REFERENCE EVAPOTRANSPIRATION ESTIMATES BY MEANS OF HARGREAVES-SAMANI AND PENMAN-MONTEITH FAO METHODS WITH MISSING DATA IN THE NORTHWESTERN MATO GROSSO DO SUL ESTIMATIVA DA EVAPOTRANSPIRAÇÃO DE REFERÊNCIA PELOS MÉTODOS HARGREAVES-SAMANI E PENMAN-MONTEITH FAO COM DADOS FALTOSOS NO NOROESTE SUL-MAT

We aimed to assess the performance of different methodologies to estimate the reference evapotranspiration (ET0) in different seasons for at 1, 3, 6 and 10 days for the Northwestern Mato Grosso do Sul. We used a set of daily data obtained from the networks of the Brazilian National Institute of Meteorology for Água Clara, Cassilândia, Chapadão do Sul, Paranaíba and Três Lagoas locations. The meteorological data used were: maximum and minimum air temperatures, wind speed, solar radiation, and relative humidity. The methodologies used for the ET0 estimation were Hargreaves-Samani and Penman-Monteith (PM-FAO) using all the meteorological data needed and missing data on relative humidity, solar radiation and wind speed. To compare ET0 values estimated by means of equations tested against the PM-FAO method we analyzed the parameters of linear regression equation “β0” and “β1”, coefficient of determination (r2), standard-error of estimate (SEE) and coefficient of performance (c). The PM-FAO method with missing data (relative humidity and wind speed) were the best alternatives to estimate ET0, followed by the PM-FAO method with missing solar radiation data in all timescales during fall and winter seasons. The HargreavesSamani method is not recommended to be used in its original form to estimate ET0 in the Northwestern Mato Grosso do Sul, in none of the timescales and seasons.


INTRODUCTION
The reference evapotranspiration (ET0) is widely used for irrigation engineering to define the crop water needs and to manage the water distribution in existing systems (DROOGERS;ALLEN;2002).
The Penman-Monteith FAO method (PM-FAO) is widely used to estimate ET0.This method is based on physical aspects of the evaporation and transpiration processes with respect to hypothetical grass reference (ALLEN et al., 1998), and it has been used worldwide (CUNHA et al., 2013).However, an important disadvantage of this method is the relative large number of input meteorological variables (DROOGERS; ALLEN, 2002;BORGES JÚNIOR et al., 2012), and then its use may become unreliable due to non-availability and low quality information at certain localities (PALARETTI et al., 2014).
In order to find alternatives when facing this situation, many researchers have been proposed estimating ET0 with methods that demand less input data (ALENCAR et al., 2011;FANAYA JUNIOR et al., 2012;CHAGAS et al., 2013;PALARETTI et al., 2014;CARVALHO et al., 2015).Among those methods, Hargreaves-Samani (HARGREAVES;SAMANI, 1985), with only minimum and maximum temperature data, is considered the simplest one to estimate the reference evapotranspiration (ALENCAR et al., 2015) and it can be used for irrigation management (NÓIA et al., 2014).
In addition, using methods that employ a small variables number is useful when there is a broken sensor in the meteorological station and/or a reading error (ALENCAR, et al., 2015;CARVALHO et al., 2015;COSTA et al., 2015;MORAIS et al., 2015).Allen et al. (1998) described alternative equations to estimate the missing data of relative air humidity, solar radiation and winds to be used in the PM-FAO equation.Sentelhas et al. (2010), when evaluating ET0 estimation with missing data in southern Ontario, Canada, found out that the PM-FAO method provided good estimates for ET0 with missing relative humidity and wind speed data.With missing solar radiation data PM-FAO method did not perform well and when only air temperature data were available, adjusted Hargreaves and modified Thornthwaite performed better.
The behavior that has been verified in Canada will not necessarily occur in other regions.Then, before indicating any method to estimate the ET0, its performance must be previously known in different locations and years, since there is great variability in weather conditions.Thus, a method may have an unsatisfactory performance in winter but it could have a good performance in summer and so it could be recommended.
