A probabilistic solution to geophysical inverse problems in complex variables and its application to complex resistivity imaging

Abstract

We introduce a novel probabilistic framework for the solution of non-linear geophysical inverse problems in complex variables. By using complex probability distributions, this approach can simultaneously account for individual errors of real and imaginary data parts, independently regularize real and imaginary parts of the complex model, and still take into account cross-sensitivities resulting from a complex forward calculation. The inverse problem is solved by means of optimization. An application of the framework to complex resistivity (CR) imaging demonstrates its advantages over the established inversion approach for CR measurements. We show that CR data, with real and imaginary parts being subject to different errors, can be fitted adequately, accounting for the individual errors and applying independent regularization to the real and imaginary part of the subsurface conductivity. The probabilistic framework itself serves as a basis for the future application of global sampling approaches, such as Markov chain Monte Carlo methods.