Calibration of empirical equations for estimating reference evapotranspiration in different climates of Iran

The accurate estimation of reference evapotranspiration (ETref) is a crucial component for modeling hydrological and ecological cycles. The goal of this study was the calibration of 32 empirical equations used to determine ETref in the three classes of temperature-based, solar radiation–based, and mass transfer–based evapotranspiration. The calibration was based on measurements taken between the years 1990 and 2019 at 41 synoptic stations located in very dry, dry, semidry, and humid climates of Iran. The performance of the original and calibrated empirical equations compared to the PM-FAO56 equation was evaluated based on model evaluation techniques including the coefficient of determination (R2), the root mean square error (RMSE), the average percentage error (APE), the mean bias error (MBE), the index of agreement (D), and the scatter index (SI). The results show that the calibrated Baier and Robertson equation for temperature-based models, the Jensen and Haise equation for solar radiation–based models, and the Penman equation for mass transfer–based models performed better than the original empirical equations. The calibrated equations had, respectively, an average R2 = 0.73, 0.67, and 0.78; RMSE = 35.14, 35.02, and 30.20 mm year-1; and MBE = − 5.6, − 3.89, and 2.57 mm year-1. The original empirical equations had values of average R2 = 0.60, 0.37, and 0.65; RMSE = 68.34, 66.98, and 52.62 mm year-1; and MBE = − 5.75, 4.26, and 8.99 mm year-1, respectively. The calibrated empirical equations for very dry climate (e.g., Zabol, Zahedan, Bam, Iranshahr, and Chabahar stations) also significantly reduced the SI value from SI > 0.3 (poor class) to SI < 0.1 (excellent class). Therefore, the calibrated empirical equations are highly recommended for estimating ETref in different climates.


Introduction
Water resources in semiarid regions are vulnerable to the impacts of climate change and human activities, and the accurate estimation of reference evapotranspiration (ET ref ) is a primary tool in the management of water resources. Also, the estimation of ET ref by using hydrological equations can be helpful in agriculture sectors (Celestin et al. 2020;Ndiaye et al. 2020;Yan et al. 2021). It has a key role in the management of water resources and the determination of crops' water demands in the semidry regions (Berti et al. 2014;Ferreira et al. 2019;dos Santos Farias et al. 2020).
The most accurate evaluation of ET ref is computed by the lysimeter method, but this method has high costs and requires complex instruments (Ahooghalandari et al. 2016;Ahooghalandari et al. 2017). Therefore, alternative techniques for indirect estimation of ET ref were developed based on empirical equations. Numerous empirical equations have been introduced to estimate ET ref . Despite the advantages of empirical equations such as ease of use, applicability due to the great variety of required parameters, and classification based on various climatic parameters, the low accuracy of some of these equations in estimating ET ref is one of the main challenges in their application. In contrast, the FAO 56 Penman-Monteith (PM-FAO 56 ) equation is the standard combination-based model used to estimate the ET ref in different climates and at different time scales (Güçlü et al. 2017;Saggi and Jain 2019;Shiri et al. 2019;Ndiaye et al. 2020;Sharafi and Mohammadi Ghaleni 2021). The accuracy of this equation is due to its consideration of all climatic parameters, including solar radiation, air temperature, wind speed, and relative humidity (Ndiaye et al. 2020). Furthermore, the equations for empirical models based on temperature, solar radiation, and mass transfer use fewer climatic parameters in the calculation of ET ref . Therefore, the development and calibration of empirical model equations in different climates can be more effective for agricultural and hydrological projects where only a few climatic variables are available (Heydari and Heydari and Heydari 2014;Gafurov et al. 2018).
