# Evaluation of Five Equations for Short-Term Reference Evapotranspiration Forecasting Using Public Temperature Forecasts for North China Plain

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## Abstract

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_{0}) are essential for real-time irrigation scheduling. Many models rely on current and historical temperature data to estimate daily ET

_{0}. However, easily accessible temperature forecasts are relatively less reported in short-term ET

_{0}forecasting. Furthermore, the accuracy of ET

_{0}forecasting from different models varies locally and also across regions. We used five temperature-dependent models to forecast daily ET

_{0}for a 7-day horizon in the North China Plain (NCP): the McCloud (MC), Hargreaves-Samani (HS), Blaney-Criddle (BC), Thornthwaite (TH), and reduced-set Penman–Monteith (RPM) models. Daily meteorological data collected between 1 January 2000 and 31 December 2014 at 17 weather stations in NCP to calibrate and validate the five ET

_{0}models against the ASCE Penman–Monteith (ASCE-PM). Forecast temperatures for up to 7 d ahead for 1 January 2015–19 June 2021 were input to the five calibrated models to forecast ET

_{0}. The performance of the five models improved for forecasts at all stations after calibration. The calibrated RPM is the preferred choice for forecasting ET

_{0}in NCP. In descending order of preference, the remaining models were ranked as HS, TH, BC, and MC. Sensitivity analysis showed that a change in maximum temperature influenced the accuracy of ET

_{0}forecasting by the five models, especially RPM, HS, and TH, more than other variables. Meanwhile, the calibrated RPM and HS equations were better than the other models, and thus, these two equations were recommended for short-term ET

_{0}forecasting in NCP.

## 1. Introduction

_{0}) forecasts are essential for real-time irrigation demand estimation [5]. The ET

_{0}models available in the literature can be broadly classified as (1) fully physically based combination models that account for mass and energy conservation principles; (2) black-box models based on artificial neural networks, empirical relationships, and fuzzy and genetic algorithms; and (3) semi-physically based models dealing with either mass or energy conservation [6].

_{0}and as a benchmark to evaluate other models [7,14]. However, the application of ASCE-PM is limited by the lack of required weather variables in many regions. Therefore, under the condition of the deficiency of climatological data, many studies have used machine learning models due to their excellent capability for tackling non-linear relationships between dependent and independent variables [15,16,17].

_{0}modeling using machine learning, such as multi-layer perceptron (MLP) [18,19], extreme learning machine (ELM) [20,21], support vector machines (SVMs) [22,23], support vector regression (SVR) [24], and extreme learning machines (ELMs). They have mainly emphasized estimating the current ET

_{0}using various computing approaches, while less attention has been paid to machine learning codes to forecast futuristic ET

_{0}. The network was performed as a black box with no possibility of providing algebraic equations for rapid deployment. Traore et al. [19] forecasted daily ET

_{0}based on gene-expression programming (GEP) with the limited public weather forecast in Jiangsu, China, and found that the GEP algorithm is a deployable tool for ET

_{0}forecasts to anticipate decisions on short-term irrigation scheduling. They proposed a deployable model formulated into mathematic equations. The above methods take fewer variables as input and can produce precise results. However, for farmers and policymakers in rural areas, the machine learning methods are difficult to understand and employ. In addition, the soft computing methods are often not free and require high-performance computers. This makes it difficult to estimate ET

_{0}, particularly in remote areas where most irrigated fields are located, specifically in the developing and emerging world. For this reason, simpler empirical equations that require fewer climatic variables for ET

_{0}calculation are utilized [25].

_{0}calculation based on forecast temperature. Trajkovic [29] found that a temperature-based radial basis function (RBF) network can be used to forecast ET

_{0}when relative humidity, radiation, and wind speed data are unavailable. Lu et al. [30] compared three temperature-based estimation models with three radiation-based estimation models and recommended the Hamon model for use in the southeastern United States. The temperature-based Hargreaves–Samani (HS) model has been used to forecast ET

_{0}in many studies and produced good results [31,32,33,34]. Other models, such as Thornthwaite (TH) and Blaney–Criddle (BC), can also be used to forecast ET

_{0}after being calibrated [5,35,36,37].

_{0}and forecast irrigation demand in the near future [19,40]. Luo et al. [41] proposed using a locally calibrated HS model and temperature forecast for short-term ET

_{0}forecasts and found that the method can provide daily ET

_{0}forecasts with certain accuracy for real-time irrigation forecast. Xiong et al. [5] forecast ET

_{0}with the BC model and temperature forecast, and the results indicate that the BC model can be an alternative and effective solution for forecast ET

_{0}in east China. Chang et al. [35] used a modified daily TH equation and temperature forecasts for ET

_{0}forecasts and found that TH produced good ET

_{0}forecast results and can be a feasible solution for short-term ET

_{0}forecasts. The temperature-based models are simple, convenient, and practical, and they are widely applicable in forecasting ET

_{0}[38,42]. Although NCP has the greatest rate of agricultural irrigation in China, there has been little research into short-term ET

_{0}forecasts for NCP. Some studies have limited their interest to one model or investigated models that were impractical for various reasons, so it seems that the real-time irrigation demand and irrigation water requirements forecasting in NCP may benefit from comparative research into five different temperature-based models for short-term ET

_{0}forecasting. Most extant models require some form of correction for the particular area they are modeling, while the lack of model comparison caused confusion about whether this model is the best one for that region. It is therefore necessary to compare five models to determine the most accurate method for short-term ET

_{0}forecasts in NCP.

