Prediction Model for Reference Crop Evapotranspiration Based on the Back-propagation Algorithm with Limited Factors

The precise estimation of reference crop evapotranspiration (ETO) is vital for regional and irrigation water resource management. It is also beneficial to the rational allocation of regional water resources and alleviates the disparity between water supply and demand. This study accurately estimates the ETO of 14 meteorological stations in southern China. Five neural network models (extreme learning machine [ELM], back-propagation neural network [BP], ant colony optimization [ACO]-BP, bird swarm algorithm [BSA]-BP, and cat swarm optimization [CSO]-BP) were introduced to predict ETO with limited factors using different methods. The results demonstrated that models involving T (average, maximum, and minimum air temperature), sunshine duration (n), and relative humidity (RH) exhibited the highest accuracy of all studied combinations; the role of T, n, RH, wind speed (U2) and average atmospheric pressure (AP) regarding ETO gradually decreased. These three biological heuristic algorithms (ACO, BSA, and CSO) each significantly enhanced the capability of the BP model. The accuracy and computational cost of the CSO-BP model are better than those built by other algorithms. Therefore, it is strongly recommended to use the CSO-BP model for ETO estimation in southern China. This result serves as a reference for a more accurate estimation of ETO for future irrigation decision-making and water resource management in southern China.


Introduction
The actual evapotranspiration of crops (ET a ) refers to the water loss caused by soil evaporation and vegetation transpiration during plant growth (Elkatoury et al. 2020). Calculating ET a is vital for irrigation water resource management. ET a can be estimated by multiplying the reference crop evapotranspiration (ET O ) by the crop factor. ET O is therefore essential for determining crop water requirements (Gocić and Arab Amiri 2021;Cui et al. 2021). To obtain accurate ET O for agricultural water management, the Food and Agriculture Organization (FAO) suggested the use of the Penman-Monteith (Ravindran et al. 2021;Maqsood et al. 2022) model . FAO-56 PM is a semi-empirical and semi-physical model for ET O calculations but requires multiple input parameters, for example, the average temperature (T ave ), average wind speed (U), maximum (T max ), minimum (T min ), relative humidity (RH), and sunshine duration (n). Nonetheless, many areas lack modern weather stations, and accessing comprehensive meteorological data can be challenging, so it is difficult to estimate the ET O accurately. Therefore, improving ET O simulation accuracy has become a research highlight of recent studies without input data.
Simple input models for ET O estimation, such as temperature-and radiation-based models, have been recently constructed in many studies (Rodrigues and Braga 2021). Despite being often used for estimating ET O , empirical models have difficulty in handling the complex and nonlinear interactions between independent and dependent variables. With the development of artificial intelligence technology, machine learning (ML) for managing nonlinear problems has improved model performance considerably (Roy 2021;Zhao et al. 2021). Zhang et al. (2018) tested the adaptability of an ET O model constructed using three ML algorithms, namely the support vector machine (SVM), back-propagation neural network (BP), and adaptive neuro-fuzzy inference system (ANFIS), and the outcomes showed that the models have good applicability for predicting ET O .  evaluated eight ML models to predict daily ET O with few meteorological parameters in various climatic regions of China. In the study, the SVM models have high accuracy and stability. Bellido-Jiménez et al. (2020) selected several temperature-based ML models to estimate ET O in the semi-arid region of Andalusia, and the outcomes revealed that extreme learning machine (ELM) models have high performance in predicting ET O .
These ML models have satisfactory performances with single input factors, and numerous studies have reported that ML models are generally more accurate than empirical models in estimating ET O . Zhu et al. (2020) successfully predicted daily ET O with limited meteorological data using different ML models along with six empirical models (Priestley-Taylor, Makkink, Imark, Hargreaves-Samani (HS), Dalton, and Trabert), achieving satisfying results. dos Santos Farias et al. (2020) studied the performance of ML and empirical models in a Brazilian agricultural frontier with limited meteorological data, and the ML model with the same climatological variables performs better than the empirical model. However, ML algorithms can be problematic, and the algorithm parameters are not necessarily optimal, limiting the constructed model's estimation accuracy. Many scholars have employed optimization algorithms for ML models. The results showed that radial basis function networks (RBFN) help ML models in estimating ET O with different input parameters.  applied four bio-inspired algorithms to improve the accuracy of ELM models. They found that hybrid ELM models showed greater improvements in predicting daily ET O . Ahmadi et al. (2021) employed a novel model via intelligent water drops and support vector regression to predict ET O . Results showed that it performed best among all ML and empirical models. The optimization algorithms have improved ML models to estimate ET O , and the hybrid optimization algorithm was recommended to further improve the ML models to estimate ET O .
Along with another optimization algorithm, research has frequently reported that the three optimization algorithms of ant colony optimization (ACO), bird swarm algorithm (BSA), and cat swarm optimization (CSO) exhibit more favorable optimization effects. Wang et al. (2020) indicated that optimization with BSA markedly improved the performance of the SVM model compared with particle swarm optimization (PSO) and the genetic algorithm (GA); however, the BSA had higher stability and robustness. Huang et al. (2020) observed that CSO resulted in higher accuracy than PSO in predicting rock fragmentation. Liu et al. (2021) selected features for estimating chlorophyll content based on ACO, and the results showed that ACO can reduce the complexity and improve the estimation performance of the model. Southern China is a crucial grain production area. This region is largely subject to a tropical monsoon climate with abundant rainfall. Its topographical conditions are complex; precipitation's climatic conditions and temporal and spatial distribution in different regions differ considerably. Therefore, the efficient use of agricultural water in the southern region is crucial. This paper inputs various parameter combinations into BP, ELM, ACO-BP, BSA-BP, and CSO-BP models to construct an ET O estimation model for typical stations in southern China. The study aims were as follows: (1) to determine the influence of different input combinations on daily ET O estimation, (2) to develop five ML models (BP, ELM, ACO-BP, BSA-BP, and CSO-BP) for daily ET O estimation with limited factors, and (3) to contrast the five models' capacity for adaptation and forecast accuracy in southern China.

