Solution strategies for two- and three-dimensional electromagnetic inverse problems

We analyse a variety of solution strategies for nonlinear two-dimensional (2D) and three-dimensional (3D) electromagnetic (EM) inverse problems. To impose a realistic parameterization on the problem, the finite-difference equations arising from Maxwell equations are employed in the forward problem. Krylov subspace methods are then used to solve the resulting linear systems. Because of the efficiencies of the Krylov methods, they are used as the computational kernel for solving 2D and 3D inverse problems, where multiple solutions of the forward problem are required. We derive relations for computing the full Hessian matrix and functional gradient that are needed For computing the model update. via the Newton iteration. Different strategies are then discussed for economically carrying out the Newton iteration for 2D and 3D problems, including the incorporation of constraints necessary to stabilize the inversion process. Two case histories utilizing EM inversion are presented. These include inversion of cross-well data for monitoring electrical conductivity changes arising From an enhanced oil recovery project and the usefulness of cross-well EM methods to characterize the transport pathways for contaminants in the subsurface.