## 3 Numerical Proof of the Theory
### Model Assessment at 1-Hour Sampling  
The results of applying Equation (8) to hDATA are assessed next. First, a brief analysis of the variables is presented. Figure 2a shows a typical picture of \(k_{d}-k_{t}\) scattering. Clearness index varies from values lower than 0.1, recorded in hours with the Sun covered by opaque clouds, to values around 0.75, recorded in clear sky hours. The points are  
Figure 1: Relative sunshine \(\sigma\) (red) and sunshine stability number SSSN (blue) recorded in each hour from the hDATA database. clustered in two regions: \(k_{d}\simeq 1\), corresponding to the Sun covered by clouds and, roughly, \(0.1<k_{d}<0.2\) corresponding to a bright Sun on the sky. The cloud transmittance has a lower influence on diffuse fraction (\(k_{d}\simeq 1\) for most values of \(k_{t}<0.4\)). Differently, in sunny conditions (high values of \(k_{t}\)), \(k_{d}\) experiences a large dispersion, indicated a strong influence of the clear-sky atmospheric transmittance on the diffuse fraction. These observations are substantiated in Figure 2b, where \(k_{d}\) is plotted against relative sunshine \(\sigma\). While, in the mostly cloudy sky hours, a low dispersion of \(k_{d}\) is noted, in the mostly sunny hours, a large dispersion of \(k_{d}\) is noted. The clear-sky atmospheric transmittance \(\overline{\tau}_{a}\) in Equation (8) is expected just to cover the scattering on mostly clear sky hours. There is a wide range of options for estimating \(\overline{\tau}_{a}\). First, only those hours in which a variable sky \(0<\sigma<1\) was recorded, are considered. The cases \(\sigma=0\) and \(\sigma=1\) are discussed separately. The two limits are neither trivial nor minimalist. The partition of hDATA looks like this: 470 data recorded in overcast conditions \(\sigma=0\), 730 in variable sky \(0<\sigma<1\) and 844 in sunny conditions \(\sigma=1\). At least in temperate climate, approximately half of daylight hours are either sunny or overcast, emphasizing the substantial role of the two limits in the overall performance of a separation model. Most of the empirical separation equations cover the entire range of relative sunshine, even if at the limits \(\sigma=0\) and \(\sigma=1\), Equation (8) is reduced to very specific forms according to the physical phenomenology. For more inside this topic, we point the reader to [32] where the limits \(\sigma=0\) and \(\sigma=1\) are analyzed in the analogue context of the Angstrom-Prescott equation. Under variable sky conditions \(0<\sigma<1\), two approximations of \(\overline{\tau}_{a}\) are discussed next. In the first case, \(\overline{\tau}_{a}\) was evaluated as an average value over the entire period. For this, a linear Angstrom-Prescott equation, \(k_{t}=a+b\sigma\), was fitted on data (Figure 2b), then \(\overline{\tau}_{a}\) being simply evaluated as \(\overline{\tau}_{a}=a+b\), i.e., \(\overline{\tau}_{a}=0.3038+0.3794=0.6833\) (see e.g. [29] for details). Figure 2b shows a typical picture of \(k_{t}\) with respect to \(\sigma\), emphasizing the linear nature of the Angstrom-Prescott equation. In the second case, \(\overline{\tau}_{a}\) was evaluated hour by hour on the basis of direct-normal solar irradiance measured data series. It is worth reminding that in this case only those hours in which \(0<\sigma<1\) were considered, so that for at least one minute the Sun shined on the sky, allowing for direct-normal solar irradiance measurement. The results are gathered in Figure 3 in the form of scatter plots. Visual inspection of Figure 3 shows the high influence of the clear-sky atmospheric transmittance on the accuracy of diffuse fraction estimation. Thus, assuming a long-term average for \(\overline{\tau}_{a}\) in Equation (8) leads to a severe underestimation of measured data (Figure 3a). If hourly averages of atmospheric transmittance are considered (Figure 3b), a high accuracy in diffuse fraction estimation is noted (\(nMBE=0.053\) and \(nRMSE=0.067\)).