# An Improved Shuffled Frog Leaping Algorithm for Electrical Resistivity Tomography Inversion
## 6 Discussion and Analysis  
The performance of the SFLA essentially depends on the synergy between parameter configuration and the evolutionary mechanism. The population size (P) directly determines the breadth of the solution space: when P is too small, the population struggles to cover key regions of the high-dimensional parameter space, leading to the omission of the global optimum; when P is too large, redundant individuals significantly increase computational costs without linearly improving the convergence probability. This contradiction requires determining the \(p\)-value based on the nonlinearity of the inversion problem, which is generally positively correlated with the degrees of freedom of the model parameters. The essence of the grouping strategy is the key link in balancing local development and global exploration. If the number of groups (m) is too small, it will weaken the independence of subpopulations, leading to the homogenization of information between  
Figure 8: Improved SFLA inversion fitness variation curve. Figure 9: Inversion results of different algorithms for Profile E1. (**a**), res2 dinv; (**b**), Improved SFLA. groups, causing the algorithm to degrade into a single population optimization; whereas if m is too large, the subpopulation size (n) will become too small, and there will be insufficient diversity within the group to support deep search. The core contradiction lies in the fact that standard grouping assigns individuals based on fitness in descending order, causing high-quality subgroups (with lower group numbers) to monopolize the update rights of the global optimum \(X_{g}\). When these subgroups fall into local optima, low-quality subgroups (with higher group numbers) lose the ability to escape because their evolutionary direction is dominated by \(X_{g}\)--this is the root cause of premature convergence. This paper breaks this limitation through a balanced grouping strategy, forcing equal contribution weights to \(X_{g}\) across all subgroups, thereby ensuring population diversity from a mechanistic perspective. However, the robustness of this strategy is influenced by the complexity of the geological model and the parameter configuration, and it may fail in extreme scenarios. The failure mode shares common characteristics with the standard SFL. When the inversion target is an extremely complex geoelectrical model or the observed data contains strong noise, all subgroups may simultaneously fall into local optima. The number of iterations within a group (Ni) regulates the \"exploitation-exploration\" transition frequency. A small Ni will interrupt local deep search, forcing the algorithm to frequently perform inefficient global jumps; whereas a large Ni will cause the subpopulation to overly focus on local areas, losing the global perspective. Ideally, Ni should decay adaptively with the iteration process, allowing sufficient local search in the early stages, and gradually decreasing in later stages to accelerate global information integration. The physical meaning of the maximum step size (Rmax) is the search radius of the solution space. The standard SFL uses a uniform random step size factor in the range [0, 1], whose constant variance characteristic contradicts the entropy reduction law of the search process. The piecewise adaptive step size designed in this paper maintains a large step size in the early iterations to promote exploration and then exponentially decays to a fine-tuning mode in the later stages. This mechanism significantly improves the convergence speed of the algorithm. The global number of iterations (\(K\)) needs to match the problem complexity. For high-dimensional non-convex problems like ERT inversion, \(K\) must at least satisfy  
\[K\geq\frac{ln(1/\varepsilon)}{ln(1-\delta)}\]  
where \(\varepsilon\) is the convergence threshold, and \(\delta\) is the improvement probability per iteration. Empirical evidence shows that when the model parameters exceed 100, \(K\) < 500 increases the risk of under-convergence by more than 60%.