An integral equation solution to the geophysical electromagnetic forward-modelling problem

We investigate the use of edge element basis vectors in an integral equation solution for three-dimensional geophysical electromagnetic modelling. Expansion of the total electric field within the region of anomalous conductivity in terms of these basis vectors gives a bilinear variation of each component of the field within a cell in the two directions perpendicular to the component (and so a divergence free but not curl free field within a cell), and continuity of the tangential electric field between two cells. In addition, we use a form of the electric field integral equation that explicitly involves the charge densities on cell faces associated with any discontinuity of the normal component of the current density. The two types of integrals in the integral equation – the volume integration of the scattering current within each cell, and the surface integration of the charge density on the faces of each cell – are computed using Gaussian quadrature. The system of equations to be solved is constructed using the Galerkin approach. In this preliminary study, we consider the simple case of a homogeneous halfspace as the background model. Comparisons with results from the literature and other codes have been promising. We include here two examples: one for a grounded electric line source at low frequency (3 Hz) on the surface of a halfspace (σ = 0.02 S/m) in which a more conductive vertical prism (σ = 0.2 S/m) is buried, and one for a magnetic dipole source-receiver combination over a conductive cube (σ = 100 S/m) in a resistive (σ = 10−4 S/m) background.