Several researchers have evaluated the dependence of different empirical ET ref equations on various meteorological parameters over different climates. Gao et al. (2017) also assessed different empirical ET ref equations in various climates and observed that the PT and Hargreaves (HG) equations worked best in dry and semidry climates, while the MK equation worked best in the humid climate of China. Sharafi and Ghaleni 2021) evaluated different empirical equations for ET ref in different climates of Iran. Their results found that the simplest regression model (MLR) based on minimum and maximum temperature data was more precise than the empirical equations. They also recommended the solar radiationbased Irmak equation as the best substitute for the PM-FAO 56 model, especially in dry and semidry climates. Rahimikhoob et al. (2012) compared four temperaturebased and solar radiation-based equations using data from eight stations in subtropical climates of Iran and confirmed the applicability of Priestley-Taylor (PT) and Hargreaves-Samani (HS) equations in those climates. The comparison results showed that the original PT and HS equations were more applicable in a humid climate. The performance of the PT and HS equations improved slightly after the calibration; however, the Trajkovic (TRAJ) and Makkink (MAKK) equations improved greatly. Tabari et al. (2013) compared different temperature-based, solar radiation-based, and mass transfer-based equations for modeling ET ref in humid climates of Iran and found that the temperature-based Blaney-Criddle (BC) and HS equations surpassed the other temperature-based models. Cross-comparison of the 31 empirical equations showed that the five best equations as compared with the PM-FAO 56 model were the two solar radiation-based equations developed, the temperature-based BC and Hargreves-M4 equations. Farzanpour et al. (2019) conducted 20 ET ref equations using daily meteorological data of 10 stations (12 years) in semidry climates of Iran. Their results revealed that the calibrated equations might be a good alternative for the PM-FAO 56 equation. Bourletsikas et al. (2018) compared 24 different equations for estimating ET ref in Greece and concluded that calibrating mass transfer-based equations is essential for improving their performances. Celestin et al. (2020) compared the 32 empirical ET ref equations with the PM-FAO 56 using data on temperature, solar radiation, and mass transfer in northwest China. They found that the World Meteorological Organization (WMO) and the Mahringer equations for the mass transfer-based model provided the best results.
Above all, in the present study, we have tried to introduce the best calibrated equations with the highest accuracy in different climates of Iran. Therefore, the goals of this study were (1) the use of monthly data of eight climatic variables measured in 41 synoptic stations over a period of 30 years (1990-2019); (2) the calibration of 32 empirical equations based on temperature-based, solar radiation-based, and mass transferbased equations in four main climates (very dry, dry, semidry, and humid) in the study area; (3) the comparison of the results of the original and calibrated equations versus the PM-FAO 56 equation using different statistical criteria; and (4) drawing an accurate map of the results of the best calibrated empirical equations in the study area.

