A simple approximate representation of the spectral response of an arbitrary kind of electromagnetic (EM) prospecting system to a small conductive target in a conductive environment has been derived. The representation contains the direct response from the layered host medium and the first-order effects of eddy current induction, current channeling, magnetic induction, and the coupling between eddy current and magnetic inductions in the anomalous body, as modified by the host medium. The only significant computational task in the representation is evaluation of a few Green's functions for the host medium.As a guide to establishing proper approximations, a fundamental study of integral equations is presented. Very simple solutions for the secondary or scattering sources which represent the EM effect of the body are obtained for a few basic cases. Equations for more general cases are complicated by additional terms in the Green's functions which represent ac interaction between scattering sources and the host medium, the effects of layering in the host, or interactions between the different types of scattering sources.Through a supplementary study of the response of a conductive disk embedded in a conductive host to an axisymmetric field, the ac interaction between the host medium and scattering sources in the disk is shown to be relatively unimportant. Hence, interaction with layering in the host is also minimal. Green's functions in the integral equation can consequently be simplified.The representation as a whole has been tested by comparing its predictions of the spectral form of a response with two published data sets, responses of a plate model in a conductive host. The fits for targets of moderate and large size, relative to the scale of the particular experiment, range from excellent to good qualitative resemblance.The representation should find application in making expensive numerical or analog model studies more useful and understandable to the field geophysicist, in converting real and model data between the time- and frequency-domain formats, in constructing practical inverse algorithms, and in predicting the form of EM effects in induced polarization and magnetometric resistivity surveys.

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