Numerical methods for establishing solutions to the inverse problem of electromagnetic induction

Publisher Website
Google Scholar
Abstract
A previous paper (Parker, 1980) sets out a theory for deciding whether solutions exist to the inverse problem of electromagnetic induction and outlines methods for constructing conductivity profiles when their existence has been demonstrated. The present paper provides practical algorithms to perform the necessary calculations stably and efficiently, concentrating exclusively on the case of imprecise observations. The matter of existence is treated by finding the best fitting solution in a least squares sense; then the size of the misfit is tested statistically to determine the probability that the value would be met or exceeded by chance. We obtain the optimal solution by solving a constrained least squares problem linear in the spectral function of the electric field differential equation. The spectral function is converted into a conductivity profile by transforming its partial fraction representation into a continued fraction, using a stable algorithm due to Rutishauser. In addition to optimal models, which always consist of delta functions, two other types of model are examined. One is composed of a finite stack of uniform layers, constructed so that the product of conductivity and thickness squared is the same in each layer. The numerical techniques developed for the optimal model serve with only minor alteration to find solutions in this class. Models of the second kind are smooth. A special form of the response is chosen so that the kernel functions of the Gel'fand‐Levitan integral equation are degenerate, thus allowing very stable and numerically efficient solution. Unlike previously published methods for finding conductivity models, these algorithms can provide solutions with misfits arbitrarily close to the smallest one possible. The methods are applied to magnetotelluric observations made by Larsen in Hawaii.