# Modeling and Performance Evaluation of Precise Positioning and Time-Frequency Transfer with Galileo Five-Frequency Observations

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## Abstract

**:**

^{−16}level at 120,000 s, and the average three-dimensional (3D) root mean square (RMS) of position and average frequency stability (120,000 s) can reach 1.82 cm and 1.18 × 10

^{−15}, respectively. The differences of 3D RMS among all models are within 0.17 cm, and the differences in frequency stabilities (in 120,000 s) among all models are within 0.08 × 10

^{−15}. Using the GRG precise product, the solution performance is slightly better than that of the GBM or WUM precise product, the average 3D RMS values obtained using the WUM and GRG precise products are 1.85 cm and 1.77 cm, respectively, and the average frequency stabilities at 120,000 s can reach 1.13 × 10

^{−15}and 1.06 × 10

^{−15}, respectively.

## 1. Introduction

## 2. Methodology

#### 2.1. GNSS Observation Model

_{r}and dt

^{s}denote receiver and satellite clock offsets in seconds, respectively; MF denotes the wet mapping function; ZWD is zenith troposphere wet delay (ZWD); I denotes the slant ionospheric delay; ${\gamma}_{j}$ denotes frequency dependent ionospheric delay amplification factors, ${\gamma}_{j}={\lambda}_{j}^{2}/{\lambda}_{1}^{2}$; N is carrier phase integer ambiguity; d denotes the receiver uncalibrated code delay (UCD) in meters; b denotes uncalibrated phase delay (UPD) in cycles; and ζ and ξ represent the pseudo orange and carrier phase observation noises, respectively.

#### 2.2. Five-Frequency UC PPP Model

_{r}

_{,12}denotes E1 and E5a DCB of the receiver side; IFB

_{uc}denotes UC IFBs;

**μ**denotes the unit vector of the component from the receiver to the satellites;

**x**denotes the vector of the receiver position increments; and hat “~” denotes the reparametrized estimate.

#### 2.3. Single Five-Frequency IF Combination (FF1) PPP Model

_{1}, e

_{2}, e

_{3}, e

_{4}and e

_{5}denote the five-frequency IF combination coefficients, e is the coefficient in the denominator.

#### 2.4. Two Quad-Frequency IF Combinations (FF2) PPP Model

_{IF}

_{1235}denotes the IFB parameters between the E1/E5a/E5b/E6 and E1/E5a/E5b/E6 code IF combinations.

#### 2.5. Three Triple-Frequency IF Combinations (FF3) PPP Model

_{IF}

_{124}and IFB

_{IF}

_{125}denote the IFB parameters between the E1/E5a/E5b, E1/E5a/E6 and E1/E5a/E5b code IF combinations.

#### 2.6. Four Dual-Frequency IF Combinations (FF4) PPP Model

_{IF}

_{13}, IFB

_{IF}

_{14}and IFB

_{IF}

_{15}denote the IFB parameters between the E1/E5b, E1/E5, E1/E6 and E1/E5a code IF combinations.

#### 2.7. Comparison of Galileo Five Frequency PPP Models

#### 2.8. Date Selection and Processing Strategies

## 3. Results

#### 3.1. Number of Visible Satellites, TDOP and Multipath Combination Noise Analysis

#### 3.2. Performance of the DF, TF, QF and FF1 PPP Models

^{−16}level in 60,000 s and 5.31 × 10

^{−16}in 120,000 s, the BRUX-USN7 link reaches 9.25 × 10

^{−16}in 120,000 s, and those of the other two links are also close to the 10

^{−16}level in 120,000 s. The average frequency stabilities at 120,000 s for the four time-frequency links surpass 1.23 × 10

^{−15}for the DF PPP model, 1.15 × 10

^{−15}for the TF PPP model, 1.23 × 10

^{−15}for the QF PPP model and 1.23 × 10

^{−15}for the FF1 PPP model.

#### 3.3. Performance of the FF1, FF2, FF3, FF4 and UC PPP Models

^{−15}for the FF1 PPP model, 1.22 × 10

^{−15}for the FF2 PPP model, 1.20 × 10

^{−15}for the FF3 PPP model and 1.24 × 10

^{−15}for the UC PPP model.