The state of Mato Grosso do Sul has a great importance for agriculture in the Brazil, since it is the second largest producer of cotton, corn, soybeans and sorghum (CONAB, 2016).These crops are generally irrigated, and so farmers depend on the estimated ET0 for a correct irrigation water management.Then, the choice of methods for such a purpose will contribute to strengthen the irrigated agricultural activity.In addition, the generated information can be used by extension workers, consultants, irrigation companies, educational institutions, and others.
The objective of this paper was to evaluate the performance of the Penman-Monteith (PM-FAO) with missing data and Hargreaves-Samani methods to estimate the ET0, for five locations in the Northwestern region of Mato Grosso do Sul.

MATERIAL AND METHODS
The meteorological data required for carrying out this study were obtained from the Brazilian National Institute of Meteorology (INMET) networks, from five meteorological stations in the northwestern of the state of Mato Grosso do Sul (Table 1 and Figure 1).The meteorological data used were: average, maximum and minimum temperatures (ºC); average, maximum and minimum relative air humidities (%); average, maximum and minimum dew point temperatures (ºC); average, maximum and minimum atmospheric pressures (kPa); wind speed at 10 m height (m s -1 ) and solar radiation (kJ m -2 d -1 ).The data were obtained from an automatic weather station that consists of the equipment WAWS 301 of the brand VAISALA, described as follows: (1) Pyranometer CM6B; (2) Pressure Sensor PMT16A; (3) Thermometer QMH102; (4) Hygrometer QMH102; (5) Pluviometer QMR102 and (6) Anemometer WAA151.The hourly meteorological data were converted to daily data.In order to make the meteorological variables data more homogeneous verifications were carried out and, subsequently, all information considered discrepant or inconsistent were eliminated, aiming to obtain more representative data set.In order to be considered inconsistent the data should present a minimum temperature greater than maximum, negative or greater than 16 h of insolation, negative or greater than 100% of relative humidity or negative or greater than 20 m s -1 for wind speed (10 m high).Proceeding this way 1.92% of the data were discarded.
b) PM-FAO using the methodology with missing data (relative humidity).The relative humidity was obtained by means of the relation between the current and saturation vapor pressures.In order to calculate the saturation vapor pressure, the maximum air temperature was used and for the current vapor pressure the minimum temperature was considered as the temperature of the dew point, according to Equation 6: (6) where: e a -actual vapor pressure (kPa) and T minminimum air temperature (°C).
c) PM-FAO with missing data (solar radiation).In this methodology, solar radiation data from the nearest station were used, thus, for Chapadão do Sul, Cassilândia's data were used and vice versa; And for Águas Claras, Três Lagoas' data were used and vice versa.For Paranaíba, which has a distance of over 50 km in the north-south position, Equation 7 was applied with Cassilândia's data. ( where: R S -solar or shortwave radiation at Paranaíba's station (MJ m -2 d -1 ); R S,reg -solar or shortwave radiation at Cassilândia's station (MJ m -2 d -1 ); R a,reg -extraterrestrial radiation at Cassilândia's station (MJ m -2 d -1 ) and R a -extraterrestrial radiation at Paranaíba's station (MJ m -2 d -1 ).
d) PM-FAO with missing data (solar radiation) and replacing with estimated data by the air temperature.In order to do that Equation 8 was applied.
e) PM-FAO with missing data (wind speed).For this method, using the meteorological data (Table 1), the average wind speed values at 2 m height were calculated for each location.The average speed values used to estimate the ET0 for Águas Claras, Cassilândia, Chapadão do Sul, Paranaíba and Três Lagoas stations were 1.34, 1.08, 1.54, 1.23 and 1.15 m s -1 , respectively.-f) Hargreaves-Samani.The original equation (Equation 9) was used, according to Hargreaves and Samani (1985).
The ET0 data obtained through different methodologies were confronted with the data obtained by the Penman-Monteith FAO method at intervals of one, three, six and ten days.Irrigation frequencies higher than one day are used.Therefore, the use of larger scales was chosen in order to improve the ET0 estimates due to the unsystematic effect.ET0 estimates were compared just within the same season (winter, fall, summer and spring), since they show distinct behavior among different seasons.
