Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T14:23:31.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability Conditions for Wave Simulation in 3-D Anisotropic Media with the Pseudospectral Method

Published online by Cambridge University Press:  20 August 2015

Wensheng Zhang
Wensheng Zhang*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box Beijing 2719, P.R. China
*
*Corresponding author.Email:[email protected]
Get access

Abstract

Simulation of elastic wave propagation has important applications in many areas such as inverse problem and geophysical exploration. In this paper, stability conditions for wave simulation in 3-D anisotropic media with the pseudospectral method are investigated. They can be expressed explicitly by elasticity constants which are easy to be applied in computations. The 3-D wave simulation for two typical anisotropic media, transversely isotropic media and orthorhombic media, are carried out. The results demonstrate some satisfactory behaviors of the pseudospectral method.

Keywords

35L0565C2065N1265T9968U20Wave simulationstability conditions3-Danisotropic mediapseudospectral method
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 3 , September 2012 , pp. 703 - 720
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alford, R. M., Kelly, K. R. and Boore, D. M., Accuracy of finite-difference modeling of acoustic wave equations, Geophysics, 39 (1974), 834–842.Google Scholar
[2]Aki, K. and Richards, P. G., Quantitative Seismology: Theory and Methods, W. H. Freeman and Company, San Francisco, 1980.Google Scholar
[3]Auld, B. A., Acoustic Fields and Elastic Waves in Solids, Wiley, 1973.Google Scholar
[4]Bécache, E., Joly, P. and Tsogka, C., An analysis of new mixed finite elements for the approximation of wave propagation problems, SIAM J. Numer. Aanl., 37 (2000), 1053–1084.CrossRefGoogle Scholar
[5]Carcione, J. M., Kosloff, D., Behle, A. and Seriani, G., A spectral scheme for wave propagation simulation in 3-D elastic-anisotropic media, Geophysics, 57 (1992), 1593–1607.Google Scholar
[6]Chung, E. T. and Engquist, B., Optimal discontinuous Galerkin methods for wave propagation, SIAM J. Numer. Anal., 44 (2006), 2131–2158.Google Scholar
[7]Chung, E. T. and Engquist, B., Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions, SIAM J. Numer. Anal., 47 (2009), 3820–3848.Google Scholar
[8]Cěrvený, V., Molotkov, I. and Pšenčik, I., Ray Methods in Seismology, University Karbva, Praha, 1977.Google Scholar
[9]Cohen, G., Joly, P., Roberts, J. E. and Tordjman, N., Higher order triangular finite elements with mass lumping for the wave equations, SIAM J. Numer. Anal., 38 (2001), 2047–2078.Google Scholar
[10]Collino, F. and Tsogka, C., Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophysics, 66 (2001), 294–307.Google Scholar
[11]Dormy, E. and Tarantola, A., Numerical simulation of elastic wave propagation using a finite volume method, J. Geophys. Res., 100 (1995), 2123–2133.Google Scholar
[12]Faria, E. L. and Stoffa, P. L., Finite-difference modeling in transversely isotropic media, Geophysics, 59 (1994), 282–289.CrossRefGoogle Scholar
[13]Fornberg, B., On a Fourier method for the integration of hyperbolic equations, SIAM J. Numer. Anal., 12 (1975), 509–528.Google Scholar
[14]Fornberg, B., The pseudospectral method: comparisons with finite differences for the elastic wave equation, Geophysics, 52 (1987), 483–501.Google Scholar
[15]Fornberg, B., The pseudospectral method: accurate representation of interfaces in elastic wave calculations, Geophysics, 53 (1988), 625–637.CrossRefGoogle Scholar
[16]Gajewski, D. and Pšenčik, I., Computation of high frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media, Geophys. J. Roy. Astr. Soc., 91 (1987), 383–411.Google Scholar
[17]Golub, G. H. and Loan, C. F. Van, Matrix Computations (3rd ed.), Johns Hopkins University Press, 1996.Google Scholar
[18]Igel, H., Mora, P. and Riollet, B., Anisotropic wave propagation through finite-difference grids. Geophysics, 60 (1995), 1203–1216.Google Scholar
