Performance of Different Models for Estimating the Global Solar Radiation in Brazil

Global solar irradiance (Qg) is an important variable of the physical environment that has been constantly used in agrometeorological models, either for climatic characterization or to give support to radiometric studies developed for irrigation planning and crop weather modeling approaches. The current study aimed to compare measured daily values of Qg with estimates of this variable by means of four different methods. For that throughout the period comprised between March 28th of 2008 and August 8th of 2011 at Ponta Grossa, PR, Brazil, a simple linear regression study confronting radiometric data measured by a pyranometer and estimates of Qg was proposed herein. Global transmittance was conditioned by atmospheric cloudiness. The models based on mean global transmittance in daily basis performed more satisfactorily and generated values of Qg with accuracy and exactness at the site in study, as confirmed by the statistical parameters employed to validate the USAge of models proposed by Angström-Prescott. However, the performance of the methodologies based on the determination of mean global transmittance under extreme atmospheric conditions, showed the highest Willmott coefficients, which was to be close to 1, reflecting then precision and reliability for the calculated values of Qg, when compared to observed values monitored at an automatic weather station.

INTRODUCTION Solar irradiance is responsible for triggering great part of the chemical, physical and biological processes in the soil-plant-atmosphere system. However, not all solar irradiance can effectively reach terrestrial surface, for by passing through optical mass and interacting with the atmospheric components it suffers action of reflection, diffusion and absorption processes. From such, approximately 51% of the extraterrestrial solar irradiance is available to be utilized in processes of the biological and physical environment (Ometto, 1981), depicting the fraction of radiation that effectively reaches soil surface or global solar radiation (Qg).
Qg is an important variable of the physical environment that is constantly employed in studies of water requirement of irrigated crops, modeling of growth and crop yield, climate changes, optimization of environmental comfort, among other applications. The great problem dealing with collection of Qg data is the high cost of the pyranometer, a radiometric equipment responsible for recording global solar irradiance at a given site (Souza et al., 2011). Moreover, the Qg quantification requires the usage of recorders or data acquisition systems, as well as skilled and specialized people making the cost of such information high at operational level.
The lack of Qg observations has been a persistent problem in studies of biophysical processes in agroecosystems. The number of weather stations recording the daily irradiance is small compared to the number of weather stations that monitor temperature and rainfall. In the United States of America the ratio of weather stations that measure solar radiation components and those that measure air temperature is greater than 1:100; in global terms such a ratio is of 1:500 (Thorton and Running, 1999). In the northeast of China, region of the country of a great agricultural importance, there are 109 weather stations, being that among those only 13 perform routine radiometric measurements (Wu et al., 2012). In the state of Paraná, Brazil, there are 37 automatic weather stations and 22 conventional weather stations responsible for the records of solar irradiance components. However, the electronic sensors are not always calibrated, showing record failures for a long period and lack of accurate radiometric information in studies related to micrometeorology and optimization of irrigation in agriculture.
Ratio between global solar irradiance and solar irradiance at the top of the atmosphere represents global solar transmittance, which in conjunction with insolation ratio might be employed in the Angström-Prescott equation to estimate the a and b empirical coefficients (which reflect the factors that affect absorption and diffusion of solar radiation) and that are considered as an input variable in estimation models of global irradiance for sites that do not count on equipment and specialized labor throughout the collection of such a variable (Paulescu et al., 2008). The Angström-Prescott equation proposed to estimate global irradiance from insolation ratio was idealized by Angström em 1924 and modified by Prescott sixteen years later with the purpose of circumventing the difficulty of obtaining the Angot value (Penman, 1948). Such equation, besides referring to a time interval higher than 70 years has demonstrated satisfactory performances for small periods all over the world and has been largely used in studies of solar radiometry (Li et al., 2011;Kolebaje and Mustapha, 2012;Wu et al., 2012;Sabziparvar et al., 2013), although some limitations are to be taken into account especially due to cloud thickness.
Recently Li et al. (2011), making use of the classical model of Angström-Prescott for estimating global solar irradiance in Tibet, China, from a series of 15 years of radiometric data at four different weather stations, obtained estimates of the variable in question with errors lower than 10% in all studied sites. Wu et al. (2012), using the same model for assessing Qg throughout the crop growing season at the northeast of China, concluded that such a calculation procedure is not only effective and reliable but also an economic means to obtain radiometric data where the collection of such an environmental variable is scanty.
Kolebaje and Mustapha (2012) obtained a high accuracy for the Qg estimation model proposed by Angström-Prescott at four different climatic regions of Nigeria. The reliability and performance of the Angström-Prescott model were also evidenced by Namrata et al. (2013), by comparing such a model to other six empirical models for estimating global irradiance at the region of Jharkhand, India. Sabziparvar et al. (2013) made use of the model proposed by Angström-Prescott to estimate global irradiance at 15 sites of Iran with the purpose of using it for the calculation of reference evapotranspiration and verified that such a classical estimation model was precise to assess Qg.
Aiming at examining the performance of different models for the calculation of monthly and daily solar irradiance at a horizontal surface in Kuala Terengganu, Malaysia, Muzathik et al. (2011) observed that by means of the mean percent error, mean squared error of the Fisher correlation test and t-test the estimation model proposed by Angström-Prescott was ascribed by low dispersion measures, revealing a high precision and reliability under the studied climatic conditions.
Scientists from all over the world have been incorporating modifications into the Angström-Prescott classical model by means of the insertion of physical variables or alterations in the mathematical expression of Qg calculation in order to improve accuracy and exactness of the estimates so that reliability and feasibility might be assured under specific atmospheric conditions at several sites of the globe (Newland 1989 In order to meet the needs of knowing solar energy availability at the region of Campos Gerais of Parana, Brazil, we come up with mathematical models developed to estimate global solar irradiance, which differ among themselves in terms of complexity and number of input variables of the model. It is important to consider that such models are in general applicable to environmental conditions where they were originally developed, showing therefore problems of transferability when not properly gauged and calibrated (Borges et al., 2010).
Faced with that, the current work aimed to determine global transmittance, evaluate and compare the performance of four different models for estimating Qg under the climatic conditions of Ponta Grossa, State of Parana, Brazil, in order to maximize crop yield at production fields with environmental protection.

II.
MATERIAL AND METHODS The field trial was carried out at the Experiment Station of the State University of Ponta Grossa, Ponta Grossa, PR, Brazil (altitude of 880m, latitude of 25°5'S and longitude of 50°3'W). The local climate was classified according to Köppen's classification as of the type Cfbhumid subtropical climate.
Global solar irradiance daily data was measured by a silica photodiode pyranometer, LI-COR, model LI-200X, with a spectral response within the interval of 0.4 and 1.2 µm, with a typical absolute error of ±3% (Federer and Tanner, 1965) from a automatic weather station throughout the period comprised between March 28 th of 2008 and August 24 th of 2011. The pyranometer was coupled to a Campbell Scientific Inc., model CR-1000, data acquisition system programmed for taking readings with a 60 s frequency and storing averages at each 30 minutes. Initially instantaneous values of Qg were integrated (W m -2 ) over the course of a day in such a way as to express them in MJ m -2 day -1 .
The photoperiod duration was determined by means of Pereira and Villa Nova (2008) and effective In order to calculate daily global transmittance daily values of global solar irradiance were divided by daily extraterrestrial solar irradiance. Afterwards the same calculation procedure was repeated so as to classify the radiometric data into cloudy and sunny days as a function of insolation ratio, which was lower than 0.3 for cloudy days and higher than 0.8 for sunny days.
To estimate Qg four different models were employed in the current research.
where, ⁄ is the insolation ratio given by the rate between effective astronomical insolation (n) and photoperiod duration (N).
The accuracy of the estimates of Qg was expressed by the coefficient of determination (R 2 ) ( Legates and McCabe, 1999). Its exactness might be observed by the dispersion of the data around the fitted line of the estimates in a 1:1 graph, which was quantified by means of the agreement index (d) proposed by Willmott et al. (1985), once the values of coefficient of correlation and determination analyzed separately can lead to interpretations not always suitable for the performance of the studied model.
In this paper, a new index c proposed by Camargo and Sentelhas (1987) was also adopted to indicate the performance of the Qg model, putting together the accuracy R and exactness d indices, being defined by the multiplication between both statistical indices.
In order to evaluate the error of estimates two statistical parameters were calculated, such as the mean absolute percentage error (MAPE) and smoothed absolute percentage error (SAPE), as described by Goodwin and Lawton (1999). Moreover, to compare observed and predicted data of global solar irradiance the following errors were adopted: mean error (ME), mean absolute error (MAE) and root mean square error (RMSE) (Borges et al., 2010).

III.
RESULTS AND DISCUSSION Daily global solar transmittance (Tg) represents the proportion of global solar irradiance determined at the extreme limit of the atmosphere that effectively reaches soil surface. Since Tg comes from the interaction of Qo with the terrestrial atmosphere it is certain that optical mass thickness varies in compliance with zenithal angle, which therefore brings about instantaneous fluctuations in Tg throughout the day, showing low values at sunrise and sunset and maximum values at noon, time of the day in which there will be the maximum incidence of solar energy at the Earth surface.
Another factor that it is to be of a great deal of concern on daily global solar transmittance of the atmosphere is cloudiness, once reflectivity of the clouds is greater than reflectivity of the atmosphere under cloudless conditions (Dantas et al., 2000).  .org/10.22161/ijaers.5.8.8  ISSN: 2349-6495(P) | 2456-1908(O) www.ijaers.comPage | 65 highest Tg values obtained during those months under the lowest cloudiness condition. Faced with a seasonal analysis on Tg calculated at Ponta Grossa, PR, it was possible to observe that Tg values under cloudy days were higher during spring/summer seasons, whereas under sunny days Tg was higher during fall/winter seasons (Table 1)  Discrepancies in the variation of Tg also might be noticed as a function of local latitude. Thus, at sites of low latitudes the incidence of solar radiation is quite nearer the perpendicularity in relation to equator plan and, therefore, the saturation period of energy diminishes with the increase in latitude. Variations in global transmittance as a function of local latitude were also observed by Yousif et al. (2013), by comparing two distinct sites of Europe: Marsaxlokk in Malta and Valladolid, in Spain.
All Qg estimation methods assessed in the current research showed values consistently close to those measured by the pyranometer (Figure 1).
By comparing the coefficients of determination (R 2 ) and the Pearson correlation (R) we noticed that all models studied demonstrated a high precision. The 4 th model showed the highest value of R 2 , with 99.8% of the measured Qg variations being explained by a simple linear regression equation, evidencing therefore a high precision for such a model. Either the coefficient of determination or Pearson correlation one bring only information about the precision of the mathematical models used, but nothing reveal about its exactness (Pereira et al., 2003;Pereira and Villa Nova, 2008). Therefore, by means of the calculation of the agreement index (d) proposed by Willmott et al. (1985) it was possible to notice that the 1 st and 2 nd models were related to values of d corresponding to 0.982 and 0.989, respectively, indicating that the estimated values of Qg were extremely close to the observed ones. Although the agreement indices obtained for the 3 rd and 4 th estimation models had been low in comparative terms, such statistical parameters were also high for such models, assuming values close to 1, given the small dispersion of the data around the fitted 1:1 line (Figure 1).
The performance index (c) of the mathematical models, defined by the multiplication between R and d, was of 0.980 for the first model, 0.979 for the second, 0.839 for the third, and 0.942 for the forth model. Taking into account the interpretation criterion of performance of agrometeorological models proposed by Camargo and Sentelhas (1997), the performances obtained by the 1 st , 2 nd , and 4 th estimation models were excellent (c> 0.85), whereas only the 3 rd model was classified as that one with a very good performance (c ranging from 0.76 to 0.85).   (Table 2).
