# Stabilization Methodology in Foundation Soils by ERT-3D Application in Estepona, South Spain
## 2 The Model Development  
The ERT method discharges an electric current into the ground and measures the potential difference at two determined points on the surface. The suitability of this method lies in the fact that irregularities in the subsoil beneath a building can be identified as contrasts or anomalies in the subsoil's electrical properties.  
This method is based on [PERSON]'s law:  
\[\text{Pa}=\text{k}(\Delta\text{V}/\text{l}) \tag{1}\]  
where: Pa = apparent resistivity [36]; k = geometric constant that depends only on the reciprocal positions of the current and potential electrodes; \(\Delta\text{V}\) = potential difference; l = intensity of the injected current.  
The apparent values of resistivity depend on the real resistivity distribution in the tested area. The true resistivity distribution can be estimated through a reverse procedure, based on minimizing an adequate function [15; 39; 44]. The solution to this problem is not unique. For the same set of data, a wide range of models can calculate the same apparent resistivity values. A preliminary lithological analysis is usually carried out on the subsoil's nature, to reduce the range of possible models, which can be incorporated into the reverse subroutine.  
The solution method used minimizes the difference between the apparent resistivities measured and those calculated by the RES3 DINV software, which uses the limited softness inversion formulation, restricting the model's change in resistivity values [15; 36; 43; 53]. This study has used RES3 DINVx64, which implements a smooth routine based on least squares and is practically the only commercial software of its type available [36; 54; 55]. The inversion routine used by the program is based on the smoothness-constrained least-squares method [56]. The basic smoothness-constrained least-squares method is based on the following equation.  
\[(\text{J}^{\text{T}}\text{J}+\lambda\text{F})\;\Delta\text{q}_{\text{k}}=\text {J}^{\text{T}}\text{g}-\lambda\;\text{F}_{\text{q}\text{-}1} \tag{2}\]  
where:  
\[\text{F}=\alpha_{\text{x}}C_{X}^{\text{T}}C_{X}+\alpha_{\text{Y}}C_{Y}^{\text{ T}}\;C_{\text{Y}}+\alpha_{Z}C_{Z}^{\text{T}}\;C_{Z} \tag{3}\]  
\(\text{J}^{\text{T}}=\text{Transpose of J}\).  
\(\text{J}=\text{Jacobian matrix of partial derivatives}\).  
\(\lambda=\text{Damping factor}\).  
\(\text{q}=\text{Disturbance vector}\).  
\(\text{k}=\text{Iteration number}\).  
\(\text{g}=\text{Data mismatch vector}\).  
\(\alpha\text{x}\), \(\alpha\text{y}\), \(\alpha\text{z}=\text{weights for roughness filters}\).  
\(\text{Cx}\), \(\text{Cy}=\text{horizontal roughness filters}\).  
\(\text{Cz}=\text{vertical roughness filter}\).  
This method is advantageous because of its versatility since the damping and roughness factor filters adjust to the different types of data. This program uses the Gauss-Newton method, which recalculates the Jacobian matrix, after each interaction [57]. The interpretations of electrical tomography profiles are made using RES3 DINVx64 [34; 36] for resistivity and induced polarization. As mentioned above, this calculation software is based on the least-squares method with forced smoothing, modified with the Quasi-Newton optimization technique. The inversion method designs a subsoil model using rectangular prisms and determines the resistivity values for each of them, minimizing the difference between the apparent resistivity values observed [15; 58]. The results of our model show an uncertainty in its acceptable magnitude. They are in a critical state with respect to their structural stability, load capacity and, under these conditions, the safety of the building's habitability. This is at an acceptable level of uncertainty associated with these increasing risks of instability [5].