## 3 Application to Resistive Driven-Rod Method: Testing with Some Examples
### Synthetic Examples with No Internal Resistance  
Let us consider first a synthetic example in which the rod has no internal resistance. Let the ground be a synthetic three-layer soil with parameters \(\rho_{1}=100\)\(\Omega\)m, \(\rho_{2}=50\)\(\Omega\)m, \(\rho_{3}=300\)\(\Omega\)m, \(h_{1}=2\) m, and \(h_{2}=3\) m, where a vertical zero-resistive rod of 9 mm radius is driven into the soil from 0.5 m up to 8 m deep. The rod is assumed to be made of a single continuous piece that is introduced in segments of 0.5 m in the ground. For every 0.5 m of drilling, the grounding resistance is calculated assuming that it is a value obtained in a field measurement using the fall-of-potential method. Table 1 shows the results where L is in meters and R is in Ohms. From data of Table 1, a MatLab-based minimization algorithm using the fminsearch routine is launched where the objective function is (17). The parameters of the model obtained from the minimization procedure are \(\rho_{1}=99.01\)\(\Omega\)m, \(\rho_{2}=49.79\)\(\Omega\)m, \(\rho_{3}=292.45\)\(\Omega\)m, h\({}_{1}=2.03\) m, and h\({}_{2}=2.94\) m. Figure 4 shows by using diamond markers the values on the Table 1. The red continuous line, labeled as Calculated 1, stands for the calculated grounding resistance from the obtained optimum solution while the dashed blue line, labeled as Calculated 2, shows the calculated grounding resistance from the true soil parameters. The subfigure in Figure 4 stands for the percentage error \(\varepsilon_{R}=(R_{true}-R_{calc})/R_{true}\) in the calculation of the grounding resistance using the estimated soil model, where \(R_{true}\) is the grounding resistance from the true soil model, whilst \(R_{calc}\) is the calculated grounding resistance from the estimated soil model. Although sixteen simulated data have been used in this synthetic example, it is possible to recover the soil structure with much less data. Let consider another example of synthetic soil \(\rho_{1}=50\)\(\Omega\)m, \(\rho_{2}=200\)\(\Omega\)m, \(\rho_{3}=10\)\(\Omega\)m, h\({}_{1}=2\) m, and h\({}_{2}=3\) m. Again, we will assume a non-zero internal resistance rod of 9 mm radius driven into the soil. As before, the rod is assumed to be made of a single continuous piece that is driven into the ground, but in this experiment the grounding resistance is simulated every 1.5 m. Only five grounding resistances corresponding to five buried electrode lengths will be simulated. Table 2 collects the simulated grounding resistance for the considered rod lengths. The parameters of the model obtained from the minimization procedure are \(\rho_{1}=49.47\)\(\Omega\)m, \(\rho_{2}=171.76\)\(\Omega\)m, \(\rho_{3}=9.59\)\(\Omega\)m, h\({}_{1}=1.85\) m, and h\({}_{2}=3.22\) m, which shows significant dif \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**L (m)** & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\ \hline
**R (\(\Omega\))** & 139.90 & 80.67 & 57.79 & 44.68 & 31.74 & 25.52 & 21.66 & 19.05 \\ \hline
**L (m)** & 4.5 & 5 & 5.5 & 6 & 6.5 & 7 & 7.5 & 8.0 \\ \hline
**R (\(\Omega\))** & 17.18 & 15.97 & 15.76 & 15.54 & 15.31 & 15.07 & 14.85 & 14.62 \\ \hline \end{tabular}
\end{table} 
Table 1: Buried rod length L (m) and the corresponding grounding resistance R (\(\Omega\)). Figure 4: Simulated and calculated grounding resistance. ferences with the values of the real soil, especially in the intermediate layer. However, the grounding resistance obtained for the five lengths considered is very close to its true value. Figure 5 shows with diamond markers the values on the Table 2. The red continuous line labeled as Calculated 1, stands for the calculated continuous grounding resistance from the solution of the minimization procedure. The blue line stands for the calculated continuous grounding resistance but using the true soil structure. The differences between the estimated soil structure and the actual structure come from coarse sampling of the soil when using the grounding resistance of the vertical rod in jumps of 1.5 m length.