## 3 Applications of the Proposed Model
### Inversion Method
Since the device used for data collection was not certified and the personnel involved in the experiment were not qualified, the data in the table are not of high quality and should be considered a rough estimate of the subsurface electrical properties, though sufficiently valid to illustrate the differences between the HCMS and EGMS models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Depth \(z\) (m)** & 0.01 & 0.25 & 0.52 & 0.74 & 1.00 \\ \hline \(V_{\text{B}}\)**(volts)** & 130 & 74 & 73 & 68 & 66 \\ \hline
**Depth \(z\) (m)** & 1.25 & 1.51 & 1.74 & 2.00 & 2.25 \\ \hline
**\(V_{\text{B}}\)**(volts)** & 80 & 88 & 104 & 85 & 91 \\ \hline
**Depth \(z\) (m)** & 2.50 & 2.74 & 3.02 & 3.25 & 3.50 \\ \hline \(V_{\text{B}}\)**(volts)** & 83 & 87 & 75 & 69 & 67 \\ \hline
**Depth \(z\) (m)** & 3.77 & 3.99 & 4.25 & 4.50 & 4.76 \\ \hline
**\(V_{\text{B}}\)**(volts)** & 76 & 71 & 69 & 77 & 68 \\ \hline
**Depth \(z\) (m)** & 5.00 & 5.24 & 5.50 & 5.76 & 6.00 \\ \hline
**\(V_{\text{B}}\)**(volts)** & 66 & 73 & 57 & 65 & 64 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Measured BRL data from the soil located at Yepes, Toledo (Spain), with an SPR tool. Figure 5: Result of BRL surveys with two models for the same soil. The colored lines in the subfigures correspond to the variable (**upper subfigure**) and constant (**lower subfigure**) resistivities in the considered layers. The inversion by minimizing expression (7), enforcing the continuity of the objective function \(V_{B}(z;\{C_{11},C_{12},\beta_{i}\}_{i=1...3})\) and using the data from Table 4, results in the EGMS model with the parameters listed in Table 5.  
The inversion using the HCMS model objective function \(V_{B}(z;\{\rho_{i},h_{i}\}_{i=1...3})\) results in a model with parameters \(\rho_{1}=99.48\)\(\Omega\)m, \(\rho_{2}=138.94\)\(\Omega\)m, \(\rho_{3}=100.78\)\(\Omega\)m, \(h_{1}=1.46\) m, and \(h_{2}=1.06\) m. Figure 6 displays the BRL data from the measurements marked with green diamond symbols. It also shows the BRL simulations for the two models obtained through inversion. The BRL simulation for the EGMS model, using the parameters from Table 5, is shown with a red solid line, while the simulation for the HCMS model, with the previously mentioned parameters, is shown with a blue solid line. The upper right inset represents the resistivity assigned according to the EGMS model, which is nearly continuous. As seen in the main figure, this model closely reproduces the measured BRL data. The lower right inset shows the resistivity according to the HCMS model, and the main figure also displays the BRL simulation for this model with the blue solid line. It is evident that this simulation reproduces the measured BRL less accurately, highlighting the need for alternative models to HCMS for the proper interpretation of BRL soundings. As a final comment, it is important to note that the proposed model is a layered model in which the resistivity has a defined functional form in each layer. From a technical standpoint, it represents an advancement over models with constant resistivity layers. However, real soil can be much more complex and may not fit into any of these models. Nevertheless, introducing variability with depth will always provide a better fit of the model to the actual resistivity profile. Figure 6: Measured BRL from the soil at Yepes and the simulated BRL from the two fitted models. The colored lines in the subfigures correspond to the variable (**upper subfigure**) and constant (**lower subfigure**) resistivities in the considered layers. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**Params.** & \(C_{11}\) & \(C_{12}\) & \(\beta_{1}\) & \(C_{21}\) & \(C_{22}\) & \(\beta_{2}\) & \(C_{31}\) & \(C_{32}\) & \(\beta_{3}\) & \(h_{1}\) & \(h_{2}\) \\ \hline
**Values** & \(0.1790\) & \(-0.0931\) & \(0.2302\) & \(0.2640\) & \(-0.1363\) & \(-0.0781\) & \(0.1300\) & \(-0.0633\) & \(0.0220\) & \(0.48\) & \(1.37\) \\ \hline \end{tabular}
\end{table} 
Table 5: Continuous EGMS model parameters for the soil from Table 4 data.