Linear regression models for estimating true subsurface resistivity from apparent resistivity data

Abstract

Simple linear regression (SLR) models for rapid estimation of true subsurface resistivity from apparent resistivity measurements are developed and assessed in this study. The objective is to minimize the processing time and computer memory required to carry out inversion with conventional algorithms. The arrays considered are Wenner, Wenner-Schlumberger and dipole-dipole. The parameters investigated are apparent resistivity (ρ a) and true resistivity (ρ t) as independent and dependent variables, respectively. For the fact that subsurface resistivity is nonlinear, the datasets were first transformed into logarithmic scale to satisfy the basic regression assumptions. Three models, one each for the three array types, are thus developed based on simple linear relationships between the dependent and independent variables. The generated SLR coefficients were used to estimate ρ t for different ρ a datasets for validation. Accuracy of the models was assessed using coefficient of determination (R 2), F-test, standard error (SE) and weighted mean absolute percentage error (wMAPE). The model calibration R 2 and F-value are obtained as 0.75 and 2286, 0.63 and 1097, and 0.47 and 446 for the Wenner, Wenner-Schlumberger and dipoledipole array models, respectively. The SE for calibration and validation are obtained as 0.12 and 0.13, 0.16 and 0.25, and 0.21 and 0.24 for the Wenner, Wenner-Schlumberger and dipole-dipole array models, respectively. Similarly, the wMAPE for calibration and validation are estimated as 3.27 and 3.49%, 3.88 and 5.72%, and 5.35 and 6.07% for the three array models, respectively. When compared with standard constraint least-squares (SCLS) inversion and Incomplete Gauss-Newton (IGN) algorithms, the SLR models were found to reduce about 80-96.5% of the processing time and memory space required to carry out the inversion with the SCLS algorithm. It is concluded that the SLR models can rapidly estimate ρ t for the various arrays accurately.

Solution of Collinearity Problem in Two-Dimensional Electrical Resistivity Tomography using Wenner Array

“…The inverse resistivity contour plot obtained after 5 iteration has RMS error of 8.7%. It indicated high resistivity zone (> 900 Ω-m) within 0 to 4 m depth and low resistivity zone (< 900 Ω-m) at depths greater than 4 m, which is in conformity with the findings of (Muhammad & Saad, 2018). The high resistivity zone defines the dry top residual soil which extends from the surface to a thickness of 4 m, whereas, the low resistivity zone corresponds to the residual soil saturated with water (Abdulrahman et al, 2016).…”

“…The juxtaposed tomographic images for the WA-AEI and WA-2EO at offline distances of 0.2, 0.4, 0.6, 0.8, 1.0, and 1.2 m using the conventional approach and newly developed approach are shown in Figure 8. All the tomograms obtained for the WA-2EO models using the conventional approach indicated high (>900 Ω-m) and low (<900 Ω-m) resistivity zones at depths below and above 4 m, respectively (Muhammad & Saad, 2018). The geometries of the true model are partially resolved in all the images of WA-2EO.…”

“…The subsurface potential contrast (V) was measured between two electrodes (often called potential electrodes, indicated by M -N) upon injecting specified amount of electric current (I) into the ground between the other two electrodes (current electrodes, A -B). Since the principle of electrical resistivity method is embedded in Ohm's law, the resistance (R) is then computed using Equation 1 (Muhammad & Saad, 2018). Subsequently, the apparent resistivity is obtained when array geometrical factor is known (Equation 2).…”