# BDS-3/Galileo Time and Frequency Transfer with Quad-Frequency Precise Point Positioning

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. General Observations

#### 2.2. Dual-Frequency IF PPP Model

#### 2.3. Quad-Frequency IF PPP Models

#### 2.4. Quad-Frequency Uncombined PPP Model

#### 2.5. Characteristic of Quad-Frequency GNSS PPP Time Transfer Models

## 3. Results and Discussion

#### 3.1. Data Acquisition and Processing Strategies

#### 3.2. Quad-Frequency BDS-3 PPP Time Transfer Solutions

^{−13}, 4.5378 × 10

^{−13}, 4.5589 × 10

^{−13}, 4.8072 × 10

^{−13}) for IF0, IF1, IF2, and UC model on DOY 15, 2019, respectively. Frequency stability are (2.0471 × 10

^{−14}, 2.589 × 10

^{−14}, 1.9685 × 10

^{−14}, 1.9147 × 10

^{−14}) at 15,360 s average time (frequency stability in long-term) on DOY 15, 2019. We can conclude that the performance of quad-frequency PPP is equal to or better than that of dual-frequency PPP both for short- and long- term frequency stability. This finding can also be proved by Figure 6 (enlarged figure), which exhibits MDEV between different BDS−3 PPP models (DOY 15) on XIA3-BRCH time-link.

#### 3.3. Quad-Frequency Galileo PPP Time Transfer Solutions

^{−13}, 3.9117 × 10

^{−13}, 3.9848 × 10

^{−13}, 3.92891 × 10

^{−13}) and (4.8298 × 10

^{−15}, 3.2336 × 10

^{−15}, 3.2903 × 10

^{−15}, 3.0274 × 10

^{−15}) for BRUX-PT11 obtained from IF0, IF1, IF2, and UC model, respectively. In addition, the frequency stability at 30 s and 15,360 s are (3.8044 × 10

^{−13},3.7898 × 10

^{−13}, 3.7668 × 10

^{−13}, 3.7603 × 10

^{−13}) and (1.4392 × 10

^{−15}, 1.4301 × 10

^{−15}, 1.4294 × 10

^{−15}, 1.3864 × 10

^{−15}) for PTBB-PT11 obtained from IF0, IF1, IF2, and UC model, respectively. Obviously, we can see that quad-frequency Galileo has the same conclusions as quad-frequency BDS-3 PPP models.

#### 3.4. Observation Residuals

#### 3.5. Positioning, Tropospheric Delay, and IFB Estimates

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Comparison of time difference of XIA3-BRCH obtained from different models on DOY 15, 2019.

**Figure 5.**Comparison of MDEV between different BDS-3 PPP models from DOY 15 to 19 on XIA3-BRCH time-link.

**Figure 9.**Comparison of time difference of BRUX-PT11 obtained from different models on DOY 277, 2019.

**Figure 10.**Comparison of time difference of PTBB-PT11 obtained from different models on DOY 277, 2019.

**Figure 15.**Analysis of the pseudorange observation residuals for all observed BDS-3 satellites for IF0, IF1, IF2, and UC model.

**Figure 16.**Analysis of the pseudorange observation residuals for all observed Galileo satellites for IF0, IF1, IF2, and UC models.

**Figure 21.**Analysis of two IFB parameters time series for BDS-3 UC models at BRCH and XIA3 stations.

**Figure 22.**Analysis of two IFB parameters time series for Galileo UC model at BRUX and PT11 stations.

e_{1} | e_{2} | e_{3} | e_{4} | Noise Amplification | ||
---|---|---|---|---|---|---|

IF0 | B1I-B3I | 2.9437 | −1.9437 | 0 | 0 | 3.5275 |

IF1 | B1I-B3I | 2.9437 | −1.9437 | 0 | 0 | 3.5275 |

B1C-B2a | 0 | 0 | 2.2606 | −1.2606 | 2.5883 | |

IF2 | B1I-B3I-B1C | 1.3877 | −1.8908 | 1.5031 | 0 | 2.7857 |

B1I-B3I-B2a | 2.3433 | −0.0893 | 0 | −1.2543 | 2.6594 | |

UC | B1I | 1 | 0 | 0 | 0 | 1 |

B3I | 0 | 1 | 0 | 0 | 1 | |

B1C | 0 | 0 | 1 | 0 | 1 | |

B2a | 0 | 0 | 0 | 1 | 1 |

e_{1} | e_{2} | e_{3} | e_{4} | Noise Amplification | ||
---|---|---|---|---|---|---|