_{0}forecasting using five models with the public weather forecasts in NCP, (2) to produce five sets of calibrated parameters for temperature-based ET

_{0}models and to recommend the best short-term ET

_{0}model in NCP using the statistical indicators as criteria, and (3) to explore error source and the sensitivity of short-term ET

_{0}forecasts to the percentage change in temperature forecast.

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Data

_{0}temperature-based forecast models against the ASCE Penman–Monteith (ASCE-PM) model. Forecast temperatures up to 7 d ahead for 1 January 2015–19 June 2021 were input to the five calibrated models to forecast ET

_{0}. Data used in the study are shown in Table 2.

#### 2.3. Method

_{0}[29,31,36,37,38,42,43]. Air-temperature-based models can accurately and reliably forecast daily ET

_{0}in China [33,34,38,42]. Penman–Monteith (ASCE-PM) [7] has been recommended as the standard model for estimating ET

_{0}and as a target to evaluate other models [7,14]. However, ASCE-PM requires several meteorological variables as input, and thus it is not practical for areas where the required daily meteorological input data is not available [5,29,36]. Figure 2 shows the workflow of the ET

_{0}forecast in this study.

#### 2.3.1. ASCE Penman–Monteith Model

_{0}(mm/d) calculated by the ASCE-PM equation; R

_{n}is the net irradiance on the crop surface (MJ/m

^{2}/d); G is the soil heat flux density (MJ/m

^{2}/d); T is the air temperature at a height of 2 m (°C); U

_{2}is the wind speed at a height of 2 m (m/s); ${\mathrm{e}}_{\mathrm{s}}$ is the saturated air vapor pressure (kPa); ${\mathrm{e}}_{\mathrm{a}}$ is the measured vapor pressure (kPa); Δ is the slope of the vapor pressure curve (kPa/°C); γ is the psychometric constant (kPa/°C).

^{2}/d); n is the sunshine duration (h); N is the maximum possible duration of sunshine or daylight hours (h); Ra is the extraterrestrial radiation (MJ/m

^{2}/d); ${\mathrm{a}}_{\mathrm{s}}$ (0.25) and ${\mathrm{b}}_{\mathrm{s}}$ (0.5) are constants recommended in ASCE-EWRI [7]. Considering that there is short grass around the China Meteorological weather station and that ET

_{0}was calculated on the daily time scale, the value of Cn and Cd are 900 and 0.34, respectively.

#### 2.3.2. McCloud Model

_{0}estimation of land in Florida. The MC equation is:

_{0}calculated by the MC equation (mm/d); K (0.254) and W (1.07) are both recommended default values [42]; T is the daily mean measured air temperature (°C) at a screen height of 2.0 m.

#### 2.3.3. Hargreaves–Samani Model

_{0}estimation is expressed as:

_{0}calculated by the HS equation (mm/d); ${\mathrm{T}}_{\mathrm{max}}$ and ${\mathrm{T}}_{\mathrm{min}}$ are the maximum and minimum air temperature (°C); Ra is the extraterrestrial irradiance (MJ/m

^{2}/d); C (0.0023) and E (0.5) are parameters with recommended values [26,33,34]. The HS model has been widely used in various climates, and many studies have found that the values of the parameters C and E vary from region to region. To ensure model accuracy, the local calibration of these two parameters is necessary [32,34].

#### 2.3.4. Blaney–Criddle Model

_{0}. The equation is:

_{0}calculated by the BC model (mm/d), and Tm is the daily mean air temperature (°C) calculated according to daily ${\mathrm{T}}_{\mathrm{max}}$ and ${\mathrm{T}}_{\mathrm{min}}$, as ${\mathrm{T}}_{\mathrm{m}}=\left({\mathrm{T}}_{\mathrm{max}}+{\mathrm{T}}_{\mathrm{min}}\right)/2$. The parameter p is the daily percentage of annual daytime hours (%).

#### 2.3.5. Adjusted Thornthwaite Model

_{0}is:

_{0}calculated by the Thornthwaite equation (mm/d); I is a thermal index representing the local normal temperature regime; ${\mathrm{T}}_{\mathrm{i}}$ is the monthly average air temperature (°C), 0 if negative; the exponent a is a function of I; ${\mathrm{T}}_{\mathrm{ef}}$ is the effective daily temperature (°C); K is a calibration coefficient with a recommended value of 0.69 for calculating monthly ET

_{0}.

#### 2.3.6. Reduced-Set Penman–Monteith Model

^{2}/d); ${\mathrm{R}}_{a}$ is the extraterrestrial irradiance (MJ/m

^{2}/d); ${\mathrm{T}}_{\mathrm{max}}$ and ${\mathrm{T}}_{\mathrm{min}}$ are the maximum and minimum air temperatures (°C); K is an adjustment coefficient: K = 0.16 for inland locations and K = 0.19 for coastal locations; ${\mathrm{e}}_{a}$ is the actual vapor pressure (kPa). The RPM is used with these values to calculate ET

_{0,RPM}.

