Regionalization of the Hargreaves-Samani Coefficients to Estimate Reference Evapotranspiration in High-Altitude Areas

Abstract

The Penman-Monteith (PM) equation is considered the most accurate method for estimating reference evapotranspiration (ETo); however, its application requires a large amount of data that is not always available. This study aimed to regionalize the coefficients of the Hargreaves-Samani (HS) equation to estimate ETo in high-altitude areas, specifically the Peruvian Altiplano (PA). The methodology included (1) evaluation of the original HS equation, (2) calibration and validation of the empirical coefficient ($CH$) and empirical exponent ($EH$) at each weather station, and (3) regionalization of the calibrated coefficients using a multiple linear regression approach. The results showed that the original HS equation had NSE values ranging from −0.57 to 0.87, PBIAS from −18.60% to 12.70%, MAE from 0.16 to 0.65 mm/d, and RMSE from 0.20 to 0.67 mm/d. After calibrating $CH$ and $EH$, performance improved significantly, achieving validation values of NSE ranging from 0.67 to 0.94, PBIAS from −0.55% to 1.37%, MAE from 0.01 to 0.05 mm/d, and RMSE from 0.13 to 0.21 mm/d. Finally, the regionalization of 0.859 and 0.744, respectively. These results indicate that the HS equation, with calibrated and regionalized coefficients, is a viable alternative for estimating ETo in regions with limited meteorological data.

1. Introduction

Improving water efficiency requires the control of the high demand for irrigated agriculture, which depends on enhancing the capacity to accurately simulate the water cycle and its components [1]. The water usage of irrigated agriculture is an important issue for various aspects of water resource management, including the planning and designing of new irrigation systems and distributing water among the existing systems. Therefore, having precise information about crop water requirements is essential to address these challenges, and estimates of reference evapotranspiration (ETo) are widely used for irrigation engineering [2].

ETo is one of the most useful indicators required for efficient irrigation management [3]. ETo is essential for estimating the crop water requirements and irrigation scheduling, as well as in the design of irrigation and drainage systems, drought management, and studies related to climate change and variability [4,5,6,7]. Crop coefficients, which depend on specific crop characteristics and local conditions, are used to convert ETo into the actual crop evapotranspiration (ETr) [8]; therefore, obtaining an accurate estimation of ETo is of great importance.

The in-situ measurement of ETo is costly, time-intensive, and subject to significant uncertainties. Due to the limitations of in situ ETo measurements, several empirical models have been developed for this estimation [9]. The empirical models for estimating the ETo found in the scientific literature are classified into (1) fully physically based combined models that describe the principles of mass and energy conservation; (2) semi-physical models that address mass or energy conservation; and (3) black-box models based on artificial neural networks, empirical relationships, genetic algorithms, and fuzzy logic [10,11,12,13].

The method recommended by the Food and Agriculture Organization of the United Nations (FAO) for estimating ETo is the FAO 56 standard Penman-Monteith (PM) equation [8,14,15,16]. The PM equation is classified as the most effective method for estimating ETo in all climate types and can be applied globally without requiring local calibration, as it incorporates physical and aerodynamic parameters, and it has been validated through tests conducted with various lysimeters [3,15].

The PM model performs well in different regions of the world using data on air temperature, relative humidity, solar radiation, and wind speed data [17]. However, in places where meteorological data is scarce, the application of the PM equation becomes limited, which is the main impediment to its widespread use. Consequently, alternative approaches are required to estimate ETo. Due to data limitations, Hargreaves and Samani [18] developed an alternative method for estimating the ETo using only air temperature and extraterrestrial solar radiation data. Therefore, when the necessary meteorological data for calculating ETo using the PM method are not available, the Hargreaves-Samani (HS) method is recommended.

Although the HS method can be applied with some standard coefficient values, several authors [3,4,8,15,16] recommended calibrating this equation in relation to the PM method in places with comparable climates. It is important to mention that the validity of the empirical equations is only applicable in the areas where they were developed and within the range of the available data [19]. In addition, local calibrations can improve the accuracy of the HS method estimates, which confirms the need to improve the accuracy of this equation [4]. Consequently, the accuracy of the equation may be improved by calibrating the coefficients of the HS equation to the local conditions.

In this regard, studies conducted under different climatic conditions have reported that calibration of air temperature and radiation-based equations improved the performance of the ETo estimates [20,21,22]. Various studies performed worldwide have adjusted the coefficients of the original HS equation empirical Hargreaves coefficient ($CH$), empirical temperature Hargreaves constant ($CT$), and empirical Hargreaves exponent ($EH$) to improve the performance of the ETo estimates.

In relation to the calibration of the $CH$ coefficient in the HS equation, various studies stand out. Xu and Singh [23] conducted a study in Northwestern Ontario, Canada; Vanderlinden et al. [24] focused on Southern Spain; while Martinez-Cob and Tejero-Juste [2] analyzed semi-arid conditions in the Ebro River valley in northeastern Spain. Other relevant studies include those by Sepaskhah and Razzaghi [25] and Tabari and Talaee [26] in Iran, as well as Ferreira et al. [27] in Minas Gerais, Brazil. Regarding the calibration of the $EH$ coefficient, research by Trajkovic [8] in the Western Balkans region of Southeastern Europe; Subburayan et al. [28] in India; and Kelso-Bucio et al. [29] in Mexico can be cited. For the simultaneous calibration of the $CH$ and $EH$ coefficients, the work of Almorox et al. [30] in Argentina; Berti et al. [3] in the flat region of Veneto, northeastern Italy; Patel et al. [31] in various climatic conditions in India; and Ferreira et al. [26] in Brazil are noteworthy. Concerning the simultaneous calibration of the $CH$, $CT$, and $EH$ coefficients, the studies by Droogers and Allen [4] at a global level; Mohawesh and Talozi [32] in Jordan; Bogawski and Bednorz [20] in Poland; and Pandey et al. [33] for the climatic conditions of Gangtok, India, can be mentioned. Other studies include those by Dorji et al. [34] in Bhutan; Cobaner et al. [19] in Turkey; Ferreira et al. [27] in Minas Gerais, Brazil; Hadria et al. [1] in central and northern Morocco; and Pandey and Pandey [35] in Manipur, India. It is also relevant to mention the regional calibration carried out by Gavilán et al. [36] in Andalusia, Spain, and by Berti et al. [3] in the flat region of Veneto, northeastern Italy.

