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

Abstract

An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homoge We have developed a finite-difference solution for neous Dirichlet condition is employed along the subsur three-dimensional (3-D) transient electromagnetic face boundaries. Numerical dispersion is alleviated by problems. The solution steps Maxwell's equations in using an adaptive algorithm that uses a fourth-order dif time using a staggered-grid technique. The time-step ference method at early times and a second-order meth ping uses a modified version of the Du Fort-Frankel od at other times. Numerical checks against analytical, method which is explicit and always stable. Both integral-equation, and spectral differential-difference so conductivity and magnetic permeability can be func lutions show that the solution provides accurate results. tions of space, and the model geometry can be arbi Execution time for a typical model is about 3.5 trarily complicated. The solution provides both elec hours on an IBM 3090/600S computer for computing tric and magnetic field responses throughout the earth. the field to 10 ms. That model contains 100 x 100 x 50 Because it solves the coupled, first-order Maxwell's grid points representing about three million unknowns equations, the solution avoids approximating spatial and possesses one vertical plane of symmetry, with derivatives of physical properties, and thus overcomes the smallest grid spacing at 10 m and the highest many related numerical difficulties. Moreover, since resistivity at 100 n . m. The execution time indicates the divergence-free condition for the magnetic field is that the solution is computer intensive, but it is valu incorporated explicitly, the solution provides accurate able in providing much-needed insight about TEM results for the magnetic field at late times. responses in complicated 3-D situations.

Simulation of Airborne Electromagnetic Measurements in Three Dimensional Environments

“…The above formulation requires that the conductivity, dielectric permittivity and magnetic permeability be computed halfway along a given cell edge in Figure 1. Wang and Hohmann (1993) showed that an average conductivity (and pennitivity) can be evaluated by tracing out a line integral of the magnetic field centered on the midpoint of the cell edge. The resulting conductivity is simply a weighted sum of conductivities of the four adjoining cells.…”

A hybrid finite-difference and integral-equation method for modeling and inversion of marine controlled-source electromagnetic data

“…The conductivities where the electric fields are located are represented by a weighted average of conductivities of the four adjoining cells based on Ampere's law (Wang and Hohmann, 1993;Alumbaugh et al, 1996).…”

Simulation of Electromagnetic Diffusion in Anisotropic Media

“…To apply the boundary condition in 3-D, I use the upwardcontinuation scheme of Wang and Hohmann [32] to compute E 1 and E 2 above the surface using E 3 on the surface:…”

“…Maaø [20] performed 3-D simulations with a FD method by using a mathematical transformation similar to the Kelvin-Voigt viscoelastic model. This model introduces a high permittivity as in the case of Wang and Hohmann [32]. Maaø [20] further reduced the computer time solving the equations in the high-frequency range by using a complexfrequency Fourier transform to filter high-frequency wave-like signals.…”

“…Wang and Hohmann [32] generalized the 2-D algorithm of Oristaglio and Hohmann [25] to three dimensions. They developed a magnetic/electric formulation of Maxwell's equations, using as a solver a staggered grid and a modified version of the Du Fort-Frankel method.…”