#### 2.3.7. Calibration Method

_{0}by the five temperature-based models with ET

_{0,P}, revising the model if necessary, comparing again, until a model is accepted (validated). Although the HS equation appears to be applicable in any region or climate, its calculation accuracy actually varies according to region, and some studies have found the accuracy increased if C and E are calibrated for regional differences. The least squares method (LSM) is used to calibrate the values of the parameters C and E in the HS equation [29,31,33].

_{0}models (MC, BC, RPM, and TH), ET

_{0}calculated by the ASCE-PM model (ET

_{0,P}) is considered to be the ground truth, and linear regression between ET

_{0,P}and ET

_{0}calculated by the other models was used to correct the models for regional calibration. The linear regression model was:

_{0}calculated by ASCE-PM (mm/d); ${\mathrm{ET}}_{0,\mathrm{T}}$ is the ET

_{0}estimated by the temperature-based models (mm/d); a and b are the correlation coefficients.

#### 2.3.8. Statistical Analysis

_{0}forecasting was similarly defined as the number of days in which the absolute error of forecast ET

_{0}was within ±1.5 mm/d as a percentage of the total number of samples [5]. Mean absolute error (MAE), root mean square error (RMSE), and the correlation coefficient (R) were also used for comparisons [5,41,42,46]. The equations for these indicators are:

_{0}; ${\mathrm{E}}_{\mathrm{i}}$ is the measured air temperature or the calculated ET

_{0}with the measured meteorological data based on ASCE-PM; i is the calculated sample ordinal number, I = 1, 2, 3…n; N is the total number of samples; $\overline{{\mathrm{E}}_{\mathrm{T}}}$ is the mean value of the calculated sequence, and $\overline{\mathrm{E}}$ is the mean value of the calculated sequence; K is T or E and is the error range for $\left|{\mathrm{E}}_{\mathrm{Ti}}-{\mathrm{E}}_{\mathrm{i}}\right|\le 2\xb0\mathrm{C}$ and $\left|{\mathrm{E}}_{\mathrm{Ti}}-{\mathrm{E}}_{\mathrm{i}}\right|\le 1.5\mathrm{mm}/\mathrm{d}$; ${\mathrm{N}}_{\mathrm{rK}}$ is the number of days for which the forecasting is correct; ${\mathrm{N}}_{\mathrm{fK}}$ is the total number of days forecast.

## 3. Results and Discussion

#### 3.1. Evaluation of Temperature Forecasts

_{0}model uses air temperature data as input to calculate ET

_{0}, so the accuracy of the air temperature forecast affects the accuracy of the ET

_{0}forecast. We used Acc, MAE, RMSE, and R to evaluate the accuracy of the minimum and maximum temperature forecasts for the 17 WS in NCP. The values of the statistical indicators are shown in Figure 3. For all stations, the range of Acc for T

_{min}for 1 d ahead was 52–85% and for 7 d ahead was 42–60%. The corresponding ranges for R were 0.87–0.99 and 0.86–0.97, respectively. MAE and RMSE for ${\mathrm{T}}_{\mathrm{min}}$ were the least for the 1 d ahead and the greatest for 7 d ahead. The range of Acc for ${\mathrm{T}}_{\mathrm{max}}$ for 1 d ahead was 42.36–56.54% and for 7 d ahead was 38–49%. R for T

_{max}was >0.85 for all stations. The greatest values of RMSE and MAE for T

_{max}were for the 7 d ahead.

_{max}was 1.6 to 2.4 °C, and average RMSE was 2.2–3.1 °C. The average Acc for ${\mathrm{T}}_{\mathrm{max}}$ was 44–52%. The average MAE for T

_{max}was 2.5–2.9 °C, and the average RMSE was 3.3–3.8 °C. The R for T

_{min}was >0.96, and for ${\mathrm{T}}_{\mathrm{max}}$, it was >0.94. Our results agreed with those of Yang et al. [42], who analyzed the 7 d temperature forecasts of 6 WS in China for 2012–2014 and found that R for both ${\mathrm{T}}_{\mathrm{max}}$ and ${\mathrm{T}}_{\mathrm{min}}$ was >0.92. We found that ${\mathrm{T}}_{\mathrm{min}}$ forecasts were more accurate than ${\mathrm{T}}_{\mathrm{max}}$ forecasts for NCP. This result is consistent with previous studies [5,38]. We concluded that the forecast accuracy of minimum and maximum air temperatures in public temperature forecasts were acceptable for use in the study of ET

_{0}forecasts in NCP [34,41].