The PA also faces the challenge of estimating ETo using the PM equation because of the limited measurement of meteorological variables; therefore, studies resort to the use of empirical methods as an alternative. However, their validity is limited to certain climatic and agronomic conditions of the location, and it is not applicable under different conditions for which they were initially developed. In this regard, in the PA, a lack of research on the calibration and validation of the coefficients for the HS equation has been identified, as well as the regionalization of these coefficients based on the geographical characteristics of the PA. Therefore, it is essential to analyze the coefficients $CH$ and $EH$ to reduce the margin of error in estimating the original HS equation, especially in high-altitude areas and under conditions that are specific to the PA.

The research focuses on determining whether the regionalization of the calibrated $CH$ and $EH$ coefficients of the HS equation can improve the accuracy of ETo estimates in high-altitude areas. The main objective of this study is to regionalize the coefficients of the HS equation in high-altitude areas, with the following specific objectives: (1) evaluate the ETo estimates using the HS equation; (2) calibrate and validate the coefficients of the HS equation; and (3) regionalize the calibrated coefficients of the HS equation based on the specific geographic characteristics for the conditions of the PA.

2. Materials and Methods

2.1. Study Area

The study area is located in the Lake Titicaca basin in Peru, an endorheic basin system that is surrounded by the eastern and western cordilleras. It borders the Amazon hydrographic region to the north, the Pacific hydrographic region to the southwest, and the Titicaca Hydrographic Region (THR) of Bolivia to the east. The altitude ranges from 3804 to 5781 m (Figure 1). Due to the low levels of precipitation, high evapotranspiration rates, and low water retention capacity of the soils, water stress is a substantial limitation for agricultural production in this region [37]. According to Peru’s climate classification, the THR (Peruvian side) has a predominantly rainy climate, characterized by dry autumns and winters [38].

The study area presents an average annual precipitation that ranges from 624.9 mm to 948.3 mm. The climatic conditions in the stations vary considerably, with the maximum air temperature ranging from 10.5 °C to 17.8 °C, while the minimum air temperature varies from −2.2 °C to 3.5 °C. Relative humidity levels range from 55.0% to 80.6%, while wind speeds range from 1.5 m/s to 5.5 m/s. The sunshine hours vary from 6.3 h to 8.4 h, and the altitudes of the stations are between 3812 and 4660 m. In addition, the aridity index (AI) for the sites of the weather stations ranges from 0.52 to 0.78. This indicates that the climatic classification of the weather station sites falls between dry subhumid and humid subhumid, based on the AI levels established for Peru by Huerta and Lavado [39].

2.2. Climatic and Terrain Data

The data utilized for this study consist of monthly values for maximum air temperature ($T_{max}$, °C), minimum air temperature ($T_{min}$, °C), relative humidity (Rh, %), wind speed ($U_{10}$, m/s) at a height of 10 m, sunshine hours (Sh, h), and precipitation (P, mm). These data were obtained from nine weather stations located in the PA (Figure 1) and cover period from 2000–2019. The data were provided by the National Meteorology and Hydrology Service of Peru (SENAMHI) (

Table 1. Location of the weather stations along with the average meteorological variables, aridity index, and climatic classification.

Station	Lat.	Lon.	Alt.	${\mathbf{T}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$	${\mathbf{T}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$	Rh	U2	Sh	ETo	P	AI	CC
Ananea (ANA)	−14.676	−69.534	4660	10.5	−1.9	80.6	2.0	6.3	2.7	658.4	0.67	Humid subhumid
Chuquibambilla (CHQ)	−14.788	−70.728	3918	16.3	−2.2	61.3	2.1	7.2	3.2	787.0	0.67	Humid subhumid
Desaguadero (DES)	−16.563	−69.037	3833	15.2	1.7	65.7	5.5	7.4	3.4	736.9	0.60	Dry subhumid
Huancané (HNE)	−15.207	−69.758	3840	15.7	0.3	58.9	2.9	7.7	3.4	650.5	0.52	Dry subhumid
Juliaca (JLC)	−15.444	−70.208	3838	17.8	−0.5	75.0	1.5	7.8	3.2	624.9	0.53	Dry subhumid
Juli (JUL)	−16.204	−69.460	3830	14.1	3.0	58.3	2.4	8.4	3.5	948.3	0.78	Humid subhumid
Lampa (LAM)	−15.361	−70.374	3866	17.1	−0.3	55.0	2.4	8.1	3.6	757.1	0.60	Dry subhumid
Puno (PNO)	−15.826	−70.012	3812	16.3	3.5	61.1	1.8	8.1	3.5	750.6	0.59	Dry subhumid
Putina (PTN)	−14.921	−69.876	3861	17.3	0.1	70.1	2.6	6.9	3.1	643.9	0.56	Dry subhumid

Lat: Southern latitude, Lon: Western longitude, Alt: Altitude (m), Rh: Relative humidity (%); U₂: Wind speed (m/s) at a height of 2 m; Sh: Sunshine hours (h); ETo: Reference evapotranspiration estimated by the PM method (mm/d), P: Annual precipitation (mm), AI: Aridity index, CC: Climate classification. From Lujano et al. [40].

