Assessment of PPP Using BDS PPP-B2b Products with Short-Time-Span Observations and Backward Smoothing Method

Abstract

The BeiDou Navigation Satellite System (BDS) offers orbit and clock corrections through the B2b signal, enabling Precise Point Positioning (PPP) without relying on ground communication networks. This capability supports applications such as aerial and maritime mapping. However, achieving high precision during the convergence period remains challenging, particularly for missions with short observation durations. To address this, we analyze the performance of PPP over short periods using PPP-B2b products and propose using the backward smoothing method to enhance the accuracy during the convergence period. Evaluation of the accuracy of PPP-B2b products shows that the orbit and clock accuracy of the BDS surpass those of GPS. Specifically, the BDS achieves orbit accuracies of 0.059 m, 0.178 m, and 0.186 m in the radial, along-track, and cross-track components, respectively, with a clock accuracy within 0.13 ns. The hourly static PPP achieves 0.5 m and 0.1 m accuracies with convergence times of 4.5 and 25 min at a 50% proportion, respectively. Nonetheless, 7.07% to 23.79% of sessions fail to converge to 0.1 m due to the limited availability of GPS and BDS corrections at certain stations. Simulated kinematic PPP requires an additional 1–4 min to reach the same accuracy as the static PPP. Using the backward smoothing method significantly enhances accuracy, achieving 0.024 m, 0.046 m, and 0.053 m in the north, east, and up directions, respectively. For vehicle-based positioning, forward PPP can achieve a horizontal accuracy better than 0.5 m within 4 min; however, during the convergence period, positioning errors may exceed 1.5 m and 3.0 m in the east and up direction. By applying the smoothing method, horizontal accuracy can reach better than 0.2 m, while the vertical accuracy can improve to better than 0.3 m.

1. Introduction

With the rapid development of Global Navigation Satellite System (GNSS) technology, the demand for high-precision real-time positioning has increased significantly. Precise Point Positioning (PPP) [1] technology enables high-precision positioning using a single receiver, overcoming the limitations of traditional Real-Time Kinematic (RTK) positioning, which relies on ground communication networks and reference stations. This technology holds broad application prospects in fields such as precise positioning in remote areas [2,3], space weather monitoring [4,5], and high-precision time and frequency transfer [6,7]. PPP relies on augmentation corrections, such as precise orbits and clocks, which are essential for achieving high precision. Additionally, tropospheric and ionospheric corrections can shorten the convergence time of the positioning [8,9]. The application of PPP based on post-processed products has been extensively validated. To meet the requirements of rapid and real-time PPP applications, researchers have conducted in-depth studies using different orbit and clock products. Byram et al. (2012) utilized the rapid and ultra-rapid products provided by the United States Naval Observatory (USNO) for PPP. The results demonstrated that the combination of GPS and GLONASS observation data significantly enhances the position accuracy [10]. Elsobeiey and Al-Harbi (2016) analyzed the real-time PPP accuracy using International GNSS Service (IGS) ultra-rapid products and real-time services. Their results showed that, compared to ultra-rapid products, real-time products achieved a 50% improvement in positioning accuracy [11]. Chun et al. (2018) analyzed the real-time orbit and clock products provided by the Centre National d’Études Spatiales (CNES), which revealed that the user accuracy of real-time products ranges between approximately 0.03 and 0.35 m [12]. Subsequent research has further evaluated the performance of real-time corrections and multi-GNSS PPP on GPS, BDS, and other systems [13,14,15]. Furthermore, Yu et al. (2023) conducted a comparative analysis of PPP based on 12 different centers and assessed its accuracy. The results indicated that products from Wuhan University showed the best performance, with static PPP over 24 h achieving an accuracy of 1.0 cm in the horizontal and vertical directions and a median convergence time of 12.0 min. The hourly kinematic PPP achieved an accuracy of 10.8 cm and 9.5 cm in the horizontal and vertical directions, respectively. The multi-GNSS combination improved the accuracy to 5.2 cm [16].

