Application of different calibration methods for the Hargreaves-Samani model in Southeast Brazil

The evapotranspiration is an important variable in the hydrological cycle and one of the main components of water balance in the soil. The use of simplified equations is a potential alternative to estimate reference evapotranspiration (ET0) when there is limited meteorological data. The objective of this study was to test different methods to estimate ET0 using the Hargreaves-Samani equation (HS) under different meteorological conditions. ET0 was calibrated with HS, adjusting the HS coefficient (HC), using different methods. Adjustment by linear regression was also performed. ET0 was also estimated using the original HS and Penman-Monteith FAO-56 methods with limited climatic data (PML). The performance of the methods (mean absolute error, mm day) to estimate evapotranspiration, based on Penman-Monteith, were: PML (1.46); HS (0.68); Vanderlinden et al. (2004) (0.81); Martí et al. (2015) (0.77); and linear regression (0.53). The PML method presented the worst performance. Adjustment by linear regression presented a better performance than the adjustments of HC, improving the ET0 estimates by up to 30%, and it is considered the most recommendable of the methods tested.


Introduction
Freshwater availability on the Earth has diminished in recent years due to a combination of more frequent droughts and intensification of demands on hydrological resources from agricultural, industrial and urban applications (WWAP-UNESCO, 2015). Evapotranspiration (ET) is an important variable of the hydrological cycle and one of the main components of the hydrological balance in soil (Carvalho, Rocha, Bonomo, & Souza, 2015). However, its direct determination is difficult and costly, requiring specialized personnel and equipment. Therefore, indirect ET estimates are fundamental to improve the use of hydrological resources in forestry and agricultural areas.
The use of the method proposed by Hargreaves and Samani (1985) (HS) is indicated as an alternative to estimate reference evapotranspiration with limited meteorological data (Allen et al., 1998;Sentelhas et al., 2010). The HS model is an empirical method initially developed in California (USA) that only requires extraterrestrial solar radiation and maximum and minimum air temperature data. However, various studies have shown that the HS method overestimates ET0 in humid regions and underestimates it in dry regions (Bezerra, Moura, Silva, Lopes, & Silva, 2014;Carvalho et al., 2015;Fanaya Júnior, Lopes, Oliveira, & Jung, 2012;Kisi, 2014;Tagliaferre, Silva, Rocha, Santos, & Silva, 2010;Trajkovic, 2007).
The objective of the present study is to apply and evaluate calibration methods for the Hargreaves-Samani equation under different meteorological conditions in the state of Espírito Santo, with limited meteorological data.

Methods
The present research is a case study of a quantitative nature. The research was developed in a natural environment, where the researcher made the analysis of the collected descriptive data. Another characteristic of this study is that it is laboratory research, since the analysis employed, which involved the obtaining and organizing data and execution of statistical adjustments, was all carried out in a computational environment, that is, the research was controlled (Pereira, Shitsuka, Parreira, & Shitsuka, 2018).

Area and data used in the study
The study was developed for the state of Espírito Santo (Figure 1), with a total area of 46,184.1 km², geographically situated between the latitudes 39°38' and 41°50' W and between the longitudes 17°52' and 21°19' S. For the state of Espírito Santo, data from nine automatic weather stations were obtained.
Eight neighboring weather stations outside the state were also included, to minimize the frontier effect, with seven automatic weather stations and one conventional one. In total, 17 weather stations were used with their spatial distribution (Fig. 1).
The data records were analyzed, with the days with failures being discarded to perform this study. The description of the weather stations, with the Köppen climate classification (Alvares, Stape, Sentelhas, Gonçalves, & Sparovek, 2013), is presented in Table 1.  In most of the state, the total mean annual rainfall ranges from 1,000 to 1,600 mm, and the annual mean air temperature ranges from 18 to 24 °C, varying according to the climate of each region (Alvares et al., 2013).

