## 3 Model Inversion and Application Case
### Model Inversion  
Assume that there are two anomalous bodies in a homogeneous half-space with a surrounding rock resistivity of \(\rho=100\)\(\Omega\cdot\mathrm{m}\). The resistivities of the low-resistivity and high-resistivity anomalous bodies are \(\rho_{1}=10\)\(\Omega\cdot\mathrm{m}\) and \(\rho_{2}=1000\)\(\Omega\cdot\mathrm{m}\), respectively, both having dimensions of \(2\)\(\mathrm{m}\)\(\ast\)\(2\)\(\mathrm{m}\)\(\ast\)\(2\)\(\mathrm{m}\). The two anomalous bodies are separated by a distance of \(2\)\(\mathrm{m}\), and the distance from their top surfaces to the ground is \(2\)\(\mathrm{m}\). Considering that, in practical situations, the electrical parameters of the lithology do not change abruptly, a linear resistivity transition is introduced between the anomalous bodies and the surrounding rock. On the surface, the electrode arrays are deployed in \"\(\square\)\", \"U\", and \"L\" shapes, with a spacing of \(1\)\(\mathrm{m}\) between adjacent electrodes, as shown in Figure 5 (the blue area represents the low-resistivity anomalous body, and the red area represents the high-resistivity anomalous body). Figure 5: The deployment of electrodes within the monitoring system and the mapping of anomalies’ locations. (**a**) “\(\square\)”-shaped electrode array; (**b**) “U”-shaped electrode array; (**c**) “L”-shaped electrode array. The apparent resistivity data for the three electrode arrays are numerically simulated. The types of devices used, the number of electrodes, and the number of simulated data points are shown in Table 1.  
A 3D inversion was performed on the simulated apparent resistivity data from the three electrode arrays, with five iterations of the inversion. The convergence curves of the inversion iterations are shown in Figure 6a. It is evident that, as the number of iterations increases, the error convergence curves for all three electrode arrays steadily decrease. Moreover, the larger the number of data points (see Table 1), the smaller the RMS values of the convergence curves, indicating better data fitting. Figure 6b-d show the data fitting results between the simulated apparent resistivity \(R_{s}\) and the apparent resistivity \(R_{c}\) calculated from the final inversion model. It can be observed that the data fitting patterns of the three electrode arrays are similar, and the intersection points between \(R_{s}\) and \(R_{c}\) are mostly distributed near the diagonal of the graph. This aligns with the typical data fitting pattern in geophysical inversion. Figure 7 shows the slice plots of the inversion results from the three electrode arrays. Comparing the inversion results of the three electrode arrays, the \"\(\square\)\"-shaped electrode \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Electrode Array** & **Number of Electrodes** & \begin{tabular}{c} **Number of Simulated** \\ **Data Points** \\ \end{tabular} \\ \hline \begin{tabular}{c} \"\(\square\)\"-shaped \\ “U”-shaped \\ “L”-shaped \\ \end{tabular} & \
\begin{tabular}{c} 64 \\ 50 \\ 36 \\ \end{tabular} &
\begin{tabular}{c} 4032 \\ 2450 \\ 1260 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters of different electrode arrays. Figure 6: Error convergence curve and data crossplot for three electrode arrays. (**a**) Error convergence curve for three electrode arrays; (**b**) data fitting plot of the “\(\square\)-shaped electrode array; (**c**) data fitting plot of the “U”-shaped electrode array; (**d**) data fitting plot of the “L”-shaped electrode array. array yields the best exploration results. The inverted planar distribution of the anomalous body matches well with its actual position (see Figure 7a).