# An Improved Shuffled Frog Leaping Algorithm for Electrical Resistivity Tomography Inversion
## 1 Introduction  
In the field of geophysical electrical exploration, high-precision resistivity data offers vital insights into the in-depth analysis of underground geological structures and the characteristics of geological bodies. A precise understanding of the resistivity distribution across various underground formations aids in comprehending the occurrence and behavior of groundwater, the distribution of aquifers, and the flow paths of groundwater. This, in turn, provides scientific support for the rational development and conservation of water resources [1, 2, 3, 4, 5]; in the field of engineering geology, it can effectively identify potential geological hazards, such as faults and karst caves [6, 7, 8, 9, 10, 11], ensuring the safety and stabilityof construction projects. It is evident that high-precision resistivity data is the cornerstone of refined detection in electrical exploration, and its importance cannot be overstated. Direct current ERT is widely popular in near-surface exploration due to its cost-effectiveness and strong resistance to interference. However, with the development of exploration demands toward greater precision, its shortcomings in inversion accuracy have gradually become more apparent [12; 13; 14; 15; 16]. ERT inversion is essentially a nonlinear problem, and traditional linear inversion methods, represented by the LS, linearize it. While this simplifies the computational process to some extent, it severely compromises the accuracy of the inversion results. Nonlinear inversion methods, represented by Genetic Algorithm (GA) and particle swarm optimization (PSO) [17; 18; 19; 20], although theoretically capable of handling nonlinear problems, often encounter the issue of getting trapped in local optima in practical applications. In addition, deep learning methods have made rapid progress in recent years, but they mainly suffer from the problem of poor generalization [21; 22; 23; 24; 25]. Given the complex and variable geological conditions, the significant differences in geological conditions across different regions often lead to the existing deep learning models being unable to adapt well to these variations, resulting in a significant reduction in the reliability and accuracy of the inversion results. In recent years, iterative inversion methods using random search or heuristic random search have been introduced into ERT inversion. However, inversion studies based on a large amount of real data have revealed their shortcomings, such as slow convergence speed and the tendency to get trapped in local extrema. In view of this, some scholars have proposed a novel heuristic optimization algorithm--SFLA--based on the foraging behavior of frogs [26; 27; 28; 29]. This algorithm combines the advantages of two population-based intelligent optimization algorithms, MA and PSO. It is characterized by a simple concept, few adjustable parameters, fast computation speed, strong global search ability, good robustness, and ease of implementation. From the algorithm design logic, SFLA first divides the entire population into several subgroups. Within each subgroup, a local search process is independently executed, which is similar to the fine evolutionary operations in MA targeting the local solution space, aiming to achieve deep exploration of the solution space [30]. Subsequently, the algorithm facilitates the transfer of high-quality solutions between different subgroups through periodic subgroup mixing and global information exchange mechanisms. This process draws on the concepts of population cooperation and global guidance from PSO [31]. This unique design allows SFLA to effectively balance local exploration and global search. It has achieved excellent results in pattern recognition, function optimization, and signal processing, and has been preliminarily applied in the field of geophysical inversion [32; 33; 34].