However, the realization of the aforementioned real-time service still relies on receiving real-time correction data via ground communication networks, as it is only accessible through the network. PPP corrections are transmitted via the B2b signal (PPP-B2b) from three BDS geosynchronous Earth orbit (GEO) satellites. These corrections include orbit and clock data for both GPS and BDS-3 satellites, along with differential code bias (DCB) corrections. When combined with the GPS LNAV and BDS CNAV1 navigation messages, the precise satellite orbit and clock can be recovered, enabling decimeter to centimeter-level positioning across China and the surrounding areas (75°E–135°E, 10°N–55°N) without relying on internet connectivity [17]. Scholars have conducted thorough evaluations on the latency and accuracy of orbit and clock products, and the positioning performance of PPP-B2b products. Nie et al. (2021) evaluated the accuracy of PPP-B2b orbit and clock corrections against the final precise products obtained from GFZ. The results show that the radial, along-track, and cross-track accuracies for BeiDou-3 satellites are 0.138 m, 0.131 m, and 0.145 m, respectively, with clock offset accuracies reaching centimeter-level precision [18]. Tao et al. (2021) further compared the PPP-B2b with CNES (Centre national d’études spatiales) products. The results demonstrated that BDS-3-only PPP-B2b kinematic PPP can achieve a centimeter-level accuracy comparable to the GPS-only results using CNES [19]. Song et al. (2023) validated the performance of PPP-B2b using different signal combinations and demonstrated that PPP-B2b can meet the positioning performance required in China [20]. Furthermore, the application of PPP-B2b in ocean environments [21], single-frequency scenarios [22], time transfer [23], and its integration with inertial navigation systems [24] has been validated, demonstrating the promising enhancements PPP-B2b offers for real-time users.

The assessments so far have primarily focused on daily solutions and simulated kinematic experiments. However, applications such as aerial mapping [25] and agriculture [26] usually require rapid static positioning during the work preparation phase or high positional accuracy during the operational phase. The convergence period, particularly at the start or after an interruption, tends to exhibit relatively poor performance due to the incomplete elimination of atmospheric errors and the reinitialization of ambiguities. Although some studies investigate the performance of PPP with short observation sessions [27,28], research on real-time PPP-B2b remains relatively limited. To address this issue, this paper evaluates the performance of PPP-B2b products and investigates the positioning accuracy achieved through both static and kinematic PPP with varying session lengths. Additionally, a backward smoothing method is applied to improve the accuracy of PPP during the convergence period, enabling high-precision positioning throughout the process.

2. Methods and Theory

The GNSS raw pseudorange and carrier phase observations across different frequencies are detailed as follows:

$$\begin{array}{c}{p}_{r,j}^s={\rho }_r^s+c\cdot \left(d{t}_r+d_{r,j}\right)-c\cdot \left(d{t}^s+d_j^s\right)+I_{r,j}^s+T_r^s+{\epsilon }_{r,p,j}^s\\ {\phi }_{r,j}^s={\rho }_r^s+c\cdot \left(d{t}_r+{\delta }_{r,j}\right)-c\cdot \left(d{t}^s+{\delta }_j^s\right)-I_{r,j}^s+T_r^s+{\lambda }_j\cdot N_{r,j}^s+{\epsilon }_{r,\phi ,j}^s\end{array}$$

(1)

where superscripts $s$ and $r$ denote a specific satellite and receiver, respectively; $p$ and $\phi$ represent the pseudorange and carrier phase observations; and $j$ denotes the frequency and the corresponding wavelength ${\lambda }_j$. Furthermore, $\rho$ is the geometric distance from the satellite to receiver, while $\mathrm{d}{t}_r$ and $\mathrm{d}{t}^s$ are clock offsets for the receiver and satellite, respectively. $c$ represents the speed of light in a vacuum; $I_r^s$ is the ionospheric delay; $T$ is the tropospheric delay; $N$ stands for ambiguity; $\epsilon$ represents the observation noise and multipath effects; $d_{r,j}$ and $d_j^s$ are the receiver-dependent and satellite-dependent instrumental delay biases in the pseudorange, respectively; and ${\delta }_{r,j}$ and ${\delta }_j^s$ are the receiver-dependent and satellite-dependent instrumental delay biases in the carrier phase observations, respectively.

2.1. Recovery of Precise Orbits and Clocks Products Using PPP-B2b Correction

According to the ICD (Interface Control Document) [29] of the BDS PPP-B2b signal, orbit corrections are provided in the radial, along-track, and cross-track directions within the satellite orbit coordinate system and denoted as $\mathit{\delta }{\mathit{O}}_{\mathit{R}\mathit{A}\mathit{C}}$. The corrections should be translated to the Earth Center Earth Fixed (ECEF) system using the following equation:

$$\delta {\mathit{X}}_{ECEF}=-\left[\begin{array}{ccc}{\displaystyle \frac{\mathit{X}}{\left|\mathit{X}\right|}}& {\displaystyle \frac{\mathit{X}}{\left|\mathit{X}\right|}}\times {\displaystyle \frac{\mathit{X}\times \mathit{V}}{\left|\mathit{X}\times \mathit{V}\right|}}& {\displaystyle \frac{\mathit{X}\times \mathit{V}}{\left|\mathit{X}\times \mathit{V}\right|}}\end{array}\right]\cdot \delta {\mathit{O}}_{RAC}$$

(2)

where $X$ and $V$ are the satellite position and velocity vectors calculated by the broadcast ephemeris, respectively; $\delta {\mathit{X}}_{ECEF}$ is the transmitted satellite orbit correction. After calculating the initial position $\mathit{d}{\mathit{X}}_{brdc}$ with the broadcast ephemerides, the precise satellite orbit $\mathit{d}{\mathit{X}}_{prec}$ can be calculated with the following:

$${\mathit{X}}_{prec}=\mathit{d}{\mathit{X}}_{brdc}+\delta {\mathit{X}}_{ECEF}$$

(3)

In addition, the precise clock $\mathrm{d}{t}_{\mathrm{prec},\mathrm{B}2b}$ can be calculated using the following:

$$\mathrm{d}{t}_{\mathrm{prec},\mathrm{B}2b}=\mathrm{d}{t}_{brdc}-C_0/C_{light}$$

(4)

where $\mathrm{d}{t}_{brdc}$ is the clock correction calculated from the broadcast ephemeris in meters; ${\mathit{C}}_{\mathbf{0}}$ is the clock corrections derived from the PPP-B2b corrections in meters and ${\mathrm{C}}_{\mathrm{light}}$ is the velocity of light.

The PPP-B2b service provides clock and orbit corrections with update intervals of 6 s and 48 s, respectively. To ensure the proper synchronization of these corrections, the “IOD Corr” parameter acts as a version identifier, aligning orbit and clock data. Additionally, the “IOD SSR” (Issue of Data, State Space Representation) parameter is used to match the PPP-B2b corrections with the broadcast ephemerides. Although the synchronization of these corrections is typically reliable, discrepancies can arise due to missing correction products for certain satellites or changes in the ephemeris IOD. Such mismatches may result in the incorrect recovery of orbit and clock information or a reduction in the number of usable satellites, thereby degrading PPP performance.

To address these challenges, a predictive correction strategy is employed, leveraging the nearest available correction data to compensate for missing corrections. This method effectively handles gaps in orbit corrections within a 300 s window and clock corrections within a 60 s window. In this way, the integrity of PPP positioning is maintained even in the presence of correction gaps from certain satellites.

Apart from the satellite orbit and clock errors, satellite and receiver biases are another main factor affecting the accuracy of PPP. The receiver code hardware delays can be absorbed by the clock parameter, and the satellite and receiver phase biases can be absorbed by the float ambiguity parameters. However, the satellite code biases persist and impact the convergence of PPP. Code biases corrections are also provided in PPP-B2b corrections and can be corrected using the following equation:

$$\widetilde{P_f}=P_f-DC{B}_f$$

(5)

where $P_f$ is the original pseudorange observations and $\widetilde{P_f}$ is the corrected observations; $DC{B}_f$ is the PPP-B2b DCB corrections on the B1Cp and B2Ap signals for corresponding signals of BDS-3 in meters.

The B1I and B3I ionosphere-free (IF) combination are generally used for the satellite orbit and clock estimation, in addition to the IF PPP at the user end. However, the real-time precise product clocks calculated with PPP-B2b corrections and the matching broadcast ephemeris are referenced to the B3 frequency. For the PPP using the IF model, the satellite hardware delay correction should be applied to convert the clocks from B3I to the B1I/B3I [30], which can be described as follows:

$$d{t}_{prec}=\mathrm{d}{t}_{\mathrm{prec},\mathrm{B}2b}-{\displaystyle \frac{f_{B1I}^2}{f_{B1I}^2-f_{B3I}^2}}{\mathrm{DCB}}_{B1I}^s$$

(6)

where $d{t}_{prec}$ is the precise satellite clock offset referred to the $\mathrm{B}1\mathrm{I}/\mathrm{B}3\mathrm{I}$ IF combination, which can be used in the same manner as the IGS standard clock products; $f_{B1I}^2$ and $f_{B3I}^2$ represent the frequencies of the $\mathrm{B}1\mathrm{I}$ and $\mathrm{B}3\mathrm{I}$ signals, respectively; and ${\mathrm{DCB}}_{B1I}^s$ is the hardware delays for the B1I signal.

