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.
Interpretation of long-offset transient electromagnetic data from Mount Merapi, Indonesia, using a three-dimensional optimization approach
