Multiple-order moments of the transient electromagnetic response of a one-dimensional earth with finite conductance – theory

Abstract

The concept of moments of the electromagnetic response is useful in electromagnetic interpretation. Analytic formulae exist for low-order moments of a few conductivity models, enabling source parameters such as time constant, depth, conductance and conductivity to be estimated from the measured moments of the electromagnetic response. However, most models for which analytic formulae exist have conductivity varying abruptly as a function of depth or position. In this paper, we have derived a procedure that allows moments of any order to be calculated for a conductivity which has finite conductance but can otherwise vary arbitrarily with depth. The horizontal loop transient electromagnetic step response is computed as a sum of residues. Integration of the step response over time yields a mathematical expression for a moment of any order. We illustrate the procedure for a Gaussian conductivity function which varies smoothly with depth. The Gaussian model produces results that agree in specific limits with the thin sheet, thick sheet and half-space cases.