## 2 Air-to-Space Channel Models
### Rain Attenuation  
The A2S wireless link is expected to suffer from excess loss due to rain attenuation, especially if frequencies above 10 GHz are adopted for their operation, according to ITU [7]. Therefore, it is pivotal to determine the rain attenuation for the A2S/S2A links at frequencies above 10 GHz, such as Ku and Ka bands, which are becoming increasingly attractive to researchers aiming to support mobile-broadband satellite services for UAVs. At these frequencies, rain is expected to compromise the channel viability. In this context, a stochastic channel model for predicting the dynamic properties of rain attenuation for A2S/S2A links is appropriately modified on the basis of [21; 22]. There are many rain-attenuation synthesizers for fixed links, such as [23; 24; 25; 26], based on different assumptions, but not for A2S and A2A links. At this point, a modified model is provided, in which particular emphasis is placed on their LOS-channel state. The fundamental assumption of the model provided is that the rain-attenuation process can be approximated by a first-order stochastic differential equation. The architecture of the rain-attenuation time-series synthesizer is presented in Figure 4.  
The methodology described above is an extension of the Maseng-Bakken stochastic model of rain attenuation for fixed links [27], which was recently acknowledged as the most promising model for reproducing actual phenomena through time-series synthesizers. It is based on the basic theory of stochastic differential equations (SDEs). Next, the appropriate choice of parameters for the first-order proposed SDE model is investigated (model parameterization). This involves the calculation of static (i.e., long-term) and dynamic (i.e., short-term) parameters of rain attenuation. A key factor in the conversion of fixed rain-attenuation statistics into mobile rain-attenuation statistics is the velocity factor \(\xi\), defined as [28; 29]:  
\[\xi=\frac{u_{R}}{|u_{M}-u_{R}\cos\varphi|}, \tag{4}\]  
where \(u_{M}\) is the amplitude of the velocity vector of the UAV terminal, \(u_{R}\) is the amplitude of the velocity vector of the raincells (advection, or front speed), and \(\varphi\) is the relative angle between these two vectors. Both velocities are provided in km/h. Figure 5 presents simulated rain-attenuation time-series in the Ku and Ka bands, produced for UAVs traveling at speeds that range from pedestrian (15 km/h) to high (300 km/h). Figure 4: Layout of the rain-attenuation time-series synthesizer. Figure 5: Simulated rain-attenuation time-series in Athens (GR), for UAVs at two different velocities compared with the fixed terminal: (**a**) Ku band; (**b**) Ka band. As one can observe in Figure 5, only frequency affects the rain, with higher losses yielded in the Ka band (up to 15 dB) than in the Ku band (up to 4 dB) for the same UAV speed (15 km/h). On the other hand, higher attenuation values are induced at lower speeds, yielding, for example, up to 4 dB for 15 km/h and up to 1.5 dB for 300 km/h, respectively, in the Ku band. This is attributable to the fact that the UAV is moving faster than the raincell. Consider, as an example, the geostationary satellite, HELLAS-SAT (39\({}^{\circ}\) E), which provides coverage to UAVs in two different European cities (Athens, GR, and Milan, IT). To simplify the analysis, let the UAV be in rectilinear motion. In addition, the worst-case assumption for the angle \(\varphi\) between the front-speed vector of the raincell movement and the velocity of the UAV is adopted, that is, when both travel in the same direction (\(\varphi=0^{\circ}\)). The implementation of the model for the Athens and Milan cases in the Ka band is illustrated in Figure 6. The average front speed of the movement of the raincells for Athens is 51.3 km/h, and for Milan, it is 30 km/h. The distribution of a fixed terminal is compared with those of a UAV terminal, moving at three different velocities (15 km/h, 200 km/h, and 300 km/h), ranging from very low to very high.