# Clustering Code Biases between BDS-2 and BDS-3 Satellites and Effects on Joint Solution

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Observations

#### 2.2. ISCB Estimation Algorithm

## 3. Characteristics of BDS, 2 and BDS, 3 ISCB

#### 3.1. Data Preparation

#### 3.2. Clustering Biases between BDS, 2 and BDS, 3

#### 3.3. ISCB Time Variation Characteristics

## 4. Results and Discussion

#### 4.1. Systematic Biases in the BDS, 2+3 Joint Solution

#### 4.2. BDS, 2+3 Joint Model with Code Bias Self-Calibration

#### 4.3. Residual Analysis

#### 4.4. Improvements in BDS, 2+3 Joint PPP

## 5. Conclusions

- The ISCB real, time estimation algorithm for single stations presented in this paper accurately stripped off the receiver, dependent ISCB and clock offset biases from the pseudorange observations simultaneously.
- We analyzed the ISCB characteristics of BDS, 2 and BDS, 3 based on the B1I + B3I signal and found there to be an obvious clustering code bias phenomenon between the BDS, 2 and BDS, 3 satellites, leading to systematic biases existing in the BDS, 2+3 joint solution.
- We proposed the BDS, 2+3 joint solution with code bias self, calibration, which can accurately strip off clustering code biases between the BDS, 2 and BDS, 3 satellites and can greatly improve the PPP convergence speed and accuracy.
- The statistics showed that the residual biases and RMS of BDS, 2+3 joint PPP improved by 36% and 15%, respectively, and the convergence time improved by approximately 35%. In the convergence stage, the positioning accuracy improved by approximately 38% and 21% in the horizontal and vertical directions, respectively.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Average and stability of the BDS, 2 and BDS, 3 satellites’ inter, satellite code biases (ISCBs) of Septentrio and Trimble.

**Figure 3.**BDS, 2 and BDS, 3 ISCB clustering of Septentrio (NKLG and REDU) and Trimble (BRST and MCHL).

**Figure 7.**Comparison of the BDS, 2+3 pseudorange residuals before and after ISCB self, calibration. RMS, root mean square.

**Figure 8.**A comparison of the horizontal and vertical convergence of BDS, 2+3 joint PPP with and without ISCB self, calibration of MCHL on day of year (DOY) 363 of 2019.

**Figure 9.**BDS, 2+3 joint PPP comparison with and without ISCB self, calibration. CS, convergence stage accuracy; PS, post, convergence stage accuracy.

**Table 1.**The BeiDou navigation satellite system (BDS) satellite’s information (www.csno-tarc.cn/en/system/constellation, up to June 2020).

PRN | SVN | NORAD ID | SVN | System | Manufacture | Notes(UTC) | Civil Signal |
---|---|---|---|---|---|---|---|

