Investigating the performance of the differential evolution algorithm in estimating soil hydraulic parameters

Introduction
The soil water curve is one of the most critical soil hydraulic characteristics. This characteristic is used to determine soil water in the field capacity point and the permanent wilting point (PWP) beside it has a vital role in the application of soil water models in the study of soil-plant-water relationships. This curve is known as the quality soil index which has an effective role in the explanation of agricultural, ecological, and environmental problems. Impressive and efficient management of soil and water resources, water flow and solute transport survey, soil pollution, and contaminant leakage into water sources are dependent upon the accurate estimation of soil water curve parameters. Moreover, this index has a functional role in applying numerical and hydrological models. On the other hand, to better identify and understand its role, different models were provided to describe this curve mathematically. The efficiency of these models depended on the accuracy of estimated parameters in the model structure that was defined. Soil water curve is known as a non-linear relationship that is used to describe the relation between soil and water content or degree of soil saturation. The soil water curve provides essential information for using irrigation methods and about soil resistance and soil mechanical properties. In this research, the performance trend of two meta-heuristic algorithms, including the differential evolution (DE) and particle swarm optimization (PSO), was studied to estimate hydraulic parameters of soil water curves based on the van Genuchten and the Brooks and Cory models in four soil texture classes; loam, silt loam, sandy loam, and sandy clay loam. Besides, this study evaluated the performance of the meta-heuristic algorithm to RETC software. This software has a non-linear square local algorithm. This study can evaluate the ability of the meta-heuristic algorithms to estimate parameters for exponential relationships and nonlinear models.
 
Materials and Methods
At the agricultural farm of the University of Birjand, a study was conducted to analyze soil water content in different texture classes. The research involved the random selection of four soil texture classes and the random sampling of 20 points from each class. The soil water content was measured using a sandbox and pressure plate device, covering a broad suction range of 0-15000 cm. In the first phase, soil water curve parameters were estimated for each soil texture using the van Genuchten model and the Brooks and Cory model in the RETC software. Subsequently, the Matlab desktop environment was utilized to apply meta-heuristic algorithms (DE and PSO) to estimate the soil water curve parameters based on the two models. An objective function was defined to minimize the Root Mean Square Error (RMSE) of the meta-heuristic algorithms' performance. Finally, the study compared the performance of the meta-heuristic algorithms (DE and PSO) with the RETC software in estimating soil water curve parameters based on the van Genuchten and Brooks and Cory models, using statistical indices such as RMSE and R2. The soil texture classes play a crucial role in influencing soil water content and nutrient retention, making them an essential factor in agricultural management and crop suitability. The study's findings can contribute to a better understanding of soil water dynamics and the development of improved agricultural practices.
 
Results and Discussion
The obtained results of the statistical indices (RMSE and R²) showed that the least value of RMSE was acquired by the differential evolution algorithm (DE) performance. The values of RMSE during the application of the DE algorithm as an estimated method based on the van Genuchten model were 0.0008, 0.0005,0.0004, and 0.0006 also based on the Brooks and Cory were 0.006, 0.006, 0.005, and 0.0005 in sandy clay loam, sandy loam, loam, and silt loam respectively. Also, the highest value of the R² index was obtained equal to 0.995, 0.996, 0.994, and 0.994 by the utilization of the DE algorithm based on the van Genuchten model in the sandy clay loam, sandy loam, loam, and silt loam respectively. The values of RMSE by the utilization of the PSO algorithm based on the van Genuchten model were 0.0021, 0.006, 0.0057, and 0.006 in the sandy clay loam, sandy loam, loam, and silt loam classes respectively. The highest and lowest values of the RMSE and R2 indices by the application of RETC software were obtained equal to 0.017 and 0.912 (sandy clay loam), 0.01and 0.963 (sandy loam), 0.085 and 0.972 (loam), and 0.01 and 0.924 (silt loam) based on the van Genuchten model.
 
Conclusion
It could be concluded that RETC software has poor performance in the estimation of soil water curve parameters in all soil texture classes studied based on the van Genuchten and Brooks and Cory models. This trend represents the weakness of the local algorithms to solve multivariable problems where an exponential relationship exists between the variables and they are influenced by each other. On the other hand, the results show the meta-heuristic algorithms have sufficient ability to estimate parameters in multivariable problems. It could be concluded that the meta-heuristic algorithms have better performance in estimating the parameters of soil hydraulic models. The DE algorithm is the best method to estimate soil hydraulic parameters. The PSO algorithm has the nearest performance to the DE algorithm but the best performance to RETC. Finally, meta-heuristic algorithms are suitable options for estimating soil water curve parameters based on various hydraulic models.