). The selected stations are representative of the PA, based on their AI (relationship between annual precipitation and annual evapotranspiration), as suggested by Huerta and Lavado [39].

Quality control of the data from the weather stations was performed, which consisted of verifying specific physical limits for the Peruvian territory and evaluating the internal and temporal consistency of the data [41]. This was analyzed by visual inspection and the absolute method, which included the non-parametric Distribution-Free Cumulative Sum (CUSUM) and Rank-Sum (RS) tests. These tests were conducted independently of the data from each weather station using the trends program (TREND) (https://toolkit.ewater.org.au/Tools/TREND: accessed on 10 August 2022).

CUSUM is a non-parametric step-change test in the mean, while RS is a non-parametric test for the difference in mean between two periods [42]. The null hypothesis for the CUSUM and RS tests states that there is no change in the mean of the data series and that there is no difference in the mean between two data periods. This null hypothesis can be accepted if the maximum deviation obtained for CUSUM and the z-statistic for RS are less than the critical value at the 5% significance level; otherwise, it is rejected. Therefore, the periods of the Rh, $U_{10}$, Sh, $T_{max}$, $T_{min}$, and P data series that did not meet the homogeneity assumption were excluded in subsequent analyses.

The missing values were completed using the Random Forest machine learning algorithm embedded in the MICE (Multivariate Imputation by Chained Equations) package of the R project [43]. After completing the missing data, the homogeneity of the data was verified using the monthly data [44,45], as homogeneity tests are generally more robust when using monthly data [45]. The wind speed at a height of 2 m ($U_2$, m/s) was calculated from the wind speed at a height of 10 m ($U_{10}$, m/s) using the methodology proposed by Allen et al. [15].

The climatic and geographic variables that affect ETo, such as when the altitude increases, precipitation, reference evapotranspiration, minimum temperature, maximum temperature, sunshine hours, and wind speed decrease, whereas the relative humidity increases for the studied weather stations [40].

To regionalize the calibrated $CH$ and $EH$ coefficients of the HS equation, the digital elevation model of the NASA Shuttle Radar Topography Mission (SRTM) was obtained from the Google Earth Engine (GEE) platform, which is available at https://earthengine.google.com/ (accessed on 15 July 2022). ID from the image CGIAR/SRTM90_V4 [46], with a spatial resolution of ~90 m.

2.3. Penman–Monteith Method

The PM method was used as a replacement for the measured ETo data, being the standard approach when no lysimeter-measured data are accessible [36]. Due to the lack of experimental ETo measurements, the results given by the PM equation have been regarded as the true values, and this equation has been utilized to calibrate the modified versions of the HS equations [15,47]. The PM method was qualified by the FAO as the standard model for estimating ETo. The standard equation is presented below:

$${ET}_{O,PM}={\displaystyle \frac{0.408∆\left(R_n-G\right)+\gamma 900 U_2/(T_{mean}+273)(e_s-e_a)}{∆+\gamma (1+0.34 U_2)}}$$

(1)

where ${ET}_{O,PM}$ is the reference evapotranspiration (mm/d), $∆$ is the slope of the saturation vapor pressure vs. air temperature curve (kPa/°C), R_n is the net radiation (MJ/m²/d), $G$ is the soil heat flux density (MJ/m²/d), $T_{mean}$ is the daily mean air temperature (°C), $U_2$ is the daily mean wind speed at 2 m height (m/s), $e_s$ is the saturation vapor pressure (kPa), $e_a$ is the actual vapor pressure (kPa), $e_s-e_a$ is the vapor pressure deficit (kPa), and $\gamma$ is the psychrometric constant (kPa/°C).

The calculation of all the required data for estimating ETo followed the recommendations given in the FAO Irrigation and Drainage Paper 56 [15].

The solar radiation ($R_s$) was calculated from the measured sunshine hours (using a Campbell-Stokes sunshine recorder) according to the Angstrom equation [15].

$$R_s=\left(a_s+b_s{\displaystyle \frac{n}{N}}\right)R_a$$

(2)

where $R_a$ is the extraterrestrial radiation (MJ/m²/d), $n$ is the actual duration of sunshine (h), $N$ is the maximum possible duration of sunshine or daylight hours (h), $a_s$ is the regression constant, which expresses the fraction of extraterrestrial radiation that reaches the Earth on overcast days $(n=0$), and $a_s+b_s$ is the fraction of extraterrestrial radiation that reaches the Earth on clear days $(n=N$).

In the absence of actual measurement data and solar radiation calibration $\left(R_s\right)$, Allen et al. [15] suggested values of $a_s=0.25$ and $b_s=0.5$0. However, these default values should not be applied to high-altitude sites, where proper calibration is required [48]. In this regard, the values of $a_s$ and $b_s$ were estimated for the studied weather stations, considering the altitude, as suggested by Ye et al. [48], and the values of $a_s=0.23$ and $b_s=0.6$0 were obtained for the stations located at an altitude of 3812 to 3918 m. These values are in agreement with those suggested by Chipana et al. [5] for high-altitude areas (3820–3950 m) calibrated for the Bolivian highlands. For altitudes above 4660 m, values with slight modifications of $a_s=0.29$ y $b_s=0.55$ were considered [40]. The values obtained for $a_s+b_s$ for the study area were higher than the values recommended by FAO 56 ($a_s+b_s=0.75$). This can be explained by the high altitude of the weather stations [48].

