1887
Volume 42, Issue 3
  • ISSN: 0812-3985
  • E-ISSN: 1834-7533

Abstract

Abstract

Most conventional finite-difference methods adopt second-order temporal and (2)th-order spatial finite-difference stencils to solve the 3D acoustic wave equation. When spatial finite-difference stencils devised from the time-space domain dispersion relation are used to replace these conventional spatial finite-difference stencils devised from the space domain dispersion relation, the accuracy of modelling can be increased from second-order along any directions to (2)th-order along 48 directions. In addition, the conventional high-order spatial finite-difference modelling accuracy can be improved by using a truncated finite-difference scheme. In this paper, we combine the time-space domain dispersion-relation-based finite difference scheme and the truncated finite-difference scheme to obtain optimised spatial finite-difference coefficients and thus to significantly improve the modelling accuracy without increasing computational cost, compared with the conventional space domain dispersion-relation-based finite difference scheme. We developed absorbing boundary conditions for the 3D acoustic wave equation, based on predicting wavefield values in a transition area by weighing wavefield values from wave equations and one-way wave equations.

Dispersion analyses demonstrate that high-order spatial finite-difference stencils have greater accuracy than low-order spatial finite-difference stencils for high frequency components of wavefields, and spatial finite-difference stencils devised in the time-space domain have greater precision than those devised in the space domain under the same discretisation. The modelling accuracy can be improved further by using the truncated spatial finite-difference stencils. Stability analyses show that spatial finite-difference stencils devised in the time-space domain have better stability condition. Numerical modelling experiments for homogeneous, horizontally layered and Society of Exploration Geophysicists/European Association of Geoscientists and Engineers salt models demonstrate that this modelling scheme has greater accuracy than a conventional scheme and has better absorbing effects than Clayton-Engquist absorbing boundary conditions.

© 2011 ASEG
Loading

Article metrics loading...

