Calibration Methods for Estimation of Reference Evapotranspiration in Morro Do Chapéu , Bahia , Brazil

The objective of this work was to calibrate and to validate the methods of Camargo, Hargreaves and Samani, and Priestley and Taylor, according to the Penman-Monteith model, for the estimation of reference evapotranspiration (ETo), in the four seasons of the year for the municipality of Morro do Chapéu, Bahia. Climatological data from the conventional meteorological station belonging to the National Institute of Meteorology (INMET) were used, in the period of 18 years (2000-2018). The first 16 years were considered to adjust the parameters. The years of 2016 and 2017 were assigned as independent data to validate the adjustments. To analyze the results, it was used the root mean square error (RMSE in mm d), coefficient of determination (R), systematic error (BIAS in mm d), and Willmott’s concordance index. After adjusting the parameters, the three methods improved their performance in the estimation of ETo, however, the Camargo method presented high values of RMSE, reaching 0.41 mm d during the spring. It is concluded that the calibrated methods of Hargreaves and Samani, and, Priestley and Taylor can be recommended for the estimation of the reference evapotranspiration, for planning and execution of irrigation projects in the municipality, regardless of the season.


Introduction
Estimation of crop water requirements is an important aspect to be considered in agricultural planning and, consequently, has constituted a research area of considerable interest by the scientists involved with studies related to irrigation management and agrometeorology worldwide (Nova et al., 2006).This is because agricultural activities are highly dependent on climatic variables, making it a determinant factor for agricultural production (Sales et al., 2018a).
Reference evapotranspiration is the way in which the water of the terrestrial surface passes to the atmosphere in the vapor state, playing an important role in the hydrological cycle, which allows to know the water balance of the soil and the evapotranspiration of crops (Bezerra et al., 2010;Fernandes et al., 2012;Sales et al., 2016).In terms of irrigation management, for the correct depth of applied water, it is required to determine the crop evapotranspiration (ETc).The ETc is given by the product between the reference evapotranspiration (ETo) and the crop coefficient (Kc), which varies according to the crop and its phenological stage (Allen et al., 1998).
In regions with specific climatic characteristics at least one method that estimates ETo with good accuracy is desirable.Therefore, the Penman-Monteith method (Allen et al., 1998) was parameterized using the FAO Bulletin 56 (Food and Agriculture Organization of the United Nations) to determine the ETo reliably.This method is widely used to estimate evapotranspiration (Srivastava et al., 2018) The evapotranspiration estimation by the Penman-Monteith method was calculated using Equation 01 (Allen et al., 1998) (Equation 1).
(1) Where,  is the slope of the vapor pressure curve (kPa °C-1 ); R n is the radiation balance (MJ m -2 d -1 ); G is the soil heat flux density (MJ m 2 d -1 );  is the psychrometric constant (kPa ºC -1 ); T mean is the mean daily air temperature in ºC; U 2 is the wind speed (daily average) at 2 m height; e s is the saturation vapor pressure (kPa); e a is the actual vapor pressure (kPa).

Camargo (CM)
The Camargo Method (1971) is a simplification of the Thornthwaite method.This method has as main advantage, the use of only data of mean daily air temperature and extraterrestrial solar radiation (Equation 2). (

2)
Where, A is a coefficient of the method; R a is extraterrestrial solar radiation (MJ m -2 d -1 ); T mean is mean air temperature in ºC.

Hargreaves and Samani (HS)
The model proposed by Hargreaves and Samani (1985) was developed for dry climate regions and is a simple method (Equation 3).
(3) Where, A and B are coefficients of the method; R a is the extraterrestrial solar radiation (MJ m -2 d -1 ); T max is daily maximum temperature (ºC); T min is daily minimum temperature (ºC); T mean is daily mean temperature (ºC).

Priestley and Taylor (PT)
The Priestley and Taylor Method (1972) is a simplification of the Penman and Penman-Monteith method.Thus, this model has the advantage of requiring less input data (Equation 4). (4) Where, γ is the psychrometric constant (kPaº C -1 ); Δ is the slope of the vapor pressure curve (kPa °C-1 ); R n is the net radiation received by the reference crop (MJ mm d -1 ); G is the soil heat flux density (mm d -1 ).

Statistical Analysis
The adjustment parameter was performed by minimizing the error square sum, obtained by comparing the estimated ETo between the alternative methods and the FAO-56 PM model.For this, the Solver application within the software Office Excel 2007® was used, besides the aid of the open source program R (R core team, 2016) to perform the statistics.
To analyze the results, the statistical indicators, root mean square error (RMSE in mm d -1 ), coefficient of determination (R 2 ), systematic error (BIAS in mm d -1 ), and Willmott's concordance index (Willmott et al., 1985), were used to evaluate the accuracy of the different models.

