# Computation of Optimized Electrode Arrays for 3-D Electrical Resistivity Tomography Surveys
## 4 Quantitative Analysis  
The quantitative comparative analysis between the resistivity models resulting in the inversion of the comprehensive and optimum data sets was made feasible using the model misfit and the layer RMS. The above-mentioned techniques were used to validate the degree of model matching on the inverted model with respect to the true model as predefined by the initial user resistivity values. The model misfit is defined as the mean squared misfit between the true parameter model \(\rho_{\text{t}}\) and the inverted parameter model \(\rho_{\text{c}}\) resistivity value [27]:  
\[\textit{Model~{}Misfit}=\left(\frac{1}{N}\sum_{i=1}^{N}[log(\rho_{t}(i))-log( \rho_{c}(i))]^{2}\right)^{0.5} \tag{1}\]  
The layer RMS is the root mean square error for each layer defined as the percentage difference in the resistivity values between the true and the inverted model using the following equation [31]:  
\[\textit{layer~{}RMS}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\frac{\rho_{c}(i)- \rho_{t}(i)}{\rho_{t}(i)}\times 100\right)^{2}} \tag{2}\]  
The results of the calculations are shown in Figure 11. All values (model misfit and layer-RMS) are calculated for the FULL array only for five specific depth model layers (D2, D3, D5, D6 and D7) where the targets exist. The model misfit values of the Optim-3D data set (green-colored line) are profoundly decreased at the superficial depths (D2, D3) then equalize with the comprehensive and Optim-2D data sets at deeper levels (D5, D6, D7). The Optim-2D data set (yellow-colored line) has a higher value of model misfit at depth D2 from the Optim-3D data set and equal value with the comprehensive data set. At depth of D3, the Optim-2D data sets have lower model misfit values in comparison with  
Figure 10: Inversion results from FULL optimum array produced by [PERSON]’s algorithm.  
the comprehensive data set and higher value in comparison with the Optim-3D data set. For the following depths (D5, D6, D7), the Optim-2D data set shares the same model misfit values with Optim-3D and comprehensive data set.  
Regarding the layer-RMS values, the comprehensive data sets (red-colored line) share the same values with the optimum data sets (Optim-2D, Optim-3D) at superficial depth layers (D2, D3, D5). For deeper layer D6, the layer-RMS value of optimum data sets is higher than the comprehensive data set: Optim-2D data set has the highest value in comparison with the Optim-3D and the comprehensive data set. At the deepest layer D7, comprehensive, Optim-2D and Optim-3D share the same layer-RMS value.  
The Res-STUM data (blue-colored line) presents a high value model misfit and layer-RMS at depth layer D2 showing a decreased efficiency of the optimized protocols to reconstruct the shallow depth layers. On the other hand, it seems to have lower values than the comprehensive data set, Optim-2D and Optim-3D approaches at deeper depth layers.  
The time to compute the 3-D optimized protocols with both the Jacobian and the [PERSON]'s method will depend on the number of electrodes, the size of the comprehensive data set and the scheme to parametrize the subsurface in homogeneous 3-D resistivity blocks. Assuming a 3-D ERT survey where the number of electrodes and the parameters is fixed, the total time to extract the optimized data sets with the two algorithms will depend on the size of the comprehensive data set. Both algorithms seem to have a linear increasing trend relating the time and the number of measurements. However, the [PERSON] method shows a relative reduction of more than 20% in the actual time needed to compile the optimized data sets, thus being significantly faster than [PERSON]'s approach (Figure 12).  
Figure 11: Model misfit (**a**) and layer RMS (**b**) values for FULL array among comprehensive (red colored line), Optim-2D (yellow colored line), Optim-3D (green colored line) and Res-STUM (blue colored line) data set. Outliers indicated with red circles.