Comparison of artificial absorbing boundaries for acoustic wave equation modelling
Yingjie Gao 1 2 Hanjie Song 1 2 Jinhai Zhang 1 3 Zhenxing Yao 1
+ Author Affiliations
- Author Affiliations
1 Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China.
2 University of Chinese Academy of Sciences, Beijing 100049, China.
3 Corresponding author. Email: zjh@mail.iggcas.ac.cn
Exploration Geophysics 48(1) 76-93 https://doi.org/10.1071/EG15068
Submitted: 17 July 2015 Accepted: 22 November 2015 Published: 23 December 2015
Journal Compilation © ASEG 2017 Open Access CC BY-NC-ND
Abstract
Absorbing boundary conditions are necessary in numerical simulation for reducing the artificial reflections from model boundaries. In this paper, we overview the most important and typical absorbing boundary conditions developed throughout history. We first derive the wave equations of similar methods in unified forms; then, we compare their absorbing performance via theoretical analyses and numerical experiments. The Higdon boundary condition is shown to be the best one among the three main absorbing boundary conditions that are based on a one-way wave equation. The Clayton and Engquist boundary is a special case of the Higdon boundary but has difficulty in dealing with the corner points in implementaion. The Reynolds boundary does not have this problem but its absorbing performance is the poorest among these three methods. The sponge boundary has difficulties in determining the optimal parameters in advance and too many layers are required to achieve a good enough absorbing performance. The hybrid absorbing boundary condition (hybrid ABC) has a better absorbing performance than the Higdon boundary does; however, it is still less efficient for absorbing nearly grazing waves since it is based on the one-way wave equation. In contrast, the perfectly matched layer (PML) can perform much better using a few layers. For example, the 10-layer PML would perform well for absorbing most reflected waves except the nearly grazing incident waves. The 20-layer PML is suggested for most practical applications. For nearly grazing incident waves, convolutional PML shows superiority over the PML when the source is close to the boundary for large-scale models. The Higdon boundary and hybrid ABC are preferred when the computational cost is high and high-level absorbing performance is not required, such as migration and migration velocity analyses, since they are not as sensitive to the amplitude errors as the full waveform inversion.
Key words: absorbing boundary condition, Higdon boundary, hybrid ABC, perfectly matched layer, sponge boundary.
References
Abarbanel, S., and Gottlieb, D., 1997, A mathematical analysis of the PML method: Journal of Computational Physics, 134, 357–363| A mathematical analysis of the PML method:Crossref | GoogleScholarGoogle Scholar |
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 |
Basu, U., and Chopra, A. K., 2004, Perfectly matched layers for transient elastodynamics of unbounded domains: International Journal for Numerical Methods in Engineering, 59, 1039–1074
| Perfectly matched layers for transient elastodynamics of unbounded domains:Crossref | GoogleScholarGoogle Scholar |
Bayliss, A., and Turkel, E., 1980, Radiation boundary conditions for wave-like equations: Communications on Pure and Applied Mathematics, 33, 707–725
| Radiation boundary conditions for wave-like equations:Crossref | GoogleScholarGoogle Scholar |
Bécache, E., Givoli, D., and Hagstrom, T., 2010, High-order absorbing boundary conditions for anisotropic and convective wave equations: Journal of Computational Physics, 229, 1099–1129
| High-order absorbing boundary conditions for anisotropic and convective wave equations: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 |
Bermúdez, A., Hervella-Nieto, L., and Prieto, A., 2007, An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems: Journal of Computational Physics, 223, 469–488
| An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems:Crossref | GoogleScholarGoogle Scholar |
Bording, R. P., 2004, Finite difference modeling – nearly optimal sponge boundary conditions: 74th Annual International Meeting, SEG, Expanded Abstracts, 1921–1924.
Cerjan, C., Kosloff, D., Kosloff, R., and Reshef, 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 |
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 |
Claerbout, J., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
Clayton, R., and Engquist, B., 1977, Absorbing boundary conditions for acoustic and elastic wave equations: Bulletin of the Seismological Society of America, 67, 1529–1540
Collino, F., 1993, High order absorbing boundary conditions for wave propagation models: straight line boundary and corner cases: Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, SIAM (Newark, Delaware), 161–171.
