Electromagnetic induction in thin sheets: integral equations and model studies in two dimensions

Abstract

The model consists of a thin sheet of variable conductance in one horizontal direction above a conducting substructure of lateral uniformity. Transition anomalies are permitted, i.e. The anomalous range of varying conductance may be bounded by uniform half sheets of different conductances. For E‐polarization the inducing external source field may be non‐uniform. Two complementary integral equations are derived in the frequency‐distance domain, to find the anomalous electric field or alternatively to find the anomalous sheet current density for each polarization. The equations involve two sets of response functions for the normal structure outside the anomalous range. A first set generates for a given source field at ground level the internal magnetic field by induction and the normal electric field which serves as an input function for the derivation of the anomalous field. A second set accounts for the inductive coupling of the anomalous fields in sheet and substrucutre. Various tests are performed: for the compatibility of complementary solutions, for the correct asymptotic behaviour at infinity in the horizontal direction, and for the accuracy of numerical integrations as controlled by the grid‐point spacing. The numerical solutions are also tested against analytical solutions and against numerical solutions by other methods, in particular with regard to the validity of the thin‐sheet approximation. A comparison with results from integral equations, which involve conventional Green's functions, reveals in the case of B‐polarization complications where a strong influx of currents exists from the substructure into the thin sheet. A source transfer function is introduced which allows the extension of the response function concept to induction by non‐uniform sources. Two types are studied: a stationary jet field source and a travelling Sq source above a sedimentary basin and a coastline. Resulting magnetotelluric and geomagnetic deep sounding responses are compared with the respective responses, when the inducing field is quasi‐uniform. Copyright © 1995, Wiley Blackwell. All rights reserved