## 3 Analysis of Channel
### Loo Model  
In 1985, [PERSON] proposed the so-called Loo model for statistical channel of land mobile satellite link [14]. The model assumes that the received signal is composed of a LOS component and a multipath component. The LOS component obeys the Lognormal distribution and the multipath component obeys the Rayleigh distribution. That is, the received signal is:  
\[r(t)=z(t)s(t)+d(t) \tag{1}\]  
where \(z(t)\) is the LOS component, \(s(t)\) is the shadow fading, and \(d(t)\) is the multipath component.  
Assuming that the amplitude \(z\) of the LOS component remains constant, the received signal envelope \(r\) obeys the Rice distribution:  
\[f_{r}(r|z)=\frac{r}{\lambda_{0}^{2}}\exp\left(-\frac{r^{2}+z^{2}}{2\lambda_{0} ^{2}}\right)l_{0}\left(\frac{rz}{\lambda_{0}^{2}}\right) \tag{2}\]  
Figure 2: Roadmap of the research.  
where \(\lambda_{0}^{2}\) is the average scattered multipath power, \(I_{0}(\cdot)\) is the first kind of zeroth order modified Bessel function.  
\(z(t)\) obeys Lognormal distribution:  
\[f(z)=\frac{1}{z\sqrt{2\pi d_{0}}}\exp\ \left(-\frac{\left(\ln z-\mu\right)^{2}}{2d_{0}}\right) \tag{3}\]  
where \(\mu\) is the mean deviation of \(\ln z\), \(d_{0}\) is the variance of \(\ln z\).  
According to the total probability formula, the probability distribution of the received signal is shown in Equation (4):  
\[f(r)=\int_{0}^{\infty}f_{r}(r|z)f(z)\mathrm{d}z=\frac{r}{\lambda_{0}^{2}\sqrt {2\pi d_{0}}}\int_{0}^{\infty}\frac{1}{z}\exp\left(-\frac{r^{2}+z^{2}}{2 \lambda_{0}^{2}}-\frac{\left(\ln z-\mu\right)^{2}}{2d_{0}}\right)\mathrm{d}z \tag{4}\]  
when the signal amplitude is much larger than \(\lambda_{0}\), \(f(r)\) obeys the Lognormal distribution:  
\[f(r)=\frac{1}{r\sqrt{2\pi d_{0}}}\exp\left(-\frac{\left(\ln r-\mu\right)^{2}}{ 2d_{0}}\right) \tag{5}\]  
When the signal amplitude is much smaller than \(\lambda_{0}\), \(f(r)\) obeys the Rayleigh distribution:  
\[f(r)=\frac{r}{\lambda_{0}^{2}}\exp\left(-\frac{r^{2}}{2\lambda_{0}^{2}}\right) \tag{6}\]  
The residual cumulative probability distribution function of \(r\) is:  
\[C_{\mathrm{R}}(r)=P(r>\mathrm{R})=\int_{\mathrm{R}}^{\infty}f(r)\mathrm{d}r=1- \int_{0}^{\mathrm{R}}f(r)\mathrm{d}r \tag{7}\]  
where R is a certain value.  
The Loo model used a helicopter to simulate satellites and flied at an elevation angle of \(15^{\circ}\) relative to the receiver in light shadowing and heavy shadowing areas where the tree coverage was about 35%. Model parameters are shown in Table 1[15].  
Figure 3 is the fitting diagram of cumulative distribution curve of the theoretical data and the measured data. Figure 3a is the cumulative distribution curve of the Loo model theoretical data and the measured data under the light shadowing, while Figure 3b shows the cumulative distribution curve under heavy shadowing conditions.  
In Figure 3, it can be observed that the received signal envelope and the LOS component are affected by the shadowing. Moreover, with the increase of cumulative probability distribution, the theoretical and measured values of the received signal envelope decrease non-linearly. In a light shadowing environment, the fitting effect between the calculation results and the measured values is average. The calculation results reflect a higher shadowing effect. In contrast, the model has a better fitting effect in heavy shadowing environments. In addition, studies have shown that land mobile satellite systems are similar to urban UAV air-to-ground links. According to the analysis of statistical data and measured data, the LOO model is a suitable starting point for improving the model [12]. Therefore, the LOO model has been improved for urban ultra-low altitude areas with heavy shadow fading.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Conditions** & **Light Shadowing** & **Heavy Shadowing** \\ \hline \(10\lg(\sqrt{d_{0}})\) & \(0.5\) & \(3.5\) \\ \(10\lg(\mu)\) & \(0.5\) & \(-17\) \\ \(10\lg(\lambda_{0}^{2})\) & \(-8\) & \(-12\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of Loo model.