# Evaluation of Empirical Equations and Machine Learning Models for Daily Reference Evapotranspiration Prediction Using Public Weather Forecasts

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## Abstract

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_{o}based on public weather forecasts have been widely used, these studies lack the comparative evaluation of different types of models and do not evaluate the seasonal variation in model prediction of daily ET

_{o}performance; this may result in the selected model not being the best model. In this study, to select the best daily ET

_{o}forecast model for the irrigation season at three stations (Yinchuan, Tongxin, and Guyuan) in different climatic regions in Ningxia, China, the daily ET

_{o}s of the three sites calculated using FAO Penman–Monteith equations were used as the reference values. Three empirical equations (temperature Penman–Monteith (PMT) equation, Penman–Monteith forecast (PMF) equation, and Hargreaves–Samani (HS) equation) were calibrated and validated, and four machine learning models (multilayer perceptron (MLP), extreme gradient boosting (XGBoost), light gradient boosting machine (LightGBM), and gradient boosting with categorical features support (CatBoost)) were trained and validated against daily observed meteorological data (1995–2015 and 2016–2019). Based on public weather forecasts and daily observed meteorological data (2020–2021), the three empirical equations (PMT, PMF, and HS) and four machine learning models (MLP, XGBoost, LightGBM, and CatBoost) were compared in terms of their daily ET

_{o}prediction performance. The results showed that the daily ET

_{o}performance of the seven models in the irrigation season with a lead time of 1–7 days predicted by the three research sites decreased in the order of spring, autumn, and summer. PMT was the best model for the irrigation seasons (spring, summer, and autumn) at station YC; PMT and CatBoost with C3 (T

_{max}, T

_{min}, and Wspd) as the inputs were the best models for the spring, autumn irrigation seasons, and summer irrigation seasons at station TX, respectively. PMF, CatBoost with C4 (T

_{max}, T

_{min}) as input, and PMT are the best models for the spring irrigation season, summer irrigation season, and autumn irrigation season at the GY station, respectively. In addition, wind speed (converted from the wind level of the public weather forecast) and sunshine hours (converted from the weather type of the public weather forecast) from the public weather forecast were the main sources of error in predicting the daily ET

_{o}by the models at stations YC and TX(GY), respectively. Empirical equations and machine learning models were used for the prediction of daily ET

_{o}in different climatic zones and evaluated according to the irrigation season to obtain the best ET

_{o}prediction model for the irrigation season at the study stations. This provides a new idea and theoretical basis for realizing water-saving irrigation during crop fertility in other arid and water-scarce climatic zones in China.

## 1. Introduction

_{o}) by the crop coefficient. The determination of ET

_{o}is not only a key factor in predicting and estimating crop water demand but also an important requirement for irrigation forecasting and irrigation decision making. Timely and accurate ET

_{o}prediction has a notable reference value for real-time irrigation decision making [1] and is very important for improving the real-time irrigation prediction accuracy, irrigation management level, crop yield, and water conservation [2].

_{o}prediction methods can be divided into direct and indirect methods [1]. Among the direct methods, time series models, such as autoregressive (AR), moving average (MA), autoregressive moving average (ARMA), and autoregressive integrated moving average (ARIMA) models, have been used to forecast the daily ET

_{o}using historical daily ET

_{o}data calculated from historical daily measured meteorological data [3]; alternatively, machine learning algorithms, such as the multilayer perceptron (MLP), multivariate relevance vector machine (MVRVM) [4], multilayer perceptron–neural network model (MLP-NNM), Kohonen self-organizing feature maps–neural network model (KSOFM-NNM), gene expression programming (GEP) [5], least-squares support vector machine (LSSVM), adaptive neuro-fuzzy inference system (ANFIS), generalized regression neural network (GRNN) [3], deep learning models (long short-term memory (LSTM), one-dimensional convolutional neural network (1D CNN), and a combination of the two previous models (CNN-LSTM)), and traditional machine learning models (artificial neural network (ANN) and random forest (RF)) [6], have been employed to predict the daily ET

_{o}and the daily ET

_{o}at a lead time of 1–7 days using historical data. In these studies, alternatives to time series models have been applied for future ET

_{o}prediction based on historical meteorological data and ET

_{o}calculated with historical meteorological data. In general, machine learning models using ET

_{o}data computed from historical meteorological data outperform those directly using historical meteorological data [3]. In addition, combined machine learning models, such as the fruit fly optimized GRNN model [7] and autoencoder–decoder bidirectional LSTM (AED-BiLSTM) [8], can predict ET

_{o}more accurately than a single machine learning model (GRNN, XGBoost). However, in all these studies, so-called ET

_{o}forecast data generated from long-term historical meteorological data were used, and since the short-term daily ET

_{o}is mainly governed by weather conditions, direct methods may not be applicable [9,10,11].

_{o}can be calculated using weather variables predicted via numerical weather prediction (NWP) or public weather forecasts, such as those obtained using the Food and Agriculture Organization (FAO)-56 Penman–Monteith (PM) equation [1,12,13,14,15,16], the Hargreaves–Samani (HS) and Priestley (PT) equations [12], or multivariate time series models [17], and the numerical weather forecasts output by forecast systems or models (COSMO, Australian Community Climate and Earth System Simulator–Global (ACCESS-G), Global Ensemble Forecast System (GEFS) model, European Centre for Medium-Range Weather Forecasts (EC), National Centers for Environmental Prediction Global Forecast System (NCEP), and United Kingdom Meteorological Office (MO) forecasts) can be used to predict the daily ET

_{o}, weekly ET

_{o}, and daily ET

_{o}with a lead time of 1–16 days. Although NWP can generate forecasts of the full range of meteorological variables needed by the FAO-56 PM equation and can be used for ET

_{o}prediction, NWP data are not available to the public in some developing countries. For example, the numerical forecast products of the China Meteorological Data Network are only available to registered users for education and research purposes, while numerical forecast products require expert downscaling and bias correction [16] or postprocessing methods [14] to improve their reliability.

_{o}prediction. For example, information retrieved from public weather forecasts has been converted into the variables needed to calculate ET

_{o}using the FAO-56 PM equation and subsequently applied in daily ET

_{o}prediction [18,19,20,21]. Given that temperature is the most accurate and only quantitative variable in public weather forecasts, many studies have used only temperature forecast data retrieved from public weather forecasts to predict ET

_{o}. Based on temperature data retrieved from public weather forecasts, the daily ET

_{o}has been predicted using locally calibrated versions of the HS equation [9,22,23], the daily ET

_{o}with a lead time of 1–7 days has been predicted using the monthly calibrated Blaney–Criddle (BC) model [24], and four ANN models (linear regression (LR), probabilistic neural network (PNN), MLP, and generalized feedforward (GFF) models) [10], and the GEP algorithm [2] have been used to predict the daily ET

_{o}with a lead time of 1–7 days at Gaoyou Station in Jiangsu Province of China. The above studies based on the adoption of public weather forecast models for daily ET

_{o}prediction have achieved reasonable results, but there exists no evaluation model to determine the seasonal variation in the daily ET

_{o}prediction performance, and there is a lack of comparative evaluation of machine learning models in daily ET

_{o}prediction based on different input combinations of weather variables retrieved from public weather forecasts. These studies are aimed at a single model in a single region (or multiple regions) of China or a comparative study of similar models in a single region (or multiple regions) of China [11]. The lack of comparative research on different types of models can lead to the inability to identify the best daily ET

_{o}forecast model in a given region.

