## 3 Optical Wireless Satellite Channel Model
### Atmospheric Power Losses: Absorption and Scattering Modeling  
The power loss of the optical feeder link is a function of the atmospheric attenuation, which depends on both absorption and scattering effects that the light signal experiences while propagating [37]. To compute this value, the atmospheric attenuation coefficient:  
\[\gamma_{\rm atm}\ =\ \alpha_{\rm m}+\alpha_{\rm a}+\beta_{\rm m}+\beta_{\rm a} \tag{12}\]  
needs to be computed, where \(\alpha_{\rm m}\) and \(\alpha_{\rm a}\) are the molecular and aerosol absorption coefficients, respectively, whereas \(\beta_{\rm m}\) and \(\beta_{\rm a}\) are the molecular and aerosol scattering coefficients, respectively. **Modelling of absorption:** At Infrared (IR) wavelengths, the principal atmospheric absorbers are the molecules of water, carbon-dioxide, and ozone. As expected, the atmospheric absorption is a wavelength-dependent phenomenon. Therefore, the operating wavelength for optical feeder link transmissions should be chosen to minimize this loss, using the atmospheric transmission windows in which the molecular and aerosol absorption is less than 0.2 dB/km for clear sky conditions [38]. In addition to the low-absorption requirements, most optical feeder links are designed to work in the 780-850 nm and 1520-1600 nm windows because there are _off-the-shelf_ lasers and detectors commercially available to work in these wavelengths. **Modeling of scattering:** Like absorption, scattering is also a phenomenon that is strongly dependent on the operating wavelength. If the size of the atmospheric particles is small in comparison to the optical feeder link wavelength, then _Rayleigh scattering_ is produced. Particles like air molecules and haze cause Rayleigh scattering [39] and affect notably optical wireless transmissions in the Visible Light (VL) and Ultraviolet (UV) regions; on the other hand, Rayleigh scattering can be neglected for optical feeder link wavelengths in the IR range (i.e., when \(\lambda\gg 1\) um). Similarly, when the atmospheric particles size is comparable with the operating wavelength, then _Mie scattering_ is produced. Aerosol particles, fog, and haze are the major contributors of Mie scattering, and this phenomenon is dominant for wavelengths in the IR range. Finally, if the atmospheric particles are much larger than the operating wavelength, the scattering is better described by geometrical optical models, which should be used in case of rain, snow, and hail weather conditions [40]. **Modeling the transmittance of the Earth's atmosphere:** Apart from the previously described \(\lambda\)-dependent effects, the specific value that the atmospheric attenuation coefficient takes depends on the concentration of molecules (and aerosols) of the Earth's atmosphere at different altitudes \(h\). Based on this assumption, the atmospheric transmittance that an optical feeder link with zenith angle \(\zeta\) experiences is given by:  
\[T_{\rm atm}(\lambda)\ =\ \exp\bigg{\{}-\sec(\zeta)\int_{h_{0}}^{H}\!\! \gamma(\lambda,h)\,dh\bigg{\}}, \tag{13}\]  
where \(h_{0}\) is the altitude of the satellite gateway (ground station) over the sea level, \(H\) is the vertical height at which the GEO satellite is placed, and \(\gamma(\lambda,h)\) is the attenuation coefficient at wavelength \(\lambda\) and altitude \(h\). Based on this formula, it is possible to see that atmospheric transmittance is increased at low zenith angles (i.e., at high elevation angles), as the fractionof the incident electromagnetic power that is transferred through the atmosphere layers is increased. **Power losses due to fog:** From the common weather conditions, fog is the one that contributes most in the absorption and scattering of the optical signal when it propagates through the Earth's atmosphere. In the presence of fog, the optical feeder link connectivity is seriously put at risk, particularly when the fog layer next to the ground station extends vertically very high, forming a fog layer that can be as thick as 400 m over the Earth's surface.