Calibration of Receiver-Dependent Pseudorange Bias and Its Impact on BDS Augmentation Positioning Accuracy

Abstract

Pseudorange bias refers to the receiver-dependent and satellite-dependent constant bias in the pseudorange resulting from the nonideal characteristics of a signal. The impact of pseudorange bias on high-precision satellite navigation services has long been ignored. This paper proposes a pseudorange bias calibration method for two collocated receivers. Then, we calibrate pseudorange biases for two types of collocated receivers at a monitoring station within China and evaluate their impact on two high-precision services: BeiDou Navigation Satellite System 3 (BDS-3) dual-frequency pseudorange augmentation and precise point precision (PPP). Theoretical analysis reveals that the calibrated pseudorange biases contribute 17.2% and 7.7% to the user equivalent ranging error (UERE) of BDS-3 and Global Positioning System (GPS) dual-frequency pseudorange augmentation, respectively, and that the convergence time of the GPS static and kinematic PPP increases from 6 min and 26 min to 19 min and 58 min, respectively. The experimental results indicate that the calibrated pseudorange biases are consistent as the receiver location and time vary. The spatial distribution consistency is generally better than 0.1 m, and the temporal consistency is better than 0.15 m. The pseudorange biases for BDS-3 B1C and B2a are approximately 0.7 m and 0.1 m, respectively, whereas those for GPS L1C/A and L2P are both approximately 0.25 m. Furthermore, The results show that after correction of the pseudorange biases, the average convergence time for BDS-3/GPS static PPP decreases from 48.83/24.03 min to 38.54/21.12 min, respectively, a decrease of approximately 21%/12%. For BDS-3/GPS/BDS-3 + GPS kinematic PPP, the average convergence time decreases from 109.53/45.10/39.15 min to 62.99/40.83/22.94 min, respectively, a decrease of approximately 42%/41%/9%. Similarly, the three-dimensional positioning accuracy for BDS-3/GPS/BDS-3 + GPS dual-frequency pseudorange augmentation improves from 3.25/3.94/2.49 m to 2.65/3.69/2.16 m, respectively, increasing by approximately 6.3%, 18.5%, and 13.3%, respectively. The above analysis and experiments demonstrate that pseudorange bias is an important error source affecting both dual-frequency pseudorange augmentation and PPP services.

1. Introduction

The transmission of global navigation satellite system (GNSS) signals is affected by various external factors, which can cause live signals to deviate from the ideal. Pseudorange bias refers to bias in the measured pseudorange by a receiver due to signal deviation from the ideal, i.e., signal distortion [1]. Since the signal anomaly of GPS satellite space vehicle number (SVN) 19 in 1993, which led to a significant degradation of positioning results for differential GPS, the issue of pseudorange bias has been the subject of considerable research and discussion [2]. Even nominally healthy satellites are affected by pseudorange biases [3,4,5]. A research team at Stanford University has conducted an in-depth study on the pseudorange bias, which is caused by satellite signal distortion [6,7].

A number of studies have demonstrated that pseudorange bias has an impact on positioning, navigation, and timing (PNT). Based on 302 the Multi-GNSS Experiment (MGEX) stations and five distinct receiver types, Shi et al. demonstrated that the correction of pseudorange bias significantly enhances the precision of wide-lane and narrow-lane phase corrections [8]. Cheng et al. examined the impact of pseudorange bias on wide-phase ambiguity [9]. Their findings indicate that the positioning accuracy can be improved by correction of the pseudorange biases and that the wide-phase ambiguity fix can be effectively improved. Ai et al. employed data from 140 International GNSS Service (IGS) stations to calculate the pseudorange biases and to examine the impact of their inconsistency on the estimation of the clock offset for diverse receiver networks [10]. The findings demonstrate that the pseudorange bias has a considerable impact on the estimation of the clock offset. Correction of the pseudorange bias can enhance the accuracy by a few picoseconds for GPS and by a minimum of 15 picoseconds for Galileo and BeiDou Navigation Satellite System (BDS). Furthermore, the correction of pseudorange biases can markedly improve the convergence performance of the static PPP and dynamic PPP by a minimum of 12% and 10%, respectively. In a study conducted by Li et al., the impact of pseudorange bias on the precision orbiting of BeiDou Navigation Satellite System 2 (BDS-2) satellites was examined [11]. The results demonstrate that the correction of pseudorange bias can enhance the overlapping arc orbiting accuracy of BDS geostationary earth orbit (GEO), inclined geosynchronous orbit (IGSO), and medium earth orbit (MEO) satellites. Specifically, this enhancement is observed to be 11%, 10%, and 4%, respectively. Pseudorange biases are strongly coupled to receiver and satellite hardware delays [12,13,14]. Hauschild et al. compared the GNSS satellite differential code biases (DCBs) derived from several different types of receivers [15]. Their results revealed that the maximum difference between the GPS satellite L1CA/L2P DCB and the BDS-2 satellite B1I/B3I DCB can reach 0.3 m because of the influence of pseudorange bias. Pseudorange biases are very stable and can be calibrated as constant depending on each satellite and receiver type if the receiver and satellite hardware are not updated [16].

As can be observed from the above studies, there has been a number of studies conducted on pseudorange bias. However, the influence of pseudorange bias on high-precision satellite navigation services has been ignored for a long time. Pseudorange augmentation and PPP are two featured high-precision services provided by BDS-3. These services are essentially achieved by broadcasting orbit and clock corrections to users of the BDS-3 basic navigation system [17,18]. The BDS-3 pseudorange augmentation service provides both single-frequency GPS pseudorange augmentation and dual-frequency multisystem pseudorange augmentation. Liu et al. concluded that the pseudorange bias introduced a constant deviation of 1.57 m in clock correction for the single-frequency GPS pseudorange augmentation, which severely deteriorates the positioning accuracy after augmentation [19]. As their experiments showed, after correcting the pseudorange biases, the positioning accuracy of single-frequency pseudorange augmentation is significantly improved. Unlike single-frequency pseudorange augmentation, dual-frequency pseudorange augmentation can avoid the influence of ionospheric delay with ionosphere-free combinations. However, this augmentation amplifies the impact of pseudorange bias, which is also experienced by the BDS-3 PPP service. To date, publications concerning the effects of pseudorange bias on BDS-3 dual-frequency pseudorange augmentation and PPP services have not been published.