_{o}[25,26,27,28,29,30,31,32,33] and monthly ET

_{o}[34]. These studies have shown that the daily ET

_{o}estimated by these three tree models is suitably accurate, with a better performance than that of other models. However, there are no studies on ET

_{o}prediction with these three tree models. Compared with estimating historical daily ET

_{o}by models and historical daily measured weather forecast data, it is more valuable to predict future daily ET

_{o}by models and public weather forecasts because accurate prediction of ET

_{o}is the key to crop water demand prediction and the premise of real-time irrigation prediction, which has important reference value and significance for real-time irrigation decision making.

_{o}using public weather forecasts are evaluated to select the best daily ET

_{o}prediction model for the crop irrigation season in the study area. The objectives of this study were as follows: (1) In model assessment based on public weather forecasts, the optimal input combinations of the MLP, XGBoost, LightGBM, and CatBoost machine learning models were determined to obtain the best daily ET

_{o}prediction performance. (2) Three empirical equations (PMT, PMF, and HS) and four machine learning models (MLP, XGBoost, LightGBM, and CatBoost) were compared by determining the seasonal variation in the daily ET

_{o}prediction performance, and the best daily ET

_{o}prediction model was recommended for all four seasons at various research sites in three climate zones. The main error sources of daily ET

_{o}predicted by the PMF equation and four machine learning models were analyzed.

## 2. Materials and Methodology

#### 2.1. Study Area and Data Collection

#### 2.1.1. Study Area

#### 2.1.2. Data Collection

_{max}), daily minimum temperature (T

_{min}), average temperature, average relative humidity (RH), sunshine hours (SDun), and average wind speed (Wspd). Daily public weather forecast data 1–7 days in advance from 1 January 2020 to 31 December 2021, at the same sites were collected from the China Weather Network (http://www.weather.com.cn (accessed on 31 December 2022)), including the daily maximum temperature (T

_{max}), daily minimum temperature (T

_{min}), wind scale, and weather type. Daily observed meteorological data from 1 January 1995 to 31 December 2015, were used to calibrate the three empirical equations and train the four machine learning models, and daily observed meteorological data from 1 January 2016 to 31 December 2019 were employed to validate the three empirical equations and four machine learning models. Daily public weather forecast data with a lead time of 1–7 days and daily observed meteorological data from 1 January 2020 to 31 December 2021 were used to test and evaluate the performance of these seven models.

#### 2.2. Methodology

#### 2.2.1. FAO Penman–Monteith (PM) Equation

_{o}calculated using the FAO-56 Penman–Monteith equation [36] recommended by the United Nations Food and Agricultural Organization (FAO) and daily observed meteorological data were used to evaluate the performance of the three empirical equations and four machine learning models to predict ET

_{o}. The FAO-56 PM equation [36] can be expressed as follows:

_{o}is the daily reference evapotranspiration [mm day

^{−1}]; R

_{n}is the net radiation at the crop surface [MJ m

^{−2}day

^{−1}]; G is the soil heat flux density [MJ m

^{−2}day

^{−1}] (G may be ignored during the day); T is the mean daily air temperature at the 2 m height [°C]; u

_{2}is the wind speed at the 2 m height [m s

^{−1}]; e

_{s}is the saturation vapor pressure [kPa]; e

_{a}is the actual vapor pressure [kPa]; e

_{s}− e

_{a}is the saturation vapor pressure deficit [kPa]; ∆ is the slope of the vapor pressure curve [kPa °C

^{−1}]; and γ is the psychrometric constant [kPa °C

^{−1}].

#### 2.2.2. Temperature Penman–Monteith (PMT) Equation

_{o}calculation using only the maximum temperature, minimum temperature, and wind speed at the 2 m height, namely, the PMT method. The main calculation process can be expressed as Equations (2)–(8) [36]. ET

_{o}was predicted by estimating the actual saturated vapor pressure (e

_{a}) using the dew point temperature (T

_{dew}) and solar radiation (R

_{s}) considering the maximum and minimum air temperatures; the wind rating in public weather forecast information was converted into the wind speed according to Table 2. However, Equation (2) was established for T

_{dew}= T

_{min}, and a correction for T

_{dew}is needed in practical applications [36]. Similarly, the adjustment coefficient k

_{Rs}in Equation (4) is an empirical value that differs between interior and coastal regions, and k

_{Rs}must also be corrected [11,37,38,39,40]:

^{o}(T

_{min}) is the saturation vapor pressure at the daily minimum temperature [kPa]; R

_{a}is the extraterrestrial radiation [MJ m

^{−2}day

^{−1}]; G

_{SC}is a solar constant = 0.0820 [MJ m

^{−2}min

^{−1}]; d

_{r}is the inverse relative distance between the Earth and Sun; ${\omega}_{s}$ is the sunset hour angle [rad]; $\phi $ is the latitude [rad]; $\delta $ is the solar declination [rad]; R

_{s}is the solar or shortwave radiation [MJ m

^{−2}day

^{−1}]; T

_{max}and T

_{min}are the daily maximum and minimum air temperatures, respectively [°C]; k

_{Rs}is the adjustment coefficient (0.16..0.19) [°C

^{−0.5}], and at interior locations, k

_{Rs}= 0.16, while at coastal locations, k

_{Rs}= 0.19; z is the station elevation above sea level [m]; R

_{so}is the clear-sky solar radiation [MJ m

^{−2}day

^{−1}]; R

_{ns}is the net solar or shortwave radiation [MJ m

^{−2}day

^{−1}]; α is the albedo or canopy reflection coefficient (α = 0.23); R

_{nl}is the net outgoing longwave radiation [MJ m

^{−2}day

^{−1}]; σ is the Stefan–Boltzmann constant [4.903 × 10

^{−9}MJ K

^{−4}m

^{−2}day

^{−1}]; T

_{max,K}is the maximum absolute temperature during the 24-h period [K = °C + 273.16]; and T

_{min,K}is the minimum absolute temperature during the 24-h period [K = °C + 273.16].

#### 2.2.3. Penman–Monteith Forecast (PMF) Equation

_{o}prediction. Cai et al. [18] proposed an analytical method, namely, the PMF method [11,20,21]. ET

_{o}can be predicted by converting the wind scale into the wind speed according to Table 2, while the weather type can be transformed into the sunshine duration coefficient according to Table 3 [18,20,42]. Moreover, the solar radiation (R

_{s}) can be calculated using Equations (9)–(11), and the actual saturated vapor pressure (e

_{a}) can be estimated with Equation (2). However, the use of the PMF method requires correction for T

_{dew}[37,38,39,40,42]:

_{s}is the regression constant, reflecting the fraction of extraterrestrial radiation reaching the Earth on overcast days (n = 0); and a

_{s}+ b

_{s}is the fraction of extraterrestrial radiation reaching the Earth on clear-sky days (n = N). Where no actual solar radiation data are available and no calibration is conducted to obtain improved a

_{s}and b

_{s}parameters, a

_{s}= 0.25 and b

_{s}= 0.50 are recommended [11,18,20,36].

#### 2.2.4. Hargreaves–Samani (HS) Equation

_{o}estimation/prediction [9,11,22,23,37,38,39,40,43,44,45,46,47,48,49]. The HS equation can be expressed as Equation (12), which has been applied in practice with corrections for parameters C and E [9,11,50,51]:

_{o}is the daily ET

_{o}calculated by the HS equation [mm day

^{−1}]; T

_{max}and T

_{min}are the daily maximum and minimum air temperatures, respectively [°C]; R

_{a}is the extraterrestrial radiation [MJ m

^{−2}day

^{−1}]; and C and E are model parameters with suggested initial values of 0.0023 and 0.5, respectively [50].

