## 5 Simulation Analysis
### Simulation Results and Analysis  
First, we obtain the number of the visible satellites at different times according to the geometric relationship between the navigation satellites and the user satellites, which is shown in Figure 3a. From the observability analysis of the system in Section 3.1, the rank of the observability discriminant matrix during the entire sampling duration is calculated to be 6 all the time, which shows that the system is observable during the entire simulation process. Then the condition number is calculated. The condition number of the observable matrix during the simulation process is shown in Figure 3b. According to the observability calculation formula in Section 3.2, the observable degree of each state component of the system is obtained. Figure 4a shows the observable degree of the position vector at different moments, and Figure 4b shows the observable degree of the velocity vector at different moments. In this paper, EKF and the improved adaptive filter are used to evaluate the overall system and each state variable. Figure 5 shows the average error of each state component in the corresponding sampling period. The filter results obtained are shown in Figure 6. Figure 6 reflects the comparison result between the residual error of EKF and adaptive filter estimation results for each state variable. Figure 7a,b reflect the comparison result between diagrams of EKF and adaptive filter position estimation error and velocity estimation error respectively. Figure 8a,b are the comparison figures of the filter position and velocity error of the period with high observability respectively. Taking 8000-10,000 sampling points as an example, Figure 9 shows the comparison result between the observability of the state variable component \(y\) and its residual error result. Figure 3: Figure (**a**) is the figure of number of observed stars. The abscissa is sampling time. Figure (**b**) is the figure of condition number of observation matrix. The abscissa is sampling time. The abscissas in follow figures are the same. According to the simulation experiment results above, this paper mainly analyzes from the following aspects:  
(1) The results of the high-orbit satellite navigation are directly related to the number of the visible satellites and changes in observability. From Figures 3a,b and 4, it can be seen that the number of the visible satellites changes constantly in the simulation period. When the number decreases, the observability matrix condition number becomes larger, and the observability of the position of each state component becomes weaker, the filter estimation error also becomes larger. When the number of the visible satellites increases, the condition number of the observability matrix becomes smaller, and the observability of the position state component becomes stronger and the filter estimation has lower error. (2) The accuracy of the adaptive filter based on observability has been significantly improved compared with conventional methods. It can be seen from Figures 5-9 that in the entire navigation process, the optimized navigation filter method can reduce the estimation error of position and velocity by about 36% and 44% respectively compared with the original EKF, which verifies the effectiveness of this method proposed in this article. It can be seen from Figure 8 that in the period when the number of visible satellites is large, the adaptive filter method will reduce the position error more significantly, and the result is reduced by nearly 70%. This shows that when there exist enough visible satellites, the observability of the overall state variable will be strong, and the proposed navigation filter optimization method will achieve better performance than the other cases. (3) The observability calculation method for state components proposed in this paper can reflect the changes in the filter accuracy of each component. It can be seen from Figure 9 that the observable  
Figure 4: (a,b) show the observable degree of position and velocity respectively.