Performance of different methods for reference evapotranspiration estimation in Jaíba , Brazil

Desempenho de diferentes métodos de estimativa da evapotranspiração de referência em Jaíba, Brasil R E S U M O O método de Penman-Monteith (PM-FAO) é considerado padrão para estimativa da evapotranspiração de referência (ET0), mas exige vários dados meteorológicos, que muitas vezes não estão disponíveis. O objetivo deste trabalho foi avaliar o desempenho do método de PM-FAO com dados meteorológicos limitados e de outros métodos como alternativas para estimativa da evapotranspiração de referência no município de Jaíba, Brasil. Foram utilizados dados meteorológicos diários de 2007 a 2016 da estação do Instituto Nacional de Meteorologia. Os valores de evapotranspiração de referência diários foram aleatorizados, sendo que, 70% desses foram utilizados para determinar os parâmetros de calibração da evapotranspiração de referência para as equações de cada método sob estudo. Os dados restantes foram utilizados para testar a calibração em relação ao método padrão. Na avaliação de desempenho foram utilizados os coeficientes de Willmott, de confiança e a raiz do erro quadrático médio. Quando houve ausência de um dado meteorológico, seja radiação solar, umidade relativa do ar ou velocidade do vento, ou na ausência simultânea de velocidade do vento e umidade relativa do ar o método de PM-FAO apresentou os melhores desempenhos, sendo assim recomendado para Jaíba. O método de PM-FAO com dois dados faltantes, sendo um deles a radiação solar, apresentou desempenho intermediário. Os métodos que utilizaram apenas dados de temperatura do ar não são recomendados para a região.


Introduction
On a vegetated surface, evapotranspiration consists in the transfer of water from the soil-plant system to the atmosphere.Adequate estimation of evapotranspiration is fundamental for a project and management of irrigation, hydrological studies, among others (Antonopoulos & Antonopoulos, 2017).
In order to know the evapotranspiration of a crop, particularly to quantify the water to be applied through irrigation, it is necessary to previously know the reference evapotranspiration (ET 0 ).The Penman-Monteith method (FAO-PM) was parameterized and then recommended as a standard by FAO (Food Agriculture Organization) to estimate ET 0 , and has been adopted in various parts of the world due to its accuracy (Allen et al., 1998).
As a disadvantage, the method requires a large amount of measured meteorological data, which are often unavailable or not reliable, especially in non-developed countries, thus demanding a simpler approach and use of other alternative methods to estimate ET 0 (Landeras et al., 2008;Martí et al., 2015).
Due to the lower accuracy of these alternative methods to estimate ET 0 , many authors calibrated ET 0 through linear regression in relation to the FAO-PM method (Fanaya Júnior et al., 2012;Cunha et al., 2013;Oliveira et al., 2015).On the other hand, only a few calibrated ET 0 values of the FAO-PM method using limited meteorological data (Carvalho et al., 2013;Alencar et al., 2015;).
The municipality of Jaíba is located in the Northern region of Minas Gerais and has the largest public irrigated perimeter in Latin America, with capacity to irrigate an area of 65,879 ha.In 2015, the irrigated area was 11,182 ha and water consumption was 334.6 million m 3 (DIJ, 2016).
Considering the importance of the water resources and the need for good use of water through correct ET 0 estimation, this study aimed to evaluate the performance of the FAO Penman-Monteith method with limited meteorological data and other methods as alternatives to the FAO Penman-Monteith method with all required data for the municipality of Jaíba-MG, Brazil.