The extraterrestrial radiation $\left(R_a\right)$ was estimated using the equation recommended by Allen et al. [15]. The values of $R_a$ for every day of the year at various latitudes can be estimated using the solar constant, solar declination, and the time of year. Then selecting the $R_a$ for the 15th day of each month was converted to monthly values using the following equation:

$$R_a={\displaystyle \frac{24\left(60\right)}{\pi }}{G}_{SC}{d}_r\left[{\omega }_s\sin \left(\varnothing \right)\sin \left(\delta \right)+\cos \left(\varnothing \right)\cos \left(\delta \right)\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_s\right)\right]$$

(3)

where $G_{SC}$ is the solar constant = 0.0820 MJ/m²/min, $d_r$ is the inverse relative distance to the sun, ${\omega }_s$ is the sunset hour angle (rad), $\mathrm{\varnothing }$ is the latitude (rad), and $\delta$ is the solar declination (rad).

The net radiation $\left(R_n\right)$ was determined by subtracting the outgoing net longwave radiation ${(R}_{nl})$ from the incoming net shortwave radiation ${(R}_{ns})$.

$$R_n=R_{ns}-R_{nl}$$

(4)

To calculate the incoming net shortwave radiation ${(R}_{ns})$, an albedo value of 0.23 was used, while the net longwave radiation ${(R}_{nl})$ was estimated using the expression postulated by the modified Stefan-Boltzmann law due to absorption and the downward radiation from the sky [15].

$$R_{nl}=\sigma \left[{\displaystyle \frac{T_{max,K}^4+T_{min,K}^4}{2}}\right]\left(0.34-0.14\sqrt{e_a}\right)\times \left(1.35{\displaystyle \frac{R_s}{R_{so}}}-0.35\right)$$

(5)

where $R_{nl}$ is the net longwave radiation (MJ/m²/d), $\sigma$ is the Stefan-Boltzmann constant (4.903 × 10⁻⁹ MJ/K⁻⁴/m²/d), $T_{max,K}^4$ is the maximum absolute temperature during the 24-h period (K = °C + 273), $T_{min,K}^4$ is the minimum absolute temperature during the 24-h period (K = °C + 273), $e_a$ is the actual vapor pressure (kPa), $R_s/R_{so}$ is the relative shortwave radiation (limited to ≤1.0), $R_s$ is the calculated solar radiation (MJ/m²/d), and $R_{so}$ is the calculated clear-sky radiation (MJ/m²/d).

The vapor pressure deficit is calculated as the difference between the saturation vapor pressure $\left(e_s\right)$ and the actual vapor pressure $\left(e_a\right)$. $e_s$ is calculated as the average of the saturation vapor pressure at $T_{max}$ and $T_{min}$. Approximations can be used to estimate $e_a$ depending on the available data. When only the monthly average daily relative humidity data ${(HR}_{mean})$ are available, $e_a$ is calculated as [15]:

$$e_a={\displaystyle \frac{{Rh}_{mean}}{100}}\left[{\displaystyle \frac{e^{\mathrm{o}}\left(T_{max}\right)+e^{\mathrm{o}}\left(T_{min}\right)}{2}}\right]$$

(6)

2.4. Hargreaves-Samani Method

This research has concentrated on utilizing the HS equation [18]. Due to its simple application for calculating ETo with just temperature data. The equation can be expressed as:

$${ET}_{O,HS}=0.0135\times K_{RS}\times R_a\times \left(T_{mean}+17.8\right){\left(T_{max}-T_{min}\right)}^{0.5}$$

(7)

where, ${ET}_{O,HS}$ is the reference evapotranspiration (mm/d), $R_a$ is the extraterrestrial radiation (mm/d); $T_{max}$, $T_{min}$, and $T_{mean}$ are the maximum, minimum, and mean temperatures (°C), respectively, 0.0135 is a unit conversion factor from the American System to the International System, $K_{RS}$ is the empirical radiation adjustment coefficient, and 17.8 is an empirical factor related to the temperature units used in the original formulations.

The adjustment coefficient $K_{RS}$ was initially set to 0.17 for arid and semiarid regions [47]. With $R_a$ in mm/d and the empirical coefficient $K_{RS}$ normally considered as 0.17, the HS equation is written as:

$${ET}_{O, HS}=0.0023\times Ra\times \left(T_{mean}+17.8\right){\left(T_{max}-T_{min}\right)}^{0.5}$$

(8)

where 0.0023 is an empirical Hargreaves coefficient ($CH$), 17.8 is an empirical temperature Hargreaves constant ($CT$), and 0.5 is an empirical Hargreaves exponent ($EH$).

2.5. Statistical Metrics of Performance

To evaluate the performance of the original HS method compared to PM across the nine weather stations, various statistical performance metrics were employed. These include the correlation coefficient (R) and the Nash-Sutcliffe efficiency coefficient (NSE), as utilized by Todorovic et al. [6] and Almorox and Grieser [47]. Additionally, Cobaner et al. [19] used percent bias (PBIAS), root mean square error (RMSE), and mean absolute error (MAE).

$$R={\displaystyle \frac{\sum _{i=1}^N({ET}_{O,PM}-{\overline{ET}}_{O,PM})({ET}_{O,HS}-{\overline{ET}}_{O,HS})}{\sqrt{{\sum }_{i=1}^N{\left({ET}_{O,PM}-{\overline{ET}}_{O,PM}\right)}^2}\sqrt{{\sum }_{i=1}^N{\left({ET}_{O,HS}-{\overline{ET}}_{O,HS}\right)}^2}}}$$

(9)

$$NSE=1.0-{\displaystyle \frac{{\sum }_{i=1}^N{\left({ET}_{O,PM}-{ET}_{O,HS}\right)}^2}{{\sum }_{i=1}^N{\left({ET}_{O,PM}-{\overline{ET}}_{O,PM}\right)}^2}}$$

(10)

$$PBIAS={\displaystyle \frac{\sum _{i=1}^N({ET}_{O,HS}-{ET}_{O,PM})}{\sum _{i=1}^n{ET}_{O,PM}}}\times 100$$

(11)

$$MAE={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N|{ET}_{O,HS}-{ET}_{O,PM}|$$

(12)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left({ET}_{O,HS}-{ET}_{O,PM}\right)}^2}$$

(13)

where ${ET}_{O,PM}$ is the estimated value using PM, ${ET}_{O,HS}$ is the calculated value, ${\overline{ET}}_{O,PM}$ is the average of the estimated value using PM, ${\overline{ET}}_{O,HS}$ is the average of the calculated value, and $N$ is the total number of data points.