#### 2.2.5. Multilayer Perceptron (MLP) with Multiple Hidden Layers

_{o}estimation/prediction [10,53,54,55,56,57,58]. However, in these studies, the number of neurons in a single hidden layer and the number of hidden layers and neurons in each hidden layer have been determined by trial and error, which can be time consuming and may not always yield the best hyperparametric results.

#### 2.2.6. Extreme Gradient Boosting (XGBoost)

#### 2.2.7. Light Gradient Boosting Machine (LightGBM)

#### 2.2.8. Gradient Boosting with Categorical Feature Support (CatBoost)

#### 2.2.9. Input Combinations and Hyperparameter Tuning Methods for the Four Machine Learning Models

_{o}[53,55,63,64], public weather forecasts include only four variables: daily maximum temperature (T

_{max}), daily minimum temperature (T

_{min}), wind scale (converted into Wspd according to Table 2), and weather type (converted into SDun according to Table 3 and Equation (10)). Four input combinations, i.e., C1 (T

_{max}, T

_{min}, Sdun, and Wspd), C2 (T

_{max}, T

_{min}, and Sdun), C3 (T

_{max}, T

_{min}, and Wspd), and C4 (T

_{max}and T

_{min}), were selected for the MLP, XGBoost, LightGBM and CatBoost models.

_{o}calibration and prediction and the four machine learning models for ET

_{o}prediction and hyperparameter tuning is shown in Figure 2.

#### 2.3. Calibration Methods for Empirical Equations

^{−1}, and the long-term daily average wind speed, where the predicted Wspd was converted from the wind scale in the public weather forecasts with a lead time of 1–7 days according to Table 2, while the constant value of 2 m s

^{−1}is the average wind speed at more than 2000 weather stations worldwide [36], and the long-term daily average wind speed was calculated from the daily observed wind speed at the corresponding site from 1 January 1995 to 31 December 2022 (the daily average wind speed at YC is 2.11 m s

^{−1}, that at TX is 3.03 m s

^{−1}, and that at GY is 2.44 m s

^{−1}). T

_{dew}was calibrated as T

_{dew}= T

_{min}− a

_{T}(a

_{T}= 0, 1, 2, 3) [11,37,38,39,40], the adjustment coefficient k

_{Rs}at YC was (0.126, 0.16, 0.17, 0.20, 0.22, 0.23, 0.24, 0.27, 0.29), the adjustment coefficient k

_{Rs}at TX was (0.13, 0.16, 0.17, 0.18, 0.19, 0.20, 0.21), and the adjustment coefficient k

_{Rs}at GY was (0.13, 0.14, 0.15, 0.16, 0.18, 0.19). Then, different combinations of the above wind speed types, T

_{dew}and k

_{Rs,}were substituted into the PMT equation (different combinations of the above wind speed types and T

_{dew}were substituted into the PMF equation), and ET

_{o}was calculated. Finally, the ET

_{o}values predicted by the PMT (PMF) equation were compared to the ET

_{o}values obtained with the FAO-56 PM equation using the daily observed meteorological data. The wind speed, T

_{dew}and k

_{Rs}combination (or the wind speed and T

_{dew}combination for the PMF equation) that yielded the minimum RMSE was adopted to obtain the final wind speed, corrected T

_{dew}and k

_{Rs}values (wind speed and corrected T

_{dew}values for the PMF equation) at each of the three meteorological sites using the trial-and-error method [11,38,39,40].

_{o}with the FAO-56 PM equation, and the observed T

_{max}and T

_{min}at the same site during this period were substituted into Equation (12). The nonlinear least-squares method was used to correct parameters C and E, and the process was programmed in MATLAB (version R2014a).

#### 2.4. Evaluation Criteria

_{o}prediction. MAE reflects the actual error situation between the predicted and observed values. RMSE reflects the degree of error dispersion between the predicted and observed values; the smaller the RMSE value is, the higher the accuracy of the predicted values relative to the observed values. RM can be expressed as the ratio of the average predicted value to the average observed value. The RM value can be greater/less than 1, reflecting the overestimation/underestimation of the observed values by the predicted values [11,42,65]. R reflects the correlation between the predicted and observed values. The closer the R value is to 1, the better the correlation between the predicted and observed values. These four statistical indicators can be calculated as follows [9,11,42,65,66]:

_{i}is the predicted value; O

_{i}is the observed value; i is the sample serial number, with i = 1, 2, …, n; n is the number of samples; $\overline{\mathrm{P}}$ is the average predicted value; and $\overline{\mathrm{O}}$ is the average observed value.

## 3. Results and Discussion

#### 3.1. Prediction Performance of the Weather Variables in the Public Weather Forecast Information

#### 3.1.1. Single-Parameter Performance

_{min}ranged from 2.73–3.10 °C and 3.51–3.90 °C, respectively. At the three sites, the mean MAE and RMSE of T

_{max}ranged from 3.62–4.12 °C and 4.67–5.24 °C, respectively, while the mean R values of T

_{min}and T

_{max}were higher than 0.92 and 0.85, respectively. The accuracy of the T

_{min}forecasts at all three sites was higher than that of the T

_{max}forecasts, and the T

_{min}and T

_{max}prediction performance decreased with increasing forecast period, which is consistent with most previous studies in China [9,10,11,20,21,23,24,42,58,63]. In addition, the mean RM values of T

_{min}and T

_{max}ranged from 1.00–1.08 and 0.98–1.01, respectively, indicating that T

_{min}was slightly overestimated at the three sites, T

_{max}was slightly overestimated at GY, and T

_{max}was slightly underestimated at YC and TX.

^{−1}and 1.98–2.19 m s

^{−1}, respectively, and the mean R and RM ranged from 0.02–0.08 and 0.92–1.77, respectively. The Wspd prediction performance was poor at all stations. Wspd was overestimated by 76.78% (YC) and 2.79% (TX) and underestimated by 8.44% (GY). Among the four variables in public weather forecasts, Wspd exhibited the worst prediction performance, which may be caused by inadequate forecasts of the wind scale in public weather forecasts and errors in the conversion of the wind scale into the wind speed [18,20,21,42].

#### 3.1.2. Seasonality of the Prediction Performance

_{max}at YC (GY) all decreased in the order of autumn, summer, winter, and spring. The average performance indicators of T

_{max}at TX decreased in the order of summer, autumn, winter, and spring. At the three sites, T

_{max}was overestimated within the range of 3.43–6.32% in spring (0.13–2.91% in summer) and underestimated within the range of 3.02–5.58% in autumn (26.21–57.24% in winter).

_{min}at YC decreased in the order of autumn, summer, winter, and spring (autumn, summer, spring, and winter for GY). The average performance indicators of T

_{min}in TX decreased in the order of summer, autumn, winter, and spring. At the three sites, T

_{min}was overestimated within the range of 32.43–85.75% in spring (4.33–5.84% in summer) and underestimated within the range of 4.15–9.13% in autumn (0.49–1.28% in winter) (except T

_{min}at GY, which was overestimated by 3.96% in winter).

