Transient electromagnetic response of a sphere in a layered medium

Abstract

The integral equation for the electromagnetic response of a sphere in a layered medium may be solved as follows. First, the unknown time harmonic electric field in the sphere is expanded in spherical vector waves. Secondly, the coefficients for these wave functions are found by a set of equations. The equations are found by multiplying the integral equation throughout by each wave function and integrating over the spherical conductor.

Once the unknown coefficients have been determined, then the transient response may be found by taking the inverse Fourier transform. In carrying out the Fourier transform one learns that for most of the time range used in prospecting, only the lowest order vector wave function is significant. A study of the singularities of the spectrum of the transient shows that, for the time range considered, only a single branch cut is significant. There are no pole type responses. That is, the field does not decay exponentially. Previous studies of a sphere in free space reported only pole type responses. That is, at the later stages, the field decays exponentially. This study shows that, in order to model satisfactorily the effect of the host rock on transient electromagnetic fields, the sphere must be placed in layered ground.