Differences in Time Comparison and Positioning of BDS-3 PPP-B2b Signal Broadcast Through GEO

Abstract

The BeiDou-3 Navigation Satellite System (BDS-3) precise point positioning (PPP) service through the B2b signal (PPP-B2b) leverages precise correction data disseminated by satellites to eliminate or mitigate key error sources, including satellite orbit errors, clock biases, and ionospheric delays, thereby enabling high-precision timing and positioning. This paper investigates the disparities in time comparison and positioning capabilities associated with the PPP-B2b signals transmitted by the BDS-3 Geostationary Earth Orbit (GEO) satellites (C59 and C61). Three stations in the Asia–Pacific region were selected to establish two time comparison links. The study evaluated the time transfer accuracy of PPP-B2b signals by analyzing orbit and clock corrections from BDS-3 GEO satellites C59 and C61. Using multi-GNSS final products (GBM post-ephemeris) as a reference, the performance of PPP-B2b-based time comparison was assessed. The results indicate that while both satellites achieve comparable time transfer accuracy, C59 demonstrates superior stability and availability compared to C61. Additionally, five stations from the International GNSS Service (IGS) and the International GNSS Monitoring and Assessment System (iGMAS) were selected to assess the positioning accuracy of PPP-B2b corrections transmitted by BDS-3 GEO satellites C59 and C61. Using IGS/iGMAS weekly solution positioning results as a reference, the analysis demonstrates that PPP-B2b enables centimeter-level static positioning and decimeter-level simulated kinematic positioning. Furthermore, C59 achieves higher positioning accuracy than C61.

1. Introduction

BDS-3 was officially commissioned in 2020, with its GEO satellites broadcasting PPP-B2b signals in real time [1,2,3,4,5,6,7,8]. BDS-3 utilizes satellite information contained in PPP-B2b messages to correct satellite orbits and clock biases, thus achieving more accurate precise point positioning and timing. Initially, the C59 and C60 GEO satellites broadcast identical PPP-B2b correction messages. However, following the introduction of the C61 satellite in late 2023, the message content was updated and now differs from that of the earlier satellites. Previous research has primarily focused on PPP-B2b corrections from C59 and C60. Xu et al. evaluated the orbit accuracy, reporting radial, tangential, and normal errors of 6.8 cm, 33.4 cm, and 36.6 cm, respectively, with clock deviation precision achieving a standard deviation (STD) of 0.2 ns [9,10,11,12,13,14,15,16]. Tao et al. verified that PPP-B2b signals match 97.5% and 91.5% of BDS-3 and GPS, respectively, in the Asia–Pacific region [17]. Sun et al. analyzed the performance of BDS-3 in terms of correction availability, clock and orbit quality and positioning accuracy with PPP-B2b messages [18]. In Ren et al., real-time precise ephemerides are compared against Multi-GNSS Experiment (MGEX) final products and the positioning performance of real-time PPP is evaluated with MGEX/iGMAS stations [19]. Zhang et al. proposed a stable PPP-B2b information matching strategy with broadcast ephemeris [20]. Nie et al. used IGS stations in the Asia–Pacific region to verify the centimeter-level positioning accuracy of PPP-B2b in both static and dynamic scenarios in three directions [21]. In Dai et al., using the observations from six stations spanning 196 days, the long-term positioning performance of BDS-3/GPS real-time PPP with PPP-B2b precise products is evaluated [22]. The average positioning accuracy can be 4.6, 0.9, and 2.5 cm (with an STD of 1.3, 0.7, and 1.7 cm), and 6.4, 3.3, and 9.5 cm (with a STD of 2.3, 1.5, and 3.1 cm) in east, north, and up directions in static and kinematic modes, respectively.

This study presents a comparative analysis of PPP-B2b correction parameters broadcast by BDS-3 GEO satellites C59 and C61, evaluating both their time transfer accuracy and positioning performance. The investigation encompasses static positioning and simulated kinematic scenarios to comprehensively assess system capabilities.

2. Correction Analysis

2.1. PPP-B2b Correction Accuracy Analysis

The clock corrections and precise orbit from PPP-B2b messages were retrieved and analyzed on 22 March 2024. Using GBM final products as reference, we evaluated the accuracy of PPP-B2b-corrected satellite orbits and clock errors. Additionally, the code bias products in PPP-B2b messages were validated against reference solutions from the Chinese Academy of Sciences. While previous research has primarily examined PPP-B2b signals from C59 and C60 satellites (which broadcast identical information), this study presents the first comprehensive comparison of PPP-B2b messages with differing content, designated as C59_PPP-B2b and C61_PPP-B2b, from the newer C61 satellite configuration.

