An effective evapotranspiration estimation scheme based on statistical indicators for sustainable environments in humid and semi-arid area of Khyber Pakhtunkhwa, Pakistan

Water scarcity and crises are significant difficulties and challenges in humid, semi-arid, and desert environments. Rapid population growth, global climate change, and drought add to the global water and food security challenges. According to the United Nations World Water Development Report 2019: Leaving No One Behind. Paris, UNESCO (Water, 2019), agriculture utilizes around 70% of global freshwater availability as the largest consumer of total water resources. Thus, sustained measures should be taken to solve water and food security concerns by applying suitable adaptation strategies and plans. Reference evapotranspiration (ET_o) estimates are frequently used in hydrology, environmental issues, climate change, irrigation, and drainage designing. ET_o has been used to coordinate water crisis management strategies, agriculture water conservation plans, and water resource operations. A hydrologic assessment is essential for irrigation administration and developing an intelligent irrigation system anywhere on the earth. Accurate figures of reference or crop evapotranspiration (ET_o, ET_c) allow for the efficient use of available water resources, such as stored water for the dry season. To calculate (ET_c), first, calculate (ET_o) and then multiply it by the crop coefficient Kc proposed in Allen et al., (1998) and Doorenbos et al., (1997). Practically, the essential classical techniques for assessing ET, such as the eddy correlation system, soil water balance, and Bowen ratio, are only available at the field scale (Tao et al., 2018). The ET_o value can be explicitly or implicitly determined using statistical models. However, some drawbacks restrict their implementation to fix predictive models, particularly at spatial scales (Kumar et al., 2012; Jing et al., 2019). Apart from that, field systems are expensive, time-consuming, and challenging to maintain. ET_o is also difficult to quantify because it is determined by the interaction of several meteorological factors, including humidity, temperature, radiation, and wind speed (Tao et al., 2018). Furthermore, the Penman-Monteith (FAO-56(PM)) equation was recommended by the United Nations International Commission for Irrigation, Drainage, Food, and Agriculture Organization as a correct protocol for estimating reference evapotranspiration and the performance assessment of many other formulas (Allen et al., 1998). Likewise, the FAO-56(PM) method is the preferred way of estimating ET_o accurately if lysimeter data is unavailable (Irmak et al., 2003; Utset et al., 2004; Gavilán et al., 2006; Nandagiri & Kovoor, 2006; Trajkovic, 2007). The FAO-56(PM) estimation method uses relative humidity, air temperature, solar radiation, and wind speed. Numerous researchers demonstrated that estimating irrigation water requirements is critical for the proper operation of a sophisticated irrigation process in various places around the world (Allen et al., 1998; Tabari & Talaee, 2011; Thepadia & Martinez, 2012; Tabari et al., 2013; Çakir et al., 2017; Gul et al., 2021). In addition, ET_o estimates are frequently used to develop, supervise, plan, and perform water assets in hydrology, agriculture water resources, irrigation, and drainage engineering (Gul et al., 2021; Valipour, 2014, 2015). On the other hand, only a small proportion of the weather stations have a comprehensive set of essential parameters. If the number of meteorological stations with data for the key parameters is limited, we cannot estimate ET_o using FAO-56 (PM). However, depending on the weather, location, and local conditions of the area, certain factors are more influential than others in computing ET_o (Debnath et al., 2015; Gupta et al., 2020). Under controlled circumstances, different authors make the argument that it is vital to identify the climatic parameters most sensitive to ET_o so that assessments of those variables can be used to develop simple empirical ET_o models (Koudahe et al., 2018; Djaman et al., 2019). Classifying and reducing the impact of climate change also involves defining sensitive variables (Nouri et al., 2017; Chatterjee et al., 2021).

