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 com puting the electromagnetic induction response of an arbitrary heterogeneous conducting sphere by external current excitation. This method is appropria te as the forward solution for the problem of determining the electrical co nductivity of the Earth's deep interior. This solution in spherical geometr y is derived from that originally presented by Mackle et al. (1994) for Car tesian 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 matri x system of equations is sparse, symmetric, real everywhere except along th e diagonal and ill-conditioned. The system is solved using the minimum resi dual conjugate gradient method with preconditioning by incomplete Cholesky decomposition of the diagonal sub-blocks of the coefficient matrix. In orde r to ensure there is zero H divergence in the solution, corrections are mad e to the H held every few iterations. In order to validate the code, we com pare our results against an integral equation solution for an azimuthally s ymmetric, buried thin spherical shell model (Kuvshinov & Pankratov 1994), a nd against a quasi-analytic solution for an azimuthally asymmetric configur ation of eccentrically nested spheres (Martinec 1998).