# An Integrity Monitoring Method for Navigation Satellites Based on Multi-Source Observation Links
## 3 Design of Integrity Monitoring System with Multi-Source Observation Links
### Optimization Strategies of Observation Links  
Large numbers of LEO satellites can make it easier to obtain full-arc observation data but also bring new problems of excess data. Optimizing strategies should be proposed to improve monitoring efficiency for navigation satellites. This study proposes concepts of minimum satellite-based observation unit and orbit determination observation unit. (1) Minimum satellite-based observation unit  
The minimum satellite-based observation unit consists of chosen satellite-based observation links that can realize on-demand return of navigation messages. The construction process is as follows. Firstly, the observation effectiveness of LEO satellites to target satellites should be evaluated. Distance \(R_{OS}\) between the target satellite and the geocenter, Distance \(R_{OL}\) between the LEO satellite and the geocenter, and distance \(R_{SL}\) between the target satellite  
Figure 2: Design of integrity monitoring system with support of multi-source observation links. and the LEO satellite can be gained through ephemeris of the LEO satellite and target satellite. The angle \(\beta\) could be described as follows:  
\[\beta=\arccos\frac{R_{SL}^{2}+R_{OS}^{2}-R_{OL}^{2}}{2R_{SL}R_{OS}} \tag{1}\]  
As shown in Figure 3, when angle \(\gamma\) between \(R_{OL}\) and \(R_{OS}\) is no more than angle \(\gamma_{0}\) which is the angle between horizontal sight point-to-geocenter line and \(R_{OS}\), the satellite-based observation link is valid if \(\beta\) is less than angle \(\beta_{0}\). Angle \(\beta_{0}\) is the half-beam angle of the target satellite. As shown in Figure 4, when angle \(\gamma\) is more than angle \(\gamma_{0}\), the satellite-based observation link is valid if \(\beta\) is less than angle \(\beta_{0}\) and more than angle \(\beta_{1}\) which is the angle between the horizontal sight point-to-satellite line and \(R_{OS}\). Secondly, the position of long-term trajectory for MEO satellites can be covered with grid points of spherical space in orbital altitude. Multiple numbers of coverage for LEO satellites to all navigation satellites can be maximally evaluated by numbers of coverage for LEO satellites to the grids. Considering that the LEO satellite is in a state of high-speed motion, multiple numbers are also dynamic. The minimum and average values in the time series can be used as evaluation indicators. The calculation formulas are as follows:  
Figure 4: The chosen valid satellite-based observation links (\(\gamma>\gamma_{0}\)). Figure 3: The chosen valid satellite-based observation links (\(\gamma\leq\gamma_{0}\)). \[\left\{ \begin{array}{l}C_{\min}=\frac{\sum\limits_{i=1}^{N}k_{i}(t)_{\min}}{N} \times 100\%\\ C_{avg}=\frac{\sum\limits_{i=1}^{N}k_{i}(t)_{avg}}{N}\times 100\%\end{array} \right. \tag{2}\]  
where \(C_{\min}\) and \(C_{avg}\) are respectively the minimum and average multiple number for LEO satellites to all points, \(k_{i}(t)_{\min}\) and \(k_{i}(t)_{avg}\) are respectively the minimum and average and average multiple number of coverage for LEO satellites to point \(i\) at time \(t\), and \(N\) represents the whole number of grids in spherical space. Finally, a minimum satellite-based observation unit can be established based on demand of coverage. That means, if \(C_{\min}\) of a LEO constellation is more than (M is the demanded natural number), the minimum combination of LEO satellites in a constellation that can meet \(C_{\min}\) equals M is the minimum satellite-based observation unit. (2) Orbit determination observation unit  
The orbit determination observation unit consists of chosen satellite-to-ground and/or satellite-based observation links that can make fast calculations of satellite orbit. Firstly, the effectiveness of observation links should be evaluated.