2.2. PPP-B2b Orbit Accuracy Analysis

2.2.1. Satellite Orbit Error Evaluation Methods

The GBM precision orbit points to the center of mass of the satellite, whereas the PPP-B2b correction values point to the phase center of the satellite. Before evaluating the orbit accuracy, it is necessary to first correct the satellite’s center of mass to its phase center position. For this correction, the IGS-published igs14.atx file is required. Formula (1) is the correction method:

$$\delta X=X_{GBM}-A\cdot \delta X_{PCO}$$

(1)

where $\delta X$ is the satellite orbit error vector; $X_{GBM}$ is the reference orbit vector obtained for GBM products; $A$ is the satellite attitude matrix; $\delta X_{PCO}$ is the correct vector for the satellite PCO.

After synchronizing the satellite orbit to the phase center of the satellite, the errors between the PPP-B2b corrected position and the reference value in the radial, tangential, and normal directions can be calculated. Formula (2) is the calculation method for errors in three directions:

$${\left.\delta O=\left[\begin{array}{c}{e}_{radial} e_{along} e_{cross}\end{array}\right.\right]}^T\cdot \delta X_{GBM-PPP-B2b}$$

(2)

where $\delta O$ is the radial, tangential, and normal error; $e$ is the direction unit vector; $\delta X_{GBM-PPP-B2b}$ is the difference between the GBM satellite orbit and the PPP-B2b satellite orbit.

2.2.2. Satellite Orbit Accuracy Analysis

Using the GBM precision orbit as a reference, five BDS-3 satellites and five GPS satellites were employed to evaluate the orbit accuracy of PPP-B2b recovery satellites. Figure 1, Figure 2 and Figure 3 illustrate the radial, tangential, and normal errors of the CNAV1 broadcast ephemeris satellite orbit and the PPP-B2b corrected satellite orbit after C59/C61 broadcast, respectively. Figure 4, Figure 5 and Figure 6 depict the uncorrected LNAV broadcast ephemeris and the results after two types of corrections.

Figure 1 and Figure 4 depict the orbit errors without PPP-B2b correction. It is evident from the figures that the orbit error sequence experiences discontinuities every hour, which is attributed to the hourly update of the broadcast ephemeris. Figure 1, Figure 2, Figure 4 and Figure 5 exhibit the orbit error curves after PPP-B2b correction, which are smoother compared to the uncorrected error curves. This suggests that both PPP-B2b broadcasted by C59 and C61 have corrected the hourly discontinuity of the orbit in the CNAV1 broadcast ephemeris.

The variation amplitudes of the orbit error sequence of BDS-3 satellites in radial, tangential, and normal components are within 0.3, 0.4, and 0.4 m, respectively. The orbital error sequence of GPS satellites varies within 0.4, 0.6, and 0.8 m in the three directions, respectively. Regarding the magnitude of changes before and after correction, the PPP-B2b broadcast by the two GEO satellites did not significantly correct the orbital errors, indicating that PPP-B2b primarily corrects the hourly discontinuity of satellite orbital errors.

For the BDS-3 satellites, regarding the smoothness of the corrected orbits of the two GEO satellites, a notable jump in orbital error occurs at 2h in Figure 3, whereas this phenomenon does not occur at the same position in Figure 2. Hence, the correction effect of C59 is marginally superior to that of C61.

For GPS satellites, both types of correction information address the hourly discontinuity in their orbits. However, as evident from the figure, C59’s correction proves to be marginally superior to C61’s. While C61’s correction still leaves a slight orbit discontinuity in the error, C59’s correction effectively eliminates this issue. Nevertheless, when compared to BDS-3 satellites, the error variation after PPP-B2b correction is notably smaller, thereby enhancing the accuracy of GPS satellite orbits.

To further assess the satellite orbit accuracy of PPP-B2b,

Table 1. Average RMS of orbit errors.

