## 3 Results
### Monthly Energy Sums  
The intra-annual variation of \(\rm H_{g,t,\beta/pg}\) in each specific SEZ region is shown in Figure 5. Figure 5a refers to all sites in SEZ-A with optimal \(\beta\) = 40\({}^{\circ}\), Figure 5b to all sites in SEZ-B with optimal \(\beta\) = 45\({}^{\circ}\), Figure 5c to the sites in SEZ-C with optimal \(\beta\) = 50\({}^{\circ}\), and Figure 5d to all sites irrespective of SEZ and optimal \(\beta\). The expressions for the lines that best fit the means and their coefficient of determination, \(\rm R^{2}\), are given in Table 3. It is seen that the R\({}^{2}\) statistic obtains high values; this allows a solar energy user or investor in Saudi Arabia to estimate the monthly energy production in any of the three SEZs in an accurate way by applying the regression equations. The graphs also contain curves for the mean \(\pm\) 1 standard deviation (\(\sigma\)). It is apparent in all graphs (and especially in those that refer to SEZ-A and SEZ-B) that two secondary \(\rm H_{g,t,\beta}\) maxima in March and October exist. These occur because of the variation of the solar elevation (or altitude, \(\gamma\)) throughout the year at a certain time during the day. To demonstrate this, Figure 6 shows the variation of \(\rm H_{g,t,40}\) during 2005-2016 at three sites in Saudi Arabia at high, mid and low latitudes, i.e., Arar (#6, 30.96 degN), Jeddah (#16, 21.49 degN), and Jazan (#17, 16.89 degN). Jazan has two energy maxima, one in April and another in October, Jeddah has two, one in March and another in October, while Arar shows a normal behaviour (maximum energy in July).  
Figure 3: Corrected distribution of the 82 selected sites in Saudi Arabia for maximum \(\rm H_{g,t,opt.\beta/pg}\). The numbers in the circles refer to those in column 1 of Table 1. The red circles correspond to sites with selected opt. \(\beta\) = 40\({}^{\circ}\) (SEZ-A), the orange with selected opt. \(\beta\) = 45\({}^{\circ}\) (SEZ-B), and the green ones with selected opt. \(\beta\) = 50\({}^{\circ}\) (SEZ-C). This figure is a reproduction of Figure 4 in [1].  
Figure 5: _Cont._  
\begin{table}
\begin{tabular}{c c c} \hline
**SEZ** & **Regression Equation** & \(\mathbf{R^{2}}\) \\ \hline A (months) & \(\mathrm{H_{g,t,40/\rho_{g}}=0.0054\,s^{6}-0.218\,s^{5}+3.4241\,s^{4}-25.977\,s^{3}+ 95.627\,s^{2}-142.65\,s+296.38}\) & 0.82 \\ A (seasons) & \(\mathrm{H_{g,t,40/\rho_{g}}=-2.0956\,s^{3}-0.5587\,s^{2}+10.341\,s+787.81}\) & 1 \\ B (months) & \(\mathrm{H_{g,t,45/\rho_{g}}=0.0049\,s^{6}-0.1809\,s^{5}+2.5754\,s^{4}-18.215\,s ^{3}+65.134\,s^{2}-90.585\,s+248.07}\) & 0.94 \\ B (seasons) & \(\mathrm{H_{g,t,45/\rho_{g}}=44.915\,s^{3}-367.88\,s^{2}+870.06\,s+216.23}\) & 1 \\ C (months) & \(\mathrm{H_{g,t,50/\rho_{g}}=0.0051\,s^{6}-0.1867\,s^{5}+2.6857\,s^{4}-19.7280 \,s^{3}+74.6630\,s^{2}-106.5200\,s+235.9800}\) & 0.98 \\ C (seasons) & \(\mathrm{H_{g,t,50/\rho_{g}}=74.807\,s^{3}-627.99\,s^{2}+1499.1\,s-174.91}\) & 1 \\ A,B,C (months) & \(\mathrm{H_{g,t,B/\rho_{g}}=0.0051\,s^{6}-0.1955\,s^{5}+2.9007\,s^{4}-21.259\,s ^{3}+77.717\,s^{2}-112.08\,s+263.4}\) & 0.94 \\ A,B,C (seasons) & \(\mathrm{H_{g,t,B/\rho_{g}}=33.967\,s^{3}-286.87\,s^{2}+683.91\,s+344.56}\) & 1 \\ \hline \end{tabular}
\end{table}
Table 3: Regression equations for the best-fit curves to the monthly/seasonal mean \(\mathrm{H_{g,t,B/\rho_{g}}}\) sums averaged over all respective sites in the period 2005–2016, together with their \(\mathrm{R^{2}}\) values; s is either month in the range 1–12 or season in the range 1–4 (1 = spring, 2 = summer, 3 = autumn, 4 = winter).