^{−1}) or the long-term daily average wind speed (2.11 m s

^{−1}at YC, 3.03 m s

^{−1}at TX, and 2.44 m s

^{−1}at GY). The average performance indicators of Wspd at YC during the four seasons decreased in the order of constant value, long-term daily average wind speed, and predicted Wspd. The average performance indicators when Wspd was defined as a constant value decreased in the order of summer, spring, winter, and autumn (summer, spring, autumn, and winter when the predicted Wspd was taken). Except in summer, the average performance indicators of Wspd at TX (GY) decreased in the order of long-term daily average wind speed, constant value, and predicted Wspd. Moreover, the average performance indicators at TX when Wspd was defined as a constant value decreased in the order of winter, autumn, spring, and summer (autumn, summer, winter, and spring when the predicted Wspd was taken). The average performance indicators at GY when Wspd was defined as a constant value and predicted Wspd decreased in the order of autumn, summer, winter, and spring. When Wspd was defined as a constant value (predicted Wspd), the predicted value at YC was overestimated by 17.92%, 24.84%, 59.04%, and 42.51% (62.03%, 59.62%, 98.80%, and 97.34%) in spring, summer, autumn, and winter, respectively, the predicted value at TX was underestimated by 39.96%, 47.71%, 36.30%, and 26.39% (overestimated by 5.13%, underestimated by 6.36%, overestimated by 4.14%, and overestimated by 14.78%) in spring, summer, autumn, and winter, respectively, and the predicted value at GY was underestimated by 40.96%, 37.80%, 31.30%, and 36.93% (11.06%, 10.99%, 3.98%, and 3.80%) in spring, summer, autumn, and winter, respectively.

_{max}and T

_{min}prediction performance occurred in autumn and summer, followed by winter and spring. The SDun prediction performance was consistent at the three sites, with the best performance in winter and spring, followed by autumn and summer, whereas the Wspd prediction performance greatly varied from site to site.

#### 3.2. Calibration and Validation of the Three Empirical Equations

_{o}prediction performance indicators of the three empirical equations (PMT, PMF, and HS) for the precalibration validation period and test period (a lead time of 1–7 days) and the postcalibration validation period and test period (a lead time of 1–7 days) are listed in Table 5. During the test period, the mean MAE, RMSE, and RM of the PMT equation at YC decreased from 0.63 mm d

^{−1}, 0.89 mm d

^{−1}, and 1.03 before calibration to 0.61 mm d

^{−1}and 0.86 mm d

^{−1}and 1.02 after calibration, respectively, and the mean R increased from 0.88 before calibration to 0.89 after calibration. Moreover, the average MAE, RMSE, and RM of the PMF equation at YC decreased from 0.91 mm d

^{−1}, 1.22 mm d

^{−1}, and 1.24 before calibration to 0.86 mm d

^{−1}, 1.17 mm d

^{−1}, and 1.21 after calibration, respectively. The average R changed only slightly before and after calibration, reaching a value of 0.88. The average MAE, RMSE, and RM of the HS equation at YC decreased from 4.91 mm d

^{−1}, 6.05 mm d

^{−1}, and 2.70 before calibration to 0.68 mm d

^{−1}, 0.97 mm d

^{−1}, and 1.09 after calibration, respectively. The average R remained almost unchanged before and after calibration, reaching a value of 0.89. After calibration, the performance of the three empirical equations in ET

_{o}prediction was improved. Among them, the average MAE, RMSE, and RM of the HS equation for ET

_{o}prediction decreased by 86.15%, 83.97%, and 59.63%, respectively. The three empirical equations attained a similar performance at TX and GY during the test period. The three empirical equations after calibration could be used to predict ET

_{o}with a lead time of 1–7 days.

#### 3.3. Performance Comparison of ET_{o} Prediction by the Four Machine Learning Algorithms under the Various Input Combinations

_{o}prediction at YC, TX, and GY with a lead time of 1–7 days under the different input combinations are provided in Figure 5, Figure 6 and Figure 7, respectively, and the daily ET

_{o}prediction performance significantly varied depending on the machine learning model, input combination of the machine learning model, and study site.

_{o}prediction under the different input combinations decreased in the order of C2, C4, C1, and C3. LightGBM was the best model for ET

_{o}prediction under the C3 input combination (MAE = 0.83 mm d

^{−1}, RMSE = 1.09 mm d

^{−1}, and R = 0.85), while CatBoost attained the best prediction performance under all input combinations (MAE range: 0.69–0.77 mm d

^{−1}; RMSE range: 0.93–0.99 mm d

^{−1}; R range: 0.84–0.87). CatBoost achieved the highest performance with the C2 input combination (MAE = 0.69 mm d

^{−1}, RMSE = 0.93 mm d

^{−1}, and R = 0.86) during the testing phase at YC.

_{o}prediction performance of MLP and LightGBM decreased in the order of C2, C1, C4, and C3, the ET

_{o}prediction performance of XGBoost decreased in the order of C4, C2, C3, and C1, and the ET

_{o}prediction performance of CatBoost decreased in the order of C3, C4, C2, and C1. Second, LightGBM attained the best prediction performance under the C1 and C2 input combinations (MAE range: 0.985–1.009 mm d

^{−1}; RMSE range: 1.357–1.379 mm d

^{−1}; R range: 0.78–0.79), while CatBoost achieved the best prediction performance under the C3 and C4 input combinations (MAE range: 0.95–0.96 mm d

^{−1}; RMSE range: 1.279–1.31 mm d

^{−1}; R range: 0.80–0.81). CatBoost was the best performing machine learning model for the C3 input combination (MAE = 0.95 mm d

^{−1}, RMSE = 1.28 mm d

^{−1}, R = 0.81) at TX during the test period.

_{o}prediction performance of the MLP under the different input combinations decreased in the order of C2, C1, C4, and C3, the ET

_{o}prediction performance of XGBoost under the different input combinations decreased in the order of C2, C4, C1, and C3, and the ET

_{o}prediction performance of LightGBM and CatBoost under the different input combinations decreased in the order of C4, C1, C2, and C3. Second, CatBoost was the best performing model for the C1, C2, and C4 input combinations (MAE range: 0.907–0.92 mm d

^{−1}; RMSE range: 1.213–1.228 mm d

^{−1}; R range: 0.699–0.707), and LightGBM was the best performing model for the C3 input combination (MAE = 0.95 mm d

^{−1}; RMSE = 1.28 mm d

^{−1}; R = 0.68). CatBoost was the best performing machine learning model for the C4 input combination (MAE = 0.907 mm d

^{−1}; RMSE = 1.213 mm d

^{−1}; R = 0.707) at GY during the test period.

_{o}prediction performance of the four machine learning models [1,11,20,31,58,63,67,68]. Since the predicted Wspd exhibits the worst prediction performance among the variables in the public weather forecasts, the addition of the predicted Wspd to the input combination could lead to a decrease in the daily ET

_{o}prediction performance of the machine learning models (except for the CatBoost model at TX), which is consistent with previous research results [42]. In addition, the MLP with two–three hidden layers performs better in predicting daily ET

_{o}than the MLP with only one hidden layer [10,56,58].

