# Performance of UAV-to-Ground FSO Communications with APD and Pointing Errors
## 2 System and Channel Modeling
### Channel Modeling
#### 2.2.1 Atmospheric Attenuation and Turbulence  
The atmospheric attenuation \(X_{l}\) caused by both molecular absorption and aerosol scattering suspended in the air. The total link loss caused by the attenuation of \(X_{l}\) can therefore be obtained as [23]  
\[X_{l}=\exp(-\chi L), \tag{11}\]  
where \(L\) and \(\chi\) are the transmission distance between the FSO Tx and the FSO Rx and the weather-dependent attenuation coefficient, respectively.  
Considering the fading channel induced by atmospheric turbulence, we use the log-normal (LN) distribution model for the case of weak to moderate turbulence and the gamma-gamma (GG) distribution model for the case of moderate to strong turbulence. The probability density function (PDF) of the channel intensity fluctuation \(X_{a}\) in the LN model can be expressed as [14]  
\[f_{\text{LN}}(X_{a})=\frac{1}{X_{a}\sigma_{\text{Ln}}\sqrt{2\pi}}\exp(-\frac{ \left(\text{ln}(X_{a})+\frac{\sigma_{\text{Ln}}^{2}}{2}\right)^{2}}{2\sigma_{ \text{Ln}}^{2}}), \tag{12}\]  
where \(\sigma_{\text{Ln}}^{2}\) represents the scintillation index that depends on the channel's characteristics and is expressed as [14]  
\[\begin{split}\sigma_{\text{Ln}}^{2}=&\exp\left( \frac{0.49\sigma_{2}^{2}}{\left(1+0.18d^{2}+0.56\sigma_{2}^{12/5}\right)^{7/6} }\right.\\ +&\left.\frac{0.51\sigma_{2}^{2}\left(1+0.69\sigma_{2 }^{12/5}\right)^{-5/6}}{\left(1+0.9d^{2}+0.62d^{2}\sigma_{2}^{12/5}\right)} \right)-1,\end{split} \tag{13}\]  
where \(d=\sqrt{kD^{2}/4L}\), \(\sigma_{2}^{2}\) is the Rytov variance. In the case of spherical wave propagation, it is given by [14]  
\[\sigma_{2}^{2}=0.492C_{n}^{2}k^{7/6}L^{11/6}. \tag{14}\]In Equation (14), \(k=2\pi/\lambda\) is the optical wave-number, \(C_{n}^{2}\approx C_{0}^{2}\exp\left(-\frac{h_{d}}{100}\right)\) (in m\({}^{-2/3}\)) denotes the index of the refraction structure parameter that represents the strength of the atmospheric turbulence, where \(h_{d}\) is the operating height of the hovering UAV and \(C_{0}^{2}=1.7\times 10^{-14}\) m\({}^{-2/3}\) is the nominal value of the refractive index at the ground [24]. In the practical system, the variance \(h_{d}\) is very small. Therefore, \(C_{n}^{2}\) is assumed to be altitude-dependent and constant along the propagation path, and typically takes values varying from \(10^{-17}\) to \(10^{-13}\) m\({}^{-2/3}\) under atmospheric turbulence conditions, accordingly.  
The Rytov variance distinguishes the various link turbulence regimes such as weak turbulence if \(\sigma_{2}^{2}<1\), moderate turbulence if \(\sigma_{2}^{2}\approx 1\) and strong turbulence if \(\sigma_{2}^{2}>1\), especially saturated turbulence if \(\sigma_{2}^{2}\rightarrow\infty\).  
On the other hand, the PDF distribution of the random variable in the GG model is given by [13]  
\[f_{\text{GG}}(X_{a})=\frac{2(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{\Gamma( \alpha)\Gamma(\beta)}X_{a}^{\frac{\alpha+\beta}{2}-1}K_{\alpha-\beta}\Big{(} 2\sqrt{\alpha\beta X_{a}}\Big{)}. \tag{15}\]  
In Equation (15), \(\Gamma(\cdot)\) is the Gamma function, \(K_{n}(\cdot)\) is the \(n\)th order-modified Bessel function of the second kind. \(\alpha\) and \(\beta\) represent the effective numbers of large-scale and small-scale eddies of the scattering process. They depend on the Rytov variance \(\sigma_{2}^{2}\) as in Equation (14) and are defined as [24]  
\[\alpha=\Bigg{[}\exp\Bigg{(}\frac{0.49\sigma_{2}^{2}}{\left(1+0.18d^{2}+0.56 \sigma_{2}^{12/5}\right)^{7/6}}\Bigg{)}-1\Bigg{]}^{-1}, \tag{16}\]  
\[\beta=\Bigg{[}\exp\Bigg{(}\frac{0.51\sigma_{2}^{2}\Big{(}1+0.69\sigma_{2}^{12 /5}\Big{)}^{-5/6}}{\left(1+0.9d^{2}+0.62d^{2}\sigma_{2}^{12/5}\right)}\Bigg{)} -1\Bigg{]}^{-1}. \tag{17}\]