C01 | C020 | 44231 | GEO-8 | BDS-2 | CAST | 17 May, 2019 | B1I/B2I/B3I |

C02 | C016 | 38953 | GEO-6 | BDS-2 | CAST | 25 Oct, 2012 | B1I/B2I/B3I |

C03 | C018 | 41586 | GEO-7 | BDS-2 | CAST | 12 June, 2016 | B1I/B2I/B3I |

C04 | C006 | 37210 | GEO-4 | BDS-2 | CAST | 1 Nov, 2010 | B1I/B2I/B3I |

C05 | C011 | 38091 | GEO-5 | BDS-2 | CAST | 25, Feb, 2012 | B1I/B2I/B3I |

C06 | C005 | 36828 | IGSO, 1 | BDS, 2 | CAST | 1, Aug, 2010 | B1I/B2I/B3I |

C07 | C007 | 37256 | IGSO, 2 | BDS, 2 | CAST | 18, Dec, 2010 | B1I/B2I/B3I |

C08 | C008 | 37384 | IGSO, 3 | BDS, 2 | CAST | 10, Apr, 2011 | B1I/B2I/B3I |

C09 | C009 | 37763 | IGSO, 4 | BDS, 2 | CAST | 27, Jul, 2011 | B1I/B2I/B3I |

C10 | C010 | 37948 | IGSO, 5 | BDS, 2 | CAST | 2, Dec, 2011 | B1I/B2I/B3I |

C11 | C012 | 38250 | MEO, 3 | BDS, 2 | CAST | 30, Apr, 2012 | B1I/B2I/B3I |

C12 | C013 | 38251 | MEO, 4 | BDS, 2 | CAST | 30, Apr, 2012 | B1I/B2I/B3I |

C13 | C017 | 41434 | IGSO, 6 | BDS, 2 | CAST | 30, Mar, 2016 | B1I/B2I/B3I |

C14 | C015 | 38775 | MEO, 6 | BDS, 2 | CAST | 19, Sep, 2012 | B1I/B2I/B3I |

C16 | C019 | 43539 | IGSO, 7 | BDS, 2 | CAST | 10, Jul, 2018 | B1I/B2I/B3I |

C19 | C201 | 43001 | MEO, 1 | BDS, 3 | CAST | 5, Nov, 2017 | B1I/B3I/B1C/B2a/B2b |

C20 | C202 | 43002 | MEO, 2 | BDS, 3 | CAST | 5, Nov, 2017 | B1I/B3I/B1C/B2a/B2b |

C21 | C206 | 43208 | MEO, 3 | BDS, 3 | CAST | 12, Feb, 2018 | B1I/B3I/B1C/B2a/B2b |

C22 | C205 | 43207 | MEO, 4 | BDS, 3 | CAST | 12, Feb, 2018 | B1I/B3I/B1C/B2a/B2b |

C23 | C209 | 43581 | MEO, 5 | BDS, 3 | CAST | 29, Jul, 2018 | B1I/B3I/B1C/B2a/B2b |

C24 | C210 | 43582 | MEO, 6 | BDS, 3 | CAST | 29, Jul, 2018 | B1I/B3I/B1C/B2a/B2b |

C25 | C212 | 43603 | MEO, 11 | BDS, 3 | SECM | 24, Aug, 2018 | B1I/B3I/B1C/B2a/B2b |

C26 | C211 | 43602 | MEO, 12 | BDS, 3 | SECM | 24, Aug, 2018 | B1I/B3I/B1C/B2a/B2b |

C27 | C203 | 43107 | MEO, 7 | BDS, 3 | SECM | 12, Jan, 2018 | B1I/B3I/B1C/B2a/B2b |

C28 | C204 | 43108 | MEO, 8 | BDS, 3 | SECM | 12, Jan, 2018 | B1I/B3I/B1C/B2a/B2b |

C29 | C207 | 43245 | MEO, 9 | BDS, 3 | SECM | 29, Mar, 2018 | B1I/B3I/B1C/B2a/B2b |

C30 | C208 | 43246 | MEO, 10 | BDS, 3 | SECM | 29, Mar, 2018 | B1I/B3I/B1C/B2a/B2b |

C32 | C213 | 43622 | MEO, 13 | BDS, 3 | CAST | 19, Sep, 2018 | B1I/B3I/B1C/B2a/B2b |

C33 | C214 | 43623 | MEO, 14 | BDS, 3 | CAST | 19, Sep, 2018 | B1I/B3I/B1C/B2a/B2b |

C34 | C216 | 43648 | MEO, 15 | BDS, 3 | SECM | 15, Oct, 2018 | B1I/B3I/B1C/B2a/B2b |

C35 | C215 | 43647 | MEO, 16 | BDS, 3 | SECM | 15, Oct, 2018 | B1I/B3I/B1C/B2a/B2b |

C36 | C218 | 43706 | MEO, 17 | BDS, 3 | CAST | 18, Nov, 2018 | B1I/B3I/B1C/B2a/B2b |

C37 | C219 | 43707 | MEO, 18 | BDS, 3 | CAST | 18, Nov, 2018 | B1I/B3I/B1C/B2a/B2b |

