# Signal-in-Space and Positioning Performance of BDS Open Augmentation Service

Authors: Liang Chen; Guangyu Zhou; Guo Chen; Weibin Sun; Luotian Pan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

As a typical augmentation satellite system for centimeter-level positioning, PPP (Precise Point Positioning) service based on PPP-B2b signal of China’s new generation BeiDou system (BDS-3) was released for Asia-Pacific area on July, 2020. This paper first introduces the recovery theory of PPP-B2b SSR (State Space Representation) messages and then implements a real-time PPP-B2b precise positioning terminal to decode and obtain the PPP-B2b SSR orbit, clock, and biases. On those bases, the coverage, signal-in-space, and positioning performance of PPP-B2b SSR are evaluated. The results show that PPP-B2b SSR of BDS-3 can cover the most area of 50°E-180°E, 30°S-70°N and improve signal-in-space accuracy significantly. The improvement rate on BDS clock offset can reach 77%, which far exceeds average 40% improvement on BDS orbit, while it is opposite for GPS with 80% improvement rate on orbit and 61% on clock offset. Average SISRE (signal-in-space range error) of PPP-B2b SSR is better than 10 cm. The 95% positioning results show that centimeter- to decimeter-level positioning can be achieved after a few minutes of convergence based on PPP-B2b SSR, which satisfies the BDS open service performance standard. Comparing to BDS PPP based on PPP-B2b SSR, BDS + GPS PPP can promote convergence significantly with average 40% improvement rate for 95% epochs and can further improve positioning accuracy with 30%-35% improvement rate in horizontal and 10%–15% in vertical direction.


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## Body

## 1. Introduction

Currently, there are two technology paths of GNSS precise positioning, i.e., GNSS OSR (Observation Space Representation) and SSR (State Space Representation) [1]. RTK (Real-Time Kinematic) is one typical OSR technology and another is PPP (Precise Point Positioning) for SSR [2]. As is known to all, PPP can obtain the absolute positions and time information for global area without relying on nearby reference stations, which is the main advantage comparing to RTK [3]. PPP realizes a user precise positioning based on precise GNSS orbit and clock messages and carrier phase observations, where the ionospheric-free (IF) combinations are usually used to eliminate ionospheric delay in signal transmission [4]. It needs about 15–30 minutes to convergence undifferenced (UD) ambiguity of PPP before obtaining a centimeter-level accuracy [5], while RTK can achieve instantaneous precise positioning. With the construction and upgrading of GNSS system, more and more GNSS provide the open precise positioning services, like the Galileo HAS (High Accuracy Service), QZSS (Quasi-Zenith Satellite System), CLAS (Centimeter Level Augmentation Service), and BDS PPP service [6]. The realization path of those open services is basically the same, i.e., receiver side can reconstruct precise augmentation messages by correcting those open SSR messages including orbit, clock, and atmospheric corrections to broadcast ephemeris.Galileo HAS, broadcasting high-accuracy SSR corrections by E6-B signal for Galileo and GPS signals, will provide initial service in 2021 and full service in 2024 [7]. The services include standard global PPP service and regional rapid positioning service by additional regional atmospheric, biases corrections, etc. [7]. QZSS CLAS, based on SSR, is an open nation-wide PPP-RTK service for Japan and has been operational since November 2018 [8], which can augment QZSS, GPS, and Galileo signals currently and will provide augmentation service for GLONASS in the future [9]. The satellite system developed by Australia and New Zealand jointly will also open the real-time PPP service through one Inmarsat satellite in 2023 [10]. GLONASS plans to have a commercial PPP service using its L3 signal in 2030 [11].BDS-3, China’s new generation satellite navigation system, was opened on July 31st, 2020, and can provide various global and regional services [12, 13]. BDS-3 constellation includes 3 GEO (Geostationary Earth Orbit), 3 IGSO (Inclined Geosynchronous Orbit), and 24 MEO (Medium Earth Orbit) satellites. BDS-3 is downward compatible with BDS-2 B1I/B3I signals, while broadcasting those new signals of B1C/B2a/B2b [13]. Currently, BDS PPP service based on PPP-B2b signal can achieve decimeter-level dynamic positioning and centimeter-level static positioning for Asia-Pacific area [14]. ISO 18197 is an international standard specification in space systems category [15] and describes those requirements for space-based centimeter-level positioning services [16]. According to this specification, an augmentation satellite system for centimeter‐level positioning consists of GNSS, augmentation satellites, augmentation satellite control stations, ground reference points, positioning augmentation centers, and user terminals as shown in Figure 1. So BDS PPP service based on BDS-3 PPP-B2b signal is a typical augmentation satellite system for centimeter‐level positioning.Figure 1
ISO 18197 typical augmentation satellite system for centimeter-level positioning [16].BDS PPP-B2b SSR messages include satellite orbit, satellite clock corrections, and DCB (differential code bias) [14], which are modulated in the PPP-B2b signal and broadcasted to receiver by 3 GEOs of BDS-3 constellation. According to the design, the structure of BDS-3 PPP-B2b signal can encode those SSR corrections of BDS, GPS, GLONASS, and Galileo [14]; however, only SSR corrections of BDS-3 and GPS are broadcasted for users in and around China currently [17]. BDS PPP-B2b SSR messages can correct the CNAV1 NAV ephemeris of BDS B1C signal and the LNAV NAV ephemeris of GPS L1C/A signal [13]. Receiver can recover those SSR corrections to real-time PPP-B2b ephemeris by matching the broadcast ephemeris and achieve precise positioning combined with carrier phase and pseudorange observations.The structure of this paper is as follows: firstly, the methodology part introduces the recovery theory of PPP-B2b SSR messages and realizes a real-time PPP-B2b precise positioning terminal to decode and obtain the PPP-B2b SSR orbit, clock, biases, etc. On those bases, the coverage of BDS PPP and BDS + GPS combined with PPP based on PPP-B2b SSR is evaluated. The orbit, clock, and SISRE are also analysed. Part of GNSS tracking stations in coverage area are used to analyse the real-time positioning performance of PPP-B2b messages by postprocessing kinematic mode, including convergence and accuracy. Experiment results and conclusions are provided in Conclusions.