Study Area and Data Sets
This study analyzed data from 14 stations (Wenjiang, Kunming, Wuhan, Shapingba, Changsha, Guiyang, Nanjing, Hefei, Hangzhou, Nanchang, Xiamen, Guangzhou, Nanning, and Haikou) in southern China , with a subtropical, tropical monsoon climate. Because of the monsoon effects, the annual precipitation variation in southern China is large. The availability of water resources is changeable; the high rainfall in summer and autumn and the low rainfall in winter and spring can induce summer waterlogging and spring droughts, respectively.
The data were extracted from the China Meteorological Administration (http:// data. cma. cn/) daily dataset from 1960 to 2019 and included T, U, n, RH, and average atmospheric pressure (AP). The datasets of the 14 stations were of high quality. China Meteorological Administration conducted data quality control, based on which we further deleted missing or abnormal data. The k-fold method is used to divide the dataset, which randomly divides the data into two subsets, i.e., the Training and Testing dataset. The k value is set to 10; that is, the data is divided into ten parts, and nine parts are taken in turn as training, one part is used for testing, and the mean of the results is used for the estimation ).

Different Machine Learning for Estimating ET O
This study uses eleven combinations (Table 1) based on different meteorological data inputs for different ML models to predict the daily ET O . Various studies have indicated that temperature factors T ( average, maximum, and minimum temperature) affect ET O (Xing et al. 2016). Therefore, T is selected for the first three inputs of the model in this study. Figure 2 depicts a flowchart of the daily ET O estimation process applied in this study. To address the shortcomings of the traditional BP algorithm, the ACO, BSA, and CSO algorithms are applied to optimize the initial weights of the BP algorithm, and the BP algorithm is established with the respective optimization models, consequently referred to as the ACO-BP, BSA-BP, and CSO-BP algorithms. The program code is written in MATLAB software, version R2020b. All the simulations were performed in a computer with Intel® Core ™ i7-10700 K CPU @ 3.80 GHz and 16 GB RAM. (1)

Back-propagation Neural Network
The BP algorithm, also known as the error back-propagation algorithm, is a multilayer forward neural network. The function of a finite number of discontinuous points is approximated through an input and output layer, and several hidden layers. Effectively

Extreme Learning Machine
The ELM is a learning algorithm based on single hidden layer feedforward neural networks that can directly approximate nonlinear mapping with input data; this is useful for many natural and artificial methods that are difficult to manage using classical parameterization methods. With the presence of neural networks in the models, the hidden layer node parameters of the algorithm can be randomly or artificially provided without adjustment; the learning speed is fast and generalizability is high. For more information about the ELM model, refer to Huang et al. (2006).

