## 4 Assessment and Analysis of LeGNSS Precise Positioning
### Statistical Analysis of Convergence Time on a Global Scale  
To fully reflect the PPP performance of the LeGNSS, we conduct the PPP computations for simulated stations with the space resolution \(5^{\circ}\) by \(5^{\circ}\) on a global scale. For each station, the data is calculated every two hours for one week in total. Thus, there are 84 computations for each station in one week. Afterwards, the averaged convergence time for each component of each station is calculated. Here, we repeat our definition of convergence time as the number of epochs from the first epoch to the epoch when the positioning errors for each component reach, and afterward remain within, \(\pm 10\) cm. In terms of the numerical analysis, both BDS-only and LEO-only cannot provide standalone positioning service in some specific areas. In this statistic analysis, we choose four system combinations of GPS-only, G/C, G/L, and G/C/L to investigate their convergence time in each component on a global scale. The distributions of the convergence time are shown in Figure 9 in units of minutes. As shown in Figure 9, the averaged convergence time of GPS-only is 33.0, 24.9, and 32.7 min for east, north, and up components, respectively. The maximum convergence time is over 100 min for all three components. For 95% of the world's area, it takes about one hour to converge. For the  
Figure 9: Histogram of convergence time in each component for GPS, G/C, G/L, and G/C/L (from top to bottom) PPP. The dashed line denotes that the convergence time of 95% of stations are smaller than the corresponding time. G/C case, the averaged convergence time is reduced by 52.1%, 57.0%, and 57.5% for east, north, and up components, respectively. Most regions in the world need about 32 min to converge for three components. With the integration of GPS and LEO satellites, the results are very astonishing with the averaged convergence time of only 8.2, 5.5, and 7.3 min for east, north, and up components, respectively. It should be pointed out that the contribution of LEO to the convergence time is much greater than that of BDS, though there are a very limited number of LEO satellites in medium and low latitude regions. For G/C/L case, the averaged convergence time is 6.1, 4.2, and 5.1 min for east, north, and up directions, respectively. The improvement is not significant compared to G/L combination. This is mainly because of the slow motion and regional availability of BDS satellites with respect to the ground stations. The convergence time of more than 10 min is mostly in regions where the visible LEO satellites are rather few. It is prospective that the convergence time can be shortened to several minutes, even seconds, if there are enough LEO satellites in the sky. As aforementioned, the fast motion of LEO satellites can indeed make a significant contribution to the convergence time of PPP. Since the geometric change of LEO satellites is much faster than that of GNSS, which can be seen from the sky plot, the sampling interval of the observation data may be another factor to the convergence time. Hence we investigate the effect of the sampling rate of observation data to the combined G/C/L PPP convergence time. The distributions of convergence time for sampling intervals of 30, 10, 5, and 1 s G/C/L PPP are shown in Figure 10. The averaged convergence time is generally 10.0 min for all the components at all sampling intervals. With higher sampling interval, the convergence time is shorter. This reason is that the LEO satellites are moving fast, every second can already achieve great geometric change with respect to the ground stations. It takes 7.9, 5.3, and 6.9 min for east, north, and up components to converge with 30 s sampling interval while 6.9, 4.9, and 5.8 min with 10 s sampling interval, respectively. Further increasing the sampling interval does not shorten the convergence time significantly anymore, which means that the satellites' geometric change with respect to the ground stations are not great enough to improve the PPP convergence time.