2.2. SISRE Assessment

To assess the accuracy of the satellite orbits and clocks, the Signal-in-Space Ranging Error (SISRE) is calculated [31], which can be expressed as follows:

$$SISRE=\sqrt{{\left(\alpha · dR-c · dt\right)}^2+{\displaystyle \frac{\left(d{A}^2+d{C}^2\right)}{\beta }}}$$

(7)

where $dR$, $dA$, $dC$ are the radial, along- track, and cross-track orbit errors in the satellite orbit coordinate system, respectively; $dt$ is the clock bias; c is the speed of light in a vacuum; and α and β are coefficients determined by the satellite orbit altitudes.

The satellite orbits calculated from PPP-B2b refer to the satellite’s antenna phase center, while the precise orbit products provided by the MGEX analysis centers refer to the satellite’s center of mass. Therefore, prior to comparing broadcast ephemeris and precise orbits, Phase Center Offset (PCO) correction must be applied according to the following formula:

$$\delta {\mathit{X}}_{B2b}={\mathit{X}}_{pre,final}-\left[\begin{array}{c}{\mathit{X}}_{pre,B2b}+\mathit{A}\cdot \delta {\mathit{X}}_{pco}\end{array}\right]$$

(8)

where $\delta {\mathit{X}}_{B2b}$ is the orbit error vector; ${\mathit{X}}_{pre,final}$ is the reference orbit vector obtained from final products; ${\mathit{X}}_{pre,B2b}$ is the real-time PPP-B2b precise orbit vector; $A$ is the satellite attitude matrix; and $\delta {\mathit{X}}_{pco}$ is the satellite PCO correction vector obtained from the latest “igs14.atx” file released by IGS.

3. Data and Experiments

The objectives of this section are to comprehensively analyze the quality of BDS PPP-B2b corrections and their accuracy for the static and kinematic PPP.

3.1. Accuracy of PPP-B2b Orbits and Clocks

To verify the accuracy of real-time PPP-B2b products, the final products provided by Wuhan University (WUM) were selected as the reference. The reported accuracy of these final products for BDS-3 satellites is 3 to 4 cm [32,33]. Figure 1 shows the error time series of the PPP-B2b precise orbit products for days 075 to 092 of 2024. Notably, in the GPS orbit error series, the radial orbit errors for satellites G22 and G28 exhibit systematic biases of approximately 0.4 m and 1 m, respectively. These biases persist until day 087, after which the errors return to levels consistent with those of other satellites. Apart from this, no significant anomalies are detected in the GPS and BDS products across different days, indicating the stable quality of PPP-B2b products. Overall, the radial component, which significantly impacts positioning errors, shows the smallest errors among the three components, with most satellites exhibiting errors within 0.3 m. In the along-track direction, the GPS products exhibit relatively larger errors, whereas in the cross-track direction, GPS and BDS exhibit comparable accuracy.

Figure 2 presents the Root Mean Square (RMS) of PPP-B2b real-time orbits errors. The average values are 0.098 m, 0.368 m, and 0.214 m for the radial, along-track, and cross-track components for GPS, respectively. In comparison, the average RMS values for the BDS-3 MEO satellites are 0.059 m, 0.178 m, and 0.186 m, respectively. These results confirm that BDS real-time orbits are more accurate than GPS. This improved accuracy is due to the additional observations from the inter-satellite link (ISL) terminals on the BDS-3 satellites, which enhance orbit determination, particularly in the along-track direction [34].

The accuracy of satellite clocks is evaluated using the double-difference method [35]. This method involves first calculating the single difference between the real-time clock and final clock. The mean of these differences at each epoch is used as the reference clock bias. Double differences are then computed by subtracting this mean from the clock error of each satellite. Since the PPP-B2b products are estimated based on regional stations, interruptions can cause discontinuities in the clock datum. To address this issue, the quality of satellite clocks is evaluated using the average standard deviation (STD) and RMS on an hourly basis, as presented in Figure 3 and Figure 4 for GPS and BDS, respectively. Overall, the averaged STD and RMS for the GPS are 0.33 ns and 2.76 ns, respectively, which are significantly higher than the corresponding values of 0.13 ns and 1.82 ns for the BDS. Several GPS satellites exhibit clock RMS values exceeding 6 ns, which degrades the precision of the pseudorange observations and increases the convergence time of PPP. However, this does not impact the accuracy after convergence, as the systematic bias is absorbed into the observation residuals and ambiguities during PPP [36].

