# An Improved Shuffled Frog Leaping Algorithm for Electrical Resistivity Tomography Inversion
## 4 Numerical Simulation
### Inversion Results
Due to the inherent variability of heuristic algorithm results, in order to ensure the reliability and statistical significance of the conclusions, each algorithm is independently repeated five times and the average value is taken. Figure 4 presents a system comparison of the performance of three inversion algorithms under typical geological models and different noise levels. The following analysis will focus on a 5% noise level as an example for detailed discussion. In Model_A (Uniform Anomaly Body Model) (Figure 4(a1-a3)), the LS inversion, although capturing the spatial distribution trend of the anomaly body, exhibits significant defects. The inversion interface diffusion results in blurred boundaries, and the convergence process oscillates severely, with the final RMSE being 1.92. The standard Mixed Frog Leaping Algorithm (SFLA) achieves a positioning error of the anomaly body's edge of up to five grid units (RMSE = 1.63). The improved SFLA, by introducing adaptive step size control and a population diversity maintenance mechanism, significantly enhances the convergence stability. The boundary error of the anomaly body is generally within two grid units (RMSE = 0.91), with the volume effect notably reduced. In Model_B (Normal Fault Structure Model) (Figure 4(b1-b3)), the differences in structural recognition capabilities are highlighted. The LS inversion produces an over-smoothing effect: the resistivity transition zone between the footwall and hanging wall of the fault is difficult to identify (RMSE = 2.18), and the positive fault offset feature is completely lost. The standard SFLA inversion shows local distortions, with a dip angle error of up to 12\({}^{\circ}\) (RMSE = 1.63). The improved SFLA successfully reconstructs the fault's  
Figure 3: Fitness value variation curve of the improved SFLA inversion. geometric features: the dip angle error is less than 5%, and the boundary between the footwall and hanging wall is clearly defined (RMSE = 0.94). Model_C (inverted fault structure model) (Figure 4(c1-c3)) further validates the robustness of the algorithm. The LS inversion fails to present the fault morphology, with an RMSE of 2.06. The standard SFLA can only reflect the high resistivity characteristics of the hanging wall (RMSE = 1.93). The improved SFLA accurately characterizes the reverse fault features, with a reasonable reconstruction of the dip angle, overall clear morphology of the footwall and hanging wall, and no oscillation artifacts, resulting in an RMSE of only 0.87. In a comprehensive quantitative analysis, under a 5% noise level, the improved SFLA achieves an RMSE of less than 1.0 in all three models, representing a reduction of 57.3 \(\pm\) 2.1% compared to the LS method and a reduction of 43.7 \(\pm\) 1.8% compared to the standard SFLA. Its advantage stems from the balanced grouping strategy, which ensures a balanced contribution weight of subgroups. Additionally, the introduction of the adaptive step size control mechanism effectively suppresses local optima traps. This algorithm provides a new technological path for the fine characterization of electrical structures in complex structural zones. To comprehensively assess the impact of noise levels on the inversion results, Figure 4(d1-i3) present the inversion effects of the three models at 10% and 15% noise levels. Table 1 quantitatively provides the RMSE values of the three inversion algorithms at different noise levels. The results show that as the noise level increases, the RMSE of all algorithms tends to rise. However, the improved SFLA exhibits a more significant advantage in noise resistance, with the highest RMSE of the three models under 15% high noise conditions still remaining at a low level of 1.18. This indicates that the improved  
Figure 4: Inversion results of the algorithms for the three models under different noise levels. (**a1–a3,b1–b3,c1–c3**) are the algorithm inversion results of the three models under 5% noise. (**d1–d3,e1–e3,f1–f3**) are the algorithm inversion results of the three models under 10% noise.