REFERENCE EVAPOTRANSPIRATION THROUGH HARGREAVES METHOD USING THE SOLAR RADIATION ESTIMATION FOR GOIÁS STATE, BRAZIL

This study evaluated the Hargreaves model (HG) with seasonal adjustments of the calibration coefficient (Krs) of the radiation equation to estimate the reference evapotranspiration (ETo) in 23 weather stations in Goiás State, Brazil, in comparison to the Penman-Monteith FAO (PM-FAO) standard method. The models were evaluated using Pearson’s correlation coefficient, Willmott’s agreement index, relative error, absolute mean error and root mean square error. The Krs values ranged from 0.146 to 0.189 ° C-0.5, while ETo PM-FAO ranged from 3.68 to 4.79 mm d-1; ETo HG from 3.99 to 5.16 mm d-1 and ETo HG-Krs from 4.15 to 5.02 mm d-1 in the annual period. Seasonal adjustments resulted in values of 0.144 to 0.205 ° C-0.5 for the dry period, from April to September, and 0.144 to 0.146 ° C-0.5 for the rainy period, from October to March. The first quarter (summer), presented Krs values from 0.150 to 0.175 ° C-0.5; the second quarter (autumn), from 0.154 to 0.218 ° C-0.5; the third quarter (winter), from 0.139 to 0.206 ° C-0.5; and, finally, the fourth quarter (spring) of 0.141 to 0.166 ° C-0.5. Thus, the use of the seasonally adjusted model proved to be viable for the estimation of ETo, in view of the simplicity and its good adherence to the standard method.


INTRODUCTION
Reference evapotranspiration (ETo) represents the water demand of a reference crop according to the weather elements of a given location. In addition, it is an extremely important parameter for the definition of the water needs of the crops. The empirical and physical models for the estimation of ETo based on the monitoring of weather elements has greatly evolved from the original works of Thornthwaite (1948) and Penman (1948) to the Penman-Monteith equation parameterized by FAO, with a hypothetical crop as a reference surface very similar to grass. Allen et al. (1998) recommend the Penman-Monteith FAO method as a method for ETo determination with weather parameters due to its consistent solid formulation, requiring data on global solar radiation, wind speed, temperature and relative air humidity.
The availability of weather information for the ETo estimate is still reduced, considering the number of stations in the Brazilian network, which currently has 573 automatic weather stations and 213 conventional surface stations (INMET, 2018), which implies in an automatic station density for almost 15,000 km². The state of Goiás has 26 automatic weather stations and nine conventional surface stations as part of the national network, which results to an automatic station density of almost 12,600 km². These stations are not evenly distributed across the Brazilian territory. Grego et al. (2017) evaluated that the sample density of weather stations in the northern region of Minas Gerais influenced the spatial variability of evapotranspiration from a distance of 80 km.
Considering the difficulty in obtaining meteorological data due to the lack of measurement or failures in the data series, it is necessary to use empirical models, such as the Hargreaves model, which estimates ETo from data of maximum and minimum air temperatures, only (HARGREAVES; SAMANI, 1985, HARGREAVES;ALLEN, 2003). Fernandes et al. (2012), Conceição (2013), Vicente et al. (2014) andLima Junior et al. (2016) pointed out the viability of the Hargreaves model for different regions in Brazil due to its simplicity. Given the above, the objective of this work was to evaluate the performance of the Hargreaves model in determining ETo for the state of Goiás, making local and seasonal adjustments to estimate global solar radiation.

