## 3 Results
### Positioning Results of Static Experiments  
To verify the reconvergence performance of PPP-RTK, a reconvergence experiment was conducted using the HBA1 station. The PPP-RTK solution utilized HBA1 observations from 04:00:00 to 12:00:00 GPST, with reconvergence performed every hour. The convergence speed and accuracy of float PPP, PPP-AR, and PPP-RTK were compared and analyzed for both the BDS single system and the combined GPS (G), Galileo (E), and BDS (C) system. In this study, once the horizontal and vertical positioning errors were less than 10 cm and 20 cm [30], respectively, and this lasted more than 5 min, the positioning was viewed as convergence. The TTFF was defined as the time difference between the beginning epoch and the epoch when a fixed solution of less than 10 cm in the horizontal and 20 cm in  
Figure 3: Slant path ionospheric delay and ZTD errors for HBA1, both obtained by subtracting the PPP-AR estimated values (true values) from the interpolated values. The bottom right corner shows the ZTD error time series graph, while the other three sub graphs show the ionospheric errors of GPS, Galileo, and BDS, respectively.  
the vertical components was achieved and lasted for at least 5 min. The success rate of ambiguity resolution was defined as follows:  
\[Pcfix=(1-\frac{N_{unfix}}{N_{all}})\cdot(1-\frac{N_{fix_{corr}}}{N_{fix}})\]  
where \(N_{all}\) represents the total number of epochs; \(N_{unfix}\) represents those that do not pass the ratio test; \(N_{fix}\) denotes the number of epochs fixed successfully; \(N_{fix_{corr}}\) indicates the number of epochs that pass the ratio test but that have horizontal or vertical positioning errors that exceed 10 or 20 cm, respectively.  
Figure 4 shows the time series of the positioning error in the north, earth, and up directions during the convergence experiment. Similar to [40], it was evident that the performance of the GEC constellations of the three positioning methods was significantly better than that of the single BDS system. Under single BDS positioning conditions, it was observed that the float PPP solution required a long convergence time. PPP-AR significantly improved the convergence speed, but it still took a considerable amount of time to achieve full convergence. PPP-RTK demonstrated the best performance, achieving rapid convergence within a few seconds. Under multi-constellation conditions, similar conclusions were drawn as with the single BDS system. The convergence speed of the PPP-AR solution was significantly improved compared with the float PPP solution, and the positioning accuracy was also further improved. The PPP-RTK demonstrated the fastest convergence speed, achieving instantaneous fixation and the highest accuracy in all four reconvergence experiments.  
Table 2 shows the statistical values of convergence time and positioning errors for PPP, PPP-AR, and PPP-RTK. It was observed that the PPP-RTK solution showed 25.22% positioning accuracy improvement in horizontal components for the single BDS scheme, compared with the PPP-AR. The TTFF decreased from 1039 s to only 25 s. In the GEC scheme, PPP-AR showed a significant enhancement in horizontal accuracy by 70.45% and vertical accuracy by 49.4% compared with the conventional PPP solution. However, in comparison to PPP-AR, the improvement in horizontal positioning accuracy with PPP-RTK was only 2.80%, along with a 11.78% enhancement in vertical accuracy. On the other hand,  
Figure 4: PPP/PPP-AR/PPP-RTK reconvergence experiment E, N, U error time series. Reconvergence was performed every half hour for a total of four times.  
when compared with single BDS PPP-RTK, the utilization of multi-constellations improved horizontal accuracy by 19.19% and vertical accuracy by 44.61%. Overall, multi-constellation observations improved the positioning performance of dual-frequency PPP-RTK and increased the accuracy and stability of PPP-RTK solutions, which enabled instantaneous ambiguity resolution for static stations.