#### 3.2. Calibration of Temperature-Based Models

_{0}for the same WS calculated by different methods can vary greatly [34,48].The ET

_{0,P}was used as the ground truth for the calibration of the temperature-based models. The calibrated parameters for each WS for the five models are shown in Table 4. The range of the calibrated coefficient for a model MC was 1.41 to 1.69, and the range of the calibrated coefficient b was 0.44 to 0.52. The corresponding ranges for BC were −0.69 to −0.02 and 0.69 to 0.86. The corresponding ranges for TH were 0.49 to 0.84 and 0.72 to 0.86. The corresponding ranges for RPM were −0.16 to 0.30 and 0.89 to 1.08. The parameters a and b for the calibrated RPM model were, respectively, close to zero and one, which indicates that the original RPM model was a good fit with the ASCE-PM. The recommended values of C and E for HS were 0.0023 and 0.50 [13]. We found that the calibrated parameter C was 0.001 or 0.002, agreeing well with the recommended value. The range of parameter E was 0.52 to 0.68, which is slightly greater than the recommended value of 0.50. These results show that the recommended values of C and E for HS are suitable for ET

_{0}modeling NCP.

_{0}estimated by the five models and ASCE-PM for the calibration and validation periods for representative WS of BT, TS, and XX. These results show that, before regional calibration, ET

_{0,MC}was significantly greater than ET

_{0,P}and that ET

_{0}estimated using RPM was closest to ET

_{0,P}. The ET

_{0,HS}and ET

_{0,TH}calculated before calibration were evenly distributed on both sides of y = x, which indicates that the calculated values of these two models were close to ET

_{0,P}. ET

_{0,BC}was generally greater than ET

_{0,P}. ET

_{0}values calculated by the five calibrated models were closer to the line y = x than the uncalibrated models. Regional calibration increased the goodness-of-fit between all of the ET

_{0}values calculated by the five models and ET

_{0,P}.

_{0}estimated by MC, RPM models, and ASCE-PM and Precipitation (Precip) for the three representative stations (for clarity, only the best model RPM and worst model MC are presented). After the calibration, although the maximum value of ET

_{0}estimated by the five models was slightly less than the ground truth value, the estimated ET

_{0}agreed well with ET

_{0,P}. That the maximum value of ET

_{0,MC}before calibration was much greater than ET

_{0,P}and that ET

_{0,MC}after calibration fitted ET

_{0 P}well. ET

_{0,RPM}fitted well with ET

_{0,P}both before and after calibration. However, the high value of ET

_{0}estimated by the RPM model in the validation period is still lower than that of ET

_{0,P}, which is the main reason why the estimation accuracy of the RPM model is up to 98%. The estimation accuracy of the HS, TH, and BC models is between MC and RPM models. The low value of ET

_{0}estimated by the BC model is closer to ET

_{0,P}than ET

_{0,MC}, making the estimation accuracy of the BC model in NCP 84% higher than that of the MC model. The comparison between the daily precipitation of representative stations with ET

_{0,P}shows that ET

_{0,P}is high on rainy days, which indicates that there is a positive correlation between precipitation and ET

_{0,P}. Liu et al. [49] compared the performance of several temperature models for north China and found that the ET

_{0}calculated by HS well fitted FAO56-PM, which is consistent with our study. Awal et al. [50] evaluated the HS equation with FAO-ET

_{0}at 88 stations of West Texas Mesonet and concluded that the monthly calibrated HS model yielded a performance (R

^{2}= 0.82). Rodrigues and Braga [51] estimated ET

_{0}during the irrigation season using nine temperature-based methods in a hot-summer Mediterranean climate. The results showed that the modified HS performed acceptably with a R

^{2}higher than 0.78. Liu [52] evaluated six temperature-based PET methods, showing that the HS equation was the closeted to the PM calculation values (correlation coefficient = 0.85), followed by the Oudin and Hamon equations, whereas the revised Thornthwaite and Baier–Robertson equations performed relatively poorly. These agreed well with results in the study, and our studies performed better in some cases. Our results indicated that calibration increased the accuracy of all of the models’ ET

_{0}estimation for the study area. Among them, the estimation accuracy of the modified MC model and the modified BC model is greatly improved, by 18% and 19% respectively, so that the calibrated models are suitable for practical use in the local area. The above results indicated that the five calibrated models can be used for ET

_{0}forecasting.

#### 3.3. ET_{0} Forecasts

_{0}forecasts for three representatives WS in NCP for the period 1 January 2015 to 20 June 2021. These results show that forecast ET

_{0}has the same trend as the measured ET

_{0,P}for all models. The minimum value of ET

_{0,MC}was greater than the minimum value of ET

_{0,P}, and there was an obvious difference between ET

_{0,BC}and ET

_{0,P}. ET

_{0,RPM}and ET

_{0,HS}were similar, and both were less than ET

_{0,P}. This is consistent with the previous findings that RPM and HS underestimate ET

_{0,P}[33,42]. Forecasts of ET

_{0,BC}were similar to those of ET

_{0,MC}, but the lower values of ET

_{0,BC}were closer to ET

_{0,P}. Predictions of ET

_{0,TH}were less accurate than the values of ET

_{0,RPM}and ET

_{0,HS}but more accurate than forecasts of ET

_{0,MC}and ET

_{0,BC}. The errors show that the forecasts of the five models differ from ET

_{0,P}, and it is evident that the accuracy of the ET

_{0}forecasting is determined by the computational model used.

