Comparative study of reference evapotranspiration estimation models based on machine learning algorithm: a case study of Zhengzhou City

Evapotranspiration is an important link in the hydrological cycle and energy conversion process (El-Shafie et al. 2014; Mattar 2018), which can transport surface water into the atmosphere while also carrying away net radiated energy from the surface, and is a key element in the global energy balance and water cycle. Since evapotranspiration is difficult to measure directly and is a relatively complex nonlinear natural process, reference evapotranspiration is often used to estimate actual evapotranspiration (Jovic et al. 2018). Therefore, accurate estimation of reference evapotranspiration is of great scientific significance for regional water resources planning and management, hydrological research, and prediction of drought and flood disasters (Shiri et al. 2014; Wu et al. 2019).

There are many methods to estimate reference evapotranspiration. Lysimeter is a direct method to measure evapotranspiration. However, due to the complex operation and high maintenance cost, this method is rarely used in practice. After that, scholars put forward empirical formulas based on meteorological data. For example, empirical formulas for estimating ET₀ based only on meteorological data such as temperature and radiation (Valipour 2014; Mattar et al. 2016): Blaney-Criddle (BC) (Blaney & Criddle 1950), Priestley-Taylor (PT), Hargreaves-Samani (Hargreaves & Samani 1985), etc. Although these formulas only use a small amount of meteorological data in the calculation process, the simulation accuracy is greatly affected by the geographical location and climate change of the study area, and the prediction results are often overestimated or underestimated (Berti et al. 2014; Feng et al. 2017; Liu et al. 2017). The Penman–Monteith (PM) equation is a calculation method based on the water vapor diffusion theory and the energy balance theory, which takes into account both radiation and aerodynamic effects and is applicable to the global scale. The PM equation has been recommended by the FAO as a standard method for estimating ET₀ and calibrating other models (Allen et al. 1998; Wang et al. 2019; Tikhamarine et al. 2020; Chen et al. 2020). However, the calculation process of this method is complex and requires a large amount of reliable and high-quality meteorological data (Roy et al. 2021), so the generalization of the PM equation in regions with incomplete meteorological data is limited (Valipour 2015; Yin et al. 2020; Wu et al. 2020a, 2020b).

Evapotranspiration is a complex nonlinear dynamic process that requires methods beyond empirical models. In recent years, the rapid development of machine learning algorithms provides new ideas for estimating ET₀ (Granata 2019; Roy et al. 2020; Ahmadi et al. 2021; Chia et al. 2021; Yan et al. 2021). Huo et al. (2012) estimated ET₀ in northwest China using the ANN model, and the results showed that relative humidity, wind speed (at a 2 m height), maximum and minimum temperature were the key meteorological factors for estimating ET₀ by the ANN model. Feng et al. (2016) used extreme learning machine (ELM), wavelet neural network (WNN), and generalized artificial neural network (GANN) models to estimate ET₀, and verified it in humid areas in southwest China. The results showed that the simulation accuracy of ELM and GANN models was better than that of the WNN model. Ferreira et al. (2019) evaluated the performance of artificial neural network (ANN) and support vector machine (SVM) for estimating daily ET₀ across Brazil. The results showed that the performance of ANN and SVM for ET₀ estimation is superior to other empirical formulas. Tang et al. (2018) used SVM and GANN to estimate the actual evapotranspiration of dryland maize with no and partial mulch coverage, respectively. The results show that the performance of the GANN model is better than the SVM model. Malik et al. (2019) predicted the monthly mean ET₀ in the central hilly region of India based on multiple linear regression (MLR), radial basis neural network (RBNN), multi-layer perceptual neural network (MLPNN), and self-organizing mapping neural network (SOMNN) models. It is found that the prediction results were better when the input parameters included three (maximum air temperature, wind speed, and solar radiation) and five (minimum and maximum air temperatures, relative humidity, wind speed, and solar radiation) variables. Dong et al. (2022) used convolutional neural network (CNN), ELM, and multiple adaptive regression spline curve (MARS) to analyze the spatio-temporal variation of ET₀ in China. The results showed that the performance of the CNN model in estimating ET₀ was better than that of ELM and MARS models.

