## 2 Characterization of Cloud Types and Their Impairments
### Attenuation by Clouds  
In general, the mechanisms underlying cloud attenuation are those that arise when cloud particles interact with electromagnetic radiation. Specifically, the loss of energy passing through the cloud volume is determined by absorption processes, which are understood as the sum of absorption and scattering effects caused by liquid water drops. These particles can be considered, in essence, as ideal dielectric spheres; thus, [PERSON]'s theory of single scattering, widely described in several textbooks such as [41; 42], provides the basis for an accurate estimate of the extinction, scattering and absorption coefficients, or the so-called cross sections, \(\sigma_{e}\), \(\sigma_{s}\), and \(\sigma_{\alpha}\), (in square meters), where \(\sigma_{e}=\sigma_{\rm s}+\sigma_{\alpha}\). From these parameters, the specific attenuation \(\gamma_{c}\) (in decibels per kilometer) caused by a combination of water spheres with different radii \(r\) (in millimeters) is provided by [43]  
\[\gamma_{c}=4.343\times 10^{3}\int\sigma_{e}(r)N(r)dr, \tag{1}\]  
where \(N(r)\) is the particle size distribution of the cloud, and the corresponding product \(N(r)dr\) determines the density of water droplets of the cloud with radius \(r\). The coefficient \(4.343\times 10^{3}\) describe the fundamental definitions for the density of a liquid in rain/fog/cloud spheres as specified in [44; 45]. The behavior of \(N(r)\) for a population of cloud drops usually modeled by a modified gamma function of the form  
\[N(r)=ar^{\kappa}\exp(-br^{\beta}), \tag{2}\]  
where the parameters \(a\), \(b\), \(\alpha\), and \(\beta\) are positive real constants. Note that the use of micro-physical cloud modeling to extract \(N(r)\) and calculate cloud attenuation using Mie theory can be simplified using the Rayleigh approximation. The validity of the Rayleigh approximation is taken as the starting point in the Recommendation ITU-R P.840-6 [46] for calculating \(\gamma_{c}\) after extensive propagation studies carried out in the 1980s on absorption by suspended water droplets [47; 48]. Using this approximation, the value of \(\gamma_{c}\) is expressed in terms of the liquid crystal content \(w\) in the cloud and the specific attenuation coefficient \(K_{l}\), measured as the ratio (dB/km)/(g/m\({}^{3}\)),  
\[\gamma_{c}=K_{l}\omega, \tag{3}\]  
where \(K_{l}\) is given by  
\[K_{l}=\frac{0.819f}{\epsilon^{\prime\prime}(\eta^{2}+1)}, \tag{4}\]  
where \(f\) is the signal frequency and \(\eta=(\epsilon^{{}^{\prime}}+2)/\epsilon^{{}^{\prime\prime}}\). The coefficient 0.918 is a part of the empirical approximation from ITU-R P.840. Curves \(K_{l}\) are shown in Figure 1 for some representative cloud drop temperatures. Here, a seven-year database collected at Adolfo Suarez Madrid-Barajas Airport was used as input for cloud attenuation estimation. The presence of clouds is determined by the Salonen and Uppala model, which is also used to calculate \(\omega\). Statistical data in the form of complementary cumulative distribution functions (CCDFs) of zenith cloud attenuation values are shown in Figure 2.  
Note that the Rayleigh approximation is most effective between 1 and 1 THz for a temperature range of \(\pm 40\,^{\circ}\)C. To obtain the values of \(\epsilon^{{}^{\prime}}\) and \(\epsilon^{{}^{\prime\prime}}\), which are the real and imaginary parts of the permittivity of water, one may utilize the following approximations [45]  
\[\epsilon^{{}^{\prime}}(f)=e_{2}+\frac{(\epsilon_{0}-\epsilon_{1})} {1+(f/f_{D})^{2}}+\frac{(\epsilon_{1}-\epsilon_{2})}{1+(f/f_{S})^{2}},\] \[\epsilon^{{}^{\prime\prime}}(f)=\frac{f(\epsilon_{0}-\epsilon_{1} )}{f_{D}[1+(f/f_{D})^{2}]}+\frac{f(\epsilon_{1}-\epsilon_{2})}{f_{S}[1+(f/f_{ S})^{2}]}, \tag{5}\]where \(T\) is expressed in Kelvin, and the remaining parameters are: \(\phi=300/T\), \(f_{D}=20.09-142(\phi-1)+294(\phi-1)^{2}\), \(f_{S}=590-1500(\phi-1)\), \(\epsilon_{1}=5.48\), \(\epsilon_{2}=3.51\), \(\epsilon_{0}=77.67+103.3(\phi-1)\).