Figure 5 illustrates the time series of SISRE for the GPS and BDS-3 satellites. The SISRE of BDS-3 outperforms that of the GPS, which is consistent with the orbit and clock accuracy assessment presented in the previous analysis. The GPS SISRE exhibits significant errors and a substantial discrepancy on DOY 087, which can be easily identified through the positioning outlier screening process in PPP due to its relatively large magnitude. Moreover, no notable discrepancies are observed among different satellites, differing from the characteristics seen in the GPS orbit evaluation results. This can be attributed to the use of consistent orbit and clock products, where radial errors are effectively compensated [37].

3.2. Experiments and Validation

Figure 6 illustrates the distribution of stations used for data processing in this study. Data from eight MGEX stations for days of year (DOY) 075 to 092 in 2024 were selected, with a data sampling rate of 5 s. The observation data were processed using the in-house developed software, NavEngine V1.1. Both GPS and BDS measurements were processed using PPP-B2b real-time orbits, clocks, and code biases, as well as the rapid products from Wuhan University (WHR). All data were processed using a dual-frequency combination and an ionosphere-free model. The receiver coordinates in the forward Kalman filter [38] were initialized using the position obtained from single-point positioning, with a variance of 30 m. Afterwards, the coordinates were estimated as white noise at 30 m for kinematic mode and as constant in the static mode. The backward smoothing method [39] utilizes the estimated parameters and variance from the forward filter as constraints, thereby enhancing the accuracy of the estimated parameters during the convergence process. Details of the processing strategies are outlined in

Table 1. Processing strategies of PPP using the PPP-B2b and WHR products.

Items	Strategies
Observation	GPS: L1/L2; BDS: B1I/B3I
Elevation mask	7°
Weight for observations	Elevation-dependent weighting
Noise for observations	0.003 m for phase and 0.6 m for code
Satellite orbit/clock	PPP-B2b and WHR
Satellite antenna phase center	Using igs20.atx for WHR PPP
Filter method	Forward Kalman and backward smoothing
Tides correction	IERS 2010 [40]
Troposphere	Zenith wet delay is estimated as a random walk
Ionosphere	Ionosphere-free combination
Ambiguity	Estimated as constant with float solution
Receiver coordinate	Constant in static processing and white noise in the kinematic processing
Receiver clocks	Estimated as white noise

.

3.2.1. Static PPP Processing

Static PPP solutions were calculated in 1 h intervals. A total of 8 stations were used over 15 days, resulting in 24 periods per day, which corresponds to 2880 h of data. Figure 7 illustrates the convergence of positioning errors over time in the horizontal direction. The green dots represent the original positioning error sequence, while the red, blue, and purple curves represent the 50%, 68%, and 95% percentile positioning error at corresponding times. The three gray lines indicate horizontal positioning accuracies of 0.5 m, 0.2 m, and 0.1 m, respectively. A comparison of the positioning errors during initialization shows that the errors for “WHR” products are around 2 m, while the initial positioning error for PPP-B2b is greater than 4 m. After initialization, the overall positioning errors for “WHR” products are significantly smaller than those for PPP-B2b. At some stations, the errors exceed 1 m even after 1 h of convergence, primarily due to differences in the availability of GPS and BDS corrections across the PPP-B2b service region.

For further quantitative analysis,

Table 2. The average convergence time required for horizontal positioning errors to reach specified thresholds at different proportions.

Product Type	Error Threshold (m)	Time for 50% Percentile (min)	Time for 68% Percentile (min)	Time for 95% Percentile (min)
PPP-B2B	0.5	4.5	7.3	19.0
	0.2	12.7	18.9	60.0
	0.1	26.4	55.5	60.0
WHR	0.5	0.4	1.3	4.2
	0.2	3.6	5.9	14.3
	0.1	8.6	13.0	34.5

compares the time required for different proportions of positioning errors to reach a desired accuracy threshold of 50 cm, 20 cm, and 10 cm. At each sampling interval, the positioning errors from all processed stations are sorted. The times at which the proportions of errors within the 50%, 68%, and 95% confidence intervals first meet these thresholds are recorded as the convergence times. The analysis reveals that for static positioning with PPP-B2b, 4.5 min, 12.7 min, and 26.4 min are required for 50% of positioning errors to converge to 0.5 m, 0.2 m, and 0.1 m, respectively. The time required for high-accuracy convergence to 0.1 m increases significantly compared to 0.2 m, primarily because achieving high-accuracy positioning requires more precise atmospheric delay information, which takes time to converge. To meet 68% of positioning errors within 0.5 m, 0.2 m, and 0.1 m, it takes 7.3 min, 18.9 min, and 55.5 min, respectively, with significant increases in time. Using “WHR” rapid orbit products, 50% of the positioning errors converge to an accuracy better than 0.1 m within 8.6 min, and it takes 13.0 min for 68% of positioning errors to reach this accuracy.