In this study, we investigate the effect of pseudorange bias on BDS-3 dual-frequency pseudorange augmentation and PPP services. In Section 2, the definitions of pseudorange bias and the calibration method are proposed. In addition, the effect of pseudorange bias on the positioning performance of pseudorange augmentation and PPP is evaluated. In Section 3, the data and strategies employed in our experiments are presented. In Section 4, the pseudorange biases of the two types of receivers used in this study are calibrated and their characteristics are analyzed. In addition, the improvement in positioning performance after pseudorange bias correction is described. In Section 5, conclusions and discussions are offered.

2. Methods

2.1. Pseudorange Bias Definition

Pseudorange biases are receiver pseudorange measurement biases resulting from signal distortions. The nominal signal deformations are different for each satellite. Meanwhile, they are related to receivers’ technical configurations, such as front-end filter bandwidths, correlator spacings, and anti-multipath algorithms [2,3,4,5]. Therefore, the pseudorange biases are dependent on both the satellite and the receiver. They differ from the differential code bias (DCB), satellite clock, and receiver clock offset, which are either dependent on the satellite or the receiver.

The pseudorange bias is a new error source that cannot be completely eliminated and cannot be expressed by current navigation parameters such as DCBs. Although highly coupled to both receiver and satellite hardware delays, they are more complex and cannot be simply separated into satellite-dependent and receiver-dependent parts. First, signal deformations differ for different satellites, and the pseudorange biases for a receiver tracking different satellites are different. Therefore, the biases cannot be fully absorbed by the receiver hardware delay. Second, the user receivers and monitoring receivers used for orbit and clock determination are always different, resulting in differences in the pseudorange biases as the receiver type varies. Therefore, the biases cannot be fully absorbed by the satellite hardware delay. Therefore, pseudorange bias is considered a new source of error. It degrades the accuracy of high-precision satellite navigation services.

2.2. Pseudorange Bias Calibration

The traditional observation equation does not consider pseudorange bias. After adding pseudorange bias, the equation is as follows:

$$\left\{\begin{array}{c}{P}_i^{r,s}\left(t\right)={\rho }^{r,s}\left(t\right)+c\left(\mathsf{\Delta }{t}^r\left(t\right)-\mathsf{\Delta }{t}^s\left(t\right)+b_i^r-b_i^s+B_i^{type\left(r\right),s}\right)+I_i^{r,s}\left(t\right)\\ +{Trop}^{r,s}\left(t\right)+{MP}^{r,s}\left(t\right)+{\epsilon }_i^{r,s}\left(t\right)\\ L_i^{r,s}\left(t\right)={\rho }^{r,s}\left(t\right)+c\left(\mathsf{\Delta }{t}^r\left(t\right)-\mathsf{\Delta }{t}^s\left(t\right)\right)+{\mathsf{\lambda }}_i{N}_i^{r,s}-I_i^{r,s}\left(t\right)\\ +{Trop}^{r,s}\left(t\right)+{MP}^{r,s}\left(t\right)+{\phi }_i^{r,s}\left(t\right)\end{array}\right.$$

(1)

where $P_i^{r,s}\left(t\right)$ and $L_i^{r,s}\left(t\right)$ represent the pseudorange and carrier phase measurements from satellite $s$ to receiver $r$ on signal $i$ in units of length, respectively; $t$ denotes the epoch time; ${\rho }^{r,s}\left(t\right)$ is the geometric range with antenna phase center and earth rotation correction; $\mathsf{\Delta }{t}^r\left(t\right)$ is the receiver clock offset while $\mathsf{\Delta }{t}^s\left(t\right)$ is the satellite clock offsets; $c$ is the light speed; $b_i^r$ is the receiver hardware delay while $b_i^s$ is the satellite hardware delay; $I_i^{r,s}\left(t\right)$ is the ionospheric delay and ${Trop}^{r,s}\left(t\right)$ is the tropospheric delay; ${MP}^{r,s}\left(t\right)$ is the multipath error; $N_i^{r,s}$ is the carrier-phase ambiguity; ${\mathsf{\lambda }}_i$ denotes the wavelength; ${\epsilon }_i^{r,s}\left(t\right)$ and ${\phi }_i^{r,s}\left(t\right)$ represent the measurement errors of pseudorange and phase, respectively; and $B_i^{type\left(r\right),s}$ is the added pseudorange bias term in this paper. It is related to the satellite, the signal of the satellite, and the receiver type.

In this study, a calibration method of pseudorange bias is proposed for two collocated receivers of different types receiving the same external time signal. The effects of the satellite and receiver clock offset, ionospheric and tropospheric delay, and satellite orbit error are identical for the two receivers at the same observation frequency.

Thus, according to Equation (1), taking the difference of observation minus calculation (O-C) between receivers $r_1$ and $r_2$ for signal $i$ of satellite $s$ at time $t$, we obtain a single difference of O-C, which is denoted as ${∆OC}_i^{r12,s}\left(t\right)$. ${∆OC}_i^{r12,s}\left(t\right)$ leaves only receiver hardware delays, pseudorange biases, and observation measurement noise. The hardware delays and pseudorange biases can be treated as constant for a day. The formula for ${∆OC}_i^{r12,s}\left(t\right)$ is as follows:

$${∆OC}_i^{r12,s}\left(t\right)={OC}_i^{r1,s}\left(t\right)-{OC}_i^{r2,s}\left(t\right)=c\left(∆b_i^{r12}+{∆B}_i^{r12,s}\right)+∆{\epsilon }_i^{r12,s}\left(t\right)$$

(2)

where

$$\left\{\begin{array}{c}\begin{array}{c}{OC}_i^{r1,s}\left(t\right)=P_i^{r1,s}\left(t\right)-{\rho }^{r1,s}\left(t\right),{OC}_i^{r2,s}\left(t\right)=P_i^{r2,s}\left(t\right)-{\rho }^{r2,s}\left(t\right)\\ b_i^{r12}=b_i^{r1}-b_i^{r2}\\ {∆B}_i^{r12,s}=B_i^{r1,s}-B_i^{r2,s}\end{array}\\ ∆{\epsilon }_i^{r12,s}\left(t\right)={\epsilon }_i^{r1,s}\left(t\right)-{\epsilon }_i^{r2,s}\left(t\right)\end{array}\right.$$