## 2. Methodology

PPP is a method based on observations of a single receiver and externally known precise augmentation messages (satellite orbit, clock offset, atmospheric delay, and biases) to resolve receiver-side unknown parameters including precise coordinates, receiver clock offset, atmosphere, and ambiguity [5]. The observation and method of PPP-B2b SSR messages recovery are introduced in this section; then a real-time PPP-B2b precise positioning terminal is realized to encode and obtain the PPP-B2b SSR orbit, clock, biases, etc.


### 2.1. Observations

GNSS raw pseudorange and carrier phase observations of different frequencies are as follows [18]:(1)pr,js=ρrs+c⋅dtr+dr,j−c⋅dts+djs+Ir,js+Trs+εr,p,js,φr,js=ρrs+c⋅dtr+δr,j−c⋅dts+δjs−Ir,js+Trs+λj⋅Nr,js+εr,φ,js,where superscripts s and r identify one satellite and one receiver, respectively. p and φ represent the pseudorange and carrier phase observations. j is the frequency and the corresponding wavelength λj. Further, ρ is the signal travelled geometric distance from satellite broadcasting to receiver reception, and dtr and dts are clock offsets in receiver- and satellite-side. c is the velocity of light in vacuum; Irs is ionosphere delay; T is troposphere. N is ambiguity. ε is observation noise and multipath. dr,j and djs are receiver-dependent and satellite-dependent instrumental delay biases in pseudorange; δr,j and δjs are receiver-dependent and satellite-dependent instrumental delay biases in carrier-phase observations, respectively.There is no atmosphere augmentation message in BDS-3 PPP-B2b SSR currently, so the ionospheric delay can be eliminated by double-frequency IF combination [4] and tropospheric delay can be estimated as a zenith wet delay parameter using a wet mapping function as shown in equation (1).


### 2.2. SSR Messages Recovery

The PPP-B2b SSR message types are shown in Table1.Table 1
SSR message types in PPP-B2b [14].
Message type (in decimal)Correction content1Satellite mask2Satellite orbit correction and user range accuracy3Differential code bias4Satellite clock correction5User range accuracy index6Clock correction and orbit correction–combination 17Clock correction and orbit correction–combination 28–62Reserved63NullA group of IOD (issue of data) parameters are designed to build the interrelationship among those different message types, including IOD SSR (IOD of SSR), IODP (IOD of satellite pseudorandom noise mask), IODN (IOD of navigation data), and IOD Corr (IOD of orbit and clock correction). Currently, only SSR corrections of BDS-3 and GPS satellites are broadcasted to correct the CNAV1 NAV message of the BDS B1C signal and the LNAV NAV message of the GPS L1C/A signal [14].The parameter of IODN is used to search the corresponding navigation ephemeris by matching with the IODC (IOD of clock) parameter in BDS CNAV1 message of B1C signal and IODC in GPS LNAV message, respectively. The process can be described as follows: firstly comparing the parameters of PPP-B2b IODN and navigation messages IODC, a failure matching means that the navigation messages have been updated, so the previous navigation message should continue to be used until the PPP-B2b IODN is updated and matches the IODC. After matching the PPP-B2b SSR correction messages and navigation messages successfully, the next step is to recover the absolute SSR messages of satellite precise orbit, clock, and differential code bias.


#### 2.2.1. Orbit Recovery

According to ICD (Interface Control Document) of BDS PPP-B2b, the orbit corrections of radial-, along-, and cross-direction in satellite orbit coordinate system are provided, which can be noted asδO. Those orbit parameters in broadcast ephemeris express the position of the satellite in Earth coordinate system, the orbit correction vector δO should be translated to Earth coordinate system by the following equation:(2)eradial=−XX,ecross=−X×VX×V,ealong=−ecross×eradial,δX=eradialealongecross⋅δO,where X and V are the satellite position and velocity vector calculated by broadcast ephemeris, respectively; (eradial, ealong, ecross) is the projection vector between those two coordinate systems; δX is satellite orbit corrections in earth coordinate system.Then the corrected orbitXprec can be obtained by (3)Xprec=Xbrdc+δX.


#### 2.2.2. Clock Recovery

Like orbit corrections, clock corrections in PPP-B2b SSR are relative to clock offset calculated by broadcast ephemeris. The corrected satellite clock can be obtained by(4)dtprec=dtbrdc−C0c,where dtprec is the corrected precise clock of PPP-B2b SSR and dtbrdc is the clock parameter from broadcast ephemeris, respectively. c is the velocity of light; C0 is the clock corrections of PPP-B2b SSR.


#### 2.2.3. DCB

The differences among signal tracking modes lead to offsets in different frequency observations. PPP-B2b corrections also provide the DCB correction parameters to eliminate the differential code bias as(5)l˜=l−DCB,where l is the original pseudorange observations and l˜ is the corrected observations. When using double-frequency signals for PPP, the IF combination DCB corrections in (6) can be used to correct the satellite clock of PPP-B2b SSR directly.(6)d˜tprec=dtprec−γ⋅DCB1−DCB2γ−1,where d˜tprec is the PPP-B2b real-time satellite clock bias corrected by DCB, γ=f12/f22; DCB1 and DCB2 are the DCB of those two frequencies.


### 2.3. PPP-B2b Decoding

Based on BDS PPP-B2b signal and methodology above, PPP-B2b decoding process at the terminal is shown in Figure2. The terminal is mainly composed of a RF (radio frequency) front-end and signal processing and information processing modules. The module of RF front-end receives the signal from antenna and obtains an intermediate frequency digital signal after downconversion, filtering, and sampling. The signal processing module demodulates the digital signal to obtain GNSS observations, broadcast ephemeris, and PPP-B2b SSR messages. The information processing module mainly recovers the orbit, clock, and bias based on PPP-B2b SSR and broadcast ephemeris and then calculates the positioning information including coordinates, clock, and troposphere.Figure 2
Frame of BDS PPP-B2b decoding process at the terminal.