/content/journals/10.1071/EG11007
2011-09-01
2026-01-23

  • From This Site

    /content/journals/10.1071/EG11007
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. Abokhodair A. A. 2009Complex differentiation tools for geophysical inversion: Geophysics 74 H1 H11 10.1190/1.3052111
     https://doi.org/10.1190/1.3052111 [Google Scholar]
  2. Bansal R. Sen M. K. 2008Finite-difference modelling of S-wave splitting in anisotropic media: Geophysical Prospecting 56 293 312 10.1111/j.1365‑2478.2007.00693.x
     https://doi.org/10.1111/j.1365-2478.2007.00693.x [Google Scholar]
  3. Bérenger J. P. 1994A perfectly matched layer for the absorption of electromagnetic waves: Journal of Computational Physics 114 185 200 10.1006/jcph.1994.1159
     https://doi.org/10.1006/jcph.1994.1159 [Google Scholar]
  4. Bohlen T. Saenger E. H. 2006Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves: Geophysics 71 T109 T115 10.1190/1.2213051
     https://doi.org/10.1190/1.2213051 [Google Scholar]
  5. Cao S. Greenhalgh S. 1998Attenuating boundary conditions for numerical modeling of acoustic wave propagation: Geophysics 63 231 243 10.1190/1.1444317
     https://doi.org/10.1190/1.1444317 [Google Scholar]
  6. Cerjan C. Kosloff D. Kosloff R. Resef M. 1985A nonreflecting boundary condition for discrete acoustic and elastic wave equations: Geophysics 50 705 708 10.1190/1.1441945
     https://doi.org/10.1190/1.1441945 [Google Scholar]
  7. Clayton R. W. Engquist B. 1977Absorbing boundary conditions for acoustic and elastic wave equations: Bulletin of the Seismological Society of America 6 1529 1540
     [Google Scholar]
  8. Dablain M. A. 1986The application of high-order differencing to the scalar wave equation: Geophysics 51 54 66 10.1190/1.1442040
     https://doi.org/10.1190/1.1442040 [Google Scholar]
  9. Emerman S. Schmidt W. Stephen R. 1982An implicit finite-difference formulation of the elastic wave equation: Geophysics 47 1521 1526 10.1190/1.1441302
     https://doi.org/10.1190/1.1441302 [Google Scholar]
  10. Engquist B. Majda A. 1977Absorbing boundary conditions for numerical simulation of waves: Mathematics of Computation 31 629 651 10.1090/S0025‑5718‑1977‑0436612‑4
     https://doi.org/10.1090/S0025-5718-1977-0436612-4 [Google Scholar]
  11. Etgen J. T. O’Brien M. J. 2007Computational methods for large-scale 3D acoustic finite-difference modeling: A tutorial: Geophysics 72 SM223 SM230 10.1190/1.2753753
     https://doi.org/10.1190/1.2753753 [Google Scholar]
  12. Falk J. Tessmer E. Gajewski D. 1998Efficient finite-difference modelling of seismic waves using locally adjustable time step sizes: Geophysical Prospecting 46 603 616 10.1046/j.1365‑2478.1998.00110.x
     https://doi.org/10.1046/j.1365-2478.1998.00110.x [Google Scholar]
  13. Fei T. Liner C. L. 2008Hybrid Fourier finite difference 3D depth migration for anisotropic media: Geophysics 73 S27 S34 10.1190/1.2828704
     https://doi.org/10.1190/1.2828704 [Google Scholar]
  14. Finkelstein B. Kastner R. 2007Finite difference time domain dispersion reduction schemes: Journal of Computational Physics 221 422 438 10.1016/j.jcp.2006.06.016
     https://doi.org/10.1016/j.jcp.2006.06.016 [Google Scholar]
  15. Finkelstein B. Kastner R. 2008A comprehensive new methodology for formulating FDTD schemes with controlled order of accuracy and dispersion: IEEE Transactions on Antennas and Propagation 56 3516 3525 10.1109/TAP.2008.2005458
     https://doi.org/10.1109/TAP.2008.2005458 [Google Scholar]
  16. Gao H. Zhang J. 2008Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modeling: Geophysical Journal International 174 1029 1036 10.1111/j.1365‑246X.2008.03883.x
     https://doi.org/10.1111/j.1365-246X.2008.03883.x [Google Scholar]
  17. Graves R. W. 1996Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences: Bulletin of the Seismological Society of America 86 1091 1106
     [Google Scholar]
  18. Hayashi K. Burns D. R. 1999Variable grid finite-difference modeling including surface topography: Society of Exploration Geophysics Expanded Abstracts 18 523 527
     [Google Scholar]
  19. Heidari A. H. Guddati M. N. 2006Highly accurate absorbing boundary conditions for wide-angle wave equations: Geophysics 71 S85 S97 10.1190/1.2192914
     https://doi.org/10.1190/1.2192914 [Google Scholar]
  20. Hestholm S. 2009Acoustic VTI modeling using high-order finite differences: Geophysics 74 T67 T73 10.1190/1.3157242
     https://doi.org/10.1190/1.3157242 [Google Scholar]