The correlation coefficient measures the relationship between the observed and estimated ETo values, with values ranging from −1.0 to 1.0. Values closer to 1.0 or −1.0 indicate a stronger correlation between the observed and estimated values.

According to Legates and McCabe [49], the NSE coefficient is a suitable tool for evaluating the goodness of fit of a model. The NSE, defined by Nash and Sutcliffe [50], ranges from negative infinity to 1.0, where higher values indicate a stronger relationship. According to Moriasi et al. [51], the classification of a model’s performance is based on the following NSE ranges: very good if NSE > 0.75, good if 0.65 < NSE ≤ 0.75, satisfactory if 0.50 < NSE ≤ 0.65, and poor if NSE ≤ 0.50. Negative NSE values indicate a deficient performance of the model [52].

The PBIAS index evaluates the average bias of the simulated data compared to the observed data. The optimal value is 0, while negative values suggest an underestimation and positive values indicate an overestimation [53]. Moriasi et al. [54] state that the statistical performance of a model is evaluated as follows: “very good” when PBIAS < ±5, “good” when ±5 ≤ PBIAS < ±10, “satisfactory” when ±10 ≤ PBIAS < ±25, and “not satisfactory” when PBIAS ≥ ±25.

According to Willmott and Matsuura [55], the MAE is a direct measure of the magnitude of the average error between the observed and calculated values. A low MAE indicates that the model has a high performance. On the other hand, the RMSE is a commonly used error index [47]. A lower RMSE reflects a closer fit of the model and implies greater accuracy in its predictions.

2.6. Calibration and Validation of the Coefficients of the HS Equation

The primary techniques employed for this purpose include calibration through simple linear regression [15,19] and adjustment of the HS equation coefficients [16,20,27]. For the present study, the calibrated Hargreaves-Samani equation (HSC) was developed by adjusting its empirical coefficients as follows:

• Simultaneous calibration of the empirical Hargreaves coefficient (CH) and empirical Hargreaves exponent (EH), keeping the empirical temperature Hargreaves (CT = 17.8) constant.

$${ET}_{O, HSC}=CH\times Ra\times \left(T_{mean}+17.8\right){\left(T_{max}-T_{min}\right)}^{EH}$$

(14)

The coefficients $CH$ and $EH$ of the HS equation were calibrated using the ETo data obtained via the PM method. This calibration was performed using the Solver tool in Microsoft Excel, which utilizes the generalized reduced gradient nonlinear optimization algorithm. The objective function was defined as the maximization of the NSE coefficient and the minimization of RMSE, and these were set as the optimization goals in the Solver tool of Microsoft Excel.

Shiri et al. [56] indicate that evaluating the calibrated model with the same calibration data can generate partially valid results. To address this issue, the observed data between 2000 and 2019 were divided into two groups: 70% for calibration and 30% for validation. However, using historical data to calibrate the HS model does not directly consider the influence of climate change over time, which can result in good model performance for the calibrated years but instability when expanding the dataset [57]. To reduce this instability, the calibration and validation periods were randomly selected [40]. To confirm the validity of the new coefficients obtained for the HS equation, they were evaluated using performance metrics such as NSE, PBIAS, MAE, and RMSE.

2.7. Regionalization of the Coefficients of the HS Equation

A Geographic Information System was employed to spatially interpolate the point values of ${CH}_C$ and ${EH}_C$ to a spatial resolution of ~90 m. In addition, a multiple linear regression (MLR) method was utilized with EH_C, ${CH}_C$, and ${EH}_C$ as the dependent variables and the longitude, latitude, and altitude as the independent variables, followed by residual values.

The longitude and latitude maps were interpolated using the inverse distance weighting (IDW) method from the centroids of the pixels of the altitude map, while the residual map was interpolated using the IDW method from the point values at each location [40]. Next, ${CH}_C$ and ${EH}_C$ were obtained using map algebra (raster calculator tool) in ArcGIS program. The values of ${CH}_{\left(x\right)}$ and ${EH}_{\left(x\right)}$ at unmeasured points are obtained according to the following equations:

$${CH}_{\left(x\right)}={b_0}^{CH}+{b_1}^{CH}\times X+{b_2}^{CH}\times Y+{b_3}^{CH}\times Z+r_{CH}$$

(15)

$${EH}_{\left(x\right)}={b_0}^{EH}+{b_1}^{EH}\times X+{b_2}^{EH}\times Y+{b_3}^{EH}\times Z+r_{EH}$$

(16)

where ${CH}_{\left(x\right)}$, and ${EH}_{\left(x\right)}$ are the predicted values of the ${CH}_C$, and ${EH}_C$ variables at point $x$, respectively; $b_0, { b}_1, { b}_2$, and $b_3$ are the coefficients of the multiple linear regression; the values of $X$, $Y$, and $Z$ are the independent variables at point $x$, where $X$ is the longitude, $Y$ is the latitude, and $Z$ is the altitude; and $r$ is the residual. The regionalized values of ${CH}_{\left(x\right)}$ and ${EH}_{\left(x\right)}$ will be replaced simultaneously in Equations (15) and (16).