_{IF13}, IFB

_{IF14}, and IFB

_{IF15}in the FF4 PPP model are 7.8 ns, 4.1 ns, and 9.9 ns, respectively; and the average STD values are 0.04 ns, 0.03 ns, and 0.15 ns, respectively. The average RMS values of IFB

_{UC3}, IFB

_{UC4}, and IFB

_{UC5}in the UC PPP model are 5.49 ns, 3.09 ns, and 5.15 ns, respectively, and the corresponding average STD values are 0.03 ns, 0.02 ns, and 0.04 ns, respectively. As mentioned, the IFB

_{IF15}STD value of the FF4 PPP model is the largest, leading to the receiver clock bias, and IFB

_{IF15}cannot be accurately separated, which will affect the time-frequency transfer performance.

#### 3.4. PPP Performance by Using Different AC Products

^{−16}level, and the frequency stability of the BRUX-USN7 link using the GRG product is the most significantly improved compared to those using WUM and GRG products, which may be related to the observation quality of USN7 and the absorption of satellite UPD by GRG clock products. The average frequency stabilities over four links using GBM, WUM and GRG products in 120,000 s are 1.18 × 10

^{−15}, 1.13 × 10

^{−15}and 1.06 × 10

^{−15}, respectively. Using WUM products, the average frequency stabilities of the DF, TF, QF, FF1, FF2, FF3, FF4 and UC PPP models are 1.12 × 10

^{−15}, 1.13 × 10

^{−15}, 1.13 × 10

^{−15}, 1.15 × 10

^{−15}, 1.12 × 10

^{−15}, 1.11 × 10

^{−15}, 1.13 × 10

^{−15}, and 1.15 × 10

^{−15}, respectively, while using GRG products, the corresponding average frequency stabilities are 1.05 × 10

^{−15}, 1.07 × 10

^{−15}, 1.05 × 10

^{−15}, 1.06 × 10

^{−15}, 1.05 × 10

^{−15}, 1.03 × 10

^{−15}, 1.05 × 10

^{−15}and 1.10 × 10

^{−15}, respectively. All these results demonstrate that the frequency stability using the GRG product is better than those using WUM and GBM products at 120,000 s. It can also be found that compared with the DF model, the multifrequency model’s frequency stabilities are not significantly improved, which may be limited by the additional estimation of IFB parameters and the accuracy of multifrequency DCB products [40].

## 4. Discussion

## 5. Conclusions

^{−15}, 1.15 × 10

^{−15}, 1.23 × 10

^{−15}, and 1.23 × 10

^{−15}, respectively. The differences in 3D RMS among the DF, TF, QF and FF1 PPP models are within 0.17 cm, and the differences in frequency stabilities are within 0.08 × 10

^{−15}. The performances of FF1, FF2, FF3, FF4 and UC PPP are also consistent with each other, the differences in 3D RMS values among the FF1, FF2, FF3, FF4 and UC PPP models are within 0.12 cm, and the differences in the frequency stabilities are within 0.02 × 10

^{−15}.

^{−15}, 1.13 × 10

^{−15}and 1.06 × 10

^{−15}, respectively. These results show that the PPP performance using the GRG product is slightly better than that of GBM and WUM products, especially in terms of the long-term frequency stability of the BRUX-UNS7 link.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Distribution of number of visible satellites and position dilution of precision (PDOP) value for GPS and Galileo constellation on days of year (DOYs) 190 to 196 in 2020. (

**a**) Number of visible GPS satellites; (

**b**) GPS PDOP values; (

**c**) number of visible Galileo satellites; and (

**d**) Galileo PDOP values.

**Figure 3.**Number of visible satellites and time dilution of precision (TDOP) values at BRUX and PTBB for Galileo on DOYs 190 to 196 in 2020. (

**a**) Number of visible satellites; and (

**b**) TDOP values.

**Figure 4.**Galileo multipath combination noise sequence and elevation at frequencies E1, E5a, E6, E5 and E5b at BRUX station on DOYs 190 to 196 in 2020. (

**a**) E08 satellite; (

**b**) E15 satellite; (

**c**) E25 satellite; and (

**d**) E33 satellite.

**Figure 5.**Positioning performance of BRUX, CEBR, PTBB, ROAG and USN7 stations for the DF, TF, QF and FF1 PPP models on DOYs 190 to 196 in 2020. (

**a**) E component RMS value; (

**b**) N component RMS value; (

**c**) U component RMS value; and (

**d**) convergence time.

**Figure 6.**Clock offset sequence at four time-frequency links for the DF, TF, QF and FF1 PPP models on DOYs 190 to 196 in 2020. (

**a**) BRUX-CEBR time-frequency link; (

**b**) BRUX-PTBB time-frequency link; (

**c**) BRUX-ROAG time-frequency link; and (

**d**) BRUX-USN7 time-frequency link.

**Figure 7.**MADEV at four time-frequency links for the DF, TF, QF and FF1 PPP solutions on DOYs 190 to 196 in 2020. (

**a**) BRUX-CEBR time-frequency link; (

**b**) BRUX-PTBB time-frequency link; (

**c**) BRUX-ROAG time-frequency link; and (

**d**) BRUX-USN7 time-frequency link.

**Figure 8.**Positioning performance of the BRUX, CEBR, PTBB, ROAG and USN7 stations by the FF1, FF2, FF3, FF4 and UC PPP solutions on DOYs 190 to 196 in 2020. (

**a**) E Component RMS value; (

**b**) N component RMS value; (

**c**) U component RMS value; and (

**d**) convergence time.

**Figure 9.**Clock offset sequences at four time-frequency links for the FF1, FF2, FF3, FF4 and UC PPP solutions on DOYs 190 to 196 in 2020. (

**a**) BRUX-CEBR time-frequency link; (

**b**) BRUX-PTBB time-frequency link; (

**c**) BRUX-ROAG time-frequency link; and (

**d**) BRUX-USN7 time-frequency link.

**Figure 10.**MADEV at four time-frequency links for the FF1, FF2, FF3, FF4 and UC PPP solutions on DOYs 190 to 196 in 2020. (

**a**) BRUX-CEBR time-frequency link; (

**b**) BRUX-PTBB time-frequency link; (

**c**) BRUX-ROAG time-frequency link; and (

**d**) BRUX-USN7 time-frequency link.

**Figure 11.**IFB sequences at five stations for the FF2, FF3, FF4 and UC PPP models on DOYs 190 to 196 in 2020. (

**a**) BRUX station; (

**b**) CEBR station; (

**c**) PTBB station; (

**d**) ROAG station; and (

**e**) USN7 station.

**Figure 12.**Average 3D RMS values and average convergence times by using the GBM, WUM and GRG precise product PPP solutions on DOYs 190 to 196 in 2020. (

**a**) 3D RMS value; (

**b**) convergence time.

**Figure 13.**Clock offset sequences of the GBM, WUM and GRG products in the PPP FF1 model. (

**a**) BRUX-CEBR time-frequency link; (

**b**) BRUX-PTBB time-frequency link; (

**c**) BRUX-ROAG time-frequency link; and (

**d**) BRUX-USN7 time-frequency link.

**Figure 14.**MADEV of the PPP solution by using GBM, WUM and GRG precision products at 120,000 s. (

**a**) BRUX-CEBR time-frequency link; (

**b**) BRUX-PTBB time-frequency link; (

**c**) BRUX-ROAG time-frequency link; and (

**d**) BRUX-USN7 time-frequency link.

**Figure 15.**MADEV at the BRUX-USN7 time-frequency link of the PPP solution by using GBM, WUM and GRG precision products on DOYs 190 to 196 in 2020. (

**a**) DF PPP model; (

**b**) TF PPP model; (

**c**) QF PPP model; (

**d**) FF1 PPP model; and (

**e**) FF2 PPP model; (

**f**) FF3 PPP model; (

**g**) FF4 PPP model; (

**h**) UC PPP model.

Models | Observed Type | E1 | E5a | E5b | E5 | E6 | Ion | Noise |
---|---|---|---|---|---|---|---|---|

