## 3 Numerical Proof of the Theory
### Extreme Cases  
The study presented in Section 3.2 was performed assuming an hourly sampling, \(\Delta t=1\) h. The following question remains: how do the results change when \(\Delta t\) deviates considerably from one hour. The results presented next answer this question, the focus being on the extreme cases \(\Delta t=0\) (instantaneous diffuse fraction) and \(\Delta t=\text{daylight\_length}\). Diffuse fraction of solar irradiance (\(\Delta t=0\)) is a very specific subject-matter, closely related to the clear-sky solar irradiance modeling. When \(\Delta t\to 0\), the physical quantities in Equation (8) change to instantaneous quantities, i.e., \(\sigma\rightarrow\text{SSN}\)(see Appendix B for SSN definition) and the mean atmospheric transmittance \(\overline{\tau}_{a}\) tends to the instantaneous value \(\tau_{a}\). Thus, Equation (8) becomes:  
\[k_{d}= \begin{cases}1-\frac{1}{k_{t}}\tau_{a}&\text{IF}\quad SSN=1\\ 1&\text{IF}\quad SSN=0\end{cases}  \tag{9}\]  
Always when the Sun is covered by clouds (SSN = 0), Equation (9) estimates \(k_{d}=1\), which is theoretically sound. When the Sun shines (SSN = 1), Equation (9) preserves  
Figure 4: Distribution of \(k_{d}\) measured in the hours from hDATA with the Sun covered by clouds, \(\sigma=0\). Figure 3: Estimated with Equation (8) vs. measured diffuse. The values of clear-sky atmospheric transmittance \(\overline{\tau}_{a}\) in Equation (8) are: (**a**) a long-term average and (**b**), hourly average. the complex dependence on the atmospheric transmittance. It should be noted that the clearness index \(k_{t}\) in Equation (9) is a measured quantity, being the common parameter of the empirical separation models. Thus, aiming to empirically develop a separation model, the atmospheric transmittance \(\tau_{a}\) may be the subject of an empirical parameterization. Figure 5 shows the instantaneous diffuse fraction evaluated based on the measurements comprised by iDATA dataset. iDATA contains 139,023 measurement points, recorded at 1-min sampling, the native resolution of BSRN. The points distribution on the \(k_{d}-k_{t}\) plane (Figure 5a) is like that observed for hourly diffuse fraction (Figure 2a), but a much larger dispersion is observed. Without any averaging, the impact on the global solar irradiance of the atmospheric transmittance (measured on the Sun direction), cloud enhancement, pyranometers uncertainties is more prominent. However, visual inspection of the density plot from Figure 5b, shows a clustering of the points in two regions: Sun covered by clouds (roughly 33% of iDATA for \(k_{d}>0.95\)) and almost clear sky (roughly 45% of iDATA for \(k_{d}<0.25\) and \(0.6<k_{t}<0.8\)). Figure 5b is consistent to other studies (e.g., [24]) but combined with Equation (9) emphasize the weight of the two clusters for \(\tau_{a}\) parameterization. The ability of Equation (9) in capturing the peculiarities of instantaneous diffuse fraction is illustrated in Figure 6a. The points corresponding to the SSN = 1 branch in Equation (9) (roughly 60%) are clustered around the first diagonal while the points corresponding to the SSN = 0 branch (roughly 40%) are distributed on a horizontal line segment, the estimated \(k_{d}=1\). The density plot from Figure 6b shows that most of the points are clustered very close to the first diagonal and the measured value \(k_{d}=1\). In terms of statistical indicators, Equation (9) applied to iDATA achieves: \(nRMSE=0.074\) and \(nMBE=0.035\). As it was developed, the inherent measurement errors, the cloud enhancement phenomena are not considered by Equation (9). An empirical parameterization of \(\tau_{a}\) could capture features from the cloud enhancement. Figure 5: (**a**) Instantaneous measured diffuse fraction \(k_{d}\) with respect to measured clearness index \(k_{t}\). The entire iDADA dataset was used; (**b**) density plot of data from the frame (**a**). Figure 6: (**a**) Estimated with Equation (9) \(k_{de}\) vs. measured \(k_{dm}\) diffuse fraction. (**b**) Density plot of data from the frame (**a**).