Ant Colony Optimization Algorithm
ACO is an intelligent optimization algorithm that simulates ant colony foraging behavior. Ants communicate through pheromones when searching for food; the shorter the path, the greater the concentration of information and the probability of choosing the path. Over time, more ants prefer short paths between food sources and nests. Based on this natural phenomenon, the ACO algorithm exhibits high robustness, parallelism, and favorable characteristics, and can determine a globally optimal solution quickly. More details about the ACO can be found in Laura et al. (2008) and Abbas and Fan (2018).

Bird Swarm Algorithm
The BSA optimizes the BP neural network through the following steps (Altay and Alatas 2020): The BSA's global search function is employed to adjust the initial weight values and threshold values of the BP neural network. Each group of decision variables is contained in the spatial position of each bird in the flock. The fitness function is used to measure the superiority of the individual's spatial position. An individual's spatial position is continuously updated using behavioral strategies, such as those related to foraging, vigilance, and flying until the foraging process of the flock is optimized. For details on the algorithm, please refer to Meng et al. (2016).

Cat Swarm Optimization Algorithm
CSO (Lin et al. 2016) is a swarm intelligence algorithm proposed by observing the behavior of cats, which comprises tracking and finding modes. It optimizes the BP neural network's input and hidden layers, the connection weights between the hidden and output layers, and the threshold for each layer. The details can be found in Chu et al. (2006).

Model Evaluation
R 2 , RMSE, MAE, NSE, and GPI are the five statistical indicators used to assess the performances of the models (Feng et al. 2018;Agrawal et al. 2022;Ahi et al. 2022).
where X i is the predicted value, Y i is the measured value, and X and Y represent their mean values, respectively; n is the number of measurements. α equals -1 for R 2 and NSE, and equals 1 for RMSE and MAE. T j is the normalized value of the RMSE, MAE, R 2 and NSE, T j is the median of T j .

Evaluation of Different Meteorological Data Combinations for Daily Reference Crop Evapotranspiration Estimation
The performance of the different daily ET O estimation models using different meteorological data combinations is presented in When an input factor is added, the model accuracy improved. The highest accuracy is C3, indicating that the influence of n is greater than that of other factors. Combined with the input of only the temperature factor, it can be known that as input, the meteorological factors that have the greatest impact on the model are in descending order of T, n, RH, U 2, and AP. Four input factors, the mean ranges of RMSE, R 2 , MAE, and NSE of the constructed model are 0.391-0.708 mm d −1 , 0.792-0.930, 0.293-0.547 mm d −1 , and 0.774-0.930, respectively. When five factors are inputted, the mean ranges of RMSE, R 2 , MAE, and NSE of the constructed model are 0.326-0.697 mm d −1 , 0.805-0.952, 0.249-0.548 mm d −1 and 0.776-0.952, respectively. A fifth input factor is introduced to generate the most accurate input combination C11, compared to other combinations; it can be seen that the contribution of n to ET O is higher than the superposition of other factors, and a consistent conclusion can be obtained for RH. Figure 3 shows the variable analysis heatmap obtained by performing a Pearson correlation analysis matrix on the input. The overall results in the southern region show that other factors except RH and AP are positively correlated with ETo. The absolute value of the correlation is the largest for T max and the smallest for AP, which complies with the results obtained by the different input models constructed above.
This study confirmed that the ML models with T, n, and RH factor inputs exhibited the optimal performance for daily ET O estimation. As input and in descending order, the meteorological factors that revealed the most influence on the model are T, n, RH, U 2, and AP. Through the different divisions of the factor input combination, it can produce less input, which is better than adding more input. It is significant in terms of reducing input requirements and computational costs. After revealing the influence of variables on the model results, consistent results can be obtained by using Pearson correlation analysis. Therefore, correlation analysis is recommended to verify variable selection, and feature selection algorithms can be used to study input factors in followup research further.  Figure 4 shows the accuracy comparison of five different ML models in the southern region. The information given in the figure shows differences in the performance of different models. Regarding RMSE and MAE, the BP model has the highest median line, followed by ELM, indicating that the ELM algorithm is better than BP. However, the optimized BP model shows better accuracy than ELM in model evaluation. The CSO-BP midline is the lowest, indicating that the results are the most satisfactory among all models. In comparison, the ACO-BP accuracy is slightly better than that of BSA-BP. R 2 and NSE are opposite to RMSE and MAE in the evaluation model; that is, the higher the median line, the better the model accuracy, and R 2 and NSE give the same results as RMSE and MAE when evaluating five models.   Table 2 lists the average calculation cost (time used for calculation) for the different models. The outcomes demonstrate that the consumption time average by the BP model with different combinations of factors is at most 0.67 s. The ELM model had the lowest time cost among the five models, and the time cost of varying input factors ranged from 0.02 to 0.03 s. Among the three hybrid models, the time costs of the ACO-BP and BSA-BP were relatively close at 32.38-45.13 s and 37.25-45.51 s, respectively; that of the CSO-BP was the lowest among the hybrid models, with an average running time of 4.14-7.25 s.

