## 2 Proposal for a Separation Model  
The separation models typically estimate the diffuse fraction \(k_{d}\), defined as the ratio of diffuse solar irradiation \(H_{d}\) to global solar irradiation \(H\):  
\[k_{d}=\frac{H_{d}}{H} \tag{1}\]  
As the introduction specifies, almost all empirical separation models express \(k_{d}\) as a function of clearness index \(k_{t}\) and other atmospheric parameters. Clearness index is defined as the ratio of global solar irradiation \(H\) measured at the ground to its counterpart at the top of the atmosphere \(H_{ext}\):  
\[k_{t}=\frac{H}{H_{ext}} \tag{2}\]  
This section introduces a semi-analytical model derived from the basic theory of modeling solar radiation at the ground level. The well-known closure relation between the solar irradiation components measured in a horizontal surface is:  
\[H=H_{b}+H_{d} \tag{3}\]  
On the basis of the results from [25], the beam solar irradiation \(H_{b}\) can be approximated as:  
\[H_{b}\cong\sigma H_{b0} \tag{4}\]  
where \(H_{b0}\) is the beam solar irradiation recorded in clear sky conditions. \(\sigma=SD/\Delta t\) denotes the relative sunshine during a time interval \(\Delta t\), i.e., the ratio between the sunshine duration \(SD\) and the greatest possible time of sunshine \(\Delta t\). By way of consequence, if the Sun shines during \(\Delta t\) then \(\sigma=1\) and if the sky is overcast then \(\sigma=0\). The consistency of the approximation was established on the basis of statistical theory [25]. Equation (4) is the sole approximation in this proof. By using Equations (2)-(4) in the diffuse fraction definition (1), after elementary calculation it is obtained as:  
\[k_{d}=1-\frac{\sigma}{k_{t}}\frac{H_{b0}}{H_{ext}} \tag{5}\]  
At a generic level, the beam solar irradiation component under clear sky can be expressed as a function of the atmospheric transmittance \(\tau_{a}\) (e.g. [26]):  
\[H_{b0}=G_{\mathrm{SC}}e_{j}\int\limits_{\Delta\omega}\tau_{a}(\omega)\sin h\;d\omega \tag{6}\]  
where \(G_{\mathrm{SC}}=1366.1\) W/m\({}^{2}\) is the solar constant, \(e_{j}\) denotes the correction of solar constant with respect to the Sun-Earth distance, \(h\) is the Sun elevation angle and \(\omega\) represents the hour angle. During a given time interval \(\Delta t\), \(H_{ext}\) can be deterministically calculated:  
\[H_{ext}=G_{\mathrm{SC}}e_{j}\int\limits_{\Delta\omega}\sin h\;d\omega \tag{7}\]  
By summing up the results, it is obtained:  
\[k_{d}=1-\frac{\sigma}{k_{t}}\overline{\tau}_{a} \tag{8}\]  
where \(\overline{\tau}_{a}=\frac{\int\limits_{\Delta\omega}\tau_{a}(\omega)\sin h\;d\omega} {\int\limits_{\Delta\omega}\sin h\;d\omega}\) represents the weighted average of the atmospheric transmittance with respect to the Sun's elevation angle. Equation (8) is applicable for any time interval \(\Delta t\) during a day. Equation (8) represents a basic relationship between the diffuse fraction and three potential estimators: clearness index \(k_{t}\), relative sunshine \(\sigma\), and the mean value of the clear-sky atmospheric transmittance \(\overline{\tau}_{a}\). The three estimators are not independent. \(k_{t}\) and \(\sigma\) are strongly related (roughly linear) through the Angstrom-Prescott equation [27; 28] (see [29] for a recent review). Both \(k_{t}\) and \(\sigma\) are not completely independent of \(\overline{\tau}_{a}\), the relationships being weaker than the Angstrom-Prescott equation but more complex. Even if Equation (8) is not directly useful in practice, it points out the parameters that should be considered in an empirical equation for diffuse fraction. Thus, along with the traditional predictors, clearness index and/or relative sunshine, the insertion into the diffuse fraction of some atmospheric parameters can contribute to improving the performance in the empirical estimation of diffuse fraction. These parameters are the same as those used in modeling the atmospheric transmittance in the clear-sky solar irradiance modeling, such as water vapor column content, Angstrom parameters, etc.