Seismic wave extrapolation using lowrank symbol approximation

We consider the problem of constructing a wave extrapolation operator in a variable and possibly anisotropic medium. Our construction involves Fourier transforms in space combined with the help of a lowrank approximation of the space-wavenumber wave-propagator matrix. A lowrank approximation implies selecting a small set of representative spatial locations and a small set of representative wavenumbers. We present a mathematical derivation of this method, a description of the lowrank approximation algorithm and numerical examples that confirm the validity of the proposed approach. Wave extrapolation using lowrank approximation can be applied to seismic imaging by reverse-time migration in 3D heterogeneous isotropic or anisotropic media.

[1]  W. B. Johnson,et al.  Extensions of Lipschitz mappings into Hilbert space , 1984 .

[2]  Tariq Alkhalifah,et al.  Acoustic approximations for processing in transversely isotropic media , 1998 .

[3]  Ming Gu,et al.  Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization , 1996, SIAM J. Sci. Comput..

[4]  Sergey Fomel,et al.  Fourier finite-difference wave propagation , 2011 .

[5]  Sergey Fomel,et al.  On anelliptic approximations for qP velocities in VTI media , 2004 .

[6]  C. Fombrun,et al.  Matrix , 1979, Encyclopedic Dictionary of Archaeology.

[7]  G. Margrave,et al.  Phase-shift time-stepping for reverse-time migration: the Marmousi data experience , 2008 .

[8]  Ekkehart Tessmer,et al.  Using the rapid expansion method for accurate time-stepping in modeling and reverse-time migration , 2011 .

[9]  Paul L. Stoffa,et al.  A pseudospectral‐finite difference hybrid approach for large‐scale seismic modeling and RTM on parallel computers , 2008 .

[10]  Avner Magen,et al.  Dimensionality Reductions That Preserve Volumes and Distance to Affine Spaces, and Their Algorithmic Applications , 2002, RANDOM.

[11]  Gene H. Golub,et al.  Matrix computations , 1983 .

[12]  Sergey Fomel,et al.  Velocity continuation and the anatomy of residual prestack time migration , 2003 .

[13]  Z. Koren,et al.  AN ACCURATE SCHEME FOR SEISMIC FORWARD MODELLING , 1987 .

[14]  Tariq Alkhalifah,et al.  Velocity analysis for transversely isotropic media , 1995 .

[15]  Lexing Ying,et al.  A fast directional algorithm for high frequency acoustic scattering in two dimensions , 2009 .

[16]  Lexing Ying,et al.  Lowrank finite-differences for wave extrapolation , 2011 .

[17]  R. P. Fletcher,et al.  Pure P-wave Propagators Versus Pseudo-acoustic Propagators for RTM in VTI Media , 2010 .

[18]  GuMing,et al.  Efficient algorithms for computing a strong rank-revealing QR factorization , 1996 .

[19]  S. Brandsberg-Dahl,et al.  The 2004 BP Velocity Benchmark , 2005 .

[20]  Sverre Brandsberg-Dahl,et al.  The Pseudo-analytical Method: Application of Pseudo-Laplacians to Acoustic And Acoustic Anisotropic Wave Propagation , 2009 .

[21]  Yu Zhang,et al.  One-step extrapolation method for reverse time migration , 2009 .

[22]  Lexing Ying,et al.  Fast Directional Multilevel Algorithms for Oscillatory Kernels , 2007, SIAM J. Sci. Comput..

[23]  Tariq Alkhalifah,et al.  Source-receiver Two-way Wave Extrapolation For Prestack Exploding-reflector Modeling And Migration , 2010 .

[24]  S. Goreinov,et al.  A Theory of Pseudoskeleton Approximations , 1997 .

[25]  Tariq Alkhalifah.  An Acoustic Wave Equation For Anisotropic Media , 1998 .

[26]  Sergey Fomel,et al.  Theory of differential offset continuation , 2003 .