytical method: Application ofpseudo-Laplacian to acoustic and acoustic anisotropic wave propagation. Expanded Abstr.,79th Ann. Internat. SEG Mtg., Houston, 28: 2552-2556.
  10. Ghrist, M., Fornberg, B. and Driscoll, T.A., 2000. Staggered time integrators for wave equations.SIAM J. Numer. Analys., 38: 718-741.
  11. Kole, J.S., 2003. Solving seismic wave propagation in elastic media using the matrix exponentialapproach. Wave Motion, 38: 279-293.
  12. Kosloff, D., Filho, A.Q., Tessmer, E. and Behle, A., 1989. Numerical solution of the acoustic andelastic wave equations by a new rapid expansion method. Geophys. Prosp., 37: 383-394.
  13. Pestana, R.C. and Stoffa, P.L., 2010. Time evolution of wave equation using rapid expansionmethod REM. Geophysics, 75: T121-T131.
  14. Song, X., Fomel, S. and Ying, L., 2010. Fourier finite difference and low rank approximation forreverse-time migration. SEG 2010 Summer Research Workshop.
  15. Soubaras, R. and Zhang, Y., 2008. Two-step explicit marching method for reverse time migration.
  16. Expanded Abstr., 78th Ann. Internat. SEG Mtg., Las Vegas, 27: 2272-2276.
  17. Tal-Ezer, H., Kosloff, D. and Koren, Z., 1987. An accurate scheme for forward seismic modeling.Geophys. Prosp., 35: 479-490.
  18. Zhang, Y. and Zhang, G., 2009. One-step extrapolation method for reverse time migration.Geophysics, 74: 29-33.
  19. Zhang, Y., Zhang, G., Yingst, D. and Sun, J., 2007. Explicit marching method for reverse timemigration. Expanded Abstr., 77th Ann. Internat. SEG Mtg., San Antonio, 26: 2300-2304.
Share
Back to top
Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing