# A Real-Time Linear Prediction Algorithm for Detecting Abnormal BDS-2/BDS-3 Satellite Clock Offsets

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Linear Clock Prediction Model and Prediction Threshold Set

^{−9}, 0.033 × 10

^{−9}. For MEO satellites in the real-time clock estimation process, the orbit error seems too small to consider setting another empirical threshold. The reason for setting different hour-boundary biases is that the real-time satellite clock is estimated with the hourly updated ultra-rapid orbit, and the IGSO/GEO satellite absorbs larger hour-boundary orbit error than the MEO satellite. Too large clock bias should be deleted since it will affect the positioning result due to large interpolation errors. Thus, this algorithm can automatically regulate the detection threshold to a reasonable range at hour-boundary epochs or other normal epochs.

#### 2.2. Linear Moving Short-Term Clock Prediction and Anomaly Detection Algorithm

## 3. Experiments, Results, and Discussion

#### 3.1. Abnormal Values in the Estimated Real-Time BDS-2/BDS-3 Clock Offsets

#### 3.2. Three Kinds of Typical Abnormal Real-Time Clock Offsets and Detection Algorithm Validation

#### 3.3. Improvement of Clock Performance after Removing Abnormal Values

#### 3.4. Performance of the Real-Time Linear Clock Prediction Algorithm

#### 3.5. PPP Validation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Flowchart of the linear moving short-term clock prediction and detection algorithm at one epoch.

**Figure 2.**Clock difference of the C05, C06, and C24 satellites between real-time estimated clock offsets and WHU final products.

**Figure 5.**Frequency modulation on the C02 satellite and abnormal clock detection results in the frequency (top) and phase (bottom) domains. The frequency modulation causes the real-time estimated clock offsets to be unavailable after about UTC 11:00.

**Figure 6.**Phase jump on the C03 satellite and abnormal clock detection results in the frequency (top) and phase (bottom) domains. The phase jump causes the real-time estimated clock offsets to be unavailable from UTC 05:00 to 06:00.

**Figure 7.**Daily clock STD accuracy improvement of the GEO satellite C05, the IGSO satellite C06, and the MEO satellite C24.

**Figure 9.**Frequency histogram of the clock STD improvement of the GEO, IGSO, and MEO satellites with the linear model.

**Figure 10.**Clock accuracy of predicting 0.5 h, 1.0 h, and 2.0 h satellite clock offsets with the linear model, the grey model, and the ARIMA model.

**Figure 11.**Average clock prediction accuracy for different prediction time lengths with the three models for different orbit types.

Orbit Type | Linear Model | GM(1,1) | ARIMA |
---|---|---|---|

GEO | 22.4% (0.048 ns) | 17.1% (0.037 ns) | 20.1% (0.048 ns) |

IGSO | 16.9% (0.014 ns) | 10.7% (0.009 ns) | 9.7% (0.009 ns) |

MEO | 5.9% (0.002 ns) | 1.5% (0.001 ns) | 2.8% (0.001 ns) |

BDS-2 | 19.7% (0.032 ns) | 13.9% (0.023 ns) | 14.9% (0.029 ns) |

BDS-3 | 5.9% (0.002 ns) | 1.5% (0.001 ns) | 2.8% (0.001 ns) |

Items | PPP |
---|---|

Observations | Undifferenced B1I/B3I phase and code ionosphere-free combination observation |

Elevation mask | 10° |

Observation weight | Elevation-dependent weight, 1 for E > 30° otherwise 2sin(E) |

Phase-windup effect | corrected |

Satellite antenna phase center and variation | igs14_2136.atx |

Receiver antenna phase center and variation | igs14_2136.atx |

Station displacement | Solid Earth tide, pole tide, ocean tide, loading: IERS Convention 2010 |

Relativistic effects | Corrected: IERS Convention 2010 |

Items | PPP |

Station | E (cm) | N (cm) | U (cm) |
---|---|---|---|

PTGG | 0.7 | 1.1 | 1.7 |

SIN1 | 1.2 | 0.1 | 3.3 |

STR1 | 0.6 | 0.1 | 0.1 |

GAMG | 0.6 | 0.2 | 0.4 |

NNOR | 0.3 | 0.3 | 3.4 |

MAL2 | 1.4 | 1.4 | 4.5 |

Average | 0. 8 | 0. 6 | 2.2 |

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**MDPI and ACS Style**

Gao, Y.; Chen, G.; Fu, W.; Chen, X.; Ma, L.; Luo, T.; Xue, D. A Real-Time Linear Prediction Algorithm for Detecting Abnormal BDS-2/BDS-3 Satellite Clock Offsets. *Remote Sens.* **2023**, *15*, 1831.
https://doi.org/10.3390/rs15071831

**AMA Style**

Gao Y, Chen G, Fu W, Chen X, Ma L, Luo T, Xue D. A Real-Time Linear Prediction Algorithm for Detecting Abnormal BDS-2/BDS-3 Satellite Clock Offsets. *Remote Sensing*. 2023; 15(7):1831.
https://doi.org/10.3390/rs15071831

**Chicago/Turabian Style**

Gao, Yaping, Guo Chen, Wenju Fu, Xi Chen, Liangliang Ma, Tong Luo, and Dongdong Xue. 2023. "A Real-Time Linear Prediction Algorithm for Detecting Abnormal BDS-2/BDS-3 Satellite Clock Offsets" *Remote Sensing* 15, no. 7: 1831.
https://doi.org/10.3390/rs15071831