#### 3.4. ET_{o} Prediction Performance Evaluation of the Seven Models Based on Public Weather Forecasts

#### 3.4.1. Daily ET_{o} Prediction Performance of the Seven Models

_{o}prediction performance indicators of the three empirical equations and four machine learning models with a lead time of 1–7 days at the three sites are shown in Figure 8 and Figure 9, respectively. First, the daily ET

_{o}prediction performance of the seven models at YC, TX, and GY decreased with increasing lead time, which is due to the decrease in the performance of the public weather forecast variables with increasing lead time, which is consistent with previous research results [1,2,9,10,11,17,20,23,42,58,63]. Second, the RM values of the 7 models at YC ranged from 0.97 to 1.22, among which the XGBoost model slightly underestimated (2.04–2.96%) the daily ET

_{o}, the PMF equation overestimated (20.74–22.03%) the daily ET

_{o}, and the other models slightly overestimated (0.24–10.68%) the daily ET

_{o}. The RM values of the seven models at TX ranged from 0.91 to 1.03, among which the HS, MLP, and LightGBM models slightly underestimated (3.39–9.38%) the daily ET

_{o}, while the other models slightly overestimated (0.46–3.43%) the daily ET

_{o}. The RM values of the seven models at GY ranged from 0.92 to 1.15. The PMF equation overestimated (13.83–15.46%) the daily ET

_{o}, and the PMT equation slightly overestimated (0.19–0.47%) the daily ET

_{o}, while the other models slightly underestimated (1.67–7.93%) the daily ET

_{o}. The PMF model overestimated the daily ET

_{o}at YC, which may be due to Wspd overestimation at this site. Under arid conditions, the drier the atmosphere is, the more small changes in wind speed could lead to large changes in the evapotranspiration rate and thus the greater the impact on evapotranspiration [36]. The daily ET

_{o}overestimation by the PMF model noted at GY may be due to T

_{max}overestimation at this site. Moreover, the overestimation percentage of T

_{max}at GY was higher than that of T

_{min}, and the saturation vapor pressure was overestimated due to T

_{max}overestimation, resulting in an increase in the saturation vapor pressure deficit and overestimation of the daily ET

_{o}at GY.

_{o}with a lead time of 1–7 days by the seven models at the three sites. Combining all indicators, first, the prediction performance of the seven models at YC was better than that at TX and GY, which is mainly due to the overall better forecast performance of the public weather forecast variables at YC than at TX and GY. Second, the top three models at YC were PMT, HS, and CatBoost, whereas PMF (the worst performing model) overestimated the daily ET

_{o}(21.35%). The top three models at TX were PMT, CatBoost, and HS, whereas MLP (the worst performing model) underestimated the daily ET

_{o}(6.05%), and the top three models at GY were PMT, CatBoost, and LightGBM, whereas PMF (the worst model) overestimated the daily ET

_{o}(14.46%).

#### 3.4.2. Seasonality of the Daily ET_{o} Prediction Performance of the Seven Models

_{o}prediction performance. The average ET

_{o}prediction performance indicators with a lead time of 1–7 days of the seven methods at the three station sites during the four seasons from 2020–2021 are shown in Figure 10, Figure 11 and Figure 12, respectively.

_{o}prediction performance of all seven models was higher at YC than at TX and GY (excluding the LightGBM model in winter), which was consistent with the results of evaluating the performance of daily ETo predicted by the seven models at three research sites (YC, TX, and GY) from a daily scale.

_{o}prediction of all seven models at YC and six models at TX increased in the order of winter, spring, autumn, and summer, and the seasonal MAE and RMSE values in winter and spring were smaller than the annual values. At GY, except for the PMF model, the seasonal MAE and RMSE values for ET

_{o}prediction of the other six models increased in the order of winter, autumn, spring, summer, and the seasonal MAE and RMSE values in winter were smaller than the annual values. This is consistent with previous studies [1,16,20]. The seasonal R values for ET

_{o}prediction of all models at the three sites were smaller than the annual values, and the seasonal R values were the smallest in summer. The seasonal R values at YC were the largest in autumn (except for the PMF model), the seasonal R values at TX were the largest in spring (the three empirical equations and the CatBoost model) and winter (MLP, XGBoost, and LightGBM), and the seasonal R values at GY were the largest in spring (MLP, XGBoost, and LightGBM), autumn (LightGBM and CatBoost), and winter (the three empirical equations). The seasonal RM values for daily ET

_{o}prediction by the four machine learning models at the three sites were less than one in spring and summer and greater than one in autumn and winter, indicating that the daily ET

_{o}in spring and summer was underestimated, while the daily ET

_{o}was overestimated in autumn and winter (except at GY, where the MLP model underestimated the daily ET

_{o}by 3.20%). The seasonal RM values for daily ET

_{o}prediction by the three empirical models at YC and GY in summer and autumn and at TX in autumn were all greater than 1, indicating that the daily ET

_{o}was overestimated during the corresponding season.

_{o}values with a lead time of 1–7 days by PMT were the best during all seasons, followed by HS (spring, autumn, and winter) and CatBoost (summer). At TX, the indicators of the predicted seasonal ET

_{o}values with a lead time of 1–7 days by PMT were the best in spring, autumn, and winter. The indicators of the predicted seasonal ET

_{o}values with a lead time of 1–7 days by CatBoost were the best in summer, followed by PMF (spring), PMT (summer), and LightGBM (autumn and winter). At GY, the indicators of the predicted seasonal ET

_{o}values with a lead time of 1–7 days by PMF were the best in spring and winter, those of the predicted seasonal ET

_{o}values with a lead time of 1–7 days by CatBoost were the best in summer, and those of the predicted seasonal ET

_{o}values with a lead time of 1–7 days by PMT were the best in autumn, followed by PMT (spring and winter), LightGBM (summer), and HS (autumn). The scatter plots of the daily ET

_{o}predicted by the PMT equations for the test period versus the daily ET

_{o}computed by the PM equations for YC station; the scatter plots of the daily ET

_{o}predicted by the PMT equations and the CatBoost model for the test period versus the daily ET

_{o}computed by the PM equations for TX station; and the scatter plots of the daily ET

_{o}predicted by the PMF equations, the CatBoost model, and the PMT equations for the test period versus the daily ET

_{o}computed by the PM equations for GY station are given in Figure 13, Figure 14 and Figure 15, respectively.

#### 3.5. Impact of Weather Variable Prediction Based on Public Weather Forecasts on Daily ET_{o} Prediction

_{o}prediction, the four observed weather variables (T

_{max}, T

_{min}, SDun, and Wspd) were replaced in sequence with corresponding forecast values for a lead time of 1–7 days, such that a large change in the forecasted daily ET

_{o}indicates that the forecasted weather variable would produce a large error in daily ET

_{o}prediction [1,13,16,20]. T

_{max}, T

_{min}, Sdun, and Wspd in public weather forecasts with a lead time of 1–7 days were used to replace the corresponding daily observed values of T

_{max}, T

_{min}, SDun, and Wspd, respectively, to form new combinations, namely, SC1 (T

_{max}, T

_{min}, SDun, and Wspd), SC2 (T

_{max}, T

_{min}, SDun, and Wspd), SC3 (T

_{max}, T

_{min}, SDun, and Wspd), and SC4 (T

_{max}, T

_{min}, SDun, and Wspd). The combination of all observations was denoted as SC (T

_{max}, T

_{min}, SDun, and Wspd). The mean values of the ET

_{o}performance indicators of the five models at the three sites under the SC-SC4 input combinations are provided in Figure 16, Figure 17 and Figure 18.

_{o}slightly differed before and after replacing the observed T

_{min}with T

_{min}of the public weather forecasts. Notably, T

_{min}exhibited the smallest prediction error, and its contribution to the error in daily ET

_{o}prediction was minimal.

_{o}(first), and for the four machine learning models, SDun and T

_{max}yielded large (second) and relatively small (third) contributions, respectively, to the error in the predicted daily ET

_{o}, which is consistent with previous findings [20]. Regarding the PMF model, T

_{max}and SDun also yielded large (second) and relatively small (third) contributions, respectively, to the error in the predicted daily ET

_{o}. At TX, among the four machine learning models, n contributed the most to the error in the predicted daily ET

_{o}(first), and Wspd and T

_{max}greatly contributed to the error in the predicted daily ET

_{o}(second or third). Moreover, T

_{max}contributed the most to the error in the predicted daily ET

_{o}by the PMF model (first), while Wspd and SDun greatly contributed to the error in the predicted daily ET

_{o}(second and third, respectively). For all models at GY, SDun contributed the most to the error in the predicted daily ET

_{o}(first), while T

_{max}and Wspd also greatly contributed (second and third, respectively) to the error in the predicted daily ET

_{o}. This is consistent with previous studies [12,13,16,20].