Time and location scales
Iran is in the northern hemisphere between 25 and 40°latitude. For this study, meteorological data recorded between 1990 and 2019 were collected from 41 synoptic stations in the country. These data were collected by the National Meteorological Organization of Iran and include the monthly mean of minimum, mean and maximum air temperature, relative humidity, wind speed measured at 2 m height, and solar radiation. The data were complete, and no data needed to be reconstructed.
According to the FAO 56 index, Iran is classified into four climatic regions: very dry, dry, semidry, and humid ( Fig. 1). Figure 1 shows the location and climate for each station used in this study. Six stations were in very dry climate, 17 stations in dry climate, 14 stations in semidry climate, and 4 stations in humid climate (Fig. 1).

Empirical ET ref equations
Based on the type and importance of input variables used in each empirical equation to calculate the ET ref , the models were divided into 4 categories: combination-based (1 equation), temperature-based (11 equations), solar radiation-based (11 equations), and mass transfer-based (10 equations). Table 1 lists the 33 empirical ET ref equations used in this study and their respective references.
To calculate the PM-FAO 56 , measurements of the amount of total solar radiation at the Earth's surface (R s , MJ m -2 d -1 ), maximum and minimum temperature, wind speed (m s -1 ), and lack of vapor pressure (VPD, kPa) are required. Due to lack of access to R s and VPD, the FAO method was used (Gholipoor 2009). Daily values of R s were obtained from Hargreaves and Samani's equation (Mehdizadeh et al. 2017) and the modified Allen et al. (2006) equation. Solar radiation reaching the land surface (R n , MJ m -2 d -1 ) was first measured above the Earth's atmosphere for each day of the year based on latitude and longitude and the solar constant (Allen et al. 2006). Then, R s was calculated using Eq. 34: where Alt is altitude (m) and K Rs is the empirical constant, considered equal to 0.16 (Gholipoor 2009). The e s calculation is obtained from the difference between the daily saturated water vapor pressure (e max ) and the actual water vapor pressure (e a ). Relative humidity at temperature was assumed to be at least 100 percent and the values for e a were obtained from Eq. (35): In very dry and dry climates, the relative humidity at the T min may never reach 100%. Therefore, it was assumed that in these regions, e a values would occur at T min > 10°C and it was observed that in this case it had a minor effect on ET ref (1-2%). As a result, the ET ref was calculated assuming that the dew point was equal to the T min . Then, the T max saturated vapor pressure during the day (e max ) depending on the T max was obtained from Eq. (36). where e s is obtained from the mean e a and e max . Finally, e s is calculated as the average between e a and e max in a part of the day when the air temperature is not at its maximum. However, other researchers have found that (e max -e a ) × 0.75 is a more accurate estimate of e a (Tanner and Sinclair 1983;Allen et al. 1998). Therefore, this method is used in this current study.
Calculations were performed using SAS software (Statistical Analysis System, Version 9.1, SAS Inst., Cary, NC). The average annual rainfall over the last 30 years in Iran was reported to be 334 mm. The highest annual rainfall ET ref reference evapotranspiration (mm day -1 ); Δ the slope of saturation vapor pressure curve (mb°C -1 ); Rn net solar radiation (MJ m -2 day -1 ); G soil heat flux density (mm day -1 ); γ psychometric constant (kPa°C -1 ); T mean, max and min mean, maximum, and minimum daily temperature (°C), respectively; u 2 wind speed measured at 2 m height (m s -1 ); R a extraterrestrial radiation (mm day -1 ); ʎ latent heat of vaporization (MJ kg -1 ); RH mean relative humidity (%); R s solar radiation (MJ m -2 day -1 ); e s saturation vapor pressure (k Pa); e a actual vapor pressure (k Pa); and (e s -e a ) saturation vapor pressure deficit (kPa) occurred at Bandar Anzali station (1791.78 mm), and the lowest annual rainfall occurred at Bam station (61.61 mm). The average annual rainfall in the very dry climate was 92.89 mm, which was 140.74, 266.41, and 1150.05 mm less than in dry, semidry, and humid climates, respectively ( Fig. 2(a)). The annual average relative humidity in Iran was reported to be 55.34% with the highest relative humidity at Bandar Anzali station (84.71%) and the lowest relative humidity at Bam station (28.08%). The relative humidity in the very dry climate was 44.64%, which was 1.02, 7.9, and 35.93% less than in the dry, semidry, and humid climates, respectively ( Fig. 2(b)).
The 30-year average air temperature in Iran was reported to be 17.54°C. The hottest and coldest stations in this study were the Bandar Abbas and Ardabil stations (26.63 and 9.14°C, respectively). The average air temperature in the very dry climate was 20.44°C, which was 3.4, 10.15, and 6.04°C warmer than in the dry, semidry, and humid climates, respectively ( Fig. 2(c)). The average solar radiation received in Iran is reported to be 7.26 MJ m -2 day -1 . The highest and lowest received solar radiations were observed in Bam and Rasht stations (9.06 and 4.17 MJ m -2 day -1 , respectively). The average solar radiation received in the very dry climate was 8.55 MJ m -2 day -1 , which increased by 0.36, 1.06, and 3.74 MJ m -2 day -1 in dry, semidry, and humid climates ( Fig.  2(d)). The average wind speed in the country during the last 30 years was reported to be 4.35 m s -1 , which has increased by about 0.52 m s -1 compared to the same period. The highest and lowest wind speeds were recorded in Zabol and Gorgan stations, respectively (10.62 and 1.45 m s -1 ). The mean wind speed in a very dry climate was 6.54 m s -1 , which increased by 2.15, 2.63, and 3.97 m s -1 in dry, semidry, and humid climates ( Fig. 2(e)).