## 2.1. Observations

GNSS raw pseudorange and carrier phase observations of different frequencies are as follows [18]:(1)pr,js=ρrs+c⋅dtr+dr,j−c⋅dts+djs+Ir,js+Trs+εr,p,js,φr,js=ρrs+c⋅dtr+δr,j−c⋅dts+δjs−Ir,js+Trs+λj⋅Nr,js+εr,φ,js,where superscripts s and r identify one satellite and one receiver, respectively. p and φ represent the pseudorange and carrier phase observations. j is the frequency and the corresponding wavelength λj. Further, ρ is the signal travelled geometric distance from satellite broadcasting to receiver reception, and dtr and dts are clock offsets in receiver- and satellite-side. c is the velocity of light in vacuum; Irs is ionosphere delay; T is troposphere. N is ambiguity. ε is observation noise and multipath. dr,j and djs are receiver-dependent and satellite-dependent instrumental delay biases in pseudorange; δr,j and δjs are receiver-dependent and satellite-dependent instrumental delay biases in carrier-phase observations, respectively.There is no atmosphere augmentation message in BDS-3 PPP-B2b SSR currently, so the ionospheric delay can be eliminated by double-frequency IF combination [4] and tropospheric delay can be estimated as a zenith wet delay parameter using a wet mapping function as shown in equation (1).


## 2.2. SSR Messages Recovery

The PPP-B2b SSR message types are shown in Table1.Table 1
SSR message types in PPP-B2b [14].
Message type (in decimal)Correction content1Satellite mask2Satellite orbit correction and user range accuracy3Differential code bias4Satellite clock correction5User range accuracy index6Clock correction and orbit correction–combination 17Clock correction and orbit correction–combination 28–62Reserved63NullA group of IOD (issue of data) parameters are designed to build the interrelationship among those different message types, including IOD SSR (IOD of SSR), IODP (IOD of satellite pseudorandom noise mask), IODN (IOD of navigation data), and IOD Corr (IOD of orbit and clock correction). Currently, only SSR corrections of BDS-3 and GPS satellites are broadcasted to correct the CNAV1 NAV message of the BDS B1C signal and the LNAV NAV message of the GPS L1C/A signal [14].The parameter of IODN is used to search the corresponding navigation ephemeris by matching with the IODC (IOD of clock) parameter in BDS CNAV1 message of B1C signal and IODC in GPS LNAV message, respectively. The process can be described as follows: firstly comparing the parameters of PPP-B2b IODN and navigation messages IODC, a failure matching means that the navigation messages have been updated, so the previous navigation message should continue to be used until the PPP-B2b IODN is updated and matches the IODC. After matching the PPP-B2b SSR correction messages and navigation messages successfully, the next step is to recover the absolute SSR messages of satellite precise orbit, clock, and differential code bias.


### 2.2.1. Orbit Recovery

According to ICD (Interface Control Document) of BDS PPP-B2b, the orbit corrections of radial-, along-, and cross-direction in satellite orbit coordinate system are provided, which can be noted asδO. Those orbit parameters in broadcast ephemeris express the position of the satellite in Earth coordinate system, the orbit correction vector δO should be translated to Earth coordinate system by the following equation:(2)eradial=−XX,ecross=−X×VX×V,ealong=−ecross×eradial,δX=eradialealongecross⋅δO,where X and V are the satellite position and velocity vector calculated by broadcast ephemeris, respectively; (eradial, ealong, ecross) is the projection vector between those two coordinate systems; δX is satellite orbit corrections in earth coordinate system.Then the corrected orbitXprec can be obtained by (3)Xprec=Xbrdc+δX.


### 2.2.2. Clock Recovery

Like orbit corrections, clock corrections in PPP-B2b SSR are relative to clock offset calculated by broadcast ephemeris. The corrected satellite clock can be obtained by(4)dtprec=dtbrdc−C0c,where dtprec is the corrected precise clock of PPP-B2b SSR and dtbrdc is the clock parameter from broadcast ephemeris, respectively. c is the velocity of light; C0 is the clock corrections of PPP-B2b SSR.


### 2.2.3. DCB

The differences among signal tracking modes lead to offsets in different frequency observations. PPP-B2b corrections also provide the DCB correction parameters to eliminate the differential code bias as(5)l˜=l−DCB,where l is the original pseudorange observations and l˜ is the corrected observations. When using double-frequency signals for PPP, the IF combination DCB corrections in (6) can be used to correct the satellite clock of PPP-B2b SSR directly.(6)d˜tprec=dtprec−γ⋅DCB1−DCB2γ−1,where d˜tprec is the PPP-B2b real-time satellite clock bias corrected by DCB, γ=f12/f22; DCB1 and DCB2 are the DCB of those two frequencies.


## 2.2.1. Orbit Recovery

According to ICD (Interface Control Document) of BDS PPP-B2b, the orbit corrections of radial-, along-, and cross-direction in satellite orbit coordinate system are provided, which can be noted asδO. Those orbit parameters in broadcast ephemeris express the position of the satellite in Earth coordinate system, the orbit correction vector δO should be translated to Earth coordinate system by the following equation:(2)eradial=−XX,ecross=−X×VX×V,ealong=−ecross×eradial,δX=eradialealongecross⋅δO,where X and V are the satellite position and velocity vector calculated by broadcast ephemeris, respectively; (eradial, ealong, ecross) is the projection vector between those two coordinate systems; δX is satellite orbit corrections in earth coordinate system.Then the corrected orbitXprec can be obtained by (3)Xprec=Xbrdc+δX.


## 2.2.2. Clock Recovery

Like orbit corrections, clock corrections in PPP-B2b SSR are relative to clock offset calculated by broadcast ephemeris. The corrected satellite clock can be obtained by(4)dtprec=dtbrdc−C0c,where dtprec is the corrected precise clock of PPP-B2b SSR and dtbrdc is the clock parameter from broadcast ephemeris, respectively. c is the velocity of light; C0 is the clock corrections of PPP-B2b SSR.


## 2.2.3. DCB

The differences among signal tracking modes lead to offsets in different frequency observations. PPP-B2b corrections also provide the DCB correction parameters to eliminate the differential code bias as(5)l˜=l−DCB,where l is the original pseudorange observations and l˜ is the corrected observations. When using double-frequency signals for PPP, the IF combination DCB corrections in (6) can be used to correct the satellite clock of PPP-B2b SSR directly.(6)d˜tprec=dtprec−γ⋅DCB1−DCB2γ−1,where d˜tprec is the PPP-B2b real-time satellite clock bias corrected by DCB, γ=f12/f22; DCB1 and DCB2 are the DCB of those two frequencies.


