## 5 Conclusions  
The relevance of the models for separating the global irradiance into its direct and diffuse components is highlighted by the large number of scientific reports on this topic. The majority of the reported separation models are built empirically, paying tribute to the location of origin. Starting from basic knowledge in modelling the atmospheric transmittance, a general equation for diffuse fraction is derived, namely Equation (8). Deeply different from previous models, the equation is derived without any empirical approach. Clearness index, relative sunshine, and the mean clear-sky atmospheric transmittance are identified as primary predictors for a separation model. A numerical proof of theory is presented in detail in Section 3. This proof assumes the knowledge of the mean atmospheric transmittance with the highest possible accuracy, being the result of direct measurement. Therefore, the calculated value of normalized root mean square error can be considered as a reliable indicator of the maximum accuracy that can be achieved by the proposed semi-analytical model. For temperate continental climate, overall results show a limit of accuracy for the estimation of diffuse fraction around 7%, measured in terms of normalized root mean square error.  
Equation (8) can be applied directly in practice as it is. But Equation (8) have a great merit in the development of an empirical separation model. It formally indicates the optimal equation and the optimal predictors necessary to explain the high variability commonly experienced by the measured diffuse fraction. While the clearness index is a mandatory predictor (global solar irradiance is always supposed to be known), there  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Model** & **[37]** & **[38]** & **[39]** & **[40]** & **[41]** & **[42]** & **[43]** & **[16]** & **[17]** \\ \hline \(nRMSE\) & 0.281 & 0.281 & 0.263 & 0.260 & 0.270 & 0.254 & 0.266 & 0.280 & 0.302 \\ \(nMBE\) & 0.102 & 0.051 & 0.066 & 0.031 & 0.062 & \(-0.013\) & \(-0.074\) & 0.102 & \(-0.153\) \\ \hline \end{tabular}
\end{table}
Table 1: Performance of different hourly diffuse fraction models applied to hDATA dataset.  
Figure 7: **(a)** Estimated with Equation (8) \(k_{de}\) vs. measured \(k_{dm}\) hourly diffuse fraction. **(b)** Estimated hourly averaged diffuse solar irradiance \(DHl\)e vs. measured hourly averaged diffuse solar irradiance \(DHl\)m.  
is plenty of room for adapting the model according to the atmospheric data availability aiming to an accurate calculation of the mean atmospheric transmittance.  
The proposed separation model can be regarded as general, allowing for temporal (from 1-min to 1-day) and spatial (site specificity) customization. This study was focused on its development, proof of the theory, and an estimation of the theoretical limit of accuracy. Many facets of research are still open. A verification at the global scale of the semi-analytical separation model (assuming different implementations of different complexities) is planned in the immediate future.  
Conceptualization, E.P. and M.P.; methodology, M.P.; validation, E.P. and M.P.; formal analysis, E.P.; data curation, M.P.; writing--original draft preparation, M.P.; writing--review and editing, E.P. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Publicly available data from BSRN ([[https://bsrn.awi.de/](https://bsrn.awi.de/)]([https://bsrn.awi.de/](https://bsrn.awi.de/))) (accessed on 1 September 2022) and AERONET ([[https://aeronet.gsfc.nasa.gov/](https://aeronet.gsfc.nasa.gov/)]([https://aeronet.gsfc.nasa.gov/](https://aeronet.gsfc.nasa.gov/))) (accessed on 1 September 2022) were processed and analyzed in this study.  
The authors declare no conflict of interest.