## 2 Air-to-Space Channel Models
### Theoretical Studies and Simulations for Channel Modeling  
The first theoretical study regarding the aeronautical channel of an A2S link was conducted in 1973 [16], emphasizing the effects of the indirect paths reflected and scattered by the Earth surface. The channel was modeled by employing three distinct coefficients. These incorporated the LOS path, which is the direct path between two aircraft, the specular reflection component, which is the shortest path between the two aircraft through the underlying terrain, and the scattered paths, which included all the scattered components from the ground apart from the specular reflection. A channel model that predicts path loss for an A2S link, studying the propagation characteristics in the THz band, was proposed in [17]. In addition to the terrestrial geometrical characteristics, the suggested model considered the atmospheric characteristics of the THz propagation channel, such as rain, clouds, and non-homogeneous molecular-absorption losses. The combined path-loss model is provided by  
\[PL(f,r)=\frac{(4\pi r_{\mathrm{as}}f)^{2}\delta_{\mathrm{rain}}(r_{r})\delta_ {\mathrm{cloud}}(r_{c})}{c^{2}G_{\mathrm{Tx}}(\theta_{\mathrm{Tx}})G_{ \mathrm{Rx}}(\theta_{\mathrm{Rx}})}e^{\int_{1}^{r_{2}}\kappa_{x}(f_{x\mathrm{ atm}})d\tau_{\mathrm{atm}}}, \tag{1}\]  
where \(G_{\mathrm{Tx}}(\theta_{\mathrm{Tx}})\) and \(G_{\mathrm{Rx}}(\theta_{\mathrm{Rx}})\), are the transmitter (Tx) and receiver (Rx) antenna gains at the pointing angles \(\theta_{\mathrm{Tx}}\) and \(\theta_{\mathrm{Rx}}\), respectively, \(r_{\mathrm{as}}\) is the direct distance between the UAV and the satellite, \(f\) is the operating frequency, and \(c\) is the speed of light. Furthermore, \(\delta_{\mathrm{rain}}(r_{r})\) and  
Figure 3: Model layout of the A2S channel [15]. \(\delta_{\text{cloud}}(r_{c})\) are the rain and cloud losses, where \(r_{r}\) and \(r_{c}\) are the distance of the path within the rain and clouds, respectively. The rain loss is calculated according to the ITU-R P.838-3 recommendation, on the basis of the following expression:  
\[\delta_{\text{rain}}(r_{r})=kR_{r}^{\alpha}r_{eff}, \tag{2}\]  
where \(k\) and \(\alpha\) can be resolved according to [18], \(R_{r}\) is the rain rate in mm/h, and \(r_{eff}\) is the effective distance. The cloud loss can be determined from the ITU-R P.840-8 model, according to  
\[\delta_{\text{cloud}}(r_{c})=K_{l}Mr_{c}, \tag{3}\]  
where \(M\) is the liquid-water density in the cloud or fog in g/m\({}^{3}\), \(K_{l}\) is the cloud-liquid-water specific attenuation coefficient in (dB/km)/(g/m\({}^{3}\)), and \(r_{c}\) is the distance of the path through clouds/fog [19]. Finally, the \(k_{a}\)(\(f\),\(r_{\text{atm}}\)) in (1) is the molecular absorption coefficient for non-homogeneous atmospheric conditions. The specific parameter depends on the pressure, the temperature, and the molecular composition of the propagation path through the atmosphere. Its calculation is not straightforward, and can be resolved from molecular absorption databases, as in [17] and references therein. For the simulation process, the airplane- and satellite-antenna diameters were set as 0.5 and 1 m, respectively, with an aperture efficiency of 70%. These resulted in a combined (airplane plus satellite) gain between 191 and 150 dBi, depending on the frequency. Two different satellite orbits were tested, a GEO (35,786 km) and a LEO (500 km), which both corresponded to elevation angles in the range of 12\({}^{\circ}\)-90\({}^{\circ}\) (from the horizon of the airplane to the satellite). The excess loss due to rain and clouds reached up to 25 dB (at about 1000 GHz). Furthermore, for an airplane flying at 11 km, the path loss was severe for both the LEO and the GEO orbit, varying, on average, between 190 and 260 dB (for frequencies between 0.1 and 3 THz), presenting peaks at specific frequency instants due to the molecular absorption. These path-loss peaks reached up to 300 dB.