The analysis procedure for calibrating the HS equation for estimating ETo in high-altitude areas: a case study in the PA, is summarized in the flowchart in Figure 2.

3. Results and Discussion

3.1. Evaluation of the Original HS Equation

Figure 3 compares the ETo estimates obtained from the original HS method and the PM method. A $K_{RS}$ value of 0.17 was considered for the HS equation for all studied weather stations [40]. The accuracy of the ETo values derived from the original equation exhibited significant differences across the humid, sub-humid, and dry sub-humid climatic regions identified in the study area. The scatter plots revealed that the correlation coefficient (R) values within the two climatic regions varied from 0.84 to 0.97, with the lowest values found at the ANA and PTN stations and the highest at the PNO and JLC stations. In contrast, the NSE values ranged from −0.57 to 0.87, with low NSE values below 0.50 indicating poor performance at the PTN, JUL, JLC, and ANA stations, whereas values above 0.75 were considered very good according to Moriasi et al. [51] and were observed at the HNE and LAM weather stations.

The PBIAS ranged from −18.60 to 12.70%, indicating that the HS equation generally underestimated the ETo values at the JUL, PNO, DES, ANA, HNE, and LAM stations with respective values of −18.60, −8.80, −8.30, −5.20, −4.90, and −1.90%. In contrast, the equation overestimated the ETo values at the PTN, JLC, and CHQ stations with values of 12.70, 11.80, and 7.30%, respectively. Underestimations of ETo were noted at stations near Lake Titicaca (LT), namely JUL, PNO, DES, and HNE, as well as at the LAM and ANA stations, which are far from LT. Meanwhile, overestimations of the ETo values were also observed at stations far from LT.

For the ANA and CHQ stations, which are classified as a humid subhumid climate, and the JLC, PTN, and LAM stations, which are classified as a dry subhumid climate, the results agree with those reported by Trajkovic [8], where the HS model tends to overestimate the ETo values in humid climate areas. In contrast, the JUL station with a humid subhumid climate and the HNE, PNO, and DES stations with a dry subhumid climate showed an underestimation of ETo values. As a result, using the HS equation in its original form was not particularly effective.

Meanwhile, as pointed out by Allen et al. [15], the HS equation also tends to underestimate ETo values in conditions of strong wind (${\mathrm{U}}_2$ > 3 m/s) and overestimate ETo under high relative humidity conditions. The present study revealed that at the DES station, which records wind speeds above 3 m/s, the ETo values were underestimated, which correlates with what was mentioned by Allen et al. [15]. However, it was also observed that at the weather stations where wind speeds below 3 m/s were recorded, there was an overestimation (CHQ, JLC, and PTN) and underestimation (ANA, HNE, JUL, LAM, and PNO) of the ETo values. In contrast, regarding relative humidity, the study revealed that the ANA station, where high relative humidity is present, tended to underestimate ETo, which is in contrast to what was suggested by Allen et al. [15].

The MAE values ranged from 0.16 to 0.65 mm/d, with the lowest values occurring at the LAM, ANA, HNE, CHQ, PUN, and DES stations. In contrast, high values were found at JLC, PTN, and JUL. Similarly, RMSE values varied between 0.20 and 0.67 mm/d, with low values occurring at the LAM, ANA, HNE, CHQ, PNO, and DES stations, while high values occurred at the JLC, PTN, and JUL stations.

3.2. Calibration and Validation of the HS Coefficients

Calibration and Validation of the CH and EH Coefficient

The simultaneous calibration and validation of the $CH$ and $EH$ coefficient values were performed. The findings indicated that using a $CH$ coefficient of 0.0023 and an $EH$ coefficient of 0.5 resulted in less accurate ETo estimates in the PA. The results of the simultaneous calibration of the HS coefficients ${CH}_C$ and ${EH}_C$ ranged from 0.0012 to 0.0027 and from 0.465 to 0.753, respectively. At most of the analyzed weather stations, the ${CH}_C$ and ${EH}_C$ values differ from the recommended values of Hargreaves and Samani [18], except at the LAM weather station, where ${CH}_C$ = 0.0023 and ${EH}_C$ = 0.499 values were obtained, which are similar to the values proposed by Hargreaves and Samani [18].

The accuracy of the ETo estimation improved substantially for each weather station by utilizing the locally calibrated ${CH}_C$ and ${EH}_C$ values. The results are consistent with studies performed by Almorox and Grieser [47], where they found that the best ETo estimates were obtained with the simultaneous calibration of $CH$ and $EH$, while considering the temperature coefficient value of 17.8 as a constant. Similarly, Shiri et al. [56] indicated that the HS equation performed better with locally calibrated coefficients. Furthermore, Almorox et al. [30] suggested values of 0.00206 and 0.49 for the Coronel Dorrego station in Argentina after calibrating both the $CH$ and $EH$ coefficients. Berti et al. [3] found values of 0.00193 and 0.517 for the flat territory of the Veneto region in northeastern Italy. Meanwhile, Patel et al. [31] suggested $CH$ values ranging from 0.00281 to 0.00259 and $EH$ values of 0.605 and 0.619 for a semi-humid climate; values of 0.0031 and 0.621 for a humid climate; and values of 0.00242 and 0.574 for a semi-arid climate in India. In contrast, Ferreira et al. [27] suggested $CH$ and $EH$ values ranging from 0.0009 to 0.0014 and 0.5881 to 0.8311, respectively, for Minas Gerais in Brazil.

Table 2. Accuracy of reference evapotranspiration (ETo) estimates using the Hargreaves-Samani (HS) equation after simultaneous calibration and validation of the $CH$ and $EH$ coefficients.

