A finitemdifference algorithm for the transient electromagnetic response of a three-dimensional body

We attempted to develop a direct time-domain, finite-difference solution for the electromag-netic response of a 3-D model. The algorithm is an extension of our 2-D modelling technique, which uses the Du Fort-Frankel finite-difference scheme. However, the vector nature of the field makes the 3-D problem much more complicated than its 2-D counterpart, and a supercomputer is required for computations. Unlike the 2-D case, where we solve for the electric field, the solution is formulated in terms of secondary magnetic field to avoid dealing with a discontinuous normal component of electric field. However, that difficulty is replaced with a problem involving the gradient of conductivity, which is discontinuous at interfaces. We experimented with both smoothing the conductivity variation, so that the gradient is well defined; and integrating the gradient terms, which results in a tangential current density contrast. Our limited experiments indicate that the magnetic field step response is computationally more stable than the impulse response, because it is a smoother function with smaller dynamic range. However, the step response takes longer to compute because its source, which involves the primary fields in the earth, requires very accurate numerical integration. Due to computer time and memory limitations on the supercomputer available to us, we were not able to develop an accurate numerical solution. We were able to carry out only a few tests on small models for which results were not in good agreement with values computed using a 3-D integral equation solution. Key words. Du Fort-Frankel finite difference, step response, 3-D modelling, transient electromagnetic method (TEM