Items	Radial (cm)	Tangential (cm)	Normal (cm)
BDS-3 CNAV1	8.34	23.05	17.46
BDS-3 C59_PPP-B2b	7.83	20.46	17.16
BDS-3 C61_PPP-B2b	8.14	19.88	18.46
GPS LNAV	15.46	22.49	19.05
GPS C59_PPP-B2b	6.83	9.73	8.94
GPS C61_PPP-B2b	9.86	11.57	15.49

presents the average root mean square error (RMSE) values for both the broadcast ephemeris orbit and the PPP-B2b real-time orbit over a seven-day period from 22 March to 28 March 2024. From the statistical values presented in the table, it can be concluded that for BDS-3, both the PPP-B2b correction values broadcasted by C59 and C61 satellites have improved the accuracy of the broadcast ephemeris orbit to a certain extent. Additionally, they have corrected the discontinuity of the orbit in the broadcast ephemeris, making the error smoother. However, for GPS satellites, the correction values of C59 and C61 have enhanced the accuracy in radial, normal, and tangential directions, with C59 exhibiting higher correction accuracy.

2.3. PPP-B2b Clock Error Accuracy Analysis

2.3.1. Evaluation Method for Satellite Clock Error

Since the system time used for GBM precision clock error correction is GPST, while the system time used for PPP-B2b is BDT, it is necessary to correct the 14 s time difference between the two systems before evaluating the accuracy of satellite clock error. For the BDS-3 system, the linear combination of B1I and B3I utilized by GBM precision clock bias serves as the frequency reference [10,11], whereas PPP-B2b relies on the B3I frequency as the reference, necessitating code deviation calibration for various observed quantities [23,24,25]. For GPS systems, since PPP-B2b does not broadcast GPS code deviation information, there is no requirement to unify the frequency reference. Formulas (3) and (4) detail the specific correction methods:

For the BDS-3 system:

$$\delta t=d{t}_{GBM}-d{t}_{PPP-B2b}-{\displaystyle \frac{f_{B1I}^2}{f_{B1I}^2-f_{B3I}^2}}\mathrm{DC}{B}_{B1I}$$

(3)

For GPS systems:

$$\delta t=d{t}_{GBM}-d{t}_{PPP-B2b}$$

(4)

where $\delta t$ is the error of satellite clock bias; $dt$ is the satellite clock bias; $f$ is the signal frequency; ${\mathrm{DC}B}_{B1I}$ is the code deviation between B1I and B3I signals.

2.3.2. Analysis of Satellite Clock Error Accuracy

Using GBM precision clock bias as a reference, we evaluated the accuracy of PPP-B2b recovered satellite clock bias for five BDS-3 satellites and five GPS satellites. Figure 7 depicts the clock error curve of the BDS-3 satellite before and after correction on 22 March 2024. For the BDS-3 satellite, the clock bias error corrected by PPP-B2b broadcasted by C59 and C61 fluctuates within 1 ns, whereas the variation range of satellite clock bias in the CNAV1 broadcast ephemeris is within 3 ns. Simultaneously, the hourly discontinuity in the broadcast ephemeris is also eliminated by PPP-B2b. From the curve in the figure, it is evident that the PPP-B2b broadcasted by C59 and C61 exhibit similar correction effects on clock bias.

The situation of GPS satellites is quite unique. As can be seen from Figure 8, the variation in clock bias error is greater after PPP-B2b correction. However, for G13 and G19 satellites, the discontinuity in clock bias is eliminated, which is due to the fact that the GPS satellite clock reference was not maintained during the generation of PPP-B2b products.

However, individual clock biases still persist within the clock bias sequence. To effectively assess the precision of the satellite clock biases corrected by PPP-B2b, it is imperative to rectify individual clock offset deviations by subtracting the average clock offset [26]. For instance, the real-time PPP-B2b clock deviation of the C23 satellite attains 10 ns. Research indicates that the system errors specific to these satellites are associated with pseudorange observations. When carrier phase observations are utilized for positioning, the system errors designated by the satellite can be absorbed by ambiguity.

To visually illustrate the difference in satellite clock bias before and after PPP-B2b correction,

Table 2. Average RMS of clock difference errors.

Items	STD (ns)
BDS-3 CNAV1	0.64
BDS-3 C59_PPP-B2b	0.22
BDS-3 C61_PPP-B2b	0.24
GPS LNAV	0.31
GPS C59_PPP-B2b	0.61

presents the clock bias STD obtained from broadcast messages and PPP-B2b for both BDS-3 and GPS systems over a span of 7 days from 22 March to 28 March 2024. The table reveals that the accuracy of satellite clock bias for Beidou-3 satellites has notably improved after PPP-B2b correction, with the C59 satellite exhibiting a superior correction effect. However, PPP-B2b did not enhance the clock bias accuracy of GPS satellites, and the correction effect of PPP-B2b broadcasted by the C61 satellite was inferior to that of the C59 satellite.

