Diffusion of electromagnetic fields into a two‐dimensional earth: A finite‐difference approach

Abstract

We describe a numerical method for time‐stepping Maxwell’s equations in the two‐dimensional (2-D) TE‐mode, which in a conductive earth reduces to the diffusion equation. The method is based on the classical DuFort‐Frankel finite‐difference scheme, which is both explicit and stable for any size of the time step. With this method, small time steps can be used at early times to track the rapid variations of the field, and large steps can be used at late times, when the field becomes smooth and its rates of diffusion and decay slow down. The boundary condition at the earth‐air interface is handled explicitly by calculating the field in the air from its values at the earth’s surface with an upward continuation based on Laplace’s equation. Boundary conditions in the earth are imposed by using a large, graded grid and setting the values at the sides and bottom to those for a haft‐space. We use the 2-D model to simulate transient electromagnetic (TE) surveys over a thin vertical conductor embedded in a half‐space and in a half‐space with overburden. At early times (microseconds), the patterns of diffusion in the earth are controlled mainly by geometric features of the models and show a great deal of complexity. But at late times, the current concentrates at the center of the thin conductor and, with a large contrast (1000:1) between conductor and half‐space, produces the characteristic crossover and peaked anomalies in the surface profiles of the vertical and horizontal emfs. With a smaller contrast (100:1), however, the crossover in the vertical emf is obscured by the halfspace response, although the horizontal emf still shows a small peak directly above the target.