Approximate semianalytical solutions for the electromagnetic response of a dipping-sphere interacting with conductive overburden

Abstract

Electromagnetic exploration methods have important applications for geologic mapping and mineral exploration in igneous and metamorphic terranes. In such cases, the earth is often largely resistive and the most important interaction is between a conductor of interest and a shallow, thin, horizontal sheet representing glacial tills and clays or the conductive weathering products of the basement rocks (both of which are here termed the “conductive overburden”). To this end, we have developed a theory from which the step and impulse responses of a sphere interacting with conductive overburden can be quickly and efficiently approximated. The sphere model can also be extended to restrict the currents to flow in a specific orientation (termed the dipping-sphere model). The resulting expressions are called semianalytical because all relevant relations are developed analytically, with the exception of the time-convolution integrals. The overburden is assumed to not be touching the sphere, so there is no galvanic interactions between the bodies. We make use of the dipole sphere in a uniform field and thin sheet approximations; however, expressions could be obtained for a sphere in a dipolar (or nondipolar) field using a similar methodology. We have found that there is no term related to the first zero of the relevant Bessel function in the response of the sphere alone. However, there are terms for all other zeros. A test on a synthetic model shows that the combined sphere-overburden response can be reasonably approximated using the first-order perturbation of the overburden field. Minor discrepancies between the approximate and more elaborate numerical responses are believed to be the result of numerical errors. This means that in practice, the proposed approach consists of evaluating one convolution integral over a sum of exponentials multiplied by a polynomial function. This results in an extremely simple algorithmic implementation that is simple to program and easy to run. The proposed approach also provides a simple method that can be used to validate more complex algorithms. A test on field data obtained at the Reid Mahaffy site in Northern Ontario shows that our approximate method is useful for interpreting electromagnetic data even when the background is thick. We use our approach to obtain a better estimate of the geometry and physical properties of the conductor and evaluate the conductance of the overburden.