## 3 Comparative Analysis of Numerical Forward Simulation and Soil Trench Modeling Experiments
### Analysis of Experimental Results  
In the numerical forward simulation, the potential value is calculated based on the Cole-Cole model introduced by the dual-frequency excitation method, combined with Equations (4) and (5) to calculate the apparent amplitude frequency and apparent resistivity, respectively. The calculated values together with the experimentally measured values make the apparent resistivity and apparent amplitude frequency characteristic curves as shown in Figure 8. It can be found that the shape of the apparent amplitude frequency \(F_{\text{s}}\) curve in the figure is similar to the inverted apparent resistivity \(\rho_{\text{s}}\) curve, and there is a good correspondence at the same measurement point. In order to further analyze the accuracy of COMSOL simulation, an error analysis is required (see Table 4). Because the pollution field in the experiment and simulation are symmetric structures, it is sufficient to take 1-30 measurement points for the error analysis. Where error = \(\mid\) (simulation value \(-\) experimental value)/experimental value \(\mid\). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & Average Value & Points 1 to 20 & Error \\ \hline \multirow{2}{*}{\(\rho_{\text{s}}/(\Omega\cdot\text{m})\)} & Experimental value & 536.27 \(\Omega\cdot\text{m}\) & \multirow{2}{*}{1.57\%} \\  & Simulation value & 544.70 \(\Omega\cdot\text{m}\) & \\ \hline \multirow{2}{*}{\(F_{\text{s}}/\%\)} & Experimental value & 1.31\% & \multirow{2}{*}{3.0\%} \\  & Simulation value & 1.27\% & \\ \hline \multicolumn{4}{c}{Average Value} & Points 21 to 27 & Error \\ \hline \multirow{2}{*}{\(\rho_{\text{s}}/(\Omega\cdot\text{m})\)} & Experimental value & 418.63 \(\Omega\cdot\text{m}\) & \multirow{2}{*}{16.7\%} \\  & Simulation value & 488.89 \(\Omega\cdot\text{m}\) & \\ \hline \multirow{2}{*}{\(F_{\text{s}}/\%\)} & Experimental value & 4.31\% & \multirow{2}{*}{15.78\%} \\  & Simulation value & 3.63\% & \\ \hline \multicolumn{4}{c}{Average Value} & Points 28 to 30 & Error \\ \hline \multirow{2}{*}{\(\rho_{\text{s}}/(\Omega\cdot\text{m})\)} & Experimental value & 255.80 \(\Omega\cdot\text{m}\) & \multirow{2}{*}{4.93\%} \\  & Simulation value & 243.20 \(\Omega\cdot\text{m}\) & \\ \hline \multirow{2}{*}{\(F_{\text{s}}/\%\)} & Experimental value & 5.89\% & \multirow{2}{*}{3.22\%} \\  & Simulation value & 6.08\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error Analysis of Experiments and Simulations. Figure 8: Comparison between experimental and simulated values. (**a**) Apparent resistivity comparison. (**b**) Apparent amplitude frequency Comparison. As can be seen from Figure 8 and Table 4, at the edge of the monitoring area (measurement points 1-20), since the soil was untreated here in the experiment, differences in soil particle properties would have perturbed the apparent resistivity values. The \(F_{\mathrm{s}}\) and \(\rho_{\mathrm{s}}\) errors are both less than 3%, which indicates that COMSOL also has a fairly high-precision detection effect for the uncontaminated area. The \(F_{\mathrm{s}}\) and \(\rho_{\mathrm{s}}\) between the experimental and simulated values reached more than 15% in the measurement points 21-27. This is because the contaminants in the simulation are only fixed in the set orientation and do not infiltrate into the surrounding area. In contrast, in the experiment, the chromium solution inside the fine sand is not completely dried and volatilized, and it will still infiltrate the surrounding soil. The apparent resistivity \(\rho_{\mathrm{s}}\) decreases gradually when the MN measurement points are gradually approaching the chromium-contaminated body. The errors of \(F_{\mathrm{s}}\) and \(\rho_{\mathrm{s}}\) between the experimental and simulated values are both less than 5% at the projection position of the chromium-contaminated body (measurement points 28-30). In conclusion, monitoring the chromium pollution field using the Cole-Cole model introduced based on the dual frequency IP method is feasible.