A Coupled Finite Difference – Gaussian Beam Method for High Frequency Wave Propagation

  • Nicolay M. Tanushev
  • Yen-Hsi Richard Tsai
  • Björn Engquist
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 82)

Abstract

Approximations of geometric optics type are commonly used in simulations of high frequency wave propagation. This form of technique fails when there is strong local variation in the wave speed on the scale of the wavelength or smaller. We propose a domain decomposition approach, coupling Gaussian beam methods where the wave speed is smooth with finite difference methods for the wave equations in domains with strong wave speed variation. In contrast to the standard domain decomposition algorithms, our finite difference domains follow the energy of the wave and change in time. A typical application in seismology presents a great simulation challenge involving the presence of irregularly located sharp inclusions on top of a smoothly varying background wave speed. These sharp inclusions are small compared to the domain size. Due to the scattering nature of the problem, these small inclusions will have a significant effect on the wave field. We present examples in two dimensions, but extensions to higher dimensions are straightforward.

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References

  1. 1.
    G. Ariel, B. Engquist, N. Tanushev, and R. Tsai. Gaussian beam decomposition of high frequency wave fields using expectation-maximization. J. Comput. Phys., 230(6):2303–2321, 2011.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    V. Červený, M. Popov, and I. Pšenčík. Computation of wave fields in inhomogeneous media - Gaussian beam approach. Geophys. J. R. Astr. Soc., 70:109–128, 1982.MATHGoogle Scholar
  3. 3.
    B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Mathematics of Computation, 31(139):629–651, 1977.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    B. Engquist and O. Runborg. Computational high frequency wave propagation. Acta Numer., 12:181–266, 2003.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. Gray, Y. Xie, C. Notfors, T. Zhu, D. Wang, and C. Ting. Taking apart beam migration. The Leading Edge, Special Section:1098–1108, 2009.Google Scholar
  6. 6.
    R. Hill. Gaussian beam migration. Geophysics, 55:1416–1428, 1990.CrossRefGoogle Scholar
  7. 7.
    R. Hill. Prestack Gaussian-beam depth migration. Geophysics, 66(4):1240–1250, 2001.CrossRefGoogle Scholar
  8. 8.
    J. Keller. Geometrical theory of diffraction. Journal of Optical Society of America, 52:116–130, 1962.CrossRefGoogle Scholar
  9. 9.
    H. Liu and J. Ralston. Recovery of high frequency wave fields for the acoustic wave equation. Multiscale Modeling & Simulation, 8(2):428–444, 2009.MathSciNetCrossRefGoogle Scholar
  10. 10.
    J. Nelder and R. Mead. A simplex method for function minimization. The Computer Journal, 7(4):308–313, 1965.MATHGoogle Scholar
  11. 11.
    A. Quarteroni, F. Pasquarelli, and A. Valli. Heterogeneous domain decomposition: principles, algorithms, applications. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk, VA, 1991), pages 129–150. SIAM, Philadelphia, PA, 1991.Google Scholar
  12. 12.
    J. Ralston. Gaussian beams and the propagation of singularities. In Studies in partial differential equations, volume 23 of MAA Stud. Math., pages 206–248. Math. Assoc. America, Washington, DC, 1982.Google Scholar
  13. 13.
    N. Tanushev. Superpositions and higher order Gaussian beams. Commun. Math. Sci., 6(2):449–475, 2008.MathSciNetMATHGoogle Scholar
  14. 14.
    N. Tanushev, B. Engquist, and R. Tsai. Gaussian beam decomposition of high frequency wave fields. J. Comput. Phys., 228(23):8856–8871, 2009.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Nicolay M. Tanushev
    • 1
  • Yen-Hsi Richard Tsai
    • 1
  • Björn Engquist
    • 1
  1. 1.Department of Mathematics and Institute for Computational Engineering and Sciences (ICES)The University of Texas at AustinAustinUSA

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