Nevertheless, the first and second models showed the lowest values of MAPE and SAPE, being lesser than 16%, whereas for the 3 rd and 4 th models such dispersion measures oscillated around 20 to 23%, reinforcing the recommendation of the two first models for radiometric and agrometeorological studies at the studied region.  The confirmation that both first and second global solar irradiance estimation models were those that most reflected reality of radiometric data measured at an automatic weather station installed in Ponta Grossa, PR, Brazil, might be evidenced by the lowest values of mean absolute error (MAE) and root mean square error (RMSE) associated to such models (Table 2).
In general all models employed to estimate Qg under the climatic conditions of Ponta Grossa, PR, demonstrated a trend to overestimating such environmental variable. The Qg values of overestimation based on MAE varied from 0.99 MJ m -2 day -1 for the first model up to 3.12 MJ m -2 day -1 for the forth model. Corroborating MAE, calculated values of RMSE also revealed a low error in the estimates of Qg from the use of models 1 and 2 (Table 2).
Bakirci (2009), using a series of radiometric data for 18 sites of Turkey, calculated different estimation errors from a comparative study between measured and estimated global solar irradiance to determine the best Qg estimation model. The same author verified that even with models showing the highest coefficients of the Pearson correlation obtained by means of a regression analysis between Qg measured and estimated values, the statistical parameters that reveal the estimation errors were high, leading to the proposition of adjustment factors to be incorporated into the models for the studied sites.
Determining which model turns out to be the best to calculate Qg as a function of the estimation errors scuppers the utilization of estimation equations indiscriminately and bereft of scientific criteria of selection at a given site. In a study carried out in Tibet, China, Li et al. (2011) confirmed the utilization feasibility of different models designed to estimate global solar irradiance as a function of the calculation of dispersion measures that express the estimation errors and obtained good approximations of Qg, with errors below 10% under the local climatic conditions.
The best performance revealed in the current study, either for the first or second models, is intimately related to the direct interference of atmospheric cloudiness conditions, expressed by global solar transmittance determined locally under extreme cloud cover situations from the monitoring of radiometric data collected at weather stations.
Fittings in the Angström-Prescott classical model to estimate Qg were proposed by Toğrul (2009) by means of the addition of geographical and meteorological data collected at the region of Bishkek, Kyrgyzstan, being able to notice that such a model showed good estimation results of global solar irradiance.
A similar study was developed by Chen and Li (2012) in the northeast of China, by analyzing radiometric data collected from 13 weather stations. The fitting of the Angström-Prescott classical model was obtained from the calibration of the a and b coefficients as a function of variations of the global transmittance at the studied site. Even under the influence of non significant differences among the installation sites of weather stations and estimation methods employed, the aforementioned authors observed that alterations in the empirical coefficients of the regression equation increased accuracy of the Qg estimation model.
Despite the third Qg estimation model has demonstrated a performance index lesser than the other models, such a model was to be quick and practical for evaluation of Qg as a function of only one radiation maximum flux density measured at solar noon, which is irrespective of daily integrations and does not require a large number of radiometric measurements in projects of solar engineering.

IV.
CONCLUSIONS Daily global transmittance was conditioned to atmospheric cloudiness, being amenable to estimate global solar irradiance for sunny days.
The Angström-Prescott classical approach might be used to calculate daily global irradiance with accuracy and exactness in the municipality of Ponta Grossa, PR, Brazil.
Models for which the a and b coefficients were locally obtained as a function of daily mean global transmittance showed estimates more close to radiometric  .org/10.22161/ijaers.5.8.8  ISSN: 2349-6495(P) | 2456-1908(O) www.ijaers.comPage | 68 data measured by pyranometers, being such models therefore more reliable for agrometeorological purposes.