# Assessment of Multi-Frequency PPP Ambiguity Resolution Using Galileo and BeiDou-3 Signals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Multi-Frequency PPP Model

#### 2.2. Triple-Frequency PPP Ambiguity Resolution

#### 2.3. Galileo/BeiDou-3 Combination Observables for Rapid PPP-AR

## 3. Data Processing

## 4. Results

#### 4.1. PPP-WAR

#### 4.2. PPP-AR

#### 4.3. Vehicle-Borne Experiment

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The left panel shows the distribution of stations used for product calculation and PPP where cycles, triangles, and stars denote stations equipped with Javad, Septentrio, and Trimble receivers, respectively. The right panel shows the distribution of the regional network established for vehicle-borne experiment.

**Figure 3.**One-hour PPP-AR solutions of SAS2 on UTC 20:00, day 183 of 2020. The dashed gray lines are ±5 cm for horizontal directions and ±10 cm for vertical direction, while the vertical dashed lines mark the convergence epoch of each solution.

**Figure 4.**Distribution of PPP-AR convergence times with respect to dual satellite clock cases. Panels (

**a**–

**c**) display the results of Group A triple-frequency, Group B triple-frequency, and dual-frequency PPP-AR, respectively, while (

**d**–

**f**) are the corresponding distributions calculated with the solutions that observed more than 10 satellites. The percentages of the solutions converging successfully in 1 h, 5 min, and 1 min are also given in the right-top corner of each panel.

**Figure 5.**The top panel shows the satellite numbers and PDOP values during experiment period. The dotted red line marks the epoch when the vehicle started moving. The bottom panel is the snapshot of the experiment environment nearing to the high voltage overhead power lines.

**Figure 6.**Positioning differences of east, north, and up components between vehicle-borne PPP-AR and short-baseline solutions. Panels (

**a**–

**c**) are solutions which were processed from the beginning of the static period while in panels (

**d**–

**f**) we only processed the epochs when the vehicle was moving. Dashed cyan and black lines represent the epochs when Group A and Group B triple-frequency PPP-AR converged successfully. Finally, dashed gray lines mark the epoch from which the vehicle began to move.

**Figure 7.**Galileo/BeiDou-3 satellites pseudorange residuals of Group A PPP-AR and the elevation angles. Cycle slips are marked with short vertical lines on each elevation curve. The number of satellites with cycle slips at each epoch is marked in the bottom panel.

**Figure 8.**Clock differences between legacy satellite L1/L2 clock and the clock on the third-frequency L3. Panels (

**a**,

**b**) denote the results of Galileo E1/E5a/E6 and E1/E5a/E5b, while panel (

**c**) gives the elevation angles of Galileo satellites at stations WTZZ. Panels (

**d**–

**f**) are the results for BeiDou-3 counterpart.

**Table 1.**GPS, Galileo, and BeiDou-2/3 signal combinations used to form ionosphere-free wide-lane observables. Noise amplification factors are given in the third column. The wavelengths of (extra-)wide-lane and narrow-lane ambiguities are listed in the last three columns.

Constellation | Signals | Noise Factor | ${\mathit{\lambda}}_{\mathit{e}\mathit{w}\mathit{l}}\left(\mathbf{m}\right)$ | ${\mathit{\lambda}}_{\mathit{w}\mathit{l}}\left(\mathbf{m}\right)$ | ${\mathit{\lambda}}_{\mathit{n}\mathit{l}}\left(\mathbf{m}\right)$ |
---|---|---|---|---|---|