In comparison, several studies have verified that perhaps the Hargreaves-Samani (HS) temperature-based formula can reliably estimate evapotranspiration for five days or longer (Jensen et al., 1997; Droogers & Allen, 2002; Hargreaves & Allen, 2003). They recommended using the HS equation when sufficient data like solar radiation, relative humidity, and wind speed are limited to solve the FAO-56(PM) equation. However, the HS equation usually overestimates ET_o at humid locations (Jensen et al., 1990; Itenfisu et al., 2003; Temesgen et al., 2005; Trajkovic, 2007; Azhar et al., 2014; Valipour, 2017). Likewise, various investigations have indicated that ET_o is significantly underestimated in drylands and semi-arid territories with diverse ecosystems (Khoob, 2008; Benli et al., 2010; Azhar & Perera, 2011). Similarly, another researcher explored HS equation coefficients’ recalibration with new factors [C_H=0.0030, E_H=0.4, HT=20, and C_H=0.0025, E_H=0.5, and HT=16.8], the performance of the HS equation has not improved significantly (Droogers & Allen, 2002). Another researcher recommends correlating the (C_H) and the average temperature range (Samani, 2000). Indeed, one more investigator demonstrated that the Samani approach modification did not affect consistency (Xu & Singh, 2002). Consequently, updating the HS equation coefficients according to the surrounding environment is an alternative to enhance its estimate.

In particular, Amatya et al., (1995), a monthly modified (C_H) was developed in Eastern North Carolina for three weather stations. Analogously, the unique (C_H) value was imposed for Rawson Lake station in northwest Ontario, Canada (C_H=0.0029), as recommended by Xu & Singh, (2002). It was asserted (Spain (C_H=0.0020)) for non-windy areas in Spain's semi-arid middle Ebron River Valley region by Martınez-Cob & Tejero-Juste, (2004). A linear relationship between modified (C_H) and the fraction of the mean temperature that falls within the low daily temperature range is implied for each weather station in Andalusia, Southern Spain (Vanderlinden et al., 2004).

Furthermore, the modified HS formula overestimates the FAO-56 using the simple linear regression algorithm for the HS model (Trajkovic, 2005). In semi-arid southern Spain, the HS formula was validated using data from 86 monitoring stations, comparing daily estimates to the FAO56PM guidelines (Gavilán et al., 2006). According to the research, regional calibration must be performed solely based on temperature and wind circumstances. Additionally, since most ETo models were calibrated and validated in temperate environments, local calibration and validation of ET_o models are more critical in semi-arid and arid regions than in temperate climate zones (DehghaniSanij et al., 2004; Mohawesh & Talozi, 2012). The above consequences encouraged the HS equation's local calibration through the HS equation's local adjustment exponent. The researchers generally use the well-known ET_o estimation method because they trust it provides excellent results in diverse regions. However, their accuracy is highly sensitive to climate. There are methods to predict air pollution that fully assimilate environmental protection work and encourage environmental monitoring efficiency (Li et al., 2018; Wang et al., 2019; Lu et al., 2020).

Numerous regions of the eastern Mediterranean, including Pakistan, are semi-arid to arid. Irrigation consumes a disproportionate share of these countries’ total water budgets (Mohawesh & Talozi, 2012). In these locations, metrological data is frequently insufficient or unavailable. As a result, it is crucial to evaluate the accuracy of techniques for calculating ET_o from sparse data and HS equations in arid and semi-arid climates. Many investigators in the past also attained a performance evaluation of ET_o estimation approaches in different parts of Pakistan by operating radiation and temperature-based methods (Rasul & Mahmood, 2009; Naheed & Rasul, 2010; Zahid & Rasul, 2012; Kumar et al., 2013; Azhar et al., 2014).

In this research paper, the main contribution is as follows: we explore the HS equation's calibration exponent on the weather stations of Khyber Pakhtunkhwa, Pakistan. An effort has been made to compare the accuracy of six widely used radiation and temperature-based procedures, including Hargreaves-Samani, modified Hargreaves, Turc, Makkink, and Priestly-Taylor, with the standard FAO-56(PM) under Pakistan's climatic conditions, and identify the most appropriate method for approximating this essential agro-climatic and agro-hydrological parameter in Khyber Pakhtunkhwa. Similarly, a comparative analysis through the concept of Q-Q plot and correlation plot and histogram between a standard model, radiation-based procedures, and the proposed calibrated temperature-based method MHargreaves is also discussed in detail. The rest of this paper is organized as follows, a brief portrayal of selected models and methodology used for estimation of reference evapotranspiration is provided in the next section. The third section explains the calibrating system, the statistic indicator's performance assessment, and the study field and data details. The performance analysis of each station is presented in the fourth section. Next, the overall discussions and conclusions are shown in the fifth and sixth sections, respectively.

