3D acoustic wave modelling with time-space domain dispersion-relation-based finite-difference schemes and hybrid absorbing boundary conditions
Yang Liu 1 3 Mrinal K. Sen 2
+ Author Affiliations
- Author Affiliations
1 State Key Laboratory of Petroleum Resources and Prospecting (China University of Petroleum, Beijing), Beijing, 102249, China.
2 The Institute for Geophysics, John A. and Katherine G. Jackson School of Geosciences, The University of Texas, 10100 Burnet Road, R2200 Austin, TX 78758, USA.
3 Corresponding author. Email: wliuyang@vip.sina.com
Exploration Geophysics 42(3) 176-189 https://doi.org/10.1071/EG11007
Submitted: 23 January 2011 Accepted: 7 July 2011 Published: 2 September 2011
Abstract
Most conventional finite-difference methods adopt second-order temporal and (2M)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 (2M)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.
Key words: absorbing boundary conditions, dispersion-relation-based finite difference, 3D acoustic wave equation, time-space domain, truncated finite difference.
References
Abokhodair, A. A., 2009, Complex differentiation tools for geophysical inversion: Geophysics, 74, H1–H11| Complex differentiation tools for geophysical inversion:Crossref | GoogleScholarGoogle Scholar |
Bansal, R., and Sen, M. K., 2008, Finite-difference modelling of S-wave splitting in anisotropic media: Geophysical Prospecting, 56, 293–312
| Finite-difference modelling of S-wave splitting in anisotropic media:Crossref | GoogleScholarGoogle Scholar |
Bérenger, J. P., 1994, A perfectly matched layer for the absorption of electromagnetic waves: Journal of Computational Physics, 114, 185–200
| A perfectly matched layer for the absorption of electromagnetic waves:Crossref | GoogleScholarGoogle Scholar |
Bohlen, T., and Saenger, E. H., 2006, Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves: Geophysics, 71, T109–T115
| Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves:Crossref | GoogleScholarGoogle Scholar |
Cao, S., and Greenhalgh, S., 1998, Attenuating boundary conditions for numerical modeling of acoustic wave propagation: Geophysics, 63, 231–243
| Attenuating boundary conditions for numerical modeling of acoustic wave propagation:Crossref | GoogleScholarGoogle Scholar |
Cerjan, C., Kosloff, D., Kosloff, R., and Resef, M., 1985, A nonreflecting boundary condition for discrete acoustic and elastic wave equations: Geophysics, 50, 705–708
| A nonreflecting boundary condition for discrete acoustic and elastic wave equations:Crossref | GoogleScholarGoogle Scholar |
Clayton, R. W., and Engquist, B., 1977, Absorbing boundary conditions for acoustic and elastic wave equations: Bulletin of the Seismological Society of America, 6, 1529–1540
Dablain, M. A., 1986, The application of high-order differencing to the scalar wave equation: Geophysics, 51, 54–66
| The application of high-order differencing to the scalar wave equation:Crossref | GoogleScholarGoogle Scholar |
Emerman, S., Schmidt, W., and Stephen, R., 1982, An implicit finite-difference formulation of the elastic wave equation: Geophysics, 47, 1521–1526
| An implicit finite-difference formulation of the elastic wave equation:Crossref | GoogleScholarGoogle Scholar |
Engquist, B., and Majda, A., 1977, Absorbing boundary conditions for numerical simulation of waves: Mathematics of Computation, 31, 629–651
| Absorbing boundary conditions for numerical simulation of waves:Crossref | GoogleScholarGoogle Scholar |
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
| Computational methods for large-scale 3D acoustic finite-difference modeling: A tutorial:Crossref | GoogleScholarGoogle Scholar |
Falk, J., Tessmer, E., and Gajewski, D., 1998, Efficient finite-difference modelling of seismic waves using locally adjustable time step sizes: Geophysical Prospecting, 46, 603–616