_{0}forecast accuracy for the 17 WS in the 1–7 d ahead forecasting period in NCP. The Acc and R decreased, and the MAE and RMSE increased as the forecast time increased. The Acc for MC for the 7 d ahead forecast was >75%, and R for the same forecast was greater than 0.65. The Acc for the 7 d ahead forecast was greater than 80%, and the R for the same forecast was greater than 0.79 for the other four models. The MAE and RMSE for 7 d ahead forecasts were greater for MC than for the other four models. The greatest average values of MAE (0.1 mm/d) and RMSE (0.2 mm/d) were for MC forecasts. The changes in the four statistical indicators show that the ET

_{0}forecasts of each model become less accurate as the forecast time increases, which is consistent with the trend of temperature forecasts. However, the ET

_{0}forecast accuracy of the five models’ maximum decreased by 3% less than the decrease in temperature forecasts.

_{0}using RPM is fundamental to irrigation decision-making, a view that has been expressed in other studies [42]. ET

_{0,HS}was second in accuracy, and the originally uncalibrated HS model’s average forecast accuracy was 84%, which indicates that the architecture of the model is suitable for ET

_{0}forecasts. The forecast accuracy of TH and BC ranked third and fourth. Xiong et al. [5] forecast ET

_{0}using a monthly calibrated BC equation using temperature forecasts in east China, and the accuracy of the ET

_{0}forecasting was acceptable (MAE varied between 0.73 and 0.82 mm/d, and R ranged from 0.74 to 0.90). According to the study of Chang et al. [35], a modified daily Thornthwaite (TH) equation can be used to forecast ET

_{0}with temperature forecasts and found that the RMSE of forecasting ranged from 0.86 to 1.01 mm/d. The results were slightly better than this study, and it may be explained that the performance of the annually calibrated BC and TH model for ET

_{0}forecast is further improved after monthly calibration. Yang et al. [42] studied representative meteorological stations in different climate zones in China and found that the calibrated TH equation can better forecast the daily ET

_{0}in different climate zones in China. It agrees well with this study that BC and MC were the fourth and worst choices for daily ET

_{0}forecasts in the NCP climate zone, compared to RPM, HS, and TH. The reason for the worse performance of BC and MC may be related to the purposes for which the models were created; the BC equation was developed to use U.S. national experimental data, and the MC equation was first used to calculate ET

_{0}for grass and golf courses in Florida. For a while, the calibrated RPM and HS models were better than the other models, and thus these two models were recommended for short-term ET

_{0}forecasting in NCP.

_{0}prediction using the public weather forecast information can be adopted for an irrigation decision support system and real-time water allocation [41,53]. Zhang et al. [54] proposed a methodology to forecast short-term daily ET

_{c}using the ‘Kc-ETo’ approach and public weather forecasts. Considering a basic database for the short-term predicted ET

_{c}of different crops in the whole growth stages and irrigation system engineering information, the irrigation decision support system can make efficient decisions for water allocation with the help of real-time, in-field moisture detection and automatic weather forecast acquisition [55]. It uses technical methods to reduce requirements of a user’s specialized knowledge and can take a user’s managerial experience into account. It can also maximize their productivity and improve irrigation water-use efficiency [56,57].

#### 3.4. Sensitivity and Error Analysis

_{0}with a 5, 10, 15, and 20% increase or decrease in maximum and minimum air temperature was calculated [35]. The percentage changes in ET

_{0}forecasts for representative WS versus the percentage changes in maximum and minimum air temperatures are shown in Figure 7.

_{0}for the five temperature-based models varied linearly with the percentage changes in T

_{max}and T

_{min}but at a greater rate, which indicates that T

_{max}has greater influence on ET

_{0}forecast than T

_{min}. Other studies have also found that ET

_{0}is most sensitive to maximum air temperature [35,58]. The preceding analysis of forecast temperature accuracy showed that forecast T

_{min}was more accurate than forecast T

_{max}, so the increased accuracy of the T

_{max}forecast will influence the accuracy of the five models.

_{0,MC}and EC

_{0,BC}decreased as T

_{min}decreased and increased as T

_{min}increased, and the change trend is consistent with that of T

_{max}. The ET

_{0,HS}, ET

_{0,Th}, and ET

_{0,RPM}behave oppositely, and HS, TS, and RPM were more sensitive to T

_{min}than BC and MC. ET

_{0,HS}, ET

_{0,TH}, and ET

_{0,RPM}changed greatly as T

_{max}changed. Although the change trend was the same as for ET

_{0,BC}and ET

_{0,MC}, the range of the change was greater. Figure 7 shows that the percentage change in ET

_{0,MC}varied from −4% to 5% with the percentage change in T

_{min}and from −7% to 9% with the percentage change in T

_{max}, which was similar to the behavior of ET

_{0,BC}. The percentage change in ET

_{0,RPM}varied from −9% to 8% with the percentage change in T

_{min}and from −29% to 32% with the percentage change in T

_{max}. The percentage change in ET

_{0,HS}varied from −8% to 7% with the percentage change in T

_{min}and from −29% to 29% with the percentage change in T

_{max}. The percentage change in ET

_{0,TH}varied from −4% to 5% with the percentage change in T

_{min}and from −25% to 29% with the percentage change in T

_{max}. This indicates that HS, TH, and RPM are much more sensitive to temperature than BC and MC.