Material and Methods
Reference evapotranspiration (ET 0 ) was estimated using daily meteorological data from an automatic station of the National Institute of Meteorology (INMET), relative to 10 years of the historical series (2007 to 2016), located in the municipality of Jaíba, Northern Minas Gerais, Brazil (15° 05' 09'' S; 44° 01' 00'' W; 454 m).The climate of the region is Aw according to Köppen's classification.
The following meteorological data were used: mean, maximum and minimum air temperature (ºC) and relative humidity, solar radiation (MJ m -2 d -1 ), atmospheric pressure (hPa) and wind speed at 10 m height (m s -1 ).Wind speed was corrected to height of 2 m by multiplying the value by the coefficient 0.7480, according to Allen et al. (1998).
The standard method used to estimate ET 0 was FAO Penman-Monteith (FAO-PM) with complete meteorological data (Eq.1)(Allen et al., 1998).ET 0 was also estimated by the FAO-PM method with absence of one, two and three meteorological variables per time, generating seven combinations.The only meteorological variable that remained in all combinations was air temperature.T -mean air temperature, °C; u 2 -wind speed at 2-m height, m s -1 ; e s -saturation water vapor pressure, kPa; e a -actual water vapor pressure, kPa; s -slope of the saturation water vapor pressure curve, kPa ºC -1 ; and, γ -psychrometric constant, kPa ºC -1 .
The studied combinations were: FAO-PM without solar radiation data (Rs); FAO-PM without relative air humidity (RH); FAO-PM without wind speed (WS); FAO-PM without Rs and RH; FAO-PM without Rs and WS; FAO-PM without WS and RH, and lastly FAO-PM without WS, RH and Rs.
In the absence of solar radiation, this variable was estimated using Eq. 2 and, in the absence of relative air humidity data, the value of e a (actual water vapor pressure in the atmosphere) was estimated by Eq. 3, according to Allen et al. (1998).In the absence of wind speed, the estimation considered the daily mean value relative to the studied 10-years period, 0.86 m s -1 , due to the better performance compared with the value proposed by Allen et al. (1998), equal to 2.0 m s -1 (Silva et al., 2010).where: R a -solar radiation at the top of the atmosphere, MJ m -2 d -1 ; T max -maximum air temperature, °C; and, T min -minimum air temperature, °C.
To improve the performance of the methods, ET 0 values were subjected to calibration through simple linear regression, according to Eq. 9.The dependent variable was the ET 0 of the evaluated method and the independent variable was the ET 0 of the standard FAO-PM method.In each method evaluated, the values of "a" and "b" were obtained and were subsequently used to calibrate the ET 0 of the respective method, according to Eq. 10.
where: ET 0 -reference evapotranspiration of the evaluated method, in mm d -1 ; a and b -calibration parameters obtained through linear regression; and, ET 0 FAO-PM -reference evapotranspiration by the FAO-Penman-Monteith method, in mm d -1 ; ET 0 cal -calibrated reference evapotranspiration, in mm d -1 .
In each method evaluated, ET 0 values were previously randomized and then divided into two subsets, calibration and test, as proposed by Shiri et al. (2015).The calibration subset, containing 70% of the ET 0 values, was used to obtain the parameters "a" and "b", whereas the test subset, containing 30% of the ET 0 values, was used to evaluate the performance of the calibrated model.
ET 0 values obtained by the different methods, in relation to the FAO-PM method, were compared using Willmott's index of agreement (Willmott et al., 1985), correlation and