Station	Original HS Equation (CH = 0.0023 y EH = 0.5)				Calibration HS Equation (Constant CT = 17.8)
					Calibrated		Calibration				Validation
	NSE	PBIAS	MAE	RMSE	CH	EH	NSE	PBIAS	MAE	RMSE	NSE	PBIAS	MAE	RMSE
ANA	0.44	−5.20	0.19	0.22	0.0013	0.753	0.70	−0.43	0.01	0.16	0.67	1.00	0.03	0.18
CHQ	0.53	7.30	0.25	0.31	0.0016	0.590	0.85	−0.43	0.01	0.17	0.86	−0.27	0.01	0.17
DES	0.62	−8.30	0.29	0.33	0.0027	0.465	0.87	−0.52	0.02	0.20	0.88	−0.55	0.02	0.18
HNE	0.75	−4.90	0.23	0.28	0.0019	0.597	0.86	0.04	0.00	0.20	0.86	1.37	0.05	0.21
JLC	0.44	11.80	0.38	0.41	0.0021	0.495	0.94	−0.01	0.00	0.14	0.93	0.42	0.01	0.15
JUL	−0.57	−18.60	0.65	0.67	0.0022	0.608	0.92	0.02	0.00	0.15	0.92	0.55	0.02	0.15
LAM	0.87	−1.90	0.16	0.20	0.0023	0.499	0.87	−0.44	0.02	0.19	0.89	0.53	0.02	0.18
PNO	0.63	−8.80	0.31	0.33	0.0022	0.552	0.94	−0.19	0.01	0.13	0.94	0.20	0.01	0.13
PTN	−0.35	12.70	0.41	0.49	0.0012	0.698	0.75	−0.29	0.01	0.21	0.77	−0.20	0.01	0.20

NSE: Nash-Sutcliffe efficiency; PBIAS: percent bias (%); MAE: mean absolute error (mm/d); RMSE: root mean square error (mm/d).

shows the accuracy of the ETo estimates after calibrating and validating the $CH$ and $EH$ coefficients for the nine analyzed weather stations. The performance indicators showed a significant improvement in the NSE values, ranging from 0.71 to 0.94 during the calibration period and from 0.67 to 0.94 during the validation period. The ANA and PTN stations recorded the lowest NSE values, reaching 0.71 and 0.75 during calibration and 0.67 and 0.77 during validation, respectively.

The PBIAS values range from −0.52 to 0.04% during the calibration period, thus indicating a slight underestimation at the weather stations. In the validation period, the values fluctuated between −0.55 and 1.37%, with slight underestimations at the CHQ, DES, and PTN stations, which presented values of −0.27, −0.55, and −0.20%, respectively. Based on the criteria established by Moriasi et al. [51], the PBIAS values remained below ±5%, which is considered “very good”.

According to the MAE, the HS equation with the new $CH$ and $EH$ coefficient values that were simultaneously calibrated and validated showed low results at all the analyzed weather stations, ranging from 0.00 to 0.02 mm/d during calibration and from 0.01 to 0.05 mm/d during validation. In addition, the RMSE values varied between 0.13 and 0.21 mm/d in both the calibration and validation periods. The accuracy indicators used are in alignment with the studies performed by Berti et al. [3], Patel et al. [31], Ferreira et al. [27], and Zhu et al. [16], who found that after calibrating the $CH$ and $EH$ values, the MAE and RMSE indicators decreased. The findings of this study encourage the use of the HS equation considering the calibrated $CH$ and $EH$ values. To estimate the ETo at the meteorological stations, the calibrated $CH$ and $EH$ values from Table 2 must be substituted into equation 14.

It is important to mention that these coefficients are highly correlated; when one increases, the other tends to decrease, and vice versa. Adjusting both coefficients was the best alternative for calibrating the HS equation, except for the JLC, JUL, and LAM stations. Lima et al. [58] concluded that when calibrating the $CH$ and $EH$ coefficients simultaneously, better results were obtained.

3.3. Regionalization of the HS Coefficients

Regionalization of the CH_C and EH_C Coefficients

The regionalization of the HS coefficients CH_C and EH_C of the HS method for the PA is suggested. The MLR method was applied to develop the regionalization model, using the CH_C and EH_C coefficients, considering the geographic characteristics of longitude, latitude, and altitude.

The correlation analysis results between ${CH}_C$ and the geographic characteristics of longitude, latitude, and altitude showed correlations of 0.238, −0.862, and −0.525, respectively.

The latitude characteristic presented the highest correlation, followed by altitude and longitude. As a result, the geographic characteristics explained 85.9% (R² = 0.859) of the variability of ${CH}_C$ (Figure 4a). The p-value of 0.0145 for ${CH}_C$ indicates that the relationship between the independent variables (longitude, latitude, and altitude) and ${CH}_C$ is statistically significant at the 95% confidence level (α = 0.05). This means that there is strong evidence that at least one of the explanatory variables is related to ${CH}_C$.

This study found that the regional equation of the coefficient ${CH}_{\left(x\right)}$ could provide a better ETo estimate in areas with scarce data in the PA and that the value of R² is satisfactory for estimating ${CH}_C$ from geographic characteristics, according to the following equation:

$${CH}_{\left(x\right)}=-0.051867-0.000525\times X-0.001003\times Y+0.0000004\times Z+residual$$

(17)

The standard error for ${CH}_C$ was 0.00024. The confidence interval for the model parameter $b_0$ ranged from −0.11224 to 0.00850, while for $b_1$ it ranged from −0.00122 to 0.00017, for $b_2$ it ranged from −0.00164 to −0.00037, and for $b_3$ it ranged from −0.000001 to 0.000002, with a 95% confidence level. The residuals of ${CH}_C$ for the station points varied between −0.00025 and 0.00027 (Figure 4b). This equation is only applicable to the PA.

