Perfectly matched layer on curvilinear grid for the second-order seismic acoustic wave equation
Sanyi Yuan 1 Shangxu Wang 1 4 Wenju Sun 1 Lina Miao 2 Zhenhua Li 3
+ Author Affiliations
- Author Affiliations
1 State Key Laboratory of Petroleum Resource and Prospecting, CNPC Key Laboratory of Geophysical Exploration, China University of Petroleum, Beijing 102249, China.
2 Department of Earth, Ocean and Atmospheric Sciences, The University of British Columbia, Vancouver V6T 1Z4, Canada.
3 Department of Physics, University of Alberta, CEB, Edmonton, Alberta T6G 2E1, Canada.
4 Corresponding author. Email: wangsx@cup.edu.cn
Exploration Geophysics 45(2) 94-104 https://doi.org/10.1071/EG13066
Submitted: 4 August 2013 Accepted: 30 January 2014 Published: 11 March 2014
Abstract
A curvilinear-grid perfectly matched layer (PML) absorbing boundary condition for the second-order seismic acoustic wave equation is presented in this paper. The rectangular grids are transformed into curvilinear grids by using a mathematical mapping to fit the curvilinear boundary, and the original wave equation is reformulated under the curvilinear coordinate system. Based on the reformulated wave equation, theoretical expressions and analysis of the curvilinear-grid PML are given. Furthermore, PML model 1 with symmetric form and PML model 2 with asymmetric form are derived from the same acoustic wave equation. By combination with the finite difference (FD) method, these two models are applied to seismic wave modelling with surface topography. The results show that the absorption effect of these two models discretised by the same second-order time difference and second-order space difference are different, and the symmetric-form PML yields better modelling results than the asymmetric-form.
Key words: acoustic wave equation, finite difference, perfectly matched layer, seismic modeling, surface topography.
References
Appelö, D., and Kreiss, G., 2006, A new absorbing layer for elastic waves: Journal of Computational Physics, 215, 642–660| A new absorbing layer for elastic waves:Crossref | GoogleScholarGoogle Scholar |
Appelö, D., and Petersson, N. A., 2009, A stable finite difference method for the elastic wave equation on complex geometries with free surfaces: Communications in Computational Physics, 5, 84–107
Basu, U., 2009, Explicit finite element perfectly matched layer for transient three-dimensional elastic waves: International Journal for Numerical Methods in Engineering, 77, 151–176
| Explicit finite element perfectly matched layer for transient three-dimensional elastic waves: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 |
Bérenger, J. P., 1996, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves: Journal of Computational Physics, 127, 363–379
| Three-dimensional perfectly matched layer for the absorption of electromagnetic waves:Crossref | GoogleScholarGoogle Scholar |
Chen, J. Y., 2011, Application of the nearly perfectly matched layer for seismic wave propagation in 2D homogeneous isotropic media: Geophysical Prospecting, 59, 662–672
| Application of the nearly perfectly matched layer for seismic wave propagation in 2D homogeneous isotropic media:Crossref | GoogleScholarGoogle Scholar |
Chew, W., and Liu, Q., 1996, Perfectly matched layers for elastodynamics: a new absorbing boundary condition: Journal of Computational Acoustics, 4, 341–359
| Perfectly matched layers for elastodynamics: a new absorbing boundary condition:Crossref | GoogleScholarGoogle Scholar |
Chew, W. C., and Weedon, W. H., 1994, A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates: Microwave and Optical Technology Letters, 7, 599–604
| A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates:Crossref | GoogleScholarGoogle Scholar |
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
| Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media:Crossref | GoogleScholarGoogle Scholar |
Cummer, S. A., 2003, A simple, nearly perfectly matched layer for general electromagnetic media: IEEE Microwave and Wireless Components Letters, 13, 128–130
| A simple, nearly perfectly matched layer for general electromagnetic media:Crossref | GoogleScholarGoogle Scholar |
Festa, G., and Vilotte, J. P., 2005, The Newmark scheme as velocity–stress time-staggering: an effcient PML implementation for spectral element simulation of elastodynamics: Geophysical Journal International, 161, 789–812
| The Newmark scheme as velocity–stress time-staggering: an effcient PML implementation for spectral element simulation of elastodynamics:Crossref | GoogleScholarGoogle Scholar |
Fornberg, B., 1988, The pseudospectral method: accurate representation of interfaces in elastic wave calculations: Geophysics, 53, 625–637
| The pseudospectral method: accurate representation of interfaces in elastic wave calculations:Crossref | GoogleScholarGoogle Scholar |
Gao, H., and Zhang, J., 2008, Implementation of perfectly matched layers in an arbitrary geometric boundary for elastic wave modeling: Geophysical Journal International, 174, 1029–1036
| Implementation of perfectly matched layers in an arbitrary geometric boundary for elastic wave modeling:Crossref | GoogleScholarGoogle Scholar |
Hastings, F. D., Schneider, J. B., and Broschat, S. L., 1996, Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation: The Journal of the Acoustical Society of America, 100, 3061–3069
| Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation:Crossref | GoogleScholarGoogle Scholar |
Hestholm, S. O., and Ruud, B. O., 1998, 3-D finite difference elastic wave modeling including surface topography: Geophysics, 63, 613–622
| 3-D finite difference elastic wave modeling including surface topography:Crossref | GoogleScholarGoogle Scholar |
Komatitsch, D., and Martin, R., 2007, An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation: Geophysics, 72, SM155–SM167
| An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation: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 |
Li, Y., and Bou Matar, O., 2010, Convolutional perfectly matched layer for elastic second-order wave equation: The Journal of the Acoustical Society of America, 127, 1318–1327
| Convolutional perfectly matched layer for elastic second-order wave equation:Crossref | GoogleScholarGoogle Scholar | 20329831PubMed |
Marcinkovich, C., and Olsen, K., 2003, On the implementation of perfectly matched layers in a three-dimensional fourth-order velocity-stress finite difference scheme: Journal of Geophysical Research, 108, 2276–2292
| On the implementation of perfectly matched layers in a three-dimensional fourth-order velocity-stress finite difference scheme:Crossref | GoogleScholarGoogle Scholar |
Matzen, R., 2011, An efficient finite element time-domain formulation for the elastic second-order wave equation: a non-split complex frequency shifted convolutional PML: International Journal for Numerical Methods in Engineering, 88, 951–973
| An efficient finite element time-domain formulation for the elastic second-order wave equation: a non-split complex frequency shifted convolutional PML:Crossref | GoogleScholarGoogle Scholar |
Nakata, N., Tsuji, T., and Matsuoka, T., 2011, Acceleration of computation speed for elastic wave simulation using a Graphic Processing Unit: Exploration Geophysics, 42, 98–104
| Acceleration of computation speed for elastic wave simulation using a Graphic Processing Unit:Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DC%2BC3MXisVKrtL4%3D&md5=5b847e5a7f11d813dcd9698790c77587CAS |
Oishi, Y., Piggott, M. D., Maeda, T., Kramer, S. C., Collins, G. S., Tsushima, H., and Furumura, T., 2013, Three-dimensional tsunami propagation simulations using an unstructured mesh finite element model: Journal of Geophysical Research: Solid Earth, 118, 2998–3018
| Three-dimensional tsunami propagation simulations using an unstructured mesh finite element model:Crossref | GoogleScholarGoogle Scholar |
Roden, J. A., and Gedney, S. D., 2000, Convolution PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media: Microwave and Optical Technology Letters, 27, 334–339
| Convolution PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media:Crossref | GoogleScholarGoogle Scholar |
Tarrass, I., Giraud, L., and Thore, P., 2011, New curvilinear scheme for elastic wave propagation in presence of curved topography: Geophysical Prospecting, 59, 889–906
Tessmer, E., and Kosloff, D., 1994, 3-D elastic modeling with surface topography by a Chebyshev spectral method: Geophysics, 59, 464–473
| 3-D elastic modeling with surface topography by a Chebyshev spectral method:Crossref | GoogleScholarGoogle Scholar |
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 |
Xiao, L. Z., and Liu, Y., 2013, Geophysical modeling: Journal of Geophysics and Engineering, 10, 1–2
| Geophysical modeling:Crossref | GoogleScholarGoogle Scholar |
Yuan, S. Y., Wang, S. X., and Tian, N., 2009, Min Fresnel zone relative to large offsets: Oil Geophysical Prospecting, 44, 387–392
Yuan, S. Y., Wang, S. X., He, Y. X., and Tian, N., 2011, Second-order wave equation modeling in time domain with surface topography using embedded boundary method and PML: 73rd Annual International Meeting, EAGE, Expanded Abstracts, 10151.
Zeng, Y. Q., He, J. Q., and Liu, Q. H., 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 |
Zeng, C., Xia, J. H., Miller, R. D., and Tsoflias, G. P., 2011, Application of the multiaxial perfectly matched layer (M-PML) to near-surface seismic modeling with Rayleigh waves: Geophysics, 76, T43–T52
| Application of the multiaxial perfectly matched layer (M-PML) to near-surface seismic modeling with Rayleigh waves:Crossref | GoogleScholarGoogle Scholar |
Zhang, J., and Gao, H., 2011, Irregular perfectly matched layers for 3D elastic wave modeling: Geophysics, 76, T27–T36
| Irregular perfectly matched layers for 3D elastic wave modeling:Crossref | GoogleScholarGoogle Scholar |
Zhang, J., and Liu, T., 1999, P-SV-wave propagation in heterogeneous media: grid method: Geophysical Journal International, 136, 431–438
| P-SV-wave propagation in heterogeneous media: grid method:Crossref | GoogleScholarGoogle Scholar |
Zhang, W., and Shen, Y., 2010, Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling: Geophysics, 75, T141–T154
| Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling:Crossref | GoogleScholarGoogle Scholar |
Zhou, H., Wang, S. X., Li, G. F., and Shen, J. S., 2010, Analysis of complicated structure seismic wave fields: Applied Geophysics, 7, 185–192
| Analysis of complicated structure seismic wave fields:Crossref | GoogleScholarGoogle Scholar |