Data Fusion for Electromagnetic and Electrical Resistive Tomography Based on Maximum Likelihood

Abstract

This paper presents a maximum likelihood based approach to data fusion for electromagnetic (EM) and electrical resistive (ER) tomography. The statistical maximum likelihood criterion is closely linked to the additive Fisher information measure, and it facilitates an appropriate weighting of the measurement data which can be useful with multiphysics inverse problems. The Fisher information is particularly useful for inverse problems which can be linearized similar to the Born approximation. In this paper, a proper scalar product is defined for the measurements and a truncated Singular Value Decomposition (SVD) based algorithm is devised which combines the measurement data of the two imaging modalities in a way that is optimal in the sense of maximum likelihood. As a multiphysics problem formulation with applications in geophysics, the problem of tunnel detection based on EM and ER tomography is studied in this paper. To illustrate the connection between the Green's functions, the gradients and the Fisher information, two simple and generic forward models are described in detail regarding two-dimensional EM and ER tomography, respectively.

Fisher information for inverse problems and trace class operators

“…One of the very few exceptions can be found in e.g., [34] where the Fisher information integral operator for an infinite-dimensional parameter function has been employed in the estimation of wave forms (or random processes). An infinite-dimensional Fisher information operator has previously been exploited in the context of inverse problems in e.g., [22,23,25], and the connection to the Singular Value Decomposition (SVD) has been established. However, so far, a rigorous treatment of Gaussian noise on infinite-dimensional Hilbert spaces has been lacking.…”