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

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 multi-physics 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 productis dened 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 multi-physics 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. Numerical examples are included to illustrate the potential impact of an imbalance between the singular values and the variance of the measurement noise when dierent imaging modalities are incorporated in the inversion. The examples furthermore illustrate the signicance of taking a statistically based weighting of the measurement data into proper account