The model employed in order to quantify the consumption of water by crops includes the concept of evapotranspiration (ET), which is the rate at which water is transferred into the air from a reference surface. The model uses the FAO-56 Penman-Monteith (FAO-56-PM) equation, with a grass height of 0.12 m, an albedo of 0.23 and a surface resistance of 70 s·m⁻¹ [1] , with uniform height, actively growing, covering the entire surface of the soil and without any water shortage.

The application of the FAO-56-PM model requires measurements of net radiation (Rn), soil heat flux, air temperature, humidity, atmospheric pressure and wind speed. Specifically in the case of Rn, there are procedures for estimating its value, as described by various authors [1] [2] . In many cases, because they are simple, empirical equations, the accuracy of the model in estimating Rn affects the value of ET. The uncertainty in estimating ET can be minimized by measuring Rn directly, which allows the measured and estimated values (Rn_measured and Rn_estimated, respectively) to be compared [3] .

The net radiometer is the most widely used instrument for the measurement of Rn. However, Rn is not often measured in weather station networks [4] [5] .

Various studies have evaluated measured or estimated ET on several times scales, Rn having been measured by different instruments or estimated by different models [6] - [9] . The choice of methodology depends on factors such as climatic conditions, accessibility of the necessary meteorological data, complexity of the model, grouping of the data considered and costs [10] [11] .

Many researchers, in the absence of experimental observations, have estimated Rn from empirical relationships based on physical considerations or other meteorological data [12] , using the correlations as a function of global solar radiation (Rs). Thus, they have evaluated surfaces such as grass [13] - [16] and crops such as cotton, beans and soybeans [17] , bananas [18] , coffee [19] , cowpeas [20] , sunflowers [21] , and grapes [22] [23] , making the estimation of Rn simpler and more reliable because of the strong correlations found by these authors.

Rn is the meteorological element that has the greatest influence on ET and represents the main source of energy used in various physical-biological processes. Furthermore, it is the main parameter used in many of the models that estimate the transfer of water from vegetated surfaces into the atmosphere. It is therefore important to measure or estimate Rn accurately. In the present study, we aimed to evaluate the accuracy of estimated evapotranspiration on a daily scale during dry and wet seasons, testing various Rn estimation formulae for non-ir- rigated grass in Brazil.

Measurements were taken at the agrometeorological station of the São Paulo State University School of Agricultural and Veterinary Sciences, located in the city of Jaboticabal, Brazil (21˚14'05''S; 48˚17'09''W, altitude: 615.01 m). The surface studied was covered by grass. According to the Köppen climate classification system [24] , the climate of the area is type Aw, which is defined as “tropical wet and dry”, or “tropical savanna” (annual climatological data: average air temperature of 22.2˚C; average relative humidity of 70.8%; rainfall of 1424.6 mm).

The experimental data were collected from August 2005 to June 2008?monthly meteorological data (Table 1) in an area of 0.56 ha (80 × 70 m) covered with non-irrigated grass (Paspalum notatum L.).

The months of January, February, March, October, November and December were considered the wettest months, whereas April, May, June, July, August and September were considered the driest months [25] .

ET by the FAO-56-PM model, as described by [1] :

T, air temperature; RH, air relative humidity; P, rainfall; U₂, wind speed―2 m; n, sunshine duration; Rn, net radiation.

where G is the soil heat flux (MJ·m⁻²·day⁻¹), γ is the psychrometric coefficient (kPa·˚C⁻¹); T is the mean temperature (˚C); U₂ is the mean wind speed at a height of 2 m (m·s⁻¹), e_s is the saturated vapor pressure (kPa), given by the expression:

e_a is the actual vapor pressure (kPa), given by the expression:

where RH is relative humidity (%) and s is the slope of the curve of vapor pressure (kPa·˚C⁻¹), given by the expression:

An automated data logger (CR10X; Campbell Scientific, Logan, UT) was installed on the reference surface in order to collect the following data (from the following instruments): temperature and relative humidity at 1.5 m above the surface (CS500 probe; Vaisala, Helsinki, Finland); Rs (CM3 pyranometer; Kipp & Zonen, Delft, The Netherlands); wind speed at 2 m above the surface (014A-L-34 wind speed sensor; Met-One Instruments, Grants Pass, OR), and Rn (NR-Lite net radiometer; Kipp & Zonen). Measurements of Rn were corrected for the effects of wind according to the manufacturer’s recommendation. The soil heat flux was obtained with a heat flux plate (HFT3; REBS Inc., Seattle, WA) installed at a depth of 3.5 cm. The atmospheric pressure was obtained by aneroid barograph (290; Lambrecht Meteorological Instruments, Göttingen, Germany), and sunshine duration was quantified with a Campbell-Stokes recorder (L-1603; Lambrecht Meteorological Instruments).