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 |
Compani-Tabrizi, B., 1986, k-t scattering formulation of the absorptive acoustic wave equation: Wraparound and edge-effect elimination: Geophysics, 51, 2185–2192
| k-t scattering formulation of the absorptive acoustic wave equation: Wraparound and edge-effect elimination:Crossref | GoogleScholarGoogle Scholar |
Diaz, J., and Joly, P., 2006, A time domain analysis of PML models in acoustics: Computer Methods in Applied Mechanics and Engineering, 195, 3820–3853
| A time domain analysis of PML models in acoustics:Crossref | GoogleScholarGoogle Scholar |
Drossaert, F. H., and Giannopoulos, A., 2007a, Complex frequency shifted convolution PML for FDTD modelling of elastic waves: Wave Motion, 44, 593–604
| Complex frequency shifted convolution PML for FDTD modelling of elastic waves:Crossref | GoogleScholarGoogle Scholar |
Drossaert, F. H., and Giannopoulos, A., 2007b, A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves: Geophysics, 72, T9–T17
| A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves:Crossref | GoogleScholarGoogle Scholar |
Engquist, B., and Majda, A., 1977, Absorbing boundary conditions for numerical simulation of waves: Proceedings of the National Academy of Sciences of the United States of America, 74, 1765–1766
| Absorbing boundary conditions for numerical simulation of waves:Crossref | GoogleScholarGoogle Scholar | 1:STN:280:DC%2BC3cngslKitA%3D%3D&md5=e3ed334dd89644b021a604f5ff40c408CAS | 16592392PubMed |
Engquist, B., and Majda, A., 1979, Radiation boundary conditions for acoustic and elastic wave calculations: Communications on Pure and Applied Mathematics, 32, 313–357
| Radiation boundary conditions for acoustic and elastic wave calculations:Crossref | GoogleScholarGoogle Scholar |
Festa, G., and Nielsen, S., 2003, PML absorbing boundaries: Bulletin of the Seismological Society of America, 93, 891–903
| PML absorbing boundaries:Crossref | GoogleScholarGoogle Scholar |
Festa, G., and Vilotte, J.-P., 2005, The Newmark scheme as velocity-stress time-staggering: an efficient PML implementation for spectral element simulations of elastodynamics: Geophysical Journal International, 161, 789–812
| The Newmark scheme as velocity-stress time-staggering: an efficient PML implementation for spectral element simulations of elastodynamics:Crossref | GoogleScholarGoogle Scholar |
Festa, G., Delavaud, E., and Vilotte, J. P., 2005, Interaction between surface waves and absorbing boundaries for wave propagation in geological basins: 2D numerical simulations: Geophysical Research Letters, 32, 1–4
| Interaction between surface waves and absorbing boundaries for wave propagation in geological basins: 2D numerical simulations:Crossref | GoogleScholarGoogle Scholar |
Gedney, S. D., and Zhao, B., 2010, An auxiliary differential equation formulation for the complex-frequency shifted PML: IEEE Transactions on Antennas and Propagation, 58, 838–847
| An auxiliary differential equation formulation for the complex-frequency shifted PML:Crossref | GoogleScholarGoogle Scholar |
Givoli, D., 2004, High-order local non-reflecting boundary conditions: a review: Wave Motion, 39, 319–326
| High-order local non-reflecting boundary conditions: a review:Crossref | GoogleScholarGoogle Scholar |
Givoli, D., and Neta, B., 2003, High-order non-reflecting boundary scheme for time-dependent waves: Journal of Computational Physics, 186, 24–46
| High-order non-reflecting boundary scheme for time-dependent waves:Crossref | GoogleScholarGoogle Scholar |
Hagstrom, T., and Warburton, T., 2004, A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first-order systems: Wave Motion, 39, 327–338
| A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first-order systems: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 |
Higdon, R. L., 1986, Absorbing boundary conditions for difference approximations to the multidimensional wave equation: Mathematics of Computation, 47, 437–459
Higdon, R. L., 1987, Numerical absorbing boundary conditions for the wave equation: Mathematics of Computation, 49, 65–90
| Numerical absorbing boundary conditions for the wave equation:Crossref | GoogleScholarGoogle Scholar |
Higdon, R. L., 1990, Radiation boundary conditions for elastic wave propagation: SIAM Journal on Numerical Analysis, 27, 831–869