## 2.3. PPP-B2b Decoding

Based on BDS PPP-B2b signal and methodology above, PPP-B2b decoding process at the terminal is shown in Figure2. The terminal is mainly composed of a RF (radio frequency) front-end and signal processing and information processing modules. The module of RF front-end receives the signal from antenna and obtains an intermediate frequency digital signal after downconversion, filtering, and sampling. The signal processing module demodulates the digital signal to obtain GNSS observations, broadcast ephemeris, and PPP-B2b SSR messages. The information processing module mainly recovers the orbit, clock, and bias based on PPP-B2b SSR and broadcast ephemeris and then calculates the positioning information including coordinates, clock, and troposphere.Figure 2
Frame of BDS PPP-B2b decoding process at the terminal.


## 3. Evaluation of BDS PPP-B2b SSR

For an open service provided by GNSS, the coverage and signal-in-space performance are key service indicators. According to basic methodology of PPP and signal beam range, the coverage of PPP-B2b SSR is shown in this part. Then the orbit and clock accuracy are evaluated independently. The SISRE (signal-in-space range error), the combination of orbit and clock error, is also analysed.


### 3.1. Coverage

From Table1, the BDS-3 can provide the PPP service for Asia-Pacific users via PPP-B2b signal broadcasted by three GEO satellites. There are many conditions constraining PPP-B2b service coverage like the coverage of BDS-3 three GEOs, the number, availability, and space geometry of visible satellites matched with SSR messages successfully. From (1), the estimated parameters include three positions, one receiver clock offset, one troposphere residual, and ambiguities in single-GNSS PPP, while an ISB (intersystem bias) parameter should be added into BDS + GPS PPP in order to realize time synchronization [5]. So minimum satellite number for single-GNSS and BDS + GPS PPP is five and six, respectively. In addition, the requirements for correction targets: the BDS radio navigation satellite service (RNSS) performance meets the “BDS open service performance standard (Version 3.0)” [13]; GPS service performance meets the requirements of “GPS Standard Positioning Service Performance Standard (Version 5.0)” [19]. Therefore, the PDOP 6 is also selected as a condition for coverage analysis.The broadcast ephemeris and PPP-B2b SSR correction messages of a regression period of BDS (7 days) are used to evaluate the coverage of BDS PPP service. The global area is divided into regular grids (such as 1° × 1°). With 300s sampling interval and 7° mask elevation angle, PDOP of each grid point in the regression period is calculated based on available satellites corrected by PPP-B2b SSR messages. Then availability percentage of PDOP less than certain conditions in each grid point is statistically analysed to evaluate the coverage of BDS PPP-B2b SSR.Figures3 and 4 give the service coverage of BDS and BDS + GPS PPP based on PPP-B2b SSR messages under the condition of PDOP ≤6 from DOY (day of year) 346 to 353, 2020, respectively. From Figure 3, we can find that the availability all over China can reach more than 90% under the condition of PDOP ≤6 and the service coverage is about 65°E–160°E, 15°S–70°N. After combining GPS, the coverage is further expanded to 50°E–180°E, 30°S–70°N.Figure 3
The service availability of BDS PPP based on PPP-B2b SSR (PDOP ≤6).Figure 4
The service availability of BDS + GPS PPP based on PPP-B2b SSR (PDOP ≤6).


### 3.2. Orbit and Clock

The basic principle of BDS-3 PPP-B2b augmentation service is to provide more precise ephemeris (orbit, clock, and biases) than broadcast ephemeris to improve SISRE accuracy. So the accuracy of satellite orbit and clock offset is the key technical indicators of PPP-B2b SSR ephemeris products, which determine the PPP positioning performance including accuracy and convergence.We need to note that the satellite orbits corrected by PPP-B2b SSR message refer to satellite’s APC (antenna phase center) [17], while the precise orbit products provided by IGS refer to satellite’s CoM (center-of-mass) [20]. Therefore, the satellite APC positions need to be converted to the satellite CoM positions by applying PCO (phase center offset) corrections [21], where GPS satellite’s PCO is provided by IGS14.atx [22] and BDS-3 satellite’s PCO is provided by BeiDou official website (https://en.beidou.gov.cn/SYSTEMS/Officialdocument/201912/P020200323536112807882.atx). The accuracy of BDS PPP-B2b clock is analysed by quadratic-difference method [4, 23, 24]. As shown in (7), one satellite is chosen as the reference to remove the datum bias of different products. After deducting datum time bias, the quadratic difference can be obtained by making difference further between PPP-B2b clock and precise clock products.(7)Δ∇dtB2b−GBMS=dtB2bS−dtB2bS_ref−dtGBMS−dtGBMS_ref.Then STD can be obtained by(8)STD=∑i=1nΔ∇dtiS−AVEΔ∇dtS2n−1,where Δ∇dtiS is the quadratic difference in i-th epoch. AVEΔ∇dtS, the average of these quadratic clock differences, includes not only initial clock offset biases caused by pseudorange but also DCB caused by different signals, which can be absorbed by float ambiguity parameter. Therefore, STD of clock quadratic difference is usually used to evaluate its accuracy. GNSS precise data processing based on IF dual-frequency combination observations leads to DCB of IF combinations existing in most of IGS clock products [25]. As we all know, the DCB can keep stable for a long time [26–28]. From (7) and (8), it is can be found that the DCB will be absorbed by AVEΔ∇dtS, and whether the dtB2bS is calculated from original equation (4) or DCB corrected equation (6) will not affect final clock accuracy analysis.In order to verify the accuracy of BDS PPP-B2b SSR messages, GBM GNSS precise orbit and clock provided by GFZ (German Research Center for Geosciences) are chosen as reference. Compared to other IGS Analysis Centers products, GBM products accuracy is about 1–3 cm for GPS and 4–7 cm for BDS non-GEO, and BDS orbit accuracy with SLR calibrating is less than 10 cm [25]. The PPP-B2b SSR messages of day of year (DOY) 346 to 353 of 2020 are analysed and the broadcast ephemeris accuracy in same time period is also evaluated as comparison.Figures5 and 6 show BDS and GPS orbit errors corrected by BDS PPP-B2b SSR in satellite orbit coordinate system, respectively. Table 2 shows the statistic results of BDS-3 and GPS broadcast ephemeris and PPP-B2b SSR message comparing to GBM precise products. It can be obtained from Figures 5 and 6 that GPS orbit of PPP-B2b SSR is better than BDS, especially in radial-direction. According to the statistic in Table 2, BDS broadcast orbit accuracy is about 30–35 cm in along- and cross-direction and 15 cm in along-direction and clock offset is about 0.9 ns. After PPP-B2b SSR correction, the BDS orbit accuracy can be improved to 22.60 cm, 16.31 cm, and 8.31 cm in along-, cross-, and radial-direction, and clock offset accuracy can achieve 0.21 ns. For GPS, the orbit accuracy is about 19.34 cm, 11.79 cm, and 6.10 cm in along-, cross-, and radial-direction, and clock offset accuracy is about 0.30 ns. It is obvious that the improvement rate of PPP-B2b SSR on BDS clock offset can reach 77%, which far exceeds about 40% improvement on BDS orbit. While it is opposite for GPS, the improvement rate on orbit can reach about 80% exceeding 61% on clock offset. We can also find from Table 2, due to the use of intersatellite links, the accuracy of BDS-3 broadcast ephemeris is better than GPS.Figure 5
BDS orbit errors corrected by BDS PPP-B2b in along-direction, cross-direction, and radial-direction.Figure 6
GPS orbit errors corrected by BDS PPP-B2b in along-direction, cross-direction, and radial-direction.Table 2
Accuracy statistics of BDS-3 and GPS broadcast ephemeris and PPP-B2b SSR message.
TypeOrbit RMS (cm)Clock bias STD (ns)AlongCrossRadialBDS-3BRDC33.3031.5515.430.90PPP-B2b SSR22.6016.318.310.21Improvement rate32%48%46%77%GPSBRDC110.0244.2338.310.77PPP-B2b SSR19.3411.796.100.30Improvement rate82%73%84%61%