After the data verification, a regression analysis that correlated ET0 estimated values, the Penman-Monteith equation was performed.The coefficients "β 0 " and "β 1 " of the respective linear regressions and the coefficient of determination (r 2 ) were considered.The best alternative was the one that showed regression coefficient "β 0 " near zero, coefficient "β 1 " near unit and higher coefficient of determination, larger than 0.60.The precision was measured through the coefficient of determination, which indicates the degree to which the regression explains the sum of the total variance.The methods accuracy was also analyzed by means of linear regression coefficients with near zero values for β 0 and β 1 near unit.
The methodology adopted for results comparison was proposed by Allen et al. (1989), and it is based on the standard-error of estimate (SEE), calculated by Equation 10.The best method to estimate ET0 was the one that presented the lowest SEE.(10) where: SEE -standard-error of estimate (mm d -1 ); X i -reference evapotranspiration estimated by the standard method (mm d -1 ); Y i -reference evapotranspiration obtained by the tested method (mm d -1 ); and n -number of observations.
The approximation of ET0 values estimated by the studied method, in relation to the values obtained using the standard method, was obtained by an index called Willmott agreement index, represented by the letter "d" where its values range from zero, where there is no agreement, to 1, for the perfect agreement (WILLMOTT et al., 1985).The agreement index (d) was calculated using the Equation 11.To validate the model, it was also obtained the Pearson's correlation coefficient (r) through Equation 12and, consequently, the reliable coefficient or performance (c) through Equation 13.
(11) (12) (13) where: d -Willmott's agreement index; X ireference evapotranspiration estimated by the standard method (mm d -1 ); Y i -reference evapotranspiration obtained by the tested method (mm d -1 ); Y -average values of reference evapotranspiration obtained by the tested method (mm d -1 ); X -average values of reference evapotranspiration obtained by standard method (mm d -1 ); n -number of observations; r -Pearson's correlation coefficient; and c -reliable coefficient or performance.

RESULTS AND DISCUSSION
Results of linear regressions with different missing data on daily reference evapotranspiration (ET0) and by Hargreaves-Samani method, along the seasons of the year are presented in Figure 2. High coefficients of determination were observed (r² > 0.92), regardless the season, when estimating ET0 for missing relative air humidity and wind speed data.This result is attributed to the average value that satisfactorily represented the wind speed in each annual season, due to the low temporal variability presented by this factor; and due to a good correlation between dew point humidity and minimum humidity, which was used to estimate the air relative humidity.These data corroborate with Morais et al. (2015) researches in the sub-mid region of Vale do São Francisco and also with Alencar et al. (2015) to several cities in the State of Minas Gerais.Sentelhas et al. (2010), in a similar study in the southern part of Ontario province, Canada, found that the Penman-Monteith method, in the absence of relative humidity or wind speed data, is a good choice for ET0 estimates.The performance of the method with missing solar radiation data presented the best performance during winter season, with higher value for r², and the coefficients β 0 and β 1 close to a line type 1:1.The worst fitting was obtained during spring and summer months, in addition to the great tendency to overestimate when ET0 shows values below 4.3 mm d -1 and to underestimate above this value.This good performance during colder months, especially winter, is due to the relative humidity and the wind speed, which are the main ET0 conditioning elements.Unlike during hotter periods when the radiation is the most responsible element for the ET0 estimation, then its absence leads to worse estimates.Similar behavior is observed for missing solar radiation and replaced with estimated data by air temperature (PM-FAO missing R S "T").However, there is a performance gain in all seasons on the basis of the r², mainly in summer time.This behavior can also be attributed to the absence of clouds during the winter, favoring the estimation of the shortwave radiation balance.Silva et al. (2011) found that the most sensitive variable in determining ET0 by Penman-Monteith method is the net solar radiation followed by relative humidity, wind speed and the average air temperature.Similarly, Langensiepen et al. (2009) found that the radiation is the most sensitive variable of this method.Westerhoff (2015) found the following order: air temperature, solar radiation and relative humidity.Thus, this work also shows lower efficiency of the PM-FAO methods with missing solar radiation data.
