## 3 The Optimal Selection Model of the Backhaul Links
### Definition of the Optimization Model  
The objective is to optimize the comprehensive system performance factor \(Q(t)\), which contains two adjustable parameters: the RST and the capacity. Both of them are defined as follows:  
(1) RST:  
Due to the high mobility of the LEO satellites along the orbit, the links between the LEOs and UAVs will be broken. Otherwise, the flexible mobility of the UAVs along different trajectories will also cause the frequent handover of the backhaul links. The more link handover occurrences, the more unstable the system will be. Therefore, RST is an important element for the reliability of the system to indicate handover times. At time \(t\), the RST of the nodes \(U_{i}\) and \(S_{j}\) can be described as \(R_{ij}(t)\), and the total remaining time is the sum of each UAV's time to be associated with the satellites. \[R_{\text{sys}}(t)=\sum_{i=1,j=f(i)}^{N}R_{ij}(t) \tag{1}\]  
(2) The capacity of the backhaul links:  
In the visible area of a UAV, the selection of backhaul links to transmit the data will influence the capacity of the system. The UAV has better link quality when it is close to the satellite. At time \(t\), the information rate of each backhaul link can be described by [PERSON]'s formula:\[C_{ij}(t) = W\log_{2}\left(1+\frac{P_{r}(d_{ij}(t))}{N_{0}}\right) \tag{2}\] \[= W\log_{2}\left(1+\frac{P_{t}G_{t}G_{r}(\frac{\lambda}{4\pi d_{ij}( t)})^{2}}{N_{0}}\right)\]  
The Friis free space formula is used to compute the receiving power, \(P_{r}(d_{ij}(t))\), at the satellite end, where \(d_{ij}(t)\) represents the connection distance between the satellite \(S_{j}\) and the UAV \(U_{i}\). The distance varies with time because of the UAV's and satellite's motion. Furthermore, \(N_{0}\) is the channel noise, which uses an additive white Gaussian noise (AWGN) model for the interpretation of the channel noise. \(W\) represents the bandwidth of each UAV, and \(P_{t}\) is the transmitting power, which is equal for each UAV. \(G_{t}\) and \(G_{r}\) represent the transmitting and receiving antenna gains, respectively. The wavelength of an electromagnetic wave is \(\lambda\). In particular, the square of the distance \(d_{ij}(t)\) determines the inverse proportionality of the receiving power. The total capacity is the sum of each UAV associated with the backhaul links at time \(t\). It can be described as follows:  
\[C_{sys}(t)=\sum_{i=1}^{N}C_{if(i)}(t) \tag{3}\]  
The two parameters refer to the different aspects of the system's performance. One of the parameters, the RST \(R_{sys}(t)\), reflects the reliability of the system. Meanwhile, the other parameter, the total capacity of the backhaul links \(C_{sys}(t)\), describes the efficiency of the system. To deal with the two attributes on different scales, we can normalize them by min-max normalization. \[R^{\prime}_{sys}(t)=\frac{R_{sys}(t)-\min\limits_{h\in H_{i}}\sum_{i=1}^{N}R_ {ih}(t)}{\max\limits_{h\in H_{i}}\sum_{i=1}^{N}R_{ih}(t)-\min\limits_{h\in H_{ i}}\sum_{i=1}^{N}R_{ih}(t)} \tag{4}\]  
\[C^{\prime}_{sys}(t)=\frac{C_{sys}(t)-\min\limits_{h\in H_{i}}\sum_{i=1}^{N}C_ {ih}(t)}{\max\limits_{h\in H_{i}}\sum_{i=1}^{N}C_{ih}(t)-\min\limits_{h\in H_{ i}}\sum_{i=1}^{N}C_{ih}(t)} \tag{5}\]  
where \(\min\limits_{h\in H_{i}}\sum_{i=1}^{N}R_{ih}(t)\) and \(\max\limits_{h\in H_{i}}\sum_{i=1}^{N}R_{ih}(t)\) are the minimum and maximum absolute value of the RST, respectively. Similarly, \(\min\limits_{h\in H_{i}}\sum_{i=1}^{N}C_{ih}(t)\) and \(\max\limits_{h\in H_{i}}\sum_{i=1}^{N}C_{ih}(t)\) are the minimum and maximum absolute value of the total capacity of the backhaul links, respectively. \(H_{i}\) is the candidate satellite set corresponding to the \(i\)th UAV. \(R^{\prime}_{sys}(t)\) and \(C^{\prime}_{sys}(t)\) are the normalized value of the RST and the capacity at time \(t\). The objective is to optimize the comprehensive influence factor \(Q(t)\), which contains two adjustable parameters: \(R^{\prime}_{sys}(t)\) and \(C^{\prime}_{sys}(t)\).