Large-scale Inversion of Magnetic Data Using Golub-Kahan Bidiagonalization with Truncated Generalized Cross Validation for Regularization Parameter Estimation


Assistant Professor, Department of Earth Physics, Institute of Geophysics, University of Tehran, Iran


In this paper a fast method for large-scale sparse inversion of magnetic data is considered. The L1-norm stabilizer is used to generate models with sharp and distinct interfaces. To deal with the non-linearity introduced by the L1-norm, a model-space iteratively reweighted least squares algorithm is used. The original model matrix is factorized using the Golub-Kahan bidiagonalization that projects the problem onto a Krylov subspace with a significantly reduced dimension. The model matrix of the projected system inherits the ill-conditioning of the original matrix, but the spectrum of the projected system accurately captures only a portion of the full spectrum. Equipped with the singular value decomposition of the projected system matrix, the solution of the projected problem is expressed using a filtered singular value expansion. This expansion depends on a regularization parameter which is determined using the method of Generalized Cross Validation (GCV), but here it is used for the truncated spectrum. This new technique, Truncated GCV (TGCV), is more effective compared with the standard GCV method. Numerical results using a synthetic example and real data demonstrate the efficiency of the presented algorithm.


Main Subjects

Boulanger, O. and Chouteau, M., 2001, Constraint in 3D gravity inversion. Geophysical prospecting, 49, 265-280.
Chung, J., Nagy, J. G. and O’Leary, D. P., 2008, A weighted GCV method for Lanczos hybrid regularization. ETNA, 28, 149-167.
Farquharson, C. G., 2008, Constructing piecewise-constant models in multidimensional minimum-structure inversions. Geophysics, 73, K1-K9.
Farquharson, C. G. and Oldenburg, D. W., 2004, A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems. Geophys. J. Int., 156, 411-425.
Gazzola, S. and Nagy, J. G., 2014, Generalized Arnoldi_Tikhonov method for sparse reconstruction. SIAM J. Sci. Comput., 36(2), B225-B247.
Golub, G. H., Heath, M. and Wahba, G., 1979, Generalized Cross Validation as a method for choosing a good ridge parameter. Technometrics, 21(2), 215-223.
Golub, G. H. and Van Loan, C., 1996, Matrix computation, 3rd edition, John Hopkins University Press, Baltimore.
Hansen, P. C., 2007, Regularization Tools: A Matlab package for analysis and solution of discrete ill-Posed problems version 4.1 for Matlab 7.3. Numerical Algorithms, 46, 189-194.
Kilmer, M. E. and O’Leary, D. P., 2001, Choosing regularization parameters in iterative methods for ill-posed problems. SIAM Journal on Matrix Analysis and Application, 22, 1204-1221.
Last, B. J. and Kubik, K., 1983, Compact gravity inversion. Geophysics, 48, 713-721.
Li, Y. and Oldenburg, D. W., 1996, 3-D inversion of magnetic data. Geophysics, 61, 394-408.
Li, Y. and Oldenburg, D. W., 2003, Fast inversion of large-scale magnetic data using wavelet transform and a logarithmic barrier method. Geophys. J. Int., 152, 251-265.
Liu, S., Hu, X., Xi, Y., Liu, T. and Xu, S., 2015, 2D sequential inversion of total magnitude and total magnetic anomaly data affected by remanent magnetization, Geophysics, 80, K1-K12.
Loke, M. H., Acworth, I. and Dahlin, T., 2003, A comparison of smooth and blocky inversion methods in 2D electrical imaging surveys. Exploration Geophysics, 34, 182-187.
Paige, C. C. and Saunders, M. A., 1982a, LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software, 8, 43-71.
Paige, C. C. and Saunders, M. A., 1982b, ALGORITHM 583 LSQR: Sparse linear equations and least squares problems. ACM Trans. Math. Software, 8, 195-209.
Pilkington, M., 1997, 3-D magnetic imaging using conjugate gradients. Geophysics, 62, 1132-1142.
Pilkington, M., 2009, 3D magnetic data-space inversion with sparseness constraint. Geophysics, 74, L7-L15.
Portniaguine, O. and Zhdanov, M. S., 1999, Focusing geophysical inversion images. Geophysics, 64, 874-887.
Portniaguine, O. and Zhdanov, M. S., 2002, 3-D magnetic inversion with data compression and image focusing. Geophysics, 67, 1532-1541.
Rao, D. B. and Babu, N. R., 1991, A rapid methods for three-dimensional modeling of magnetic anomalies. Geophysics, 56, 1729-1737.
Renaut R. A., Vatankhah, S. and Ardestani V. E., 2017, Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems. SIAM Journal on Scientific Computing, 39, 2, B221-B243.
Sun, J. and Li, Y., 2014, Adaptive Lp inversion for simultaneous recovery of both blocky and smooth features in geophysical model. Geophys. J. Int., 197, 882-899.
Vatankhah, S., Ardestani V. E. and Renaut, R. A., 2014, Automatic estimation of the regularization parameter in 2-D focusing gravity inversion: application of the method to the Safo manganese mine in northwest of Iran. Journal of Geophysics and Engineering, 11, 045001.
Vatankhah, S., Ardestani V. E. and Renaut, R. A., 2015, Application of the  principle and unbiased predictive risk estimator for determining the regularization parameter in 3D focusing gravity inversion. Geophys. J. Int., 200(1), 265-277.
Vatankhah, S., Renaut, R. A. and Ardestani, V. E., 2017, 3-D Projected L1 inversion of gravity data using truncated unbiased predictive risk estimator for regularization parameter estimation. Geophys. J. Int., 210 (3), 1872-1887.
Voronin, S., 2012, Regularization of linear systems with sparsity constraints with application to large scale inverse problems. Ph.D. thesis, Princeton University, U.S.A.
Wohlberg, B. and Rodriguez, P., 2007, An iteratively reweighted norm algorithm for minimization of total variation functionals. IEEE Signal Processing Letters, 14, 948-951.