First of all, the quality of RTS orbits and clocks for different ACs was evaluated and compared. Then, the static and kinematic positioning performance of BDS-2 and BDS-3 joint RT UU-PPP was investigated and compared with corresponding RT IF-PPP solutions. Two key indicators of positioning accuracy and convergence time were used for the quantitative analysis.
4.1. Accuracy Analysis of RTS Products from Different ACs
The RMS of orbit errors and STD of clock errors for each BDS satellite from five ACs are presented in
Figure 3. “CNE” and “SHA” are abbreviations for CNES and SHAO, respectively. The results of the BDS-2 satellites (C06–C16) and BDS-3 satellites (C19–C46) are depicted on the left and right sides of the solid black line, respectively. The blue and green bar charts denote the IGSO and MEO satellites, respectively. The orbit errors exceeding 1.5 m in any direction are considered as gross errors and need to be removed in assessment, while for clock evaluation, the gross error is defined as 3.0 ns. It should be noted that the GEO satellites (C01–05) of BDS are excluded from this study. One reason is that the orbit and clock accuracy of GEO satellites are far inferior to IGSO and MEO satellites, which may generate an adverse impact on RT PPP [
8]. Another issue is that GEO satellites are only visible in the Asia–Pacific region, but this contribution mainly focuses on the global positioning performance of BDS. From
Figure 3a–c, we can see that the orbit accuracy of MEO satellites is commonly better than that of IGSO satellites in all directions for all ACs. In the radial direction, the MEO orbit accuracy of GFZ, SHAO, and WHU is superior to that of CAS and CNES. The RMS of orbit errors for CAS C23–26 satellites and CNE C25–26 satellites can be up to 20 cm and exceeds that of other MEO satellites by 2 to 4 times. The observation weights of these satellites should be reduced in RT PPP because the positioning performance is highly dependent on the radial error of orbits. The RMS of radial orbit errors for most IGSO satellites is about 10–15 cm, while the corresponding accuracies of GFZ C38, C39, and C40 satellites can reach up to 120.6, 141.3, and 53.3 cm, respectively, and similar gross errors also appear in the along-track and cross-track directions. This may be due to GFZ mistakenly using PCO values of C38–40 satellites when solving the SSR orbit corrections. Therefore, the C38–40 satellites should be excluded from the experiment of GFZ-drive RT PPP in this study. In terms of the along-track orbit errors, the RMS of C23–24, C36–37, and C45–46 satellites for CAS is far higher than that of other MEO satellites and can even be up to 38 cm. Similar situations also occur in the C37 satellite of SHAO. CNES and GFZ have five (C6–8, C16, C38) and three IGSO (C38–40) satellites with along-track orbit accuracy exceeding 30 cm, respectively. By comparison, WHU has the best orbit performance in the along-track direction, and its RMS error for most satellites can be better than 10 cm. There is no significant difference in the accuracy of MEO satellites between the five ACs regarding the cross-track orbit errors. All MEO satellites have stable cross-track orbit accuracy, but some outliers can be found in the IGSO satellites, such as the CNES C6–8, C16, and C38 satellites.
As to clock accuracy (see in
Figure 3d), the CAS has the worst performance among all ACs, and the STD of clock errors exceeds 0.4 ns for 60% of satellites. Except for C39–40 satellites with an STD of up to 1.4 ns, the clock accuracy of GFZ is comparable to that of SHAO and WHU for all satellites and can be stable within 0.4 ns. If both MEO and IGSO satellites are considered, SHAO and WHU have similar and the best clock accuracy.
According to the property and service stage of satellites, all BDS satellites can be divided into four parts, including IGSO (C06–10, C13, C16, C38–40), MEO (C11–12, C14, C19-C30, C32–37, C41–46), BDS-2 (C6–14, C16), and BDS-3 (C19–30, C32–46) satellites. The mean RMS of orbit errors and mean STD of clock errors for each part are shown in
Figure 4. It is worth noting that the C38–40 satellites of GFZ were defined as outliers and excluded from this assessment. We can see that the orbit accuracy decreases sequentially from the radial (R), cross-track (C), and along-track (A) directions. Regardless of ACs, the orbit accuracy of MEO satellites is superior to that of IGSO satellites in all directions. As for MEO satellites, the mean RMS of radial orbit errors for GFZ, SHAO, and WHU is at the same level with about 4.5 cm, while for CAS and CNES, their mean RMS may increase to over 7 cm. The mean RMS of IGSO orbit errors in the radial direction is about 11–15 cm for all ACs. In terms of the along-track and cross-track orbit errors, the mean RMS of CNES is larger than that of other ACs and is even up to 29.2 and 40.1 cm for IGSO satellites, respectively. Conversely, no matter the IGSO and MEO satellites, the orbit accuracy of WHU in both the along-track and cross-track directions is the best among all ACs. From the perspective of BDS-2 and BDS-3 satellites, the orbit accuracy of BDS-3 satellites is better than that of BDS-2 satellites in all directions. The mean RMS of radial orbit errors for the BDS-3 satellites provided by SHAO and WHU is about 5.5 cm and better than that of GFZ (6.5 cm), CNES (8.0 cm), and CAS (8.6 cm). While for the BDS-2 satellites, the radial orbit accuracy of GFZ is the highest with a mean RMS of 7.9 cm. In the along-track direction, the orbit accuracy of WHU is the best and reaches 11.1 and 9.0 cm for the BDS-2 and BDS-3 satellites, respectively. When it comes to cross-track orbit accuracy for the BDS-2 satellites, CAS, GFZ, and WHU have a similar RMS of about 12 cm, but the corresponding accuracy for the BDS-3 satellites can be improved to no more than 9 cm. Compared with the results of previous research using RTS products in 2021 [
8,
10], the orbit accuracies of CAS, CNES, GFZ, and WHU are improved at the end of 2022, especially for GFZ products and IGSO satellites.
In addition to radial orbit errors, the quality of satellite clock errors is closely related to the RT PPP performance. The clock accuracy of IGSO satellites for CAS, CNES, and GFZ is at the same level with a mean RMS of about 0.4 ns, while for SHAO and WHU, the corresponding accuracy can be improved to 0.36 ns. In comparison with IGSO satellites, the mean RMS of MEO clock errors for CNES, GFZ, SHAO, and WHU can be improved by 32.5%, 47.6%, 38.9%, and 44.4% to 0.27, 0.22, 0.22, and 0.20 ns, respectively. There is an abnormal phenomenon in that the clock accuracy of CAS has decreased from 0.39 ns for IGSO satellites to 0.46 ns for MEO satellites, a decrease of 18.0%. The mean RMS of BDS-2 clock errors is 0.40, 0.38, 0.28, 0.35, and 0.32 ns for CAS, CNES, GFZ, SHAO, and WHU, respectively. Except for CAS, the BDS-3 clock accuracy of other ACs is higher than that of BDS-2 clock accuracy, and its mean RMS of 0.28, 0.26, 0.24, and 0.21 ns can be achieved for CNES, GFZ, SHAO, and WHU, respectively. Compared to the clock accuracy of the BDS-2 (about 0.5 ns) and BDS-3 satellites (about 0.35 ns) in 2021 [
8,
9], the quality of RTS clock products at the end of 2022 has been improved for both the BDS-2 and BDS-3 satellites.
4.2. Performance of BDS-2 and BDS-3 Joint Real-Time Undifferenced and Uncombined PPP in Static Mode
Figure 5 gives the convergence curves of BDS-2 and BDS-3 joint static RT UU-PPP at the 68% confidence level. To measure the performance of this RT UU-PPP, the time series of convergence for BDS-2 and BDS-3 joint RT IF-PPP in static mode using the same datasets are also displayed. First of all, the absolute positioning errors of 240 daily solutions (20 stations × 12 days) for each epoch are sorted in ascending order. Next, we select a value for each epoch that is lower than 68% of all sorted positioning errors. Finally, the values of each epoch are concatenated in the first 90 min. Please note that the horizontal error denotes the combination of north (N) and east (E) errors. We can see that whether it is UU-PPP or IF-PPP, the convergence speed of the vertical error is generally faster than that of the horizontal error. Since the CAS has the worst clock accuracy among all ACs, especially for MEO satellites, both the horizontal and vertical convergence curves of CAS in RT IF-PPP were significantly higher than those of other ACs during the 10–40 min. However, this situation did not occur in the horizontal component of RT UU-PPP and was not particularly evident in the vertical component. After 40 min, the horizontal convergence curve of all ACs showed good consistency for both UU-PPP and IF-PPP. Thanks to the great orbit and clock accuracies of WHU, after converging to 0.2 m, its convergence curve always remains at the best level in the horizontal and vertical components of both UU-PPP and IF-PPP.
