# Estimation of Direct Normal Irradiance at Antarctica for Concentrated Solar Technology
## 2 Materials and Methods
### Fine-Tuning Model
This phenomenon is inevitable considering that the radiative attenuation in the atmosphere depends on many other meteorological variables, which are all unpredictable and difficult to measure or estimate, such as: humidity, wind, temperature, aerosol content, and various others. Usually, the above-mentioned correlation is mostly studied through average hourly data, for which the variability is obviously less accentuated than the measures adopted in this study, carried out at a frequency of every 5 min. The same consideration also applies to the dispersion graph that considers the second pair of parameters (\(K_{\mathrm{T}}\), \(K_{\mathrm{bn}}\)) in Figure 7.  
(2) Usually the estimation of the regression parameters is carried out using the least squared method, which is computationally easy if the proposed correlation is linear. Therefore, until regression analysis can be applied to find the parameters \(\alpha\) and \(\beta\), it is necessary to transform Equation (11) in its linear expression, which contains the needed parameters that now can be estimated. Following the above, we obtain:  
Figure 5:0  
The following parameters pair (\(K_{\mathrm{bn}}\), \(k\)) in Figure 8 better represent the measurements with the approximate fitting second order polynomial curve than the other two parameters pairs. On the  
Figure 5:1 Dispersion graph of hourly parameter pairs of (\(K_{\mathrm{T}}\), \(k\)), where \(K_{\mathrm{T}}\) is global transmission coefficient and \(k\) is the fraction of the diffused radiation with respect to global radiation with their correlation Equation (11) proposed by Reference [12] (in red). Figure 5:2 Dispersion graph of hourly parameter pairs (\(K_{\mathrm{T}}\), \(K_{\mathrm{bn}}\)), where \(K_{\mathrm{T}}\) is global transmission coefficient and \(K_{\mathrm{bn}}\) is the transmission coefficient for normal direct radiation with their correlation Equation (12) (in red). graph we observe that the measurement points are gathered more strongly and continuously on the fitting curve. In this case, the candidate second order polynomial is:  
Figure 5:3  
In conditions of zero direct radiation, \(K_{\text{bn}}=0\), and the global radiation becomes entirely composed of the diffused component; consequently the diffused fraction must be unitary, \(k=1\). Therefore, in Equation (14) the constant \(c\) is \(1\). When these assumptions are taken into account and the least squares method is applied, the regression on the parabola (14) produces the following estimates (\(a=0.994\), \(b=-1.93\), \(R^{2}=0.98\)). To interpret the parabolic correlation of Equation (14) in the Boland-Ridley formulas, it is necessary to report on a Cartesian reference of the abscissa \(K_{\text{T}}\) and the ordinate \(k\), both the dependence \(k=f(K_{\text{T}})\) obtained by the polynomial expression (14) and also the dependence found in expression (11) of Boland-Ridley with different parameters pairs (\(\alpha\), \(\beta\)). Afterwards, we are able to estimate the final coefficients by regression. Proceeding in this way we obtain:  
\[ \begin{split}\alpha&=-9.18\\ \beta&=14.28\end{split}  \tag{15}\]  
The two final coefficients (15) are estimated from Equation (11) considering the 65 days of observed data from the Antarctic region. Figures 9 and 10 show the final dispersion graphs between the three coefficients (\(K_{\text{T}}\); \(k\), \(K_{\text{bn}}\)) and the estimated interpolation curves. The Boland-Ridley model uses the \(\alpha\) and \(\beta\) coefficients for the European (blue) and Australian (violet) regions [13], while the model developed in this study uses the calculated coefficients (15) for the Antarctic region (ENEA). Figure 8: Dispersion graphs of hourly parameter pairs (\(K_{\text{bn}}\), \(k\)), where and \(K_{\text{bn}}\) is the transmission coefficient for normal direct radiation and \(k\) is the fraction of the diffused radiation with respect to global radiation with their correlation Equation (14) (in red).