Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints

Geophysical Journal International

Volumn 180, Issue 2, 2010, Pages 871-882

  Gholami, A ^a   Siahkoohi, H R ^a  

^a UNIVERSITY OF TEHRAN   (Iran)

Author keywords

Inverse theory; Tomography; Wavelet transform

Indexed keywords

ALGORITHM; FINITE DIFFERENCE METHOD; INVERSE PROBLEM; LINEARITY; NONLINEARITY; NUMERICAL MODEL; PARAMETERIZATION; PERFORMANCE ASSESSMENT; SEISMIC TOMOGRAPHY; SMOOTHING; TRAVEL TIME; WAVELET ANALYSIS;

EID: 74549114255     PISSN: 0956540X     EISSN: 1365246X     Source Type: Journal    
DOI: 10.1111/j.1365-246X.2009.04453.x     Document Type: Article

References (43)

1. Ajo-Franklin JB, Minsley BJ, Daley TM. Applying compactness constraints to differential traveltime tomography. Geophysics 2007, 72(4):R67-R75.
2. Bertete-Aguirre H, Cherkaev E, Oristaglio M. Non-smooth gravity problem with total variation penalization functional. Geophys. J. Int. 2002, 149:499-507.
3. Bregman L. The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex optimization. USSR Comput. Math. Math. Phys. 1967, 7:200-217.
4. Candes EJ, Guo F. New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction. Signal Process 2002, 82:1519-1543.
5. Chambolle A, DeVore RA, Lee NY, Lucier BJ. Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process 1998, 7:319-335.
6. Charbonnier P, Blanc-Feraud L, Aubert G, Barlaud M. Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process 1997, 6:298-311.
7. Constable SC, Parker RL, Constable CG. Occam's inversion: a practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics 1987, 52:289-300.
8. Daubechies I. Ten Lectures on Wavelets 1992, SIAM, Philadelphia
9. Daubechies I, Defriese M, De Mol C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure appl. Math. 2004, LVII:1413-1457.
10. Donoho DL. Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 1995a, 2:101-126.
11. Donoho DL. De-noising by soft thresholding. IEEE, Trans. Inf. Theory 1995b, 41:613-627.
12. Donoho DL, Johnstone IM. Minimax estimation via wavelet shrinkage. Ann. Statist. 1998, 26:879-921. as, technical report, Department of Stat., Stanford University, 1992
13. Durand S, Froment J. Artifact free signal denoising with wavelets. 2001, in, May 7-11, Salt Lake City, Utah
14. Farquharson CG, Oldenburg DW. Non-linear inversion using general measures of data misfit and model structure. Geophys. J. Int. 1998, 134:213-227.
15. Gholami A, Siahkoohi HR. A two-step wavelet-based regularization for linear inversion of geophysical data. Geophys. Prospect. 2009a, 57:847-862.
16. Gholami A, Siahkoohi HR. Simultaneous constraining of model and data smoothness for regularization of geophysical inverse problems. Geophys. J. Int. 2009b, 176:151-163.
17. Goldstein T, Osher S. The Split Bregman method for L1 regularized problems. 2008, UCLA, CAAM Report 08-29
18. Hansen P, O'Leary D. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Scient. Comput. 1993, 14:1487-1503.
19. Herrmann FJ, Moghaddam PP, Stolk CC. Sparsity- and continuity-promoting seismic image recovery with curvelet frames. Appl. Comput. Harmonic Anal. 2008a, 24:150-173.
20. Herrmann FJ, Wang W, Hennenfent G, Moghaddam PP. Curvelet-based seismic data processing: a multiscale and nonlinear approach. Geophysics 2008b, 73:A1.
21. Kelley CT. Iterative Methods for Optimization 1999, SIAM, Philadelphia
22. Kim SJ, Koh K, Lustig M, Boyd S, Gorinevsky D. An interior-point method for large-scale l1-regularized least squares. IEEE Trans. Sel. Topics Signal Proc. 2007, 1:606-617.
23. Klann E, Maass P, Ramlau R. Two-step regularization methods for linear inverse problems. J. Inverse Ill-Posed Problems 2006, 14:583-607.
24. Li X, Sacchi MD, Ulrych TJ. Wavelet transform inversion with a priori scale information. Geophysics 1996, 61:1379-1385.
25. Loris I, Nolet G, Daubechies I, Dahlen FA. Tomographic inversion using l1-norm regularization of wavelet coefficients. Geophys. J. Int. 2007, 170:359-370.
26. Lustig M, Donoho DL, Pauly JM. Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn. Resonan. Med. 2007, 58:1182-1195.
27. Mallat SG. A Wavelet Tour of Signal Processing 1998, Academic Press, San Diego
28. Meyer CD. Matrix Analysis and Applied Linear Algebra 2000, SIAM, Philadelphia
29. Osher S, Burger M, Goldfarb D, Xu J, Yin W. An iterative regularization method for total variation-based image restoration. Multiscale Model. Simulat. 2005, 4:460-489.
30. Ramlau R, Teschke G. Tikhonov replacement functionals for iteratively solving nonlinear operator equations. Inverse Problems 2005, 21:1571-1592.
31. Ramlau R, Teschke G. A projection iteration for nonlinear operator equations with sparsity constraints. Num. Math. 2006, 104:177-203.
32. Routh PS, Qu L, Sen MK, Anno PD. Inversion for non-smooth models with physical bounds. SEG, Expanded Abstracts 2007, 26(1):1795-1799.
33. Sacchi MD, Ulrych TJ. High resolution velocity gathers and offset space reconstruction. Geophysics 1995, 60:1169-1177.
34. Starck JL, Donoho DL, Candes E, Laine A, Unser MA, Aldroubi A. Very high quality image restoration by combining wavelets and curvelets. 2001, in, SPIE Vol. 4478, eds, SPIE, San Diego, CA
35. Tarantola A. Inverse Problem Theory and Methods for Model Parameter Estimation 2005, SIAM, Philadelphia
36. Teschke G, Ramlau R. An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector valued regimes and an application to color image inpainting. Inverse Problems 2007, 23:1851-1870.
37. Tikhonov AN, Arsenin VY. Solutions of Ill-posed Problems 1977, VH Winston and Sons, Washington, DC
38. van Wijk K, Scales JA, Navidi W, Tenorio L. Data and model uncertainty estimation for linear inversion. Geophys. J. Int. 2002, 149:625-632.
39. Wahba G. Spline models for observational data. 1990, in, Vol. 59, SIAM, Philadelphia, PA
40. Wang Y. Sparseness-constrained least-squares inversion: application to seismic wave reconstruction. Geophysics 2003, 68:1633-1638.
41. Wang Y, Yin W, Zhang Y. A fast algorithm for image deblurring with total variation regularization. 2007, CAAM Technical Reports TR07-10
42. Youzwishen CF, Sacchi MD. Edge preserving imaging. J. Seism. Explor. 2006, 15:45-58.
43. Zelt CA, Barton PJ. 3D seismic refraction tomography: a comparison of two methods applied to data from the Faeroe Basin. J. geophys. Res. 1998, 103:7187-7210.