## 5 Inversion Test of an Ideal Model
### One-Dimensional Three-Layer Model Test  
Inversion tests were conducted on the regularization factors (\(\lambda_{1}\sim\lambda_{7}\)) using a three-layer H-type resistivity model to select the regularization weight corresponding to the threshold value of each layer thickness. The inversion was performed using a segmented regularization model, where the layer-like low-resistance distribution of the three H-type models was located at depths of 70-80 M, 250-300 M, and 500-600 M with corresponding inner layer thickness thresholds of, respectively, J\({}_{1}\) (2-25 M), J\({}_{2}\) (5-50 M), and J\({}_{3}\) (10-100 M). Additionally, to verify the stability of the regularization factors under different resistivity ratios of the intermediate layer and the background layer (\(\sigma_{r}\)), tests were conducted using \(\sigma_{r}=5\) and \(\sigma_{r}=40\). In the three-layer H-type model,  
\[\sigma_{r}=(\rho_{1}+\rho_{3})/(2\rho_{2}) \tag{15}\]  
where \(\rho\) represents the resistivity of each layer. Considering the probability distribution properties of the algorithm, we selected the results corresponding to the median of D-MSE in 11 inversions. The results are presented in Table 4. The data comparison in Table 4 reveals that the regularization factor (\(\lambda_{min}\)) associated with the minimum MF-MSE for each row is approximately located around \(\lambda_{3}\) and \(\lambda_{5}\) when \(\sigma_{r}\) is 5 and 40, respectively. This indicates that the size of \(\sigma_{r}\) significantly influences the selection of the regularization factor under the specified threshold of layer thickness. The inversion results for \(\lambda_{1}\), \(\lambda_{7}\), and \(\lambda_{min}\) are depicted in Figures 7 and 8. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Without Noise**} & \multicolumn{2}{c}{**5\% Noise**} & \multicolumn{2}{c}{**10\% Noise**} \\ \cline{2-7}  & **Fitness Value** & **D-MSE** & **Fitness Value** & **D-MSE** & **Fitness Value** & **D-MSE** \\ \hline ISSA & 0.0031 & 0.1182 & 0.0065 & 0.1298 & 0.0123 & 0.1893 \\ SSA & 0.0065 & 0.1966 & 0.0095 & 0.2255 & 0.0165 & 0.2563 \\ JADE & 0.0103 & 0.2299 & 0.01424 & 0.2177 & 0.0224 & 0.2388 \\ SAPSO & 0.0059 & 0.1945 & 0.0086 & 0.2013 & 0.0157 & 0.2255 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of averages of model evaluation parameters for 11 inversion results. Figure 6: Comparison of one-dimensional inversion results of various algorithm. (**a**) Comparison of inversion models; (**b**) plot of adaptation with number of iterations. From the figures, it is evident that the curve of \(\lambda_{7}\) exhibits poor smoothness in resistivity at multiple points. This suggests that the non-uniqueness of the inversion is exacerbated, and the results may be inadequate when the regularization constraint is too weak. Conversely, the curve of \(\lambda_{1}\) corresponds to a situation where the regularization constraint is overly strong. In Figure 7, where \(\sigma_{r}=5\), not only is there a significant shift in the boundary of the target body but also the overall inversion performance needs improvement. In Figure 8, as \(\sigma_{r}\) increases, the fitting of the inversion to low resistivity improves to some extent, but the curve of the inversion result becomes excessively smooth, leading to reduced accuracy. The curve of \(\lambda_{min}\) strikes a relative balance and does not exhibit significant false anomalies. The inversion results accurately reflect the low resistivity characteristics of the