## 2 Materials and Methods
### Methodology
#### 2.1.2 Calculation of the Unsworth-Monteith Turbidity Coefficient, \(T_{\rm UM}\)  
[PERSON] and [PERSON] [8] introduced their turbidity coefficient, which expresses the absorption of solar rays by a dust-laden atmosphere relative to a dust-free one, with both atmospheres having a specified water-vapor content. \(T_{\rm UM}\) typically varies in the range 0 \(<T_{\rm UM}\)\(\leq\) 1. According to Unsworth-Monteith,  
\[B_{e}=B_{e}^{*}\cdot sin\gamma\cdot exp(-m^{\prime}\,\cdot T_{\rm UM}) \tag{8}\]  
from which  
\[T_{\rm UM}=\frac{lnB_{e}^{*}-lnB_{e}+lnsin\gamma}{m^{\prime}} \tag{9}\]  
where \(B_{e}^{*}\) is the direct solar radiation in a dust-free atmosphere with a specified water-vapor content. This quantity is given by [PERSON] and Hulstrom [36; 37]:  
\[B_{e}^{*}=S\cdot G_{e,o}\cdot T_{r}\cdot T_{o}\cdot T_{g}\cdot T_{w} \tag{10}\]  
where the \(T_{\rm i}\) values are the atmospheric transmitances (i = r, o, g, w) due to the Rayleigh scattering (T\({}_{\rm r}\)), attenuation by the ozone (\(T_{\rm o}\)), mixed gases (\(T_{\rm g}\)), and water-vapor (\(T_{\rm w}\)) content in the atmosphere. The general transmittance function \(T_{\rm i}\) for seven main atmospheric gases (water vapor, ozone, and mixed gases; i.e., CO\({}_{2}\), CO, N\({}_{2}\)O, CH\({}_{4}\) and O\({}_{2}\)), can be expressed by the equation proposed by [PERSON] et al. [38; 39; 40; 41]:  
\[T_{i}=1-\frac{A_{1}\cdot m^{\prime}u_{i}}{\left(1+A_{2}\cdot m^{\prime}u_{i} \right)^{A3}+A_{4}\cdot m^{\prime}u_{i}} \tag{11}\]where the \(A_{i}\) values (\(i=1\dots\) 4) are numerical coefficients that depend on the specific extinction process given in Table 1 as proposed by [PERSON] et al. (2018); [PERSON] et al. (2019); [PERSON] et al. (2020); [PERSON] et al. (2020); \(u_{i}\) implies the absorbed amount in a vertical column. More specifically, \(u_{w}\) (in cm) is the content of water vapor, \(u_{o}\) (in atm-cm) is the content of ozone, and \(u_{i}\) (in atm-cm) is the content of mixed gases ([PERSON] et al., 2019; [PERSON] et al., 2020). The broadband transmittance function due to the total absorption by the uniformly mixed gases can then be calculated by ([PERSON] et al., 2019; [PERSON] et al., 2020)  
\[T_{g}=T_{CO2}\cdot T_{CO}\cdot T_{N2O}\cdot T_{CH4}\cdot T_{O_{2}} \tag{12}\]  
where the transmittances \(T_{CO2}\), \(T_{CO}\), \(T_{N2O}\), \(T_{CH4}\) and \(T_{O2}\) are given by Equation (11) using the appropriate coefficients, as proposed by [PERSON] et al. (2019). The transmittance corresponding to the Rayleigh scattering is calculated according to the method of [PERSON] et al. (2019):  
\[T_{r}=exp\left[-0.1128\cdot m^{\prime\;0.8346}\left(0.9341-m^{\prime\;0.9868}+ 0.9391\cdot m^{\prime}\right)\right] \tag{13}\]  
For the estimation of the total amount of water vapor in a vertical column (the so-called precipitable water, in cm), the following expression proposed by [PERSON] (2018) was used:  
\[u_{w}=\frac{0.493\cdot e_{m}}{t} \tag{14}\]  
where \(e_{m}\) is the partial water-vapor pressure (in hPa) given by  
\[e_{m}=\frac{e_{s}\cdot RH}{100} \tag{15}\]  
where \(RH\) is the relative humidity at the station's height (in %), and \(e_{s}\) is the saturation water-vapor pressure (in hPa), given by [PERSON] (2018):  
\[e_{s}=exp\left(22.329699-49.140396\cdot t_{o}{}^{-1}-10.921853\cdot t_{o}{}^{ -2}-0.39015156\cdot t_{o}\right) \tag{16}\]  
with \(t_{\rm o}=t/100\) and \(t\) being the air temperature at the station's height (in K). For the estimation of the ozone-column content \(u_{o}\) (in atm-cm or in DU (Dobson Units); 1 DU = 0.001 atm-cm), the adjusted van Heuklon ozone model ([PERSON], 2018) for the Europe area, based on TOMS (Total Ozone Mapping Spectrometer) data in the period 1975-2005, and proposed by [PERSON] (2018), was used: