Fréchet Derivatives calculation for electrical resistivity imaging using forward matrix method

Fréchet derivatives calculation or sensitivity matrix is an integral part of every non-linear inversion process. The sensitivity values indicate the variation of the forward response with respect to the variation of model parameters. Sensitivity patterns are also a criterion to assess the reliability of inverted models and to design optimum resistivity surveys. In this study, a numerical approach based on the forward matrix calculation in the framework of the 2.5D finite difference electrical resistivity forward modeling is presented. First, using the potential distribution in the Fourier space obtained from the forward calculation and the derivatives of the coupling coefficients with respect to the conductivity distribution, the sensitivity values in the wavenumber domain are computed. Then, these values are transformed into the space domain using an inverse Fourier technique. To verify and analyze the proposed numerical method, the sensitivity distributions assuming the homogeneous and inhomogeneous media for commonly used electrical resistivity tomography configurations (e.g. pole-pole, pole-dipole, dipole-dipole, and the Wenner arrays) are computed. The numerical experiments reveal that the sensitivity patterns vary spatially throughout the model depending not only on the resistivity distribution but also on the electrode configuration. It is also concluded that the sensitivity analysis can be used as a supplementary tool for any optimum electrical tomography survey design.