## 3 Integrity Monitoring Method of Space-Ground Integration
### Simulation Results and Sensitivity Analysis
#### 3.4.2 Simulation of GNSS Integrity Service Performance Under LEO Constellation Enhancement
Since the probabilities of most fault modes are extremely low, only a subset of fault modes needs to be monitored, which are numbered from 1 to N.  
For each monitored fault mode, the corresponding navigation solution is calculated based on the healthy satellites under that fault mode. Then, fault detection is performed based on solution separation, as shown below. Let the subscript \(q=1,\ 2,\ 3\) represent the east, north, and up directions, respectively. If the following equation holds true for any \(1\leq\ k\leq\ N\) and \(1\leq\ q\leq\ 3\), it is considered that the navigation system is fault-free; otherwise, it is deemed that the navigation system is faulted:  
\[\left|\widetilde{x}_{q}^{(k)}-\widetilde{x}_{q}^{0}\right|<T_{q}^{(k)} \tag{1}\]  
Figure 12: Pseudocode for integrity performance evaluation under CAT-I. where \(\hat{x}_{q}^{(k)}\) represents the navigation solution corresponding to the \(k\)-th fault mode, \(\hat{x}_{q}^{(0)}\) represents the navigation solution calculated using all the visible satellites, and \(T_{q}^{(k)}\) represents the test threshold. The calculation of the test threshold is given by  
\[T_{q}^{(k)}=Q^{-1}\bigg{(}\frac{P_{FA,q}}{2\cdot N}\bigg{)}\cdot\sigma_{ss,q}^{ (k)} \tag{2}\]  
In this equation, \(Q^{-1}(p)\) represents the \((1-p)\) quantile of a standard Gaussian distribution with zero mean, \(P_{FA,q}\) denotes the false alert probability allocated to the \(q\)-th direction, and \(\sigma_{ss,q}^{(k)}\) represents the standard deviation of the test statistic \((\hat{x}_{q}^{(k)}-\hat{x}_{q}^{(0)})\). Finally, the protection level is computed as follows (using VPL as an example):  
\[2Q\bigg{(}\frac{VPL}{\sigma_{3}^{(0)}}\bigg{)}+\sum_{k=1}^{N}Q\bigg{(}\frac{ VPL-T_{3}^{(k)}}{\sigma_{3}^{(k)}}\bigg{)}\cdot P(H_{k})=PHMI_{VERT}\bigg{(}1- \frac{P_{NM}}{PHMI}\bigg{)} \tag{3}\]  
where \(Q(u)\) represents the tail probability of a standard Gaussian distribution with zero mean; \(\sigma_{3}^{(k)}\) denotes the error standard deviation of \(\hat{x}_{3}^{(k)}\) (for \(0\leq k\leq N\)), which is related to the geometry matrix G and the URA values of each satellite; \(T_{3}^{(k)}\) represents the test threshold, which is related to the geometry matrix G, the URE values of each satellite, and the false alert probability; \(P(H_{k})\) indicates the prior probability of fault mode \(k\), which is related to constellation fault probabilities and satellite fault probabilities; \(P_{NM}\) is the sum of the prior probabilities of unmonitored fault modes; \(PHMI\) represents the target integrity risk; and \(PHMI_{VERT}\) denotes the target integrity risk allocated to the vertical position, derived from navigation performance requirements. Based on the parameter settings and algorithms above, Figure 13 presents the protection level world map while considering the BDS-3 and GPS. The figure caption provides the global coverage value under CAT-I and the average protection levels. The results indicate that when only considering the BDS-3 and GPS, the CAT-I global coverage rate is only approximately 67%. Figure 14 gives the protection level world map while considering LEO satellites as space monitoring stations. Comparing Figures 13 and 14 reveals that employing LEO satellites as space monitoring stations can lead to a significant reduction in both the VPL and HPL, while the global coverage rate for CAT-I increases from 67% to 78%. This benefits from the fact that if LEO satellites act as space monitoring stations, employing LEO satellites can reduce the average duration of BeiDou satellite and constellation faults. Figure 13: Protection level world map with BDS-3 and GPS. Figure 15 presents the protection level world map while considering LEO satellites as navigation satellites, and the parameter settings for LEO satellites are given by Simulation Case 1 in Table 11. In this scenario, both the LEO satellite fault probability and the LEO constellation fault probability are set to \(10^{-5}\), representing the most ideal conditions.