Three-Dimensional Electromagnetic Inversion Combining a Finite Difference Forward Solver With Integral Equations Jacobians

Abstract

In an attempt to overcome long computer run times in direct calculation of resistivity parameter Jacobians in 3-D EM inversion, we combine forward calculations for the staggered grid finite difference (FD) method with Jacobians estimated from the forward results using the integral equations (IE) method. The second-order electric field formulation in the FD method yields the electric field everywhere in the earth efficiently. The IE formulation for the Jacobians utilizes the electric field just within the parameter of interest to define a Born term for the Jacobian, plus a depolarization term reflecting the incremental change in the electric field in all parameters. The amplitude of the depolarization decreases rapidly with distance from the parameter allowing a lumped cell approach to setting up the IE expression to be solved for the Jacobian. The lumped cells may grow geometrically from the parameter in accord with EM scaling, requiring only 100-200 cells to be considered for each parameter. Although this constitutes a sizeable pre-multiplier, the expression for computational cost of the total Jacobian set is only linearly proportional to the number of parameters.