| Radiation boundary conditions for elastic wave propagation:Crossref | GoogleScholarGoogle Scholar |
Higdon, R. L., 1991, Absorbing boundary conditions for elastic waves: Geophysics, 56, 231–241
| Absorbing boundary conditions for elastic waves:Crossref | GoogleScholarGoogle Scholar |
Hustedt, B., Operto, S., and Virieux, J., 2004, Mixed-grid and staggered-grid finite-difference methods for frequency-domain acoustic wave modelling: Geophysical Journal International, 157, 1269–1296
| Mixed-grid and staggered-grid finite-difference methods for frequency-domain acoustic wave modelling:Crossref | GoogleScholarGoogle Scholar |
Katsibas, T. K., and Antonopoulos, C. S., 2002, An efficient PML absorbing medium in FDTD simulations of acoustic scattering in lossy media: Proceedings - IEEE Ultrasonics Symposium, 1, 551–554
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 |
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 |
Kristek, J., Moczo, P., and Galis, M., 2009, A brief summary of some PML formulations and discretizations for the velocity-stress equation of seismic motion: Studia Geophysica et Geodaetica, 53, 459–474
| A brief summary of some PML formulations and discretizations for the velocity-stress equation of seismic motion:Crossref | GoogleScholarGoogle Scholar |
Kuzuoglu, M., and Mittra, R., 1996, Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers: IEEE Microwave and Guided Wave Letters, 6, 447–449
| Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers:Crossref | GoogleScholarGoogle Scholar |
Lan, H., Chen, J., Zhang, Z., Liu, Y., and Zhao, J., 2013, Application of the perfectly matched layer in numerical modeling of wave propagation with an irregular free surface: 83rd Annual International Meeting, SEG, 3515–3520.
Liao, Z. P., 1996, Extrapolation non-reflecting boundary conditions: Wave Motion, 24, 117–138
Liao, Z, Huang, K, Yang, B, and Yuan, Y, 1984, A transmitting boundary for transient wave analyses: Scientia Sinica (Series A), 27, 1063–1076
Liu, Q., 1998, The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media: IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 45, 1044–1055
| The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media:Crossref | GoogleScholarGoogle Scholar | 1:STN:280:DC%2BD1c%2Fos1Oisg%3D%3D&md5=e20314bec414148f9b608e902370fed2CAS |
Liu, Q., 1999, Perfectly matched layers for elastic waves in cylindrical and spherical coordinates: The Journal of the Acoustical Society of America, 105, 2075–2084
| Perfectly matched layers for elastic waves in cylindrical and spherical coordinates:Crossref | GoogleScholarGoogle Scholar |
Liu, Y., and Sen, M. K., 2010, 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., 2012, A hybrid absorbing boundary condition for elastic staggered-grid modeling: Geophysical Prospecting, 60, 1114–1132
| A hybrid absorbing boundary condition for elastic staggered-grid modeling:Crossref | GoogleScholarGoogle Scholar |
Liu, Q., and Tao, J., 1997, The perfectly matched layer for acoustic waves in absorptive media: The Journal of the Acoustical Society of America, 102, 2072–2082
| The perfectly matched layer for acoustic waves in absorptive media:Crossref | GoogleScholarGoogle Scholar |
Luebbers, R. J., and Hunsberger, F., 1992, FDTD for Nth-order dispersive media: IEEE Transactions on Antennas and Propagation, 40, 1297–1301
| FDTD for Nth-order dispersive media:Crossref | GoogleScholarGoogle Scholar |
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 - Solid Earth, 108, 18-1–18-16
| On the implementation of perfectly matched layers in a three-dimensional fourth-order velocity-stress finite difference scheme:Crossref | GoogleScholarGoogle Scholar |
Martin, R., Komatitsch, D., Gedney, S. D., and Bruthiaux, E., 2010, A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using Auxiliary Differential Equations (ADE-PML): Computer Modeling in Engineering & Sciences, 56, 17–41
Nataf, F., 2013, Absorbing boundary conditions and perfectly matched layers in wave propagation problems: Direct and Inverse Problems in Wave Propagation and Applications, 14, 219–231
Pasalic, D., and McGarry, R., 2010, Convolutional perfectly matched layer for isotropic and anisotropic acoustic wave equations: 80th Annual International Meeting, SEG, Expanded Abstracts, 2925–2929.