2.4. PPP-B2b Code Deviation Accuracy

After comparing, it was found that the inter-symbol deviation correction values in PPP-B2b signals broadcasted by the C59 and C61 GEO satellites are identical. Figure 9 and Figure 10 compare the satellite code deviations published by the Chinese Academy of Sciences (CAS) with the code deviations broadcasted in real time by PPP-B2b from 22 March to 28 March 2024. The comparison frequencies are B1C(P) [27] and B3I, versus B1I and B3I. Since PPP-B2b currently only broadcasts code deviations for BDS-3 satellites, this section focuses solely on the BDS-3 satellites.

From the two figures, it is evident that the code deviations of PPP-B2b and CAS exhibit consistency, with minimal systematic deviation. Regarding the deviation between B1C(P) and B3I, the STD of the difference between PPP-B2b and CAS is 0.535 ns. For the deviation between B1I and B3I, the STD is 0.763 ns, which is comparable to the satellite clock error. Since precise single-point positioning heavily relies on carrier phase observations for positioning, systematic errors can be absorbed by ambiguity [28].

3. PPP-B2b Time Comparison

3.1. PPP-B2b Time Comparison Processing Strategy

A time comparison study was conducted on the BDS-3 B1I/B3I + GPS L1/L2 ionosphere-free combination using BDS-3 PPP-B2b. The Sinan K803 receiver was employed to capture and store the original binary messages of PPP-B2b transmitted by GEO satellites between 22 March and 28 March 2024. These binary messages were then decoded based on the “Precision Single Point Positioning Service Signal PPP-B2b” document. The study utilizes receiver observation data from the XIA6, SE22, and JLJI stations at the National Time Service Center (NTSC) of the Chinese Academy of Sciences (CAS) [29].

Table 3. Station information of PPP-B2b time transfer.

Station Name	Station Category	Receiver Model	Antenna Model	External Clock
XIA6	NTSC	Sept Polarx5TR	SEPCHOKE_B3E6	UTC(NTSC)
SE22	NTSC	Sept Polarx5TR	SEPCHOKE_B3E6	UTC(NTSC)
JLJI	NTSC	TRIMBLE ALLOY	SEPCHOKE_B3E6	H-Maser

details the GNSS receivers involved in the PPP-B2b time comparison. XIA6 and SE22 are GNSS receiver stations equipped with a shared atomic clock and antenna within the NTSC laboratory. Both XIA6 and SE22 are connected to the local UTC (NTSC), while JLJI is externally connected to a local hydrogen clock.

The experimental design initially employs the BDS-3 + GPS dual system for zero-baseline time comparison, aiming to validate the accuracy of the PPP-B2b time comparison method introduced in the preceding text. Subsequently, it utilizes the GBM post-processed precise product as a benchmark for conducting experimental analysis on long baseline time comparison.

The processing strategy for PPP-B2b is outlined in

Table 4. Error terms and processing models of PPP-B2b time transfer.

Error Terms	Processing Model
Precise ephemeris and clock bias	PPP-B2b Precision Orbit and Clock Error
Ionospheric delay	Dual-frequency ionosphere-free combination
Tropospheric time delay	ZTD estimation
Observations	B1I/B3I, L1/L2 code pseudorange and carrier phase
Elevation mask	15°
Solid tide correction	IERS 2010
Antenna phase center	igs14.atx
Parameter estimation	Extended Kalman filter
Receiver position model	Static model
Receiver clock bias model	White noise
Observation value sampling interval	30 s

. The precise satellite position and clock bias are achieved by correcting the satellite position and clock bias, which are calculated from navigation messages, using PPP-B2b products. The ionospheric delay is mitigated through a dual-frequency ionosphere-free combination, eliminating ionospheric effects. Additionally, the extended Kalman filter is employed for parameter estimation. This recursive filtering algorithm estimates the system state and updates it based on measurement data, correcting errors accordingly. Furthermore, igs14.atx is utilized to eliminate the deviation between the antenna phase center and the centroid. IERS 2010 parameters were utilized to ensure consistency between satellite orbit, clock bias, and receiver coordinates within a unified reference framework.