Evaluation performance criteria
Until now, many performance criteria have been used to evaluate the results of the model for prediction of ET ref . The equations were assessed for each station by means of six statistical measures used to evaluate the accuracy of each model in estimating the ET ref : the coefficient of determination (R 2 ), the root mean square error (RMSE), the average percentage error (APE), the mean bias error (MBE), the index of agreement (D), and the scatter index (SI). The explanations for the statistical measures appear in Table 2. These criteria are widely reported in the literature (Kisi 2014;Samaras et al. 2014;Celestin et al. 2020).
The R 2 coefficient, acquired by the least squared regression analysis, is a commonly used correlation measure. For the absolute and/or relative errors' estimation, RMSE, APE, and MBE indices were also evaluated. The descriptive index of agreement (D) was used for the correlation between the equations, expressing the degree to which an equation's predictions are error-free (Willmott 1982). According to Li et al. (2013), the range of SI for the accuracy of the models is excellent (SI < 0.1), good (0.1 < SI < 0.2), fair (0.2 < SI < 0.3), and poor (SI > 0.3).

Empirical equation calibration
The basis of empirical equations used in estimating ET ref is the regression relationship between the ET ref equation as a dependent variable and meteorological parameters as independent variables. In the process of developing each of the empirical equations, one of two modifications may be made, either a change in meteorological parameters or a change in the coefficients of the equation. In this study, modification (optimization) of constant coefficients in empirical equations is the basis for increasing the accuracy of ET ref estimation in different climates. The objective function of that change has been to minimize the RMSE error criterion by optimizing the constant coefficients of the equations as decision variables. For instance, in the HASA equation, the two coefficients a and b in Eq. (43) are optimized to minimize the amount of error between the estimated ET ref and the PM-FAO 56 .
The accuracy of empirical equations in estimating ET ref before and after calibration was evaluated using error evaluation criteria separately for various empirical equations (temperature-based, solar radiation-based, and mass transferbased) and for very dry, dry, semidry, and humid climates. (37) Coefficient of determination (R 2 )  Ma and Iqbal (1984) (38) Root mean square error Ma and Iqbal (1984)  Li et al. (2013)  Cal. Ori. Cal. Ori. Cal. Ori. Cal. Ori. Cal. Ori. Cal. Ori. Cal. Ori. Cal.

Accuracy evaluation of empirical equations
To assess the 32 empirical equations for temperature-based, solar radiation-based, and mass transfer-based models in different climates, meteorological datasets from 1990 to 2019 were evaluated.  (Table 3). For solar radiation-based methods, the maximum R 2 in very dry, dry, and humid climates derived by the original HARG equation was 0.6, 0.47, and 0.45, respectively, but the best RMSE in very dry climate obtained by the original IRMA1 equation was 67.51, and in dry and humid climates as obtained by the original JEHA equation was 65.92 and 66.77, respectively. The result for calibrated equations showed that in very dry (R 2 = 0.84 and RMSE = 33.13), dry (R 2 = 0.74 and RMSE = 34.9), and semidry (R 2 = 0.63 and RMSE = 34.36) climates, the OUDI, ABTE2, and TATA3 equations yielded reliable estimates. In the humid climate, calibrated PRTA and MAKK equations showed the maximum R 2 = 0.71 and the minimum RMSE = 33.71 (Table 3).
The results from mass transfer-based methods showed that the values of R 2 for the original PEMN equation in very dry, dry, semidry, and humid climates were acceptable (0.74, 0.7, 0.53, and 0.64, respectively). The values of RMSE for the    (Table 3).
Radar charts in Fig. 3  A reduction in values of APE for the calibrated empirical equations was found in very dry, dry, semidry, and humid climates in temperature-based (1, 1.5, 2.2, and 3.2%), solar radiation-based (1, 1.4, 1.9, and 3.1%), and mass transferbased (1, 1.3, 1.6, and 2.8%) methods when compared to the original equations (Fig. 3). This indicates the great effect calibration has relative to other empirical equations on increasing the accuracy of temperature-based methods. This increase confirms the accuracy of calibrated empirical equations in estimating ET ref in humid climate. At the same time, the accuracy of ET ref estimation for all empirical equations decreased from very dry to humid climates, indicating that the process of ET ref estimation in humid climate is more complex due to its greater dependence on multiple climatic parameters.