IF0 | E1-E5a | 2.2606 | −1.2606 | 0 | 0 | 2.5883 |

IF1 | E1-E5a | 2.2606 | −1.2606 | 0 | 0 | 2.5883 |

E1-E5b | 2.4220 | −1.4220 | 0 | 0 | 2.8086 | |

E1-E5 | 2.3380 | −1.3380 | 0 | 0 | 2.6938 | |

IF2 | E1-E5a-E5b | 2.3149 | −0.8363 | −0.4787 | 0 | 2.5075 |

E1-E5a-E5 | 2.2929 | −0.7340 | 0 | −0.5589 | 2.4715 | |

UC | E1 | 1 | 0 | 0 | 0 | 1 |

E5a | 0 | 1 | 0 | 0 | 1 | |

E5b | 0 | 0 | 1 | 0 | 1 | |

E5 | 0 | 0 | 0 | 1 | 1 |

Station Name | Timing Lab | Receiver | Antenna | External Clock | |
---|---|---|---|---|---|

BDS-3 | BRCH | PTB | CETC-54-GMR-4016 | GNSS-750 | UTC(PTB) |

XIA3 | NTSC | CETC-54-GMR-4016 | NOV750.R4 | UTC(NTSC) | |

Galileo | PTBB | PTB | SEPT POLARX4TR | LEIAR25.R4—LEIT | UTC(PTB) |

PT11 | PTB | SEPT POLARX5TR | LEIAR25.R4—LEIT | UTC(PTB) | |

BRUX | ROB | SEPT POLARX4TR | JAVRINGANT_DM—NONE | UTC(ROB) |

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

Observations | BDS-3: undifferenced and uncombined observations in B1I, B3I, B1C, and B2a frequency Galileo: undifferenced and uncombined observations in E1, E5a, E5b, and E5 frequency |

Weight of observation | Elevation-dependent |

Cutoff angle | 10° |

Sampling rate | 30 s |

PCO and PCV | Corrected |

Station displacement | Corrected [35] |

Wind up | Corrected [36] |

Relativistic effect | Corrected |

Estimator | Kalman filter |

Phase ambiguity | Estimated as constants at each arc (float values) |

IFB | Estimated as random walk (RW) noise [18] |

Receiver position | Estimated as constants in static model |

Receiver clock offset | Estimated as white noise |

Ionospheric delay | IF0, IF1, IF2 models: first-order effects removed with IF combination UC model: estimated as white noise |

Tropospheric delay | Wet delay: estimated as random walk noise Dry delay: corrected by Saastamoinen model Mapping function: GMF [37] |

**Table 5.**Mean and STD values of time difference with different Galileo PPP models for BRUX-PT11 (ns).

DOY | IF0 | IF1 | IF2 | UC | ||||
---|---|---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | Mean | STD | |

277 | −90.28 | 0.10 | −90.29 | 0.09 | −91.17 | 0.10 | −90.27 | 0.10 |

278 | −90.46 | 0.03 | −90.42 | 0.04 | −91.34 | 0.03 | −90.44 | 0.03 |

279 | −90.57 | 0.09 | −90.60 | 0.11 | −91.44 | 0.09 | −90.57 | 0.09 |

280 | −90.77 | 0.05 | −90.77 | 0.05 | −91.65 | 0.05 | −90.78 | 0.05 |

281 | −90.85 | 0.05 | −90.86 | 0.06 | −91.74 | 0.06 | −90.87 | 0.05 |

282 | −90.79 | 0.06 | −90.79 | 0.08 | −91.68 | 0.07 | −90.81 | 0.06 |

**Table 6.**Mean and STD values of time difference with different Galileo PPP models for PTBB-PT11 (ns).

DOY | IF0 | IF1 | IF2 | UC | ||||
---|---|---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | Mean | STD | |

277 | −146.60 | 0.03 | −146.67 | 0.05 | −144.60 | 0.04 | −146.60 | 0.03 |

278 | −146.61 | 0.03 | −146.66 | 0.04 | −144.62 | 0.04 | −146.61 | 0.03 |

279 | −146.60 | 0.03 | −146.65 | 0.06 | −144.59 | 0.05 | −146.59 | 0.04 |

280 | −146.66 | 0.03 | −146.69 | 0.04 | −144.63 | 0.04 | −146.65 | 0.03 |

281 | −146.66 | 0.03 | −146.72 | 0.05 | −144.70 | 0.05 | −146.64 | 0.03 |

282 | −146.63 | 0.04 | −146.68 | 0.06 | −144.65 | 0.05 | −146.61 | 0.04 |

**Table 7.**The RMS values of the position error for BRCH and XIA3 stations with different BDS-3 PPP models (cm).

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

IF0 | IF1 | IF2 | UC | IF0 | IF1 | IF2 | UC | IF0 | IF1 | IF2 | UC | ||

BRCH | 15 | 2.03 | 0.71 | 0.65 | 1.03 | 0.51 | 0.29 | 0.59 | 0.59 | 2.76 | 2.25 | 2.72 | 2.13 |

16 | 1.84 | 0.56 | 0.41 | 0.56 | 0.77 | 0.31 | 0.69 | 0.48 | 2.87 | 2.46 | 2.67 | 2.79 | |

