The inverse problem of electromagnetic induction: Existence and construction of solutions based on incomplete data

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Abstract
A theory is described for the inversion of electromagnetic response data associated with one‐dimensional electrically conducting media. The data are assumed to be in the form of a collection of (possibly imprecise) complex admittances determined at a finite number of frequencies. We first solve the forward problem for conductivity models in a space of functions large enough to include delta functions. Necessary and sufficient conditions are derived for the existence of solutions to the inverse problem in this space. The approach relies on a representation of real‐part positive functions due to Cauer and an application of Sabatier's theory of constrained linear inversion. We find that delta‐function models are fundamental to the problem. When existence of a solution has been established for a given set of data, actual conductivities fitting the measurements may be explicitly constructed for various special classes of functions. For a solution in delta functions or homogeneous layers a development as a continued fraction is the essential idea; smoothly varying models are found with an adaption of Weidelt's analytic solution.

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