In order to quantitatively analyze the specific convergence time of different static RT PPP solutions, the convergence criterion is defined as the time when the above convergence curve is below 0.2 m, and the statistical results of each AC are shown in
Figure 6. Due to the lower clock accuracy of MEO satellites for CAS compared to other ACs, the convergence time of CAS-drive RT IF-PPP is the longest in both the horizontal and vertical components and exceeds 30 min. With the introduction of CNES RT-VTEC constraints, the horizontal convergence time of CAS-drive RT UU-PPP can be improved by 10.9% to 28.5 min. However, this phenomenon of external ionospheric constraints improving convergence performance did not appear in the results of other ACs. The main reason is that the accuracy of CNES RT-VTEC products is limited and cannot precisely eliminate ionospheric errors like the IF-PPP model. If the external ionospheric model is accurate enough, the convergence speed of UU-PPP is fully capable of surpassing that of IF-PPP [
26]. The difference in convergence time between UU-PPP and IF-PPP for all ACs is summarized in
Table 3. We can see that SHAO has the maximum decline rates in horizontal convergence time and is up to 36.6% from 20.5 min of RT IF-PPP to 28.0 min of RT UU-PPP, while for the vertical convergence time, the maximum decline rate appears in the results of CNES, which does not exceed 20%. Only focusing on RT UU-PPP, the horizontal convergence time of CAS and SHAO is about 28 min and worse than that of other ACs with no more than 25 min. There are apparent differences in the vertical convergence time among different ACs. CNES, GFZ, and WHU are in the first level and less than 20 min. CAS and SHAO have relatively long vertical convergence times of over 32 and 25 min, respectively.
The positioning errors after 2 h in the daily solution are collected to calculate the RMS positioning accuracy. In total, 240 daily RMS values can be obtained from 20 MGEX stations in the period of DoY 305–316. The boxplot of the RMS positioning accuracy for BDS-2 and BDS-3 joint RT UU-PPP and RT IF-PPP in static mode is shown in
Figure 7, and the numbers in this figure denote the median of all RMS values. The red plus sign “+” in
Figure 7 represents outlier. It should be noted that 2D (two-dimensional) represents the horizontal component, and 3D (three-dimensional) represents the combination of N, E, and U (up) errors. We can see that the positioning accuracy of the N component is significantly better than that of the E component for all ACs regardless of RT UU-PPP or RT IF-PPP. The RMS of both N and E positioning errors for CAS is clearly larger than that for other ACs, which is caused by the poor quality of CAS clock products, but it has no impact on the U component. GFZ has the worst positioning accuracy in the U component with a median RMS of about 8.5 cm. Except for CAS, the horizontal positioning accuracy of other ACs is at the same level and can be better than 4 cm. There are no evident differences in 3D positioning accuracy between all ACs, whose median RMS is no more than 10 cm. Taking the static RT IF-PPP as references, the improvement rates of positioning accuracy of RT UU-PPP for different ACs are summarized in
Table 4. With the help of CNES RT-VTEC constraints, RT UU-PPP has slightly better positioning accuracy in both horizontal and vertical components after convergence. The main reason is that the observation noise of UU-PPP is lower than that of IF-PPP and the ionospheric errors can be completely removed since the ionospheric parameters are estimated very accurately after convergence. For different ACs, the 3D positioning accuracy of RT UU-PPP can only be improved by 2.0% to 8.7% in comparison with RT IF-PPP in static mode.