_{o}prediction at YC in an arid region is Wspd transformed from the wind scale in public weather forecasts. Due to regional differences, as well as the influences of the location of weather stations and local topographic features [42], wind speed is one of the most difficult parameters to accurately predict [20,69]. Cai et al. [18] stated that it is acceptable to estimate the wind speed from the wind scale in public weather forecasts, but the error in such estimates could be large in arid regions [20]. George et al. [70] showed that the discrepancy between predicted and measured reference crop evapotranspiration values largely resulted from erroneous predictions of the mean wind speed. Li and Beswick [71] reported that wind speed represents a more serious source of error than solar radiation in ET

_{o}estimation. Except in arid and windy areas, the effect of the wind speed on the estimated ET

_{o}values is relatively limited [72].

_{o}prediction. First, since the weather type in the public weather forecasts was converted into SDun using the sunshine duration coefficient values listed in Table 3, obtained using 2004 measured solar radiation data for Daxing District, Beijing, which is located in a warm temperate semihumid continental monsoon climate zone, the application of sunshine duration coefficients derived from one region to other regions with different climate types could result in different degrees of error due to the climatic differences between these regions [11,21]. Second, the values of a

_{s}and b

_{s}in Equation (11) can vary depending on atmospheric conditions (humidity and dust) and solar declination (latitude and month), so the improved a

_{s}and b

_{s}parameters must be calibrated when calculating R

_{s}with Equation (11) [36]; in contrast, we directly used the recommended values of 0.25 and 0.5 without calibration, which could also cause errors. Perera et al. [1] found that the largest error source between predicted and observed ET

_{o}values is the prediction accuracy of the incident solar radiation. The results of Medina et al. [13] showed that the error in radiation prediction imposed the greatest influence on ET

_{o}prediction, followed by the error in wind prediction. Fan et al. [16] noted that in the temperate continental zone (TCZ), temperate monsoon zone (TMZ), and other climate zones, the contribution of the R

_{s}(solar radiation) prediction error to the ET

_{o}prediction error was the greatest.

## 4. Conclusions

_{o}prediction at three stations in three climatic regions of Ningxia, China, was evaluated using public weather forecasts with a lead time of 1–7 days. The prediction performance of weather variables in public weather forecasts and the prediction performance of the daily ET

_{o}of the seven models were analyzed and evaluated on daily and seasonal scales, respectively. The optimal input combination of the machine learning models was determined, and the weather forecast variables contributing to the error in the predicted daily ET

_{o}were clarified. The main conclusions of this study are as follows:

- (1)
- At all stations, in the public weather forecast variables with a lead time of 1–7 days during the four seasons of spring, summer, autumn and winter from 2020–2021, T
_{max}followed the order of autumn > summer> winter > spring (or summer > autumn > winter > spring), T_{min}followed the order of autumn > summer > winter > spring (or summer > autumn) > spring > winter), SDun followed the order of winter > spring > autumn > summer, and the average forecast performance decreased in sequence. The performance of wind speeds predicted by public weather forecasts was the worst, and their prediction performance showed great variation depending on the station. It was recommended that the model predict the daily ET_{o}by considering the wind speed to take a constant value (2 m s^{−1}) or the daily average wind speed over a long period; - (2)
- The calibrated PMT equation (k
_{Rs}= 0.16, T_{dew}= T_{min}− 1) with public weather forecasts of T_{max}, T_{min,}and 2 m s^{−1}of constant wind speed as inputs was recommended as the best model for the irrigation season at YC station for the predicted lead time of 1–7 day ET_{o}for all three stations. The calibrated PMT equation (k_{Rs}= 0.21, T_{dew}= T_{min}− 3) with public weather forecasts of T_{max}, T_{min}, and constant wind speed of 2 m s^{−1}as inputs and CatBoost with C3 (T_{max}, T_{min}, Wspd) as inputs were the best-performing models for the spring and autumn irrigation seasons and the summer irrigation season at station TX, respectively. The calibrated PMF equation (T_{dew}= T_{min}) with T_{max}, T_{min}, SDun, and 2 m s^{−1}constant wind speed from the public weather forecast as inputs, CatBoost with C4 (T_{max}, T_{min}) as inputs, and the calibrated PMT equation (kr = 0.18, T_{dew}= T_{min}−2) with T_{max}, T_{min}, and 2 m s^{−1}constant wind speed from the public weather forecast as inputs were the optimal models for the spring irrigation season, summer irrigation season, and autumn irrigation season, respectively, at the GY station; - (3)
- The four machine learning models MLP, XGBoost, LightGBM, and CatBoost are recommended to select C2 (T
_{max}, T_{min}, SDun) or C4 (T_{max}, T_{min}) as the input combinations to predict the daily ET_{o}of the three stations. However, for station YC located in the arid zone, the error in the model prediction of daily ET_{o}is mainly caused by Wspd from the public weather forecast, followed by SDun. SDun from the public weather forecast is the main source of error in the model prediction of daily ET_{o}for station TX located in the semiarid zone and station GY located in the semimoisture zone, followed by Wspd. Therefore, the combination of Wspd from public weather forecasts and SDun should be considered carefully when adding them to the input combination of the model.

_{o}model for the irrigation season recommended in this study has been successfully used in the irrigation forecast of 1–3 days in advance for 134 ha of Lycium barbarum crop in Tongxin, Ningxia, China. Compared with the local conventional irrigation, the irrigation forecasting of 1–3 days in advance using the ET

_{o}model of the optimal prediction day of the irrigation season realized the improvement of quality and yield, and water-saving irrigation of field Lycium barbarum, and improved the utilization of the limited rainfall during the fertility period of the Lycium barbarum crop.

_{o}during the irrigation season at the study stations were obtained, some stations had lower model prediction accuracy during certain irrigation seasons (e.g., the best models for the summer season at the TX and GY stations), and the combination of these machine learning models with optimization algorithms (ant colony optimization algorithms, cuckoo search algorithms, flower pollination algorithms, etc.) could be explored to optimize the model’s hyperparameters through optimization algorithms to further enhance the performance of the model to predict ET

_{o}and improve the accuracy of the model to predict ET

_{o}.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Statistics of the forecast performance indicators of the daily scale Tmax, Tmin, SDun, and Wspd forecasts obtained from public weather forecasts with a lead time of 1–7 days at the three research sites from 2020–2021 ((

**a**) MAE, (

**b**) RMSE, (

**c**) RM, and (

**d**) R).

**Figure 4.**Average statistics of the performance indicators of the predicted weather variables (T

_{max}, T

_{min}, three types of wind speed, and sunshine duration) at the three stations by the public weather forecasts with a lead time of 1–7 days during each season from 2020–2021 (T

_{max}/spring, T

_{min}/spring, Wspd/spring, and SDun/spring denote the performance of the public weather forecast in spring predicting T

_{max}, T

_{min}, wind speed, and sunshine hours at the stations (YC, TX, and GY), respectively. Wspd/2/spring and Wspd/2.11 (3.3 and 2.44)/spring denote the performance of wind speeds at sites (YC, TX, and GY) when the predicted wind speed in spring is taken to be constant at 2 m s

^{−1}and the long-term average wind speed, respectively. The other symbols have the same meaning). ((

**a**) T

_{min}and T

_{max}at station YC, (

**d**) SDun at station YC, (

**g**) Wspd at station YC; (

**b**) T

_{min}and T

_{max}at station TX, (

**e**) SDun at station TX, (

**h**) Wspd at station TX; (

**c**) T

_{min}and T

_{max}at station GY, (

**f**) SDun at station GY, (

**i**) Wspd at station GY).

**Figure 5.**Average statistics of the predictions with a lead time of 1–7 days by the MLP, XGBoost, LightGBM, and CatBoost models at YC station under the different input combinations during training, validation, and testing (C1 denotes T

_{max}, T

_{min}, n, WSpd, C2 denotes T

_{max}, T

_{min}, n, C3 denotes T

_{max}, T

_{min}, WSpd, C4denotes T

_{max}, T

_{min}, and MLP/C1 denotes the MLP model using (T

_{max}, T

_{min}, n, WSpd) as inputs to predict the day ET

_{o}, and the other symbols are the same).

