Geoelectromagnetic induction in a heterogeneous sphere: a new three-dimensional forward solver using a conservative staggered-grid finite difference method

A conservative staggered-grid finite difference method is presented for computing the electromagnetic induction response of an arbitrary heterogeneous conducting sphere by external current excitation. This method is appropriate as the forward solution for the problem of determining the electrical conductivity of the Earth's deep interior. This solution in spherical geometry is derived from that originally presented by Mackle et al. (1994) for Cartesian geometry. The difference equations that we solve are second order in the magnetic field H, and are derived from the integral form of Maxwell's equations on a staggered grid in spherical coordinates. The resulting matrix system of equations is sparse, symmetric, real everywhere except along the diagonal and ill-conditioned. The system is solved using the minimum residual conjugate gradient method with preconditioning by incomplete Cholesky decomposition of the diagonal sub-blocks of the coefficient matrix. In order to ensure there is zero H divergence in the solution, corrections are made to the H held every few iterations. In order to validate the code, we compare our results against an integral equation solution for an azimuthally symmetric, buried thin spherical shell model (Kuvshinov & Pankratov 1994), and against a quasi-analytic solution for an azimuthally asymmetric configuration of eccentrically nested spheres (Martinec 1998).