C38 | C220 | 44204 | IGSO, 1 | BDS, 3 | CAST | 20, Apr, 2019 | B1I/B3I/B1C/B2a/B2b |

C39 | C221 | 44337 | IGSO, 2 | BDS, 3 | CAST | 24, Jun, 2019 | B1I/B3I/B1C/B2a/B2b |

C40 | C224 | 44709 | IGSO, 3 | BDS, 3 | CAST | 4, Nov, 2019 | B1I/B3I/B1C/B2a/B2b |

C41 | C227 | 44864 | MEO, 19 | BDS, 3 | CAST | 16, Dec, 2019 | B1I/B3I/B1C/B2a/B2b |

C42 | C228 | 44865 | MEO, 20 | BDS, 3 | CAST | 16, Dec, 2019 | B1I/B3I/B1C/B2a/B2b |

C43 | C226 | 44794 | MEO, 21 | BDS, 3 | SECM | 23, Nov, 2019 | B1I/B3I/B1C/B2a/B2b |

C44 | C225 | 44793 | MEO, 22 | BDS, 3 | SECM | 23, Nov, 2019 | B1I/B3I/B1C/B2a/B2b |

C45 | C223 | 44543 | MEO, 23 | BDS, 3 | CAST | 22, Sep, 2019 | B1I/B3I/B1C/B2a/B2b |

C46 | C222 | 44542 | MEO, 24 | BDS, 3 | CAST | 22, Sep, 2019 | B1I/B3I/B1C/B2a/B2b |

C59 | C217 | 43683 | GEO, 1 | BDS, 3 | CAST | 1, Nov, 2018 | B1I/B3I |

C60 | C229 | 45344 | GEO, 2 | BDS, 3 | CAST | 9, Mar, 2020 | B1I/B3I |

C61 | C230 | 45807 | GEO, 3 | BDS, 3 | CAST | 23, Jun, 2020 | B1I/B3I |

Manufacturer | Type | Firmware | Antenna | Site | BDS Signal |
---|---|---|---|---|---|

Septentrio | POLARX5TR | 5.3.0 | SEPCHOKE_B3E6 | CEBR | B1I/B2I/B3I |

POLARX5 | 5.3.0 | SEPCHOKE_B3E6 | KIRU, KOUR, REDU | ||

TRM59800.00 | NKLG | ||||

Trimble | ALLOY | 5.37 5.37 | TRM57971.00 | BRST | B1I/B2I/B3I |

TRM59800.00 | MCHL | ||||

5.42 | TRM57971.00 | UNB3 | |||

5.43 | LEIAR25.R3 | KIR8, MAR7 | |||

NETR9 | 5.42 | TRM59800.00 | METG | ||

TRM115000.00 | POAL, POVE, SALU, SAVO, TOPL, UFPR |

**Table 3.**Processing strategy, model, and parameters of the inter, satellite code bias (ISCB) estimation.

Parameters | Model | |
---|---|---|

Observations | UD IF pseudorange/carrier phase | |

Prior information | Pseudorange 1.0 m; carrier phase 0.02 cycles | |

Cut, off elevation | 7° | |

Observation weights | p = 1, elev. >30° p = 2sin(elev.), elev. ≤30° | |

Interval | 30s | |

PCO/PCV | Satellite | GPS, Galileo PCO: IGS14.atx [28]; BDS, 2 GEO PCO: IGS M, GEX; BDS, 2 IGSO/MEO PCO: ESA Mode [29]; BDS, 3 MEO: www.beidou.gov.cn GPS PCV: IGS14.atx; BDS, Galileo PCV: Uncorrected |

Receiver | GPS PCO: IGS14.atx; BDS, Galileo PCO: Same as GPS; GPS PCV: IGS14.atx; BDS, Galileo PCV: Same as GPS | |

Adjustment method | Square root information filtering [30] | |

Troposphere delay | Saastamoinen model + GMF mapping function random, walk process for each epoch | |

Receiver clock offset | Estimated as white noise | |

Ambiguity | Float | |

ISCB | Constant estimation with zero, mean condition every day [25] |

**Table 4.**The average ISCBs of BDS, 2 and BDS, 3 and the inter, system bias (ISB) and standard deviation (STD) of the Septentrio and Trimble receivers (m).

