Acoustic inverse scattering via Helmholtz operator factorization and optimization

Journal of Computational Physics

Volumn 229, Issue 22, 2010, Pages 8445-8462

  Wang, Shen ^a   de Hoop, Maarten V ^a   Xia, Jianlin ^a  

^a PURDUE UNIVERSITY   (United States)

Author keywords

Helmholtz solver; Imaging; Inverse scattering; Optimization; Structured multifrontal method; Wave equation tomography

Indexed keywords

ACOUSTIC WAVE SCATTERING; DIGITAL STORAGE; FACTORIZATION; FINITE DIFFERENCE METHOD; HELMHOLTZ EQUATION; INVERSE PROBLEMS; TOMOGRAPHY;

FINITE FREQUENCIES; HELMHOLTZ; HELMHOLTZ SOLV; INVERSE-SCATTERING; MULTIFRONTAL METHODS; OPTIMISATIONS; OPTIMIZATION PROBLEMS; SCATTERING FREQUENCY; STRUCTURED MULTIFRONTAL METHOD; WAVE-EQUATION TOMOGRAPHY;

OPTIMIZATION;

EID: 77956415161     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.07.027     Document Type: Article

References (58)

1. Marfurt K. Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations. Geophysics 1984, 49:533-549.
2. J. Xia, M. Gu, Robust structured multifrontal factorization and preconditioning for discretized PDEs, preprint, http://math.purdue.edu/~xiaj/work/mfhssr.pdf.
3. J. Xia, M. Gu, Robust structured factorization and preconditioning for spd matrices, SIAM J. Matrix Anal. Appl., submitted for publication. http://math.purdue.edu/~xiaj/work/schurmonospd.pdf.
4. Elman H., Ernst O., O'Leary D. A multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations. SIAM J. Scient. Comp. 2001, 23:1291-1315.
5. Plessix R.-E. A helmholtz iterative solver for 3D seismic-imaging problems. Geophysics 2007, 72:SM185-SM194.
6. Erlangga Y., Vuik C., Oosterlee C. On a class of preconditioners for solving the helmholtz equation. Appl. Numer. Math. 2004, 50:409-425.
7. Erlangga Y., Oosterlee C., Vuik C. A novel multigrid based preconditioner for heterogeneous helmholtz problems. SIAM J. Scient. Comput. 2006, 27:1471-1492.
8. Riyanti C., Erlangga Y., Plessix R.-E., Mulder W., Vuik C., Oosterlee C. A new iterative solver for the time-harmonic wave equation. Geophysics 2006, 71:E57-E63.
9. Operto S., Virieux J., Amestoy P., L'Excellent J., Giraud L., Hadj Ali H. A helmholtz iterative solver for 3D seismic-imaging problems. Geophysics 2007, 72:SM185-SM194.
10. Larsson E. A domain decomposition method for the helmholtz equation in a multilayer domain. SIAM J. Scient. Comp. 1999, 20:1713-1731.
11. Sirgue L., Pratt R. Efficient waveform inversion and imaging: a strategy for selecting temporal frequencies. Geophysics 2004, 69:231-248.
12. Mulder W., Plessix R.-E. How to choose a subset of frequencies in frequency-domain finite-difference migration. Geophys. J. Int. 2004, 158:801-812.
13. Shin C., Ha W. A comparison between the behavior of objective functions for waveform inversion in the frequency and laplace domains. Geophysics 2008, 73:VE119-VE133.
14. Virieux J., Operto S. An overview of full-waveform inversion in exploration geophysics. Geophysics 2009, 74:WCC1-WCC26.
15. Lailly P. The seismic inverse problem as a sequence of before stack migrations. Proceedings of the International Conference on Inverse Scattering, Theory and Applications 1983, 206-220. SIAM, Tulsa, OK.
16. Tarantola A. Inverse Problem Theory 1987, Elsevier, New York.
17. Kühl H., Sacchi M. Least-squares wave-equation migration for avp/ava inversion. Geophysics 2003, 68:262-273.
18. Pratt R. Seismic waveform inversion in the frequency domain, part 1: Theory and verification in a physical scale model. Geophysics 1999, 64:888-901.
19. Ravaut C., Operto S., Improta L., Virieux J., Herrero A., Dell'Aversana P. Multiscale imaging of complex structures from multifold wide-aperture seismic data by frequency-domain full-waveform tomography: application to a thrust belt. Geophys. J. Int. 2004, 159:1032-1056.
20. Stolk C., De Hoop M. Microlocal analysis of seismic inverse scattering in anisotropic elastic media. Commun. Pure Appl. Math. 2002, 55:261-301.
21. De Hoop M., Van der Hilst R., Shen P. Wave-equation reflection tomography: annihilators and sensitivity kernels. Geophys. J. Int. 2006, 167:1332-1352.
22. Xie X., Yang H. The finite-frequency sensitivity kernel for migration residual moveout and its applications in migration velocity analysis. Geophysics 2008, 73:S241-S249.
23. Duchkov A., De Hoop M., Sá Barreto A. Evolution-equation approach to seismic image, and data, continuation. Wave Motion 2008, 45:952-969.
24. S. Wang, M. De Hoop, Illumination analysis of wave-equation imaging with "curvelets", submitted for publication.
