# Rapid Estimation of Undifferenced Multi-GNSS Real-Time Satellite Clock Offset Using Partial Observations

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Multi-GNSS Satellite Clock Offset Estimation Model

#### 2.2. Satellite Clock Offset Estimation Using Partial Observations

## 3. Results

#### 3.1. Experiment Setup

#### 3.2. Computation Efficiency

#### 3.3. Clock Offset Accuracy

#### 3.4. PPP Validation

## 4. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Number of dimensions of satellite clock offset estimation using all and partial observations with 50, 60, 70 and 80 stations.

**Figure 7.**Elevation mask for clock offset estimation using partial observations with 50, 60, 70 and 80 stations.

**Figure 8.**Elapsed time of clock offset estimation for each epoch using all and partial observations with different stations.

**Figure 9.**Clock offset accuracy of each satellite using all and partial observations with different stations.

Items | Strategies |
---|---|

Observations | GPS: L1/L2; Galileo: E1/E5a; BDS-3: B1I/B3I; GLONASS: G1/G2 |

Weight | E ≤ 30°, P = 2sinE; E > 30°, P = 1; |

Satellite orbit/ERP | Center for Orbit Determination in European (CODE) [30] |

Relativistic effect | Corrected [31] |

Wind-up | Corrected [32] |

Satellite DCB | CAS [33] |

Satellite PCO/PCV | igs14.atx |

Station coordinates | Fixed to IGS weekly solutions |

Station displacement | Solid tide, ocean tide, pole tide: IERS Convention 2010 [34] |

Receiver PCO/PCV | igs14.atx |

Estimator | Sequential least square adjustment [20] |

Satellite and receiver clock offset | White noise |

ISB | White noise |

Troposphere | Saastamoinen model [35] + GMF [36], estimated as a piece-wise constant every hour |

Ambiguity | Float solution, estimated as a constant for a station-satellite pair if no cycle slip |

**Table 2.**The mean number of dimensions of satellite clock offset estimation using all and partial observations with 50, 60, 70 and 80 stations.

Number of Stations | All Observations | Partial Observations | Reduction Rate |
---|---|---|---|

50 | 1871 | 1871 | 0% |

60 | 2221 | 1932 | 13.0% |

70 | 2575 | 1901 | 26.2% |

80 | 2919 | 1890 | 35.3% |

**Table 3.**Mean elapsed time of clock offset estimation for each epoch using all and partial observations with different stations (unit: second).

Number of Stations | All Observations | Partial Observations | Reduction Rate |
---|---|---|---|

50 | 2.86 | 2.86 | 0% |

60 | 6.80 | 3.10 | 54.4% |

70 | 7.93 | 2.97 | 62.6% |

80 | 12.04 | 3.14 | 73.9% |

Number of Stations | GPS | BDS-3 | Galileo | GLONASS | ||||
---|---|---|---|---|---|---|---|---|

All | Partial | All | Partial | All | Partial | All | Partial | |

50 | 0.0360 | 0.0360 | 0.0381 | 0.0381 | 0.0285 | 0.0285 | 0.1215 | 0.1215 |

60 | 0.0350 | 0.0351 | 0.0375 | 0.0378 | 0.0262 | 0.0265 | 0.1173 | 0.1202 |

70 | 0.0338 | 0.0358 | 0.0376 | 0.0416 | 0.0261 | 0.0275 | 0.1157 | 0.1255 |

80 | 0.0322 | 0.0338 | 0.0376 | 0.0455 | 0.0256 | 0.0278 | 0.1155 | 0.1512 |

**Table 5.**Positioning accuracy using the clock offset from different schemes at the 95% confidence level (unit: cm).

Number of Stations | All Observations | Partial Observations | ||||
---|---|---|---|---|---|---|

E | N | U | E | N | U | |

50 | 2.30 | 1.90 | 5.35 | 2.30 | 1.90 | 5.35 |

60 | 2.32 | 1.87 | 5.35 | 2.28 | 1.88 | 5.44 |

70 | 2.40 | 1.87 | 5.24 | 2.60 | 1.99 | 5.52 |

80 | 2.29 | 1.91 | 5.24 | 2.68 | 2.15 | 5.58 |

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**MDPI and ACS Style**

Xie, W.; Huang, G.; Fu, W.; Du, S.; Cui, B.; Li, M.; Tan, Y. Rapid Estimation of Undifferenced Multi-GNSS Real-Time Satellite Clock Offset Using Partial Observations. *Remote Sens.* **2023**, *15*, 1776.
https://doi.org/10.3390/rs15071776

**AMA Style**

Xie W, Huang G, Fu W, Du S, Cui B, Li M, Tan Y. Rapid Estimation of Undifferenced Multi-GNSS Real-Time Satellite Clock Offset Using Partial Observations. *Remote Sensing*. 2023; 15(7):1776.
https://doi.org/10.3390/rs15071776

**Chicago/Turabian Style**

Xie, Wei, Guanwen Huang, Wenju Fu, Shi Du, Bobin Cui, Mengyuan Li, and Yue Tan. 2023. "Rapid Estimation of Undifferenced Multi-GNSS Real-Time Satellite Clock Offset Using Partial Observations" *Remote Sensing* 15, no. 7: 1776.
https://doi.org/10.3390/rs15071776