3.2. PPP-B2b Time Comparison Analysis

3.2.1. Zero-Baseline PPP-B2b Time Comparison

The zero-baseline common clock experiment utilizes observation data from two external UTC (NTSC) receivers at NTSC CAS. Continuous daily zero-baseline common clock time comparisons can reflect the uncertainty of receiver noise and product time comparisons. Figure 11 illustrates the PPP-B2b time comparison results for zero-baseline clocks XIA6 and SE22, utilizing BDS-3 + GPS. As evident from Figure 11, the time comparison noise for both C59_PPP-B2b and C61_PPP-B2b fluctuates within 0.5 ns. Specifically, the STD for the time comparison of C59_PPP-B2b is 0.071 ns, while that of C61_PPP-B2b is 0.094 ns. The presence of automatic internal delay calibration in XIA6, coupled with the absence of such calibration in SE22, leads to a constant deviation of approximately 26 ns in the PPP-B2b dual-frequency ionosphere-free combined zero-baseline clock time comparison results between the two receivers. Furthermore, Figure 11 reveals that C61_PPP-B2b exhibits poorer availability compared to C59_PPP-B2b during the final time period, which subsequently results in slightly inferior performance in its zero-baseline time transfer STD.

Figure 12 illustrates the corrected Allan variances for C59_PPP-B2b and C61_PPP-B2b during zero-baseline time transfer, demonstrating a high degree of consistency between the two. Overall, C59_PPP-B2b performs slightly better than C61_PPP-B2b. At 1000 s, the corrected Allan variances for the zero-baseline common clock PPP-B2b time comparison of C59_PPP-B2b and C61_PPP-B2b were both better than 1 × 10⁻¹³. For 100,000 s, the corrected Allan variances reached 3.56 × 10⁻¹⁶ and 3.67 × 10⁻¹⁶, respectively. This verifies the reliability of PPP-B2b time comparison, paving the way for PPP-B2b long-baseline time comparison.

3.2.2. Long-Baseline PPP-B2b Time Comparison

Figure 13 illustrates the time comparison results of the BDS-3 + GPS dual-system C59_PPP-B2b long baseline between the XIA6 and JLJI stations from 22 March to 28 March 2024. The reference depicted in the figure is derived from the precise single-point positioning time comparison using GBM’s post-production BDS-3 + GPS dual system. Evidently from Figure 13, there is minimal systematic deviation between the PPP-B2b dual-system dual-frequency ionosphere-free combination and the GBM dual-system ionosphere-free combination. Subtracting these results allows for a clearer observation of the PPP-B2b time comparison residuals. Figure 14 presents the obtained time difference curve, where the time difference fluctuations remain below 2 ns, and the STD of the time comparison and reference value difference stands at 0.544 ns.

The two curves in Figure 15 depict the corrected Allan variance of the receiver clock bias for XIA6 and JLJI, obtained through the BDS-3 + GPS PPP-B2b dual-frequency ionosphere-free combination and using GBM products for BDS-3 + GPS. The graph reveals that the stability of GBM time transfer surpasses that of C59_PPP-B2b, yet the stability trend of C59_PPP-B2b mirrors that of GBM. As time progresses, the disparity in stability between C59_PPP-B2b and GBM progressively narrows. At 10,000 s, the stability difference between the two is merely 2.1 × 10⁻¹⁵. The stability of C59_PPP-B2b in XIA6 and JLJI, when compared over 10,000 s, attains 3.87 × 10⁻¹⁵.

Figure 16 illustrates the time comparison results of BDS-3 + GPS C61_PPP-B2b long baseline between the XIA6 and JLJI stations from 22 March to 28 March 2024. The reference depicted in the figure is the time comparison result of precise single-point positioning obtained using the GBM post-processed product BDS-3 + GPS. As evident from Figure 16, both curves exhibit no systematic deviation. However, there is a period of unavailability for C61_PPP-B2b, indicating a certain gap in its availability compared to C59_PPP-B2b. Subtracting the two curves allows for a clearer observation of the residuals in the time comparison of C61_PPP-B2b. Figure 17 presents the obtained time difference curve, where the time difference fluctuations remain within 2 ns, and the STD of the time comparison difference is 0.511 ns.