### 3.3. Signal-in-Space Accuracy

From (1), we can find that users pay more attention on projection error in the line-of-sight direction from the satellite to receiver, which effects residuals in normal equation. Signal-in-Space Range Error (SISRE) is a statistical parameter of precise ephemeris projection errors [4], and calculation formula is as follows [29]:(9)SISRE=α⋅dR−c⋅dt2+dA2+dC2β,where dR, dA, dC are radial-, along-, and cross-direction orbit error in satellite orbit coordinate system, respectively; dt is the clock bias; c is the velocity of light in vacuum. According to methodology of SISRE, the different satellite orbit altitudes determinate the coefficient α and β, which are shown in Table 3. Equation (9) and the coefficients of Table 3 show the radial-direction orbit error and clock bias are main factors affecting SISRE. dR and dt influence each other and keep self-consistent for unified track and clock difference products.Table 3
SISRE coefficients of BDS and GPS.
SatelliteOrbit Altitude (km)αβBDSGEO/IGSO357860.99127MEO215280.9854GPS202000.9849Figures7 and 8 show SISRE time series and average of PPP-B2b SSR message from DOY 346 to 353, 2020. From Figure 7, most of BDS and GPS SISRE are less than 20 cm and can keep stable over time. According to statistic in Figure 8, the average SISRE of BDS-3 and GPS is 7.10 cm and 7.58 cm, respectively.Figure 7
BDS and GPS SISRE time series of BDS PPP-B2b SSR message.Figure 8
Average of BDS-3 and GPS SISRE of PPP-B2b SSR message.


## 3.1. Coverage

From Table1, the BDS-3 can provide the PPP service for Asia-Pacific users via PPP-B2b signal broadcasted by three GEO satellites. There are many conditions constraining PPP-B2b service coverage like the coverage of BDS-3 three GEOs, the number, availability, and space geometry of visible satellites matched with SSR messages successfully. From (1), the estimated parameters include three positions, one receiver clock offset, one troposphere residual, and ambiguities in single-GNSS PPP, while an ISB (intersystem bias) parameter should be added into BDS + GPS PPP in order to realize time synchronization [5]. So minimum satellite number for single-GNSS and BDS + GPS PPP is five and six, respectively. In addition, the requirements for correction targets: the BDS radio navigation satellite service (RNSS) performance meets the “BDS open service performance standard (Version 3.0)” [13]; GPS service performance meets the requirements of “GPS Standard Positioning Service Performance Standard (Version 5.0)” [19]. Therefore, the PDOP 6 is also selected as a condition for coverage analysis.The broadcast ephemeris and PPP-B2b SSR correction messages of a regression period of BDS (7 days) are used to evaluate the coverage of BDS PPP service. The global area is divided into regular grids (such as 1° × 1°). With 300s sampling interval and 7° mask elevation angle, PDOP of each grid point in the regression period is calculated based on available satellites corrected by PPP-B2b SSR messages. Then availability percentage of PDOP less than certain conditions in each grid point is statistically analysed to evaluate the coverage of BDS PPP-B2b SSR.Figures3 and 4 give the service coverage of BDS and BDS + GPS PPP based on PPP-B2b SSR messages under the condition of PDOP ≤6 from DOY (day of year) 346 to 353, 2020, respectively. From Figure 3, we can find that the availability all over China can reach more than 90% under the condition of PDOP ≤6 and the service coverage is about 65°E–160°E, 15°S–70°N. After combining GPS, the coverage is further expanded to 50°E–180°E, 30°S–70°N.Figure 3
The service availability of BDS PPP based on PPP-B2b SSR (PDOP ≤6).Figure 4
The service availability of BDS + GPS PPP based on PPP-B2b SSR (PDOP ≤6).


