Approximate inverse mappings in DC resistivity problems

Abstract

In an E-SCAN DC resistivity experiment a large amount of common source pole-pole potential data from many sources over a pre-designed survey grid are collected. The large number of DC potentials and the spatial configuration in which they are collected invite the development of new imaging and inversion methods. We present three one-pass inversion algorithms. They are based upon the linearized equation for the DC potential under the Born approximation and the assumption that the conductivity model consists of perturbations to a uniform background conductivity. Two algorithms are presented for imaging and inverting for 2-D conductivity models. The first is based upon the theory of charge accumulation in the DC resistivity experiment, in which a sparse representation of the accumulated charge densities due to all current sources is recovered so as to image the structural boundaries. The second method inverts directly for an approximate conductivity model having a minimum structure. Both methods employ linear programming techniques. For 3-D problems, a fast algorithm is developed using a convolutional representation of the surface pole-pole apparent resistivity. It decomposes the 3-D inverse problem into a sequence of 1-D inversions in the wavenumber domain. These 1-D problems are independent of eachother and can be solved efficiently. The final spatial domain conductivity model is obtained by inverse Fourier transforming. Despite the Born approximation, this algorithm seems to work for large conductivity contrasts and a certain amount of near-surface variation in conductivity. All algorithms are tested on synthetic data but we also present results from the 3-D inversion applied to field data.