Although neural network models have been proven to be effective in estimating ET₀, there are still some problems in practical use. In the process of neural network training, the input data and output data establish a corresponding relationship, which mainly shows the generalization ability of the model. However, the general neural network model will have the problem of overfitting in the training process, which will reduce the generalization ability of the model and produce the suboptimal solution (EL-Shafie et al. 2014). In addition, due to the different sizes and structures of the available data input from different models (Granata 2019), problems such as slow operation speed, weak global search ability, and easy convergence to local extremum will also occur in the running process of the model (Sun et al. 2016), which will affect the overall performance of the model. To overcome these problems, an improved Levenberg–Marquardt (L-M) model based on neural networks is introduced in this study. The L-M model can be regarded as an improved form of the Gauss-Newton method, which has both the local characteristics of the Gauss-Newton method and the global characteristics of the gradient method (Sajedi et al. 2021). It can adjust the iteration speed and accuracy in the training process, so as to have more powerful generalization ability. Compared with the general neural network algorithm, it has a faster iterative convergence speed and higher accuracy (Ali et al. 2021). In recent years, the L-M model is usually used in artificial intelligence computing (Vidmar et al. 2020), electromagnetic simulation (Darisma & Marwan 2019), new energy battery development, and medical trials (Smirnova et al. 2019; Cheema et al. 2020). All these results show that the L-M model has good predictive ability. In addition, the GA-BP model is proposed as a competitive alternative to the L-M model, a BP neural network optimized by a genetic algorithm. Through the implementation of the genetic algorithm's global optimization capability, optimal initial weights and thresholds for the BP neural network were attained. These optimal values were then utilized as the initial weights and thresholds of the BP neural network to avoid falling into local minima during subsequent training (Zheng et al. 2018; Tu et al. 2020; Yin et al. 2022). The GA-BP model has the advantages of faster convergence speed, fewer iterations, and higher simulation accuracy, and is widely used in the prediction of long-term weather temperature (Dou & Sun 2021), local scour depth of bridge pier (Dong et al. 2020), concrete strength (Tu et al. 2020), flow and pressure of urban water supply network (Xia et al. 2021). It is worth noting that L-M and GA-BP models are rarely used in predicting ET₀.

In general, no specific model is suitable for all situations, and the performance of different models depends on how much effective information is provided by the input data and the structure of the input data (Granata 2019). The purpose of this study is to use the improved L-M algorithm and the GA-BP algorithm based on the traditional neural network to estimate ET₀, respectively. By inputting different combinations of meteorological variables and data structures, the accuracy and robustness of two different models to estimate daily ET₀ under different input scenarios are compared and analyzed. In addition, this study is based on the data of seven weather stations in Zhengzhou City, in order to obtain a model that can use fewer meteorological parameters, it does not need to consider hydrological principles, and can accurately estimate ET₀.

In this study, the daily data of seven meteorological stations in the Zhengzhou Administrative region were used. The meteorological stations were evenly distributed in the area of Zhengzhou, including both the western mountain climate and the eastern plain climate, which was a good representation. Daily meteorological data of seven meteorological stations from 2000 to 2019 were selected, including wind speed (U₂, m/s), hours of sunshine (SS_H, h), relative humidity (R_H, %), maximum temperature (T_max, °C), and minimum temperature (T_min, °C). The daily meteorological data (2000–2019) collected were divided into two datasets: the training dataset (2000–2017) and the validation dataset (2018–2019). The model parameters are trained on the training dataset and the weights are estimated. The validation dataset further verifies the accuracy of the training model.

≥ 0.8 shows a high positive correlation between two variables; 0.5 ≤ < 0.8 indicating that the two variables are moderately correlated; < 0.5 indicating low correlation between the two variables. In this paper, variables with high or medium correlation were selected as the input of the prediction algorithm (Table 1).