The Figure 2 also shows, from the standard line (1:1), that the Hargreaves-Samani method overestimates the ET0 practically throughout the year, especially when the Penman-Monteith FAO method shows estimates below 5.8 mm d -1 , which can also be seen by the high values of the "β 0 " coefficient.The worst fitting was observed in winter (r² = 0.4057).These results can be explained based on the climate of the studied localities (Table 1) which are of type Aw (tropical hot and humid), with a rainy weather in summer and a dry weather in winter, according to the Köppen classification (KOTTEK et al., 2006), unlike the semi-arid conditions in which the Hargreaves-Samani method was derived.Tanaka et al. (2016), working with data from 28 locations in the State of Mato Grosso found the worst statistical indicators to estimate ET0 at daily scale using Hargreaves-Samani method.They do not recommend its use.Several other papers show the ET0 overestimations using this method, in daily scale, for humid or subhumid climate regions (YODER et al., 2005;LIMA et al., 2013;ALENCAR et al., 2011;ALENCAR et al., 2015).
Table 2 shows ET0 estimates for the scales of three, six and ten days for different seasons of the year, aiming to identify the performance of methods in a larger timescale.Pereira et al. (2009) highlighted that before applying a method to a particular location or region, it is necessary to check its performance in relation to the standard method, in different timescales and, when necessary, to make calibrations in order to minimize estimation errors.It turns out that, for the PM-FAO method with missing relative humidity or wind speed data, there was an excellent independent fitting for 3, 6 or 10 days timescales according to r², where all of them were greater than or equal to 0.92.Moreover, the coefficient "β 0 " tended to zero and "β 1 " near unit, showing an efficient ET0 estimate, regardless the timescale and the time of year.
Table 2.The coefficients of linear regression "β 0 " and "β 1 " values, coefficient of determination (r 2 ) for the regression between the reference evapotranspiration (ET0) obtained by the Penman-Monteith FAO method compared with ET0 values obtained by Penman-Monteith FAO method calculated with missing data and for the Hargreaves-Samani method on scales of three, six and ten days in different seasons The calculated ET0 for PM-FAO with missing R S "R a ", as noted in the daily scale, had a better performance in winter with a value of r² above 0.97 and coefficients "β 0 " and "β 1 " tending to the 1:1 line, for all periods.Fall also showed a good fitting, having a higher value for the r² in the 10 days scale.For spring and summer we had the worst fitting.We can see by the r 2 and by the regression coefficients that ET0 estimates by PM-FAO with missing R S "T" presented better settings in relation to the PM-FAO missing R S "R a " methodology.However, the trends were the same and we observed better performances for winter, fall, spring and summer, respectively.
The timescale increase has favored almost every ET0 estimation method with missing data, based only the r², except for PM-FAO with missing R S "T", for spring and summer time; and PM-FAO with missing wind speed data for spring time, considered 6 and 10 days, which showed a slight decrease compared to the daily scale.Bragança et al. (2010) studied several methods in different timescales in the rainy weather for the localities Venda Nova do Imigrante, Sooretama and Cachoeiro de Itapemirim in the State of Espírito Santo, and they found higher values for the coefficient of determination (r 2 ), coefficient of correlation (r), the Willmott agreement index (d) and the confidence index or performance (c) for the three timescales of three, five and seven days, compared to the daily timescale.
The Hargreaves-Samani method, as well as daily scale, adjusted itself poorly based on r² for all the tested time scales.By analyzing the coefficients, especially the "β 0 " (positively far from zero), there was a tendency to keep the overestimates as samples in the daily scale (Figure 2).However, summer and fall seasons showed higher r² for all timescales, in relation to the daily scale.Pilau et al. (2012) tested the Hargreaves-Samani method for the localities of Frederico Westphalen and Palmeira das Missões, in Rio Grande do Sul State, in different timescales (5, 10, 15 and 30 days) and verified better coefficients of correlation, accuracy and performance for the tested scales in relation to the daily scale.In similar work, Mendonça et al. (2003) tested the Hargreaves-Samani method for the location of Campos de Goytacazes, in Rio de Janeiro State, which has similar climatic conditions to the studied regions for this paper and found that this method offers better results for ET0 estimated values collected in 7 and 10 days.
