Efficient electric resistivity inversion using adjoint state of mixed finite-element method for Poisson’s equation

We propose an electric resistivity inversion method that is similar to the reverse time migration technique applied to seismic data. For calculating model responses and inversion, we use the mixed finite-element method with the standard P1 − P0 pair for triangular decompositions, which makes it possible to compute both the electric potential and the electric field vector economically. In order to apply the adjoint state of the Poisson equation in the resistivity inverse problem, we introduce an apparent electric field defined as the dot product between the computed electric field vector and a weighting factor and then defining a virtual source to compute the partial derivative of the electric field vector. We exploit the adjoint state (the symmetry of Green’s function) of matrix equations derived from solving the Poisson equation by the mixed finite-element method, for the calculation of the steepest descent direction of our objective function. By computing the steepest descent direction by a dot product of backpropagated residual and virtual source, we can avoid the cumbersome and expensive process of computing the Jacobian matrix directly. We calibrate our algorithm on a synthetic of a buried conductive block and obtain an image that is compatible with the limits of the resistivity method.