Pearson correlation coefficients of T_min, T_max, R_H, SS_H, and U₂ were all above 0.5, belonging to medium and high correlation levels. It is worth noting that the correlation coefficient of R_H is negative, indicating that R_H hinders the evapotranspiration process. In order to reflect the actual evapotranspiration process more truly, the influence of R_H on the actual evapotranspiration was also considered in this study. In addition, according to relevant research results (Traore et al. 2010; Yang et al. 2019), ET₀ value is greatly affected by temperature, so temperature is a necessary input when selecting input meteorological factors. Therefore, the model input combination is divided into six scenarios (Table 2). Scenarios 1–3 have three input factors, scenarios 4–5 have four input factors, and scenario 6 has five input factors. The input meteorological factors in this study are all parameters easily obtained by meteorological stations, which to some extent solves the problems of backward monitoring facilities and difficulty in obtaining complex parameters in some regions.

The commonly used neural network algorithm is the gradient descent method. The parameters move in the opposite direction of the error gradient to reduce the error function until the minimum value is obtained. The complexity of the calculation is mainly caused by the calculation of partial derivatives. However, the linear convergence rate based on the gradient descent method is very slow, and the L-M algorithm is an improved form of the Gauss-Newton method, which has both the local characteristics of the Gauss-Newton method and the global property of the gradient method. The L-M algorithm is much faster than the gradient method because it uses the approximate second derivative information. The following is a brief description of the L-M algorithm.

Genetic algorithm is an iterative optimization model based on Darwinian evolution theory and genetic evolution theory. Each species undergoes natural evolution through successive generations, inheriting genetic traits from their parents while also experiencing certain genetic changes. In each generation, individuals with genetic traits better suited for the environment have a higher chance of survival. Through the process of natural selection and evolution over multiple generations, the offspring that remain possess the genetic makeup that best fits their environment. As a chromosome, it is a survival of the fittest problem-solving process, through the evolution of chromosome generations, to obtain the optimal or satisfactory solution to the problem. The BP neural network algorithm is a local search optimization method, the essence of which is the gradient descent method. The advantage lies in strong adaptive ability, good generalization ability, and fault tolerance ability. The disadvantage is that the algorithm is extremely sensitive to the initial weight and threshold, its results are easy to converge to local minima, and the convergence speed is slow (Liu et al. 2019). Combined with BP neural network and genetic algorithm, the population search method is used to optimize the weight and threshold of the neural network, which can better overcome the local defect of the BP neural network which tends to the local optimal solution (Saleh et al. 2016).

In this study, both L-M and GA-BP models are improved algorithms based on neural network, including input layer, hidden layer, and output layer in the process of machine learning. The number of input layer and output layer is relatively easy to determine, while the number of hidden layer is a very important factor in the operation process (Faris et al. 2019). Improper selection of hidden layer will not only affect the efficiency of machine learning but also directly affect the simulation results. However, in many pieces of literature (El-Shafie et al. 2014; Maroufpoor et al. 2020), there is no specific rule to determine the number of hidden layers. In this study, the number of hidden layer neurons is determined by trial and error to ensure the simulation effect, operation efficiency, and robustness of the model. The L-M model and GA-BP model adopt the same input structure. It is worth noting that the iteration types of the L-M model and GA-BP model are different, so the number of iterations is also different. It is found (Maroufpoor et al. 2020) that when the number of iterations reaches a certain number, increasing the number of iterations has little impact on improving the simulation accuracy of the model, and even increases the running burden of the model. Through experiments, the iteration times of L-M model and GA-BP are 1,500 times and 10 times, respectively.

In this study, the performance of the model was evaluated according to the different numbers of input data, including scenarios 1–3 with three meteorological data input, scenarios 4–5 with four meteorological data input, and scenario 6 with five meteorological data input.

The same results were obtained when four meteorological parameters were input. For different types of models, the simulation effect of the L-M model is still better than that of the GA-BP model in the same input scenario. The RMSE, MAE, and NSE of L-M4 were 0.2897 mm, 0.2175 mm, and 0.9655, respectively. Compared with GA-BP4, RMSE and MAE decreased by 14.72 and 14.37%, respectively, while NSE increased by 1.36%. For the same model, compared with L-M5, RMSE and MAE of L-M4 decreased by 20.56 and 25.25%, respectively, while NSE increased by 2.10%. Compared with GA-BP5, RMSE and MAE of GA-BP4 decreased by 17.13 and 19.65%, respectively, while NSE increased by 2.33%. This may be related to the inclusion of both temperature and wind speed in the input scenario (Maroufpoor et al. 2020).

