## 4 Contribution
### Construction of SUUIL Model for Urban Environment Ultra-Low Altitude
#### 4.1.1 Analysis of the Ultra-Low Altitude Received Signal Envelope  
When the UAV collects and transmits data in the ultra-low altitude environment, the received signal consists of LOS component and multipath component. Assume that the shadow only affects the LOS component. In the urban environment, the shadow fading is obvious. Considering the influence of the shadow in the communication channel, it is assumed that the LOS component obeys logarithmically distributed. The received signal is:  
\[R(t)=W(t)e^{jF(t)}+A(t)e^{jF_{0}} \tag{16}\]  
where \(W(t)\) is the amplitude of the scatter component, which obeys the Rayleigh distribution, \(F(t)\) is the phase of the multipath component, which is uniformly distributed in the interval \([-\pi,\pi)\). \(A(t)\) is the amplitude of the LOS component, which obeys the Lognormal distribution, \(F_{0}\) is the deterministic phase of the LOS component. \(W(t)\), \(F(t)\), \(A(t)\) are independent of each other. Assuming that \(A(t)\) remains unchanged, the conditional probability density function of received signal envelope \(R(t)=|R(t)|\), which obeys the Rice distribution is:  
\[f_{R|A}(r|a)=\frac{r}{\lambda_{0}^{2}}\exp(-\frac{r^{2}+a^{2}}{2\lambda_{0}^{2 }})I_{0}\left(\frac{ar}{\lambda_{0}^{2}}\right) \tag{17}\]  
where \(2\lambda_{0}^{2}\) is the average scattered power, and it satisfies \(2\lambda_{0}^{2}=E\big{[}W^{2}\big{]}\). When \(\alpha\) tend to 0, the power of the received signal consists of multipath component power, and there is no LOS component. Set \(x=\frac{ar}{\lambda_{0}^{2}}\), \(v=1\):  
\[I_{0}\left(\frac{ar}{\lambda_{0}^{2}}\right)\approx\frac{1}{\Gamma(1)}=1 \tag{18}\]  
Substitute Equation (18) into Equation (17):  
\[f_{R|A}(r|a)=\frac{r}{\lambda_{0}^{2}}\exp(-\frac{r^{2}+a^{2}}{2\lambda_{0}^{ 2}}) \tag{19}\]  
At this time, \(W\) obeys the Rayleigh distribution:  
\[f_{W}(w)=\frac{w}{\lambda_{0}^{2}}\exp(-\frac{w^{2}}{2\lambda_{0}^{2}})\qquad, \qquad\qquad w\geq 0 \tag{20}\]  
The conditional mathematical expectation of the received signal envelope is calculated by the total probability equation \(f_{R}(r)=E_{A}[f_{R|A}(r|a)]\):  
\[f_{R}(r)=\frac{r}{\lambda_{0}^{2}}\int_{0}^{\infty}\exp(-\frac{r^{2}+a^{2}}{2 \lambda_{0}^{2}})I_{0}\left(\frac{ar}{\lambda_{0}^{2}}\right)f_{A}(a)\text{d }a, \tag{21}\]\[f_{A}(a)=\frac{1}{a\sqrt{2\pi d_{0}}}\exp\left(-\frac{\left(\ln a-\mu\right)^{2}}{2 d_{0}}\right) \tag{22}\]  
where \(\mu=E[\ln(A)]\), \(d_{0}=Var[\ln(A)]\). The probability density function of the envelope of the received signal affected by shadow fading suitable for the Loo model is calculated:  
\[f_{R}(r)=\frac{r}{\lambda_{0}^{2}\sqrt{2\pi d_{0}}}\int_{0}^{\infty}\frac{1}{a }\exp\left(-\frac{\lambda_{0}^{2}(\ln a-\mu)^{2}+d_{0}\left(r^{2}+a^{2}\right) }{2\lambda_{0}^{2}d_{0}}\right)\,l_{0}\left(\frac{ar}{\lambda_{0}^{2}}\right) \mathrm{d}a \tag{23}\]  
The channel conditions are estimated in real time through the data collected by the UAV. The parameters in Equation (21) are associated with the actual parameters \(\left(d,\sigma_{X}^{2},K_{r}\right)\) in order to calculate the shadow variance \(\sigma_{X}^{2}\)[17] and the Rice factor \(K\)[18]. Firstly, \(K\) is derived from the probability rate density function of \(K\). The calculated probability density function of K in the channel is as follows:  
\[f_{K}(k)=\frac{1}{2k\sqrt{2\pi d_{0}}}\exp\left(-\frac{\left(\ln\lambda_{0} \sqrt{2k}-\mu\right)^{2}}{2d_{0}}\right) \tag{24}\]  
The Equation (24) is a probability density function of a lognormal variable.