## 4 Selected Optimization Methods  
Using an analytical soil model, we searched for the soil data that fit the measured data best and, therefore, we solved an optimization problem. The parameters' determination procedure must be repeated for each subarea shown in Figure 4c. Therefore, it is important that the selected optimization is suitable for the presented problem. The objective function (OF) used for the determination of the soil parameters is written in (8). \[f=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{\rho_{\text{c,i}}-\rho_{\text{m,i}}}{ \rho_{\text{m,i}}}\right|\cdot 100\ \ (\%), \tag{8}\]  
where \(\rho_{\text{c,i}}\) are the calculated apparent resistivity values based on the analytical soil model and \(\rho_{\text{m,i}}\) are the measured apparent resistivity values based on the Wenner method measurement. \(n\) is the number of different distances \(d\) that were used for the measurements. The term \(1/n\) is included in the objective function to achieve comparability of results for measurements with different numbers of used distances \(d\). Various optimization methods can be used for determining soil parameters. In [20, 22, 29], standard optimization methods were used, and in [18, 25, 26, 27, 28], metaheuristics were used, as in the presented work. Metaheuristics are based on natural selection and the survival of the fittest, which makes them robust and not prone to becoming stuck in the local minima. Different optimization methods have different success rates for different optimization problems. In the past, a genetic algorithm [18, 25, 26, 27] or differential evolution [28] was used for determining soil parameters, but different methods were not compared with each other. In the presented work, different metaheuristics are compared with each other using a three-layered soil model. The goal is to determine the most suitable among the selected ones for the presented problem. We tested Differential Evolution [28, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] using three different strategies, namely DE/rand/1/exp, DE/rand/2/exp, and DE/best/1/bin. DE has two internal parameters, which are step size and the crossover probability constant. Based on our experiences using optimization methods, for all calculations, the step size (\(F\)) was set to 0.6 and the crossover probability (\(CR\)) was set to 0.8. The same step size and crossover probability were used in [43, 44], where, similarly, the inverse problem was solved by finding the appropriate  
Figure 6: Two-dimensional cross-section of the FEM test model at the ground surface with marked dimensions. parameters of the magnetization curve in [43] and the DC engine parameters in [44], and the selected DE parameters were proven to be appropriate. Teaching-Learning-Based Optimization (TLBO) [47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57] was also tested. The number of Objective Function Evaluations (OPEs) performed was determined as OFEs = (2 \(\times\) population size \(\times\) iterations). This approach of calculating OFEs was possible because the implementation was used with no duplication elimination phase. TLBO has no internal parameters. Artificial Bee Colony (ABC) was also used, based on good previous experience [58; 59; 60; 61; 62]. It has three phases: foragers, onlookers, and scout. In the implementation, we used [61], and the number of OFEs was determined dynamically, as the scout phase is not proceeded for every iteration. Many practical problems were solved using the ABC method [63; 64; 65; 66; 67]. The internal parameter limit was set to 100, because such a setting of this parameter had achieved good results on similar problems. For the sake of clarity, the basic equations of the optimization methods used are written down [68]. For different DE strategies, the variation is first determined by (9) for DE/rand/1/exp, by (10) for DE/rand/2/exp, and by (11) for DE/best/1/bin.