## 3.2. Orbit and Clock

The basic principle of BDS-3 PPP-B2b augmentation service is to provide more precise ephemeris (orbit, clock, and biases) than broadcast ephemeris to improve SISRE accuracy. So the accuracy of satellite orbit and clock offset is the key technical indicators of PPP-B2b SSR ephemeris products, which determine the PPP positioning performance including accuracy and convergence.We need to note that the satellite orbits corrected by PPP-B2b SSR message refer to satellite’s APC (antenna phase center) [17], while the precise orbit products provided by IGS refer to satellite’s CoM (center-of-mass) [20]. Therefore, the satellite APC positions need to be converted to the satellite CoM positions by applying PCO (phase center offset) corrections [21], where GPS satellite’s PCO is provided by IGS14.atx [22] and BDS-3 satellite’s PCO is provided by BeiDou official website (https://en.beidou.gov.cn/SYSTEMS/Officialdocument/201912/P020200323536112807882.atx). The accuracy of BDS PPP-B2b clock is analysed by quadratic-difference method [4, 23, 24]. As shown in (7), one satellite is chosen as the reference to remove the datum bias of different products. After deducting datum time bias, the quadratic difference can be obtained by making difference further between PPP-B2b clock and precise clock products.(7)Δ∇dtB2b−GBMS=dtB2bS−dtB2bS_ref−dtGBMS−dtGBMS_ref.Then STD can be obtained by(8)STD=∑i=1nΔ∇dtiS−AVEΔ∇dtS2n−1,where Δ∇dtiS is the quadratic difference in i-th epoch. AVEΔ∇dtS, the average of these quadratic clock differences, includes not only initial clock offset biases caused by pseudorange but also DCB caused by different signals, which can be absorbed by float ambiguity parameter. Therefore, STD of clock quadratic difference is usually used to evaluate its accuracy. GNSS precise data processing based on IF dual-frequency combination observations leads to DCB of IF combinations existing in most of IGS clock products [25]. As we all know, the DCB can keep stable for a long time [26–28]. From (7) and (8), it is can be found that the DCB will be absorbed by AVEΔ∇dtS, and whether the dtB2bS is calculated from original equation (4) or DCB corrected equation (6) will not affect final clock accuracy analysis.In order to verify the accuracy of BDS PPP-B2b SSR messages, GBM GNSS precise orbit and clock provided by GFZ (German Research Center for Geosciences) are chosen as reference. Compared to other IGS Analysis Centers products, GBM products accuracy is about 1–3 cm for GPS and 4–7 cm for BDS non-GEO, and BDS orbit accuracy with SLR calibrating is less than 10 cm [25]. The PPP-B2b SSR messages of day of year (DOY) 346 to 353 of 2020 are analysed and the broadcast ephemeris accuracy in same time period is also evaluated as comparison.Figures5 and 6 show BDS and GPS orbit errors corrected by BDS PPP-B2b SSR in satellite orbit coordinate system, respectively. Table 2 shows the statistic results of BDS-3 and GPS broadcast ephemeris and PPP-B2b SSR message comparing to GBM precise products. It can be obtained from Figures 5 and 6 that GPS orbit of PPP-B2b SSR is better than BDS, especially in radial-direction. According to the statistic in Table 2, BDS broadcast orbit accuracy is about 30–35 cm in along- and cross-direction and 15 cm in along-direction and clock offset is about 0.9 ns. After PPP-B2b SSR correction, the BDS orbit accuracy can be improved to 22.60 cm, 16.31 cm, and 8.31 cm in along-, cross-, and radial-direction, and clock offset accuracy can achieve 0.21 ns. For GPS, the orbit accuracy is about 19.34 cm, 11.79 cm, and 6.10 cm in along-, cross-, and radial-direction, and clock offset accuracy is about 0.30 ns. It is obvious that the improvement rate of PPP-B2b SSR on BDS clock offset can reach 77%, which far exceeds about 40% improvement on BDS orbit. While it is opposite for GPS, the improvement rate on orbit can reach about 80% exceeding 61% on clock offset. We can also find from Table 2, due to the use of intersatellite links, the accuracy of BDS-3 broadcast ephemeris is better than GPS.Figure 5
BDS orbit errors corrected by BDS PPP-B2b in along-direction, cross-direction, and radial-direction.Figure 6
GPS orbit errors corrected by BDS PPP-B2b in along-direction, cross-direction, and radial-direction.Table 2
Accuracy statistics of BDS-3 and GPS broadcast ephemeris and PPP-B2b SSR message.
TypeOrbit RMS (cm)Clock bias STD (ns)AlongCrossRadialBDS-3BRDC33.3031.5515.430.90PPP-B2b SSR22.6016.318.310.21Improvement rate32%48%46%77%GPSBRDC110.0244.2338.310.77PPP-B2b SSR19.3411.796.100.30Improvement rate82%73%84%61%


## 3.3. Signal-in-Space Accuracy

From (1), we can find that users pay more attention on projection error in the line-of-sight direction from the satellite to receiver, which effects residuals in normal equation. Signal-in-Space Range Error (SISRE) is a statistical parameter of precise ephemeris projection errors [4], and calculation formula is as follows [29]:(9)SISRE=α⋅dR−c⋅dt2+dA2+dC2β,where dR, dA, dC are radial-, along-, and cross-direction orbit error in satellite orbit coordinate system, respectively; dt is the clock bias; c is the velocity of light in vacuum. According to methodology of SISRE, the different satellite orbit altitudes determinate the coefficient α and β, which are shown in Table 3. Equation (9) and the coefficients of Table 3 show the radial-direction orbit error and clock bias are main factors affecting SISRE. dR and dt influence each other and keep self-consistent for unified track and clock difference products.Table 3
SISRE coefficients of BDS and GPS.
SatelliteOrbit Altitude (km)αβBDSGEO/IGSO357860.99127MEO215280.9854GPS202000.9849Figures7 and 8 show SISRE time series and average of PPP-B2b SSR message from DOY 346 to 353, 2020. From Figure 7, most of BDS and GPS SISRE are less than 20 cm and can keep stable over time. According to statistic in Figure 8, the average SISRE of BDS-3 and GPS is 7.10 cm and 7.58 cm, respectively.Figure 7
BDS and GPS SISRE time series of BDS PPP-B2b SSR message.Figure 8
Average of BDS-3 and GPS SISRE of PPP-B2b SSR message.


## 4. Performance of PPP-B2b Precise Positioning

The coverage and signal-in-space performance of BDS PPP-B2b SSR have been analysed above. Part of GNSS tracking stations in coverage area are used to assess the real-time kinematic positioning performance of PPP-B2b messages by postprocessing analysis, including convergence and accuracy.


### 4.1. Data and Strategy

The mission of iGMAS (international GNSS monitoring and assessment system) is to build global tracking stations to monitor and evaluate GNSS service performance [30]. Part stations of iGMAS in the coverage of BDS PPP-B2b signal are selected to evaluate the BDS-3 PPP service performance, which are shown in Figure 9 and can track all the open service signals of GNSS. Those data of DOY 340–353, 2020 are analysed following the processing strategy shown in Table 4. As mentioned above, the precise orbits recovered by PPP-B2b messages refer to the APC, so there is no need to correct the satellite-dependent PCO in precise positioning process based on PPP-B2b SSR.Figure 9
Distribution of tracking station.Table 4
Estimation strategy of precise positioning receiver based on BDS PPP-B2b.
ParametersModelObservationsUD IF carrier-phase + pseudorangeBDS-3: B1I + B3IGPS: L1 + L2Interval: 30 sElevation mask: 10°Prior informationPseudorange 1.0 m; carrier-phase 0.02 cyclesObservation weightp = 1, elevation >30°p = 2sin (elevation), elevation ≤30° [31]Ocean tidesIERS conventions 2003Earth solid tidesPhase wind-upCorrection [32]Relativistic effectsCorrectionAdjustment methodSquare-root information filter [4]Receiver clock offsetEstimated as white noiseAmbiguityFloatTroposphere delaySaastamoinen + GMFSingle epoch random-walk processISBEstimated as random-walk process [5]


### 4.2. Positioning Performance

Bases on those station’s positioning results in different days, previous-4-hours including convergence period are analysed based on the following rules:(1)
The previous-4-hours start from the first epoch which can achieve positioning based on PPP-B2b SSR(2)
All those positioning results in the previous-4-hours regardless the different stations and epoch time formed a data pool(3)
The positioning errors of the data pool at same epoch in the previous-4-hours are sorted from small to large(4)
Then count the positioning error of mean, 68% and 95% epoch to analyse the positioning convergence and accuracyFigures10 and 11 show the previous-4-hours BDS and BDS + GPS PPP convergence statistic results based on BDS PPP-B2b SSR, where the red, blue, and black lines are the mean, 68%, and 95% statistical results, respectively. The convergence condition, coming from newest BDS open service performance standard [13], is shown as red dotted line, which is 30 cm in horizontal and 60 cm in vertical direction for BDS PPP and 20 cm and 40 cm for BDS + GPS PPP, respectively. Table 5 shows the convergence comparison of BDS PPP and BDS + GPS combined with PPP based on BDS PPP-B2b SSR messages under those two conditions. From Figures 10 and 11 and Table 5, we can find that BDS PPP based PPP-B2b SSR can converge to 30 cm in horizontal direction and 60 cm in vertical direction after an average of 5 minutes and 2.5 minutes, separately. BDS + GPS PPP based on PPP-B2b SSR can converge to 20 cm in horizontal direction and 40 cm in vertical direction after an average of 10 minutes and 3.5 minutes, separately. 68% statistic results show it takes 8.5 minutes in horizontal and 3.5 minutes in vertical direction to achieve the BDS PPP convergence described in BDS OS PS and 15.5 minutes and 7 minutes for BDS + GPS PPP.Figure 10
Previous-4-hours convergence statistic of BDS PPP based on BDS PPP-B2b SSR.Figure 11
Previous-4-hours convergence statistic of BDS + GPS PPP based on BDS PPP-B2b SSR.Table 5
Convergence statistic of BDS PPP and BDS + GPS PPP based on BDS PPP-B2b SSR.
ModeConditionConvergence (min)Improvement rateMean68%95%Mean68%95%BDSH (20 cm)11.520.5118—V (40 cm)4.56.554.5H (30 cm)58.558V (60 cm)2.53.523.5BDS + GPSH (20 cm)1015.566.513.04%24.39%43.64%V (40 cm)3.573722.22%−7.69%32.11%H (30 cm)6831−20.00%5.88%46.55%V (60 cm)2415.520.00%−14.29%34.04%Average—39.09%As a comparison, the convergence of BDS PPP and BDS + GPS PPP under same condition is also analysed and those improvements are shown in Table5. From Table 5, there is a difference of several epochs for the two positioning modes in normal situations. However, for most cases like 95%, GPS corrections of PPP-B2b SSR can promote convergence significantly with average 40% improvement rate for 95% epochs.Table6 provides accuracy statistic of BDS PPP and BDS + GPS combined with PPP based on BDS PPP-B2b SSR messages after convergence. From Table 6, we can find that BDS PPP based on PPP-B2b SSR can achieve an average of 7.93 cm in horizontal and 12.43 cm in vertical direction after convergence. The 68% statistic results, the common statistic percentage, show that the accuracy of BDS PPP is 10.83 cm and 18.51 cm in horizontal and vertical direction, while the BDS + GPS PPP is 7.01 cm and 16.49 cm, separately. The 95% statistic results show that the two positioning modes can satisfy the BDS open service performance standard, i.e., 30 cm in horizontal and 60 cm in vertical direction for BDS PPP and 20 cm and 40 cm for BDS + GPS PPP. BDS + GPS PPP based on PPP-B2b SSR can improve positioning accuracy compared to BDS PPP significantly, and the improvement rate is about 34% in horizontal and 12% in vertical direction.Table 6
Accuracy statistic of BDS PPP and BDS + GPS PPP based on BDS PPP-B2b SSR.
ModeAccuracy (cm)Improvement rateMean68%95%Mean68%95%BDSH7.9310.8319.41—V12.4318.5134.99BDS + GPSH5.247.0113.0833.92%35.27%32.61%V10.616.4931.1914.72%10.91%10.86%


## 4.1. Data and Strategy

The mission of iGMAS (international GNSS monitoring and assessment system) is to build global tracking stations to monitor and evaluate GNSS service performance [30]. Part stations of iGMAS in the coverage of BDS PPP-B2b signal are selected to evaluate the BDS-3 PPP service performance, which are shown in Figure 9 and can track all the open service signals of GNSS. Those data of DOY 340–353, 2020 are analysed following the processing strategy shown in Table 4. As mentioned above, the precise orbits recovered by PPP-B2b messages refer to the APC, so there is no need to correct the satellite-dependent PCO in precise positioning process based on PPP-B2b SSR.Figure 9
Distribution of tracking station.Table 4
Estimation strategy of precise positioning receiver based on BDS PPP-B2b.
ParametersModelObservationsUD IF carrier-phase + pseudorangeBDS-3: B1I + B3IGPS: L1 + L2Interval: 30 sElevation mask: 10°Prior informationPseudorange 1.0 m; carrier-phase 0.02 cyclesObservation weightp = 1, elevation >30°p = 2sin (elevation), elevation ≤30° [31]Ocean tidesIERS conventions 2003Earth solid tidesPhase wind-upCorrection [32]Relativistic effectsCorrectionAdjustment methodSquare-root information filter [4]Receiver clock offsetEstimated as white noiseAmbiguityFloatTroposphere delaySaastamoinen + GMFSingle epoch random-walk processISBEstimated as random-walk process [5]


## 4.2. Positioning Performance

Bases on those station’s positioning results in different days, previous-4-hours including convergence period are analysed based on the following rules:(1)
The previous-4-hours start from the first epoch which can achieve positioning based on PPP-B2b SSR(2)
All those positioning results in the previous-4-hours regardless the different stations and epoch time formed a data pool(3)
The positioning errors of the data pool at same epoch in the previous-4-hours are sorted from small to large(4)
Then count the positioning error of mean, 68% and 95% epoch to analyse the positioning convergence and accuracyFigures10 and 11 show the previous-4-hours BDS and BDS + GPS PPP convergence statistic results based on BDS PPP-B2b SSR, where the red, blue, and black lines are the mean, 68%, and 95% statistical results, respectively. The convergence condition, coming from newest BDS open service performance standard [13], is shown as red dotted line, which is 30 cm in horizontal and 60 cm in vertical direction for BDS PPP and 20 cm and 40 cm for BDS + GPS PPP, respectively. Table 5 shows the convergence comparison of BDS PPP and BDS + GPS combined with PPP based on BDS PPP-B2b SSR messages under those two conditions. From Figures 10 and 11 and Table 5, we can find that BDS PPP based PPP-B2b SSR can converge to 30 cm in horizontal direction and 60 cm in vertical direction after an average of 5 minutes and 2.5 minutes, separately. BDS + GPS PPP based on PPP-B2b SSR can converge to 20 cm in horizontal direction and 40 cm in vertical direction after an average of 10 minutes and 3.5 minutes, separately. 68% statistic results show it takes 8.5 minutes in horizontal and 3.5 minutes in vertical direction to achieve the BDS PPP convergence described in BDS OS PS and 15.5 minutes and 7 minutes for BDS + GPS PPP.Figure 10
Previous-4-hours convergence statistic of BDS PPP based on BDS PPP-B2b SSR.Figure 11
Previous-4-hours convergence statistic of BDS + GPS PPP based on BDS PPP-B2b SSR.Table 5
Convergence statistic of BDS PPP and BDS + GPS PPP based on BDS PPP-B2b SSR.
ModeConditionConvergence (min)Improvement rateMean68%95%Mean68%95%BDSH (20 cm)11.520.5118—V (40 cm)4.56.554.5H (30 cm)58.558V (60 cm)2.53.523.5BDS + GPSH (20 cm)1015.566.513.04%24.39%43.64%V (40 cm)3.573722.22%−7.69%32.11%H (30 cm)6831−20.00%5.88%46.55%V (60 cm)2415.520.00%−14.29%34.04%Average—39.09%As a comparison, the convergence of BDS PPP and BDS + GPS PPP under same condition is also analysed and those improvements are shown in Table5. From Table 5, there is a difference of several epochs for the two positioning modes in normal situations. However, for most cases like 95%, GPS corrections of PPP-B2b SSR can promote convergence significantly with average 40% improvement rate for 95% epochs.Table6 provides accuracy statistic of BDS PPP and BDS + GPS combined with PPP based on BDS PPP-B2b SSR messages after convergence. From Table 6, we can find that BDS PPP based on PPP-B2b SSR can achieve an average of 7.93 cm in horizontal and 12.43 cm in vertical direction after convergence. The 68% statistic results, the common statistic percentage, show that the accuracy of BDS PPP is 10.83 cm and 18.51 cm in horizontal and vertical direction, while the BDS + GPS PPP is 7.01 cm and 16.49 cm, separately. The 95% statistic results show that the two positioning modes can satisfy the BDS open service performance standard, i.e., 30 cm in horizontal and 60 cm in vertical direction for BDS PPP and 20 cm and 40 cm for BDS + GPS PPP. BDS + GPS PPP based on PPP-B2b SSR can improve positioning accuracy compared to BDS PPP significantly, and the improvement rate is about 34% in horizontal and 12% in vertical direction.Table 6
Accuracy statistic of BDS PPP and BDS + GPS PPP based on BDS PPP-B2b SSR.
ModeAccuracy (cm)Improvement rateMean68%95%Mean68%95%BDSH7.9310.8319.41—V12.4318.5134.99BDS + GPSH5.247.0113.0833.92%35.27%32.61%V10.616.4931.1914.72%10.91%10.86%


## 5. Conclusions

The BDS PPP service based on PPP-B2b SSR is introduced in this paper firstly, including the recovery theory of PPP-B2b SSR messages and decoding process. Then the signal-in-space coverage and augmentation accuracy of BDS PPP-B2b SSR are evaluated. Finally the positioning performance of PPP-B2b SSR is analysed based on GNSS tracking stations. The experiment results show the following:(1)
The BDS PPP-B2b signal can provide PPP service for the most area of 65°E–160°E, 15°S–70°N. After combining GPS, the coverage is further expanded to 50°E–180°E, 30°S–70°N.(2)
BDS PPP-B2b SSR can improve signal-in-space accuracy significantly with 40% improvement on BDS orbit, 77% improvement on BDS clock offset, 80% on GPS orbit, and 61% on GPS clock offset. The average SISRE of BDS-3 and GPS is 7.10 cm and 7.58 cm, respectively.(3)
The 95% positioning results show that centimeter- to decimeter-level positioning can be achieved after a few minutes of convergence based on PPP-B2b SSR. Combining GPS corrections can reduce about 40% convergence time compared to single BDS PPP and can further improve positioning accuracy.As an open service, the BDS PPP-B2b service will further promote mass high-precision applications in mass-market, including autonomous vehicles and precision agriculture.



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*Source: 1112646-2022-04-01.xml*