Tomographic inversion using L1-norm regularization of wavelet coefficients

We propose the use of $ell_1$ regularization in a wavelet basis for the solution of linearized seismic tomography problems $Am=data$, allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background. An iterative method is used to find a sparse solution $m$ that contains no more fine-scale structure than is necessary to fit the data $data$ to within its assigned errors.

[1]  Thierry Blu,et al.  Orthogonal Hilbert transform filter banks and wavelets , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

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

[3]  Wolfgang Osten,et al.  Introduction to Inverse Problems in Imaging , 1999 .

[4]  Guust Nolet,et al.  Three‐dimensional sensitivity kernels for surface wave observables , 2004 .

[5]  G. Laske,et al.  A shear - velocity model of the mantle , 1996, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[6]  Roel Snieder,et al.  Model Estimations Biased by Truncated Expansions: Possible Artifacts in Seismic Tomography , 1996, Science.

[7]  Roel Snieder,et al.  Obtaining smooth solutions to large, linear, inverse problems , 1994 .

[8]  L. Landweber.  An iteration formula for Fredholm integral equations of the first kind , 1951 .

[9]  D. Donoho.  For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution , 2006 .

[10]  Bradford H. Hager,et al.  Large‐scale heterogeneities in the lower mantle , 1977 .

[11]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[12]  R. Hilst,et al.  On Sensitivity Kernels for 'Wave-Equation' Tomography , 2002 .

[13]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[14]  G. Nolet,et al.  Slab temperature and thickness from seismic tomography: 2. Izu‐Bonin, Japan, and Kuril subduction zones , 1999 .

[15]  Richard Baraniuk,et al.  The dual-tree complex wavelet transform , 2005, IEEE Signal Processing Magazine.

[16]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[17]  R. Parker,et al.  Occam's inversion; a practical algorithm for generating smooth models from electromagnetic sounding data , 1987 .

[18]  Hugh G. Gauch,et al.  Scientific method in practice , 2002 .

[19]  N. Kingsbury.  Image processing with complex wavelets , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[20]  Onur G. Guleryuz,et al.  Weighted overcomplete denoising , 2003, The Thrity-Seventh Asilomar Conference on Signals, Systems & Computers, 2003.

[21]  Wim Spakman,et al.  Imaging algorithms, accuracy and resolution in delay time tomography , 1988 .

[22]  N. Kingsbury.  Complex Wavelets for Shift Invariant Analysis and Filtering of Signals , 2001 .

[23]  Guust Nolet,et al.  Slab temperature and thickness from seismic tomography: 1. Method and application to Tonga , 1999 .

[24]  Mario Bertero,et al.  Introduction to Inverse Problems in Imaging , 1998 .

[25]  Michael S. Zhdanov,et al.  Focusing geophysical inversion images , 1999 .

[26]  Richard Baraniuk,et al.  The Dual-tree Complex Wavelet Transform , 2007 .

[27]  B. Kennett.  Seismic wave propagation and seismic tomography , 1998 .

[28]  A. Dziewoński,et al.  Global Images of the Earth's Interior , 1987, Science.

[29]  Guust Nolet,et al.  Slab temperature and thickness from seismic tomography , 1999 .