The two curves in Figure 18 depict the corrected Allan variance of the receiver clock bias for XIA6 and JLJI, obtained through the BDS-3 + GPS PPP-B2b dual-frequency ionosphere-free combination and using GBM products for BDS-3 + GPS. The graph indicates that the overall stability of GBM time transfer remains superior to that of C61_PPP-B2b, with a relatively larger time difference in the short term. As time progresses, C61_PPP-B2b demonstrates a higher stability rate than GBM. At 10,000 s, the stability difference between the two curves is merely 0.2 × 10⁻¹⁵. Specifically, the stability of C61_PPP-B2b in XIA6 and JLJI reached 1.99 × 10⁻¹⁵ at a 10,000 s time ratio, surpassing the stability achieved by C59_PPP-B2 at the same duration.

3.2.3. Comparison of Time Transfer Links

The BDS-3 + GPS combination was utilized to investigate the accuracy of time comparison between C59_PPP-B2b and C61_PPP-B2b.

Table 5. PPP-B2b time transfer accuracy statistics.

Items	PPP-B2b	Time Comparison STD (ns)	Correct Allen’s Variance
XIA6-SE22	C59	0.071	3.56 × 10−16
	C61	0.094	3.67 × 10−16
XIA6-JLJI	C59	0.544	3.87 × 10−15
	C61	0.511	1.99 × 10−15

outlines two performance indicators, STD and modified Allan variance, pertaining to time comparison across zero and long baselines. These indicators reveal that the time comparison accuracy of C59_PPP-B2b and C61_PPP-B2b is comparable, yet C59_PPP-B2b exhibits superior stability and availability compared to C61_PPP-B2b.

4. PPP-B2b Positioning

4.1. PPP-B2b Positioning Processing Strategy

Five iGMAS/IGS observation stations in the Asia–Pacific region were selected to conduct PPP-B2b precise point positioning research based on BDS-3/GPS. The experiment employed the BDS-3 B1I/B3I ionosphere-free combination and the BDS-3 B1I/B3I + GPS L1/L2 ionosphere-free combination for PPP-B2b positioning studies of C59 and C61. Detailed information about each experimental station is presented in

Table 6. Station information for PPP-B2b positioning research.

Station Name	Station Category	Receiver Model	Antenna Model	External Clock
GAMG	IGS	Sept Polarx5TR	LEIAR25.R4 LEIT	Internal
GUA1	IGMAS	MGR_iGMAS	Geodetic-GNSS	Rubidium
JFNG	IGS	TRIMBLE ALLOY	TRM59800.00	Internal
SHA1	IGMAS	Unicore UB4B0	NOV750.R4 NOVS	Rubidium
USUD	IGS	Sept Polarx5	AOAD/M_T JPLA	H-Maser

.

The processing strategy for PPP-B2b is outlined in

Table 7. Error terms and processing models of PPP-B2b positioning.

Error Terms	Processing Model
Precise ephemeris and clock bias	PPP-B2b Precision Orbit and Clock Error
Ionospheric delay	Dual-frequency ionosphere-free combination
Tropospheric time delay	ZTD estimation
Observations	B1I/B3I, L1/L2 code pseudorange and carrier phase
Elevation mask	15°
Solid tide correction	IERS 2010
Antenna phase center	igs14.atx
Parameter estimation	Extended Kalman filter, least squares estimation
Receiver position model	Constant model
Receiver clock bias model	White noise
Observation value sampling interval	30 s

. Precise satellite positions and clock biases are achieved by correcting the satellite positions and clock biases calculated from navigation messages using PPP-B2b products. To eliminate ionospheric effects, the ionospheric delay employs a dual-frequency ionosphere-free combination. Additionally, extended Kalman filtering and least squares estimation are utilized for parameter estimation. This recursive filtering algorithm estimates the system state, updates it based on measurement data, and corrects errors. Static PPP-B2b positioning calculations were conducted on a daily basis using experimental data. Since precise single-point positioning methods require a certain convergence time, the solution results corresponding to this convergence time were excluded from the daily data used for positioning.

4.2. BDS-3 PPP-B2b Positioning Research

Figure 19 illustrates the average number of satellites in the static solution epoch at the stations. For these five stations, the average number of BDS satellites involved in PPP-B2b calculations per epoch is 7.85. Among the five stations, the USUD observation station has the fewest satellites, resulting in slightly poorer PPP-B2b positioning performance.