_{0}forecast accuracy order of the five models for the NCP. Our results are consistent with those of Yang et al. [42], who found that, when the same input data is used, differences in forecast accuracy clearly reflect the quality of the model architecture. Figure 7g shows the influence of variables on the ASCE-PM model, including T

_{max}, T

_{min}, wind speed (U

_{2}), mean relative humidity (RH

_{mean}), and sunshine hours (n). These results show T

_{max}has a greater impact on ET

_{0,PM}than T

_{min}. According to the study of G. T. Patle and D. K. Singh [58], the more sensitive the model is to the maximum temperature, the better the ET

_{0}forecast performance would be. The ET

_{0}forecast error factors might be divided into two categories. The errors in temperature forecasts directly influenced the ET

_{0}forecast accuracy, and the effects of wind and relative humidity are not taken into account when forecasting ET

_{0}[51]. With an increase in the temperature forecast accuracy, especially in maximum temperature, the model performance can be improved.

## 4. Conclusions

_{0}for a 1 to 7 d ahead period for 1 January 2015 to 20 June 2021 using public weather forecasts from 17 WS in the NCP. We investigated the accuracy of temperature forecasts and analyzed the sensitivity of each model to air temperature. The main conclusions are as follows.

_{min}were more accurate than forecasts of T

_{max}, and forecast temperature accuracy decreased by 14% as the forecast period increased. The accuracy of ET

_{0}forecasts by the five models similarly decreased by 3% as the period of the forecast increased. The forecast accuracy of the five calibrated models was all greater than 75%. All five calibrated temperature-based models can be used for short-term ET

_{0}forecasts for NCP, but RPM, HS, and TH are, in that order, more accurate than BC or MC. Therefore, the calibrated RPM and HS models were better than the other models and thus these two equations were recommended for short-term ET

_{0}forecasting in NCP.

_{max}had greater influence on the ET

_{0}forecast than T

_{min}. Changes in temperature had the greatest influence on RPM, followed in descending order by HS, TH, MC, and BC. The ET

_{0}forecast error factors might be in two categories. The errors in temperature forecasts directly influenced the ET

_{0}forecast accuracy, while other weather factors not considered in the temperature-based model also caused ET

_{0}forecast errors. With an increase in the temperature forecast accuracy, especially in maximum temperature, the model performance can be improved.

_{0}forecasts to be used as an irrigation scheduling decision support tool for direct field applications in crop cultivation in the NCP. Although we have taken the results provided by the ASCE-PM equation as the benchmark values, it should be noted that this is also a calculation model, which may be inadequate for some cases. It would be therefore much more reliable to attempt to improve five equations based on actual lysimeter measurements.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of the 17 weather stations used in this study. The red sites Xinxiang (XX), Botou (BT), and Tangshan (TS) are representative of the north–south and east–west geographic locations.

**Figure 3.**Statistical indices for daily minimum temperature forecasts: (

**a**) Accuracy, (

**b**) MAE, (

**c**) RMSE, and (

**d**) R; for daily maximum temperature forecasts: (

**e**) Accuracy, (

**f**) MAE, (

**g**) RMSE, and (

**h**) R at 17 stations.

**Figure 4.**Scatterplots for ET

_{0}calculated by the temperature-based models and the PM model for original, calibrated, and validated models in stations BT, TS, and XX; (

**a**–

**c**) MC model, (

**d**–

**f**) HS model, (

**g**–

**i**) BC model, (

**j**–

**l**) TH model, (

**m**–

**o**) RPM model; Fit-O, Fit-C, and Fit-V are respectively the best fit lines for the original, calibrated, and validated models.

**Figure 5.**Variation in ET

_{0}estimated by the MC and RPM temperature-based models and PM and Precipitation (Precip) for stations BT, TS, and XX; (

**a**–

**c**) MC model and the (

**d**–

**f**) RPM model.

**Figure 6.**Comparisons of ET

_{0}forecasts between MC and RPM temperature-based models and PM for BT, TS, and XX; (

**a**–

**c**) 1 d ahead, (

**d**–

**f**) 4 d ahead, and (

**g**–

**i**) 7 d ahead.

**Figure 7.**Percentage change in ET

_{0}forecasts with respect to the percentage change in temperature for stations BT, TS, and XX; T

_{min}(

**a**,

**c**,

**e**), T

_{max}(

**b**,

**d**,

**f**); percentage change in ET

_{0,PM}with respect to the percentage change in climatic variables (

**g**).

No. | WMO No. | Station | Province | Longitude | Latitude | Elevation (m) |
---|---|---|---|---|---|---|