  21. Holberg O. 1987Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena: Geophysical Prospecting 35 629 655 10.1111/j.1365‑2478.1987.tb00841.x
     https://doi.org/10.1111/j.1365-2478.1987.tb00841.x [Google Scholar]
  22. Hu W. Abubakar A. Habashy T. M. 2007Application of the nearly perfectly matched layer in acoustic wave modeling: Geophysics 72 SM169 SM175 10.1190/1.2738553
     https://doi.org/10.1190/1.2738553 [Google Scholar]
  23. Igel H. Mora P. Riollet B. 1995Anisotropic wave propagation through finite-difference grids: Geophysics 60 1203 1216 10.1190/1.1443849
     https://doi.org/10.1190/1.1443849 [Google Scholar]
  24. Kelly K. R. Ward R. Treitel W. S. Alford R. M. 1976Synthetic seismograms: A finite-difference approach: Geophysics 41 2 27 10.1190/1.1440605
     https://doi.org/10.1190/1.1440605 [Google Scholar]
  25. Keys R. G. 1985Absorbing boundary conditions for acoustic media: Geophysics 50 892 902 10.1190/1.1441969
     https://doi.org/10.1190/1.1441969 [Google Scholar]
  26. Komatitsch D. Tromp J. 2003A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation: Geophysical Journal International 154 146 153 10.1046/j.1365‑246X.2003.01950.x
     https://doi.org/10.1046/j.1365-246X.2003.01950.x [Google Scholar]
  27. Kosloff R. Kosloff D. 1986Absorbing boundaries for wave propagation problems: Journal of Computational Physics 63 363 376 10.1016/0021‑9991(86)90199‑3
     https://doi.org/10.1016/0021-9991(86)90199-3 [Google Scholar]
  28. Krüger O. S. Saenger E. H. Shapiro S. 2005Scattering and diffraction by a single crack: An accuracy analysis of the rotated staggered grid: Geophysical Journal International 162 25 31 10.1111/j.1365‑246X.2005.02647.x
     https://doi.org/10.1111/j.1365-246X.2005.02647.x [Google Scholar]
  29. Larner K. Beasley C. 1987Cascaded migrations – improving the accuracy of finite-difference migration: Geophysics 52 618 643 10.1190/1.1442331
     https://doi.org/10.1190/1.1442331 [Google Scholar]
  30. Lele S. K. 1992Compact finite difference schemes with spectral-like resolution: Journal of Computational Physics 103 16 42 10.1016/0021‑9991(92)90324‑R
     https://doi.org/10.1016/0021-9991(92)90324-R [Google Scholar]
  31. Li Z. 1991Compensating finite-difference errors in 3-D migration and modeling: Geophysics 56 1650 1660 10.1190/1.1442975
     https://doi.org/10.1190/1.1442975 [Google Scholar]
  32. Liao Z. Wong H. Yang B. Yuan Y. 1984A transmitting boundary for transient wave analysis: Scientia Sinica Series A 27 1063 1076
     [Google Scholar]
  33. Liu Y. Sen M. K. 2009 a A practical implicit finite-difference method: examples from seismic modeling: Journal of Geophysics and Engineering 6 231 249 10.1088/1742‑2132/6/3/003
     https://doi.org/10.1088/1742-2132/6/3/003 [Google Scholar]
  34. Liu Y. Sen M. K. 2009 b An implicit staggered-grid finite-difference method for seismic modeling: Geophysical Journal International 179 459 474 10.1111/j.1365‑246X.2009.04305.x
     https://doi.org/10.1111/j.1365-246X.2009.04305.x [Google Scholar]
  35. Liu Y. Sen M. K. 2009 c A new time-space domain high-order finite-difference method for the acoustic wave equation: Journal of Computational Physics 228 8779 8806 10.1016/j.jcp.2009.08.027
     https://doi.org/10.1016/j.jcp.2009.08.027 [Google Scholar]
  36. Liu Y. Sen M. K. 2009 d Numerical modeling of wave equation by a truncated high-order finite-difference method: Earth Science 22 205 213 10.1007/s11589‑009‑0205‑0
     https://doi.org/10.1007/s11589-009-0205-0 [Google Scholar]
  37. Liu Y. Sen M. K. 2010 a A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation: Geophysics 75 A1 A6 10.1190/1.3295447
     https://doi.org/10.1190/1.3295447 [Google Scholar]
  38. Liu Y. Sen M. K. 2010 b Acoustic VTI modeling with a time-space domain dispersion-relation-based finite-difference scheme: Geophysics 75 A11 A17 10.1190/1.3374477
     https://doi.org/10.1190/1.3374477 [Google Scholar]
  39. Madariaga R. 1976Dynamics of an expanding circular fault: Bulletin of the Seismological Society of America 66 639 666
     [Google Scholar]
  40. Opršal I. Zahradník J. 1999Elastic finite-difference method for irregular grids: Geophysics 64 240 250 10.1190/1.1444520
     https://doi.org/10.1190/1.1444520 [Google Scholar]
  41. Pratt R. G. Shin C. Hicks G. J. 1998Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion: Geophysical Journal International 133 341 362 10.1046/j.1365‑246X.1998.00498.x
     https://doi.org/10.1046/j.1365-246X.1998.00498.x [Google Scholar]
