## 2 Materials and Methods
### Parameters to Estimate the Inversion Reliability  
The inversion of the calculated data was performed using the RES2 DINVx64 [51] and RES3 DINVx64 [52] programs. To perform a quantitative comparison, the inversion parameters were set to be the same for all the inversions made so that the results could be correctly compared and discussed. An \(L_{1}\)-norm robust model inversion constraint [46] was used to obtain models capable of highlighting sharp boundaries with a high resistivity contrast. This is the most appropriate method for delineating the shape of the anomalous block. In this case, the discrepancy between the predicted and observed data was expressed as the absolute error through the \(L_{1}\)-norm normalized data misfit \(L_{1N}\). We set a convergence limit for the relative change in the \(L_{1N}\) misfit equal to 2% and a maximum number of iterations for the inversion equal to 10. For all simulations, the mesh sizes were chosen to be equal to half the electrode distance for both the forward problem and the inverse problem to take into account the abrupt resistivity changes of the models. The evaluation of the results had to include a quantitative analysis of the defined parameters indicating the ability of the inverse model to approach the real situation. Therefore, to evaluate the overall effectiveness of each inversion, some simple comparison parameters were considered. The first parameter is given by  
\[\mathbf{P}_{1}=\frac{\rho_{prism}(l,s,d)}{\rho_{prism}(l,\infty)}, \tag{7}\]  
where \(\rho_{prism}(l,s,d)\) is the average value of the electrical resistivity calculated in the area of the anomaly (the one in which the resistivity exceeded the background one by 15% was considered), \(l\) is the square section size of the prism, \(s\) is its orthogonal size, \(d\) is the orthogonal distance from the center of the prism at which the survey is carried out and \(\rho_{prism}(l,\infty)\) is the corresponding value when \(s\rightarrow\infty\) (tunnel model shown in Figure 3). Figure 3: Simulation of a 2D ERT survey (\(10^{5}\)\(\Omega\)-m) in a background rock (100 \(\Omega\)-m). Figure 2: Simulation of a 3D ERT survey on a model of a prism-shaped resistive block (\(10^{5}\)\(\Omega\)-m) with the orthogonal width \(s\) not equal to the longitudinal and vertical sizes \(l\) in a background rock (100 \(\Omega\)-m). The size \(l\) of the square section is equal to 8 m. The depth of the top of the prism is 4 m. The thick blue line indicates the considered 2D ERT survey carried out over the prism center in the orthogonal direction to the length \(s\) of the prism. Basically, \(\mathbf{P}_{1}\) gives the ratio between the resistivity anomaly caused by a 3D block with a given vertical section and a given electrical resistivity and that caused by a 2D prism with the same vertical section and resistivity. Consequently, this reflects how much the three-dimensionality affects the electrical resistivity value of the anomaly obtained from a 2D inversion, decreasing the estimate. To calculate the anomalous area, the threshold of 15% was chosen following some tests to exclude small variations in resistivity in the section, which may have been caused by noise or inaccuracy of the inversion. The second parameter is given by  
\[\mathbf{P}_{2}=\frac{\rho_{prism}(l,s,d)}{\rho_{prism}(l,0)}, \tag{8}\]  
where the section centered on the prism is considered for the denominator (\(d=0\)) and \(\mathbf{P}_{2}\) indicates how much the non-centrality of the 2D ERT with respect to the resistive block affects the inversion. It was calculated only for the model of the cubic resistive block. The third parameter considered is defined as  
\[\mathbf{P}_{3}=\frac{\sum_{anom}(l,s,d)}{\sum_{prism}(l,s,d)}, \tag{9}\]  
where \(\sum_{anom}(l,s,d)\) is the area of the resistive anomaly (in which the resistivity exceeds the background one by 15%) as shown by the inverse model and \(\sum_{prism}(l,s,d)\) is the actual area of the prism section.