25. V. Brytik, M. De Hoop, M. Salo, Sensitivity analysis of wave equation tomography: a multi-scale approach, Journal of Fourier Analysis and Applications, submitted for publication.
26. W.W. Symes, Lecture notes: Short Course on Mathematical Foundations of Reflection Seismology, T.R.I.P. consortium.
27. Stolk C., De Hoop M., Symes W. Kinematics of shot-geophone migration. Geophysics 2009, 74:WCA19-WCA34.
28. Nolan C., Symes W. Global solution of a linearized inverse problem for the wave equation. Commun. Partial Differ. Equat. 1997, 22:919-952.
29. Levy B., Esmersoy C. Variable background born inversion by wavefield backpropagation. SIAM J. Appl. Math. 1988, 48:952-972.
30. C. Stolk, M. De Hoop, T. Op't Root, Inverse scattering of seismic data in the reverse time migration (rtm) approach, preprint. http://www.math.purdue.edu/~mdehoop/10_topics/rtm.pdf.
31. Zhao L., Jordan T., Chapman C. Three-dimensional fréchet differential kernels for seismic delay times. Geophys. J. Int. 2000, 141:558-576.
32. De Hoop M., Van der Hilst R. On sensitivity kernels for 'wave-equation' transmission tomography. Geophys. J. Int. 2005, 160:621-633.
33. Liu Q., Tromp J. Finite-frequency kernels based upon adjoint methods. Bull. Seism. Soc. Amer. 2006, 96:2383-2397.
34. Turkel E., Yefet A. Absorbing pml boundary layers for wave-like equations. Appl. Numer. Math. 1998, 27:533-557.
35. Singer I., Turkel E. A perfectly matched layer for the Helmholtz equation in a semi-infinite strip. J. Comput. Phys. 2004, 201:439-465.
36. Bamberger A., Joly P., Roberts J. Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem. SIAM J. Numer. Anal. 1990, 27:323-352.
37. Singer I., Turkel E. High-order finite difference methods for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 1998, 163:343-358.
38. Lele S. Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 1992, 103:16-42.
39. Bai Z., Demmel J., Dongarra J., Ruhe A., Van den Vorst H. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide 2000, SIAM, Philadelphia, PA.
40. Saad Y. Iterative methods for sparse linear systems 2003, SIAM, Philadelpha, PA. second ed.
41. Xia J., Chandrasekaran S., Gu M., Li X. Superfast multifrontal method for large structured linear systems of equations. SIAM J. Matrix Anal. Appl. 2009, 31:1382-1411.
42. George J. Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 1973, 10:345-363.
43. Hoffman A., Martin M., Rose D. Complexity bounds for regular finite difference and finite element grids. SIAM J. Numer. Anal. 1973, 10:364-369.
44. Duff I., Reid J. The multifrontal solution of indefinite sparse symmetric linear equations. ACM Trans. Math. Softw. 1983, 9:302-325.
45. Liu J. The multifrontal method for sparse matrix solution: theory and practice. SIAM Rev. 1992, 34:82-109.
46. Eisenstat S., Liu J. The theory of elimination trees for sparse unsymmetric matrices. SIAM J. Matrix Anal. Appl. 2005, 26:686-705.
47. Gilbert E.N.J.R. Predicting Structure in Nonsymmetric Sparse Matrix Factorizations: Graph Theory and Sparse Matrix Computation 1993, Springer-Verlag.
48. Liu J. The role of elimination trees in sparse factorization. SIAM J. Matrix Anal. Appl. 1990, 18:134-172.
49. Schreiber R. A new implementation of sparse gaussian elimination. ACM Trans. Math. Softw. 1982, 8:256-276.
50. Eisenstat S.C., Liu J.W.H. A tree based dataflow model for the unsymmetric multifrontal method. Electron. Trans. Numer. Anal. 2005, 21:1-19.
51. Amestoy P., Puglisi C. An unsymmetrized multifrontal lu factorization. SIAM J. Matrix Anal. Appl. 2002, 24:553-569.
52. Bebendorf M. Efficient inversion of galerkin matrices of general second-order elliptic differential operators with nonsmooth coefficients. Math. Comp. 2005, 74:1179-1199.
53. ∞ coefficients. Numer. Math. 2003, 95:1-28.
54. Chandrasekaran S., Dewilde P., Gu M., Somasunderam N. On the numerical rank of the off-diagonal blocks of schur complements of discretized elliptic PDES. SIAM J. Matrix Anal. Appl. 2010, 31:2261-2290.
55. Chandrasekaran S., Dewilde P., Gu M., Lyons W., Pals T. A fast solver for HSS representations via sparse matrices. SIAM J. Matrix Anal. Appl. 2006, 29:67-81.
56. J. Xia, S. Chandrasekaran, M. Gu, X. Li, Fast algorithms for hierarchically semiseparable matrices, Numer. Linear Algebra Appl. (2010), in press, doi:10.1002/nla.691.
57. Chandrasekaran S., Gu M., Pals T. A fast ulv decomposition solver for hierarchically semiseparable representations. SIAM J. Matrix Anal. Appl. 2006, 28:603-622.
58. Erlangga Y., Herrmann F. An iterative multilevel method for computing wavefields in frequency-domain seismic inversion. SEG Tech. Prog. Expanded Abstr. 2008, 27:1956-1960.