GPS/QZSS | L1/L2/L5 | 109.98 | 5.86 | 0.86 | 0.11 |

Galileo | E1/E5a/E6 | 67.03 | 2.93 | 0.75 | 0.11 |

E1/E5a/E5b | 172.29 | 9.77 | 0.75 | 0.11 | |

E1/E5a/E5 | 331.04 | 19.54 | 0.75 | 0.11 | |

BeiDou-2 | B1I/B2I/B3I | 114.37 | 4.88 | 0.85 | 0.11 |

BeiDou-3 | B1C/B2a/B3I | 71.23 | 3.26 | 0.75 | 0.11 |

B1C/B2a/B2b | 172.29 | 9.77 | 0.75 | 0.11 | |

B1C/B1I/B2a | 620.04 | 20.93 | 0.78 | 0.10 |

Receiver Type | Stations | Signal |
---|---|---|

Javad TRE_3 DELTA | 15 | Galileo: E1/E5a/E5b/E6 BeiDou-3: B1I/B1C/B2a/B2b/B3I |

Septentrio PolaRX5 | 6 | Galileo: E1/E5a/E5b/E6 BeiDou-3: B1I/B1C/B2a/B3I |

Trimble Alloy | 1 | Galileo: E1/E5a/E5b/E6 BeiDou-3: B1I/B2a/B2b/B3I |

Item | Strategy |
---|---|

Observation used | Galileo: E1/E5a/E5b/E6 BeiDou-3: B1I/B1C/B2a/B2b/B3I |

Filter | Square root information filter |

Elevation cut-off angle | 7° |

Weighting strategy | Elevation-dependent; a priori noise of 3 mm and 0.3 m for carrier-phase and pseudorange, respectively |

Troposphere delay | Saastamoinen model [31]; estimated hourly based on global mapping function [32] with a process noise of $2cm/\sqrt{hour}$ |

Ionosphere delay | Estimated as random-walk-like parameters with process noise of $25m/\sqrt{30\mathrm{s}}$ |

Receiver clock | Estimated as white-noise-like parameter |

Satellite clock | Estimated as white-noise like parameter |

Inter-system bias | Estimated as a constant |

Ambiguity resolution | Searched by LAMBDA with a ratio test threshold of 3; |

Partial ambiguity resolution [33] | Maximum exclude: 4 Minimum reserve: 4 |

Data processing length | Clock estimation: 1 day PPP-WAR/PPP-AR: 1 h |

**Table 4.**PPP-WAR convergence times and positioning precisions for one month of solutions. Mean, 50th, and 90th percentiles of the convergence time are given and delimited by two slashes. “-“ denotes those solutions that did not converge within 1 h. Note that only the first ten minutes of each solution was used to calculate the positioning RMS.

Strategy | Legacy Satellite Clocks | Dual Satellite Clocks | |||
---|---|---|---|---|---|

Convergence Time (min) | Positioning RMS (m) | Convergence Time (min) | Positioning RMS (m) | ||

Group A | Float PPP | 31.9/28.5/- | 0.15/0.14/0.23 | 31.8/28.5/- | 0.15/0.14/0.23 |

PPP-WAR | 26.3/20.0/- | 0.06/0.08/0.14 | 18.2/13.5/44.0 | 0.05/0.07/0.13 | |

Group B | Float PPP | 32.0/28.5/- | 0.15/0.14/0.23 | 31.9/28.5/- | 0.15/0.14/0.23 |

PPP-WAR | 25.1/19.0/- | 0.07/0.09/0.17 | 25.0/19.5/- | 0.08/0.09/0.17 |

**Table 5.**Convergence times of PPP-AR over all solutions. Mean, 50th, and 90th percentiles are given, delimited by two slashes. The fourth column is the improvement of solutions using dual clocks compared with dual-frequency counterparts.

Strategy | Convergence Time (Minutes) | Rates (%) | |
---|---|---|---|

Legacy Satellite Clocks | Dual Satellite Clocks | ||

All Solutions | |||

Dual. PPP-AR | 11.6/7.0/29.0 | -- | -- |

Group A triple. PPP-AR | 8.1/2.0/21.5 | 5.6/1.0/16.0 | 31% |

Group B triple. PPP-AR | 9.0/3.5/26.5 | 8.8/3.5/25.0 | 2% |

More than 10 satellites | |||

Group A triple. PPP-AR | 5.1/1.0/15.0 | 3.1/0.5/10.0 | 39% |

Group B triple. PPP-AR | 5.3/2.0/15.0 | 5.2/2.0/14.5 | 2% |

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**MDPI and ACS Style**

Guo, J.; Zhang, Q.; Li, G.; Zhang, K.
Assessment of Multi-Frequency PPP Ambiguity Resolution Using Galileo and BeiDou-3 Signals. *Remote Sens.* **2021**, *13*, 4746.
https://doi.org/10.3390/rs13234746

**AMA Style**

Guo J, Zhang Q, Li G, Zhang K.
Assessment of Multi-Frequency PPP Ambiguity Resolution Using Galileo and BeiDou-3 Signals. *Remote Sensing*. 2021; 13(23):4746.
https://doi.org/10.3390/rs13234746

**Chicago/Turabian Style**

Guo, Jiang, Qiyuan Zhang, Guangcai Li, and Kunlun Zhang.
2021. "Assessment of Multi-Frequency PPP Ambiguity Resolution Using Galileo and BeiDou-3 Signals" *Remote Sensing* 13, no. 23: 4746.
https://doi.org/10.3390/rs13234746