| Efficient finite-difference modelling of seismic waves using locally adjustable time step sizes:Crossref | GoogleScholarGoogle Scholar |
Fei, T., and Liner, C. L., 2008, Hybrid Fourier finite difference 3D depth migration for anisotropic media: Geophysics, 73, S27–S34
| Hybrid Fourier finite difference 3D depth migration for anisotropic media:Crossref | GoogleScholarGoogle Scholar |
Finkelstein, B., and Kastner, R., 2007, Finite difference time domain dispersion reduction schemes: Journal of Computational Physics, 221, 422–438
| Finite difference time domain dispersion reduction schemes:Crossref | GoogleScholarGoogle Scholar |
Finkelstein, B., and Kastner, R., 2008, A comprehensive new methodology for formulating FDTD schemes with controlled order of accuracy and dispersion: IEEE Transactions on Antennas and Propagation, 56, 3516–3525
| A comprehensive new methodology for formulating FDTD schemes with controlled order of accuracy and dispersion:Crossref | GoogleScholarGoogle Scholar |
Gao, H., and Zhang, J., 2008, Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modeling: Geophysical Journal International, 174, 1029–1036
| Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modeling:Crossref | GoogleScholarGoogle Scholar |
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
Hayashi, K., and Burns, D. R., 1999, Variable grid finite-difference modeling including surface topography: Society of Exploration Geophysics Expanded Abstracts, 18, 523–527
Heidari, A. H., and Guddati, M. N., 2006, Highly accurate absorbing boundary conditions for wide-angle wave equations: Geophysics, 71, S85–S97
| Highly accurate absorbing boundary conditions for wide-angle wave equations:Crossref | GoogleScholarGoogle Scholar |
Hestholm, S., 2009, Acoustic VTI modeling using high-order finite differences: Geophysics, 74, T67–T73
| Acoustic VTI modeling using high-order finite differences:Crossref | GoogleScholarGoogle Scholar |
Holberg, O., 1987, Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena: Geophysical Prospecting, 35, 629–655
| Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena:Crossref | GoogleScholarGoogle Scholar |
Hu, W., Abubakar, A., and Habashy, T. M., 2007, Application of the nearly perfectly matched layer in acoustic wave modeling: Geophysics, 72, SM169–SM175
| Application of the nearly perfectly matched layer in acoustic wave modeling:Crossref | GoogleScholarGoogle Scholar |
Igel, H., Mora, P., and Riollet, B., 1995, Anisotropic wave propagation through finite-difference grids: Geophysics, 60, 1203–1216
| Anisotropic wave propagation through finite-difference grids:Crossref | GoogleScholarGoogle Scholar |
Kelly, K. R., Ward, R., Treitel, W. S., and Alford, R. M., 1976, Synthetic seismograms: A finite-difference approach: Geophysics, 41, 2–27
| Synthetic seismograms: A finite-difference approach:Crossref | GoogleScholarGoogle Scholar |
Keys, R. G., 1985, Absorbing boundary conditions for acoustic media: Geophysics, 50, 892–902
| Absorbing boundary conditions for acoustic media:Crossref | GoogleScholarGoogle Scholar |
Komatitsch, D., and Tromp, J., 2003, A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation: Geophysical Journal International, 154, 146–153
| A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation:Crossref | GoogleScholarGoogle Scholar |
Kosloff, R., and Kosloff, D., 1986, Absorbing boundaries for wave propagation problems: Journal of Computational Physics, 63, 363–376
| Absorbing boundaries for wave propagation problems:Crossref | GoogleScholarGoogle Scholar |
Krüger, O. S., Saenger, E. H., and Shapiro, S., 2005, Scattering and diffraction by a single crack: An accuracy analysis of the rotated staggered grid: Geophysical Journal International, 162, 25–31
| Scattering and diffraction by a single crack: An accuracy analysis of the rotated staggered grid:Crossref | GoogleScholarGoogle Scholar |
