# Estimation of Direct Normal Irradiance at Antarctica for Concentrated Solar Technology
## 2 Materials and Methods
### Fine-Tuning Model  
The 10,641 triplet (GHI, DNI, Diff.HI) observed daily and validated records, related to a period of 65 consecutive days, cannot be considered sufficient to extrapolate an estimate of the radiation in Antarctica. Indeed, the period that extends for the entire calendar year of 365 days, cannot be considered sufficient for the multi-year characterization of Sun exposure on the ground in Antarctica. However, the observed period can provide an adequate sample if the purpose is to model the radiative phenomenon through the correlations between the three forms of radiation. In particular, the relationship between DNI and GHI is of particular interest in this study, where:  
\[\mathrm{DNI}=\mathrm{function}\ (\mathrm{GHI})\] \begin{table}
\begin{tabular}{c c} \hline Total Number of Records & 18,709 \\ (65 days) & 18,545 \\ Number of “daily” validated records & 10,641 \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of observed data. Figure 5: Validation of observation criteria. The quality of the relationship between the radiation components (3) lies in the fact that numerous expeditions in Antarctica have enabled the acquisition of a multi-decade historical series of hourly GHI, both at Eneide station and at the weather stations scattered around the Mario Zucchelli base. The availability of GHI data allows the derivation of the DNI through the previous relationship and consequently to fully characterize the area of the Mario Zucchelli base. In the literature, numerous correlations between the irradiance components are attested, established in different geographical areas for different time intervals (hourly, daily and so on), as in References [7; 8; 9; 10; 11] and others. Historically, the most studied correlation is the dependence of the diffused component \(k\) on the global atmospheric transmission coefficient \(K_{\mathrm{T}}\):  
\[k=f(K_{\mathrm{T}}) \tag{6}\]  
The global atmospheric transmission coefficient (\(K_{\mathrm{T}}\)) is defined as the ratio between the global irradiance observed on the ground (\(I\)) and the extraterrestrial irradiance (\(I_{0}\)), both referred to the horizontal plane:  
\[K_{T}=\frac{I}{I_{0}}=\frac{\mathrm{GHI}}{\mathrm{EHI}} \tag{7}\]  
while the fraction of the horizontal diffused radiation (\(k\)) with respect to the global irradiance is given by the ratio:  
\[k=\frac{I_{\mathrm{d}}}{I}=\frac{\mathrm{D}_{\mathrm{iff}}\mathrm{HI}}{ \mathrm{GHI}} \tag{8}\]  
where \(I_{\mathrm{d}}\) is diffuse horizontal irradiance (Diff.HI), Thus, the transmission coefficient for normal direct radiation (\(K_{\mathrm{bn}}\)) is defined as:  
\[K_{\mathrm{bn}}=\frac{I_{\mathrm{bn}}}{I_{\mathrm{on}}}=\frac{\mathrm{ DNI}}{\mathrm{ENI}} \tag{9}\]  
where \(I_{\mathrm{on}}\) is the extraterrestrial normal irradiance. \(K_{\mathrm{bn}}\) is primary relevant for the calculation of DNI, where could be derived from \(K_{\mathrm{T}}\) and \(k\) through the relationship:  
\[K_{\mathrm{bn}}=K_{\mathrm{T}}(1-k) \tag{10}\]  
An excellent correlation \(k=f(K_{\mathrm{T}})\) established by Reference [12] is as follows:  
\[k=\frac{1}{1+\mathrm{e}^{\alpha+\beta K_{\mathrm{T}}}} \tag{11}\]  
where, \(\alpha\) and \(\beta\) are the Boland-Ridley parameters defined for certain specific geographic locations [12]. Therefore, the relationship between DNI and GHI is derived from:  
\[\frac{K_{\mathrm{bn}}}{K_{\mathrm{T}}}=\frac{1}{1+\mathrm{e}^{-\alpha-\beta K _{\mathrm{T}}}} \tag{12}\]  
There are two non-negligible contraindications which need to be satisfied in order to obtain a direct correlation between the two parameter pairs; (\(K_{\mathrm{T}}\), \(k\)) and (\(K_{\mathrm{T}}\), \(K_{\mathrm{bn}}\)):  
(1) The dispersion graph of the parameters pair (\(K_{\mathrm{T}}\), \(k\)) in Figure 6 presents the excessive variability of the data around the hypothetical correlation curve.