Double-difference traveltime tomography with edge-preserving regularization and a priori interfaces

S U M M A R Y Conventional traveltime seismic tomography methods with Tikhonov regularization (L2 norm) typically produce smooth models, but these models may be inappropriate when subsurface structure contains discontinuous features, such as faults or fractures, indicating that tomographic models should contain sharp boundaries. For this reason, we develop a doubledifference (DD) traveltime tomography method that uses a modified total-variation regularization scheme incorporated with a priori information on interfaces to preserve sharp property contrasts and obtain accurate inversion results. In order to solve the inversion problem, we employ an alternating minimization method to decouple the original DD tomography problem into two separate subproblems: a conventional DD tomography with Tikhonov regularization and a L2 total-variation inversion. We use the LSQR linear solver to solve the Tikhonov inversion and the split-Bregman iterative method to solve the total-variation inversion. Through our numerical examples, we show that our new DD tomography method yields more accurate results than the conventional DD tomography method at almost the same computational cost.

[1]  Richard M. Jones Future of NSF to be reviewed , 1992 .

[2]  D. Oldenburg,et al.  3-D inversion of gravity data , 1998 .

[3]  Clifford H. Thurber,et al.  Development and Applications of Double-difference Seismic Tomography , 2006 .

[4]  M. Sambridge Geophysical inversion with a neighbourhood algorithm—I. Searching a parameter space , 1999 .

[5]  Monica Maceira,et al.  Joint Inversion of Body-Wave Arrival Times and Surface-Wave Dispersion for Three-Dimensional Seismic Structure Around SAFOD , 2014, Pure and Applied Geophysics.

[6]  W. Menke Geophysical data analysis : discrete inverse theory , 1984 .

[7]  D. Oldenburg,et al.  Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method , 2003 .

[8]  Gene H. Golub,et al.  Matrix computations , 1983 .

[9]  Tom Goldstein,et al.  The Split Bregman Method for L1-Regularized Problems , 2009, SIAM J. Imaging Sci..

[10]  Michael A. Saunders,et al.  Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems , 1982, TOMS.

[11]  A. Tikhonov,et al.  Numerical Methods for the Solution of Ill-Posed Problems , 1995 .

[12]  Colin Farquharson,et al.  Constructing piecewise-constant models in multidimensional minimum-structure inversions , 2008 .

[13]  C. Wijns,et al.  Interactive geophysical inversion using qualitative geological constraints , 2007 .

[14]  P. Hansen Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[15]  Lixin Shen,et al.  Proximity algorithms for the L1/TV image denoising model , 2011, Advances in Computational Mathematics.

[16]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[17]  Junfeng Yang,et al.  A New Alternating Minimization Algorithm for Total Variation Image Reconstruction , 2008, SIAM J. Imaging Sci..

[18]  R. Glowinski,et al.  Constrained optimization in seismic reflection tomography: a Gauss-Newton augmented Lagrangian approach , 2006 .

[19]  Clifford H. Thurber,et al.  Local earthquake tomography with flexible gridding , 1999 .

[20]  Peter G. Lelièvre,et al.  Gradient and smoothness regularization operators for geophysical inversion on unstructured meshes , 2013 .

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

[22]  Brendt Wohlberg,et al.  UPRE method for total variation parameter selection , 2010, Signal Process..

[23]  J. S. Hunter,et al.  Statistics for experimenters : an introduction to design, data analysis, and model building , 1979 .

[24]  Douglas W. Oldenburg,et al.  Integrating geological and geophysical data through advanced constrained inversions , 2009 .

[25]  Changsoo Shin,et al.  Waveform inversion using a logarithmic wavefield , 2006 .

[26]  A. Tarantola Inversion of seismic reflection data in the acoustic approximation , 1984 .

[27]  Remko Scharroo,et al.  Generic Mapping Tools: Improved Version Released , 2013 .

[28]  Michael Elad,et al.  Sparse and Redundant Representations - From Theory to Applications in Signal and Image Processing , 2010 .