Due to the fact that PPP-B2b corrections primarily cover the Asia–Pacific region, different stations experience variations in the number of visible satellites and DOP values, which in turn affect the positioning accuracy. To further analyze this impact, we examined the station-specific convergence time needed for PPP to reach 0.5 m and 0.1 m accuracy, as presented in Figure 8. The average convergence time needed to achieve a 0.5 m positioning accuracy is 6 min for PPP-B2b corrections and 2 min for WHR products, with the average difference between stations being within 2 min. To achieve a 0.1 m positioning accuracy, the average convergence time required is 24.8 min for PPP-B2b corrections and 16.6 min for WHR products, with the average difference between stations being within 5 min and 8 min, respectively. However, it is important to note that, in some cases, static PPP fails to converge to 0.1 m even after the mean convergence time or beyond 50 to 60 min, as indicated by the station-specific statistics in

Table 3. Proportion of convergence time exceeding the average, 50 min and 60 min for each station.

Station	BIK0	GAMG	JFNG	POL2	ULAB	URUM	WUH2
>average	39.23%	36.66%	39.02%	38.46%	37.18%	44.48%	42.32%
>50 min	27.33%	9.65%	11.74%	19.55%	12.50%	27.05%	18.26%
>60 min	23.79%	7.07%	7.57%	16.98%	10.26%	19.57%	12.45%

. This is primarily attributed to the observation geometry and the limited number of available GPS and BDS corrections from PPP-B2b.

Figure 9 further depicts the station-specific hourly three-dimensional positioning RMS and its corresponding average number of satellites. The average RMS is approximately 0.1 m with hourly data at different stations, with the number of satellites ranging from 12.5 to 14.5. Stations with a lower average number of satellites typically exhibit longer convergence times. The statistical analysis reveals that 95% of the positions fall within 0.065 m, 0.143 m, and 0.148 m for the hourly positioning RMS in the N/E/U directions, respectively.

3.2.2. Simulated Kinematic PPP Processing

Based on selected data, static stations are estimated using the simulated kinematic mode, where receiver coordinates are re-initialized each epoch, and ambiguities are continuously estimated unless a cycle slip is detected. Figure 10 shows the convergence times needed for the hourly dynamic PPP to achieve accuracies of 0.5 m and 0.1 m using PPP-B2b and WHR products. The average times needed to reach a 0.5 m accuracy are 9.9 min with PPP-B2b and 2.1 min with WHR, while for a 0.1 m accuracy, the times are 25.9 min and 13.2 min, respectively. Compared to static PPP, dynamic PPP using PPP-B2b products requires an additional 1–4 min to achieve the same level of accuracy.

To further characterize the station-specific positioning accuracy, Figure 11 shows the relationship between the RMS of 3D positioning errors after convergence to 0.1 m accuracy and the number of visible satellites. The blue data points represent the RMS for each hourly session, plotted against an average number of visible satellites, while the red line shows the RMS average grouped by the integer number of visible satellites. The results indicate that the RMS of 3D positioning errors varies significantly after convergence to 0.1 m accuracy, ranging from 0.05 m to 0.3 m. This can be attributed to factors such as the number of visible satellites, the orbit accuracy of different satellites, and the data quality of different stations. Some stations experience periods with fewer than 10 usable satellites in the combined GPS + BDS system. As the average number of visible satellites exceeds 10, the average 3D positioning error gradually decreases.

To eliminate the inaccuracies during the convergence period and fully utilize observations from dynamic scenarios over a short time span, backward filtering was conducted based on forward Kalman filtering. The average positioning RMS values in the N/E/U directions using PPP-B2b corrections are 0.024 m, 0.046 m, and 0.053 m, respectively, while using WHR products, they are reduced to 0.009 m, 0.021 m, and 0.027 m. Figure 12 shows the distribution of positioning error in the N/E/U directions using PPP-B2b products (left) and WHR products (right) for each station. Specifically, the proportions of hourly PPP-B2b solutions with positioning errors less than 0.1 m in the N/E/U directions are 94.03%, 73.00%, and 60.79%, respectively, while those with errors less than 0.2 m are 98.53%, 91.00%, and 88.89%, respectively. When using WHR products, the proportions of positioning errors less than 0.1 m in the N/E/U directions are 100.00%, 97.61%, and 96.88%, respectively; the proportions with positioning errors less than 0.2 m are 100.00%, 100.00%, and 99.82%, respectively. The comparison reveals that the larger positioning errors obtained with PPP-B2b products are mainly due to the accuracy of the PPP-B2b products rather than the quality of the station observation.