1 | 53,798 | Xingtai (XT) | Hebei | 114.5° E | 37.1° N | 77.3 |

2 | 54,518 | Bazhou (BZ) | Hebei | 116.4° E | 39.2° N | 8.9 |

3 | 54,534 | Tangshan (TS) | Hebei | 118.1° E | 39.7° N | 23.2 |

4 | 54,539 | Laoting (LT) | Hebei | 118.9° E | 39.4° N | 8.5 |

5 | 54,602 | Baoding (BG) | Hebei | 115.5° E | 38.7° N | 16.8 |

6 | 54,606 | Raoyang (RY) | Hebei | 115.7° E | 38.2° N | 19.0 |

7 | 54,618 | Botou (BT) | Hebei | 116.6° E | 38.1° N | 13.2 |

8 | 54,624 | Huanghua (HH) | Hebei | 117.3° E | 38.4° N | 4.5 |

9 | 54,705 | Nangong (NG) | Hebei | 115.4° E | 37.4° N | 27.4 |

10 | 53,986 | Xinxiang (XX) | Henan | 113.9° E | 35.3° N | 73.2 |

11 | 54,416 | Miyun (MY) | Beijing | 116.9° E | 40.4° N | 71.8 |

12 | 54,511 | Beijing (BJ) | Beijing | 116.5° E | 39.8° N | 31.3 |

13 | 54,525 | Baodi (BI) | Tianjin | 117.3° E | 39.7° N | 5.1 |

14 | 54,527 | Tianjin (TJ) | Tianjin | 117.1° E | 39.1° N | 3.5 |

15 | 54,715 | Lingxian (LX) | Shandong | 116.6° E | 37.3° N | 18.6 |

16 | 54,725 | Huimin (HM) | Shandong | 117.5° E | 37.5° N | 11.7 |

17 | 54,808 | Shenxian (SX) | Shandong | 115.6° E | 36.2° N | 37.8 |

Data Type | Data | Time Period | Data Source |
---|---|---|---|

Meteorological data | T_{max}, T_{min}, RH_{mean}, SH, U_{2} | 1 January 1970–20 June 2021 | http://data.cma.gov.cn |

Temperature forecast | T_{max}, T_{min} | 1 January 2015–19 June 2021 | http://www.weather.com.cn |

_{max}= daily maximum air temperature; T

_{min}= daily minimum air temperature; RH

_{mean}= average relative humidity; SH = sunshine duration; U

_{2}= wind speed at 2 m.

Lead Time (d) | T_{min} | T_{max} | ||||||
---|---|---|---|---|---|---|---|---|

ACC (%) | MAE (°C) | RMSE (°C) | R | ACC (%) | MAE (°C) | RMSE (°C) | R | |

1 | 72 | 1.6 | 2.2 | 0.98 | 52 | 2.5 | 3.3 | 0.95 |

2 | 68 | 1.8 | 2.3 | 0.97 | 51 | 2.6 | 3.5 | 0.95 |

3 | 66 | 1.9 | 2.4 | 0.97 | 50 | 2.6 | 3.4 | 0.95 |

4 | 61 | 2.0 | 2.7 | 0.97 | 49 | 2.7 | 3.5 | 0.95 |

5 | 58 | 2.1 | 2.7 | 0.97 | 47 | 2.8 | 3.6 | 0.95 |

6 | 55 | 2.3 | 2.9 | 0.96 | 46 | 2.8 | 3.6 | 0.94 |

7 | 52 | 2.4 | 3.1 | 0.96 | 44 | 2.9 | 3.8 | 0.94 |

M | 62 | 2.0 | 2.6 | 0.97 | 49 | 2.7 | 3.5 | 0.95 |

Station | MC | HS | BC | TH | RPM | |||||
---|---|---|---|---|---|---|---|---|---|---|

