SOLAR RADIATION ESTIMATED FROM EMPIRICAL MODELS FOR THE NORTH OF MINAS GERAIS

Estimating daily solar radiation (Rs) provides an important alternative in situations where it cannot be measured by conventional pyranometers. This study used meteorological data from nine cities in the north of the Minas Gerais state, Brazil, for the period from 2008 to 2010 with the aim of evaluate the accuracy and applicability of some simple models to help regions where Rs is impossible to be measured. Five models were evaluated for their estimates of Rs based on simple available data. For each city studied, the equations were previously calibrated. Meteorologically based empirical models to estimate daily global solar radiation are an appropriate tool if the parameters can be calibrated for different locations. These models have the advantage of using meteorological data, which are commonly available. Based on the overall results, we conclude that the accuracy of estimation by available meteorological data is acceptable and comparable with the accuracy of classical models. Considering the greater availability of air temperature data and application in studies that do not require great accuracy in estimating Rs, all models were adequate for use. The accuracy of Rs was only slightly improved by adding rainfall records as input variable. Therefore, in the region studied, the choice of simpler models, having as input the daily maximum and minimum air temperature would not imply large error in the estimates. For most sites, Bristow and Campbell model had the best estimate of Rs with a RMSE of 2.69 MJ m -2 and R= 0.69, with the possibility to calibrate with available temperature data, becoming a practical and reliable model. Hargraves model should be avoided due to its lower performance compared to the other models applied.


INTRODUCTION
Local daily solar radiation data (R s ) is extremely important for studies involving the surface energy-balance, thermal load on buildings analysis, solar energy collecting systems, crop growth models and studies of the water requirement of irrigated crops (THORNTON;RUNNING, 1999;ROBBA, 2009).However, R s data are not available in some places, due to the absence of instruments for their measurement.Thereby, empirical equations were developed to estimate R s from variables normally available at a majority of weather stations such as sunshine duration (Ångström, 1924), air temperature range (HARGREAVES, 1981;BRISTOW;CAMPBELL, 1984;CHEN et al., 2004), air temperature range and rainfall (DE JONG;STEWART, 1993;HUNT et al., 1998;LIU;SCOTT, 2001), air temperature and water vapor pressure (ALMOROX et al., 2011) or based on day of year (BULUT, 2003;LI et al., 2010).Those models vary with numbers of variables and complexity.It is generally recognized that sunshine duration based models yield best results (WU et al., 2007;BAKIRCI, 2009).However, sunshine duration is not commonly observed at all standard meteorological stations compared to air temperature and precipitation.
Although empirically derived and conceptually simple, the air temperature-based model is founded on theoretical concepts for energy exchange in the surface boundary layer (GOODIN et al., 1999).This model assumes radiation loading as the predominant forcing mechanism for diurnal air temperature variation.Bristow and Campbell (1984) found that the model provided accurate estimates and could account for 70-90% of R s at three sites in the U.S.A.
In Brazil, despite the large increase in automatic weather stations network, R s data are not commonly available at agricultural areas, where it is essential for the reference evapotranspiration used in irrigation management.Therefore, it is important to check simple R s estimation models, especially in northern of Minas Gerais, which has 46.075ha in four irrigated perimeters (Gorutuba, Lagoa Grande, Pirapora and Jaíba) (CODEVASF, 2012).The aim of this study was to evaluate the accuracy and applicability of some simple models for estimating daily values of solar radiation to the north region of Minas Gerais State, Brazil, to determinate which models are more reliable to be used in sites where R s cannot be measured.

Meteorological data
The hourly meteorological data were obtained from nine automatic weather stations located in the cities listed in  Data sets were collected from 2008 to 2010.The limited series of data was due to the automatic weather stations from National Institute of Meteorology in Brazil were acquired in 2006 but the data were available only from 2008.For each site, a series of two years data were used in the study: the first to calibrate and the last to validate the models.Firstly, data reported at hourly intervals were reduced to daily values, obtaining the values of maximum and minimum temperatures (T max , T min ) and daily solar radiation (R s ) and total rainfall (P).Data were subjected to a screening to verify their integrity and consistency.Criterions for the elimination of data, proposed by Liu et al. (2009) were used.Data were discarded in the case of: a) missing data for any of the elements T max , T min or R s ; b) T max <T min ; c) R s /R a >1.The percentage of omitted days was calculated by the period considered.

Radiation models
Most models used in this study required the daily total extraterrestrial radiation (R a , MJ m -2 ).Therefore, R a was calculated using the equations detailed by Allen et al. (1998).The only input required to calculate these daily values, for a specific day of the year, is the latitude of the location.(1) 360 1 0.033cos 365 where d r is the eccentricity correction factor of the Earth's orbit, hn the hour angle of the sun at sunrise (radian), ∅ the latitude of the site (radian, south negative), δ is the solar declination (radian) and NDA is the day of year (1 for January first and 365 or 366 at December 31).
Hargreaves Model (Ha) Hargreaves (1981) elaborated a simple equation to estimate daily R s which requires only the air temperature range and R a : (5) where a is an empirical coefficient and ∆T 1 is the daily maximum (T max ) minus minimum (T min ) air temperature.The coefficient a must be derived for the site where data measurements are available.This model has served as the initial basis for daily solar radiation prediction by temperature-based models.Allen et al. (1998) recommended use a=0.16 for interior locations, where land mass dominates and air masses are not strongly influenced by a large water body.Therefore, in order to evaluate how the Ha model with an uncalibrated coefficient (a=0.16) would affect the model performance, we include it denoting as Ha-fixed.Chen et al. Model (Ch) Similar to the Hargreaves model, Chen et al. (2004) proposed the estimation of daily R s from air temperature and R a , but using a logarithmic relationship with two coefficients: ( ) where a and b are empirical coefficients.
Bristow and Campbell Model (B-C) Bristow and Campbell (1984) also developed a simple equation to estimate solar radiation based on the range of air temperature (∆T 2 ) in which R s is an exponential function of ∆T 2 with three coefficients: where a, b and c are empirical coefficients.To help reduce the effect of large-scale hot or cold air masses which may move through the area, ∆T 2 , is calculated as the difference between maximum and average minimum air temperature of the two consecutive days as: where i is the current day and i+1 is the next day.
These empirical coefficients have some physical explanation.The coefficient a represents the maximum solar transmittance which can be expected on a clear day and the coefficients b and c determine how soon the maximum R s is achieved as ∆T 2 increases (BRISTOW; CAMPBELL, 1984).

Li et al. Model (Li)
Since R s is a quasi-periodic phenomenon on a yearly cycle due to seasonal effects, sinusoidal correlations give excellent fitting (LI et al., 2010).The model uses sine and cosine wave correlations as follows: where a, b, c, d, e, f and g are empirical coefficients and NDA is the day of year.

De Jong and Stewart Model (J-S)
Commonly, for empirical models, more input variables promote greater chance of an improved fit of the observed data (LIU; SCOTT, 2001).De Jong and Stewart (1993) used rainfall combined with ∆T 1 for estimating R s as follows: where a, b, c and d are empirical coefficients, P is the daily rainfall (mm), and ∆T 1 is defined as in Eq. (5).

Calibration and statistical evaluation
In the period analyzed (three years), one year was used to calibrate the coefficients of models applying the nonlinear least square fitting method.The fitting process was performed with free R statistical software, version 2.13.1 through NLS function (nonlinear least-square) that outputs coefficient values and residual standard error (RSE) of the model which is the estimate of standard deviation of model error.Subsequently, the fitted models were validated using the second set of data (another year) for the same station.To ensure stability of the coefficients, solved by an iterative method using the R software, a range of coefficient values were used.Models performance was evaluated in terms of the following statistical parameters: coefficient of determination R-squared (R 2 ), root mean-square error (RMSE), the mean bias error (MBE) and the intercept (a) and slope (b) of the least-squares regression.These parameters are the most commonly applied in comparing models of solar radiation estimations (YORUKOGLU; CELIK, 2006).

