Electromagnetic modeling of 3-D sources over 2-D inhomogeneities in the time domain

Abstract

A numerical method is presented by which the transient electromagnetic response of a two‐dimensional (2-D) conductor, embedded in a conductive host rock and excited by a rectangular current loop, can be modeled. This 2.5-D modeling problem has been formulated in the time domain in terms of a vector diffusion equation for the scattered magnetic induction, which is Fourier transformed into the spatial wavenumber domain in the strike direction of the conductor. To confine the region of solution of the diffusion equation to the conductive earth, boundary values for the components of the magnetic induction on the ground surface have been calculated by means of an integral transform of the vertical component of the magnetic induction at the air‐earth interface. The system of parabolic differential equations for the three magnetic components has been integrated for 9 to 15 discrete spatial wavenumbers ranging from [Formula: see text] to [Formula: see text] using an implicit homogeneous finite‐difference scheme. The discretization of the differential equations on a grid representing a cross‐section of the conductive earth results in a large, sparse system of linear equations, which is solved by the successive overrelaxation method. The three‐dimensional (3-D) response has been computed by an inverse Fourier transformation of the cubic spline interpolated scattered magnetic induction in the wavenumber domain using a digital filtering technique. To test the algorithm, responses have been computed for a two‐layered half‐space and a vertical prism embedded in a conductive host rock. These examples were then compared with results obtained analytically or numerically using frequency‐domain finite‐element and time‐domain integral equation methods. The new numerical procedure gives satisfactory results for a wide range of 2-D conductivity distributions with conductivity ratios exceeding 1:100, provided the grid is sufficiently refined at the corners of the conductivity anomalies.