FOR ESTIMATING REFERENCE SURFACE NET RADIATION FROM SOLAR RADIATION

Net radiation (Rn) of reference surface is important information that has many applications, but its measurement is rare due to the high cost of the sensor and the complexity involved on the measurement. Therefore, estimate Rn from another variable is desirable, as from solar radiation (Rs); however, standard methods used are complex, making interesting the use of simplified methodologies. Considering these aspects, the present study aimed to set two empirical methods to estimate Rn from Rs for Dourados region, Mato Grosso do Sul, Brazil. One method was based on mathematical modeling (Gauss Method). The other one was a more simplified and practical approach (Practical Method) comprising the determination of fixed monthly conversion factors. It was used daily Rs data of a 12-years database. With these, there were estimated Rn values by the standard method recommended by FAO. Gauss Method was set using Table Curve 2D 5.01 software. Modeling consisted in defining the values of the equation parameters. On Practical Method, we developed monthly coefficients of the ratio Rn/Rs. In order to validate both methods it was measured Rs and Rn during two years using high precision sensors. Both estimating methods showed satisfactory results, with relative mean absolute error values lower than 5.8%.


INTRODUCTION
In agrometeorology, reference surface consists of a large grass field, which is under full growth, completely shadowing the ground, with approximately 12 cm height, maintained without water and soil fertility restrictions, besides of being free of pests, diseases and weeds (Allen et al., 1998).
In tropical countries, this theoretical definition is replicated by grass fields cropped with Paspalum notatum Flügge.This reference surface standardization has great importance in order to ensure the quality of the data obtained, besides of allowing a reliable comparison among different weather stations.
For some applications in agrometeorology and irrigation, measurements of net radiation (Rn) from this surface are required.Rn corresponds to the difference between the total energy that reaches the surface and the amount that leave it.This difference constitutes the amount of energy that is used by the reference surface, essentially for three processes: evapotranspiration, air and soil warming.For example, one of the most important application of Rn relies on its use for estimating the reference surface evapotranspiration (ETo).Nowadays, the main model used to estimate ETo, known as ASCE Penman-Monteith (Allen et al., 2005), requires Rn as an input.In this sense, it is necessary to remember that ETo is an important information required in many studies or practical works, such as for rainfed agriculture (definition of best sowing periods and crop zoning) or for irrigated agriculture (design and management of irrigation systems).
Although Rn is important information, its measurement on weather stations is rare due to the high cost of the sensor and the complexity involved on the measurement.This complexity can be attributed to errors in sensor operation, but mainly due to the fact that is relatively normal that weather stations do not receive the well-deserved care to ensure the surface can really be considered a reference surface, as demonstrated by Allen (1996) and Allen et al. (2005).If the weather station surface is not managed accordingly to the protocol to match a reference surface, Rn data are certainly influenced by this mismanagement, producing Rn values that differ from those that should occur in practice.Examples of mismanagement are: lack of grass field irrigation in arid regions or during severe dry seasons in humid regions; use of mixed grass species or other different species than Paspalum notatum Flügge; nutritional deficiency; grass field that do not completely shadow the ground or the absence of a grass field (use of bare soil or a cemented floor).
Different alternatives have been applied to allow Rn estimation of different surfaces, each one presenting positive or negative characteristics.Sophisticated alternatives allow large area estimates, but it evolves acquisition, treatment, and analyses of satellite images, and consequent application in models such as in the study of Santos et al. (2015).They used MODIS sensor images and the models SEBAL, METRIC and one proposed by Bisht et al. (2005).Bisht & Bras (2010), also using MODIS sensor, states that the methodology proposed in their study can relies exclusively in remote sensing, when in the absence of ground auxiliary observations.Considering these difficulties, Allen et al. (2005) recommended Rn values required for ETo calculation should, preferentially, be estimated from measured solar radiation (Rs), accordingly to the methodology described in the document.This is a high-quality method, but it also has a high level of complexity, because requires some astronomic calculations, besides of requiring as input data the geographical coordinates and measured data of air temperature and relative humidity.
The use of empirical methods is an alternative as a practical solution where data are obtained from the systematic observation of the phenomenon under evaluation.These methods have limited validity, been applicable for situations similar where were calibrated, but with satisfactory results when used on these criteria.Its use in other regions can be feasible, since its validity is previously certified.Conceição (2006), when studying Ilha Solteira region, Brazil, obtained an empirical relation which says that reference surface Rn represents 65.3% of measured Rs.Heldwein et al. (2012) evaluating potato crop endorsed this perception, by stating that Rn of any vegetation can be estimated from Rs measurements with good accuracy for modeling purposes.For Piracicaba region, Brazil, Pereira et al. (2007) states that Rn of the reference surface can be assessed by a strong relationship between Rn and Rs, being Rn 57.4% of the Rs.Sentelhas & Nascimento (2003), also in Piracicaba, found same strong relationship having a linear dependence and being Rn equal to 54% of Rs.These authors also remind there is a seasonal variability in this relationship, considering a maximum value (59.5%) occurred in February, while a minimum occurred in May (47.3%).
Therefore, this study aimed to establish two empirical methods to estimate Rn of the reference surface (Paspalum notatum Flügge) from Rs measurements for the region of Dourados, State of Mato Grosso do Sul, Brazil.One of the methods was set considering a precise purpose (Gauss Method), because it constitutes a mathematical modelling, while the other was set considering a more practical purpose (Practical Method), because it considers the utilization of fixed monthly conversion coefficients.