Turc recommended a simple equation for estimating ET_o using only solar radiation, relative humidity, and mean temperature. The model was initially developed for Western Europe (Turc, 1961). Similarly, a well-known researcher also discussed that the Turc model is appropriate and is reliable in humid conditions (Jensen et al., 1990). The equation can be given as:

The eight weather stations selected for this study are located in the Khyber Pakhtunkhwa region: Balakot, Cherat, Chitral, Kakul, Parachinar, Peshawar, Risalpur, and Saidusharif (Gul et al., 2018). The Khyber Pakhtunkhwa province's economy is based on agriculture, even though the mountainous topography is unfavorable to extensive agronomy. Irrigation is carried out on about one-third of the province's cultivated land, where major crops are wheat, corn (maize), sugarcane, and tobacco. The mountain ranges experience cold winters and moderate summers, and temperatures rise significantly to the south. Precipitation over the province is capricious but averages roughly (400 mm) annually, with considerable occurring from January to April (Khan, 2010). The mountain slopes in the north region are full of natural scenic assets, having five rivers, and it is exceptionally rich in charming and alluring landscape, exotic valleys, and condensed pine forest. The mean monthly air temperature, wind speed, and relative humidity data are collected at these stations under standard grass height and well-irrigated environment. The meteorological ten-year data (2000–2009) were obtained from the Pakistan Meteorological Department, Islamabad (Gul et al., 2020). The description of the different weather stations is presented in Table 1 and graphically displayed in Figure 1.

The basic statistic of the collected climatic parameters such as precipitation, air temperature, wind speed, relative humidity, and solar radiation are shown in Table 2. The mean annual maximum and minimum temperature for Balakot is 25.70 C° and 11.65 C°, while the relative humidity is 70.65% and 66.34%, respectively. The mean annual wind speed (U₂) is 1.24 meters per second, and the mean annual solar radiation is 18.06. The mean annual extra-terrestrial radiation is 30.27 [MJm⁻² day⁻¹]. Furthermore, the minimum mean annual precipitation was observed in Chitral's concern region, with 15.33 mm, while the maximum precipitation was confirmed (121.86 mm) in Balakot station. Similarly, variances, skewness, and kurtosis are also shown in Table 2 to convey more information about the density and distribution shape of the study parameters. The rest of the research study locations statistics and details are shown in Table 2.

Tables 3 and 4 show the different statistical measures errors of five estimated reference evapotranspiration methods in Pakistan's KP region. Similarly, Figures 2 and 3 offer the Pearson product-moment correlation coefficient, commonly called correlation coefficient r between x and y, and the performance of different methods for estimating reference evapotranspiration for each station given in the following subsections.

The over- or underestimation performance is tested by statistical indicator CRM; for each of the five selected models in this study at Balakot Station the results are given in Table 3. The coefficient of residual mass in Column 4 indicates the over or underestimation of included ET_o models. It is noticeable that Priestly-Taylor, HS, and modified Hargreaves overestimated monthly mean ET_o values by (−0.05, −1.82, −0.17) mm per day, respectively. The Turc equation tended to underestimate ET_o by (0.89) mm per day, and Makkink shows excellent results with CRM (0.11) mm per day. In Row 4, the observed MAE for HS and modified Hargreaves, respectively (6.34, 0.79) mm per day, indicates better MHargreaves versus HS equation estimates. If data (such as wind speed, relative humidity, and solar radiation) are available, Priestly Taylor's results were remarkable in Balakot. Otherwise, the temperature-based proposed MHargreaves equation is an appropriate alternate choice for this location.

Moreover, a comparison between our benchmark model FAO-56(PM) and other ET_o estimations models is shown in Figure 3. Thus, from the output, the correlation coefficient between our proposed calibrated MHargreaves and FAO-56(PM) is 91%, indicating a highly significant value level and encouraging modified Hargreaves in data deficient regions Balakot. However, if the radiation data is available, then 96% of the correlation between FAO-56(PM) and Makkink shows the high significance and encourages applying for ET_o estimation purposes.

Similarly, the CRM values for Priestly-Taylor, Turc, and Makkink are respectively (0.21, 0.90, 0.34) mm per day tended to underestimate the ET_o values in Cherat. The HS equation overestimates ET_o with CRM (−1.03) mm per day, but modified Hargreaves performed better with CRM (0.07) mm close to zero, indicating better performance against HS. The MAE for HS and modified Hargreaves, respectively (4.07, 0.66) mm per day, shows excellent performance of modified Hargreaves for the concern station. Similarly, from the output in Figure 3, the correlation coefficient between our proposed calibrated MHargreaves and FAO-56(PM) is 90% shows a high level of significant value. It encourages the use of modified Hargreaves in data deficient regions. Nevertheless, if the radiation data is available, an 89% correlation between FAO-56(PM) and Makkink shows high significance and inspiration for ET_o estimation purposes.