It can be seen from the above analysis that scenarios 3, 4, and 6 are the best input combinations with different numbers of meteorological parameters for L-M and GA-BP models, respectively. Therefore, this study mainly conducts model comparison and analysis of the above three input scenarios.

According to the above analysis, the simulation effect of the L-M model is better than that of the GA-BP model. The main reasons are as follows: on the one hand, the operation structure of the model is different, which may be because the GA-BP model only optimized the selection of initial weights and thresholds, and didn't consider the problem of overfitting in the iterative process. However, the L-M model combines Gauss-Newton and the steepest descent method in the operation process and adaptively adjusts the damping factor to achieve the convergence characteristics, which has a high iterative convergence speed. At the same time, the problems such as overfitting in the operation process are solved (El-Shafie et al. 2014), and the performance of estimating ET₀ value is improved. On the other hand, models have different sensitivities to input data. Under the same input combination, the L-M model is more sensitive to meteorological parameters. As can be seen from Table 3 and Figures 3–8, all evaluation indexes under the same input combination are better than the GA-BP model, and simulation accuracy improvement of different input combination models is greater in the L-M model than in the GA-BP model.

For the planning and management of the regional hydrological cycle and water resources, accurate estimation of reference evapotranspiration is crucial. If given the right meteorological information, machine learning algorithms are a potent tool that can reliably estimate ET₀. To estimate the reference evapotranspiration of Zhengzhou in the usual temperate monsoon climate, respectively, better L-M and GA-BP models based on machine learning algorithms were introduced in this study. Six alternative input scenarios made up of typical meteorological factors were presented using the Pearson correlation analysis. It compares, analyzes, and assesses how well different models execute ET₀ estimate under various input conditions. The results lead to the following conclusions:

• (1)

  The L-M model combines Gauss-Newton and the steepest descent method. In all input scenarios, the estimation accuracy, applicability, and robustness of the L-M model are better than those of the GA-BP model.

• (2)

  The L-M model is more sensitive to the input meteorological combination. When the three meteorological parameters are input, the combination of T_max, T_min, and U₂ has the best simulation effect, and the correlation coefficient R² of L-M3 and GA-BP3 is 0.9285 and 0.9168, respectively. When four meteorological parameters are input, the combination of T_max, T_min, SS_H, and U₂ has the best simulation effect, and the correlation coefficient R² of L-M4 and GA-BP4 is 0.9675 and 0.9542, respectively. This indicates that temperature and wind speed can provide more effective information when the model estimates ET₀.

• (3)

  When the number of input meteorological parameters is different, the GA-BP model also shows good prediction performance, but the L-M model shows the optimal prediction ability. Among them, L-M3 (T_min, T_max, U₂), L-M4 (T_min, T_max, SS_H, U₂), and L-M6 (T_min, T_max, SS_H, U₂, R_H) have better simulation results, with R² values of 0.9285, 0.9675, and 0.9925, respectively. Therefore, within the allowable error range, the L-M model can be used as a reliable model to estimate ET₀ with a small number of meteorological parameters.

Due to data limitations, further research is needed to determine the applicability of the L-M model in other regions of the world. However, this study has shown that the L-M model has good prediction ability, and its accuracy and robustness are better than those of the GA-BP model. The effects of human activities and climate change on the model's performance were also taken into consideration. Therefore, while the applicability of the L-M model in other regions of the world should be studied, it is also recommended to improve the model's performance by combining it with a deep learning model to increase its reliability.

For this research paper with several authors, a short paragraph specifying their individual contributions was provided. C.N. developed the original idea and contributed to the research design for the study. S.J. gave some revised opinions for the article. S.L. and C.L. were responsible for data collection and charting. S.S. provided some guidance for the writing of the article. C.H. provided guidance and improved suggestions. All authors have read and approved the final manuscript.

This work was supported by the National Natural Science Foundation of China, Project No. U2243219, Key projects of National Natural Science Foundation of China (51979250, 51739009), and Key Research and Promotion Projects (technological development) in Henan Province, grant number 222102320455.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.