Three-dimensional visco-acoustic modeling using a renormalized integral equation iterative solver

We present a frequency-domain renormalized integral equation formulation for solving a three-dimensional visco-acoustic medium using an iterative solver. Upon applying this special renormalization, the resulting integral equation operator can be proven to have a contraction property. Hence, solving the linear-system of equations using a Krylov optimization method, will result in a good convergence rate. Furthermore since the matrix-vector multiplication can be done using a Fast-Fourier transform (FFT) technique, its operation is of the order of O N log N , where N is the size of the discretization grid. This technique also allows us to use matrix-free implementation. Hence, the memory usage is about O N . Numerical tests show that the computational time and memory usage of this renormalized integral equation approach can be quite competitive with the frequency-domain finite difference iterative solver. Further, the numerical examples demonstrate that it is possible to solve a problem with over 100 million unknowns using an integral equation approach.

[1]  Jean Virieux,et al.  An overview of full-waveform inversion in exploration geophysics , 2009 .

[2]  B. Sh. Singer,et al.  Integral equation approach based on contraction operators and krylov subspace optimisation , 2003 .

[3]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[4]  J. Richmond.  Scattering by a dielectric cylinder of arbitrary cross section shape , 1965 .

[5]  René-Édouard Plessix,et al.  Separation-of-variables as a preconditioner for an iterative Helmholtz solver , 2003 .

[6]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[7]  Cornelis Vuik,et al.  A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems , 2005, SIAM J. Sci. Comput..

[8]  Jianlin Xia,et al.  On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver , 2011 .

[9]  P. M. Berg,et al.  Iterative forward and inverse algorithms based on domain integral equations for three-dimensional electric and magnetic objects , 2004 .

[10]  K. Marfurt.  Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations , 1984 .

[11]  W. Press,et al.  Numerical Recipes in Fortran: The Art of Scientific Computing.@@@Numerical Recipes in C: The Art of Scientific Computing. , 1994 .

[12]  William H. Press,et al.  Book-Review - Numerical Recipes in Pascal - the Art of Scientific Computing , 1989 .

[13]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[14]  Martin H. Gutknecht,et al.  Variants of BICGSTAB for Matrices with Complex Spectrum , 1993, SIAM J. Sci. Comput..

[15]  Dmitry B. Avdeev,et al.  High-Performance Three-Dimensional Electromagnetic Modelling Using Modified Neumann Series. Anisotropic Earth , 1997 .

[16]  P. M. van den Berg,et al.  Seismic applications of acoustic reciprocity , 1993 .

[17]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[18]  Aria Abubakar,et al.  An effective perfectly matched layer design for acoustic fourth-order frequency-domain finite-difference scheme , 2012 .

[19]  J. Carcione,et al.  Seismic modelingSeismic modeling , 2002 .

[20]  Cornelis Vuik,et al.  A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation , 2007, J. Comput. Phys..

[21]  Dianne P. O'Leary,et al.  A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations , 2001, SIAM J. Sci. Comput..

[22]  Timothy A. Davis,et al.  An Unsymmetric-pattern Multifrontal Method for Sparse Lu Factorization , 1993 .

[23]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[24]  E. B. Fainberg,et al.  Generalization of the iterative dissipative method for modeling electromagnetic fields in nonuniform media with displacement currents , 1995 .

[25]  Lexing Ying,et al.  Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers , 2010, Multiscale Model. Simul..

[26]  René-Édouard Plessix,et al.  A Helmholtz iterative solver for 3D seismic-imaging problems , 2007 .

[27]  Azzam Haidar,et al.  Seismic wave modeling for seismic imaging , 2009 .

[28]  S. Operto,et al.  3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study , 2007 .