In Chitral, based on CRM values, the Makkink and Priestly-Taylor are (0.04, 0.03) mm per day, respectively, indicating perfect estimation. Turc and MHargreaves tend to underestimate ET_o, but HS showed overestimated results. MAE values for modified Hargreaves and HS are (1.07, 5.46) mm per day, respectively. It offers a better performance of MHargreaves as compared with the HS equation. Priestly-Taylor, Makkink, and modified Hargreaves show low RMSE (0.64, 1.03, 1.63) mm per day versus Hargreaves and Turc (6.04, 5.16) mm per day, respectively. Besides, from Figure 3, our standard equation and Priestly-Taylor is 98%, which shows a high level of significant results. On the other hand, our proposed calibrated MHargreaves correlates 95% with the standard FAO-56(PM). Thus the work motivates the use of the proposed calibrated Hargreaves equation.

In Kakul, three models (Priestly-Taylor, HS, and modified Hargreaves) with negative CRM values (−0.08, −2.04, −0.28) mm per day gains overestimated results. MAE values for HS and modified Hargreaves are (6.20, 0.87) mm per day, respectively, indicating modified Hargreaves’ excellent performance against HS. Priestly-Taylor, Makkink, and modified Hargreaves showed low RMSE (0.50, 0.51, 1.02) mm per day versus HS and Turc with (6.7, 3.04) mm per day, respectively. If sufficient data is available, then radiation-based methods are suitable. Kakul station is humid; therefore, the proposed modified Hargreaves is a proper ET_o estimation model. Next, according to the output in Figure 2, the correlation between our standard equation and Makkink is 99%, showing a high level of significant results. On the other hand, our proposed calibrated MHargreaves show a correlation of 94% with the standard FAO-56(PM). Thus the work persuades the use of our proposed calibrated MHargreaves equation in the study region.

In Parachinar (Priestly-Taylor, HS and MHargreaves) with negative CRM (−0.002, −0.01, −0.01) mm per day overestimated the ET_o values, the Turc method with positive CRM (0.99) mm per day underestimated ET_o values and Makkink with CRM (0.00) produce perfect estimated ET_o. The MAE for HS is 48.25 mm per day, while MHargreaves is 8.41 mm per day, which shows excellent performance of MHargreaves against HS in Parachinar. Priestly-Taylor modified Hargreaves, and Makkink shows low RMSE (8.17, 11.38, 15.10) mm per day versus HS and Turc, with (52.12, 62.50) mm per day, respectively. Furthermore, our standard equation and Priestly-Taylor correlation is 96%, which shows a high level of significant results, while our proposed calibrated MHargreaves show a correlation of 91% with the standard FAO-56(PM). Therefore, the work stimulates the use of our proposed calibrated MHargreaves equation in Parachinar.

Three models (Priestly-Taylor, Makkink, and Turc) with positive CRM values (0.21, 0.38, 0.91) mm per day indicate underestimation in Peshawar. HS with negative CRM (−1.30) mm per day tends to overestimate the ET_o, but the MHargreaves accurately estimates with CRM (0.00) mm per day. The MAE values for Hargreaves and modified Hargreaves are respectively (5.90, 0.88) mm per day, showing the modified Hargreaves equation's excellent performance. It explains in row 25 that modified Hargreaves, Priestly-Taylor and Makkink produced low RMSE (1.19, 1.65, 2.49) mm per day while HS and Turc show high RMSE (6.27, 4.90) mm per day. Moreover, as pointed out from Figure 2, the correlation between our standard equation FAO-56(PM) and Turc is 96% shows a high level of significant results. Similarly, our proposed calibrated MHargreaves show a correlation of 93% with the standard FAO-56(PM). Thus the outcome encourages the use of the proposed calibrated MHargreaves equation in the region.