In addition to the Rn_measured, Rn_estimated was obtained by combining the Angström-Prescott equations for shortwave radiation components with the Brunt equation for the longwave radiation component emitted by the atmosphere. Thus, the Rn_estimated values were obtained using four models that take into account the effects of cloud cover and a fifth model involving linear regression with Rs as the predictor variable:

Rn_BRUNT: original equation of Brunt [26] :

where Rs is the global solar radiation (MJ·m⁻²·day⁻¹), r is the reflection coefficient of grass, T is the mean temperature (K), ea is the actual vapor pressure in the air (mmHg), n is the number of hours of sunshine duration (h), and N is the photoperiod (h).

Rn_FAO-24W and Rn_FAO-24D: the Brunt equation, as adapted, in two forms [2] :

Rn_FAO-56: FAO-56 equation [1] :

where

where Rso is the global solar radiation without the presence of clouds (MJ·m⁻²·day⁻¹), z is the altitude (m), Ra is the extraterrestrial radiation (MJ·m⁻²·day⁻¹), G_sc is the solar constant (0.0820 MJ·m⁻²·min⁻¹), d_r is the relative Earth-Sun distance (rad), δ is the solar declination (rad), φ is latitude (rad), ω_s is the solar hour angle (rad), and J is the Julian day of the year (1 to 365 or 366).

Rn_Rs: equation for estimating the Rn at Jaboticabal linear regression with global solar radiation as the predictor variable, as proposed by André and Volpe [27] :

where Rs is the global solar radiation (MJ·m⁻²·day⁻¹).

We adopted the cloud cover classification system proposed [28] using the clearness index (K_T) which is the ratio between incident global solar radiation and extraterrestrial radiation: K_T < 0.35 (overcast), 0.35 ≤ K_T < 0.55 (broken clouds), 0.55 ≤ K_T ≤ 0.65 (scattered clouds) and K_T > 0.65 (clear sky).

We compared the estimation of ET based on Rn_measured with that based on Rn_estimated using the five models mentioned previously, through the statistical indicators simple linear regression analysis through the origin (y = bx), index of agreement (d), mean relative error (MRE) and efficiency (EF) [29] :

where d is the index of agreement, Pi is the ET estimated by the FAO-56-PM model with Rn_estimated via the model in question (Rn_BRUNT; Rn_FAO-24W; Rn_FAO-24D; Rn_FAO-56; or Rn_Rs), O_i is the ET estimated by the FAO-56-PM model with Rn_measured (the standard), is the mean ET obtained by the FAO-56-PM model with Rn_estimated via the alternative model in question, is the mean ET obtained by the FAO-56-PM model with Rn_measured, and n is the number of observations.

Analyzing the mean monthly ET values shown in Figure 1, which were obtained from the daily mean values, and comparing the Rn_measured ET with the Rn_estimated ET from the various models, we can see that the models in which the Rn_estimated most closely approximated the Rn_measured were the Rn_FAO-24D and Rn_FAO-56 models. In addition, the Rn_Rs equation was shown to have overestimated Rn, whereas there was an underestimation of Rn when the Brunt and FAO-24D equations were applied.

Initially, we analyzed the dry and wet months separately to determine the effect of seasonality of rainfall (Ta- ble 2).

Despite the similarity of the equations applied in the Rn_FAO-24D and Rn_FAO-56 models, which differ only in the effect of cloud cover, there were significant differences between those two models. When we analyzed the dry months separately from the wet months, the Rn_FAO-56 model underestimated the cloud cover, by 8.5% in the dry months and 22.9% in the wet months, resulting in the estimated ET being 1.6% and 2.8% higher in the dry and wet months, respectively, relative to the estimates obtained with the Rn_FAO-24D model. When we analyzed the dry and wet months together, the Rn_FAO-56 model underestimated the cloud cover by 15.2%, increasing the estimated ET by 4.9% (Table 3). However, no statistically significant differences were observed between the separate and

N, number of observations; b, slope of the regression line; R², coefficient of determination; d, index of agreement; MRE, mean relative error; EF, efficiency. Means followed by the same letter in the same column do not differ at the 5% level by t-test.

