zbMATH — the first resource for mathematics

Regularized inversion of multi-frequency EM data in geophysical applications. (English) Zbl 06981865
Ortegón Gallego, Francisco (ed.) et al., Trends in differential equations and applications. Selected papers based on the presentations at the XXIVth congress on differential equations and applications/XIVth congress on applied mathematics, Cádiz, Spain, June 8–12, 2015. Cham: Springer (ISBN 978-3-319-32012-0/hbk; 978-3-319-32013-7/ebook). SEMA SIMAI Springer Series 8, 357-369 (2016).
Summary: The purpose of this work is to detect or infer, by non destructive investigation of soil properties, inhomogeneities in the ground or the presence of particular conductive substances such as metals, minerals and other geological structures. A nonlinear model is used to describe the interaction between an electromagnetic field and the soil. Starting from electromagnetic data collected by a ground conductivity meter, we reconstruct the electrical conductivity of the soil with respect to depth by a regularized Gauss-Newton method. We propose an inversion method, based on the low-rank approximation of the Jacobian of the nonlinear model, which depends both on a relaxation parameter and a regularization parameter, chosen by automatic procedures. Our numerical experiments on synthetic data sets show that the algorithm gives satisfactory results when the magnetic permeability in the subsoil takes small values, even when the noise level is compatible with real applications. The inversion problem becomes much harder to solve if the value of the permeability increases substantially, that is in the presence of ferromagnetic materials.
For the entire collection see [Zbl 1350.35002].

65 Numerical analysis
Full Text: DOI
[1] Björck, A.A.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)
[2] Boaga, J., Ghinassi, M., D’Alpaos, A., Deidda, G.P., Rodriguez, G., Cassiani, G.: Unravelling the vestiges of ancient meandering channels in tidal landscapes via multi-frequency inversion of Electro-Magnetic data (2016, submitted)
[3] Borchers, B., Uram, T., Hendrickx, J. M.: Tikhonov regularization of electrical conductivity depth profiles in field soils. Soil Sci. Soc. Am. J. 61, 1004-1009 (1997)
[4] Deidda, G.P., Fenu, C., Rodriguez, G.: Regularized solution of a nonlinear problem in electromagnetic sounding. Inverse Prob. 30, 125014 (2014) · Zbl 1308.35294
[5] Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998) · Zbl 0890.65037
[6] Hansen, P.C., Jensen, T.K., Rodriguez, G.: An adaptive pruning algorithm for the discrete L-curve criterion. J. Comput. Appl. Math. 198, 483-492 (2007) · Zbl 1101.65044
[7] Hendrickx, J.M.H., Borchers, B., Corwin, D.L., Lesch, S.M., Hilgendorf, A.C., Schlue, J.: Inversion of soil conductivity profiles from electromagnetic induction measurements. Soil Sci. Soc. Am. J. 66, 673-685 (2002)
[8] McNeill, J.D.: Electromagnetic terrain conductivity measurement at low induction numbers. Technical Report TN-6 Geonics Limited (1980)
[9] Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970) · Zbl 0241.65046
[10] Reichel, L., Rodriguez, G.: Old and new parameter choice rules for discrete ill-posed problems. Numer. Algorithm. 63, 65-87 (2013) · Zbl 1267.65045
[11] Ward, S.H., Hohmann, G.W.: Electromagnetic theory for geophysical applications. In: Nabighian, M.N. (ed.) Electromagnetic Methods in Applied Geophysics, Volume 1. Theory, Volume 3 of Investigation in Geophysics, pp. 131-311. Society of Exploration Geophysicists, Tulsa, OK (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.