Manufacturer | Station | ISCB Average | ISB between BDS, 2 and BDS, 3 | ISCB STD | ||
---|---|---|---|---|---|---|

BDS, 2 | BDS, 3 | BDS, 2 | BDS, 3 | |||

Septentrio | CEBR | 0.906 | −0.476 | 1.382 | 0.477 | 0.191 |

KIRU | 0.853 | −0.534 | 1.387 | 0.210 | 0.153 | |

KOUR | 1.267 | −0.248 | 1.515 | 0.417 | 0.197 | |

REDU | 0.905 | −0.612 | 1.517 | 0.383 | 0.225 | |

NKLG | 1.105 | −0.515 | 1.620 | 0.523 | 0.191 | |

Average | 1.007 | −0.477 | 1.484 | 0.402 | 0.191 | |

Trimble | BRST | −1.664 | 1.484 | −3.148 | 0.217 | 0.340 |

MCHL | −2.979 | 3.276 | −6.255 | 0.232 | 0.202 | |

UNB3 | −0.766 | 0.142 | −0.907 | 0.325 | 0.163 | |

KIR8 | −0.168 | 0.091 | −0.259 | 0.208 | 0.135 | |

MAR7 | −0.106 | 0.069 | −0.176 | 0.245 | 0.166 | |

METG | 0.497 | −0.467 | 0.964 | 0.233 | 0.176 | |

POAL | −0.885 | 0.223 | −1.108 | 0.418 | 0.255 | |

POVE | −0.686 | 0.172 | −0.858 | 0.416 | 0.289 | |

SALU | −0.669 | 0.167 | −0.836 | 0.551 | 0.427 | |

SAVO | −0.339 | 0.085 | −0.424 | 0.478 | 0.244 | |

TOPL | −0.633 | 0.158 | −0.792 | 0.408 | 0.211 | |

UFPR | −0.683 | 0.171 | −0.854 | 0.429 | 0.225 | |

Average | −0.757 | 0.464 | −1.221 | 0.347 | 0.236 |

**Table 5.**Convergence comparisons and improvements in the different convergence conditions of BDS, 2+3 joint PPP with and without ISCB self, calibration.

Model | Convergence Condition | Average | ||||
---|---|---|---|---|---|---|

40 cm | 30 cm | 20 cm | 10 cm | |||

Convergence time in the horizontal direction (min) | BDS | 52 | 68 | 91.5 | 133 | / |

BDS+ISCB | 18.5 | 38 | 58.5 | 99.5 | / | |

Improvement rate | 64% | 44% | 36% | 25% | 42% | |

Convergence time in the vertical direction (min) | BDS | 31.5 | 40 | 64 | / | / |

BDS+ISCB | 18.5 | 32.5 | 48.5 | / | / | |

Improvement rate | 41% | 19% | 24% | / | 28% |

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## Share and Cite

**MDPI and ACS Style**

Chen, L.; Li, M.; Zhao, Y.; Zheng, F.; Zhang, X.; Shi, C.
Clustering Code Biases between BDS-2 and BDS-3 Satellites and Effects on Joint Solution. *Remote Sens.* **2021**, *13*, 15.
https://doi.org/10.3390/rs13010015

**AMA Style**

Chen L, Li M, Zhao Y, Zheng F, Zhang X, Shi C.
Clustering Code Biases between BDS-2 and BDS-3 Satellites and Effects on Joint Solution. *Remote Sensing*. 2021; 13(1):15.
https://doi.org/10.3390/rs13010015

**Chicago/Turabian Style**

Chen, Liang, Min Li, Ying Zhao, Fu Zheng, Xuejun Zhang, and Chuang Shi.
2021. "Clustering Code Biases between BDS-2 and BDS-3 Satellites and Effects on Joint Solution" *Remote Sensing* 13, no. 1: 15.
https://doi.org/10.3390/rs13010015