# A New GNSS Single-Epoch Ambiguity Resolution Method Based on Triple-Frequency Signals

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

- i—the i-th frequency, e.g., GPS L1, L2 or L5;
- f—frequency (Hz);
- P—pseudo-range measurements (meter);
- φ—carrier phase measurements (cycle);
- λ—wavelength (meter);
- ρ—geometric distance from satellite to receiver (meter);
- N—carrier phase ambiguity;
- K—the parameter of the first-order ionospheric delay, K = 40.28 TEC; TEC represents the Total Electron Content;
- T—tropospheric delay (meter);
- ${\epsilon}_{i,P}\text{},{\epsilon}_{i,\varphi}$—measurement noise of pseudo range and carrier phase, respectively.

## 3. GNSS Single-Epoch Ambiguity Resolution

#### 3.1. Ambiguity Resolution of BDS Triple-Frequency Signals

#### 3.2. GNSS Ambiguity Resolution under the Constraint of BDS Triple-Frequency Observation

_{1}, V

_{2}are the residual of BDS and the GNSS observations, respectively. ${X}^{\prime}$ is the vector of unknown parameters of the coordinate. $\nabla \u2206{N}_{e}$ and $\nabla \u2206{N}_{h}$ are the ambiguities of the BDS B1 observation and the other GNSS L1 observations. L

_{1}, L

_{2}are the carrier phase observation of BDS (B

_{1}) and the GNSS observations (L

_{1}), respectively. A

_{1}, A

_{2}and C

_{1}, C

_{2}are the corresponding coefficient matrices. Generally, Equation (28) can be simplified to

#### 3.3. GNSS Single-Epoch Ambiguity Resolution Based on Robust Estimation

## 4. Experiments and Discussion

#### 4.1. Ambiguity Resolution of BDS Triple-Frequency Signals

#### 4.2. Ambiguity Resolution of GPS/GLONASS Signals

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Structure of the new single-epoch Global Navigation Satellite Systems ambiguity resolution (GNSS AR) method for a short-range baseline based on triple-frequency signals.

**Figure 3.**Variation of distribution of the difference (DF) of three kinds of BDS satellites. GEO = Geostationary Earth Orbit; IGSO = Inclined Geosynchronous Satellite Orbit; MEO = Medium Earth Orbit. (

**a**) GEO; (

**b**) IGSO; (

**c**) MEO.

**Figure 5.**Deviation of coordinate (North). (

**a**) without robust estimation; (

**b**) with robust estimation.

**Figure 6.**Deviation of coordinate (East). (

**a**) without robust estimation; (

**b**) with robust estimation.

Without Robust Estimation | With Robust Estimation | |||||
---|---|---|---|---|---|---|

0–0.03 cm | 0.03–0.05 cm | >0.05 cm | 0–0.03 cm | 0.03–0.05 cm | >0.05 cm | |

North | 99.0 | 0.2 | 0.8 | 99.9 | 0.1 | 0 |

East | 99.0 | 0.5 | 0.5 | 99.9 | 0.1 | 0 |

Up | 98.2 | 0.6 | 1.2 | 99.4 | 0.6 | 0 |

Time to Fix (epoch) | S1 | S3 |
---|---|---|

1 | 100.00% | 97.53% |

2 | 99.41% | |

3 | 99.65% | |

4 | 99.69% | |

5 | 100.00% |

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**MDPI and ACS Style**

Wang, S.; Deng, J.; Lu, X.; Song, Z.; Xu, Y.
A New GNSS Single-Epoch Ambiguity Resolution Method Based on Triple-Frequency Signals. *ISPRS Int. J. Geo-Inf.* **2017**, *6*, 46.
https://doi.org/10.3390/ijgi6020046

**AMA Style**

Wang S, Deng J, Lu X, Song Z, Xu Y.
A New GNSS Single-Epoch Ambiguity Resolution Method Based on Triple-Frequency Signals. *ISPRS International Journal of Geo-Information*. 2017; 6(2):46.
https://doi.org/10.3390/ijgi6020046

**Chicago/Turabian Style**

Wang, Shengli, Jian Deng, Xiushan Lu, Ziyuan Song, and Ying Xu.
2017. "A New GNSS Single-Epoch Ambiguity Resolution Method Based on Triple-Frequency Signals" *ISPRS International Journal of Geo-Information* 6, no. 2: 46.
https://doi.org/10.3390/ijgi6020046