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

Abstract

We attempted to develop a direct time-domain, finite-difference solution for the electromagnetic 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.

Novel inverse methods for wire fault detection and diagnosis

“…Because it is a time domain method, solutions can cover a wide frequency range with a single simulation run [7].…”

A finite‐difference, time‐domain solution for three‐dimensional electromagnetic modeling

“…Based on the method, Oristaglio and Hohmann (1984) presented a solution to a 2-D problem. Later, Adhidjaja and Hohmann (1989) applied it to a 3-D problem in which they attempted without success to solve the second-order differential equations for the magnetic field.…”

“…To prevent the fictitious displacement current from dominating the diffusive EM field, we must constrain the length of the time step. One can show that the field retains its diffusive nature if Hohmann, 1984 andHohmann, 1989) ( ~m in at) 1/2 si« -6 -Amin , (20) where a can be taken as the minimum conductivity in the model. In practice, one can use a time step…”

“…First, we interpolate, using a bicubic spline function (see also Adhidjaja and Hohmann, 1989), the nonuniform grid spanned by the discrete b z values on the ground surface to a constant grid with grid spacing 8. Then we Fourier transform b ; into the wavenumber domain and multiply the results by the operating coefficients in the equations.…”

“…For example, Adhidjaja and Hohmann (1989) solved the second Solving three-dimensional (3-D) transient electromagnetic order equations obtained by eliminating the electric field (TEM) problems is important in understanding the physics of from Maxwell's equations, and found serious difficulty in observed responses, and in providing insight for data inter accurately evaluating the derivatives of the physical proper pretation. This paper describes a finite-difference solution to ties.…”