It was considered one weather station of Embrapa
Western Agriculture, which is located at Dourados, State of Mato Grosso do Sul, Brazil (latitude of 22º16'S; longitude of 54º49'W and 408 m of altitude).The weather station was managed accordingly to the protocols (Allen et al., 2005;Allen et al., 1998) that are required to ensure and it can be considered as a reference surface.It is located in an area of 1.4 ha, cropped with Paspalum notatum Flügge grass and positioned in such a way that the shortest distance from the border has 37 m, ensuring the minimization of any problem related to advection, shadowing and windbreak (Figure 1).Grass cover was periodically mowed to keep its height between 9 and 15 cm, besides of adequate crop management and irrigation (when under severe drought), which was done in order to promote no water restriction.According to Kottek et al. (2006), the climate is an equatorial monsoon (Am), being characterized as a transition zone between the equatorial savannah with dry winter (Aw), to the north, and the warm temperate fully humid with hot summer (Cfa), to the south.As a rule, it has hot and rainy summer, while the winter is dry and cold (Fietz & Fisch, 2008).The Dourados weather station data (EMBRAPA, 2014) for the period of 1980 to 2012 (33 years), presents an average annual precipitation of 1,389 mm, being December the rainiest month, with 178 mm, and July the driest one, with 44 mm.Average annual temperature is 22.5 ºC, being July the coldest month (18 ºC), and December and January the hottest ones (25.4 ºC).
The study was divided into two phases: calibration and validation.On the calibration phase, daily Rs data of 12 consecutive years were used (01/01/2001 to 12/31/2012).Database is almost free of missing data and it presents 99.5% of the data for this period.All Rs data were previously scrutinized regarding its quality, according to Allen (1996) and Allen et al. (2005) criteria.
Rn values were estimated accordingly to the standardization presented in ASCE-EWRI (2005).In addition to the Rs, it was also necessary to use air temperature and relative humidity measurements, and these data were obtained from the same weather station and same period.
Calibration procedure consisted of defining two methods (Gauss Method and Practical Method) by analyzing the relation that exists between the Rs and the Rn values, it mean, the ratio Rn/Rs in %.For example, a relation of 60% for a day or a month indicates from the total energy that reached surface (Rs), 60% was used by the surface for its different process (Rn).
The Gauss equation (Equation 1) coefficients were calibrated using the software Table Curve 2D 5.01; and the mathematical model is described as follow: (1) being, y -variable to be predicted; a, b, c and d -equation coefficients to be set, and x -predictor variable.
Modeling was done in such way to establish a y value, which corresponds to the relation Rn/Rs, for every x value that corresponds to the day of the year (day 1 to day 366 in leap year).Therefore, modeling work consisted in setting the values of the coefficients a; b; c and d of the Equation 1 that best fit the relation between y and x.
On the other hand, fixed monthly coefficients of the ratio Rn/Rs, instead of daily, were determined for the Practical Method, requiring no mathematical modelling.This was achieved through a simple calculation of the mean values for each month of the year.
In order to represent each of the 366 days in the Gauss Method case, or each of the 12 months in the Practical Method case, mean values of all 12 years were used.
Validation of both models consisted in analyze the quality of Rn estimates obtained from the Gauss and Practical Methods, using as input daily measurements of Rs.Therefore, we used Rs measurements obtained by a high precision sensor (pyranometer CMP3 of Kipp & Zonen ® ).This sensor was installed on the Dourados weather station during two years (07/01/2013 to 06/30/2015).During this period, Rn was also determined by a high precision sensor (net radiometer NR-Lite of Kipp & Zonen ® ).
Rn obtained by estimates of both methods from measurements of Rs was compared against the measurements of Rn, using as analysis criteria linear regression.In addition, estimates were evaluated by methods of simple agreements analysis between modeled and measured values (mean error "ME", root mean squared error "RMSE", mean absolute error "MAE" and relative mean absolute error "RMAE").Normalized methods were also used (modeling efficiency "EF", Pearson correlation coefficient "r", and Willmott agreement index "d").Finally, a last analysis was performed by a method based on the adoption of a limited value (percentage of errors lower than the limited established "PELLE", assuming a maximum error of 10%).All analyses were implemented as disposed in Wallach et al. (2006).