(3)

where ${\rho }^{r1,s}\left(t\right)$ and ${\rho }^{r2,s}\left(t\right)$ are calculated with the precise orbit and clock products from IGS and the precise coordinates of the receivers. Since observation measurement noise is characterized by white noise, we remove it by taking the expectation of ${∆OC}_i^{r12,s}\left(t\right)$ in the day and denote it as ${dif1}_i^{r12,s}$. The formula for ${dif1}_i^{r12,s}$ is as follows:

$${dif1}_i^{r12,s}=E\left[{∆OC}_i^{r12,s}\left(t\right)\right]=c\left(∆b_i^{r12}+{∆B}_i^{r12,s}\right)$$

(4)

Then, one satellite’s ${dif1}_i^{r12,s}$ is selected as the reference and denoted as ${dif1}_i^{r12,ref}$. We can obtain the pseudorange bias for signal $i$ of satellite $s$ by subtracting ${dif1}_i^{r12,ref}$ from ${dif1}_i^{r12,s}$, where the receiver hardware delay $∆b_i^{r12}$ is eliminated. The calibrated pseudorange bias is denoted as ${dif2}_i^{r12,s},$ as follows:

$${dif2}_i^{r12,s}={dif1}_i^{r12,s}-{dif1}_i^{r12,ref}=c({∆B}_i^{r12,s}-{∆B}_i^{r12,ref})$$

(5)

The aforementioned methodology can be employed to calibrate pseudorange biases between monitoring receivers and a range of mainstream receivers for users, thereby enhancing the positioning performance of the pseudorange augmentation and PPP services.

2.3. Pseudorange Bias Analysis for the BDS-3 High-Precision Service

2.3.1. Dual-Frequency Pseudorange Augmentation Service

The BDS-3 pseudorange augmentation services include ground-based and satellite-based augmentation. They improve the real-time positioning accuracy of pseudorange users by broadcasting precise orbit and clock corrections. In this section, we analyze the effect of the pseudorange bias on positioning performance of the dual-frequency pseudorange augmentation service.

Positioning accuracy depends on the user equivalent ranging error (UERE). For dual-frequency ionosphere-free (IF) users, the UERE is as follows:

$$UERE=\sqrt{{SISRE}^2+{\delta Trop}^2+{MP}^2+{{(B}_{if})}^2}$$

(6)

where $SISRE$ represents the signal-in-space ranging error, which includes satellite clock and orbit errors. $\delta Trop$ represents the error of the tropospheric delay model. $MP$ represents the error of pseudorange multipath. $B_{if}$ represents the pseudorange bias of dual-frequency IF combinations.

The SISREs of the GPS and BDS-3 navigation messages are 0.49 m and 0.35 m, respectively [20]. After orbital and clock corrections of the dual-frequency pseudorange augmentation service, the SISRE for the BDS and GPS satellite orbit and clock corrections are approximately 0.13 m [21]. The widely used troposphere delay models include EGNOS, UNB3, UNB3m, GPT, GPT2, and GPT2w, etc., whose errors are approximately 4–7 cm [22,23]. The errors of multipath and noise, which are amplified by the dual-frequency IF combination, can reach approximately 0.7 m [7]. The pseudorange biases are also considered here for the BDS-3 and GPS dual-frequency IF combinations. The pseudorange bias errors for the BDS-3 and GPS dual-frequency IF combinations can be obtained by taking the average of the absolute values of the pseudorange biases of each dual-frequency combination in Section 4.1.1, which are 0.5 m and 0.3 m, respectively.

The errors contributing to UERE in the dual-frequency fundamental service (RNSS) and pseudorange augmentation services for BDS-3 and GPS are compared and demonstrated in

Table 1. Factors affecting UERE for dual-frequency service.

	BDS-3 (B1C/B2a)		GPS (L1/L2)
	Fundamental Service	Pseudorange Augmentation Service	Fundamental Service	Pseudorange Augmentation Service
SISRE	0.35 m	0.13 m	0.49 m	0.13 m
Error of tropospheric delay model	<0.10 m	<0.10 m	<0.10 m	<0.10 m
Multipath errors	0.70 m	0.70 m	0.70 m	0.70 m
UERE without pseudorange biases	0.79 m	0.72 m	0.86 m	0.72 m
Pseudorange biases	0.50 m	0.50 m	0.30 m	0.30 m
UERE with pseudorange biases	0.93 m	0.87 m	0.91 m	0.78 m
The proportion of pseudorange bias in UERE	15.10%	17.20%	5.50%	7.70%

.

The table shows that in the fundamental service, the magnitude of the pseudorange bias is similar to that of the SISRE and accounts for a significant proportion of the UERE. Owing to the satellite orbit and clock corrections, the SISRE can be greatly reduced in the pseudorange augmentation service. However, the pseudorange bias accounts for a larger portion of the UERE and becomes a major source of error in the dual-frequency pseudorange augmentation service.

2.3.2. PPP Service

Both PPP-B2b and the ground-based augmentation system provide PPP services. The provision of precise orbit and clock offset corrections enables users to obtain real-time PPP services with decimeter-level and centimeter-level accuracy in kinematic and static modes, respectively. Although PPP users primarily utilize carrier-phase observations, the convergence time of initial carrier-phase ambiguity estimation is highly dependent on the accuracy of the pseudorange observations. This section explores the effect of pseudorange bias on the convergence time of the PPP service.

Similar to pseudorange augmentation, the calculation of PPP service corrections uses the same type of receivers so that pseudorange biases are absorbed by satellite clock corrections. According to Equation (1), after parameter recombination, the equations for pseudorange and phase observations of dual-frequency IF combination are as follows:

$$\left\{\begin{array}{c}{P}_{if}^{r,s}\left(t\right)={\rho }^{r,s}\left(t\right)+c\left(\tilde{\mathsf{\Delta }}{t}^r\left(t\right)-\tilde{\mathsf{\Delta }}{t}^s\left(t\right)\right)+{Trop}^{r,s}\left(t\right)+{MP}^{r,s}\left(t\right)+{\epsilon }_p\\ L_{if}^{r,s}\left(t\right)={\rho }^{r,s}\left(t\right)+c\left(\tilde{\mathsf{\Delta }}{t}^r\left(t\right)-\tilde{\mathsf{\Delta }}{t}^s\left(t\right)\right)+{\lambda }_N{\tilde{N}}_{if}^{r,s}+{Trop}^{r,s}\left(t\right)+{\epsilon }_{\varphi }\end{array}\right.$$

(7)

where

$$\left\{\begin{array}{c}\tilde{\mathsf{\Delta }}{t}^r\left(t\right)=\Delta t^r\left(t\right)+b_{if}^r\\ \tilde{\mathsf{\Delta }}{t}^s\left(t\right)=\Delta t^s\left(t\right)+b_{if}^s-B_{if}^s\\ {\tilde{N}}_{if}^{r,s}=N_{if}^{r,s}+\left(b_{if}^s-b_{if}^r-B_{if}^s\right)/{\lambda }_N\end{array}\right.$$

(8)

In the equations above, the parameters with subscripts $if$ represent the dual-frequency ionosphere-free combination values. Compared with pseudorange observations, phase observations have much better measurement accuracy. Therefore, in the PPP process, carrier-phase observations are always given a dominant weight. However, the estimation of phase ambiguities depends on the initialization and constraints of the pseudorange observation. Equation (8) shows that the ambiguities ultimately absorb the pseudorange bias, thus the convergence of the ambiguity resolution is affected.

To validate the effect of pseudorange bias on a PPP service, a PPP experiment is conducted. GPS observations of the IGS WHIT station are used, as are the precise satellite clock and orbit products provided by IGS. Except for the satellite orbits and clock, other PPP strategies are the same as those used in the experiment in Section 3. The pseudorange biases calibrated in Section 4.1 are added to the pseudorange observations for simulation.. The convergence conditions are set such that all three directions (north, east, and up (NEU)) are less than 10 cm and are sustained for 30 min.

Table 2. Comparison of convergence times in PPP with and without pseudorange biases (unit: minutes).

PPP	Convergence Time
	Without Pseudorange Biases	With Pseudorange Biases
Static PPP	6	19
Kinematic PPP	26	58

and Figure 1 show the detailed information on convergence. For static PPP, the convergence times are 6 min and 19 min without and with pseudorange biases, respectively. For kinematic PPP, the convergence times are 26 min and 58 min without and with pseudorange biases, respectively.

On the basis of the above analysis and experiments, the pseudorange biases significantly prolong the convergence time of both static and kinematic PPP, indicating that the pseudorange bias is a vital factor affecting the performance of the PPP service.

3. Data and Strategies

To evaluate the impact of pseudorange bias on the performance of BDS-3 dual-frequency pseudorange augmentation and PPP services, observations from 24 widely and evenly distributed monitoring stations of the BeiDou satellite-based augmentation systems (BDSBAS) from 1 December to 31 December 2022 are used for analysis. The 24 monitoring stations are located in Heilongjiang, Inner Mongolia, Liaoning, Beijing, Shandong, Shanghai, Hunan, Shanxi, Guangdong, Guangxi, Yunnan, Sichuan, Xizang, Qinghai, Gansu, and Xinjiang, China. The 24 monitoring stations were equipped with two types of receivers, referred to as receiver Type A and Type B. The two types of collocated receivers at the Beijing monitoring station, which receive the same time signals, are selected to calibrate the pseudorange biases with the method in Section 2.2. Furthermore, 12 of the monitoring stations with Type A receivers were selected to calculate orbit and clock corrections, whereas the remaining 12 stations with both Type A and Type B receivers were used as user terminals to evaluate the impact of pseudorange bias on the performance of the BDS-3 high-precision augmentation service.

Both the BDS-3 dual-frequency pseudorange augmentation and PPP services reduce the SISRE by broadcasting orbit and clock corrections on the basis of fundamental services. As analyzed in Section 2.3, when orbit and clock corrections are calculated for the dual-frequency pseudorange augmentation and PPP services, the pseudorange biases for the two types of receivers are absorbed by either the clock corrections or the pseudorange residuals, whereas the orbit corrections remain unaffected. Therefore, the differences between the precise orbits restored by users through orbit corrections and those computed by the IGS analysis center are negligible. However, there are significant differences between the precise clocks restored by the clock corrections and those provided by IGS because of the pseudorange biases. Therefore, in this study, IGS MGEX precise orbits are directly adopted as the source of orbital corrections for pseudorange augmentation and PPP services, whereas BDS/GPS precise satellite clock offsets are calculated via 12 monitoring stations with Type A receivers for clock corrections. The detailed strategy for calculating BDS/GPS precise satellite clock offsets is given in

Table 3. Strategy for calculating BDS/GPS precise satellite clock offsets.

	Strategy
Frequency	BDS-3: B1C/B2a GPS: L1/L2
Observations	Dual-frequency IF combinations of pseudorange and carrier-phase observation.
Elevation angle	Cut-off: 7°.
Orbits	Fix with precise satellite orbits of IGS MGEX.
Satellite/Receiver PCO and PCV	BDS-3: Provided by manufacturers; GPS: igs14.atx [24].
Station displacement	Correcting solid Earth tide, ocean tide, pole tide [25].
Relativity	Corrected [25].
Troposphere delay	Saastamoinen model [26] for wet and dry hydrostatic delay with the GMF [27] without the gradient model; residual tropospheric delay as a random-walk process.
Ionosphere delay	The first-order ionosphere delay is eliminated by the dual-frequency combinations.
Receiver clock	Estimating as white noise per epoch.
Satellite clock	Estimating as white noise.
Carrier-phase ambiguities	Estimating as floats and constant parameters.
Estimation method	Kalman filter.

.

In this way, the precise satellite clock offsets are calculated. Finally, post-processed precise orbits of the IGS MGEX and the generated satellite clock offsets are used to calculate the orbit and clock corrections for the BDS-3 pseudorange augmentation and PPP services.

The remaining 12 stations with both Type A and Type B receivers are adopted as user terminals for dual-frequency pseudorange augmentation positioning, static, and kinematic PPP. In accordance with the satellite navigation system of the observations, each experiment is conducted in three modes: BDS-3, GPS, and the BDS-3/GPS dual-system. The positioning accuracy (95% confidence level) and convergence time will be assessed for dual-frequency pseudorange augmentation and PPP service, respectively. The detailed strategy for dual-frequency pseudorange augmentation positioning and PPP is summarized in

Table 4. Strategy for dual-frequency pseudorange augmentation positioning.

	Strategy
Frequency	BDS-3: B1C/B2a GPS: L1/L2
Observations	Dual-frequency IF combinations of pseudorange observation.
Elevation angle	Cut-off: 7°.
Satellite orbits	BDS-3: Orbit in CNAV broadcasted navigation message + orbit correction messages. GPS: Orbit in LNAV broadcasted navigation message + orbit correction messages.
Satellite clock	BDS-3: Clock in CNAV broadcasted navigation message + clock correction messages. GPS: Clock in LNAV broadcasted navigation message + clock correction messages.
Relativity	Corrected [25].
Troposphere delay	Correcting with empirical model recommended in RTCA DO-229F [28] and ED-259 [29].
Ionosphere delay	Eliminating through the dual-frequency IF combinations.
Receiver clock	Estimating as white noise per epoch, the dual-system needs to estimate ISB.
Receiver coordinate	Estimating as white noise per epoch.
Estimation method	Least squares estimation.

and

Table 5. Strategy for PPP.

	Strategy
Frequency	BDS-3: B1C/B2a GPS: L1/L2
Observations	Dual-frequency IF combinations of pseudorange and carrier-phase observation.
Elevation angle	Cut-off: 7°.
Satellite orbits	BDS-3: Orbit in CNAV broadcasted navigation message + orbit correction messages. GPS: Orbit in LNAV broadcasted navigation message + orbit correction messages.
Satellite clock	BDS-3: Clock in CNAV broadcasted navigation message + clock correction messages. GPS: Clock in LNAV broadcasted navigation message + clock correction messages.
Relativity	Corrected [25].
Station displacement	Correcting solid Earth tide, ocean tide, pole tide [25].
Troposphere delay	Saastamoinen model [26] for wet and dry hydrostatic delay with the GMF [27] without the gradient model; residual tropospheric delay as a random-walk process.
Ionosphere delay	The first-order ionosphere delay is eliminated by the dual-frequency combinations.
Receiver clock	Estimating as white noise per epoch, the dual-system needs to estimate ISB.
Receiver coordinate	Estimating as constant parameters for Static PPP and as white noise for Kinematic PPP.
Carrier-phase ambiguities	Estimating as constant parameters.
Estimation method	Kalman filter.

, respectively.

4. Results

4.1. Pseudorange Bias Spatial and Temporal Variation Characteristics

4.1.1. Temporal Variation Characteristics

The method described in Section 2.2 is employed to compute pseudorange biases for multiple GPS/BDS-3 signals. The pseudorange biases are calculated with observations from two collocated receivers deployed at the Beijing monitoring station. The two receivers received the same time signals. The GPS satellites were referenced to G01, whereas the BDS-3 satellites were referenced to C19.

The time series of one-day pseudorange bias estimations with the two receivers is illustrated in Figure 2 and Figure 3. Figure 2 presents the pseudorange bias for the B1C and B2a signals of three BDS-3 satellites, whereas Figure 3 shows the pseudorange bias for the L1C/A and L2P signals of three GPS satellites. Figure 4 and Figure 5 present the standard deviation (STD) and the mean of the pseudorange biases, respectively. Figure 4a and Figure 5a present the results for B1C and B2a signals of the BDS-3 satellites, whereas Figure 4b and Figure 5b show the results for L1C/A and L2P signals of the GPS satellites.

With respect to the variation characteristics of the pseudorange bias calibration with respect to time, as illustrated in Figure 2 and Figure 3, the pseudorange bias series for different signals of the two systems are stable. The pseudorange bias dispersion of BDS-3 satellites is marginally greater than that of GPS satellites. The maximum variation amplitude of the pseudorange bias for most BDS-3/GPS satellites does not exceed 30/20 cm. Correspondingly, Figure 4 shows that the standard deviations of the pseudorange bias series for two signals from BDS-3 satellites are generally greater than those for two signals from GPS satellites. Except for the B2a signals of satellites C26, C30, and C32, which exhibit a slightly greater STD of approximately 0.1 m, the remaining signals of BDS-3 satellites exhibit an STD of less than 0.1 m, with the majority clustered at approximately 0.06 m. The GPS satellite signals exhibit a STD of less than 0.1 m. Except for satellites G10, G15, and G32, the standard deviation for all other satellites does not exceed 0.06 m, with the majority concentrated at approximately 0.03 m.

From the average pseudorange bias estimations, as shown in Figure 5, there is a strong correlation between the pseudorange bias and the satellites, as well as the signals. For BDS-3, the absolute values of the pseudorange bias for the B1C signals of satellites are much larger than those for the B2a signals of the satellites. For the B1C signal, the maximum difference among the satellites is approximately 1.3 m, whereas the maximum difference among the satellites for the B2a signal is smaller, at approximately 0.2 m. For GPS, the absolute magnitudes of the pseudorange biases for L1C/A and L2P signals are comparable.

In summary, the average pseudorange biases for the signals of BDS-3 and GPS, from largest to smallest, are B1C, L1C/A and L2P, B2a. The standard deviation shows good consistency of the pseudorange bias over time.

4.1.2. Spatial Variation Characteristics

In the previous section, the temporal characteristics of the variation in the pseudorange bias were analyzed. To further analyze the spatial characteristics, the same collocated receivers meeting the requirements of the method in Section 2.2 in Xi’an were used for pseudorange bias calibration. The calibrated pseudorange bias is compared with that calibrated at the Beijing station. Figure 6 presents a comparison of the pseudorange biases calibrated at the two locations.