  42. Ravaut C. Operto S. Improta L. Virieux J. Herrero A. Dell’Aversana P. 2004Multiscale imaging of complex structures from multifold wide-aperture seismic data by frequency-domain full-waveform tomography: application to a thrust belt: Geophysical Journal International 159 1032 1056 10.1111/j.1365‑246X.2004.02442.x
     https://doi.org/10.1111/j.1365-246X.2004.02442.x [Google Scholar]
  43. Reynolds A. C. 1978Boundary conditions for the numerical solution of wave propagation problems: Geophysics 43 1099 1110 10.1190/1.1440881
     https://doi.org/10.1190/1.1440881 [Google Scholar]
  44. Ristow D. Ruhl T. 1994Fourier finite-difference migration: Geophysics 59 1882 1893 10.1190/1.1443575
     https://doi.org/10.1190/1.1443575 [Google Scholar]
  45. Ristow D. Ruhl T. 19973-D implicit finite-difference migration by multiway splitting: Geophysics 62 554 567 10.1190/1.1444165
     https://doi.org/10.1190/1.1444165 [Google Scholar]
  46. Robertsson J. O. A. 1996A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography: Geophysics 61 1921 1934 10.1190/1.1444107
     https://doi.org/10.1190/1.1444107 [Google Scholar]
  47. Robertsson J. O. A. Blanch J. Symes W. 1994Viscoelastic finite-difference modeling: Geophysics 59 1444 1456 10.1190/1.1443701
     https://doi.org/10.1190/1.1443701 [Google Scholar]
  48. Saenger E. H. Gold N. Shapiro S. A. 2000Modeling the propagation of elastic waves using a modified finite-difference grid: Wave Motion 31 77 92 10.1016/S0165‑2125(99)00023‑2
     https://doi.org/10.1016/S0165-2125(99)00023-2 [Google Scholar]
  49. Tessmer E. 2000Seismic finite-difference modeling with spatially varying time steps: Geophysics 65 1290 1293 10.1190/1.1444820
     https://doi.org/10.1190/1.1444820 [Google Scholar]
  50. Tian X. B. Kang I. B. Kim G. Y. Zhang H. S. 2008An improvement in the absorbing boundary technique for numerical simulation of elastic wave propagation: Journal of Geophysics and Engineering 5 203 209 10.1088/1742‑2132/5/2/007
     https://doi.org/10.1088/1742-2132/5/2/007 [Google Scholar]
  51. Virieux J. 1984SH-wave propagation in heterogeneous media: velocity-stress finite-difference method: Geophysics 49 1933 1957 10.1190/1.1441605
     https://doi.org/10.1190/1.1441605 [Google Scholar]
  52. Virieux J. 1986P-SV wave propagation in heterogeneous media: Velocity stress finite difference method: Geophysics 51 889 901 10.1190/1.1442147
     https://doi.org/10.1190/1.1442147 [Google Scholar]
  53. Wang Y. Schuster G. T. 1996Finite-difference variable grid scheme for acoustic and elastic wave equation modeling: Society of Exploration Geophysics Expanded Abstracts 15 674 677
     [Google Scholar]
  54. Wang T. Tang X. 2003Finite-difference modeling of elastic wave propagation: A nonsplitting perfectly matched layer approach: Geophysics 68 1749 1755 10.1190/1.1620648
     https://doi.org/10.1190/1.1620648 [Google Scholar]
  55. Zeng Y. He J. Liu Q. 2001The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media: Geophysics 66 1258 1266 10.1190/1.1487073
     https://doi.org/10.1190/1.1487073 [Google Scholar]
  56. Zhang W. Chen X. 2006Traction image method for irregular free surface boundaries in finite difference seismic wave simulation: Geophysical Journal International 167 337 353 10.1111/j.1365‑246X.2006.03113.x
     https://doi.org/10.1111/j.1365-246X.2006.03113.x [Google Scholar]
  57. Zhang G. Zhang Y. Zhou H. 2000Helical finite-difference schemes for 3-D depth migration: Society of Exploration Geophysics Expanded Abstracts 19 862 865
     [Google Scholar]
  58. Zhou B. Greenhalgh S. A. 1992Seismic scalar wave equation modelling by a convolutional differentiator: Bulletin of the Seismological Society of America 82 289 303
     [Google Scholar]
  59. Zhou H. B. McMechan G. A. 2000Rigorous absorbing boundary conditions for 3-D one-way wave extrapolation: Geophysics 65 638 645 10.1190/1.1444760
     https://doi.org/10.1190/1.1444760 [Google Scholar]
/content/journals/10.1071/EG11007
Loading
3D acoustic wave modelling with time-space domain dispersion-relation-based finite-difference schemes and hybrid absorbing boundary conditions
Exploration Geophysics 42, 176 (2011); https://doi.org/10.1071/EG11007
/content/journals/10.1071/EG11007
/content/journals/10.1071/EG11007
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): 3D acoustic wave equation; absorbing boundary conditions; dispersion-relation-based finite difference; time-space domain; truncated finite difference

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error