Overall, the results indicate that the positioning accuracy in the north direction is better than that in the east direction, with a higher proportion of smaller errors. However, some stations still exhibit hourly positioning errors exceeding 0.4 m in the E and U directions. Figure 13 further illustrates the positioning time series of station BIK0 on DOY 075 in the north, east, and up directions, along with the variations in the number of satellites (Nsat) and Geometric Dilution of Precision (GDOP) values. It confirms that the backward smoothing PPP achieves better consistency in the north direction, while the east and up directions exhibit greater variations across different sessions. Notably, the number of satellites decreases to fewer than 10 from 12:00 to 13:00, during which the positioning errors of the forward PPP increase to approximately 0.6 m in all three components. Although the backward smoothing PPP improves the positioning accuracy during this session, it still performs worse than in other sessions. As only the dual-system GPS and BDS-3 can be used with PPP-B2b, despite the availability of multi-GNSS observations from the receiver, a high positioning accuracy cannot be maintained when the number of satellites is insufficient. A reasonable solution is to partially combine the PPP-B2b with other correction sources, such as those from the Galileo HAS service [41], using a proper weighting model or an equivalent transformation model.

3.2.3. Vehicle-Based Dynamic PPP

To further validate the accuracy of dynamic PPP, we conducted a vehicle dynamic experiment. The vehicle’s trajectory, as shown in the left panel of Figure 14, begins at the point marked “1” and ends at the point marked “2”. Data were collected using a multi-constellation, multi-frequency GNSS positioning module (UM980) manufactured by Unicore, along with a corresponding multi-frequency GNSS antenna mounted on the roof of the vehicle. The experiment took place on the school campus from 08:07:44 to 08:31:00 (GPS time) on 15 July 2024, with a sampling interval of 0.5 s. The corresponding time series of positioning errors in the N/E/U directions, which are converted into a local coordinate system based on the coordinates at the final epoch, are shown in the right panel of Figure 14. The vehicle remained stationary for approximately five minutes before initiating dynamic motion, followed by another stationary period from 08:26:21 to 08:28:58, with brief movement thereafter. Due to the 40 km distance between the base station and the rover, the RTK ambiguity resolution rate was only 93.2%, as indicated by the float ambiguities represented by the yellow points. The positioning errors observed during stationary periods show that the RTK positioning accuracy achieved was better than 0.05 m, which is sufficient for assessing the accuracy of the PPP results.

First, simulated kinematic PPP using observations from the reference station was used to evaluate the accuracy of PPP using two separate software tools, namely PRIDE PPP-AR [42] and NavEngine. PRIDE utilizes a total least squares algorithm, while NavEngine uses forward Kalman filtering combined with backward smoothing. Both tools used rapid products provided by Wuhan University. Figure 15 compares the time series of positioning errors, revealing that both software tools produced stable error sequences. However, systematic biases of varying magnitudes were observed across the three components.

Table 4. Averaged RMS and STD using different processing modes.

	RMS (m)			STD (m)
	North	East	Up	North	East	Up
Pride_WHR	0.051	0.111	0.014	0.004	0.004	0.011
NavEngine_WHR	0.061	0.086	0.039	0.005	0.008	0.010
NavEngine_B2b	0.055	0.076	0.049	0.006	0.024	0.045

summarizes the statistical results of positioning errors. The analysis shows that the accuracy, measured by the STD, is similar across both software, achieving millimeter-level precision, while the vertical precision is slightly lower, with errors around 1 cm. The RMS analysis reveals systematic errors of 0.1 m in the east direction and 0.06 m in the north direction. This discrepancy is primarily due to the short observation duration of the reference station’s data, which lasted only two hours, leading to the incomplete convergence of the dynamic solution’s positioning accuracy.

Furthermore, a comparison of the positioning errors observed using NavEngine with WHR and PPP-B2b products, as shown in Figure 16, reveals that the systematic biases exhibit good consistency, with RMS differences of less than 0.01 m, which is reasonable within the kinematic positioning noise of PPP. However, the STD accuracy of the PPP-B2b solution is larger than that of WHR, particularly in the east and up directions. Overall, the positioning errors obtained using PPP-B2b products at the reference station can achieve centimeter-level accuracy when utilizing the backward smoothing method.

