Compressive sensing approach for two‐dimensional magnetotelluric inversion using wavelet dictionaries

We present a 2D inversion scheme for magnetotelluric data, where the conductivity structure is parameterized with different wavelet functions that are collected in a wavelet-based dictionary. The inversion model estimate is regularized in terms of wavelet coefficient sparsity following the compressive sensing approach. However, when the model is expressed in the basis of a single wavelet family only, the geometrical appearance of model features reflects the shape of the wavelet functions. Combining two or more wavelet families in a dictionary provides greater flexibility to represent model structure, permitting for example for the simultaneous occurrence of smooth and sharp anomalies within the same model. We show that the application of the sparsity regularization scheme with wavelet dictionaries provides the user with a number of different model classes that may explain the data to the same extent. For a real data example from the Dead Sea Transform, we show that the use of such a scheme can be beneficial to evaluate the geometries of conductivity anomalies and to understand the effect of regularization on the model estimate. This article is protected by copyright. All rights reserved

[1]  A. Gholami,et al.  Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints , 2010 .

[2]  M. Rosas Carbajal,et al.  Focused time-lapse inversion of radio and audio magnetotelluric data , 2012, 1701.02529.

[3]  Paul A. Bedrosian,et al.  Lithology-derived structure classification from the joint interpretation of magnetotelluric and seismic models , 2007 .

[4]  Ignace Loris,et al.  Global seismic tomography with sparsity constraints: Comparison with smoothing and damping regularization , 2013 .

[5]  Michael S. Zhdanov,et al.  Minimum support nonlinear parametrization in the solution of a 3D magnetotelluric inverse problem , 2004 .

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

[7]  William Rodi,et al.  Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion , 2001 .

[8]  Ute Weckmann,et al.  Geophysical images of the Dead Sea Transform in Jordan reveal an impermeable barrier for fluid flow , 2003 .

[9]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[10]  Felix J. Herrmann,et al.  Modified Gauss-Newton full-waveform inversion explained — Why sparsity-promoting updates do matter , 2016 .

[11]  Xiang Li,et al.  Efficient least‐squares imaging with sparsity promotion and compressive sensing , 2012 .

[12]  James H. McClellan,et al.  Sparse-promoting Full Waveform Inversion based on Online Orthonormal Dictionary Learning , 2015, ArXiv.

[13]  P. Bedrosian,et al.  Characterizing a large shear‐zone with seismic and magnetotelluric methods: The case of the Dead Sea Transform , 2005 .

[14]  I. Daubechies,et al.  Tomographic inversion using L1-norm regularization of wavelet coefficients , 2006, physics/0608094.

[15]  Felix J. Herrmann,et al.  Randomized sampling and sparsity: Getting more information from fewer samples , 2010 .

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

[17]  S. Constable,et al.  Occam's inversion to generate smooth, two-dimensional models from magnetotelluric data , 1990 .

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

[19]  Seong Kon Lee,et al.  MT2DInvMatlab - A program in MATLAB and FORTRAN for two-dimensional magnetotelluric inversion , 2009, Comput. Geosci..

[20]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[21]  Michael Becken,et al.  Inversion of magnetotelluric data in a sparse model domain , 2016 .

[22]  Gary D. Egbert,et al.  An efficient data-subspace inversion method for 2-D magnetotelluric data , 2000 .