Results and Discussion
ET 0 calibration parameters obtained through linear regression and the coefficient of determination (r 2 ) for each of the evaluated methods, in relation to the standard FAO Penman-Monteith method (FAO-PM), are described in Table 2.
The methods that showed lower coefficients of determination (r 2 ) were those that incorporated the estimated solar radiation (Rs) (Table 2).The same result was found by Sousa et al. (2010) and Alencar et al. (2015).This is due to the fact that the main factor contributing to the evapotranspiration is solar radiation (Allen et al., 1998).Therefore, the proposed equation is inadequate to estimate Rs for the studied site.Similar results for the Priestley-Taylor and Hicks-Hess methods were found by Cunha et al. (2013) andFerronato et al. (2016).The other methods tended to overestimate ET 0 in comparison to the standard method (Figures 1C, H, J and L), corroborating the results of Fanaya Júnior et al. ( 2012), Cunha et al. (2013) and Oliveira et al. (2015).
Figure 2 shows the graphical relationship between ET 0 values estimated by the alternative methods after calibration.It is possible to note that calibration avoids under-or overestimation of ET 0 , improving the accuracy, but without altering the coefficient of determination (r 2 ) (Figure 2).
Unlike the present research, Carvalho et al. ( 2013) used an exponential equation to test ET 0 values, but obtained only a slight increase (0.8%) in the coefficient c.Another alternative to improve the ET 0 estimation method is the calibration of the equation through the addition of coefficients, but such  procedure does not avoid under-or overestimation of ET 0 , decreasing the accuracy (Fernandes et al., 2012).
The most recommended methods for Jaíba or with best performances ("excellent" classification) were FAO-PM with one missing meteorological variable and with simultaneous absence of WS and RH data (Table 3).In the absence of Rs and more one or two meteorological variables, the performance of the FAO-PM method was compromised (Table 3), corroborating the results of Alencar et al. (2015).
The methods Jensen-Haise, Hicks-Hess, Priestley-Taylor and Tanner-Pelton were classified as "Very good", and can still be used.On the other hand, Cunha et al. (2013) found "tolerable" and "intermediate" performance for these methods for Chapadão do Sul-MS.Probably, the tropical megathermal climate in Jaíba favors the use of these methods with greater success, compared with the humid tropical climate of Chapadão do Sul.
Relatively, in the present study, the worst performances ("Good") occurred for methods that used only air temperature  Samani).These methods also exhibited the highest values of RMSE (0.72 and 0.76 mm d -1 ) and, therefore, are not recommended (Table 3).
It is worth pointing out that a certain method should be selected in such a way to adapt to the available meteorological data and, at the same time, exhibit good performance in ET 0 estimation.Lastly, special attention should be paid to the confidence coefficient.
The methods classified as "excellent" showed the highest and lowest values of c and RMSE, respectively.The method FAO-PM without RH showed the lowest RMSE (0.30 mm d -1 ), whereas the highest RMSE (0.76 mm d -1 ) was obtained by the Hargreaves-Samani method (Table 3).
Evaluating various methods of ET 0 estimation, Sousa et al. (2010) concluded that those closest to the Penman-Monteith method were, in sequence, Jensen-Haise, Priestley-Taylor and Hargreaves, as observed in the present study.Alencar et al. (2015) concluded, for various cities of Minas Gerais, that when only air temperature data are available, the performance of the FAO Penman-Monteith method was superior to that of Hargreaves-Samani.However, in the present study, these two methods showed very similar confidence coefficient and were thus considered as with the same performance ("Good").
The Hargreaves-Samani method was recommended by Allen et al. (1998) in situations with availability of only air temperature data.However, its performance for Jaíba-MG was not satisfactory (Cunha et al., 2013;Alencar et al., 2015).

Conclusions
1.In the absence of only solar radiation, relative humidity or wind speed data, or even simultaneous absence of relative humidity and wind speed data, the FAO Penman-Monteith method showed the best performances.
2. The FAO Penman-Monteith method in the absence of measured solar radiation data and more one meteorological variable showed intermediate performance in ET 0 estimation.
4. The methods that used only measured air temperature data are not recommended for Jaíba-MG, Brazil, even after calibration of ET 0 values.
reference evapotranspiration, mm d -1 ; R n -net radiation on the surface, MJ m -2 d -1 ; G -heat flux in the soil, MJ m -2 d -1 (considered as null for daily scale); -Taylor and Hicks-Hess tended to underestimate ET 0 values in comparison to the standard method (Figures 1A, B, D, E, F, G, I and K, respectively).

Figure 1 .
Figure 1.ET 0 values estimated by the methods FAO-PM without Rs (A); FAO-PM without RH (B); FAO-PM without WS (C); FAO-PM without Rs and RH (D); FAO-PM without Rs and WS (E); FAO-PM without RH and WS (F); FAO-PM without Rs, WS and RH (G); Hargreaves-Samani (H); Priestley-Taylor (I); Jensen-Haise (J); Hicks-Hess (K) and Tanner-Pelton (L), in comparison to those estimated by FAO-PM before calibration

Table 1 .
Criteria to interpret the confidence coefficient "c"

Table 2 .
Calibration parameters and coefficient of determination for each method evaluated FAO-PM -FAO Penman-Monteith; Rs -Solar radiation; WS -Wind speed; RH -Relative air humidity

Table 3 .
Root-mean-square error (RMSE), Willmott's index of agreement (d) and confidence coefficient (c) with the respective classification for the alternative methods with calibrated ET 0 values FAO-PM -FAO-Penman-Monteith method; Rs -Solar radiation; WS -Wind speed; RH -Relative air humidity data (FAO-PM without WS, RH and Rs, and Hargreaves-