Three-dimensional induction in a non-uniform thin sheet at the surface of a uniformly conducting earth

Abstract

Summary. A new method for solving problems in three-dimensional electromagnetic induction in which the Earth is represented by a uniformly conducting half-space overlain by a surface layer of variable conductance is presented. Unlike previous treatments of this type of problem the method does not require the fields to be separated into their normal and anomalous parts, nor is it necessary to assume that the anomalous region is surrounded by a uniform structure; the model may approach either an E- or a B-polarization configuration at infinity. The solution is expressed as a vector integral equation in the horizontal electric field at the surface. The kernel of the integral is a Green’s tensor which is expressed in terms of elementary functions that are independent of the conductance. The method is applied to an illustrative model representing an island near a bent coastline which extends to infinity in perpendicular directions. None of the methods presently available for solving local problems in electromagnetic induction by a uniform source in a region of the Earth where the conductivity varies in three dimensions may be regarded as completely satisfactory. The finitedifference method of Lines & Jones (1973a,b) requires a great deal of computer time and storage and although it can be used to analyse models that approach a two-dimensional configuration at large distances in the two directions parallel to the electric field of the uniform inducing source, it is not capable of handling similar models that have a two-dimensional limit in the B-polarization rather than in the E-polarization mode. More efficient procedures have been developed (Raiche 1974; Weidelt 1975)’ but they are somewhat more restrictive in their applicability since they require the anomalous region to be entirely surrounded by a ‘normal’ (Le. layered) structure. It appears that no method has yet been devised for the treatment of the completely general problem in which the model may tend to either an E-polarization limit in the directions parallel to the electric field of the source, or a B-polarization limit in the directions parallel to the magnetic field of the source, or even both together in the same model as illustrated schematically in Fig. 1. Configurations of this type are not far-fetched;