In Saidu-Sharif, all the models with negative CRM tend to overestimate ET_o. The modified Hargreaves equation with MAE (15.34) mm per day shows a better performance than HS (51.38) mm per day at Saidu Sharif weather station. The modified Hargreaves, Priestly-Taylor, and Makkink produce low RMSE (23.43, 27.84, 29.27) mm per day, while HS and Turc show high RMSE (56.49, 33.31) mm per day at Saidu-Sharif. If sufficient data is available, Priestly-Taylor should use it for ET_o estimation. Otherwise, modified Hargreaves will be an excellent alternative equation in Saidu-Sharif. According to the result provided in Figure 2, the correlation between our standard equation and Priestly-Taylor is 90%, which shows a high level of significant impact. On the other hand, our proposed calibrated MHargreaves show a correlation of 82% with the standard FAO-56(PM), which encourages using the proposed calibrated modified Hargreaves equation.

Risalpur, Priestly-Taylor, and HS's CRM values are (−0.16, −0.84) mm per day, indicating overestimation. Turc and modified HS tend to underestimate ET_o, and only Makkink with CRM (0.01) mm per day shows better performance. The modified Hargreaves equation performs well with MAE (2.36) mm per day against the original HS with (7.01) mm per day. The radiation-based models are in good contact with FAO-56(PM). In contrast, temperature-based methods are not suitable for Risalpur. Furthermore, according to Figure 2, the correlation between our standard equation and the Priestly-Taylor is 99%, showing a high level of significant result to be used if data is available. On the other hand, the proposed calibrated MHargreaves show a correlation of 57% with the standard FAO-56(PM).

Various studies have been conducted in various regions of the globe to ascertain the optimised ET_o model (Farzanpour et al., 2019; Jing et al., 2019; Khodayvandie et al., 2020; Salam et al., 2020; Goh et al., 2021; Sharafi & Ghaleni, 2021; Trajkovic & Gocic, 2021; Vishwakarma et al., 2022). Tables 3 and 4 compare the five estimated reference evapotranspiration methods based on statistical errors for eight stations of Khyber Pakhtunkhwa province in Pakistan. The test data set had a total of 120 months that were used for assessment. The Priestly-Taylor model provides the best estimates among all other evapotranspiration methods in Balakot, Chitral, and Parachinar stations. Similarly, based on SI-modified Hargreaves, Cherat, Peshawar, and Saidu Sharif stations, Makkink fit for Risalpur. As pointed out from Table 3, for the Kakul station, the Priestly-Taylor and Makkink methods produced almost the same amount of error. The performance of the Priestly-Taylor and Makkink methods for Kakul station was the same. We also noted that HS, Makkink, and Turc's had the highest estimated errors for all stations, among other methods. As a result, the Hargreaves-Samani process gave a very high standard error estimate and absolute error at all stations. SEE shows HS varied from 2.71 mm per day at Balakot to 56.33 mm per day for Saidu Sharif.

In contrast, the SEE of the modified Hargreaves method varied from 0.57 mm per day at Kakul to 22.61 mm per day at Saidu Sharif. Although both MAE and RMSE are similar measures, reporting both indices in many cases is appropriate. MAE is less sensitive to significant prediction errors and preferred for small or limited data sets, while RMSE is practical as it shows the errors in the same unit and scale. From Table 3, MAE ranged from 4.07 mm per day at Cherat to 51.38 mm per day for Saidu Sharif compared to the modified Hargreaves equation with 0.66 mm per day in Cherat 15.34 mm per day in Saidu Sharif. The analysis shows that the modified Hargreaves agreed with the standard FAO-56(PM) at all locations. The HS equation, up to some extent, overestimated FAO-56(PM) values at Balakot, Kakul, Parachinar, Cherat, Chitral, and Saidusharif.

Similarly, the modified Hargreaves equation showed satisfying results in ET_o, compared with the standard FAO-56(PM). However, the Hargreaves-Samani equation overestimated ET_o values almost in all stations, ranging from −1.03 mm per day at Cherat to −2.04 mm per day in Kakul. Moreover, the graphical representation of SEE estimates for different stations, including radiation-based and temperature-based models, is shown in Figure 6. Interestingly, the average monthly (ET_o) determined for 10 years at the study area using FAO-56 (PM) mechanisms, HS, and MHargreaves can be seen in Figure 5. The results indicate overestimated ET_o values in all locations. The modified Hargreaves equation estimates are very close to FAO-56(PM) estimates.

