Acoustic inverse scattering via Helmholtz operator factorization and optimization

We present a joint acoustic/seismic inverse scattering and finite-frequency (reflection) tomography program, formulated as a coupled set of optimization problems, in terms of inhomogeneous Helmholtz equations. We use a higher order finite difference scheme for these Helmholtz equations to guarantee sufficient accuracy. We adapt a structured approximate direct solver for the relevant systems of algebraic equations, which addresses storage requirements through compression, to yield a complexity for computing the gradients or images in the optimization problems that consists of two parts, viz., the cost for all the matrix factorizations which is roughly O(rN) (for example O(rNlogN) when d=2) times the number of frequencies, and the cost for all solutions by substitution which is roughly O(N) (for example O(Nlog(rlogN)) when d=2) times the number of frequencies times the number of sources (events), where N=n^d if n is the number of grid samples in any direction, and r is a parameter depending on the preset accuracy and the problem at hand. With this complexity, the multi-frequency approach to inverse scattering and finite-frequency tomography becomes computationally feasible with large data sets, in dimensions d=2 and 3.

[1]  E. Turkel,et al.  Absorbing PML boundary layers for wave-like equations , 1998 .

[2]  Changsoo Shin,et al.  A comparison between the behavior of objective functions for waveform inversion in the frequency and Laplace domains , 2008 .

[3]  JIANLIN XIA,et al.  ROBUST STRUCTURED MULTIFRONTAL FACTORIZATION AND PRECONDITIONING FOR DISCRETIZED PDES , 2011 .

[4]  Cornelis Vuik,et al.  A new iterative solver for the time-harmonic wave equation , 2006 .

[5]  Bernard C. Levy,et al.  Variable Background Born Inversion by Wavefield Backpropagation , 1988 .

[6]  Joseph W. H. Liu.  The role of elimination trees in sparse factorization , 1990 .

[7]  Patrick Amestoy,et al.  An Unsymmetrized Multifrontal LU Factorization , 2000, SIAM J. Matrix Anal. Appl..

[8]  J. Tromp,et al.  Finite-Frequency Kernels Based on Adjoint Methods , 2006 .

[9]  Stanley C. Eisenstat,et al.  A TREE-BASED DATAFLOW MODEL FOR THE UNSYMMETRIC MULTIFRONTAL METHOD , 2005 .

[10]  MAARTEN V. DE HOOP,et al.  INVERSE SCATTERING OF SEISMIC DATA IN THE REVERSE TIME MIGRATION ( RTM ) APPROACH , 2011 .

[11]  Maarten V. de Hoop,et al.  Wave-equation reflection tomography: annihilators and sensitivity kernels , 2006 .

[12]  Shivkumar Chandrasekaran,et al.  A Fast Solver for HSS Representations via Sparse Matrices , 2006, SIAM J. Matrix Anal. Appl..

[13]  Shivkumar Chandrasekaran,et al.  On the Numerical Rank of the Off-Diagonal Blocks of Schur Complements of Discretized Elliptic PDEs , 2010, SIAM J. Matrix Anal. Appl..

[14]  Joseph W. H. Liu,et al.  The Multifrontal Method for Sparse Matrix Solution: Theory and Practice , 1992, SIAM Rev..

[15]  E. Ng,et al.  Predicting structure in nonsymmetric sparse matrix factorizations , 1993 .

[16]  M. Sacchi,et al.  Least‐squares wave‐equation migration for AVP/AVA inversion , 2003 .

[17]  R. Pratt.  Seismic waveform inversion in the frequency domain; Part 1, Theory and verification in a physical scale model , 1999 .

[18]  William W. Symes,et al.  Global solution of a linearized inverse problem for the wave equation , 1997 .

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

[20]  Jianlin Xia,et al.  Fast algorithms for hierarchically semiseparable matrices , 2010, Numer. Linear Algebra Appl..

[21]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[22]  Maarten V. de Hoop,et al.  On sensitivity kernels for ‘wave-equation’ transmission tomography , 2005 .