Results and Discussion
The values of the original and the adjusted coefficients for each of the evaluated methods can be visualized by season of the year, in relation to the FAO-56 PM standard method for the municipality of Morro do Chapéu, Bahia (Table 1).When evaluating the adjustment of each method per season of the year, it is observed that the values followed very close to each other, except for HS, in which the values of B coefficient ranged from 29.0 to 82.0.Sales et al. (2018b) when adjusting the HS method for the region of São Mateus, ES, found a value of B of 5.59, which is lower than that found in this study.does native ethod when input and a itions es for the HS method in the Agreste Alagoano.However, they recommend the method for its use, since that adjustment of the equation is done.
For the CM model, it is expected that under aridity conditions, the method presents underestimate in relation to the FAO-56 PM (Fernandes et al., 2010).The high BIAS values observed with the CM method reflected higher RMSE values (Table 2), especially in the spring and summer seasons, which presented high errors with a low agreement between the data.The RMSE values found by the original HS were the lowest among the three evaluated methods, with values varying from 0.22 to 0.29 mm d -1 , according to the evaluated season.Therefore, the potentiality of the HS method in the region was evident, since this method only uses air temperature, and yet it showed good results, with low RMSE values and high values of d.This result may be linked to the model's foundation, which was developed for semi-arid conditions in California, and Morro do Chapéu municipality is part of the semi-arid region of the State of Bahia (Bastos et al., 2012), conferring low rainfall indices and high temperatures, favoring the model.
It is observed in all methods that, the seasons of spring and summer were the ones that presented the biggest estimative errors of ETo.These stations represent the rainy season of the municipality, as well as the incidence of higher temperatures, suggesting that these meteorological variables may have provided greater errors to the methods used.
After adjusting the parameters (Table 2), the CM, HS, and PT methods had the errors (RMSE) of their estimates reduced from 16.00 to 133.00% (Dif RMSE ).The PT method, although a simplified version of the FAO-56 PM method, was the most benefited by the adjustment, with RMSE ranging from 0,06 mm d -1 for autumn, to 0,09 mm d -1 in spring.In other words, the adjustment reduced the error by more than 95% and increased the d index, in such a way to present values of 0.99, for the four stations evaluated.
Among the methods that only use air temperature as an independent variable (CM and HS), it can be observed after the adjustment that the CM method obtained higher increases in the d index, with values varying from 12.77 to 54.05 (Dif d ).However, only in the seasons of autumn and winter, a d value greater than 0.90 was observed, while the HS obtained values of d ≥ 0.99, during all seasons of the year.
The CM results followed the same trend identified by Hallal et al. (2017), when estimating the ETo daily, quinquidial, monthly, and seasons of the year for the climatic conditions of Pelotas, RS.Where was verified for the seasons of the year that the d index ranged from 0.54 to 0.78.However, this method was classified by these authors as bad and poor performance for all seasons.
Although the adjustment benefited all methods, the CM presented higher RMSE values when compared to the other models, reaching 0.41 mm d -1 in the spring.Bad results in methods based only on air temperature have been reported in the literature, and the low temperature variation is indicated as the main cause (Fanay a Júnior et al., 2012;Santos et al., 2017;Sales et al., 2018b).
The HS method after the adjustment showed RMSE ≤ 0.07 mm d -1 and a high concordance index using only the air temperature as an independent variable.Silva et al. (2005) when working with the municipality of Petrolina, PE, verified that it is possible to indicate the original HS method for the estimation of ETo.Since it presented satisfactory results when compared to the FAO-56 PM standard method.
Thus, such methods can be an alternative, since many study areas have simple meteorological stations, equipped only with sensors of rain and air temperature.However, their low accuracy must be taken into account at the time of its application, recognizing the limitation of each method.
For the validation of the adjusted methods, the RMSE error coefficient and the Wilmmott's concordance index were presented in Figures 2, 3, and 4. The values refer to the statistical performance of the different methods used for the daily estimation of ETo, during the years of 2016 and 2017, for the municipality of Morro do Chapéu, BA.When applying the ETo methods, with adjusted parameters in this study, for a series of independent climatic data, it was observed that there was a variation of 0.08 to 0.43 mm d -1 between the methods and the seasons of the year.
In Figure 3, it is noteworthy that the spring (A) and summer (B) seasons presented the greatest errors for CM, as well as the lowest values of Wilmmott's concordance index.However, even the autumn (C) and winter (D) seasons showed better values after calibration, where errors of 0.35 and 0.29 mm d -1 for the seasons were observed, respectively.Corroborating with Hallal et al. (2017), when estimating the ETo for the seasons of the year in Pelotas, verified that in spring and summer they had the highest standard error of estimate (EPE) (1.04 mm d -1 ) and (1.17 mm d -1 ), respectively.It is verified that these stations for the municipality under study are characterized by increased rainfall and air temperature, which may contribute to the results found.
The Camargo method proved to be inefficient to estimate the ETo by seasons of the year in the municipality of Morro do Chapéu.Since it presented unsatisfactory results, with little agreement and high errors, mainly during the spring and summer.It should be noted that the Camargo method was initially proposed to determine the ETo for periods after seven days, which contributed to its low accuracy (Paz et al., 2018).Less precise results by the CM method were observed by Sales et al. (2018b), when adjusting different methods for the region of São Mateus, ES, with RMSE showing deviations of 24% and agreement of 0.81 when compared to FAO-56 PM.
Figure 3. F , but it requires a large number of

Table 2 .
Evaluation of the ETo estimates (mm d -1 ) obtained in the four seasons of the year from 2000 to 2015, with different methods in relation to the standard method, FAO-56 PM, before (original) and after (adjusted) adjustment of the parameters