## 2 Theoretical Model  
### Channel Model  
The satellite-to-ground link combined with the joint atmospheric turbulence model is considered, including Kolmogorov turbulence and the non-Kolmogorov turbulence model [17]. According to the actual situation of the free space wireless communication link, the channel model of the turbulence could be divided into two layers based on the altitude. The first layer is 0-6 km in the troposphere, and the second layer is above 6 km in the stratosphere. A bulk of experimental results and theoretical investigations have shown that the turbulence model in the troposphere meets the Kolmogorov turbulence, which meets the non-Kolmogorov turbulence in the stratosphere. Thus, joint atmospheric turbulence refraction spectra density function \(\Phi_{n}(\kappa,\alpha)\) can be described as follows:  
\[\Phi_{n}(\kappa,\alpha)=\Phi_{n1}(\kappa,\alpha)+\Phi_{n2}(\kappa,\alpha), \tag{1}\]  
where function \(\Phi_{n1}(\kappa,\alpha)\) is the atmospheric turbulence refraction spectra density of the troposphere, and function \(\Phi_{n2}(\kappa,\alpha)\) is the atmospheric turbulence refraction spectra density of the stratosphere. The symbol \(\kappa\) is the magnitude of the spatial frequency vector in rad/m units, and the symbol \(\alpha\) denotes the spectral power-law exponent. By introducing the spatial filter function \(G(\kappa,\alpha)\), spectra density function in Equation (1) could expand into  
\[\left\{ \begin{array}{l}\Phi_{n1}(\kappa,\alpha)=0.033C_{n}^{2}(h)\kappa^{-11 /3}G(\kappa,\alpha)\\ \Phi_{n2}(\kappa,\alpha)=0.036\overline{C}_{n}^{2}(h)\kappa^{-5}G(\kappa, \alpha)\end{array} \right., \tag{2}\]  
where \(C_{n}^{2}(h)\) is the generalized atmospheric refractive index structure parameter for Kolmogorov turbulence, which represents the measurement of the refractive index fluctuation with the variety of the altitude \(h\), and \(\overline{C}_{n}^{2}(h)\) is the generalized atmospheric refractive index structure parameter for non-Kolmogorov turbulence, which is directly associated with turbulence strength. Many influencing factors of atmospheric turbulence, temperature gradient, humidity fluctuations, and the correlation between temperature and humidity fluctuations should be considered. According to References [18; 19], the atmospheric refractive index structure parameter \(\overline{C}_{n}^{2}(h)\) can be estimated in terms of the Hufnagel-Valley model:  
\[ \begin{array}{ll}\overline{C}_{n}^{2}(h)&=0.00594\left(\frac{\omega}{27} \right)^{2}\left(10^{-5}h\right)^{10}\exp\left(-\frac{h}{1000}\right)\\ &+2.7\times 10^{-16}\exp\left(-\frac{h}{1500}\right)+C_{n}^{2}(0)\exp\left(- \frac{h}{100}\right)\end{array}  \tag{3}\]  
where \(\omega\) is the wind speed, which is usually taken as a value of 20 m/s, and \(C_{n}^{2}(0)\) represents the altitude \(h=0\). Because the effects of atmospheric turbulence can be considered as large-scale eddies and small-scale eddies, the individual eddies are independent and do not interact with each other. Then we could get the following expression:  
\[G(\kappa,\alpha)=G_{x}(\kappa,\alpha)+G_{y}(\kappa,\alpha) \tag{4}\]  
Based on the extended Rytov theory [20], the symbols \(G_{x}(\kappa,\alpha)\) and \(G_{y}(\kappa,\alpha)\) represent the large-scale and small-scale filter functions for weak-to-strong turbulence, respectively, and have the following forms:  
\[\left\{ \begin{array}{l}G_{x}(\kappa,\alpha)=\exp\left(-\kappa^{2}/\kappa_{ x}^{2}\right)\\ G_{y}(\kappa,\alpha)=\kappa^{\alpha}/\left(\kappa^{2}+\kappa_{y}^{2}\right)^{ \frac{\alpha}{2}}\end{array} \right., \tag{5}\]  
where \(\kappa_{x}\) is the cutoff spatial frequencies under weak-to-strong fluctuations of large-scale eddies, and \(\kappa_{y}\) is the cutoff spatial frequencies under weak-to-strong fluctuations of small-scale eddies. According to Reference [18], the spatial cutoff frequency for large-scale and small-scale filter functions can be further expressed as  
\[