Additionally, the overall steps of the models and calibration are shown in the Data Flow chart in Figure 4. The graphical representation and comparative analysis of the modified Hargreaves equation presented in Figures 2 and 3 show that the proposed method results are much more reliable than HS and radiation-based models like Turc, Makkink, and PT. Furthermore, the correlation between FAO-56(PM) with other models is shown in the upper off diagonals. The histograms and probability density curves for selected models are given in the diagonals. However, the lower off-diagonal showed the scatter plots for different ET_o estimation models. At the same time, the ranking of the multiple equations is based on SSE values. Similarly, the best fitting reference evapotranspiration models are based on statistical indicators shown in Table 4.

Furthermore, the graphical representation of SEE estimates for the different stations is shown in Figure 6, based on Bayesian Kriging as an interpolation procedure. On the other hand, the comparison of ET_o estimates through standard method FAO-56(PM), Hargreaves-Samani (HS), and modified Hargreaves are shown in Figure 5. However, Table 5 shows the SEE-based ranking of our proposed ESS scheme. According to the results provided in the table, at Balakot, Priestly-Taylor ranked first. Besides, our proposed MHargreaves showed the lowest SEE value compared to other models. Similarly, at Cherat, Peshawar, and Saidu Sharif, our proposed MHargreaves equation is ranked first based on their related SEE (0.78, 1.22, and 22.61). While in Chitral, Kakul, Parachinar, and Risalpur, our proposed MHargreaves ranked second based on SEE values.

Moreover, Figures 2 and 3 show the histogram, correlation coefficient, and Q-Q plot between our standard models FAO-56(PM) versus other reference evapotranspiration models during the whole study period in the entire KP region. The correlation coefficient between the benchmark method FAO-56(PM) and modified Hargreaves is high in all the stations, indicating the significant correlation between MHargreaves and standard FAO-56(PM). Similarly, the scatter plots for the modified Hargreaves and FAO-56(PM) are provided in Figure 7. Hence, these results strongly suggest that our proposed MHargreaves equation can be recommended as an alternative temperature-based method for ET_o estimation in data deficient regions like Khyber Pakhtunkhwa, Pakistan.

The primary purpose of this research was to validate various temperature and radiation combination-based ET_o models for Khyber Pakhtunkhwa meteorological and agro-ecological diversity. The meteorological data sets such as solar radiation, wind speed, and relative humidity are limited in the data-sparse region to solve the FAO-56(PM) equation. In such situations, the temperature-based equation is recommended for estimating ET_o. However, the Hargreaves-Samani equation generally overestimates ET_o at humid locations. The present study used the five different ET_o estimation equations in eight selected regions of Khyber Pakhtunkhwa. The results obtained by the Hargreaves-Samani equation indicate that an adjustment of the Hargreaves-Samani equation exponent (E_H) is essential. Thus, in the present research, a value of (0.16) is proposed as the Hargreaves exponent instead of the original (0.5). The modified Hargreaves equation's performance was evaluated by comparing ET_o estimates with the FAO-56(PM) equation using different statistical indicators and correlation coefficients. The modified Hargreaves with average MAE and SEE for eight locations are (3.797, 5.668) mm per day compared to those with (16.827, 15.962) mm, respectively.

The calibration method has some advantages: (1) The calibration technique can be used to precisely measure ET_o in different humid and semi-arid conditions throughout Khyber Pakhtunkhwa. (2) Despite the constraints of accessibility and reliability of climatological data, the derived formulas reported in this research are proposed as feasible techniques for estimating ET_o in humid and sub-irrigated areas when the standard FAO56-PM equation cannot be used due to the limitations stated earlier. (3) the equation is preferred for humid and subtropical regions above the other primary temperature or radiation-based approaches assessed in this study. However, certain drawbacks of the proposed scheme exist: (1) The methodology's main disadvantage is that it cannot be applied in all environments. We must recalibrate the equation regularly to account for factors such as climate and altitude. (2) The calibration must have ten or more years of data to assess the Hargreaves-Sammani performance (Gavilán et al., 2006). Further research must be commenced to assess the proposed modified Hargreaves equation's applicability to other locations in the country. The approach presented in this paper could be applied to other regions for the regional calibration of the Hargreaves-Samani equation.

NA.

NA.

The author declares no conflict of interest.

This research is supported by the National Natural Science Foundation of China (No. 52071298) and ZhongYuan Science and Technology Innovation Leadership Program (No. 214200510010).

We are also appreciative to the Pakistan Meteorological Department, Islamabad, for providing the necessary data.

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