Original Calibrated a-2) Dry
-20 0 20 40        empirical temperature-based methods in very dry, semidry, and humid climates. The highest accuracy of empirical equations for temperature-based and solar radiation-based methods for estimating ET ref in dry climate ( Fig. 4(a-2) and (b-2)) is obtained when the MBE in this climate for all empirical equations is less than 20 mm year -1 . Figure 4( (Fig. 4(a)). The MBE values for most original and calibrated equations were overestimated in very dry ( Fig. 4(a-1)) and dry ( Fig. 4(a-3)) climates. Also, it is noteworthy that the highest overestimation and underestimation were observed in humid climate, which was reported in MAHR (39.9 mm year -1 ) and BARO (−34.3 mm year -1 ) original equations, respectively ( Fig. 4(a-4), 4(c-4)). The overestimation of the in original equations varied from 25.27 mm year -1 to 26.15 mm year -1 in very dry and semidry climates, respectively. The overestimation of the temperaturebased equations was found by Trajkovic (2007) and Landeras et al. (2008). Furthermore, according to Temesgen et al. (2005), higher wind speed combined with lower humidity resulted in lower values of temperature-based equations compared to PM-FAO 56 , especially in drier climates.

a-4) Humid
For solar radiation-based methods, the negative MBE values are observed for PRTA, IRMA2, TATA3, and OUDI original equations and PRTA calibrated equation in very dry climate, and ABTE2 original and calibrated equations in humid climate. Also, most equations in very dry, dry, and semidry climates showed a good response to calibration. In humid climate, due to the influence of more factors on ET ref , more fluctuations were observed. However, the calibrated MAKK, HARG, and ABTE1 equations showed better results in humid climate ( Fig. 4(b-4)). Analyses by Trajkovic and Kolakovic (2009) showed the solar radiation-based equations had a tendency to overestimate ET ref in humid climates of Serbia. For mass transfer-based equations, most of the original and calibrated equations exhibited underestimation in very dry, dry, and semidry climates. According to the MBE values, the original and calibrated equations of ROHW in the very dry climate, PENM in the dry climate, DALT in the semidry climate, and ROMA in the humid climate showed better results ( Fig. 4(c)). In general, the comparative results revealed that the mass transfer-based equations had the best performances among the ET ref equations evaluated in very dry and semidry climates. Also, the temperature-based and solar radiation-based equations were the good equations for the dry climates ( Fig. 4(a) and (b)).

Correlation between PM-FAO 56 and empirical equations
Based on the results presented in Table 3, the highest accuracy of empirical equations for temperature-based, solar radiationbased, and mass transfer-based methods of estimating ET ref in different climates is determined by the BARO, JEHA, and PENM equations, respectively. Figure 5 shows the values of D for the best equations of temperature-based, solar radiationbased, and mass transfer-based methods in different climates.
A better fit between the estimated ET ref and PM-FAO 56 appears in the calibrated empirical equations when compared to the original empirical equations in all empirical equations and climates.
The best fit between the PM-FAO 56 and estimated ET ref values is related to the calibrated PENM equation in humid climate and is equal to 0.94 (Fig. 5(c-4)). Generally, the best and worst fit between the PM-FAO 56 and estimated ET ref values were related to the mass transfer-based (Fig. 5(c)) and the solar radiation-based methods (Fig. 5(b)). Figure 6 shows  (Fig. 6(c-3)).

c-4) Humid
Time ( (Fig. 6(a-1-4)). The radiationbased equations overall performed better than the mass transfer equations, since a more important role is expected for R s when estimating ET ref in humid climates (Irmak et al. 2006). The calibrated JEHA equation exhibited the highest R 2 and lowest RMSE (Table 3). Additionally, this equation presented the D index from 0.46 in very dry climate to 0.85 in semidry climate ( Fig. 5(b-1-4)). Therefore, it can be concluded that the JEHA equation of this class can be suitable for estimating of ET ref values in surveyed climates ( Fig. 6(b-1-4)).
These results for JEHA are in contrast to Tabari et al. (2013) (Fig. 6).  Figure 7 shows that calibration at stations with very dry climate, such as Zabol, Zahedan, Bam, Iranshahr, and Chabahar stations, had a greater effect on the accuracy of ET ref estimation based on the SI value in the excellent class (SI < 0.1). The highest amount of error in the SI index is related to stations with humid climates, such as Rasht and Nowshahr. This is due to the complexity of the ET ref process in humid climate.