MATERIAL AND METHODS
The Penman-Monteith FAO (PM-FAO) and Hargreaves (HG) methods were used to estimate the ETo for the state of Goiás. The 23 automatic meteorological stations in the network of the National Institute of Meteorology ( Figure 1) were used based on in their complete daily data as shown in Table 1.
The equation proposed by Hargreaves and Samani (1985) (Equation 2) was tested for the climatic conditions in the State of Goiás. (2) Where R s -global solar radiation, mm d -1 . The radiation values can also be given in MJ m -2 d -1 , as it is more common.
The R s is estimated from extraterrestrial solar radiation (R a ) and thermal amplitude (HARGREAVES; SAMANI, 1982), according to Equation (3), as it follows. (3) Where, K rs -calibration coefficient, °C -0.5 ; T xmaximum air temperature, °C; T n -minimum air temperature, °C. The difference between T x and T n indirectly accounts for the cloudiness effects (HARGREAVES; ALLEN, 2003).   The K rs adjustment coefficient of Equation (3) was estimated at each location for different periods of time, using non-linear regressions with the Microsoft Excel® Solver® tool. This adjustment has been used for different regions in Brazil (CONCEIÇÃO, 2013;LIMA JUNIOR et al., 2016;BARROS et al., 2017;THEBALDI, 2018). The tool was used in this study in order to minimize the error between the estimated values of R s and that measured at the weather station.
In order to guarantee the data integrity and to allow the comparison between the models, the evaluations were made only for situations where it was possible to obtain complete daily data from the weather station. The results obtained were compared with the ET o from the application of the standard PM-FAO method. The comparisons had their performance evaluated using simple linear regression, by means of Pearson's correlation coefficient (r), according to Equation (5).
The performance index (c), as stipulated by Camargo and Sentelhas (1997) is obtained by the product between Pearson's correlation coefficient (r) and Willmott's Agreement index (d). The estimation of the error between the models was given by three measures, namely: the relative error (RE); the mean absolute error (MAE); and the root of the mean square error (RMSE); given by Equation (9), (10) and (11), respectively.
(8) (9)     Although the amplitude of this coefficient is not high, it was possible to demarcate four very different regions for the annual period ( Figure 2a) and for the dry period (Figure 2b), with a very similar pattern with continuous and well-defined regions. For the rainy season ( Figure 2c), a defined pattern was not found, showing a more erratic variation of the K rs coefficient, which may occur due to a greater daily variation of cloudiness in the rainy season, as pointed out by Baratto et al. (2017) when analyzing the variability of this coefficient at different temporal scales for different stations in the country.