4.2.1. BDS-3 PPP-B2b Static Positioning

Taking the GAMG station on 22 March 2024 as an example, Figure 20 and Figure 21 demonstrate the static positioning results in the east/north/up (E/N/U) direction over time. Station coordinates for GAMG were computed utilizing IGS weekly solution products. From top to bottom, the E/N/U direction error sequences represent the positioning of PPP-B2b broadcasted by the C59 satellite and the C59 satellite, respectively. To provide a more intuitive view of the converged positioning effect, the half-hour convergence process has been omitted from the images. The calculated RMS errors in the E/N/U direction after static positioning convergence for C59_PPP-B2b are 3.21/1.37/4.47 cm, respectively, while for C61_PPP-B2b, they are 4.43/1.95/5.49 cm. After static PPP-B2b positioning convergence at the GAMG station, the static positioning errors of both C59_PPP-B2b and C61_PPP-B2b fluctuate within 0.03 m.

Figure 22 illustrates the time-varying changes in the number of visible satellites and the positioning dilution of precision (PDOP) during BDS-3 PPP-B2b positioning conducted by the GAMG station on 22 March 2024. The average number of visible satellites for precise single-point positioning calculations stands at approximately 8. The PDOP serves as an indicator of the geometric distribution of satellites involved in these calculations. Typically, a lower PDOP signifies higher positioning accuracy. Notably, there are four time points throughout the day where PDOP experiences significant fluctuations, accompanied by a decrease in the number of visible satellites, suggesting a correlation between PDOP and the number of visible satellites. Apart from these four time points, PDOP values remain below 3 for all other time periods.

Table 8. BDS-3 Static positioning error RMS.

Station	C59_PPP-B2b			C61_PPP-B2b
	E (cm)	N (cm)	U (cm)	E (cm)	N (cm)	U (cm)
GAMG	0.68	0.87	2.62	0.89	1.16	2.81
GUA1	2.53	0.88	1.93	2.31	0.84	1.92
JFNG	0.87	1.7	7.07	1.03	1.92	6.86
SHA1	2.72	2.05	4.98	2.51	2.04	5.23
USUD	5.66	3.72	8.51	5.99	3.86	8.81

presents the average RMS error values for the static Beidou single system PPP-B2b in the E/N/U direction across five IGS stations over a span of 7 days from 22 March to 28 March 2024. The RMS values of the coordinates serve as an indicator of the degree of difference between each point in the calculated coordinate set and the mean. In Table 8, the positioning errors RMS for C59_PPP-B2b and C61_PPP-B2b in the E direction are primarily clustered within the range of 1–3 cm. Meanwhile, the positioning errors RMS in the U direction are slightly inferior to those in the E and N directions, predominantly falling between 2 cm and 8 cm. In terms of positioning error RMS, the GAMG station demonstrates the best performance. However, due to the limited number of observation satellites, the USUD station exhibits a noticeable gap in positioning accuracy compared to other stations.

The average RMS values of the static positioning errors in the E/N/U directions for the Beidou-3 system C59_PPP-B2b are 2.49/1.84/5.02 cm, respectively. Similarly, the average RMS values for C59_PPP-B2b in the E/N/U directions are 2.54/1.96/5.12 cm, respectively. Both C59_PPP-B2b and C61_PPP-B2b are capable of achieving centimeter-level static positioning accuracy. Based on the positioning results over these 7 days, C59_PPP-B2b demonstrates slightly superior positioning accuracy in all directions compared to C61_PPP-B2b.

4.2.2. BDS-3 PPP-B2b Kinematic Positioning

PPP-B2b kinematic positioning is a simulated dynamic approach that utilizes static observation data but employs a dynamic PPP model. Unlike static PPP, which estimates coordinate parameters as constants, it incorporates white noise and re-performs PPP calculations on satellite observation data to achieve a simulated dynamic effect.

Kinematic positioning has broadened the application scope of precise single-point positioning, allowing for the real-time high-precision positioning of mobile receivers in areas like vehicle navigation, aircraft navigation, and mobile measurement. In kinematic positioning, motion models and filtering algorithms are introduced, enabling precise single-point positioning to account for the motion state of the receiver. This allows for the continuous estimation of the receiver’s position and velocity, yielding more accurate and stable positioning results.

