On closed-form expressions for the approximate electromagnetic response of a sphere interacting with a thin sheet — Part 2: Theory in the moment domain, validation, and examples

Abstract

Fully analytical formulas are derived for the approximate electromagnetic response of a sphere interacting with a thin sheet in the moment domain. The moment-domain expressions are found to be expressed as simple polynomials of hyperbolic functions. These are significantly simpler to evaluate than the frequency- and time-domain expressions and therefore provide an attractive alternative for modeling. An efficient procedure is outlined for generalizing the moment-domain expression to bipolar-repetitive waveforms. This procedure is validated on a synthetic test example and field data from the Reid-Mahaffy test site, in northern Ontario. Here, results are found in agreement with the work of previous studies. The analytical time-domain procedure is validated through synthetic test examples. The asymptotic formulas for the time-domain expressions are found to significantly reduce the number of required function evaluations, especially for models in which the sphere is not too shallow or not too big or conductive. For example, for a sphere (and overburden) of conductivity (and conductance) of [Formula: see text] (and [Formula: see text]), if the sphere radius is three times smaller than the depth to the top, the amount of required function evaluations is halved; when the ratio is two or less, the asymptotic formulas do not reduce the amount of function evaluations, for the tolerances chosen here. Less-strict tolerances will lead to a further reduction of the required function evaluations.