RESULTS AND DISCUSSION
As expected, the second and third quarters (Figures 2e and 2f) showed a K rs coefficient pattern with continuous and delimited regions similar to the dry semester, as it covers the same time period. For the first and fourth quarters (Figures 2d and  2g), the variation in the coefficient also followed a distribution close to that of the rainy season.
For the annual period, the highest K rs value was found in Posse, 0.189 °C -0.5 , and the lowest value, 0.146 °C -0.5 , was found in Itapaci. Allen (1997) proposed K rs equal to 0.17°C -0.5 for inland regions and 0.20°C -0.5 for coastal regions, including an adjustment according to the altitude. Allen et al.
(1998) proposes values of 0.19°C -0.5 for coastal regions and 0.16 °C -0.5 for inland regions. But Todorovic et al. (2013) argues on the value of K rs to be adopted, considering that its variation may be lower in temperate climates, and large where the climate is tropical and subtropical, as in Brazil. Table 5 shows the analysis of the adjustments based on ET o , using the Penman-Monteith FAO method (PM-FAO) as a reference for comparison with the Hargreaves method (HG) and the application of K rs coefficients (HG-Krs). LEITE, C. V. et al. It is underlined that only five stations did not present a better adjustment to the HG -K rs model to estimate ET o in relation to the standard model PM-FAO, namely: Alto Paraíso de Goiás, Cristalina, Goiânia, Monte Alegre de Goiás and Niquelândia. However, these stations are not grouped in any specific region neither are close to each other. All the other 18 stations showed a better adjustment for the model with the K rs coefficient, evaluated mainly by means of the lowest error value (RE, MAE and RMSE) and by the highest values of the Willmott agreement (d) and performance indexes (c).
Afterwards, the seasonality analysis of this coefficient was carried out, estimating its values for semiannual periods. This division is justified as an approach of the dry (April to September) and rainy (October to March) periods. Considering the dry period, the highest K rs value was found in Posse, 0.205°C -0.5 , and the lowest value, 0.144°C -0.5 , in Itapaci. In relation to the rainy season, the values of this coefficient varied from 0.146°C -0.5 in Alto Paraíso de Goiás up to a maximum of 0.169°C -0.5 in Posse.
Tables 6 and 7 show the analysis of the HG and HG -K rs models compared to the PM-FAO method, in order to assess the quality of these adjustments. Four stations did not show improvement in the model with the seasonal adjustment for K rs for the dry period, namely: Caiapônia, Goiás, Monte Alegre de Goiás and São Simão, as it can be seen from the values of the d and c indexes. Nevertheless, all stations showed better results for the d and c indices for the rainy season, in addition to a less error (RE, MAE and RMSE) for the HG -K rs model, when compared to the standard PM-FAO method.   Finally, the K rs coefficient was analyzed for quarterly periods, which is a reasonable approach of the summer (January to March), autumn (April to June), winter (July to September) and spring (October to December) seasons.
For the first quarter, the highest K rs was found in Cristalina, 0.175 ºC -0.5 , and the lowest in Alto Paraíso de Goiás, 0.150 ºC -0.5 . For the second quarter, the maximum and the minimum values of the coefficient was 0.218 ºC -0.5 for Niquelândia and 0.154 ºC -0.5 for Itapaci, respectively. As for the third quarter, the coefficient ranged from 0.206 ºC -0.5 in Posse to 0.139 ºC -0.5 in Itapaci. In the fourth and last quarter, the highest and the lowest K rs values was 0.166 ºC -0.5 for Posse and 0.141ºC -0.5 for Itapaci.
Tables 8, 9 10 and 11 show the analyses of the Hargreaves model (HG) without and with K rs coefficient (HG -K rs ) compared to the standard method (PM-FAO) for assessing the adjustment quality.
REFERENCE EVAPOTRANSPIRATION THROUGH HARGREAVES METHOD USING THE SOLAR RADIATION...    Only three stations showed no improvement in the model with the seasonal adjustment of K rs for the first quarter, namely: Cristalina, Monte Alegre de Minas and Posse. For the second quarter, only 9 stations showed improvement in the model with the seasonal adjustment of the K rs coefficient; Alto Paraíso de Goiás, Goiânia, Itapaci, Jataí, Mineiros, Morrinhos, Posse, São Simão and Silvânia. In relation to the third quarter, four stations would not show improvement in the model with the adjustment, that is, in the stations of Aragarças, Goiás, Itumbiara and São Simão, it was proved to be better to apply the Hargreaves model in its classic configuration, without adjustments. In the fourth and last quarter, only the Cristalina and Posse stations did not perform better when a seasonal calibration of the Krs coefficient was applied. Martí et al. (2015) found better results for the Hargreaves model adjusted on larger time scales. The estimate of this work for K rs was better due to the data on temperature, longitude, altitude and qualitative assessments of wind speed, although they recognize that the introduction of more variables impairs the application of the model. Carbone et al. (2016) found better results for ET o through the Hargreaves model using measured solar radiation rather than the estimated one as a function of maximum and minimum temperature and extraterrestrial solar radiation (R a ). The authors attributed this fact to the relevance of radiation for estimating ET o . Several authors chose not to estimate the K rs adjustment coefficient, proceeding only with the linear regression of ET o values between the HG and PM-FAO models, as it was done by Bachour et al. (2013), Mendicino and Senatore (2013), Vicente et al. (2014), Berti et al. (2014). Valiantzas (2018), however, despite using nonlinear regression, makes only statistical adjustments for calibration of a new model of global solar radiation as a function of temperature and air relative humidity, finding good results for this estimate.

CONCLUSIONS
• The use of the Hargreaves model to estimate the reference evapotranspiration for the state of Goiás was shown to be a feasible alternative due to the reduced need for climatic data and its good adherence to the standard Penman-Monteith model parameterized by FAO.
• The local and seasonal calibration of the model based on the estimate of global solar radiation using the Hargreaves equation also improved the quality of the model.
• It is recommended the evaluation of the tests performed in this study for other stations and, preferably, for longer time scales such as quarterly or monthly periods.