## 2 Materials and Methods
### Methodology
#### 2.1.3 Classification of the Sky Conditions  
[PERSON] (2017) gives an account of ten possible ways to identify clear skies from available meteorological and/or solar radiation measurements. Nevertheless, he states that the most-adopted technique is the clearness index, \(k_{\rm t}\), which is defined as the ratio \(G_{\rm e}/G_{\rm e,extra}\)-\(\sin\)\(\gamma\). As regards this index, [PERSON] et al. (2018) have proposed values of \(k_{\rm t}>0.6\) and \(k_{\rm t}<0.2\) for clear and cloudy skies, respectively. [PERSON] and [PERSON] (2019) and [PERSON] et al. (2020) used values of \(0<k_{\rm t}\leq 0.15\), \(0.15<k_{\rm t}\leq 0.7\) and \(k_{\rm t}>0.7\) for overcast, intermediate and clear skies, respectively, in Hong Kong. [PERSON] and [PERSON] (2018) used \(k_{\rm t}>0.65\) and \(0.12\)\(\leq k_{\rm t}\leq 0.35\) for very clear and cloudy skies, respectively, to classify the sky conditions at Port Harcourt, Nigeria. As \(k_{\rm t}\) depends on \(m\), [PERSON] et al. (2019) have defined a modified clearness index, \(k^{\prime}_{\rm t}\), independent of the air mass:  
\[k^{\prime}_{t}=\frac{k_{t}}{0.1+1.031\cdot exp\left[-1.4/\left(0.9+\frac{9.4} {m}\right)\right]} \tag{18}\]  
This work adopts the modified \(k_{\rm t}\) using the following limits: \(0<k^{\prime}_{\rm t}\leq 0.3\), \(0.3<k^{\prime}_{\rm t}\leq 0.65\), and \(0.65\)\(<k^{\prime}_{\rm t}\leq 1\) for overcast, intermediate and clear skies, respectively.  
Another index defining the sky conditions is the diffuse fraction, \(k_{\rm d}\) (e.g., (1992, 1993, 1994, 1995)), which is the ratio \(D_{\rm e}/G_{\rm e}\). This sky-condition parameter also depends on air mass through the solar radiation components. To avoid this dependency, the present work defines a modified diffuse fraction, \(k^{\prime}_{\rm d}\), for the first time worldwide, following the formulation of \(k^{\prime}_{\rm t}\) in Equation (18):  
\[k^{\prime}_{d}=\frac{k_{d}}{0.1+1.031\cdot exp\left[-1.4/\left(0.9+\frac{9.4} {m}\right)\right]} \tag{19}\]  
Figure 1 shows an example of the variation of \(k^{\prime}_{\rm t}\) vs. \(k^{\prime}_{\rm d}\) for the site of Alexandroupoli (see Table 2 for site information). It is seen that the dependence of \(k^{\prime}_{\rm t}\) on \(k^{\prime}_{\rm d}\) has an exponential decay shape (Equation (20)) and not a linear one as is the case of the original indices; for example, \(k_{\rm t}=0.860\)-\(1.015\)\(k_{\rm d}\) in Hijazin (1993). It is worth mentioning that both paradigms of Alexandroupoli and Hijazin refer to hourly values of the clearness index and the diffuse fraction. The best-fit curve to the hourly data points of Alexandroupoli in Figure 1 has the following form:  
\[k^{\prime}_{t}=1.1430\cdot exp\left(-1.1030k^{\prime}_{d}\right)\ -\ 0.4346\cdot exp\left(-5.5970k^{\prime}_{d}\right) \tag{20}\]  
Expressions of \(k^{\prime}_{\rm t}\) vs. \(k^{\prime}_{\rm d}\) for seven other sites have also been derived (see Section 3.3). Further, the characterization of the sky conditions by using \(k^{\prime}_{\rm d}\) alone, as done with \(k^{\prime}_{\rm t}\), is risky, because \(k^{\prime}_{\rm d}\) depends also upon \(D_{\rm e}\), which is not always measured in the same manner as \(G_{\rm e}\). However, the derivation of a relationship of the form \(k^{\prime}_{\rm d}=\) f(\(k^{\prime}_{\rm t}\)) may be useful as it can provide information about \(D_{\rm e}\), if \(G_{\rm e}\) measurements are available in a location.  
Figure 2 provides information about the distribution of clear, intermediate and overcast skies over eight sites in Greece following the \(k^{\prime}_{\rm t}\) limitations. The other sites in Table 2 show similar patterns (not shown here).  
Figure 1: Graph of hourly \(k^{\prime}_{\rm t}\) vs. hourly \(k^{\prime}_{\rm d}\) values over one complete year for the site of Alexandroupoli; the limits of overcast, intermediate and clear skies are shown. The solid black line is the best-fit curve to the data points from Equation (20) with R\({}^{2}\) = 0.82.  
Figure 2: _Cont._