The three-dimensional (3-D) electromagnetic scattering problem is first formulated in the frequency domain in terms of an electric field volume integral equation. Three-dimensional responses are then Fourier transformed with sine and cosine digital filters or with the decay spectrum. The digital filter technique is applied to a sparsely sampled frequency sounding, which is replaced by a cubic spline interpolating function prior to convolution with the digital filters. Typically, 20 to 40 frequencies at five to eight points per decade are required for an accurate solution. A calculated transient is usually in error after it has decayed more than six orders in magnitude from early to late time. The decay spectrum usually requires ten frequencies for a satisfactory solution. However, the solution using the decay spectrum appears to be less accurate than the solution using the digital filters, particularly after early times. Checks on the 3-D solution include reciprocity and convergence checks in the frequency domain, and a comparison of Fourier-transformed responses with results from a direct time-domain integral equation solution.
The galvanic response of a 3-D conductor energized by a large rectangular loop is substantial when host currents are strong near the conductor. The more conductive the host, the longer the galvanic responses will persist. Large galvanic responses occur if a 3-D conductor is in contact with a conductive overburden. For a thin vertical dike embedded within a conductive host, the 3-D response is similar in form but differs in magnitude and duration from the 2-D response generated by two infinite line sources positioned parallel to the strike direction of the 2-D structure.
We have used the 3-D solution to study the application of the central-loop method to structural interpretation. The results suggest variations of thickness of conductive overburden and depth to sedimentary structure beneath volcanics can be mapped with one-dimensional inversion. Successful 1-D inversions of 3-D transient soundings replace a 3-D conductor by a conducting layer at a similar depth. However, other possibilities include reduced thickness and resistivity of the 1-D host containing the body. Many different 1-D models can be fit to a transient sounding over a 3-D structure. Near-surface, 3-D geologic noise will not permanently contaminate a central-loop apparent resistivity sounding. The noise is band-limited in time and eventually vanishes at late times.