Efficient inversion of 2.5D electrical resistivity data using the discrete adjoint method

Abstract

We have developed a memory and operation-count efficient 2.5D inversion algorithm of electrical resistivity (ER) data that can handle fine discretization domains imposed by other geophysical (e.g, ground penetrating radar or seismic) data. Due to numerical stability criteria and available computational memory, joint inversion of different types of geophysical data can impose different grid discretization constraints on the model parameters. Our algorithm enables the ER data sensitivities to be directly joined with other geophysical data without the need of interpolating or coarsening the discretization. We have used the adjoint method directly in the discretized Maxwell’s steady state equation to compute the data sensitivity to the conductivity. In doing so, we make no finite-difference approximation on the Jacobian of the data and avoid the need to store large and dense matrices. Rather, we exploit matrix-vector multiplication of sparse matrices and find successful convergence using gradient descent for our inversion routine without having to resort to the Hessian of the objective function. By assuming a 2.5D subsurface, we are able to linearly reduce memory requirements when compared to a 3D gradient descent inversion, and by a power of two when compared to storing a 2D Hessian. Moreover, our method linearly outperforms operation counts when compared with 3D Gauss-Newton conjugate-gradient schemes, which scales cubically in our favor with respect to the thickness of the 3D domain. We physically appraise the domain of the recovered conductivity using a cutoff of the electric current density present in our survey. We evaluate two case studies to assess the validity of our algorithm. First, on a 2.5D synthetic example, and then on field data acquired in a controlled alluvial aquifer, where we were able to match the recovered conductivity to borehole observations.