RESULTS AND DISCUSSION
Gauss Method was satisfactorily calibrated as shown by high coefficient of determination (R 2 = 0.9171), low standard error (± 1.8721%) and graphical adjustment (Figure 2).There can be noted that the relation Rn/Rs varies over the year, being higher during months with higher temperature and more rain, summer (graphic extremes), and lower during the months of lower temperatures and less rain, winter (center of the graph).In a similar study conducted in the northwest region of São Paulo State, Brazil, Conceição (2006) found mean annual value of 65.3% for the relation Rn/Rs, being this higher than the mean value observed in the present study (57%).Besides, the value found by Conceição (2006) exceeds even the maximum monthly value of the present study, which was observed in January (64.4%).Although there is no mention provided by the author, the difference might be due to the fact that only positive Rn values were used, neglecting the negative ones, what certainly result in higher Rn/Rs ratio, as demonstrated by Sentelhas & Nascimento (2003).Another important observation regarding Conceição (2006) is that the author presents a fixed relation to be used during the entire year, what might produce less accurate estimates, considering that the present study demonstrated a real necessity of recognize the variability of the relation Rn/Rs during the year (Figure 2 and Table 1).Pereira et al. (2007) recommended for the region of Piracicaba, São Paulo, Brazil, that values of Rn can be predicted from measurements of Rs by considering Rn as a fraction of 57.4% of Rs.This value is very close to the observed in the present study (57%).However, we still recommend the importance of recognizing the variability of this relation during the year.Silva et al. (2007) as well as Pereira et al. (2007) also obtained values (53%) for the relation Rn/Rs of Paspalum notatum Flügge close to that obtained in the present study.
During the two years of the validation the modeled trend for both methods followed the measured relation Rn/Rs, showing lower values during the winter and higher in the summer (Figure 3).However, there can be observed during the summer the modeled relation seems to be slightly and systematically higher as compared to the measured.
When analyzing the entirety of the data obtained during the validation years, there can be noted that both estimate methods showed performance considered satisfactory for estimating Rn from Rs.This is evidenced by a good agreement between the data pair to the 1:1 line, showing high R 2 values and slopes close to 1 (Figure 4).Other statistics used to compare the Gauss and Practical Methods demonstrated adequate and similar performance between then, with a slightly advantage for Gauss Method (Table 2).The statistics of simple agreements analysis between modeled and measured values demonstrated results with a slightly advantage for the Gauss Method, considering the lower RMSE, MAE and RMAE.The exception was ME, which was closer to 0 in the Practical Method.Negative ME reveals a trend of both methods in overestimate the Rn.However, this overestimation is small, remaining close to 0.1 MJ m -2 day -1 .Nevertheless, as pointed out by Wallach et al. (2006), analyze only ME is not adequate to explain the extent of the errors, because they can be originated from small errors that cancels each other, or they can be big errors that also cancels each other.Therefore, analyzing MAE and RMAE helps when demonstrate the real extent of these errors.In this way, the MAE is 0.53 and 0.55 MJ m -2 day -1 for the Gauss and Practical Methods, respectively, resulting in errors that are really small and equals 5.6 and 5.8%, respectively.
With respect to the normalized methods of analysis (EF, r and d), results demonstrate high quality of both methods (values close to 1) and also their similarity (equal values for both methods).However, when analyzing PELLE statistic, which is centered on the acceptance of errors contained in a pre-established limit, it was observed a small advantage of the Gauss Method, considering that 85.2% of the estimates by this method produced errors lower than 10% (accepted limit).For the Practical Method the result was slightly worst, because 83.5% of the estimates were within the pre-established limit.