Larner, K., and Beasley, C., 1987, Cascaded migrations – improving the accuracy of finite-difference migration: Geophysics, 52, 618–643
| Cascaded migrations – improving the accuracy of finite-difference migration:Crossref | GoogleScholarGoogle Scholar |
Lele, S. K., 1992, Compact finite difference schemes with spectral-like resolution: Journal of Computational Physics, 103, 16–42
| Compact finite difference schemes with spectral-like resolution:Crossref | GoogleScholarGoogle Scholar |
Li, Z., 1991, Compensating finite-difference errors in 3-D migration and modeling: Geophysics, 56, 1650–1660
| Compensating finite-difference errors in 3-D migration and modeling:Crossref | GoogleScholarGoogle Scholar |
Liao, Z., Wong, H., Yang, B., and Yuan, Y., 1984, A transmitting boundary for transient wave analysis: Scientia Sinica Series A, 27, 1063–1076
Liu, Y., and Sen, M. K., 2009a, A practical implicit finite-difference method: examples from seismic modeling: Journal of Geophysics and Engineering, 6, 231–249
| A practical implicit finite-difference method: examples from seismic modeling:Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DC%2BD1MXivFeitbk%3D&md5=7b9bbf854344aef0588beaad100b7e55CAS |
Liu, Y., and Sen, M. K., 2009b, An implicit staggered-grid finite-difference method for seismic modeling: Geophysical Journal International, 179, 459–474
| An implicit staggered-grid finite-difference method for seismic modeling:Crossref | GoogleScholarGoogle Scholar |
Liu, Y., and Sen, M. K., 2009c, A new time-space domain high-order finite-difference method for the acoustic wave equation: Journal of Computational Physics, 228, 8779–8806
| A new time-space domain high-order finite-difference method for the acoustic wave equation:Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DC%2BD1MXht1GnsLvL&md5=5dc143a2309b04c18dc70a132c7b7ea9CAS |
Liu, Y., and Sen, M. K., 2009d, Numerical modeling of wave equation by a truncated high-order finite-difference method: Earth Science, 22, 205–213
| Numerical modeling of wave equation by a truncated high-order finite-difference method:Crossref | GoogleScholarGoogle Scholar |
Liu, Y., and Sen, M. K., 2010a, A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation: Geophysics, 75, A1–A6
| A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation:Crossref | GoogleScholarGoogle Scholar |
Liu, Y., and Sen, M. K., 2010b, Acoustic VTI modeling with a time-space domain dispersion-relation-based finite-difference scheme: Geophysics, 75, A11–A17
| Acoustic VTI modeling with a time-space domain dispersion-relation-based finite-difference scheme:Crossref | GoogleScholarGoogle Scholar |
Madariaga, R., 1976, Dynamics of an expanding circular fault: Bulletin of the Seismological Society of America, 66, 639–666
Opršal, I., and Zahradník, J., 1999, Elastic finite-difference method for irregular grids: Geophysics, 64, 240–250
| Elastic finite-difference method for irregular grids:Crossref | GoogleScholarGoogle Scholar |
Pratt, R. G., Shin, C., and Hicks, G. J., 1998, Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion: Geophysical Journal International, 133, 341–362
| Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion:Crossref | GoogleScholarGoogle Scholar |
Ravaut, C., Operto, S., Improta, L., Virieux, J., Herrero, A., and Dell’Aversana, P., 2004, Multiscale 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
| Multiscale imaging of complex structures from multifold wide-aperture seismic data by frequency-domain full-waveform tomography: application to a thrust belt:Crossref | GoogleScholarGoogle Scholar |
Reynolds, A. C., 1978, Boundary conditions for the numerical solution of wave propagation problems: Geophysics, 43, 1099–1110
| Boundary conditions for the numerical solution of wave propagation problems:Crossref | GoogleScholarGoogle Scholar |
