On the electrical reflectivity tensor in d.c. electric field modelling

Abstract

ABSTRACTWe use the integral equation for a d.c. electric field, published in the literature, to introduce the concept of the electrical reflectivity tensor into d.c. electric field modelling. It is shown that in d.c. electric field modelling, the electric reflectivity tensor can be obtained in exactly the same way as in electromagnetic modelling. As a result, for a d.c. electric field, the quasi‐linear and the quasi‐analytical approximations, as well as the quasi‐analytical series, can be constructed in exactly the same way as in electromagnetic modelling. If the primary field is uniform, and if the anomalous body is a uniform circular cylinder or a uniform sphere, the reflectivity tensor is zero order (constant), relating to the free surface charge density. Thus, for some homogeneous bodies that have simple shapes and are embedded in a uniform primary field, the electrical reflectivity tensor is not only a mathematical mechanism for obtaining approximate solutions, but also a physical reality. Indeed, the free surface charge density is defined as the change of the electric displacement vector across the boundary surface under consideration. If the primary field is caused by a point source, and if the anomalous body is a uniform sphere, the reflectivity tensor is second order, varying slowly within the sphere. The relationship to the free surface charge density can be established only when both the reflectivity tensor and the free surface charge density are approximated by the first terms of their series solutions. If the point source is far from the centre of the sphere, the corresponding reflectivity tensor reduces to zero order, and is independent of the observation position within the sphere, i.e. it is a constant. Therefore, the basic idea of the quasi‐analytical approximation, i.e. taking the reflectivity tensor outside the integral operator, is justified in the case considered here.