N, number of observations; b, slope of the regression line; R², coefficient of determination; d, index of agreement; MRE, mean relative error; EF, efficiency. Means followed by the same letter in the same column do not differ at the 5% level by t-test.

joint analyses of dry and wet months, for any of the models. This, together with the values for slope, coefficient of determination, index of agreement, MRE and efficiency (Table 1), made it apparent that the separate analyses were not necessary, and we were able to group the data for the entire period (Table 3), thus simplifying the analysis.

When comparing the mean Rn_measured ET for the entire period (4.1 mm·day⁻¹, Table 3) with the mean Rn_estimated ET for the entire period obtained via the Rn_BRUNT and Rn_FAO-24W models, we found that those two models unde-

restimated the ET by 10.8% and 7.9%, respectively, whereas ET was overestimated by 2.4%, 4.9% and 24.4%, respectively, when we used the models Rn_FAO-24D, Rn_FAO-56 and Rn_Rs. A slope of the regression equation that is closer to 1 indicates a better estimate of the ET. The best fit (slope = 1.020) was obtained with the Rn_FAO-24D model, as confirmed by t-test (Table 3) and scatterplot (Figure 2). According to the coefficient of determination, index of agreement, mean absolute error (MAE) and efficiency values, the best estimates of ET were obtained via the Rn_FAO-24D model, followed by the models Rn_FAO-56 models, Rn_FAO-24W, Rn_BRUNT and Rn_Rs. In the present study, the strongest correlation with the Rn_measured was achieved via the Rn_FAO-24D model. For the study period as a whole, the MRE of Rn (MJ·m⁻²·day⁻¹) was 1.6, 1.5, 0.5, 0.9 and 4.4, respectively, for the models Rn_BRUNT, Rn_FAO-24W, Rn_FAO-24D, Rn_FAO-56 and Rn_Rs (Table 3).

Some authors, such as [6] and [8] , recommend using the FAO-56 equation [1] to calculate Rn when obtaining estimates of ET via the FAO-56-PM model, given that some other authors, such as [30] and [31] , encountered difficulties with respect to the estimation of net longwave radiation in studies employing the FAO-24 equation. This is because the estimation of Rn requires the evaluation of several meteorological variables, including sunshine duration, which is not always possible due to the absence of measurements [32] , which could make it more costly to estimate Rn than to measure it directly.

Given that estimates of ET obtained via the FAO-56-PM model are affected by the method employed in obtaining Rn, [15] recommend that Rn be obtained with the NR-Lite (Kipp & Zonen) sensor, whereas [33] stated that Rn can be obtained with either the NR-Lite (Kipp & Zonen) sensor or the Q-7.1 (REBS) sensor, provided that the sensor employed is properly calibrated against a CNR1 (Kipp & Zonen) sensor, which is considered the standard for its high accuracy [34] .

For the Jaboticabal region, the Rn_Rs model André and Volpe [27] overestimated ET by 28.8% and 22.6% for dry and wet months, respectively, and by 24.4% for the study period as a whole (Table 2 and Table 3). This is explained in part by the fact that the regression equation was devised in 1988, with Rn measured by a net exchange radiometer (Packard Bell model TCN-188), without dome and with ventilation, and Rs measured with an Eppley thermopile pyranometer (model 8-48), instruments quite different from those currently used. Most current net radiometers consist of thermopile covered by a dome of polyethylene to eliminate natural ventilation and reduce thermal convection from the body of the device. In view of the disadvantages regarding maintenance and operation of a net radiometer with a dome, the “dome-less” NR-Lite [34] , in which the dome has been replaced by a black Teflon coating [35] , is now widely used.

In the present study, despite the high correlation between Rn and Rs, Rs overestimated Rn by 48.4% according to the methodology of André and Volpe [27] and, consequently, the FAO-56-PM model with Rn estimated by the Rn_Rs model overestimated the daily ET by 24.4% (Table 3). Many authors have identified such overestimation of ET, ranging from 6% to 29%, at various locations [15] [36] - [44] .

Significant errors can be made in estimating the ET when Rn is not correctly measured or estimated, with differences of as much as 2.2 MJ·m⁻²·day⁻¹ [45] . In the present study, when we analyzed the different cloud cover conditions, we found that the MRE for Rn_estimated was 0.2 - 5.1 MJ·m⁻²·day⁻¹ for clear sky days, 0.01 - 3.5 MJ·m⁻²·day⁻¹ for days with scattered clouds, 0.4 - 3.1 MJ·m⁻²·day⁻¹ for days with broken clouds and 1.3 - 2.7 MJ·m⁻²·day⁻¹ for overcast days (Table 4).