Following the simulated kinematic validation, the positioning accuracy of the vehicle dynamic station was analyzed by comparing it against the RTK solution. Figure 17 presents the positioning errors of the forward PPP using WHR and PPP-B2b products in NavEngine software. The results indicate that PPP achieves a horizontal accuracy of 0.5 m within 4 min of convergence. Afterward, the horizontal positioning errors continue to decrease, with WHR and PPP-B2b products converging to horizontal accuracies of 0.05 m and 0.23 m, respectively. The vertical positioning error requires about 12 min to converge to a 0.5 m accuracy, and a constant bias persists even after the convergence.

In comparison, Figure 18 shows the positioning error time series using the WHR and PPP-B2b products with backward smoothing. Although the trends are similar due to variations in the reference RTK solution, significant biases exist between the products. The WHR products exhibit better accuracy in the horizontal direction, with average RMS values of 0.062 m and 0.152 m for the north and east directions, respectively, compared to 0.139 m and 0.163 m for the PPP-B2b products. Conversely, PPP-B2b products demonstrate superior accuracy in the vertical direction, achieving an RMS value of 0.137 m compared to 0.354 m for the WHR products, respectively. This discrepancy arises because the solution using PPP-B2b products continues to converge vertically during the forward filter, whereas the WHR products reach a stable state earlier. Although backward filtering can enhance the PPP accuracy during the convergence period, the overall accuracy largely depends on the parameter convergence precision achieved through the forward filtering. The convergence period is partially influenced by the quality control strategy implemented in the software, as well as the accuracy of the products.

4. Conclusions

BDS provides precise orbit, clock, and bias corrections through PPP-B2b signals, enabling PPP in the Asia–Pacific region without the need for internet connectivity. To achieve a high positioning accuracy with short observation periods, the accuracy of static PPP using PPP-B2b was evaluated, and a backward smoothing method was proposed to improve the accuracy during the initialization period. The performance was validated using the static and vehicle observations. The following conclusions can be drawn:

(1)

The accuracy of the orbits and clocks recovered from the PPP-B2b is assessed, indicating that BDS products outperform GPS. The improvement is largely due to the BDS inter-satellite link ranging, which enhances orbit performance.

(2)

The static hourly PPP results show that the average convergence times required to achieve horizontal accuracies better than 0.5 m and 0.1 m are approximately 4.5 min and 25 min, respectively. However, a proportion of the sessions, ranging from 7.07% to 23.79%, fail to converge to 0.1 m due to the limited availability of GPS and BDS satellites. For sessions that do converge, the average 3D RMS is 0.1 m.

(3)

The simulated kinematic PPP results indicate that the average times required to achieve horizontal accuracies of 0.5 m and 0.1 m are approximately 9.9 min and 25.9 min, respectively. The average positioning RMS values in the N/E/U directions using backward smoothing PPP are 0.024 m, 0.046 m, and 0.053 m. Similar to the case of static PPP, convergence to 0.1 m accuracy cannot always be achieved due to variations in the observation quality at different stations. Using the backward smoothing method, the proportions of positioning errors less than 0.1 m in the north, east, and up directions are 94.03%, 73.00%, and 60.79%, respectively.

(4)

The vehicle experiments show that forward PPP can achieve a horizontal accuracy better than 0.5 m within 4 min, with steady improvement over longer observation periods. However, large positioning errors of 1.5 m in the east direction and 3.0 m in the vertical direction are observed during the convergence period. Using the backward smoothing method, an RMS of 0.139 m, 0.163 m, and 0.137 m is achieved in the north, east, and up directions, respectively.

Overall, the results demonstrate that using PPP-B2b can achieve decimeter-level positioning with different observation durations, without relying on an external base station. The backward smoothing method further enhances accuracy during the convergence period. However, the accuracy observed in the vehicle experiments is lower than that of the simulated real-time results due to the effects of the observation environment. For applications such as Unmanned Aerial Vehicles (UAVs) and marine navigation, where the observation environment typically offers open-sky conditions, the results are expected to be comparable. Furthermore, at the boundary of the PPP-B2b service region, the number of available satellites decreases, which may degrade the positioning accuracy. The integration of PPP-B2b with the High-Accuracy Service (HAS) could be further explored to improve global positioning accuracy.