# An Improved Shuffled Frog Leaping Algorithm for Electrical Resistivity Tomography Inversion
## 5 The Practical Application of the Improved Hybrid Frog Leaping Algorithm  
The study area is located 5 km northeast of Dongchang Street, Biancheng Town, Jurong City, Jiangsu Province (Figure 5). The survey line E1 crosses borehole ZK2 (with a bedrock depth of 6.7 m) and ZK3 (with a bedrock depth of 12.0 m). The bedrock is limestone (resistivity theoretical value: 10\({}^{2}\)\(\sim\)10\({}^{4}\)\(\Omega\cdot\)m), with overlying plain fill soil (50-500 \(\Omega\cdot\)m). The stratigraphic sequence revealed by the borehole lithological logs is shown in Figures 6 and 7. In the improved hybrid frog leaping algorithm inversion, the total number of frogs is set to 6400 (divided into 80 groups, with 80 frogs per group). The algorithm is run for 1000 iterations, with a resistivity constraint range of 2 to 10\({}^{4}\)\(\Omega\cdot\)m (after natural logarithmic transformation, this corresponds to a range of 0.693 to 9.210). The fitness variation curve in Figure 8 shows that at iteration 9, the fitness value decreases from 4.237 to 0.612; at iteration 400, it further decreases to 0.277; and after iteration 733, it stabilizes and converges to 0.189. This reflects that the algorithm has both fast global search capabilities and fine local optimization abilities.  
The inversion results comparison in Figure 9 shows that although the LS inversion (Figure 9a) can identify the trend of bedrock burial depth (indicated by the black dashed line in the figure), the bedrock interface is blurred, and the root mean square error (RMSE) reaches 2.920. The improved SFLA inversion (Figure 9b) shows a significant enhancement in boundary clarity (indicated by the black dashed line in the figure): at a horizontal distance of 55 m (corresponding to ZK3 location), the inversion depth is 14.05 m (measured 12.0 m, error 17%); at a horizontal distance of 96 m (corresponding to ZK2 location), the inversion depth is 6.9 m (measured 6.7 m, error 3.0%); the inversion resistivity of the limestone layer is 8.2 \(\times\) 10\({}^{3}\)\(\Omega\cdot\)m, which aligns with the expected geological range; and the root mean square error is reduced to 1.053 (a decrease of 64%). The algorithm accurately captures the undulating morphology of the bedrock--in the horizontal distance range of 20-58 m, a continuous downward trend is observed; between 58 and 78 m, a rising feature appears; from 78 to 136 m, a descending trend is observed, followed by a gradual uplift. This aligns with the depth variations revealed by the boreholes, indicating the accuracy and reliability of the inversion results.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Inversion Algorithm**} & \multicolumn{3}{c}{**Model\_A**} & \multicolumn{3}{c}{**Model\_B**} & \multicolumn{3}{c}{**Model\_C**} \\ \cline{2-10}  & **5\%** & **10\%** & **15\%** & **5\%** & **10\%** & **15\%** & **5\%** & **10\%** & **15\%** \\ \hline The LS & 1.920 & 2.300 & 2.680 & 2.058 & 2.420  
It should be noted that the 17% error level in this inversion meets the relevant criteria for geophysical exploration and the requirements of the engineering task. In the field of geophysical exploration in China, for resistivity inversion in complex geological areas such as deep transitional zones and weathered fractured regions, the commonly accepted error threshold is typically \(\leq\)20%. Additionally, the task book for the Baota Mountain mining area clearly specifies that an inversion error within \(\pm\)20% for bedrock depth is considered acceptable. In accordance with the task book's requirement to randomly place at least one borehole every 100 m along the survey line, two borehole data sets were provided for comparison and verification.  
Figure 5: Location of the study area and layout of the measurement lines.  
Figure 6: Borehole ZK2 lithological log.  
Figure 7: Borehole ZK3 lithological log.