Taking the GAMG station on 22 March 2024 as an example, Figure 23 and Figure 24 depict the kinematic positioning results in the E/N/U direction over time. From top to bottom, the figures represent the error sequences of kinematic positioning in the E/N/U direction for C59_PPP-B2b and C61_PPP-B2b, respectively. To visually illustrate the convergence effect, the daily convergence process has been omitted from the images. The calculated RMS values of the E/N/U directional errors after the kinematic positioning convergence for C59_PPP-B2b are 7.21 cm, 7.33 cm, and 12.44 cm, respectively. Similarly, for C61_PPP-B2b, the RMS values are 8.12 cm, 7.92 cm, and 14.22 cm, respectively. After achieving dynamic PPP-B2b positioning convergence, the positioning errors for both C59_PPP-B2b and C61_PPP-B2b fluctuate within a range of ±0.3 m at the GAMG station.

Table 9. BDS-3 Kinematic positioning error RMS.

Station	C59_PPP-B2b			C61_PPP-B2b
	E (cm)	N (cm)	U (cm)	E (cm)	N (cm)	U (cm)
GAMG	0.68	0.87	2.62	0.89	1.16	2.81
GUA1	2.53	0.88	1.93	2.31	0.84	1.92
JFNG	0.87	1.7	7.07	1.03	1.92	6.86
SHA1	2.72	2.05	4.98	2.51	2.04	5.23
USUD	5.66	3.72	8.51	5.99	3.86	8.81

presents the average RMS error values for the dynamic single BeiDou PPP-B2b E/N/U directions across five IGS stations over a span of 7 days from 22 March to 28 March 2024. In Table 9, the RMS errors of kinematic positioning for C59_PPP-B2b and C61_PPP-B2b in the E and N directions primarily fall between 5 cm and 15 cm. Conversely, the RMS errors in the U direction are predominantly between 15 cm and 25 cm, indicating a slight inferiority compared to the positioning errors in the E and N directions.

The average RMS values of the kinematic positioning errors in the N/E/U direction for the Beidou-3 system C59_PPP-B2b are 10.43/7.53/16.49 cm, respectively. Similarly, the average RMS values of the kinematic positioning errors in the N/E/U directions for C61_PPP-B2b are 16.45/8.42/18.75 cm, respectively. Regarding positioning errors in different directions, both C59_PPP-B2b and C61_PPP-B2b exhibit high consistency between dynamic and static positioning.

5. Conclusions

Regarding satellite orbits, PPP-B2b primarily rectifies the orbital discontinuity resulting from hourly updates of broadcast ephemeris, and it also enhances the orbit accuracy to a certain degree. As for the clock bias of BeiDou satellites, it not only rectifies the hourly discontinuity in the broadcast ephemeris but also elevates the accuracy. Nevertheless, the PPP-B2b product fails to enhance the clock bias accuracy of GPS. Concerning the real-time code deviation information contained in PPP-B2b, its deviation from the code deviation product released by the Chinese Academy of Sciences does not exceed 2 ns. Overall, the correction efficacy of C59 broadcast PPP-B2b on satellite orbits and clock errors surpasses that of C61.

In terms of PPP-B2b time comparison, the zero-baseline time comparison results fluctuate within a range of 0.5 ns, with STD better than 0.1 ns, and a frequency stability of 100,000 s exceeding 4 × 10⁻¹⁶. In long-baseline time comparisons, using the time comparison results of GBM products as a reference, the residuals for C59_PPP-B2b time comparisons fluctuate within 2 ns, achieving a frequency stability of 3.87 × 10⁻¹⁵ over 100,000 s. Similarly, the residuals for C61_PPP-B2b time comparisons also fluctuate within 2 ns, attaining a frequency stability of 1.99 × 10⁻¹⁵ over the same time period. In terms of accuracy, C59_PPP-B2b and C61_PPP-B2b are comparable in time comparisons, yet C59_PPP-B2b excels in usability and stability compared to C61_PPP-B2b.

When PPP-B2b is statically positioned, the average RMS values of errors in the N/E/U direction for C59_PPP-B2b and C61_PPP-B2b are 2.49/1.84/5.02 cm and 2.54/1.96/5.12 cm, respectively. When dynamically positioned, the average RMS values of errors in the N/E/U direction for C59_PPP-B2b and C61_PPP-B2b are 10.43/7.53/16.49 cm and 16.45/8.42/18.75 cm, respectively. Comprehensive analysis indicates that C59_PPP-B2b exhibits superior positioning performance compared to C61_PPP-B2b. For C61_PPP-B2b, future improvements can be made in terms of accuracy, primarily by enhancing the continuity of orbit and clock error cycles. Additionally, C61_PPP-B2b has potential for further enhancement in terms of GPS clock error correction effectiveness.