[23]  R. Shipp,et al.  Seismic waveform inversion in the frequency domain, Part 2: Fault delineation in sediments using crosshole data , 1999 .

[24]  Jean Virieux,et al.  Multiscale imaging of complex structures from multifold wide-aperture seismic data by frequency-domain full-waveform tomography: application to a thrust belt , 2004 .

[25]  R. Plessix,et al.  Frequency-domain finite-difference amplitude-preserving migration , 2004 .

[26]  Shen Wang,et al.  Illumination analysis of wave-equation imaging with curvelets , 2010 .

[27]  A. George.  Nested Dissection of a Regular Finite Element Mesh , 1973 .

[28]  Cornelis Vuik,et al.  On a Class of Preconditioners for Solving the Helmholtz Equation , 2003 .

[29]  Joseph W. H. Liu,et al.  The Theory of Elimination Trees for Sparse Unsymmetric Matrices , 2005, SIAM J. Matrix Anal. Appl..

[30]  Patrick Joly,et al.  Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem , 1990 .

[31]  Robert Schreiber,et al.  A New Implementation of Sparse Gaussian Elimination , 1982, TOMS.

[32]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[33]  Thomas H. Jordan,et al.  Three‐dimensional Fréchet differential kernels for seismicdelay times , 2000 .

[34]  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 .

[35]  Mario Bebendorf,et al.  Mathematik in den Naturwissenschaften Leipzig Existence of H-Matrix Approximants to the Inverse FE-Matrix of Elliptic Operators with L ∞-Coefficients , 2003 .

[36]  Maarten V. de Hoop,et al.  Sensitivity Analysis of Wave-equation Tomography: A Multi-scale Approach , 2010 .

[37]  R. Gerhard Pratt,et al.  Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies , 2004 .

[38]  I. Singer,et al.  High-order finite difference methods for the Helmholtz equation , 1998 .

[39]  Elisabeth Larsson,et al.  A Domain Decomposition Method for the Helmholtz Equation in a Multilayer Domain , 1999, SIAM J. Sci. Comput..

[40]  Maarten V. de Hoop,et al.  Microlocal analysis of seismic inverse scattering in anisotropic elastic media , 2002 .

[41]  A. Morelli.  Inverse Problem Theory , 2010 .

[42]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[43]  I. Singer,et al.  A perfectly matched layer for the Helmholtz equation in a semi-infinite strip , 2004 .

[44]  D. Rose,et al.  Complexity Bounds for Regular Finite Difference and Finite Element Grids , 1973 .

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

[46]  Jianlin Xia,et al.  Superfast Multifrontal Method for Large Structured Linear Systems of Equations , 2009, SIAM J. Matrix Anal. Appl..

[47]  Mario Bebendorf,et al.  Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients , 2004, Math. Comput..

[48]  John K. Reid,et al.  The Multifrontal Solution of Indefinite Sparse Symmetric Linear , 1983, TOMS.

[49]  Shivkumar Chandrasekaran,et al.  A Fast ULV Decomposition Solver for Hierarchically Semiseparable Representations , 2006, SIAM J. Matrix Anal. Appl..

[50]  René-Édouard Plessix,et al.  How to choose a subset of frequencies in frequency-domain finite-difference migration , 2004 .

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

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

[53]  Jack Dongarra,et al.  Templates for the Solution of Algebraic Eigenvalue Problems , 2000, Software, environments, tools.

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

[55]  Felix J. Herrmann,et al.  An Iterative Multilevel Method For Computing Wavefields In Frequency-domain Seismic Inversion , 2008 .

[56]  Maarten V. de Hoop,et al.  Evolution-equation approach to seismic image, and data, continuation ☆ , 2008 .

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

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

[59]  Christiaan C. Stolk,et al.  Kinematics of shot-geophone migration , 2009 .

[60]  Hui Yang,et al.  The finite-frequency sensitivity kernel for migration residual moveout and its applications in migration velocity analysis , 2008 .