RESULTS AND DISCUSSION
Li model did not fit the set of data used in this work and its results were omitted.In this model, during the fitting process, different initial values resulted in different coefficients.Li model uses only day of the year like input, which makes it more suitable to describe the normal variation of R s along of the year.Therefore, model calibrations that use only one year of data do not express cyclical component in the dataset.Li et al. (2010) used series of at least 10 years to calibrate the model, generating results more promising.Other models using the same approach like Bulut (2003) and Kaplanis and Kaplani (2007) would have the same restriction, requiring more than a year of data for proper calibration procedure.The B-C model resulted in improved fit, with average RSE of 2.81 MJ m-2, and the Ha had the worst, with an average RSE of 3.22 MJ m-2.The Ch and the J-S models presented similar average RSE of about 2.96 MJ m-2.Despite the differences, all RSE values are similar, indicating that all models similarly describe the variation of R s .
For the Ha model, the coefficient varied from 0.149 to 0.179 with 0.162 in average and close to 0.16 suggested by Allen et al. (1998) without calibration.The Ch model coefficients ranged from a = 0.302 to 0.404 and b= -0.389 to -0.186 with averaged values of a = 0.343 and b = -0.285,being close to the original calibration (a = 0.28 and b = -0.15).
For the B-C model, a ranged from 0.697 to 0.789 with an average of 0.724.Typical values for a is 0.7 (MEZA; VARAS, 2000).Liu et al. (2009) observed in China a trend of larger a coefficient in areas with higher altitude and lower rainfall (drier climate), where the values of a increased from 0.11 (lower altitudes) to 0.29 (higher altitudes).This sensitivity depends upon the local partitioning of solar energy that varies with altitude and season (LIU et al., 2009).In the present work, b ranged between 0.006 and 0.046 with an average of 0.020 and c ranged between 1.272 and 2.127 with an average of 1.879.In general, the coefficients remained within the range of calibrations obtained for other locations according researches of Liu and Scott (2001), Liu et al. (2009) and Almorox et al. (2011).For the J-S model, low values for coefficients c and d indicate reduced influence of rainfall as compared to air temperature on estimation of R s .Therefore, the inclusion of rainfall improves slightly the model, which is evident by the lower value of RSE.In part, this is explained by the fact that the number of days without rain was greater for all locations studied, making rainfall less important.

Model performance
The models that use air temperature data only (B-C, Ch and Ha) had a mean R2 value close to 0.60 (Table 3).For most sites, the B-C model resulted in the highest R2 (0.69) compared to the other models.There is no reference values for R2, but higher values indicate a more parsimonious model.Models with R2 higher than 0.60 showed a good quality of adjustment with researchers who used same models (YORUKOGLU, CELIK, 2006;ABRAHA, SAVAGE, 2008;ALMOROX et al., 2011).When using both rainfall and air temperature data (J-S), R2 was 0.61.This suggests that despite the inclusion of rainfall in the J-S model, it resulted in little improvement in R2.For RMSE, a better performance for all sites was obtained using the B-C model, with a mean value of 2.69 MJ m-2, followed by the models of Ch (2.76 MJ m-2), J-S (2.78 MJ m-2) and Ha (2.99 MJ m-2).The RMSE for each location and model followed the same sequence as for the RSE values from calibration, except for the São Romão station, where improved performance was obtained using the J-S model.Similar trends in RMSE and RSE is an indication of similar data distribution in calibration and performance data set.Generally, the values of the RMSE decrease as the R2 increase (YORUKOGLU, CELIK, 2006).
The results for MBE were similar for the various models ranging from -1.627 to 0.418 MJ m-2 for B-C model, -1.609 to 0.418 MJ m-2 for Ch model, -1.271 to 0.418 MJ m-2 for J-S model and -1.340 to 0.397 MJ m-2 d-1 for the Ha model.In general, there was an underestimation in R s .The greater absolute MBE values for air temperature based models were found for São Romão and Salinas sites, probably due to lower air temperature range in validation data set compared to the calibration set.Many factors besides R s could affect levels of maximum and minimum air temperature, especially on a daily basis, e.g.cloudiness, wind speed, atmospheric water vapor content, availability of soil water for evaporation, elevation, precipitation, aerosol, frontal weather systems and others (ALLEN, 1997).These factors confound the relationship used in air temperature based solar radiation models.Improvement in model performance was evident for increased elevation and days with clear sky events.Larger ∆T generally results in better predictive accuracy (Liu et al., 2009).Abraha and Savage (2008) found that the B-C model was improved with higher elevations than with lower ones.This could be due to reduced attenuation of R s and therefore more heating of the air.
The intercept (a) and slope (b) of linear regression provide information about trends of models throughout the observed R s .The B-C model had lower values of a (2.44-8.19) and b was closer to 1 (0.63-0.80) when compared with other models at all sites.Otherwise, Ha and Ha-fixed models had the highest values of a (7.57-10.96)and the lowest for b (0.48-0.61).High values for parameter a and lower for b can lead to an overestimate of lower values of R s and an underestimate of higher values of R s .In the north of Minas Gerais, most of rainfed crops are cultivated in the beginning of the rainy season.It coincides with longer period of cloudy days and hence a greater number of days with lower values of R s .Therefore, Ha and Ha-fixed models may be inappropriate in crop yield and evapotranspiration simulation models because in cloudy days the observed R s had lower values.
Results for the Ha-fixed model were very closed to the Ha model, confirming that it can be used without calibration.However, it showed poor performance.The Ha model is the best-known air temperature model due to its simplicity, as uses only one coefficient.However, this leads to less adjustment freedom.Borges et al. (2010), on the other hand, found that the Ha model had the best performance of three uncalibrated air temperature models (R2=0.68,RMSE of 4.76 MJ m -2 ) in Cruz das Almas, Brazil, using data from the years of 2004 to 2006, also from an automatic meteorological station of INMET.However, they did not evaluate Bristow-Campbell model.

CONCLUSIONS
The accuracy of estimation via available meteorological data was acceptable and comparable with the accuracy of classical models.
The accuracy of R s was only slightly improved by adding rainfall records as input variable.Therefore, in the region studied, the choice of simpler models, having as input the daily maximum and minimum air temperature would not imply large error in the estimates.
For the set of data used in the current study, the model based only in the number of the year was not acceptable.In this case, datasets with a large period of time could generate more satisfactory results, given the characteristics of this model.
For most sites, Bristow and Campbell model had the best estimate of R s with a RMSE of 2.69 MJ m -2 and R 2 = 0.69, with the possibility to calibrate with available temperature data, becoming a practical and reliable model.
Hargraves model should be avoid due to its lower performance compared to the other models applied.

Table 1 .
Table 1 and are shown in Figure 1.These cities are in the north of the Minas Gerais state.The stations are part of National Meteorology Institute (Instituto Nacional de Meteorologia -INMET).The INMET represents Brazil in the World Meteorological Organization.These weather stations were acquired in 2006 but the data were only available in 2008.It was used a CM6B pyranometer (Kipp & Zonen, Netherlands, 5% of accuracy) for measurements of daily R s .Geographic location of meteorological stations used in the study and the mean and range of daily solar radiation (R s ), period and percentage of omission data records for each station.
Distribution of stations in the north of Minas Gerais state.

Table 3 .
Performance of models by R2, RMSE, MBE errors and the intercept (a) and slope (b) of the linear regression between observed and estimated daily solar radiation for the North of Minas Gerais State for 2009 or 2010.