[29]  Monica Maceira,et al.  Joint inversion of surface wave velocity and gravity observations and its application to central Asian basins shear velocity structure , 2009 .

[30]  C. Vogel Computational Methods for Inverse Problems , 1987 .

[31]  D. Oldenburg,et al.  Incorporating geological dip information into geophysical inversions , 2000 .

[32]  Michael A. Saunders,et al.  LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares , 1982, TOMS.

[33]  Dianne P. O'Leary,et al.  The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems , 1993, SIAM J. Sci. Comput..

[34]  Clifford H. Thurber,et al.  Double-Difference Tomography: The Method and Its Application to the Hayward Fault, California , 2003 .

[35]  Valéria C. F. Barbosa,et al.  Interactive 2D magnetic inversion: A tool for aiding forward modeling and testing geologic hypotheses , 2006 .

[36]  D. Oldenburg,et al.  A comprehensive study of including structural orientation information in geophysical inversions , 2009 .

[37]  Douglas W. Oldenburg,et al.  3-D inversion of magnetic data , 1996 .

[38]  Paul A. Rodríguez,et al.  Total Variation Regularization Algorithms for Images Corrupted with Different Noise Models: A Review , 2013, J. Electr. Comput. Eng..

[39]  L. Bregman The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[40]  Colin Farquharson Constructing Piecewise-constant Models In Multi-dimensional Minimum-structure Inversions , 2006 .

[41]  Ignace Loris,et al.  Iterative algorithms for total variation-like reconstructions in seismic tomography , 2012, 1203.4451.

[42]  J. Berryman Weighted least-squares criteria for seismic traveltime tomography , 1989 .

[43]  Shu-Huei Hung,et al.  A data‐adaptive, multiscale approach of finite‐frequency, traveltime tomography with special reference to P and S wave data from central Tibet , 2011 .

[44]  Wotao Yin,et al.  Bregman Iterative Algorithms for (cid:2) 1 -Minimization with Applications to Compressed Sensing ∗ , 2008 .

[45]  C. Farquharson,et al.  Geologically constrained gravity inversion for the Voisey's Bay ovoid deposit , 2008 .

[46]  James Foster,et al.  High-resolution locations of triggered earthquakes and tomographic imaging of Kilauea Volcano's south flank , 2010 .

[47]  Ignace Loris,et al.  Nonlinear regularization techniques for seismic tomography , 2008, J. Comput. Phys..

[48]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[49]  Jean-Paul Chilès,et al.  Wiley Series in Probability and Statistics , 2012 .

[50]  F. Waldhauser,et al.  A Double-Difference Earthquake Location Algorithm: Method and Application to the Northern Hayward Fault, California , 2000 .

[51]  Clifford H. Thurber,et al.  Parameter estimation and inverse problems , 2005 .

[52]  Michel Chouteau,et al.  3D gravity inversion using a model of parameter covariance , 2003 .

[53]  Michael K. Ng,et al.  A Fast Total Variation Minimization Method for Image Restoration , 2008, Multiscale Model. Simul..

[54]  Wei Lin,et al.  Fast MR Image Reconstruction for Partially Parallel Imaging With Arbitrary $k$ -Space Trajectories , 2011, IEEE Transactions on Medical Imaging.

[55]  M. Sambridge Geophysical inversion with a neighbourhood algorithm—II. Appraising the ensemble , 1999 .

[56]  Brendt Wohlberg,et al.  An Iteratively Reweighted Norm Algorithm for Minimization of Total Variation Functionals , 2007, IEEE Signal Processing Letters.

[57]  Åke Björck,et al.  Numerical methods for least square problems , 1996 .

[58]  Douglas W. Oldenburg,et al.  A 3D total magnetization inversion applicable when significant, complicated remanence is present , 2009 .

[59]  Ling-Yun Chiao,et al.  Multiscale seismic tomography , 2001 .

[60]  Steven J Franke,et al.  Critical level interaction of a gravity wave with background winds driven by a large-scale wave perturbation , 2009 .