4.3. Performance of BDS-2 and BDS-3 Joint Real-Time Undifferenced and Uncombined PPP in Kinematic Mode
Using statistical methods similar to static RT PPP, the convergence curves of BDS-2 and BDS-3 joint kinematic RT UU-PPP and RT IF-PPP at the 68% confidence level can be seen in
Figure 8. Different from the vertical convergence curves, the convergence curves of the horizontal component for all ACs have good consistency in both RT UU-PPP and RT IF-PPP. The horizontal positioning errors of all ACs can be less than 0.2 m after around 60 min. The vertical convergence curve of CAS in RT IF-PPP is higher than that of other ACs during the 10–30 min, which is caused by the worse accuracy of CAS clock products. However, the corresponding convergence performance in RT UU-PPP can be substantially improved by adopting the CNES RT-VTEC constraints. As for RT UU-PPP, CNES exhibits the slowest convergence speed in the vertical component, which may be related to the worst IGSO orbit accuracy of CNES. It is important to note that the vertical convergence performance of SHAO is optimal in most periods regardless of UU-PPP or IF-PPP.
Figure 9 shows the convergence time of BDS-2 and BDS-3 joint RT UU-PPP and RT IF-PPP in kinematic mode for all ACs when positioning errors converge to 0.2 m. Compared with static RT PPP, the convergence time of kinematic RT PPP in both the horizontal and vertical components has increased by at least 2 times. In this kinematic case of poor convergence performance, the convergence time of RT UU-PPP can be shortened through CNES RT-VTEC constraints, especially for the CAS solutions with relatively longer convergence times. The main reason is that the ionospheric errors of UU-PPP can be precisely calculated, thereby achieving the same effect as the IF-PPP model. Meanwhile, the observation noise of UU-PPP is much smaller than that of IF-PPP, which is beneficial for fast convergence.
Table 5 summarizes the improvement rates of the convergence speed for RT UU-PPP compared to RT IF-PPP in kinematic mode. It can be seen that the horizontal and vertical convergence time of CAS-drive RT UU-PPP can be improved by 31.7% and 22.9%, respectively. Such a significant increase also appears in the horizontal convergence time of CNES. Interestingly, the adoption of CNES RT-VTEC constraints has little positive effect on the convergence performance of GFZ-, SHAO-, and WHU-drive RT UU-PPP in both the horizontal and vertical components. This may be due to the limited accuracy of CNES RT-VTEC products. If more precise ionospheric models like global ionospheric maps (GIMs) are used, the convergence speed of RT UU-PPP may be further improved. In terms of RT UU-PPP, the convergence time of CAS, GFZ, and SHAO is at the same level, with about 55 min in the horizontal and 60 min in the vertical, while for the horizontal component of WHU and the vertical component of CNES, their convergence time is significantly longer than other ACs.
Consistent with the statistical method for positioning accuracy in static RT PPP, the boxplot of RMS positioning accuracy for BDS-2 and BDS-3 joint kinematic RT UU-PPP and RT IF-PPP solutions is shown in
Figure 10. As expected, CAS has the worst positioning accuracy among all ACs under the influence of poor clock quality, with median RMS in the horizontal and vertical components of approximately 14 and 19 cm, respectively. While for other ACs, the horizontal and vertical positioning accuracy can be better than 9 and 14 cm, respectively. Except for CAS, there is not much difference in the positioning accuracy among different ACs. By comparison, SHAO has the best positioning accuracy, and its median RMS can achieve 7 and 10 cm in the horizontal and vertical components, respectively.
Table 6 summarizes the improvement rates of positioning accuracy for RT UU-PPP compared to RT IF-PPP in the kinematic mode. It can be seen that the application of CNES RT-VTEC constraints has a limited impact on improving the positioning accuracy of RT UU-PPP; the maximum improvement rate of 3D components only reaches 8.2%. The 3D positioning accuracy of WHU has even slightly decreased by 1.4% in RT UU-PPP. At present, the kinematic cm level accuracy can only be achieved in the horizontal component, and the 3D positioning accuracy is around 15 cm for most ACs. It is noteworthy that BDS-2 and BDS-3 joint kinematic RT PPP using RTS products still cannot obtain cm level vertical positioning accuracy.