## 3 Results
### Variation of \(k^{\prime}{}_{d}\) vs. \(k^{\prime}{}_{t}\) over Greece  
An expression of \(k^{\prime}{}_{d}\) as function of \(k^{\prime}{}_{t}\) is very useful as \(D_{\rm e}\) can be derived from \(k^{\prime}{}_{d}\) provided there are available \(G_{\rm e}\) measurements in a location, as already mentioned in Section 2.1.3. As shown in Figure 1, the hourly values provide a wide dispersion due to extreme weather effects. In order to smooth out any such unwanted effect on the \(k^{\prime}{}_{d}\) vs. \(k^{\prime}{}_{t}\) relationship, monthly mean values of these indices were considered for the eight selected sites, following the philosophy of the eight representative sites in the four climatic zones of Greece. On the other hand, monthly values are sufficient for engineering applications. Table 4 provides the coefficients for the linear \(k^{\prime}{}_{d}\) vs. \(k^{\prime}{}_{t}\) equations under all-sky conditions. On the other hand, by grouping the monthly mean \(k^{\prime}{}_{d}\) vs. \(k^{\prime}{}_{t}\) data pairs of the stations which belong to the same climatic zone, new best-fit lines were obtained and are shown in Figure 8. It is seen from the linear regression equations that the best-fit lines of the zonal pairs A, B, C, and D are very close to each other. This gives the freedom to derive an equation for all climatic zones (Figure 9). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Climatic Zone**} & **Site** & **a** & **b** & **R\({}^{2}\)** \\ \hline B & \multicolumn{1}{c}{Agrinio} & 1.8616 & \(-\)2.1402 & 0.93 \\ \hline C & \multicolumn{1}{c}{Alexandroupoli} & 1.7406 & \(-\)1.9311 & 0.92 \\ \hline A & \multicolumn{1}{c}{Irakleio} & 1.7201 & \(-\)1.9172 & 0.99 \\ \hline A & \multicolumn{1}{c}{Kalamata} & 1.6518 & \(-\)1.7932 & 0.91 \\ \hline D & \multicolumn{1}{c}{Kastoria} & 2.0177 & \(-\)2.3781 & 0.94 \\ \hline B & \multicolumn{1}{c}{Lesvos} & 1.7920 & \(-\)1.9125 & 0.98 \\ \hline D & \multicolumn{1}{c}{Serres} & 1.8999 & \(-\)2.1945 & 0.94 \\ \hline C & \multicolumn{1}{c}{Tripoli} & 1.8737 & \(-\)2.1123 & 0.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coefficients and R\({}^{2}\) of the relationship \(k^{\prime}{}_{d}=\) a + b \(k^{\prime}{}_{t}\) for the eight selected sites in Greece. The relationships have been derived from monthly mean \(k^{\prime}{}_{d}\), \(k^{\prime}{}_{t}\) values under all-sky conditions during TATY. Figure 7: Month-hour diagrams of the \(T_{\rm UM}\) values in TATY under clear-sky conditions in climatic zone A: first raw, Irakleio (left), Kalamata (right); climatic zone B: second raw, Agrinio (left), Lesvos (right); climatic zone C: third raw, Alexandroupoli (left), Tripoli (right); climatic zone D: fourth raw: Kastoria (left), Serres (right). The color code refers to the \(T_{\rm UM}\) levels, which have been kept in the range \(0<T_{\rm UM}\)\(\leq 0.3\). The 06.00–20.00 h LST interval limitation of the graphs is due to the adopted criterion \(\gamma\geq 5^{\circ}\). Figure 8: Monthly mean \(K_{\rm d}\) vs. \(K_{\rm t}\) data pairs for sites belonging to the same climatic zone under all-sky conditions in TATY and their best-fit lines. Linear regression equations for (i) climatic zone A: \(K_{\rm d}\) = 1.730\(-\)1.925 \(K_{\rm t}\), R\({}^{2}\) = 0.97; (ii) climatic zone B: \(K_{\rm d}\) = 1.774\(-\)1.989 \(K_{\rm t}\), R\({}^{2}\) = 0.96; (iii) climatic zone C: \(K_{\rm d}\) = 1.851\(-\)2.097 \(K_{\rm t}\), R\({}^{2}\) = 0.92; (iv) climatic zone D: \(K_{\rm d}\) = 1.839\(-\)2.054 \(K_{\rm t}\), R\({}^{2}\) = 0.90. The lower R\({}^{2}\) values in the climatic zones C and D—and especially in D—are expected because of the fewer data points and harsher weather conditions at these sites in comparison to those in the other two climatic regions. Figure 9: Monthly mean \(K_{\rm d}\) vs. \(K_{\rm t}\) data pairs for all 33 sites belonging to the four climatic zones under all-sky conditions in TATY and their best-fit line. Linear regression equation: \(K_{\rm d}^{\prime}\) = 1.777\(-\)1.9991 \(K_{\rm t}\), R\({}^{2}\) = 0.94.