Variational Bayesian inversion of synthetic 3D controlled-source electromagnetic geophysical data

Inversion of controlled-source electromagnetic data is dealt with for a geophysical application. The goal is to retrieve a map of conductivity of an unknown body embedded in a layered underground from measurements of the scattered electric field that results from its interaction with a known interrogating wave. This constitutes an inverse scattering problem whose associated forward problem is described by means of electric field domain integral equations. The inverse problem is solved in a Bayesian framework in which prior information is introduced via a Gauss-Markov-Potts model. This model describes the body as being composed of a finite number of different materials distributed into compact homogeneous regions. The posterior distribution of the unknowns is approached by means of the variational Bayesian approximation as a separable distribution that minimizes the Kullback-Leibler divergence with respect to the posterior law. Thus, we get a parametric model for the distributions of the induced currents, the conductivity contrast, and the various parameters of the prior model that are obtained following a semisupervised iterative approach. This method is applied to multifrequency synthetic data corresponding to a 3D crosswell configuration in which the sought body is made of two separated anomalies, a conductive heterogeneity and a resistive one, and its results are compared with that given by the classic contrast source inversion (CSI). The method succeeds in retrieving compact homogeneous regions that correspond to the two anomalies whose shape and conductivities are obtained with a good precision compared with that obtained with CSI.