## 3 Applications of the Proposed Model
### Inversion Method  
As mentioned in the second item of the previous section, the task now is to determine the parameters of a specific EGMS model using the results from a BRL. For the simple SPR device shown in Figure 1, these initial data correspond to \(\phi_{A,B}^{meas}(z)\) for a given set of depths \(z\) from the ground surface. The goal is to minimize expressions (6) or (7), where \(\phi_{A,B}(z;\{C_{i1},C_{i2},\beta_{i}\}_{i=1..3})\) represents the potential value at depth z for the three-layer EGMS model, i.e., the objective function. For the conventional HCMS model, the objective function is represented by \(\phi_{A,B}(z;\{\rho_{i},h_{i}\}_{i=1..3})\). It is important to note that if the resistivity of a terrain fits well with a model, the simulation of the BRL with that model will closely resemble the BRL obtained with a direct measurement device, and expressions (6) or (7) will have very small values. However, the inverse method does not guarantee that an experimental BRL will yield a model that fits well with the data. At best, approximations may be obtained depending on the initial data. This is illustrated by Figure 3, where a stepped BRL shape suggests an HCMS soil model, while the presence of curves or slopes in the graph suggests another type of model, such as EGMS. This can also be observed in Figure 4, where two types of models are tested for the same soil. In this case, both soil models result in very similar BRL curves. Several application examples will be analyzed next. Some of these involve simulations with synthetic soils illustrating the validity of the presented model. However, due to the high cost of BRL surveys, available data of this type are very scarce. Therefore, we are forced to use more abundant Wenner-type VES surveys. By using inversion procedures with the apparent resistivity profile as the objective function, we can obtain model parameters. This approach was previously used to develop the case shown in Figure 4.  
Here, a new example is presented in which, starting from the apparent resistivity of a Wenner sounding applied to a synthetic soil, inversion is performed with apparent resistivity as the objective function in an EGMS model, also imposing continuity of resistivity at the interfaces. Table 3 shows the parameter values of the model, which exhibits continuity at the interfaces, as illustrated in the upper inset of Figure 5.  
Figure 4: Result of the BRL surveys with the two models for the same soil. The colored lines in the subfigure correspond to the variable resistivities in the considered layers. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Params.** & \(\mathbf{C_{11}}\) & \(\mathbf{C_{12}}\) & \(\mathbf{\beta_{1}}\) & \(\mathbf{C_{21}}\) & \(\mathbf{C_{22}}\) & \(\mathbf{\beta_{2}}\) & \(\mathbf{C_{31}}\) & \(\mathbf{C_{32}}\) & \(\mathbf{\beta_{3}}\) & \(\mathbf{h_{1}}\) & \(\mathbf{h_{2}}\) \\ \hline
**Values** & 0.0501 & \(-\)0.0101 & 0.1002 & \(-\)0.0218 & 0.0787 & 0.0201 & 0.0203 & 0.0310 & 0.0511 & 2.00 & 2.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Continuous EGMS model parameters for the soil depicted in the graph shown in the inset of Figure 5.  
The equivalent HCMS model obtained through inversion is shown in the lower inset of Figure 5 and has the parameters \(\rho_{1}=569.16\)\(\Omega\)m, \(\rho_{2}=386.61\)\(\Omega\)m, \(\rho_{3}=0.71\)\(\Omega\)m, \(h_{1}=1.03\) m, and \(h_{2}=24.79\) m. The main body of Figure 5 displays the BRL results using these two soil models. As can be seen, this soil also exhibits a close equivalence between its HCMS and EGMS models for BRL surveys. However, this case, similarly to the one represented in Figure 4, is quite rare. Typically, there is often incompatibility between the models considered. Next, we will analyze the only example with measured data that we have available, demonstrating the utility of the proposed model in this work. Table 4 presents the results of a BRL survey using a device similar to the one shown in Figure 1 at a site located in Yepes, Toledo (Spain).