**Figure 6.**Average statistics of the predictions with a lead time of 1–7 days by the MLP, XGBoost, LightGBM, and CatBoost models at the TX station under the different input combinations during training, validation, and testing (symbols have the same meaning as in Figure 5).

**Figure 7.**Average statistics of the predictions with a lead time of 1–7 days by the MLP, XGBoost, LightGBM, and CatBoost models at the GY station under the different input combinations during training, validation, and testing (symbols have the same meaning as in Figure 5).

**Figure 8.**Statistics of performance indicators MAE and RMSE for ET

_{o}prediction with a lead time of 1–7 days using the three empirical equations and four machine learning methods at the three sites.

**Figure 9.**Statistics of performance indicators RM and R for ET

_{o}prediction with a lead time of 1–7 days using the three empirical equations and four machine learning methods at the three sites.

**Figure 10.**Mean statistics of the performance indicators for ET

_{o}prediction with a lead time of 1–7 days by the seven methods at YC station during the four seasons from 2020–2021.

**Figure 11.**Mean statistics of the performance indicators for ET

_{o}prediction with a lead time of 1–7 days by the seven methods at the TX station during the four seasons from 2020–2021.

**Figure 12.**Mean statistics of the performance indicators for ET

_{o}prediction with a lead time of 1–7 days by the seven methods at GY station during the four seasons from 2020–2021.

**Figure 13.**Scatter plots of daily ET

_{o}predicted by PMT equations at YC station during the test period versus daily ET

_{o}calculated by PM equations. ((

**a**) 1-day lead time, (

**b**) 4-day lead time, (

**c**) 7-day lead time).

**Figure 14.**Scatter plots of daily ET

_{o}predicted by the PMT equations (

**a**–

**c**) and CatBoost model (

**d**–

**f**) at the TX station during the test period versus daily ET

_{o}calculated by the PM equations. ((

**a**,

**d**) 1-day lead time, (

**b**,

**e**) 4-day lead time, (

**c**,

**f**) 7-day lead time).

**Figure 15.**Scatter plots of daily ET

_{o}predicted by PMF equations (

**a**–

**c**), CatBoost model (

**d**–

**f**) and PMT equations (

**g**–

**i**) at GY station during the test period versus daily ET

_{o}calculated by PM equations. ((

**a**,

**d**,

**g**) 1-day lead time, (

**b**,

**e**,

**h**) 4-day lead time, (

**c**,

**f**,

**i**) 7-day lead time).

**Figure 16.**Mean statistics of the ET

_{o}prediction performance indicators for a lead time of 1–7 days of the five models at YC stations under the five input combinations when replacing the observed weather variables with predicted weather variables based on public weather forecasts one at a time ((

**a**) PMF equation, (

**b**) MLP model, (

**c**) XGBoost model, (

**d**) LightGBM model, and (

**e**) CatBoost model).

**Figure 17.**Mean statistics of the ET

_{o}prediction performance indicators for a lead time of 1–7 days of the five models at the TX stations under the five input combinations when replacing the observed weather variables with predicted weather variables based on public weather forecasts one at a time ((

**a**) PMF equation, (

**b**) MLP model, (

**c**) XGBoost model, (

**d**) LightGBM model, and (

**e**) CatBoost model).

**Figure 18.**Mean statistics of the ET

_{o}prediction performance indicators for a lead time of 1–7 days of the five models at GY stations under the five input combinations when replacing the observed weather variables with predicted weather variables based on public weather forecasts one at a time (

**a**) PMF equation, (

**b**) MLP model, (

**c**) XGBoost model, (

**d**) LightGBM model, and (

**e**) CatBoost model).

**Table 1.**Location characteristics and annual mean values of the meteorological data at each of the three weather stations in this study.

WMO Number | Station | Latitude | Longitude | Elevation | T_{max} | T_{min} | T_{mean} | u_{2} | RH_{mean} | Sdun | ET_{o} | Climate Zone |
---|---|---|---|---|---|---|---|---|---|---|---|---|

(N) | (E) | (m) | (°C) | (°C) | (°C) | (m s^{−1}) | (%) | (h) | (mm d^{−1}) | |||

53614 | Yinchuan (YC) | 38°29′ | 106°13′ | 1111.4 | 17.3 | 4.8 | 10.5 | 1.54 | 51.11 | 7.55 | 3.05 | Arid |

53810 | Tongxin (TX) | 36°58′ | 105°54′ | 1339.3 | 17.5 | 4.2 | 10.1 | 2.28 | 52.47 | 8.02 | 3.37 | Semiarid |

53817 | Guyuan (GY) | 36°00′ | 106°16′ | 1753.0 | 13.9 | 2.6 | 7.7 | 1.87 | 58.51 | 6.95 | 2.68 | Subhumid |

**Table 2.**Beaufort wind scale (GB/T 35227—2017, 2017) [41].

Wind Scale | Designation | u_{10} (ms^{−1}) | |
---|---|---|---|

Range | Average (WSpd) | ||

0 | Calm | 0.0–0.2 | 0.0 |

1 | Light | 0.3–1.5 | 1.0 |

2 | Slight | 1.6–3.3 | 2.0 |

3 | Gentle | 3.4–5.4 | 4.0 |

4 | Moderate | 5.5–7.9 | 7.0 |

5 | Fresh | 8.0–10.7 | 9.0 |

6 | Strong wind | 10.8–13.8 | 12.0 |

7 | High wind | 13.9–17.1 | 16.0 |

8 | Gale | 17.2–20.7 | 19.0 |

9 | Strong gale | 20.8–24.4 | 23.0 |

10 | Whole gale | 24.5–28.4 | 26.0 |

11 | Storm | 28.5–32.6 | 31.0 |

12 | Hurricane | 32.7–36.9 | 35.0 |

Weather Type | Sunny | Clear to Overcast | Cloudy | Overcast | Rainy | Snow | Dust | Haze |
---|---|---|---|---|---|---|---|---|

Coeffient (α) | 0.9 | 0.7 | 0.5 | 0.3 | 0.1 | 0.1 | 0.2 | 0.2 |

**Table 4.**Best estimates of the corrected parameter for the three empirical equations at the three study sites.

Station | PMT | PMF | HS | ||||
---|---|---|---|---|---|---|---|

k_{Rs} | T_{dew} | WSpd | T_{dew} | WSpd | C | E | |

Yinchuan | 0.16 | T_{dew} = T_{min} − 1 | 2 ms^{−1} | T_{dew} = T_{min} | 2 ms^{−1} | 0.00096673 | 0.48475 |

Tongxin | 0.21 | T_{dew} = T_{min} − 3 | 2 ms^{−1} | T_{dew} = T_{min} − 1 | 2 ms^{−1} | 0.00089671 | 0.53594 |

Guyuan | 0.18 | T_{dew} = T_{min} − 2 | 2 ms^{−1} | T_{dew} = T_{min} | 2 ms^{−1} | 0.00082219 | 0.55182 |

**Table 5.**Statistics based on the performance metrics of ET

_{o}predicted with three empirical equations during the precalibration validation and test periods and the postcalibration validation and test periods.