Penman-Monteith FAO-56 Method (PM)
The standard Penman-Monteith FAO-56 equation (Allen et al., 1998) was applied to estimate evapotranspiration, the values (ET0 PM ) of which were used in this study as a reference for all the adjustments and comparisons undertaken. The calculation to obtain the ET0 PM was performed in a standardized manner according to Allen et al. (1998). Heat flow in soil (G) was equal to zero and the albedo of the reference culture was equal to 0.23. Research, Society and Development, v. 9, n. 8, e368984811, 2020 (CC BY 4.

Penman-Monteith Method with limitation of meteorological data (PML)
For reasons of comparison, the realization of the Penman-Monteith method was evaluated, only using air temperature data to determine ET0. Allen et al. (1998), in the FAO bulletin N°56, presented the methodology that should be used in the absence of wind speed, global solar radiation and relative humidity data.
Wind speed estimates depend on local climate and the season of the year. Where wind speed data is unavailable, the value 2 m s -1 can be used as a provisional estimate. This value is a mean from more than 2,000 weather stations studied around the world (Allen et al., 1998).
In the absence of relative humidity data or when data quality is unreliable, the estimate of actual vapour pressure (ea) can be obtained assuming that the dew point temperature (Tpo) value is close to minimum temperature (Tmin) (Allen et al., 1998). This implicitly presupposes that at sunrise, when the temperature is close to Tmin the air is practically saturated, that is, the relative air humidity is nearly 100%. Therefore, ea was estimated using Tmin to represent Tpo in Tetens' equation.
For the calculation of the solar radiation (Rs, MJ m -2 day -1 ), Eq. 1 is used, proposed by Hargreaves and Samani (1985), that estimates global solar radiation as a function of the maximum (Tmax) and minimum daily air temperature (thermal amplitude).
Where Ra is the extraterrestrial radiation (MJ m -2 day -1 ), and kRs is the adjustment coefficient (°C -0.5 ).
The kRs value varies from 0.16 to 0.19, with the value 0.16 used for interior regions (further than 20 km from the ocean) and 0.19 for coastal regions (up to 20 km from the ocean) (Hargreaves & Samani, 1985). Ra values were also calculated according to the recommendations of Allen et al. (1998). Research, Society and Development, v. 9, n. 8, e368984811, 2020 (CC BY 4.

Hargreaves-Samani Method (1985) (HS)
The method proposed by Hargreaves and Samani (1985) Where  is the latent heat of water vaporization (2.45 MJ kg -1 ), and Tmean is the mean daily air temperature (°C) at 2 m height (mean between Tmax and Tmin).
The HS equation was adjusted for the study area using different methods: Hargreaves-Samani coefficient (HC) adjustment, using the Vanderlinden et al. (2004) and Martí et al. (2015) methods; and the ET0 HS adjustment using simple linear regression. These methodologies are described in the following items.

Vanderlinden et al. (2004) method for the adjustment of HS equation
The method proposed by Vanderlinden et al. (2004) (Eq. 4) to adjust HC in Espírito Santo, was tested. Firstly, the daily HC values observed were determined for each weather station using Eq. 5. Based on the daily HC values, their mean was calculated for each station.
With the mean air temperature, thermal amplitude and mean HC data for each station, a regression equation was adjusted (k1 and k2) to obtain the regional HC for the state.

Martí et al. (2015) method for adjustment of the HS equation
The methodology proposed by Martí et al. (2015), which takes new input variables and mean air temperature and thermal amplitude into account, was used to adjust HC. The input variables were: latitude (), longitude (), altitude (z), mean air temperature (Tmean) and thermal amplitude (∆T). All the information for these variables was obtained from the stations studied. With the input variables and the mean HC for each station, the multiple linear regression equations were used to adjust the regional HC. Different combinations of the input variables were adopted to adjust the models. Firstly, models were proposed using only Tmean and ∆T. Following this, the temperature inputs were combined with the latitude, longitude and altitude.