Peng, C, and Toksöz, M. N., 1995, An optimal absorbing boundary condition for elastic wave modeling: Geophysics, 60, 296–301
Qi, Q., and Geers, T. L., 1998, Evaluation of the perfectly matched layer for computational acoustics: Journal of Computational Physics, 139, 166–183
| Evaluation of the perfectly matched layer for computational acoustics:Crossref | GoogleScholarGoogle Scholar |
Rabinovich, D., Givoli, D., and Bécache, E., 2010, Comparison of high-order absorbing boundary conditions and perfectly matched layers in the frequency domain: International Journal for Numerical Methods in Biomedical Engineering, 26, 1351–1369
| Comparison of high-order absorbing boundary conditions and perfectly matched layers in the frequency domain: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 |
Roden, J. A., and Gedney, S. D., 2000, Convolutional PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media: Microwave and Optical Technology Letters, 27, 334–339
| Convolutional PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media:Crossref | GoogleScholarGoogle Scholar |
Sochacki, J., Kubichek, R., George, J., Fletcher, W., and Smithson, S., 1987, Absorbing boundary conditions and surface waves: Geophysics, 52, 60–71
| Absorbing boundary conditions and surface waves:Crossref | GoogleScholarGoogle Scholar |
Sun, W., 2003, Finite difference modeling for elastic wave field in complex media and global optimization method research [in Chinese]: Tsinghua University.
Wagner, R. L., and Chew, W. C., 1995, An analysis of Liao’s absorbing boundary condition: Journal of Electromagnetic Waves and Applications, 9, 993–1009
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 |
Wang, X, and Tang, S, 2010, Analysis of multi-transmitting formula for absorbing boundary conditions: International Journal for Multiscale Computational Engineering, 8, 207–219
Xie, Z., Komatitsch, D., Martin, R., and Matzen, R., 2014, Improved forward wave propagation and adjoint-based sensitivity kernel calculations using a numerically stable finite-element PML: Geophysical Journal International, 198, 1714–1747
| Improved forward wave propagation and adjoint-based sensitivity kernel calculations using a numerically stable finite-element PML:Crossref | GoogleScholarGoogle Scholar |
Xu, Y., and Zhang, J., 2008, An irregular-grid perfectly matched layer absorbing boundary for seismic wave modeling: Chinese Journal of Geophysics, 51, 1520–1526
Yang, D., Song, G., and Lu, M., 2007, Optimally accurate nearly analytic discrete scheme for wave-field simulation in 3D anisotropic media: Bulletin of the Seismological Society of America, 97, 1557–1569
| Optimally accurate nearly analytic discrete scheme for wave-field simulation in 3D anisotropic media: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 |
Zhang, L, and Yu, T, 2012, A method of improving the stability of Liao’s higher-order absorbing boundary condition: Progress in Electromagnetics Research, 27, 167–178
Zhang, J., Wang, W., Wang, S., and Yao, Z., 2010, Optimized Chebyshev Fourier migration: a wide-angle dual-domain method for media with strong velocity contrasts: Geophysics, 75, S23–S34
| Optimized Chebyshev Fourier migration: a wide-angle dual-domain method for media with strong velocity contrasts:Crossref | GoogleScholarGoogle Scholar |
Zhou, B., 1988, On: “k-t scattering formulation of the absorptive acoustic wave equation: Wraparound and edge-effect elimination” by B. Compani-Tabrizi (Geophysics, 51, 2185–2192, December 1986): Geophysics, 53, 564–565
| On: “k-t scattering formulation of the absorptive acoustic wave equation: Wraparound and edge-effect elimination” by B. Compani-Tabrizi (Geophysics, 51, 2185–2192, December 1986):Crossref | GoogleScholarGoogle Scholar |