Statistical Performance of Different Machine Learning Models for Daily Reference Crop Evapotranspiration Estimation
In the BP algorithm, the gradient descent method requires multiple iterations to modify the weights and thresholds, and the training speed is therefore slow. Zhang et al. (2018) combined remote sensing data with an ML algorithm and applied the BP machine algorithm to establish an ET O estimation model of spatial distribution. The results revealed that the BP algorithm had lower estimation accuracy for ET O than models, such as the ANFIS, SVM, and artificial neural network. The BP model sinks into the local optimum easily and cannot reach the global minimum. The ELM algorithm does not require weights or threshold adjustments during the training process. Still, it only adjusts the number of neurons in the hidden layer to obtain the only optimal solution. Therefore, the accuracy of the ELM model is higher than that of the BP model. Optimization algorithms can markedly improve the simulation accuracy of ML models. Arora et al. (2021) used the ANIFS algorithm to optimize the GA to predict flood sensitivity, significantly improving the model's accuracy with the optimized algorithm.
This study used ACO, BSA, and CSO to optimize the BP model. Among the three hybrid models, CSO-BP had the highest accuracy. These optimization algorithms show satisfactory optimization results for the BP model. CSO-BP is more accurate than the other two optimization algorithms, mainly because the CSO algorithm has a dynamic grouping mechanism to hinder the algorithm from falling into local optimum. The optimization algorithm has the advantages of fewer control parameters, fast convergence speed, and high robustness, and the optimized model has higher accuracy. The other two optimization algorithms may face being trapped in local optima and slow to converge, affecting their performance ).

Comparison of Reference Crop Evapotranspiration Models by GPI
To use as little meteorological data as possible to build a prediction model for ET O Dong et al. (2022) verified that the machine learning model is superior to the empirical model in estimating ETo, and the convolutional neural network model performed best. The study shows satisfactory accuracy with less data input. It can better predict ET O in southern China and recommends the CSO-BP (T, n, and RH) model because of its accuracy, stability, and computational efficiency. However, this study only covers part of China, and the global promotion of the established model needs to be studied. The study shows that the estimation accuracy of sites with different climatic characteristics is quite different. There is no free lunch theorem proving that a single algorithm cannot perform satisfactorily on all problems. Therefore, the next step can apply this model to different climate zones and regions to find more suitable climatic conditions. Furthermore, by optimizing the optimization model, the performance of the model regarding the accuracy and computational cost can be improved. Future research can integrate the climate models constructed in different regions into the irrigation decision-making system to provide new references for agricultural water resource management.

Conclusion
In this paper, three optimization algorithms are used to optimize the estimation model of ET O , compare the time cost of different models, and select the model with the best performance. Correlation analysis is performed to verify the results of feature combination, revealing the effects of different input features on ET O estimation in southern China, and providing a reference for ET O estimation in regions lacking meteorological data. The results demonstrated the following: 1. With the input of T, n, and RH factors, ET O estimation models achieved the highest accuracy. The meteorological factors as input with the most influence on the model were, in descending order, T, n, RH, U 2, and AP. 2. Compared to the unoptimized BP model, the accuracy of the ELM model was greater.
The ELM model had the lowest time costs of all five estimation models (0.02-0.03 s). The three algorithms (ACO, BSA, and CSO) exerted satisfactory optimization effects on the BP model. The CSO-BP model had the highest accuracy, with mean values of 0.326 to 0.586, 0.842 to 0.952, 0.249 to 0.445, 0.842 to 0.952, and 0.329 to 1.703 for RMSE, R 2 , MAE, NSE, and GPI, respectively. The time cost of the CSO-BP model was much lower than that of the other two hybrid models. 3. Five ET O models (BP, ELM, ACO-BP, CSO-BP, and BSA-BP) exhibited high accuracy at most of the 14 stations in southern China, particularly in Wenjiang and Shapingba; in coastal areas (Xiamen and Guangzhou), however, they had slightly lower accuracy.