Figure 6 presents that the pseudorange biases calibrated for the B1C and B2a signals of the BDS-3 satellites and for the L1C/A and L2P signals of the GPS satellites are highly consistent at the two locations. Compared with those of the GPS satellites, the calibrated pseudorange biases of the BDS-3 satellites are more consistent at the two sites, with the B1C signal showing the best consistency. The differences in the calibrated pseudorange biases at the two locations for the B1C signal of each satellite are no more than 0.05 m. This is similar to the B2a signal, with only satellites C21, C33, and C43 slightly exceeding the value. For the L1C/A signals of all the GPS satellites, the differences do not exceed 20 cm, whereas for the L2P signals of most of the GPS satellites, the differences do not exceed 10 cm, with the exceptions of G27 and G29. Therefore, the calibrated pseudorange biases are in good spatial agreement.

4.2. Dual-Frequency Pseudorange Augmentation Positioning Service

4.2.1. The Impact of Pseudorange Bias on Dual-Frequency Pseudorange Augmentation Positioning

Figure 7, Figure 8 and Figure 9 compare the dual-frequency pseudorange augmentation positioning accuracy of the Type A and Type B receivers at each station for the GPS, BDS-3, and BDS-3/GPS dual-system, respectively. The figures demonstrate that the positioning accuracies of the BDS-3, GPS, and BDS-3/GPS dual-system for the Type B receiver are consistently lower than those for the Type A receiver. This discrepancy can be attributed to the pseudorange biases between the Type B receiver and the Type A receiver employed for the correction calculation. As previously stated, these pseudorange biases are amplified and become a significant error source in dual-frequency pseudorange augmentation positioning. A comparison of Figure 7 and Figure 8 reveals that the discrepancy in the positioning accuracy of BDS-3 between Type A and Type B receivers is greater than that of the GPS. This corroborates the findings presented in Table 1, which indicate that the relative pseudorange bias of the two receiver types for BDS-3 contributes a greater proportion of UERE than that for GPS. Furthermore, a comparison of Figure 7 and Figure 8 with Figure 9 reveals that the discrepancy in the positioning accuracy of the two receiver types for the BDS-3/GPS dual-system is less obvious than that for BDS-3 but greater than that for the GPS.

4.2.2. User-Side Pseudorange Bias Correction

In this part, the pseudorange biases of the two receiver types calibrated in Section 4.1 are used as user-side corrections in dual-frequency pseudorange augmentation positioning for further study.

Table 6. GPS dual-frequency pseudorange augmentation positioning accuracy (unit: meters).

	Without Correction			With Correction
Station	Horizontal	Vertical	3D	Horizontal	Vertical	3D
Heilongjiang	2.68	5.90	6.14	2.66	5.68	5.93
Shanghai	1.20	2.95	3.08	1.16	2.91	3.04
Fujian	1.25	2.70	2.83	1.23	2.59	2.73
Guangdong	1.73	4.29	4.48	1.73	4.07	4.28
Hunan	1.31	2.92	3.05	1.27	2.74	2.89
Hainan	1.34	3.04	3.18	1.36	3.09	3.21
Sichuang	2.43	4.94	5.17	2.40	4.87	5.11
Tibet	2.04	3.87	4.13	1.99	3.77	4.00
Yunnan	1.93	4.05	4.24	1.93	4.07	4.27
Qinghai	1.64	3.38	3.54	1.60	3.11	3.29
Gansu	1.26	2.57	2.70	1.20	2.29	2.42
Xinjiang	2.45	4.55	4.84	1.64	2.93	3.12
Mean	1.77	3.76	3.94	1.68	3.51	3.69

presents a comparison of the positioning accuracies of GPS dual-frequency pseudorange augmentation with and without the pseudorange bias correction for Type B receivers. This finding indicates that the positioning accuracy of Type B receivers is improved after pseudorange bias correction. The average positioning errors are 1.68 m, 3.51 m, and 3.69 m in the horizontal, vertical, and three-dimensional directions, respectively. The counterparts without pseudorange bias corrections are 1.77 m, 3.76 m, and 3.94 m, respectively.

Table 7. BDS-3 dual-frequency pseudorange augmentation positioning accuracy (unit: meters).

	Without Correction			With Correction
Station	Horizontal	Vertical	3D	Horizontal	Vertical	3D
Heilongjiang	1.76	3.07	3.30	1.46	2.33	2.54
Shanghai	1.23	2.06	2.19	0.76	1.39	1.46
Fujian	1.29	2.28	2.41	0.88	1.61	1.70
Guangdong	1.55	2.99	3.14	1.26	2.41	2.56
Hunan	1.34	2.42	2.56	0.96	1.78	1.89
Hainan	1.34	2.57	2.68	0.98	1.86	1.97
Sichuang	2.13	3.76	4.02	1.90	3.36	3.65
Tibet	1.95	3.75	3.97	1.69	3.34	3.52
Yunnan	1.88	3.76	3.95	1.64	3.24	3.46
Qinghai	1.69	2.98	3.22	1.36	2.55	2.71
Gansu	1.31	2.31	2.52	0.83	1.60	1.70
Xinjiang	2.39	4.72	5.09	2.06	4.37	4.64
Mean	1.65	3.05	3.25	1.31	2.48	2.65

presents a comparison of the positioning accuracies of BDS-3 dual-frequency pseudorange augmentation with and without pseudorange bias correction for Type B receivers. This illustrates that the correction of the pseudorange bias results in notable positioning accuracy improvements in the horizontal, vertical, and three-dimensional directions for Type B receivers at all stations. The average accuracies of all the stations increase from 1.65 m, 3.05 m, and 3.25 m to 1.31 m, 2.48 m, and 2.65 m, respectively. This represents a 20.6%, 18.7%, and 18.5% improvement, respectively.

Table 8. BDS-3/GPS dual-system dual-frequency pseudorange augmentation positioning accuracy (unit: meters).

