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

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.

Geophysical images of the deep crust: the Iapetus suture

“…The H+ solutions provided by Parker (1980) offer a means of constructing layered models with a χ 2 misfit that can be made as close as possible to the minimum value. A systematic family of H+ models possessing two distinct end members can thus be generated.…”

“…Such algorithms possess an inherent weakness in their ability to provide systematic inferences about real Earth structure. The difficulties of model construction can be overcome using the fully non-linear solution to the MT problem given by Parker (1980Parker ( , 1983 and Parker & Whaler (1981). The solutions provide three classes of model, of which we consider here only two, referred to as D+ and H+.…”

“…The theory presented by Parker (1980) establishes the fact that the optimum (best-fit) model we can generate from the data is a delta-function (D+) model. This model consists of a finite number of delta-functions (layers of zero thickness but finite conductance), separated by perfect insulators.…”

The use of the D+ solution in magnetotelluric interpretation

“…Using equation ( 2 ), Parker (1980) provides a complete theory for the existence of valid solutions and constructs the ideal geoelectric profile called the DPLUS (D+) solution. This class of solution comprises a finite number of delta functions (layers of zero thickness but finite conductance) separated by perfect insulators.…”

“…A brief background to the D+ solution due to Parker (1980) and Parker and Whaler (1981) is given. For the data set considered, adequate o+ solutions misfits can be obtained for the vast majority of soundings.…”