Stage/Methods | Yinchuan | Tongxin | Guyuan | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | RMSE | RM | R | MAE | RMSE | RM | R | MAE | RMSE | RM | R | ||

(mm d^{−1}) | (mm d^{−1}) | (mm d^{−1}) | (mm d^{−1}) | (mm d^{−1}) | (mm d^{−1}) | ||||||||

Uncalibrated | |||||||||||||

Training | PMT | 0.40 | 0.60 | 0.92 | 0.96 | 0.44 | 0.66 | 0.93 | 0.96 | 0.37 | 0.54 | 0.94 | 0.96 |

PMF | 0.70 | 0.86 | 1.20 | 0.97 | 0.69 | 0.84 | 1.18 | 0.98 | 0.68 | 0.83 | 1.23 | 0.98 | |

HS | 4.39 | 5.35 | 2.44 | 0.95 | 4.35 | 5.21 | 2.30 | 0.95 | 3.79 | 4.59 | 2.41 | 0.94 | |

PMT | 0.39 | 0.56 | 0.91 | 0.96 | 0.51 | 0.73 | 0.90 | 0.96 | 0.35 | 0.50 | 0.93 | 0.96 | |

PMF | 0.70 | 0.88 | 1.22 | 0.98 | 0.63 | 0.77 | 1.14 | 0.97 | 0.63 | 0.77 | 1.00 | 1.00 | |

HS | 4.63 | 5.63 | 2.56 | 0.95 | 4.26 | 5.11 | 2.22 | 0.94 | 3.86 | 4.64 | 2.45 | 0.95 | |

testing/the average of 1–7 days lead time | |||||||||||||

PMT | 0.63 | 0.89 | 1.03 | 0.88 | 0.99 | 1.36 | 0.90 | 0.80 | 0.92 | 1.22 | 0.92 | 0.72 | |

PMF | 0.91 | 1.22 | 1.24 | 0.88 | 1.00 | 1.38 | 1.06 | 0.80 | 1.02 | 1.39 | 1.16 | 0.72 | |

HS | 4.91 | 6.05 | 2.70 | 0.89 | 4.30 | 5.40 | 2.24 | 0.82 | 3.72 | 4.77 | 2.35 | 0.73 | |

Calibrated | |||||||||||||

Validation | PMT | 0.36 | 0.53 | 0.93 | 0.97 | 0.53 | 0.76 | 1.11 | 0.96 | 0.32 | 0.47 | 1.04 | 0.96 |

PMF | 0.80 | 1.00 | 1.25 | 0.97 | 0.58 | 0.72 | 1.11 | 0.96 | 0.64 | 0.79 | 1.22 | 0.98 | |

HS | 0.46 | 0.62 | 1.04 | 0.95 | 0.54 | 0.76 | 0.95 | 0.94 | 0.38 | 0.51 | 0.99 | 0.95 | |

testing/the average of 1–7 days lead time | |||||||||||||

PMT | 0.61 | 0.86 | 1.02 | 0.89 | 0.90 | 1.26 | 1.02 | 0.83 | 0.89 | 1.20 | 1.00 | 0.73 | |

PMF | 0.86 | 1.17 | 1.21 | 0.88 | 0.95 | 1.31 | 1.03 | 0.81 | 0.99 | 1.36 | 1.14 | 0.72 | |

HS | 0.68 | 0.97 | 1.09 | 0.89 | 0.95 | 1.31 | 0.96 | 0.81 | 0.94 | 1.26 | 0.95 | 0.72 |

**Table 6.**Optimal input combinations and model tuning information of the four machine learning methods at the three meteorological stations.

Station | MLP | XGBoost | LightGBM | CatBoost | ||
---|---|---|---|---|---|---|

Inputs | Structure | Parameter Value | Inputs | Inputs | Inputs | |

Yinchuan | C2 | 3-100-100-100-1 | Lr = 0.0088 | C2 | C2 | C2 |

Tongxin | C2 | 3-85-85-1 | Lr = 0.0088 | C4 | C2 | C3 |

Guyuan | C2 | 3-62-62-1 | Lr = 0.0055 | C2 | C4 | C4 |

**Table 7.**Mean statistics of the performance indicators for ET

_{o}prediction with a lead time of 1–7 days by the seven methods at the three sites.

Method | Yinchuan | Tongxin | Guyuan | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | RMSE | RM | R | MAE | RMSE | RM | R | MAE | RMSE | RM | R | |

(mm d^{−1}) | (mm d^{−1}) | (mm d^{−1}) | (mm d^{−1}) | (mm d^{−1}) | (mm d^{−1}) | |||||||

PMT | 0.61 | 0.862 | 1.02 | 0.89 | 0.90 | 1.257 | 1.02 | 0.83 | 0.89 | 1.20 | 1.00 | 0.73 |

PMF | 0.86 | 1.17 | 1.21 | 0.88 | 0.949 | 1.307 | 1.03 | 0.807 | 0.99 | 1.36 | 1.15 | 0.72 |

HS | 0.675 | 0.971 | 1.09 | 0.89 | 0.949 | 1.309 | 0.96 | 0.814 | 0.94 | 1.26 | 0.95 | 0.72 |

MLP | 0.70 | 0.938 | 1.02 | 0.86 | 0.991 | 1.362 | 0.94 | 0.786 | 0.94 | 1.25 | 0.93 | 0.69 |

XGBoost | 0.70 | 0.936 | 0.98 | 0.86 | 0.989 | 1.363 | 1.00 | 0.789 | 0.923 | 1.234 | 0.94 | 0.70 |

LightGBM | 0.72 | 0.938 | 1.01 | 0.86 | 0.985 | 1.358 | 0.91 | 0.793 | 0.911 | 1.217 | 0.98 | 0.70 |

CatBoost | 0.691 | 0.933 | 1.01 | 0.86 | 0.954 | 1.279 | 1.01 | 0.811 | 0.908 | 1.214 | 0.98 | 0.71 |

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## Share and Cite

**MDPI and ACS Style**

Liang, Y.; Feng, D.; Sun, Z.; Zhu, Y.
Evaluation of Empirical Equations and Machine Learning Models for Daily Reference Evapotranspiration Prediction Using Public Weather Forecasts. *Water* **2023**, *15*, 3954.
https://doi.org/10.3390/w15223954

**AMA Style**

Liang Y, Feng D, Sun Z, Zhu Y.
Evaluation of Empirical Equations and Machine Learning Models for Daily Reference Evapotranspiration Prediction Using Public Weather Forecasts. *Water*. 2023; 15(22):3954.
https://doi.org/10.3390/w15223954

**Chicago/Turabian Style**

Liang, Yunfeng, Dongpu Feng, Zhaojun Sun, and Yongning Zhu.
2023. "Evaluation of Empirical Equations and Machine Learning Models for Daily Reference Evapotranspiration Prediction Using Public Weather Forecasts" *Water* 15, no. 22: 3954.
https://doi.org/10.3390/w15223954