Regarding the results of the correlation analysis between the coefficient ${EH}_C$ and geographic characteristics of longitude, latitude, and altitude, the results showed a correlation of 0.103, 0.644, and 0.674, respectively.

The characteristics of altitude and latitude exhibited higher correlations than that of longitude. Thus, the evaluated geographic characteristics managed to explain 74.4% (R² = 0.744) of the variability of ${EH}_C$ (Figure 5a). The value of p = 0.0611 is close to the common significance threshold (α = 0.05), indicating that the relationship between the independent variables (longitude, latitude, and altitude) and ${EH}_C$ is marginally not significant. This suggests that there may be a relationship, but it is not strong enough with the current data. Although the p-value is not significant, the confidence interval (CI) suggests that the true values lie within a reasonable range (Figure 5b).

The regional equation of the coefficient ${EH}_{\left(x\right)}$ could provide a more accurate ETo estimate for areas with scarce data in the PA, and the R² value is satisfactory for estimating ${EH}_C$ in the PA based on geographic characteristics, according to the following equation:

$${EH}_{\left(x\right)}=11.503632+0.1242313\times X+0.1485293\times Y+0.0000143\times Z+residual$$

(18)

The standard error for ${EH}_C$ was 0.061. The confidence interval for the model parameter $b_0$ ranged from −3.81129 to 26.81855, while for $b_1$ it ranged from −0.05279 to 0.30125, for $b_2$ it ranged from −0.01224 to 0.30930, and for $b_3$ it ranged from −0.00031 to 0.00033, with a 95% confidence level. The residuals of ${EH}_C$for the station points ranged from −0.057 to 0.084 (Figure 5b). This equation is only applicable for the PA.

Figure 6 illustrates the spatial distribution of the coefficients ${CH}_{\left(x\right)}$ and ${EH}_{\left(x\right)}$, which are essential in the HS equation to estimate the ETo values in the PA. In Figure 6a, the distribution of coefficient ${CH}_{\left(x\right)}$ is observed, with values ranging from 0.0009 to 0.0042. Similarly, Figure 6b represents the distribution of the coefficient ${EH}_{\left(x\right)}$, with values fluctuating between 0.35 and 0.80.

The spatial distribution of the coefficients ${CH}_{\left(x\right)}$ and ${EH}_{\left(x\right)}$ reveals notable regional variability. This variability suggests that ETo is not uniform in the PA and is influenced by local factors, such as altitude, proximity to water bodies, and the specific climatic characteristics of each region. In Figure 6a, the lowest values of ${CH}_{\left(x\right)}$ are predominantly found in the north, while the highest values are located in the south. On the other hand, Figure 6b shows that the lowest values of ${EH}_{\left(x\right)}$ are located in the south, while the highest values are distributed toward the north. Although at first glance ${CH}_{\left(x\right)}$ and ${EH}_{\left(x\right)}$ (Figure 6) appear to exhibit a similar spatial distribution, a more detailed analysis reveals significant differences. In Figure 6a, the highest values of ${CH}_{\left(x\right)}$ are found in the south, whereas in Figure 6b, the highest values of ${EH}_{\left(x\right)}$ are found in the north. This difference suggests that while both are related to ETo, they respond to different climatic and geographic factors.

In summary, Figure 6 illustrates that the simultaneous calibration of the coefficients ${CH}_{\left(x\right)}$ and ${EH}_{\left(x\right)}$ provides a detailed insight into the variability of the ETo in the PA. These results can be used to improve ETo estimation models, thereby facilitating more precise planning and management of water resources based on specific local conditions.

The regional equation of Hargreaves-Samani (HSR) for estimating ETo in the PA, based on the coefficients ${CH}_{\left(x\right)}$ and ${EH}_{\left(x\right)}$, is as follows:

$${ET}_{O,HSR}={CH}_{\left(x\right)}\times R_a\times (T_{mean}+17.8){\left(T_{max}-T_{min}\right)}^{{EH}_{\left(x\right)}}$$

(19)

where ${CH}_{\left(x\right)}$ and ${EH}_{\left(x\right)}$ represent the estimated values at any location in the PA.

4. Conclusions

The HS model, which relies on the air temperature, is the simplest and most practical approach for estimating ETo. The ETo estimates using the HS method were evaluated against PM ETo estimates in the PA. This evaluation covered the period from January 2000 to December 2019 on a daily and monthly basis. The ETo values calculated using the PM and HS models exhibited a strong correlation but also significant bias, leading to notable underestimations and overestimations of ETo. This indicates that the model requires local calibration before use. Therefore, it was necessary to make changes to the HS equation for its applicability in the PA study area.

The calibration process followed by validation substantially improved the performance of the calibrated HS equation for each weather station. In addition to calibrating the $CH$ and $EH$ coefficients, the calibrated HS models outperformed the original HS model for each weather station, obtaining better ETo estimates with the calibrated HS, mainly by eliminating biases. New values of $CH$ and $EH$ coefficients of the empirical HS model for the PA were presented.

The multiple linear regression model was used to regionalize the $CH$ and $EH$ coefficients. The HS equation, along with the regionalized $CH$ and $EH$ coefficients, could significantly improve the ETo estimation in regions with limited information within the PA. However, in the absence of complete climate data measurements, more conclusive results could be achieved with the availability of a larger number of weather stations recording the necessary climatic data that it uses, the standard PM equation, in the area of southwestern PA. Therefore, it is recommended to exercise caution when applying the HS equation with the new values of the regionalized coefficients $CH$ and $EH$ in areas distant from the weather stations. Furthermore, it is suggested to conduct more studies to analyze the applicability of the proposed approach, considering the influence of other additional climatic factors.