# A Method to Accelerate the Convergence of Satellite Clock Offset Estimation Considering the Time-Varying Code Biases

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Undifferenced Model

#### 2.2. Models for Extracting Code Biases

#### 2.2.1. Undifferenced-SCB Model

#### 2.2.2. Undifferenced-RCB Model

#### 2.2.3. Undifferenced-SRCB Model

#### 2.3. Processing Strategies

## 3. Results

#### 3.1. Residual Analysis

#### 3.2. Code Bias Analysis

#### 3.3. Satellite Clock Validation

#### 3.4. Kinematic PPP Validation

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Overview of the experimental stations. Red circles denote 70 stations for satellite clock estimation, and blue triangles denote 57 stations for PPP experiment. The observation data are obtained from the IGS MGEX networks.

**Figure 8.**RMS and STD values of clock estimates compared with the IGS final products on DOY 031. The mean values are also plotted near the right edge of each panel. G01 is the reference satellite.

**Figure 9.**Time series of the clock biases with respect to the IGS final products on DOY 031. Different colors correspond to different satellites, and mean values are removed.

**Figure 10.**Comparison of positioning accuracy and convergence time with different ratio values on DOY 031.

**Figure 11.**Time series of the positioning errors using different satellite clocks for station ABPO on DOY 031.

**Figure 12.**RMS of positioning errors for different PPP solutions when compared to IGS weekly solutions at 57 stations from DOY 031 to 037.

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

Observations | GPS code and phase |

Priori noises | 0.3 and 0.003 m for raw code and phase observations in clock estimation, respectively; in PPP, 0.003 m for phase and priori noises for code discussed in Section 3 |

Cut-off elevation | 10° |

Differential code bias | CODE P1-C1 products |

Relativistic effect | Corrected |

Phase wind-up | Corrected |

PCOs/PCVs | igs14.atx |

Tidal displacements | Solid earth tide, ocean tide loading and pole tide |

Station coordinates | Fixed to IGS weekly solutions in clock estimation, and estimated as white noises in PPP |

Satellite orbits | Fixed to IGS final products |

Earth rotation effects | Fixed to IGS final products |

Receiver clocks | Estimated as white noises |

Satellite clocks | Estimated as white noises in clock estimation, and fixed to estimated clocks or IGS final products in PPP |

Zenith troposphere delays | Estimated as random-walk noises (${10}^{-7}{\mathrm{m}}^{2}/\mathrm{s}$) with respect to Saastamoinen model ^{1}, and NMF ^{2} is used |

Ambiguities | Estimated as constants over each continuous session |

Code biases | Estimated as white noises |

Estimator | Least square filter |

Models | Code Residuals (m) |
---|---|

Undifferenced | 1.033 |

Undifferenced-SCB | 1.014 |

Undifferenced-RCB | 0.997 |

Undifferenced-SRCB | 0.999 |

**Table 3.**The mean RMS and STD of clock estimates by different models when compared to the IGS final products on DOY 031.

Models | RMS (ns) | STD (ns) |
---|---|---|

Undifferenced | 0.73 | 0.15 |

Undifferenced-SCB | 1.24 | 0.06 |

Undifferenced-RCB | 0.72 | 0.15 |

Undifferenced-SRCB | 1.33 | 0.08 |

**Table 4.**The RMS and STD of clock estimates using different models when compared to the IGS final products from DOY 031 to 037.

DOY | Undifferenced Model | Undifferenced-SCB Model | ||
---|---|---|---|---|

RMS (ns) | STD (ns) | RMS (ns) | STD (ns) | |

031 | 0.73 | 0.15 | 1.24 | 0.06 |

032 | 0.69 | 0.14 | 0.85 | 0.07 |

033 | 0.71 | 0.14 | 0.82 | 0.09 |

034 | 0.73 | 0.15 | 1.12 | 0.07 |

035 | 0.70 | 0.15 | 0.85 | 0.08 |

036 | 0.70 | 0.15 | 1.13 | 0.08 |

037 | 0.71 | 0.14 | 0.75 | 0.06 |

Mean | 0.71 | 0.15 | 0.97 | 0.07 |

**Table 5.**The mean RMS (cm) for different PPP solutions when compared to IGS weekly solutions at 57 stations from DOY 031 to 037.

PPP Solutions | East | North | Up |
---|---|---|---|

IGS solution | 2.43 | 2.03 | 5.36 |

Undifferenced solution | 6.61 | 4.49 | 12.14 |

Undifferenced-SCB solution | 3.77 | 3.44 | 7.78 |

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**MDPI and ACS Style**

Liu, S.; Yuan, Y.
A Method to Accelerate the Convergence of Satellite Clock Offset Estimation Considering the Time-Varying Code Biases. *Remote Sens.* **2021**, *13*, 2714.
https://doi.org/10.3390/rs13142714

**AMA Style**

Liu S, Yuan Y.
A Method to Accelerate the Convergence of Satellite Clock Offset Estimation Considering the Time-Varying Code Biases. *Remote Sensing*. 2021; 13(14):2714.
https://doi.org/10.3390/rs13142714

**Chicago/Turabian Style**

Liu, Shuai, and Yunbin Yuan.
2021. "A Method to Accelerate the Convergence of Satellite Clock Offset Estimation Considering the Time-Varying Code Biases" *Remote Sensing* 13, no. 14: 2714.
https://doi.org/10.3390/rs13142714