CONCLUSIONS
For Dourados City, located in the State of Mato Grosso do Sul, Brazil, both adjusted methods are adequate for estimating Rn from Rs measurements.These authors would like to recommend the use of the Gauss Method, especially in more rigorous applications, where care concerning the estimated value is more pronounced.On the other hand, for ordinary applications, with less rigidity, we recommend the use of the Practical Method, due to its easier calculation.
Although both methods were empirically set to be used in the studied region, these authors presumes that they are also valid for other regions that have similar climatic and latitude conditions, such as other regions in the States of Mato Grosso do Sul, Paraná, São Paulo, and even maybe in Paraguay.

FIGURE 1 .
FIGURE 1. Layout of the weather station of Embrapa Western Agriculture located at Dourados, State of Mato Grosso do Sul, Brazil.Dimensions in meters.

FIGURE 2 .
FIGURE 2. Adjustment of Gauss Method for determination of Rn/Rs (net radiation/solar radiation) daily values, for the region of Dourados, State of Mato Grosso do Sul, Brazil.

FIGURE 3 .
FIGURE 3. Measured Rn/Rs (net radiation/solar radiation) values for the region of Dourados, State of Mato Grosso do Sul, Brazil, during the models validation period (07/01/2013 to 06/30/2015) and its comparison against the values parameterized for the Gauss and Practical Methods.

FIGURE 4 .
FIGURE 4. Linear regression analysis comparing Rn (net radiation) estimated by the Gauss and Practical Methods from Rs (solar radiation) against measured Rn values during the validation period (07/01/2013 to 06/30/2015) at Dourados, State of Mato Grosso do Sul, Brazil.
the calibration of the Practical Method the mean annual relation Rn/Rs is 57%.From October to March this relation is higher than the annual mean, while from April to September is lower (Table1).The highest Rn/Rs is observed in January (64.4%) and the lowest one in June (47.7%),showingadifference between these extremes of 16.7%.The values presented in Table1constitute the monthly adjustment coefficients of the Practical Method. Considering

TABLE 2 .
Validation of the Gauss and Practical Methods for estimating Rn (net radiation) from Rs (solar radiation) measurements.The validation consisted in the agreement analysis between values estimated by both methods against the measured ones.