Electromagnetic induction in thin sheet conductivity anomalies at the surface of the earth

Lateral variations in the Earth´s conductivity complicate considerably the calculation of the electromagnetic response of the Earth to an external inducing field which is uniform and horizontal. Although analytic solutions have been found for a few simple two-dimensional models in which the conductivity varies in one horizontal direction only, it is necessary, in general, to resort to numerical methods. If the conductivity variations of interest are confined to a surface layer it is often possible to represent the Earth mathematically as a uniform conducting half-space covered by an infinitely thin sheet of variable surface conductance. This simplification effectively reduces by one the number of dimensions over which the field equations need to be integrated numerically. It is shown that for a two-dimensional model the horizontal component of the electric field satisfies an integral equation on the surface of the thin sheet, which can be solved numerically for arbitrary sheet conductance. The accuracy of the numerical procedure is confined by applying it to E- and B-polarization induction in two adjacent half-sheets and then comparing the solution obtained with known analytic solutions of the same problem. In three-dimensions the two horizontal components of the surface electric field satisfy a coupled pair of double integral equations which can also be solved numerically for an arbitrarily varying conductance of the surface sheet.