Regularized solution of a nonlinear problem in electromagnetic sounding

Non destructive investigation of soil properties is crucial when trying to identify inhomogeneities in the ground or the presence of conductive substances. This kind of survey can be addressed with the aid of electromagnetic induction measurements taken with a ground conductivity meter. In this paper, starting from electromagnetic data collected by this device, we reconstruct the electrical conductivity of the soil with respect to depth, with the aid of a regularized damped Gauss–Newton method. We propose an inversion method based on the low-rank approximation of the Jacobian of the function to be inverted, for which we develop exact analytical formulae. The algorithm chooses a relaxation parameter in order to ensure the positivity of the solution and implements various methods for the automatic estimation of the regularization parameter. This leads to a fast and reliable algorithm, which is tested on numerical experiments both on synthetic data sets and on field data. The results show that the algorithm produces reasonable solutions in the case of synthetic data sets, even in the presence of a noise level consistent with real applications, and yields results that are compatible with those obtained by electrical resistivity tomography in the case of field data.

[1]  C. G. Broyden.  A Class of Methods for Solving Nonlinear Simultaneous Equations , 1965 .

[2]  I. J. Won,et al.  GEM‐2: A New Multifrequency Electromagnetic Sensor , 1996 .

[3]  M. A. Pérez-Flores,et al.  Imaging of 3D electromagnetic data at low-induction numbers , 2012 .

[4]  Marlis Hochbruck,et al.  On the convergence of a regularizing Levenberg–Marquardt scheme for nonlinear ill-posed problems , 2010, Numerische Mathematik.

[5]  Lea Fleischer,et al.  Regularization of Inverse Problems , 1996 .

[6]  V. A. Morozov,et al.  Methods for Solving Incorrectly Posed Problems , 1984 .

[7]  P. M. van den Berg,et al.  An apparent‐resistivity concept for low‐frequency electromagnetic sounding techniques , 2000 .

[8]  R. W. Groom,et al.  Vertical Spatial Sensitivity and Exploration Depth of Low‐Induction‐Number Electromagnetic‐Induction Instruments , 2007 .

[9]  I. J. Won,et al.  Automated anomaly picking from broadband electromagnetic data in an unexploded ordnance (UXO) survey , 2003 .

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

[11]  Per Christian Hansen,et al.  Regularization Tools version 4.0 for Matlab 7.3 , 2007, Numerical Algorithms.

[12]  M. Dabas,et al.  Detection of resistive features using towed slingram electromagnetic induction instruments , 2009 .

[13]  Toke Koldborg Jensen,et al.  An adaptive pruning algorithm for the discrete L-curve criterion , 2007 .

[14]  Lothar Reichel,et al.  Regularization parameter determination for discrete ill-posed problems , 2015, J. Comput. Appl. Math..

[15]  C. Vogel.  Non-convergence of the L-curve regularization parameter selection method , 1996 .

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

[17]  J. D. Mcneill.  Electromagnetic Terrain Conduc-tivity Measurement at Low Induction Numbers , 1980 .

[18]  T. Uram,et al.  Tikhonov regularization of electrical conductivity depth profiles in field soils , 1997 .

[19]  Lothar Reichel,et al.  Old and new parameter choice rules for discrete ill-posed problems , 2013, Numerical Algorithms.

[20]  G. Rodriguez,et al.  An algorithm for estimating the optimal regularization parameter by the L-curve , 2005 .

[21]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[22]  M. Hanke.  A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems , 1997 .

[23]  F. A. Monteiro Santos,et al.  Inversion of Multiconfiguration Electromagnetic (DUALEM‐421) Profiling Data Using a One‐Dimensional Laterally Constrained Algorithm , 2010 .

[24]  Claudio Estatico,et al.  Preconditioned iterative regularization in Banach spaces , 2013, Comput. Optim. Appl..

[25]  M. Hanke.  Limitations of the L-curve method in ill-posed problems , 1996 .

[26]  Gregory M. Schultz,et al.  Inversion of inductive electromagnetic data in highly conductive terrains , 2005 .

[27]  Qinian Jin.  On a class of frozen regularized Gauss-Newton methods for nonlinear inverse problems , 2010, Math. Comput..

[28]  J. Nagy,et al.  Quasi-Newton approach to nonnegative image restorations , 2000 .

[29]  L. Pellerin.  Applications Of Electrical And Electromagnetic Methods For Environmental And Geotechnical Investigations , 2002 .

[30]  Brian R. Spies,et al.  5. Electromagnetic Sounding , 1991 .

[31]  Hilda Patricia Martinelli,et al.  Small-loop electromagnetic induction for environmental studies at industrial plants , 2010 .

[32]  R. D. Murphy,et al.  Iterative solution of nonlinear equations , 1994 .