It is worth to show that the mere adoption of the r 2 as the only rule for defining the quality of methods is not suitable, since this method does not establish the type and magnitude of differences between a default value and a value predicted by models estimate or other mechanisms of different measure (BARROS et al., 2009).Then, we used the criteria listed in Table 3 to assist in interpreting the results which presented the standard-error of estimate (SEE) and the Camargo and Sentelhas performance (c), obtained from the correlations between ET0 values by the Penman-Monteith FAO method and those obtained by the studied methods.It was observed that the best methods for estimating the ET0 in Northwestern Mato Grosso do Sul were the PM-FAO with missing relative humidity and wind speed data, for all timescales and seasons.The methodologies for ET0 estimates with missing solar radiation data, regardless the timescale, have also presented optimal performance according to "c" classification (CAMARGO; SENTELHAS, 1997) for fall and winter showing low SEE values, ranging from 0.0894 to 0.3888.
The second best performance category is seen for the PM-FAO (missing R S "T") when applied during spring and summer in all timescales and for PM-FAO (missing R S "R a ") in spring (timescales of 6 and 10 days) and in summer (10 days).It showed very good performance according to Camargo and Sentelhas (1997) index and SEE of 0.3; 0.4 and 0.5.The PM-FAO missing R S "R a " method received the rating "good".These results allow classifying the PM with missing relative humidity or wind speed data as the best options among the proposed models, followed by PM-FAO (missing R S "T") and PM-FAO (missing R S "R a "), respectively.These results corroborate with those obtained by Alencar et al. (2015) for 20 locations in the State of Minas Gerais, Morais et al. (2015) in the sub-mid region of Vale do São Francisco and Carvalho et al. (2015) in the States of Espírito Santo and Rio de Janeiro, also studied with the Penman-Monteith FAO method with missing data.
The Hargreaves-Samani method presented the worst performance by varying from the classifications of bad, terrible to poor in the "c" classification proposed by Camargo and Sentelhas (1997).From the SEE it presented higher values ranging from 0.9844 to 1.4127, mainly in winter and spring.Thus, the Hargreaves-Samani method should not be used in the Northwest region of Mato Grosso do Sul in its original form for the studied timescales.Several authors have tested this method in many locations in Brazil and they have not found good results either, like Reis et al. (2007) for Venda Nova, in Espírito Santo, Rigoni et al. (2013) for Aquidauana, in Mato Grosso do Sul, Alencar et al. (2015) for 20 locations in Minas Gerais and Morais et al. (2015) in the Sub-mid region of São Francisco Valley.However, the method needs only maximum and minimum temperature data, and that presented a reasonable value of r² in summer and fall.Another fact is that the method can easily be used by producers for irrigation management (NÓIA et al., 2014).

CONCLUSIONS
The Penman-Monteith FAO method with missing relative humidity and wind speed data is the best alternative to estimate the reference evapotranspiration, followed by the methods of Penman-Monteith FAO with missing solar radiation data in all timescales for fall and winter.
The Hargreaves-Samani method is not recommended to be used in its original form to estimate the reference evapotranspiration in the Northwest region of Mato Grosso do Sul, for none of the timescales and periods of the year.

Figure 2 .
Figure 2. Reference evapotranspiration (ET0) values obtained by the Penman-Monteith FAO method compared with the ET0 values obtained by the Penman-Monteith FAO method calculated with missing data and from Hargreaves-Samani in daily timescale for different seasons.

Table 1 .
Information regarding the meteorological stations of Mato Grosso do Sul used for estimating the reference evapotranspiration (KOTTEK et al., 2006)ing to Köppen classification(KOTTEK et al., 2006).

Table 3 .
The standard-error of estimate (SEE) and the Camargo and Sentelhas performance coefficient (c) obtained from correlations between the values of reference evapotranspiration by Penman-Monteith FAO method with the tested methods in different scales and seasons