## 3 Applications of the Proposed Model
### Direct Method  
Figure 1 schematically shows the simplest Electrical Resistivity Tool (ERT) for collecting BRL data. It is the so-called Single-Point Resistance (SPR) Tool or Single-Point Log (SPL). Through contact A, a current I is injected into the ground, while contact B records the electric potential. The assembly is moved vertically along the well, probing the soil at different depths. For this device, the measured quantity is the potential at B at various depths, \(\phi_{A,B}(z)\), as described by Equation (4). The points M and N are considered sufficiently distant so as not to disturb the current and potential fields, ensuring that what is measured at B is the absolute electric potential generated by a current point source at A. It is also assumed that the borehole does not significantly disturb the current field associated with point A, meaning that the measurement results are not different from those associated with a point current source [19]. A typical pattern of a BRL using the device shown in an ideal multilayered soil can be seen in the right panel of Figure 1. This image was generated by simulating the survey using a conventional 5-layer HCMS model with the resistivities indicated in the figure. The horizontal distance between electrodes A and B is fixed at \(d_{AB}^{h}=5\,\mathrm{cm}\), while the vertical distance is \(d_{AB}^{v}=10\,\mathrm{cm}\). It is worth noting that the SPL data acquisition device shown in Figure 1 is not the only possible configuration. Figure 2 shows a simplified schematic of another possible ERT used for BRL surveys. It is called the Normal Resistivity Log (NRL). In this case, the quantity considered as the survey result is the potential difference \(\Delta V_{BN}(z)=\phi_{A,B}(z)-\phi_{A,N}(z)\). The same figure also displays the simulated survey result for the same soil used in Figure 1. In this ERT device, the potential probe N is located 10 cm above B. Focusing on three-layer models for the reasons stated at the beginning of this section, Figure 3 shows the appearance of a BRL for two different soils using the same SPR measuring device shown in Figure 1. The left panel shows the simulated BRL in three-layer HCMS soil, with the resistivity parameters labeled on the figure itself and the widths marked with horizontal dashed lines. The right panel shows the BRL result for an EGMS model in the main plot, while the functional form of the resistivity in each layer can be seen in the inscribed plot. Table 1 lists the values of the model parameters. The horizontal dashed lines in the main plot indicate the layer boundaries, as do the vertical dashed lines in the inscribed plot. Figure 2: The NRL device for data acquisition and the simulated result image for the same 5-layer HCMS soil used in Figure 1.  
In the following example, some results from the paper [18] will be used, where the objective function is the apparent resistivity obtained from Wenner soundings. For this, we start with a three-layer stratified soil with parameters \(\rho_{1}=100\) f\(\Omega\)m, \(\rho_{2}=50\) f\(\Omega\)m, \(\rho_{3}=200\) f\(\Omega\)m, \(h_{1}=3\) m, and \(h_{2}=2\) m. A Wenner sounding is simulated over this soil, providing a defined pattern of apparent resistivity \(\rho_{app}^{meas}(a)\) that will be considered as the result of the sounding as a function of the electrode spacing \(a\) in the array. This sounding is then subjected to an inversion, with the objective function being the apparent resistivity expression in a three-layer EGMS model \(\rho_{app}(a;\left\{C_{i1},C_{i2},\beta_{i}\right\}_{i=1...3})\)[18], resulting in the parameter set shown in Table 2.  
The graphs obtained from performing a BRL with these two models for the same soil are shown in Figure 4. The inset presents an image of the resistivity in each layer of the EGMS model. As can be seen, the BRL survey is very similar for both soil models, with the exception of the central region, where the models show a discrepancy.