Back projection resistivity fast imaging technique for 2D electrical resistivity data

Electrical resistivity techniques are well-established and applicable to a wide range of geophysical problems. 2D resistivity measurements can give information about both the lateral and vertical variations of the subsurface resistivity and can be used in a qualitative fashion for the identification of the structure and depth of masses. The resistivity inverse problem involves constructing an estimate of a subsurface resistivity distribution, which is consistent with the experimental data. This is a fully non-linear problem and its treatment involves iterative full matrix inversion algorithms, which can give good quality results.
The back-projection resistivity technique (BPRT) can be applied to a set of apparent resistivity measures to quickly obtain an approximate image of the resistivity distribution of the investigated volume. This technique is based on the consideration that a resistivity perturbation in a point element (voxel) of a bounded region produces a change in voltage thus an apparent resistivity anomaly at the surface of the region, according to a sensitivity coefficient. The value of the coefficient is dependent on the position of the voxel considered in respect of both the current and the voltage dipoles, in agreement with the sensitivity theorem of Geselowitz. This consideration suggests that it is possible to correlate all the measured resistivity values, weighted by the appropriate sensitivity coefficients to each voxel of the investigated volume and to estimate the resistivity value of each cell of the model using a weighted summation of the apparent resistivity measurements.
The BPRT considering a two-step approach. Initially, a damped least squares solution is obtained after a full matrix inversion of the linearized geoelectrical problem. Furthermore, on the basis of the results, a subsequent filtering algorithm is applied to the Jacobian matrix, aiming at reducing smoothness, and the linearized damped least square inversion is repeated to get the final result. This fast imaging technique aims at increasing the resistivity contrasts, and practically, since it does not require a parameter set optimization, it can be used to easily obtain fast and preliminary results.
The procedure proposed in this work consists of four steps:
(1) Evaluation of sensitivity matrix B,
(2) Inversion of matrix B using a damped LSQR solution,
(3) Recalculation of a filtered Jacobian matrix B‘ obtained by means of a correlation filter,
(4) Inversion of the filtered sensitivity matrix.
The proposed technique is tested on resistivity synthetic data from the Schlumberger, Wenner, Dipole-dipole and Pole-pole arrays, the objective of which is to find the optimal parameter set. The synthetic tests carried out with 2D data suggested that a good compromise for 2D inversions is to choose 𝜆 for the Schlumberger, Wenner, Dipole-dipole and Pole-pole arrays, 0.1, 20, 1 and 0.5, respectively. Furthermore, all the synthetic tests carried out with 2D data suggested that a good compromise for 2D inversions is to choose 𝜒 ≈ 5. The approximate images using the BPRT inverse modeling for all synthetic data, with and without random noise, is compared with the least square inversion by RES2DINV software. Finally, a field case is discussed, and the comparison between the back-projection and inversion is shown.