## 2 Materials and Methods
### Design of Synthetic Models  
The design of synthetic models was based on simplicity criteria aimed at quantitative evaluation of the interpretative distortions that can arise when three-dimensional structures are investigated with two-dimensional resistivity surveys. The two-dimensional approach inevitably leads to an incorrect estimate of the size and electrical resistivity of the anomalous zone. It was therefore decided in the first phase to simulate resistive blocksof simple and regular shapes to obtain inverse models that would allow a quantitative parametric approach and a systematic comparison between 3D resistivity models and the corresponding 2D inverse models when 2D-ERT surveys were simulated.  
For the 3D forward modeling, RES3 DMOD software was used [47], considering the finite element method and a grid of rectangular meshes in which 48 \(\times\) 21 electrodes space at 2 m in the \(x\) and \(y\) directions, respectively, were considered. For each designed model, the apparent resistivity measurements related to a 3D data set were calculated. The 3D electrical resistivity survey was simulated while considering the inline dipole-dipole array, with the dipole length \(a\) ranging from 1 to 12 times the electrode spacing and the dipole-dipole distance \(n\) ranging from 1 to 12 times the electrode spacing. This array configuration allows a good resolution and reliability for the survey according to a not-too-long acquisition time [7; 48; 49]. The dipole-dipole array allows, thanks to modern multi-channel instruments, simultaneously acquiring many voltage measures with the same current dipole input, thus allowing short acquisition times and, consequently, a large number of measures which are difficult to acquire in reasonable times using Wenner or Wenner-Schlumberger arrays.  
From these data sets, some parallel 2D data sets related to the parallel profiles were extracted and separately inverted to study and quantify the effects of 3D structures on 2D inversion, parametrically estimating the differences between 2D imaging and the corresponding section of the 3D inverse model.  
The first and simplest model considered was a resistive cube (10\({}^{5}\)\(\Omega\cdot\)m) centered under the grid of a size \(l\) = 8 m in a background rock of 100 \(\Omega\cdot\)m (Figure 1). The depth of the upper face of the cube was \(h\) = 4 m. From the obtained 3D data set of calculated apparent resistivities, some 2D data subsets were extracted, simulating many parallel 2D surveys along the \(x\) direction placed at variable distance \(d\) between the center of the survey line and the projection on the surface of the center of the cube.  
The second test (Figure 2) regarded a series of prism-shaped cavities with a square vertical section of a size \(l\) = 8 m and a length in the orthogonal \(y\) direction that ranged from \(s\) = 8 m (the same as the first model) to \(s\rightarrow\infty\), the latter being a 2D tunnel model of the squared section (Figure 3) simulated with the RES2 DMOD software [50]. In these cases, the 3D electrical resistivity surveys were simulated while considering the same electrode arrays and measurement sequences as those in the previous case. From each of these 3D data sets, a 2D subset was extracted, simulating a survey that passed over the center of the cavity and crossed it perpendicular to its width \(s\) (\(x\) direction).  
Figure 1: Simulation of a 3D ERT survey on a model of a cube-shaped resistive block (10\({}^{5}\)\(\Omega\cdot\)m) buried in a background rock (100 \(\Omega\cdot\)m). The blue points indicate the electrode positions. The size \(l\) of the cube is equal to 8 m. The depth of the top of the cube is 4 m. The thick blue line indicates the considered 2D-ERT survey, carried out at an orthogonal distance \(d\) from the projection on the surface of the cube’s center.