Evapotranspiration adjustment of the HS equation using linear regression
In the FAO bulletin n°56, Allen et al. (1998), recommend that the ET0 adjustment estimated by the HS equation be performed using a simple linear regression equation. In this case, the PM method was used as an independent variable to perform the ET0 adjustment on a daily timeframe (Eq. 6).
The adjustment of the empirical parameters (a and b) was performed using the data from all the weather stations simultaneously (Table 1).

Evaluation of the performance of models
For the evaluation of the results, five statistical indices were used: the absolute mean error (MAE), mean error (MBE), root square mean error (RMSE) (Willmott, 1982), relative RMSE (RRMSE) (Martí et al., 2015) and the coefficient of determination of linear regression between observed and estimated ET0 (r²).
Where ŷ is the ET0 estimated by the several methods evaluated, y is the ET0 obtained by PM (reference), and n is the number of data.
The evaluation of the performance of the models was performed individually for each of the 16 weather stations used in the study. In the evaluation of the performance of the model adjusted by linear regression (Eq. 6), the leave-one-out cross-validation was used: for each station evaluated, the model was adjusted using the data from the other 15 stations, that is, the model was evaluated with data not used in its adjustment, seeking to obtain a non-tendentious evaluation.
As the Vanderlinden et al. (2004) and Martí et al. (2015) models were adjusted using the data from all the stations simultaneously, as was done by the authors themselves. The evaluation of the performance of these models was consequently performed with the same data used in their adjustment. This procedure was carried out like this to allow comparison of the results from the present study with those obtained by the aforementioned authors.  The k1 and k2 values found in this study were close to the values found by Vanderlinden et al. (2004) in southern Spain. However, these authors observed an r 2 of 0.90, significantly higher to that obtained in this study. The superior adjustment of the HC by the authors can be explained, probably, due to southern Spain having a homogeneous climate, different to Espírito Santo, which contains different climates (Table 1) (Table 2). Research, Society and Development, v. 9, n. 8, e368984811, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i8.4811 Table 2. Adjustment of the models proposed by Martí et al. (2015) to estimate the coefficient of Hargreaves-Samani (HC) in the state of Espírito Santo, Brazil. Temperature, latitude, longitude, altitude HC = 6.61 x 10 -4 + 3.43 x 10 -5 Tmean -1.13 x 10 -4 ∆T + 1.40 x 10 -6  -4.11 x 10 -5 φ -1.14 x 10 -7 z 0.046 0.56 0.00007 HC: adjusted Hargreaves-Samani coefficien, Tmean: mean air temperature, ΔT: thermal amplitude, z: altitude, : latitude, φ: longitude, RRMSE: relative root mean square error (dimensionless), r 2 : adjusted coefficient of determination, MAE: mean absolute error (dimensionless). Source: Elaborated by the authors. Table 2 presents the models adjusted for HC, using the Martí et al. (2015) method, using multiple linear regression. Several combinations were performed, with model 1 depending only on temperature data, whereas models 2 to 8 are combinations of the temperature data with geographic information.

