## 4 Practical Implementation: Case Study  
Equation (8) is general, there being almost unlimited ways to customize it. The general equation can be substituted purely with empirical models based on the predictors formalized in the Equation (8) inference. Another option could be keeping the form of Equation (8) and expressing \(\overline{\tau}_{a}\) as a function of the available atmospheric parameters. There are a large variety of clear-sky solar irradiance models that can be integrated for \(\overline{\tau}_{a}\)[34]. The accuracy of \(\overline{\tau}_{a}\) estimation depends on the clear-sky solar irradiance model and it can be conditioned by several factors, such as the atmospheric parameters availability, the measurements quality, and so on.  
This section presents a case study focused on the latter version. The assumptions underlying the study are the following:  
1. The aim is to estimate the hourly mean diffuse solar irradiance at the station Magurele, Romania, briefly described in Section 3.1;
2. The estimation of diffuse fraction is made based on Equation (8), using the PS\(\alpha\) clear-sky solar irradiance model [35] for evaluating \(\overline{\tau}_{a}\).
3. Global solar irradiance and sunshine duration are measured in situ, enabling the calculation of the clearness index and relative sunshine, respectively. In particular, records from hDATA dataset are used;
4. For running PS\(\alpha\), the aerosol optical depth (AOD), the Angstrom exponent \(\alpha\) and the water vapor column content \(w\) are retrieved from the Aerosol Robotic Network (AERONET) [36] at daily resolution. The Angstrom turbidity factor \(\beta\) is evaluated based on AOD and \(\alpha\) (see e.g., [35] and the references therein for details).
5. For the ozone column content, a mean value of 0.35 cm-atm was assumed.  
The results of testing this specific implementation of Equation (8) are collected in Figure 7. Visual inspection of Figure 7a shows a relatively large scatter of estimated vs. measured diffuse fraction (\(nRMSE=0.245\)). No bias is observed (\(nMBE=-0.015\)). However, such a large scattering of data is characteristic of the separation models. Table 1 shows the results of running different models for the diffuse fraction reported over time. The equations are various: triple segmented linear [37; 38; 39], triple segmented polynomials of order three [40] and four [41; 42; 43], exponential [16] and double segmented [17]. There is also a variety of places of origin: North America, South America, Australia, Europe, and Asia. The models performance ranges between \(nRMSE=0.254\) for a polynomial model of order 4 and \(nRMSE=0.302\) for the double segmented model. The specific implementation of Equation (8) combined with PS\(\alpha\) keeps a small advance in front of these models. While all the models considered in this comparison hold frozen numerical coefficients, Equation (8) exhibit a large flexibility in implementation. There are many other possibilities of computing \(\overline{\tau}_{a}\) that can lead more or less to better results.  
The final outcome of the separation process is the hourly diffuse solar irradiance. This is evaluated in Figure 7b, also as a scatter plot. Comparing with the diffuse fraction, a slight decreasing in the estimation accuracy can be observed (\(nRMSE=0.313\) and \(nMBE=-0.046\)). It is worth emphasizing that the majority of points are clustered in a lenticular domain around the first diagonal. There is a very small number of points where the diffuse irradiance is estimated with large errors. Since \(nRMSE\) is a statistical indicator that penalizes errors more drastically as they increase, the weight of this small number of errors has a major contribution in establishing the \(nRMSE\) value. Looking at the proposed model precision, measured by the percentage \(p\) [%] of estimates accurate to within a given tolerance interval \(TOL\) [%], it appears very good: \(p\) [\(TOL=10\%\)] = 36.7% while \(p\) [\(TOL=20\%\)] = 59.6%.