Ristow, D., and Ruhl, T., 1994, Fourier finite-difference migration: Geophysics, 59, 1882–1893
| Fourier finite-difference migration:Crossref | GoogleScholarGoogle Scholar |
Ristow, D., and Ruhl, T., 1997, 3-D implicit finite-difference migration by multiway splitting: Geophysics, 62, 554–567
| 3-D implicit finite-difference migration by multiway splitting:Crossref | GoogleScholarGoogle Scholar |
Robertsson, J. O. A., 1996, A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography: Geophysics, 61, 1921–1934
| A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography:Crossref | GoogleScholarGoogle Scholar |
Robertsson, J. O. A., Blanch, J., and Symes, W., 1994, Viscoelastic finite-difference modeling: Geophysics, 59, 1444–1456
| Viscoelastic finite-difference modeling:Crossref | GoogleScholarGoogle Scholar |
Saenger, E. H., Gold, N., and Shapiro, S. A., 2000, Modeling the propagation of elastic waves using a modified finite-difference grid: Wave Motion, 31, 77–92
| Modeling the propagation of elastic waves using a modified finite-difference grid:Crossref | GoogleScholarGoogle Scholar |
Tessmer, E., 2000, Seismic finite-difference modeling with spatially varying time steps: Geophysics, 65, 1290–1293
| Seismic finite-difference modeling with spatially varying time steps:Crossref | GoogleScholarGoogle Scholar |
Tian, X. B., Kang, I. B., Kim, G. Y., and Zhang, H. S., 2008, An improvement in the absorbing boundary technique for numerical simulation of elastic wave propagation: Journal of Geophysics and Engineering, 5, 203–209
| An improvement in the absorbing boundary technique for numerical simulation of elastic wave propagation:Crossref | GoogleScholarGoogle Scholar |
Virieux, J., 1984, SH-wave propagation in heterogeneous media: velocity-stress finite-difference method: Geophysics, 49, 1933–1957
| SH-wave propagation in heterogeneous media: velocity-stress finite-difference method:Crossref | GoogleScholarGoogle Scholar |
Virieux, J., 1986, P-SV wave propagation in heterogeneous media: Velocity stress finite difference method: Geophysics, 51, 889–901
| P-SV wave propagation in heterogeneous media: Velocity stress finite difference method:Crossref | GoogleScholarGoogle Scholar |
Wang, Y., and Schuster, G. T., 1996, Finite-difference variable grid scheme for acoustic and elastic wave equation modeling: Society of Exploration Geophysics Expanded Abstracts, 15, 674–677
Wang, T., and Tang, X., 2003, Finite-difference modeling of elastic wave propagation: A nonsplitting perfectly matched layer approach: Geophysics, 68, 1749–1755
| Finite-difference modeling of elastic wave propagation: A nonsplitting perfectly matched layer approach:Crossref | GoogleScholarGoogle Scholar |
Zeng, Y., He, J., and Liu, Q., 2001, The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media: Geophysics, 66, 1258–1266
| The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media:Crossref | GoogleScholarGoogle Scholar |
Zhang, W., and Chen, X., 2006, Traction image method for irregular free surface boundaries in finite difference seismic wave simulation: Geophysical Journal International, 167, 337–353
| Traction image method for irregular free surface boundaries in finite difference seismic wave simulation:Crossref | GoogleScholarGoogle Scholar |
Zhang, G., Zhang, Y., and Zhou, H., 2000, Helical finite-difference schemes for 3-D depth migration: Society of Exploration Geophysics Expanded Abstracts, 19, 862–865
Zhou, B., and Greenhalgh, S. A., 1992, Seismic scalar wave equation modelling by a convolutional differentiator: Bulletin of the Seismological Society of America, 82, 289–303
Zhou, H. B., and McMechan, G. A., 2000, Rigorous absorbing boundary conditions for 3-D one-way wave extrapolation: Geophysics, 65, 638–645
| Rigorous absorbing boundary conditions for 3-D one-way wave extrapolation:Crossref | GoogleScholarGoogle Scholar |