17 | 1.67 | 0.41 | 0.70 | 0.82 | 0.92 | 0.43 | 0.77 | 0.75 | 2.10 | 2.67 | 2.14 | 2.05 | |

18 | 1.64 | 0.30 | 0.64 | 0.90 | 0.84 | 0.31 | 0.61 | 0.67 | 2.54 | 2.52 | 2.34 | 2.50 | |

19 | 0.85 | 0.30 | 0.22 | 0.55 | 0.85 | 0.35 | 0.51 | 0.84 | 2.42 | 2.38 | 2.15 | 2.41 | |

XIA3 | 15 | 0.75 | 0.67 | 0.90 | 0.59 | 0.93 | 0.50 | 0.52 | 0.84 | 3.91 | 3.68 | 3.48 | 3.70 |

16 | 1.59 | 2.25 | 1.61 | 1.91 | 0.71 | 0.38 | 0.27 | 0.62 | 2.53 | 3.39 | 2.91 | 2.49 | |

17 | 2.79 | 2.75 | 2.62 | 2.03 | 0.45 | 0.19 | 0.22 | 0.41 | 3.35 | 3.53 | 3.00 | 3.20 | |

18 | 2.86 | 2.81 | 2.84 | 2.86 | 0.52 | 0.19 | 0.33 | 0.43 | 3.85 | 3.10 | 3.79 | 3.57 | |

19 | 2.63 | 2.60 | 2.75 | 2.25 | 0.45 | 0.18 | 0.35 | 0.43 | 3.61 | 3.60 | 3.57 | 3.57 |

**Table 8.**The RMS values of the position error for BRUX, PT11, and PTBB stations with different Galileo PPP models (ns).

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

IF0 | IF1 | IF2 | UC | IF0 | IF1 | IF2 | UC | IF0 | IF1 | IF2 | UC | ||

BRUX | 277 | 0.42 | 0.14 | 0.15 | 0.36 | 0.15 | 0.17 | 0.12 | 0.08 | 0.96 | 1.17 | 1.51 | 0.95 |

278 | 0.35 | 0.09 | 0.11 | 0.35 | 0.19 | 0.20 | 0.15 | 0.08 | 0.99 | 1.16 | 1.18 | 0.91 | |

279 | 0.46 | 0.21 | 0.23 | 0.40 | 0.20 | 0.20 | 0.15 | 0.09 | 0.94 | 0.88 | 0.88 | 0.90 | |

280 | 0.39 | 0.16 | 0.17 | 0.35 | 0.23 | 0.22 | 0.18 | 0.13 | 0.86 | 0.86 | 0.87 | 0.81 | |

281 | 0.34 | 0.14 | 0.15 | 0.32 | 0.24 | 0.24 | 0.20 | 0.14 | 0.89 | 0.83 | 0.84 | 0.85 | |

282 | 0.35 | 0.16 | 0.18 | 0.31 | 0.24 | 0.25 | 0.20 | 0.14 | 0.88 | 0.85 | 0.87 | 0.85 | |

PT11 | 277 | 0.45 | 0.46 | 0.13 | 0.45 | 0.13 | 0.12 | 0.13 | 0.13 | 1.10 | 1.09 | 1.21 | 1.06 |

278 | 0.41 | 0.42 | 0.11 | 0.40 | 0.11 | 0.08 | 0.11 | 0.11 | 0.90 | 0.88 | 1.00 | 0.77 | |

279 | 0.39 | 0.39 | 0.10 | 0.33 | 0.10 | 0.07 | 0.10 | 0.10 | 0.78 | 0.76 | 0.81 | 0.79 | |

280 | 0.34 | 0.34 | 0.08 | 0.37 | 0.08 | 0.06 | 0.08 | 0.08 | 0.80 | 0.80 | 0.82 | 0.75 | |

281 | 0.34 | 0.33 | 0.08 | 0.31 | 0.08 | 0.06 | 0.08 | 0.08 | 0.77 | 0.79 | 0.77 | 0.77 | |

282 | 0.35 | 0.34 | 0.12 | 0.31 | 0.12 | 0.06 | 0.12 | 0.12 | 0.80 | 0.83 | 0.83 | 0.73 | |

