AccScience Publishing / JSE / Article
Submit to JSE
Cite this article
1
Download
44
Views
Journal Browser
Volume | Year
Issue
Search
Forthcoming Issue
Current Issue
View All
News and Announcements
View All
ARTICLE

Time-domain formulation of a perfectly matched layer for the second-order elastic wave equation with VTI media

JAEJOON LEE1 ,  CHANGSOO SHIN2
Show Less
1 Computational Science and Technology Program, Seoul National University, Seoul, South Korea.,
2 Department of Energy System Engineering, Seoul National University, Seoul, South Korea.,
JSE 2015, 24(3), 231–257;
Submitted: 7 November 2014 | Accepted: 12 April 2015 | Published: 1 July 2015
© 2015 by the Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Download PDF
Cite
XML
Abstract

Lee, J. and Shin, C.S., 2015. Time-domain formulation of a perfectly matched layer for the second-order elastic wave equation with VTI media. Journal of Seismic Exploration, 24: 231-257. In this study, we introduce the unsplit Perfectly Matched Layer (PML) for the 2D and 3D second-order elastic wave equations with isotropic and transversely isotropic vertical axis of symmetry (VTI) media in the time domain. The introduced PML formulations are successfully applied to practical applications in terms of efficiency and stability. The PML formulations require less than or an equal number of auxiliary variables than other formulations, thereby decreasing the computational power necessary to calculate the solution in the PML zone. Derived directly from the second-order wave form, the PML formulation demonstrates an improved stability compared to first-order PMLs or second-order PMLs that are derived from first-order systems. Numerical examples demonstrate that the bulk waves and strong surface waves are perfectly damped out without introducing instability for an isotropic material in both 2D and 3D. The derived formulation also provides effective absorption with strong VTI materials, including zinc and apatite, that cause instability problems in other PML formulations.

