## 2 System and Channel Models
### Channel Models  
The OGS-to-satellite FSO link is assumed to follow a Gamma-Gamma distribution with atmospheric attenuation and pointing error. The probability density function (PDF) of instantaneous SNR \(\gamma_{GS}^{f}\) is given by ([2], Equation (30))  
\[f_{\gamma_{GS}^{f}}(\gamma)=\frac{{\xi_{1}}^{2}}{b\gamma\Gamma(\alpha_{1}) \Gamma(\beta_{1})}G_{1,3}^{3,0}\bigg{[}\frac{\alpha_{1}\beta_{1}{\xi_{1}}^{2} }{{\xi_{1}}^{2}+1}\bigg{(}\frac{\gamma}{\mu_{1}}\bigg{)}^{\frac{1}{b}}\bigg{|} \frac{{\xi_{1}}^{2}+1}{{\xi_{1}}^{2},\,\,\alpha_{1},\,\,\beta_{1}}\bigg{]} \tag{9}\]  
where \(\alpha_{1}\) and \(\beta_{1}\) are parameters that indicate large-scale and small-scale irradiance fluctuations, \(\xi_{1}\) is the pointing error coefficient, which is used to measure the severity of beam misalignment, \(\mu_{1}\) is the average electrical SNR defined as \(\mu_{1}=\overline{\gamma}_{GS}^{f}\) for HD and \(\mu_{1}=\frac{\alpha_{1}\beta_{1}\xi_{1}^{2}(\xi_{1}^{2}+2)}{(\alpha_{1}+1)( \beta_{1}+1)\big{(}\xi_{1}^{2}+1\big{)}^{2}}\overline{\gamma}_{GS}^{f}\) for IM/DD technology. The cumulative distribution function (CDF) of \(\gamma_{GS}^{f}\) can be obtained by using ([28], Equation (07.34.21.0084.01)) shown as  
\[F_{\gamma_{GS}^{f}}(\gamma)=F^{1}G_{b+1,3b+1}^{3b,\,1}\bigg{(}\frac{{D_{1}}^{ b}\gamma}{b^{2}\mu_{1}}\bigg{|} \begin{matrix}1&E_{1}\\ E_{2}&0\end{matrix} \bigg{)} \tag{10}\]  
where \(F^{1}=\frac{{\xi_{1}}^{2}{b\mu_{1}+\beta_{1}-2}}{(2\pi)^{b-1}\Gamma(\alpha_{1 })\Gamma(\beta_{1})}\), \(D_{1}=\frac{\alpha_{1}\beta_{1}\xi_{1}^{2}}{\xi_{1}^{2}+1}\), \(E_{1}=\Big{\{}\frac{{\xi_{1}}^{2}+1}{b},\ldots,\frac{{\xi_{1}}^{2}+b}{b}\Big{\}}\), \(E_{2}=\Big{\{}\frac{{\xi_{1}}^{2}}{b},\ldots,\frac{{\xi_{1}}^{2}+b-1}{b},\ldots,\frac{{\xi_{1}}}{b},\ldots,\frac{{\xi_{1}}+b-1}{b},\frac{{\xi_{1}}}{b}, \ldots,\frac{{\xi_{1}}+b-1}{b}\Big{\}}\). The satellite-to-HAP FSO re  
Figure 1: Hybrid FSO/RF systems in SAGIN with parallel transmissions. lay link is assumed to follow Gamma-Gamma distribution with angle-of-arrival (AOA) fluctuations and the PDF of \(\gamma_{SH}^{f}\) is given by ([17], Equation (23))  
\[ \begin{array}{l}f_{\gamma_{SH}^{f}}(\gamma)=\exp\left(-\frac{\theta_{2 eV}^{2}} {2\
u_{angle,\mu}^{2}}\right)\delta(\gamma)+\left[1-\exp\left(-\frac{\theta_{2 eV}^{2}}{2\
u_{angle,\mu}^{2}}\right)\right]\\ \times\frac{\xi_{2}^{2}}{b_{\gamma}\Gamma(\alpha_{2})\Gamma(\beta_{2})}G_{1,3} ^{3}\bigg{[}\frac{\alpha_{2}\beta_{2}\xi_{2}^{2}}{\xi_{2}^{2}+1}\bigg{(}\frac{ \gamma}{\alpha_{2}}\bigg{)}^{\frac{1}{2}}\bigg{|}\begin{array}{c}\xi_{2}^{2} +1\\ \xi_{2}^{2},\ \alpha_{2},\ \beta_{2}\end{array}\bigg{]}\end{array}  \tag{11}\]  
where \(\theta_{FOV}\) is the field-of-view angle, \(\sigma_{angle,\mu}^{2}\) is the variance of AOA, \(\mu_{2}=\overline{\gamma}_{SH}^{f}\) for HD technology, \(\mu_{2}=\frac{\alpha_{2}\beta_{2}\xi_{2}^{2}(\zeta_{2}^{2}+2)}{(\alpha_{2}+1)( \beta_{2}+1)(\xi_{2}^{2}+1)^{2}}\overline{\gamma}_{SH}^{f}\) for IM/DD technology. Similar to (10), the CDF of \(\gamma_{SH}^{f}\) is obtained as  
\[\begin{array}{l}F_{\gamma_{SH}^{f}}(\gamma)=\exp\left(-\frac{\theta_{FOV}^{2} }{2\sigma_{angle,\mu}^{2}}\right)+\left[1-\exp\left(-\frac{\theta_{EeV}^{2}} {2\sigma_{angle,\mu}^{2}}\right)\right]F^{2}G_{b+1,3b+1}^{3b,1}\left(\frac{D_ {2}^{b}\gamma}{b^{2b}\mu_{2}}\bigg{|} \begin{array}{cc}1&E_{3}\\ E_{4}&0\end{array} \right)\\ \text{where}\quad F^{2}=\frac{\xi_{2}^{2}\mu_{2}+\beta_{2}^{2}-2}{(\alpha_{2})^{ -1}\Gamma(\alpha_{2})\Gamma(\beta_{2})}\text{, }D_{2}=\frac{\alpha_{2}\beta_{2}\zeta_{2}^{2}}{\xi_{2}^{2}+1}\text{, }E_{3}=\left\{\frac{\xi_{2}^{2}+1}{b},\ldots,\frac{\xi_{2}^{2}+b}{b}\right\} \text{, }E_{4}=\left\{\frac{\xi_{2}^{2}}{b},\ldots,\frac{\xi_{2}^{2}+b-1}{b},\ldots, \frac{\alpha_{2}}{b},\ldots,\frac{\alpha_{2}+b-1}{b},\frac{\beta_{2}}{b},\ldots,\frac{\beta_{2}+b-1}{b}\right\}\text{.