[33]  L. Sambuelli,et al.  Study of riverine deposits using electromagnetic methods at a low induction number , 2007 .

[34]  Patricia Martinelli,et al.  Laterally filtered 1D inversions of small-loop, frequency-domain EMI data from a chemical waste site , 2008 .

[35]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

[36]  Induction-response functions for frequency-domain electromagnetic mapping system for airborne and ground configurations , 2007 .

[37]  R. Barker,et al.  Practical techniques for 3D resistivity surveys and data inversion1 , 1996 .

[38]  J. Stafford,et al.  Depth sounding with the EM38-detection of soil layering by inversion of apparent electrical conductivity measurements. , 2007 .

[39]  I. J. Won,et al.  Real‐time resistivity sounding using a hand‐held broadband electromagnetic sensor , 2003 .

[40]  T. Dahlin,et al.  A comparison of smooth and blocky inversion methods in 2D electrical imaging surveys , 2001 .

[41]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[42]  R. Barker,et al.  Rapid least-squared inversion of apparent resisitivity pseudosections by a quasi-Newton method , 1996 .

[43]  T. Dahlin,et al.  A numerical comparison of 2D resistivity imaging with 10 electrode arrays , 2004 .

[44]  E. Bonomi,et al.  Inversion of electrical conductivity data with Tikhonov regularization approach: some considerations , 2003 .

[45]  R. Yao,et al.  Quantitative evaluation of soil salinity and its spatial distribution using electromagnetic induction method. , 2010 .

[46]  James G. Nagy,et al.  An Efficient Iterative Approach for Large-Scale Separable Nonlinear Inverse Problems , 2009, SIAM J. Sci. Comput..

[47]  Jeffrey G. Paine,et al.  Determining salinization extent, identifying salinity sources, and estimating chloride mass using surface, borehole, and airborne electromagnetic induction methods , 2003 .

[48]  Hassane Sadok,et al.  A new L-curve for ill-posed problems , 2008 .

[49]  Scott M. Lesch,et al.  Spatial Prediction of Soil Salinity Using Electromagnetic Induction Techniques: 1. Statistical Prediction Models: A Comparison of Multiple Linear Regression and Cokriging , 1995 .

[50]  Harry Vereecken,et al.  Efficient loop antenna modeling for zero-offset, off-ground electromagnetic induction in multilayered media , 2010 .

[51]  Teresa Reginska,et al.  A Regularization Parameter in Discrete Ill-Posed Problems , 1996, SIAM J. Sci. Comput..

[52]  P. Dietrich,et al.  Noninvasive Monitoring of Soil Static Characteristics and Dynamic States: A Case Study Highlighting Vegetation Effects on Agricultural Land , 2012 .

[53]  Ana Osella,et al.  3D electrical imaging of an archaeological site using electrical and electromagnetic methods , 2005 .

[54]  J. P. Greenhouse,et al.  THE USE OF RECONNAISSANCE ELECTROMAGNETIC METHODS TO MAP CONTAMINANT MIGRATION , 1983 .

[55]  Yutaka Sasaki,et al.  Multidimensional inversion of loop-loop frequency-domain EM data for resistivity and magnetic susceptibility , 2010 .

[56]  Jan M. H. Hendrickx,et al.  Inversion of Soil Conductivity Profiles from Electromagnetic Induction Measurements , 2002 .

[57]  I. J. Won,et al.  Coaxial coil towed EMI sensor array for UXO detection and characterization , 2007 .

[58]  M. Pastorino,et al.  A Novel Microwave Imaging Approach Based on Regularization in $L^{p}$ Banach Spaces , 2012, IEEE Transactions on Antennas and Propagation.

[59]  HansenPer Christian.  The truncated SVD as a method for regularization , 1987 .

[60]  Andrew Binley,et al.  Electrical resistance tomography. , 2000 .

[61]  A. Osella,et al.  EMI data from an archaeological resistive target revisited , 2006 .

[62]  Dennis L. Corwin,et al.  Characterizing soil spatial variability with apparent soil electrical conductivity , 2005 .

[63]  Irma Shamatava,et al.  Theoretical analysis and range of validity of TSA formulation for application to UXO discrimination , 2004, IEEE Transactions on Geoscience and Remote Sensing.

[64]  M. Saunders,et al.  Towards a Generalized Singular Value Decomposition , 1981 .

[65]  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..

[66]  Walter L. Anderson,et al.  Computer program; numerical integration of related Hankel transforms of orders O and 1 by adaptive digital filtering , 1979 .

[67]  Frederick D. Day-Lewis,et al.  Inversion of multi-frequency electromagnetic induction data for 3D characterization of hydraulic conductivity , 2011 .