The model used in obtaining Rn and, specifically, the way in which the effect that cloud cover has on the longwave component is calculated, can cause significant errors in the estimation of daily ET by the Penman- Monteith model [15] [46] .

The Rn depends heavily on the Rs, which is in turn dependent on other factors, such as the effect of cloud cover, increases in cloud cover decreasing the Rs and Rn fluxes and consequently decreasing the ET. This is because the clear sky condition reveals the dependence of Rn on cloud cover [47] .

When ET is estimated by the Penman-Monteith model on the basis of Rn estimated by the Rn_Rs model, the effect of cloud cover is embedded in the term Rs, but varies slightly in comparison with that of the Rn estimation models in which the effect of cloud cover is taken into account. This is because cloud cover has a major influence on variations in net longwave radiation and consequently on estimates of ET. The Rn_Rs model limits variations that other models allow, because it sets fixed values for the seasons. [12] recommend that parameters such as Rs, surface albedo, K_T and air temperature, normally used to estimate Rn, be incorporated into new elements, such as Rs and pressure of water vapor, to improve the credibility of the estimates. As cloud cover de creases, the net longwave radiation balance becomes more negative and therefore has a greater effect on the calculation of the Rn_estimated, bringing it into closer proximity with the Rn_measured (Table 5).

N, number of observations; b, slope of the regression line; R², coefficient of determination; d, index of agreement; MRE, mean relative error; EF, efficiency. Means followed by the same letter in the same column do not differ at the 5% level by t-test.

As can be seen in Table 5, on days with overcast skies, all four of the Rn estimation models that took cloud cover into consideration overestimated the Rn in relation to the Rn_measured, because net shortwave radiation is directly dependent on Rs, the proportional contribution of net shortwave radiation increasing in parallel with increases in Rs. Determining the net longwave radiation depends on indices that correct for the effects of cloud

K_T, clearness index, Rs, global solar radiation; Rns, net shortwave radiation; Rnl, net longwave radiation, Rn, net radiation.

cover and pressure of water vapor. In the Rn_BRUNT, Rn_FAO-24W and Rn_FAO-24D models, net longwave radiation has the same effect as cloud cover, changing only the indices that correct for pressure of water vapor in the air, which decreases the size of its effect, the index values being 0.47 for Equation (5), compared with 0.31 for Equation (6) and 0.20 for Equation (7).

Under conditions of clear sky, scattered clouds and broken clouds, the Rn_estimated values obtained with the Rn_FAO-24D and Rn_FAO-56 models were comparable to the Rn_measured. Under overcast conditions, none of the models employed was able to adequately represent the Rn_measured or the ET obtained therefrom (Table 4). The Rn_estimated depends on the proportional contribution of net shortwave radiation and net longwave radiation. Under overcast conditions, the net longwave radiation share corresponded to only 15% - 22% of that of net shortwave radiation, which affected the estimation of Rn, because the net shortwave radiation was more prominent. Under conditions of clear sky, the net longwave radiation share corresponded to 30% - 49% of the net shortwave radiation share, having an even greater effect on the estimation of Rn (Table 5).

According to the coefficient of determination, index of agreement, MAE and efficiency values, the best estimates of ET were obtained via the Rn_FAO-24D model, followed by the models Rn_FAO-56, Rn_FAO-24W, Rn_BRUNT and Rn_Rs. The Rn_estimated obtained with the Rn_FAO-24D and Rn_FAO-56 models more closely approximated the Rn_measured than did that obtained with the other models. Despite the similarity of the equations applied in the Rn_FAO-24D and Rn_FAO-56 models, which differ only in the effect of cloud cover, there were significant differences between the two models. The Rn_FAO-56 model underestimated the cloud cover, thereby increasing the estimated ET.

Under conditions of clear sky, scattered clouds and broken clouds, the Rn_estimated values obtained with the Rn_FAO-24D and Rn_FAO-56 models were comparable to the Rn_measured value. As cloud cover decreases, the net longwave radiation balance becomes more negative and therefore has a greater effect on the calculation of the Rn_estimated, bringing it into closer proximity with the Rn_measured.

The Rn is the meteorological element that has the greatest influence on ET and can cause significant errors in the estimation of ET when not correctly measured or estimated.

This study received financial support from the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP, São Paulo Research Foundation; Grant No. 05/59535-4).