	Without Correction			With Correction
Station	Horizontal	Vertical	3D	Horizontal	Vertical	3D
Heilongjiang	1.32	3.23	3.35	1.22	2.55	2.67
Shanghai	0.80	1.85	1.92	0.65	1.27	1.34
Fujian	0.85	1.85	1.94	0.72	1.4	1.48
Guangdong	1.09	2.42	2.53	1.01	2.07	2.17
Hunan	0.93	1.89	1.98	0.79	1.51	1.60
Hainan	0.93	2.12	2.19	0.83	1.86	1.92
Sichuang	1.53	2.88	3.05	1.48	2.66	2.85
Tibet	1.36	2.59	2.74	1.27	2.52	2.67
Yunnan	1.31	2.67	2.80	1.22	2.46	2.59
Qinghai	1.15	2.16	2.30	1.05	1.96	2.07
Gansu	0.86	1.86	1.95	0.76	1.42	1.50
Xinjiang	1.62	3.01	3.19	1.57	2.89	3.07
Mean	1.14	2.37	2.49	1.04	2.04	2.16

presents a comparison of the positioning accuracies of BDS-3/GPS dual-system dual-frequency pseudorange augmentation with and without correction of the pseudorange bias for Type B receivers. Moreover, the corrections of the pseudorange bias result in notable improvements in the horizontal, vertical, and three-dimensional positioning accuracies of Type B receivers at all stations. The average accuracies of all the stations increase from 1.14 m, 2.37 m, and 2.49 m to 1.04 m, 2.04 m, and 2.16 m, respectively. This represents an 8.8%, 13.9%, and 13.3% improvement, respectively.

Figure 10 and Figure 11 show the comparison of UERE with and without the pseudorange bias correction for two GPS and BDS-3 satellites at Hunan Station, respectively. The average UERE of both the GPS and BDS-3 satellites decreases significantly after correction. Specifically, the average UEREs of satellites G07 and G27 decreased by 0.58 m and 0.82 m, respectively; whereas the average UEREs of satellites C32 and C46 decreased by 0.78 m and 1.33 m, respectively. Figure 12 shows the RMS UERE for all the type B receivers on 2 December 2022. After correction, the RMS of each station’s UERE for GPS decreases by approximately 0.15 m, and the RMS of the UERE for BDS-3 decreases by approximately 0.4 m. The percentage reduction in UERE after correction is in general agreement with the improvement in the three-dimensional positioning accuracy of the GPS and BDS-3 satellites.

In conclusion, pseudorange bias correction can serve to further reduce the UERE and improve the user positioning accuracy for the dual-frequency pseudorange augmentation service. The magnitude of the pseudorange bias for the BDS-3 B1C/B2a dual-frequency pseudorange combination is greater than that for the GPS L1/L2 combination. Consequently, the improvement in the positioning accuracy for BDS-3 after pseudorange bias correction is more obvious than that for GPS, with an improvement of nearly 20%. Pseudorange bias correction can result in an improvement of approximately 14% in three-dimensional accuracy for the BDS-3/GPS dual-system.

4.3. PPP Service

4.3.1. The Impact of Pseudorange Bias on PPP

Figure 13 shows the static PPP convergence times of Type A and Type B receivers at each station for the GPS, BDS-3, and BDS-3/GPS dual-system. For BDS-3, the convergence time of Type A receivers is generally shorter than that of Type B receivers, which is consistent with the findings of Section 4.2.1 and indicates that the Type B receiver is different from the Type A receiver, resulting in relative pseudorange biases between the user and the server end. However, for the GPS, the convergence times of the two receiver types are relatively similar. This can be attributed to the fact that the pseudorange bias introduced by the two types of receivers for GPS is smaller than that for BDS-3. For the BDS-3/GPS dual-system, the overall convergence times for the two receiver types are similar. The convergence speed is significantly better than that of BDS-3 and slightly better than that of GPS.

Figure 14 shows the kinematic PPP convergence times of Type A and Type B receivers at each station for the GPS, BDS-3, and BDS-3/GPS dual-system. For BDS-3, similar to static PPP, Type B receivers generally require significantly more convergence time than Type A receivers do because of the influence of the relative pseudorange bias. In the case of the GPS, the overall convergence times of Type B receivers are relatively comparable to those of Type A receivers, as the effect of the relative pseudorange bias on Type B receivers is relatively minor. However, in the case of the BDS-3/GPS dual-system, in contrast to static PPP, Type A receivers demonstrate notably shorter convergence times than Type B receivers do.

4.3.2. User-Side Pseudorange Bias Correction

To assess the impact of pseudorange biases on the PPP service, the pseudorange biases calibrated in Section 4.1 are used as the user-end corrections.

Table 9. Convergence statistics of Type B receivers in static PPP (unit: minutes).

	BDS		GPS		BDS + GPS
Station	Without Correction	With Correction	Without Correction	With Correction	Without Correction	With Correction
Heilongjiang	39.07	27.74	27.90	28.74	28.90	35.57
Shanghai	59.06	41.23	23.73	19.90	30.06	22.90
Fujian	42.56	27.73	24.39	22.56	21.23	15.40
Guangdong	39.56	37.73	27.73	24.40	14.39	17.06
Hunan	64.39	64.90	28.89	19.56	20.56	20.90
Hainan	44.23	40.23	35.73	14.06	28.39	18.40
Sichuang	33.06	34.73	16.73	19.73	16.06	19.56
Tibet	29.57	25.74	15.57	17.40	16.73	18.40
Yunnan	51.56	46.90	20.89	14.06	17.06	15.73
Qinghai	55.56	52.40	29.23	31.23	20.06	22.90
Gansu	72.56	37.40	18.39	21.40	19.23	19.06
Xinjiang	54.73	25.73	19.23	20.40	14.56	17.06
Mean	48.83	38.54	24.03	21.12	20.60	20.25

presents a comparison of the static PPP convergence times with and without pseudorange bias correction for the Type B receiver at each station in the cases of the BDS-3, GPS, and BDS-3/GPS dual-system. This demonstrates that there is an improvement in the convergence speed after pseudorange bias correction for BDS-3. The average convergence times of all the stations without and with pseudorange bias correction are 48.83 min and 38.54 min, respectively. This represents a 21% improvement in convergence speed after correction. In the case of the GPS, there is a slight improvement in the convergence speed for Type B receivers after pseudorange bias correction. The average convergence times without and with correction are 24.03 min and 21.12 min, respectively, resulting in a 12% improvement in the convergence speed. In the case of the BDS-3/GPS dual-system, there is a minimal difference in convergence times without and with pseudorange bias correction, with average convergence times of 20.60 min and 20.25 min, respectively.

Table 10. Convergence statistics of Type B receivers in kinematic PPP (unit: minutes).