[19]Jerri, A., The Shannon sampling theorem –its various extensions and applications: a tutorial review, Proceedings of the IEEE, 65 (1977), 1565–1596.CrossRefGoogle Scholar
[20]Kelly, K. R., Ward, R. W., Treitel, S. and Alford, R. M., Synthetic seismograms: a finite-difference approach, Geophysics, 41 (1976), 2–27.CrossRefGoogle Scholar
[21]Kosloff, D. D. and Baysal, E., Forward modeling by a Fourier method, Geophysics, 47 (1982), 1402–1412.Google Scholar
[22]Marfurt, K. J., Accuracy of finite-difference and finite-element modelling of the scalar and elastic wave equations, Geophysics, 49 (1984), 535–549.CrossRefGoogle Scholar
[23]Miller, J. J. H., On the location of zeros of certain classes of polynomials with applications to numerical analysis, IMA J. Appl. Math., 8 (1971), 397–406.CrossRefGoogle Scholar
[24]Mulder, W. A., A comparison between high-order finite elements and finite differences for solving the wave equation, Proc. 2nd ECCOMAS Conf. on Numerical Methods in Engineering, John Wiley & Sons, Paris, 1996, 344-350.Google Scholar
[25]Nilsson, S., Petersson, N. A., Sjörn, B. and Kreiss, H., Stable difference approximations for the elastic wave equation in second order formulation, SIAM J. Numer. Anal., 45 (2007), 1902–1936.Google Scholar
[26]Orszag, S. A., Spectral methods for problems in complex geometries, J. Comp. Phys., 39 (1980), 70–92.Google Scholar
[27]Patera, A. T., A spectral element method for fluid dynamics: Laminar flow in a channel expansion, J. Comp. Phys., 54 (1984), 468–488.Google Scholar
[28]Payton, R. G., Elastic wave propagation in transversely isotropic media, Martinus Nijhoff Publishers, 1983.Google Scholar
[29]Reshef, M., Kosloff, D., Edwards, M. and Hsiung, C., Three-dimensional acoustic modeling by the Fourier method, Geophysics, 53 (1988), 1175–1183.Google Scholar
[30]Reshef, M., Kosloff, D., Edwards, M. and Hsiung, C., Three dimensional elastic modeling by the Fourier method, Geophysics, 53 (1988), 1184–1193.Google Scholar
[31]Sayers, C. M. and Ebrom, D. A., Seismic traveltime analysis of azimuthally anisotropic media: Theory and experiment, Geophysics, 62 (1997), 1570–1582.CrossRefGoogle Scholar
[32]Schoenberg, M. and Helbig, K., Orthorhombic media: Modelling elastic wave behavior in a vertically fractured earth, Geophysics, 62 (1997), 1954–12974.CrossRefGoogle Scholar
[33]Seriani, G. and Priolo, E., Spectral element method for acoustic wave simulation in heterogeneous media, Finite Elements in Analysis and Design, 16 (1994), 337–348.Google Scholar
[34]Thomas, J. W., Numerical Partial Difference Equations: Finite Difference Methods, Springer Verlag New York Inc., 1995.CrossRefGoogle Scholar
[35]Tsvankin, I., Anisotropic parameters and P-wave velocity for orthorhombic media, Geophysics, 62 (1997), 1292–1309.Google Scholar
[36]Tsingas, C., Vafidis, A. and Kanasewich, E. R., Elastic wave propagation in transversely isotropic media using finite differences, Geophys. Prosp., 38 (1990), 933–949.Google Scholar
[37]Virieux, J., P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method, Geophysics, 51 (1986), 889–901.Google Scholar
[38]Yang, D., Liu, E., Zhang, Z. and Teng, J., Finite-difference modelling in two-dimensional anisotropic media using a flux-corrected transport technique, Geophys. J. Int., 148 (2002), 320–328.Google Scholar
[39]Yang, D., Song, G. and Lu, M., Optimally accurate nearly analytic discrete scheme for wave-field simulation in 3D anisotropic media, Bulletin of the Seismological Society of America, 97 (2007), 1557–1569.Google Scholar
[40]Zienkiewicz, O. C., Taylor, R. L. and Zhu, J. Z., The Finite Element Method –its Basis and Fundamentals (6th ed.), Elsevier Butterworth-Heinemann, 2005.Google Scholar
[41]Zhang, J. F., Elastic wave modeling infractured media with an explicit approach, Geophysics, 70 (2005), T75–T85.CrossRefGoogle Scholar
[42]Zhang, W. S. and He, Q. D., Pseudo-spectral forward modeling in transversally isotropic medium, Oil Geophysical Prospecting, 33 (1998), 310–319 (in Chinese).Google Scholar