PTBB | 277 | 0.10 | 0.39 | 0.11 | 0.10 | 0.31 | 0.58 | 0.47 | 0.49 | 0.85 | 0.80 | 0.90 | 0.51 |

278 | 0.36 | 0.80 | 0.34 | 0.26 | 0.13 | 0.21 | 0.12 | 0.11 | 1.12 | 0.82 | 0.92 | 0.73 | |

279 | 0.63 | 0.63 | 0.19 | 0.07 | 0.11 | 0.19 | 0.13 | 0.12 | 1.43 | 0.75 | 0.98 | 0.98 | |

280 | 0.20 | 0.52 | 0.20 | 0.18 | 0.15 | 0.18 | 0.13 | 0.09 | 1.28 | 0.99 | 0.89 | 0.82 | |

281 | 0.15 | 0.27 | 0.07 | 0.14 | 0.18 | 0.12 | 0.08 | 0.12 | 1.31 | 0.99 | 0.80 | 0.83 | |

282 | 0.10 | 0.08 | 0.11 | 0.07 | 0.19 | 0.09 | 0.08 | 0.09 | 1.33 | 0.89 | 0.78 | 0.82 |

**Table 9.**The RMS values of ZTD errors for BRCH and XIA3 stations with different BDS-3 PPP models (cm).

DOY | IF0 | IF1 | IF2 | UC | |
---|---|---|---|---|---|

BRCH | 15 | 1.3 | 1.32 | 1.32 | 1.31 |

16 | 0.80 | 0.80 | 0.83 | 0.84 | |

17 | 1.20 | 1.40 | 1.40 | 1.21 | |

18 | 1.20 | 1.02 | 1.02 | 0.64 | |

19 | 1.00 | 0.88 | 0.88 | 0.74 | |

XIA3 | 15 | 1.76 | 1.82 | 1.82 | 1.81 |

16 | 2.06 | 2.11 | 2.11 | 2.01 | |

17 | 2.17 | 2.20 | 2.19 | 2.01 | |

18 | 1.66 | 1.93 | 1.94 | 1.95 | |

19 | 1.72 | 1.78 | 1.78 | 1.78 |

**Table 10.**The RMS values of ZTD errors for BRUX, PT11, and PTBB stations with different Galileo PPP models (cm).

DOY | IF0 | IF1 | IF2 | UC | |
---|---|---|---|---|---|

BRUX | 277 | 0.98 | 1.00 | 1.04 | 0.88 |

278 | 0.61 | 0.76 | 0.70 | 0.57 | |

279 | 0.60 | 0.92 | 0.76 | 0.64 | |

280 | 0.52 | 0.63 | 0.63 | 0.52 | |

281 | 0.83 | 1.04 | 0.96 | 0.81 | |

282 | 0.77 | 0.80 | 0.86 | 0.71 | |

PT11 | 277 | 0.97 | 0.96 | 0.98 | 0.84 |

278 | 0.60 | 0.81 | 0.73 | 0.58 | |

279 | 0.59 | 0.72 | 0.58 | 0.54 | |

280 | 0.40 | 0.48 | 0.47 | 0.39 | |

281 | 0.65 | 0.74 | 0.69 | 0.64 | |

282 | 0.79 | 0.80 | 0.84 | 0.78 | |

PTBB | 277 | 0.86 | 0.90 | 0.91 | 0.72 |

278 | 0.53 | 0.63 | 0.62 | 0.50 | |

279 | 0.75 | 0.88 | 0.90 | 0.70 | |

280 | 0.53 | 0.68 | 0.57 | 0.51 | |

281 | 0.8 | 0.91 | 1.05 | 0.83 | |

282 | 0.60 | 0.90 | 0.87 | 0.56 |

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**MDPI and ACS Style**

Ge, Y.; Cao, X.; Shen, F.; Yang, X.; Wang, S.
BDS-3/Galileo Time and Frequency Transfer with Quad-Frequency Precise Point Positioning. *Remote Sens.* **2021**, *13*, 2704.
https://doi.org/10.3390/rs13142704

**AMA Style**

Ge Y, Cao X, Shen F, Yang X, Wang S.
BDS-3/Galileo Time and Frequency Transfer with Quad-Frequency Precise Point Positioning. *Remote Sensing*. 2021; 13(14):2704.
https://doi.org/10.3390/rs13142704

**Chicago/Turabian Style**

Ge, Yulong, Xinyun Cao, Fei Shen, Xuhai Yang, and Shengli Wang.
2021. "BDS-3/Galileo Time and Frequency Transfer with Quad-Frequency Precise Point Positioning" *Remote Sensing* 13, no. 14: 2704.
https://doi.org/10.3390/rs13142704