	BDS		GPS		BDS + GPS
Station	Without Correction	With Correction	Without Correction	With Correction	Without Correction	With Correction
Heilongjiang	69.07	41.40	52.07	51.90	24.90	28.40
Shanghai	109.73	57.56	35.90	34.56	21.56	15.23
Fujian	54.98	41.23	44.56	41.56	24.23	16.06
Guangdong	260.23	88.06	62.73	56.73	63.90	17.40
Hunan	57.06	58.73	68.90	42.90	40.73	18.90
Hainan	60.23	51.40	58.40	31.40	66.23	33.06
Sichuang	62.73	50.90	53.56	39.56	40.40	25.06
Tibet	80.24	61.99	23.74	35.49	17.07	22.74
Yunnan	189.23	50.56	36.23	41.73	71.56	23.56
Qinghai	67.23	64.23	48.06	37.06	45.23	32.06
Gansu	132.23	27.23	27.90	32.06	33.73	18.56
Xinjiang	171.40	162.56	29.23	45.06	20.23	24.23
Mean	109.53	62.99	45.10	40.83	39.15	22.94

presents a comparison of the kinematic PPP convergence times with and without pseudorange bias correction for the Type B receiver at each station in the cases of the BDS-3, GPS, and BDS-3/GPS dual-system. In the case of the BDS-3 and BDS-3/GPS dual-system, there is a significant overall improvement in the convergence speed after pseudorange bias correction. For BDS-3, the average convergence times without and with pseudorange bias correction are 109.53 min and 62.99 min, respectively. This represents a 42% improvement in convergence speed. Similarly, for the BDS-3/GPS dual-system, the average convergence times without and with pseudorange bias correction are 39.15 min and 22.94 min, respectively. This results in a 41% improvement in convergence speed. However, in the case of the GPS, the improvement in convergence speed is relatively small. The average convergence times without and with pseudorange bias correction are 45.10 min and 40.83 min, respectively. This results in a 9% improvement. These experimental results are in accordance with the analysis presented in Section 4.3.1 regarding the impact of pseudorange bias on kinematic PPP in the PPP-B2b service.

Pseudorange bias correction can reduce the convergence time for both static and kinematic PPP in the BDS-3 PPP service. Since the pseudorange biases for the BDS-3 dual-frequency pseudorange combination are greater than those for the GPS, the correction of the pseudorange bias can lead to a more obvious improvement in the convergence speed for BDS-3. The improvements are 21% and 42% for BDS-3 static and kinematic PPP, respectively. In BDS-3/GPS dual-system kinematic PPP, the implementation of pseudorange bias correction leads to an improvement in the convergence speed of approximately 9%.

5. Conclusions and Discussions

In this paper, we study the impact of the pseudorange bias on high-precision BDS-3 services. This work proposes a method for calibrating pseudorange bias and evaluating the effect of pseudorange bias on the BDS-3 dual-frequency pseudorange augmentation and PPP service. The conclusions of the paper are summarized as follows:

First, the pseudorange bias calibration method proposed in this paper is employed to calibrate the relative pseudorange bias of two collocated receivers for BDS-3 and GPS satellites. The results demonstrate that the calibrated pseudorange biases exhibit good spatial and temporal consistency.

Second, the effect of pseudorange bias on the BDS-3 dual-frequency pseudorange augmentation and PPP services is analyzed. For dual-frequency pseudorange augmentation services, the pseudorange biases contribute 17.2% and 7.7% to the UERE for BDS-3 and GPS, respectively. This indicates that this error is a significant error source that cannot be ignored in pseudorange augmentation services. In the simulation experiments, the convergence time of both static and kinematic PPP increases from 6 min and 26 min, respectively, to 19 min and 58 min, respectively. This finding indicates that pseudorange biases significantly increase the convergence time of the PPP service.

Finally, high-precision service product calculations and user positioning are performed by utilizing monitoring station observations and calibrated pseudorange biases. After the pseudorange bias is corrected, the average three-dimensional positioning accuracy of dual-frequency pseudorange augmentation increases by 18.5%, 6.3%, and 13.3% for the BDS-3, GPS, and BDS-3/GPS dual-system, respectively. The convergence times of static PPP decrease by 21% and 12% for BDS-3 and GPS, respectively, whereas the convergence times of kinematic positioning decrease by 42%, 9%, and 41% for the BDS-3, GPS, and BDS-3/GPS dual-system, respectively.

In conclusion, the methodology proposed in this work is effective in calibrating pseudorange biases for diverse receiver types. The pseudorange bias represents a significant error source for BDS-3 pseudorange augmentation and PPP services. After the pseudorange bias correction, both the positioning accuracy of the dual-frequency pseudorange augmentation service and the convergence performance of the PPP service are significantly improved.

Notably, the calibration results of the pseudorange bias in this contribution are based on two different types of receivers and do not cover all receiver configurations, such as the correlator spacing, front-end filter bandwidth, and anti-multipath algorithms. However, it is sufficient to indicate the impact of pseudorange bias on the accuracy of the dual-frequency pseudorange augmentation and PPP service.

The results and analyses presented in this article demonstrate the significant potential of pseudorange bias correction in enhancing the precision of the BDS-3 high-precision service. In order to fully eliminate the impact of pseudorange bias in the BDS-3 high-precision service, it is essential that the user-side receiver and the server-side receiver responsible for generating the orbit and clock correction are identical. Nevertheless, in practice, this is challenging to achieve. This is due to the fact that there are numerous varieties of user-side receivers, and it is not feasible to restrict the specific type that the user employs. In light of the aforementioned considerations, two recommendations are put forth with the aim of mitigating the impact of pseudorange bias. The first recommendation is to publish the brand, model, and configuration of the receiver utilized by the server-side of BDS-3 high-accuracy service in the interface control document (ICD). This would enable the user to make the requisite adjustments to their receiver configuration to align it as closely as possible with that of the server-side receiver. Second, the pseudorange biases between the service-side receiver and some mainstream user-side receivers are calibrated by the pseudorange bias calibration method, such as the method proposed in this paper, and are disclosed to the user as products. In this way, most users of the BDS-3 high-precision service can realize the correction of the pseudorange bias.