## 2 Measurements and Quality Control Processes
### Parameterizing Daytime Longwave Downward Radiation
#### 3.7.2 Estimation of Downward Irradiance under All-Sky Conditions  
[PERSON] and [PERSON] [42] proposed the following equation for the estimation of the daylight downward longwave irradiance under all-sky conditions (\(L_{dnc}\)):  
\[L_{dnc}=L_{dnc}[1+p(1-G/G_{c})^{q}] \tag{16}\]  
where \(L_{dnc}\) is the daylight downward longwave irradiance under clear-sky conditions (Equations (14) and (15)), \(G/G_{c}\) is the ratio of the measured global solar horizontal irradiance to clear-sky global horizontal irradiance, and \(p\) and \(q\) are empirical coefficients. The above equation was selected because the experimental station proposed by the researches is an inland one with a latitude of about 41\({}^{\circ}\) N i.e., closed to our site of measurements. Clear-sky hourly horizontal irradiance was estimated from the following equation obtained from [PERSON] [43]:  
\[G_{c}=1098*\cos(\theta_{z})*\exp(-0.057/\cos(\theta_{z})) \tag{17}\]  
where \(\theta_{z}\) is the solar zenith angle.  
The calibration of the coefficients \(p\) and \(q\) of Equation (16) was based on the training data set (2013-2014) and its evaluation on the independent validation data set (2015). The final estimated local coefficients of \(p\) and \(q\) had the following values [9]:  
\(p=0.212\) and \(q=1.608\) for the Idso model, and  
\(p=0.238\) and \(q=1.613\) for the Brunt model.  
Similar values of \(p\) coefficients were obtained in Valladolid (Spain) [42], but the \(q\) coefficients at Athalassa are almost twice higher than those in Spain.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Sky Conditions** & **Model** & **Variable** & **Slope** & **R2** & **RMSE (W/m2)** \\ \hline \multirow{2}{*}{Clear sky} & Idso & \(L_{nalc}\) & 1.017 & 0.997 & 21.819 \\  & Brunt & \(L_{nalc}\) & 1.019 & 0.997 & 19.347 \\ \hline \multirow{2}{*}{All-sky} & Idso \(L_{nalc}\) & \(L_{nalc}\) & 1.070 & 0.992 & 30.597 \\  & Brunt \(L_{nalc}\) & \(L_{nalc}\) & 1.061 & 0.993 & 29.009 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of statistical estimators of the two calibrated daylight downward longwave irradiance models under (a) clear-sky and (b) all-sky conditions at Athalassa using the current data set (2016–2017).  
Using the current data set, and the coefficients reported earlier, the results of this exercise are presented in Table 6. As can be seen the slopes and the coefficients of determination are close to unity for both models for clear-sky conditions. On the other hand, both models showed slightly higher RMSE values than those obtained with clear conditions. Similar results were obtained by [PERSON] and [PERSON] [42] in Valladolid, Spain. The fitted line of the estimated versus the measured hourly daylight downward longwave irradiances under all-sky conditions, using the [PERSON]'s and [PERSON]'s clear-sky models are shown in Figure 16. As seen, the data points are within the 95% prediction intervals. Additionally, the coefficients of determination are very close to each other, i.e., 71% and 72%, respectively, for each model.  
Figure 16: Comparison of measured and estimated hourly daylight downward longwave irradiances under all-sky conditions for the calibrated coefficients of (**a**) [PERSON]’s and (**b**) [PERSON]’s models at Athalassa using the current dataset (2016–2017).