Model
The use of the geographic coordinates that are easily accessible for any station or location, did not present a significant difference in relation to the model only temperaturebased. In eastern Spain, an improvement of the models to estimate HC with geographic coordinates was observed by Martí et al. (2015). These authors also observed a better adjustment of the models, in relation to this study, with RRMSE of up to 0.037, which could also be explained by the homogeneity of the climate in eastern Spain. To evaluate the adjustment of Martí et al. (2015) in the present study, model 7 was adopted, due to its ease of application and better performance. These results are presented and compared with the other models later on.
When adjusting the ET0 of the HS equation using linear regression (Eq. 6), the Eq. (12) was obtained, with r 2 = 0.78 significant at 1% probability by F test (p<0.01).
HS 00 ET 1.08 ET 0.81 =− (12) Figure 3 shows the dispersion graphs demonstrating the quality of the adjustment of the methods evaluated in this study, in relation to the Penman-Monteith method with complete data, applied simultaneously with all stations. The general performance of the models evaluated in this study, by the statistical indices, is presented in Table 3. Generally, in Espírito Santo, the HS method overestimated ET0 by Development, v. 9, n. 8, e368984811, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i8.4811 16 0.47 mm day -1 (MBE). Similar results were observed by Reis et al. (2007) and Carvalho et al. (2015), in the same state. Even though this value seems low, it is a daily mean error, that is, a daily hydrological sequence was performed showing an accumulative error in the ET0 of 172 mm year -1 . The Martí et al. (2015) and Vanderlinden et al. (2004) methods presented coefficients of determination (r 2 ) of 0.54, with the lower value found between the methods tested. In their better model to estimate HC in Spain, Martí et al. (2015) encountered RRMSE of 0.19 and r 2 of 0.87, significantly higher than that found in this study using the same method.
The PML method presented the worst performance of the methods, differing on average by 1.46 mm day -1 from the PM method. Using 72 weather stations in Florida, Martinez and Thepadia (2010), obtained similar performance for the PML and HS methods, in which the HS method provided better ET0 estimates than the PML method.
The adjustment of the HS equation, using linear regression, corrects the ET0 without altering the r 2 values, as illustrated in Figure 4. This adjustment alters the inclination and the vertical position of the line of tendency, without altering the data dispersion around it, resulting in null MBE values (this detail only can be seen putting ET0 HS on the x-axis.). In general, adjustment by linear regression presented better performance than the HC adjustments, which improved the ET0 estimation by up to 30%, in relation to the original HS method. It is also worth noting that in the present study, independent stations were used to test the adjustment by linear regression, different to the adjustment methods for the HC. estimates depending on climatic condition (Allen et al., 1998;Fernandes et al., 2012;Alencar et al., 2015;Shiri et al., 2015). The RMSE values referent to the PML method, in Espírito Santo, varied between 1.07 and 2.08 mm day -1 . These values were higher than those found for daily estimates in Bulgaria, Tunisia and in the Mediterranean, by Popova, Kercheva and Pereira (2006) (0.52-0.58 mm day -1 ), Jabloun and Sahli (2008) (0.41-0.80 mm day -1 ) and Todorovic, Karic and Pereira (2013) (0.60-0.65 mm day -1 ), respectively. In southeastern Brazil, (Carvalho et al., 2015) found values between 0.05 and 0.85 mm day -1 .

The adjustment methods by linear regression presented better performance in Serra dos
Aimorés-MG, and worse performance in Presidente Kennedy-ES. In the location of its worst performance, the adjustment methods by regression underestimated ET0, presenting MAE of up to 0.78 mm day -1 .
Even with a 5-year meteorological data series, adjustments by linear regression provided satisfactory ET0 estimates. However, new studies to calibrate the Hargreaves and Samani (1985) equation by linear regression should be performed in other regions of Brazil, with a longer meteorological data series, to provide better ET0 estimates.

Final Considerations and Suggestions
Adjustments by linear regression obtained better performance than the HC adjustments with the Vanderlinden et al. (2004) and Martí et al. (2015) methods.
The general adjustment method by linear regression is considered the most recommendable of the methods tested due to its ease of application.
In order of increasing efficiency for the evapotranspiration estimation methods with limited meteorological data, we have: PML, Martí et al. (2015), Vanderlinden et al. (2004), Hargreaves and Samani (1985), and general adjustment by linear regression.
Future research may be more promising using the method proposed by Zanetti et al.
(2019) to calibrate empirical temperature-based models to estimate ET0. Using this method, on hotter and drier days (with greater thermal amplitude), when ET0 is higher, it can be estimated more accurately using a specific calibration for each day's condition. It is a simple method that can improve results. This method can be applied to calibrate any empirical methods to estimate ET0. Also, future research may improve the quality of results if, in addition to temperature, other variables are added to the empirical models, such as relative humidity. Currently, the relative humidity can also be obtained in a practical and inexpensive way, using automatic psychrometers built with sensors such as thermistors and thermocouples, using low-cost platforms such as Arduino  .