SI map
The SI map demonstrates that the original temperature, radiation, and mass transfer-based equations generally did not have excellent class (SI < 0.1), except BARO and PENM equations in Bam and Iranshahr stations, respectively. In general, the results of the present study confirmed that temperature-based equation had the more accuracy than solar radiation and mass transfer-based equations. Similar results were also reported by Farzanpour et al. (2019). Their results cleared that the temperature-and radiation-based equations generally have had similar SI values, giving more accurate simulations than the mass transfer-based equations. This might show the importance of temperature parameters on ET ref estimating in the very dry and dry stations, as well as its superiority to the solar radiation-based. The impact of input climatic parameters seems to be very low as the mass transferbased equations gave the undesirable results in these climates. However, this situation should be used with cautiousness because a comprehensive study should be conducted to determinate the portion of each parameter on the empirical ET ref magnitudes, which is beyond the scope of this research.

Conclusion
The accurate estimation of ET ref by empirical equations can be helpful for water resources management, crops' water demand, and irrigation scheduling. This study attempted to investigate and calibrate 32 empirical equations classified in three categories (temperature-based, solar radiation-based, and mass transfer-based) in main climates (very dry, dry, semidry, and humid) of Iran. The results show that most of the calibrated empirical equations had good accuracy in estimating ET ref in all studied climates. However, the accuracy of the ET ref estimate before and after calibration depended on the classification of the equation in the type and number of input data and the type of climate under study. In other words, each climatic region has its own superior empirical equation. Also, with the complexity of climatic variables, the accuracy of various empirical equations is associated with change.
According to the results of the best values of R 2 and RMSE in the temperature-based BARO equation for very dry (0.82 and 32.79), dry (0.73 and 42.12), semidry (0.65 and 30.34), and humid (0.71 and 35.29), climates were observed in the calibrated BARO equation. Also, the calibrated JEHA equation showed reliable estimates in very dry (R 2 = 0.78 and RMSE = 32.39), dry (R 2 = 0.69 and RMSE = 37.25), semidry (R 2 = 0.60 and RMSE = 34.23), and humid (R 2 = 0.68 and RMSE = 41.86) climates. Finally, the best values of R 2 and RMSE were reported for the calibrated PENM equation in very dry (0.85 and 27.87), dry (0.79 and 31.83), semidry (0.71 and 29.38), and humid (0.77 and 31.71) climates, respectively. The results of APE, MBE, and D criteria show that the accuracy of the empirical equations after calibration increased significantly when compared to their original values. At the same time, the results of SI criterion and the effect of factors such as high relative humidity and the balance between air temperature and rainfall mean that the estimation of ET ref is more complex. Considering the dependence of the ET ref process on fewer meteorological parameters, we can conclude that in very dry climates the empirical equations before and after calibration are more accurate.
Considering the limitations associated with the availability and reliability of the climatological data, especially in developing countries, the good performance of empirical models must be emphasized, since they deal with a very simple equation. Further studies are needed in order to evaluate the performance of the calibrated equations in other areas in the world with different climates. Also, evaluation is needed for the performance of the empirical equations on a different time scale (daily). Therefore, more studies might use empirical equations and stations as well as other calibration scenarios for assessing these results in various climates.
Availability of data and material All data used in this article have been prepared from the Meteorological Organization of Iran and, after validation, have been used. In this study, meteorological information was used that lacked outdated data.
Code availability The software used in this research will be available (by the corresponding author), upon reasonable request.
Authors' contributions All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Saeed Sharafi and Mehdi Mohammadi Ghaleni. The first draft of the manuscript was written by Saeed Sharafi, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Funding This research was funded by the Agricultural Experiment Station of Arak University, Iran.

Declarations
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