Keywords
perfectly matched layer
time-domain seismic wave modeling
second-order elastic wave equation
VTI media
References
  1. Abarbanel, S. and Gottlieb, D., 1997. A mathematical analysis of the PML method. J. Computat.Phys., 134: 357-363.
  2. Abarbanel, S., Gottlieb, D. and Hesthaven, J.S., 2002. Long time behavior of the perfectly matchedlayer equations in computational electromagnetics. J. Scientif. Comput., 17: 405-422.
  3. Appelé, D. and Petersson, N.A., 2009. A stable finite difference method for the elastic waveequation on complex geometries with free surfaces. Communic. Computat. Phys. , 5: 84-107.
  4. Assi, H. and Cobbold, R.S., 2013. Perfectly matched layer for second-order time-domain elasticwave equation: formulation and stability. arXiv preprint arXiv: 1312.3722.
  5. Bécache, E., Fauqueux, S. and Joly, P., 2003. Stability of perfectly matched layers, group velocitiesand anisotropic waves. J. Computat. Phys., 188: 399-433.PML FOR SECOND-ORDER ELASTIC EQUATION 255
  6. Berenger, J.P., 1994. A perfectly matched layer for the absorption of electromagnetic waves. J.Computat. Phys., 114: 185-200.
  7. Carcione, J.M., Kosloff, D. and Kosloff, R., 1988. Wave-propagation simulation in an elasticanisotropic (transversely isotropic) solid. Quart. J. Mechan. Appl. Mathemat., 41: 319-346.
  8. Chang, W.F. and McMechan, G.A., 1987. Elastic reverse-time migration. Geophysics, 52:1365-1375.
  9. Chew, W.C. and Liu, Q.H., 1996. Perfectly matched layers for elastodynamics: a new absorbingboundary condition. J. Computat. Acoust., 4: 341-359.
  10. Chew, W.C. and Weedon, W.H., 1994. A 3D perfectly matched medium from modified Maxwell’sequations with stretched coordinates. Microw. Opt. Technol. Lett., 7: 599-604.
  11. Collino, F., and Tsogka, C., 2001. Application of the perfectly matched absorbing layer model tothe linear elastodynamic problem in anisotropic heterogeneous media. Geophysics, 66:294-307.
  12. Crase, E., Pica, A., Noble, M., McDonald, J. and Tarantola, A., 1990. Robust elastic nonlinearwaveform inversion: Application to real data. Geophysics, 55: 527-538.
  13. De Basabe, J.D., Sen, M.K. and Wheeler, M.F., 2008. The interior penalty discontinuous Galerkinmethod for elastic wave propagation: grid dispersion. Geophys. J. Internat., 175: 83-93.
  14. Duru, K., and Kreiss, G., 2012. A well-posed and discretely stable perfectly matched layer forelastic wave equations in second order formulation. Communic. Computat. Phys., 11: 1643.
  15. Grote, M.J. and Sim, I., 2010. Efficient PML for the wave equation, arXiv preprint arXiv:0319.
  16. Hastings, F.D., Schneider, J.B. and Broschat, S.L., 1996. Application of the perfectly matchedlayer (PML) absorbing boundary condition to elastic wave propagation. J. Acoust. Soc. Am.,100: 3061-3069.
  17. Kim, Y., Min, D.J. and Shin, C., 2011. Frequency-domain reverse-time migration with sourceestimation. Geophysics, 76: S41-S49.
  18. Komatitsch, D. and Martin, R., 2007. An unsplit convolutional perfectly matched layer improvedat grazing incidence for the seismic wave equation. Geophysics, 72: SM155-SM167.
  19. Komatitsch, D. and Vilotte, J.P., 1998. The spectral element method: an efficient tool to simulatethe seismic response of 2D and 3D geological structures. Bull. Seismol. Soc. Am., 88:368-392.
  20. Komatitsch, D. and Tromp, J., 2003. A perfectly matched layer absorbing boundary condition forthe second-order seismic wave equation. Geophys. J. Internat., 154: 146-153.
  21. Kosloff, D., Queiroz Filho, A., Tessmer, E. and Behle, A., 1989. Numerical solution of theacoustic and elastic wave equations by a new rapid expansion method. Geophys. Prosp., 37:383-394.
  22. Ledbetter, H.M., 1977. Elastic properties of zinc: A compilation and a review. J. Phys. Chem. Ref.Data, 6: 1181-1203.
  23. Li, Y. and Matar, O.B., 2010. Convolutional perfectly matched layer for elastic second-order waveequation. J. Acoust. Soc. Am., 127: 1318-1327.
  24. Liu, Q.H. and Tao, J., 1997. The perfectly matched layer for acoustic waves in absorptive media.J. Acoust. Soc. Am., 102: 2072-2082.
  25. Marfurt, K.J., 1984. Accuracy of finite-difference and finite-element modeling of the scalar andelastic wave equations. Geophysics, 49: 533-549.
  26. Meza-Fajardo, K.C. and Papageorgiou, A.S., 2008. A nonconvolutional, split-field, perfectlymatched layer for wave propagation in isotropic and anisotropic elastic media: Stabilityanalysis. Bull. Seismol. Soc. Am., 98: 1811-1836.
  27. Mora, P., 1987. Nonlinear two-dimensional elastic inversion of multioffset seismic data. Geophysics,52: 1211-1228.
  28. Pratt, R.G., Shin, C. and Hick, G.J., 1998. Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion. Geophys. J. Internat., 133: 341-362.
  29. Qi, Q. and Geers, T.L., 1998. Evaluation of the perfectly matched layer for computationalacoustics. J. Computat. Phys., 139: 166-183.256 LEE & SHIN
  30. Shin, C. and Cha, Y.H., 2008. Waveform inversion in the Laplace domain. Geophys. J. Internat.,173: 922-931.
  31. Shin, C. and Cha, Y.H., 2009. Waveform inversion in the Laplace-Fourier domain. Geophys. J.Internat., 177: 1067-1079.
  32. Sun, R. and McMechan, G.A., 2001. Scalar reverse-time depth migration of prestack elastic seismicdata. Geophysics, 66: 1519-1527.
  33. Tarantola, A., 1986. A strategy for nonlinear elastic inversion of seismic reflection data.Geophysics, 51: 1893-1903.
  34. Thomsen, L., 1986. Weak elastic anisotropy. Geophysics, 51: 1954-1966.
  35. Yan, J. and Sava, P., 2008. Isotropic angle-domain elastic reverse-time migration. Geophysics, 73:9-S239.
Share
Back to top
Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here
Accept