a | b | C | E | a | b | a | b | a | b | |

XT | 1.45 | 0.45 | 0.002 | 0.56 | −0.69 | 0.82 | 0.62 | 0.79 | 0.03 | 1.01 |

XX | 1.54 | 0.44 | 0.002 | 0.54 | −0.55 | 0.79 | 0.68 | 0.77 | 0.08 | 0.97 |

MY | 1.55 | 0.48 | 0.001 | 0.64 | −0.02 | 0.69 | 0.57 | 0.73 | 0.14 | 0.89 |

BJ | 1.69 | 0.49 | 0.002 | 0.54 | −0.18 | 0.77 | 0.84 | 0.78 | 0.30 | 1.00 |

BZ | 1.46 | 0.49 | 0.002 | 0.61 | −0.49 | 0.80 | 0.55 | 0.81 | −0.002 | 1.02 |

BI | 1.55 | 0.48 | 0.002 | 0.60 | −0.09 | 0.70 | 0.66 | 0.72 | 0.18 | 0.90 |

TJ | 1.58 | 0.46 | 0.002 | 0.55 | −0.26 | 0.75 | 0.73 | 0.77 | 0.16 | 0.99 |

TS | 1.50 | 0.50 | 0.002 | 0.57 | −0.30 | 0.76 | 0.60 | 0.80 | 0.06 | 1.02 |

LT | 1.42 | 0.51 | 0.002 | 0.52 | −0.26 | 0.73 | 0.58 | 0.80 | 0.07 | 1.02 |

BG | 1.41 | 0.46 | 0.002 | 0.63 | −0.55 | 0.79 | 0.56 | 0.79 | 0.01 | 1.00 |

RY | 1.47 | 0.52 | 0.001 | 0.64 | −0.51 | 0.82 | 0.49 | 0.82 | −0.08 | 1.01 |

BT | 1.53 | 0.50 | 0.002 | 0.64 | −0.52 | 0.83 | 0.58 | 0.83 | −0.04 | 1.05 |

HH | 1.61 | 0.49 | 0.002 | 0.61 | −0.45 | 0.83 | 0.66 | 0.84 | 0.06 | 1.08 |

NG | 1.52 | 0.52 | 0.002 | 0.64 | −0.60 | 0.86 | 0.49 | 0.86 | −0.16 | 1.07 |

LX | 1.48 | 0.51 | 0.002 | 0.58 | −0.55 | 0.82 | 0.52 | 0.82 | −0.05 | 1.01 |

HM | 1.61 | 0.49 | 0.001 | 0.68 | −0.39 | 0.80 | 0.62 | 0.82 | 0.01 | 1.03 |

SX | 1.47 | 0.46 | 0.001 | 0.62 | −0.48 | 0.77 | 0.53 | 0.77 | −0.04 | 0.96 |

**Table 5.**Average values of the statistical indicators of five temperature-based models’ ET

_{0}forecast accuracy for 17 stations in 1–7 d ahead forecasts for the NCP.

Statistical Indicator | Lead Time | MC | HS | BC | TH | RPM |
---|---|---|---|---|---|---|

Acc (%) | 1 | 75 | 87 | 81 | 84 | 87 |

2 | 75 | 86 | 81 | 83 | 86 | |

3 | 75 | 86 | 81 | 83 | 86 | |

4 | 75 | 85 | 81 | 82 | 85 | |

5 | 75 | 85 | 81 | 82 | 85 | |

6 | 75 | 84 | 81 | 82 | 84 | |

7 | 75 | 84 | 81 | 81 | 84 | |

M | 75 | 85 | 81 | 82 | 85 | |

MAE (mm/d) | 1 | 1.11 | 0.74 | 0.87 | 0.83 | 0.75 |

2 | 1.11 | 0.76 | 0.88 | 0.85 | 0.77 | |

3 | 1.11 | 0.76 | 0.88 | 0.85 | 0.76 | |

4 | 1.12 | 0.78 | 0.88 | 0.86 | 0.78 | |

5 | 1.12 | 0.79 | 0.88 | 0.87 | 0.79 | |

6 | 1.12 | 0.80 | 0.89 | 0.88 | 0.80 | |

7 | 1.13 | 0.81 | 0.89 | 0.88 | 0.81 | |

M | 1.12 | 0.78 | 0.88 | 0.86 | 0.78 | |

RMSE (mm/d) | 1 | 1.42 | 1.01 | 1.14 | 1.10 | 1.00 |

2 | 1.43 | 1.03 | 1.16 | 1.12 | 1.03 | |

3 | 1.43 | 1.03 | 1.15 | 1.12 | 1.02 | |

4 | 1.43 | 1.05 | 1.16 | 1.13 | 1.05 | |

5 | 1.44 | 1.06 | 1.16 | 1.15 | 1.06 | |

6 | 1.45 | 1.08 | 1.16 | 1.16 | 1.07 | |

7 | 1.45 | 1.10 | 1.17 | 1.17 | 1.09 | |

M | 1.44 | 1.05 | 1.16 | 1.13 | 1.05 | |

R | 1 | 0.67 | 0.86 | 0.80 | 0.82 | 0.86 |

2 | 0.66 | 0.85 | 0.80 | 0.81 | 0.85 | |

3 | 0.66 | 0.85 | 0.80 | 0.81 | 0.85 | |

4 | 0.66 | 0.84 | 0.80 | 0.81 | 0.84 | |

5 | 0.66 | 0.84 | 0.79 | 0.80 | 0.84 | |

6 | 0.65 | 0.84 | 0.79 | 0.80 | 0.83 | |

7 | 0.65 | 0.83 | 0.79 | 0.79 | 0.83 | |

M | 0.66 | 0.84 | 0.80 | 0.81 | 0.84 |

_{0}forecasts by five temperature-based models for 17 stations in the NCP.

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**MDPI and ACS Style**

Zhang, L.; Zhao, X.; Ge, J.; Zhang, J.; Traore, S.; Fipps, G.; Luo, Y. Evaluation of Five Equations for Short-Term Reference Evapotranspiration Forecasting Using Public Temperature Forecasts for North China Plain. *Water* **2022**, *14*, 2888.
https://doi.org/10.3390/w14182888

**AMA Style**

Zhang L, Zhao X, Ge J, Zhang J, Traore S, Fipps G, Luo Y. Evaluation of Five Equations for Short-Term Reference Evapotranspiration Forecasting Using Public Temperature Forecasts for North China Plain. *Water*. 2022; 14(18):2888.
https://doi.org/10.3390/w14182888

**Chicago/Turabian Style**

Zhang, Lei, Xin Zhao, Jiankun Ge, Jiaqi Zhang, Seydou Traore, Guy Fipps, and Yufeng Luo. 2022. "Evaluation of Five Equations for Short-Term Reference Evapotranspiration Forecasting Using Public Temperature Forecasts for North China Plain" *Water* 14, no. 18: 2888.
https://doi.org/10.3390/w14182888