## 2 Materials and Methods  
### Basic Electrical Resistivity Tomography, Forward Modeling and Inversion  
Electrical resistivity techniques allow for estimating the distribution of the electrical resistivity of the subsoil. They are essentially based on [PERSON]'s two laws, from which we can derive the following equation in vector form for the current flow in a continuous medium:  
\[\mathbf{J}=\sigma\
abla\Phi, \tag{1}\]  
where \(\mathbf{J}\) is the current density, \(\Phi\) is the electric field potential and \(\sigma\) is the conductivity of the medium, which is equal to the reciprocal of the resistivity \(\rho\). Equation (1) describes the relationships between the potential of the electric field generated by an electromotive force, the current flowing through the ground and the resistance of the subsoil. The latter depends on the distribution of the electrical resistivity in the subsoil and on the positions of the electrodes. To perform a resistivity measurement, an electric current \(I\) is generated in the ground by applying an electromotive force through two current electrodes. The trend of the electric field is observed by measuring the potential differences \(\Delta V\) with electrodes appropriately arranged on the surface. Considering the subsoil as a homogeneous half-space, the apparent resistivity \(\rho_{a}\) is calculated by the following formula [30]:  
\[\rho_{a}=k\Delta V/I, \tag{2}\]  
where \(k\) is a geometric factor which depends on the positions and shapes of the electrodes. Over the years, different electrode arrays have been conceived and used for resistivity surveys, each having different characteristics and being suitable for different modeling (1D, 2D or 3D). The advantages and disadvantages of each of these arrays depend on many factors, including the sensitivity, signal-to-noise ratio, depth of investigation [31; 32]and lateral coverage [33]. Recently, the use of multichannel systems has allowed connecting up to several hundreds of electrodes at once, greatly reducing the survey time. This feature has led to the need to choose arrays that allow the contemporary voltage measurements of many potential dipoles for each current dipole, such as the Wenner-Schlumberger or linear dipole-dipole arrays. Furthermore, optimized arrays for multi-channel acquisitions have been proposed [2; 3; 4; 5; 7]. The forward problem of calculating the electric potential due to a point source located in \((x_{s},y_{s},z_{s})\) consists of the solution of the following equation [34]:  
\[\
abla\Big{[}\frac{1}{\rho(x,y,z)}\
abla\Phi(x,y,z)\Big{]}=-\frac{\partial j_{ c}}{\partial t}\delta(x-x_{S})\delta(y-y_{S})\delta(z-z_{S}), \tag{3}\]  
where \(\rho\) is the electrical resistivity at the coordinate point \((x,y,z)\), \(\Phi\) is the potential at the same point, \(j_{c}\) is the electrical charge density and \(\delta\) is the Dirac delta function. Generally, forward modeling uses approximate approaches through finite difference [34; 35; 36] or finite element [37] methods in which the subsurface is represented by a grid of meshes, inside which the electrical resistivity is constant or varies linearly. In ERT, the inverse problem is not linear and ill-posed, and it is generally faced by solving the forward problem iteratively, starting from a simple initial model \(m_{0}\) (usually a homogeneous half-space) and minimizing the misfit between the measured and calculated data.