DF | E1/E5a | 2.261 | −1.261 | / | / | / | / | 2.588 |

TF | E1/E5a/E5b | 2.315 | −0.836 | −0.479 | / | / | / | 2.507 |

QF | E1/E5a/E5b/E5 | 2.317 | −0.606 | −0.274 | −0.437 | / | / | 2.450 |

FF1 | E1/E5a/E5b/E5/E6 | 2.217 | −0.680 | −0.351 | −0.512 | 0.326 | / | 2.423 |

FF2 | E1/E5a/E5b/E5 | 2.317 | −0.606 | −0.274 | −0.437 | / | / | 2.450 |

E1/E5a/E5b/E6 | 2.255 | −0.904 | −0.545 | 0.193 | / | / | 2.497 | |

FF3 | E1/E5a/E5b | 2.315 | −0.836 | −0.479 | / | / | / | 2.507 |

E1/E5a/E5 | 2.293 | 0.734 | −0.559 | / | / | / | 2.471 | |

E1/E5a/E6 | 2.269 | −1.244 | −0.024 | / | / | / | 2.588 | |

FF4 | E1/E5a | 2.261 | −1.261 | / | / | / | / | 2.588 |

E1/E5b | 2.422 | / | −1.422 | / | / | / | 2.809 | |

E1/E5 | 2.338 | / | / | −1.338 | / | / | 2.694 | |

E1/E6 | 2.931 | / | / | / | −1.931 | / | 3.510 | |

UC | E1 | 1.000 | / | / | / | / | 1.000 | 1.000 |

E5a | / | 1.000 | / | / | / | 1.793 | 1.000 | |

E5b | / | / | 1.000 | / | / | 1.703 | 1.000 | |

E5 | / | / | / | 1.000 | / | 1.747 | 1.000 | |

E6 | / | / | / | / | 1.000 | 1.518 | 1.000 |

Stations | Receiver | Antenna | Clock | Country | Distance |
---|---|---|---|---|---|

BRUX | SEPT POLARX5TR | JAVRINGANT_DM | UTC (ORB) | Belgium | / |

CEBR | SEPT POLARX5TR | SEPCHOKE_B3E6 | H-MASER | Spain | 1331.6 km |

PTBB | SEPT POLARX5TR | LEIAR25.R4 | UTC (PTB) | Germany | 454.6 km |

ROAG | SEPT POLARX5TR | LEIAR25.R4 | UTC (ROA) | Spain | 1796.3 km |

USN7 | SEPT POLARX5TR | TPSCR.G5 | UTC (USNO) | America | 5991.2 km |

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

Solutions | DF/TF/QF/FF1/FF2/FF2/FF3/FF4/UC PPP models |

Observations | E1/E5a/E5b/E5/E6 observations |

Elevation cutoff | 7° |

Orbits and clock | GBM, WUM and GRG precise products |

Satellite DCB | CAS BSX products [32]$,\mathrm{E}1\mathrm{I}:-{\beta}_{12}\cdot DC{B}_{12}$$,\mathrm{E}5\mathrm{a}:{\alpha}_{12}\cdot DC{B}_{12}$, E5b:${\alpha}_{12}\cdot DC{B}_{12}-DC{B}_{23}$, E5:${\alpha}_{12}\cdot DC{B}_{12}-DC{B}_{24}$, E6:${\alpha}_{12}\cdot DC{B}_{12}-DC{B}_{25}$ |

Earth rotation | Corrected [33] |

Relativistic effect | Corrected [34] |

Phase windup effect | Corrected [35] |

Tide effect | Corrected [36] |

PCO/PCV | IGS14.atx |

Station coordinates | Estimated as constants |

Receiver clock | Estimated as white noises (10^{4} m^{2}/s) |

Ionospheric delay | UC: Estimated as random walk process DF/TF/QF/FF1/FF2/FF3/FF4: Eliminated first-order by IF combinations |

Tropospheric delay | Dry component: GPT add Saastamoinen model [37,38] Wet component: Estimated as random walk part (10 ^{−8} m^{2}/s), GMF mapping function [39] |

Ambiguities | Estimated as constant, float solution |

**Table 4.**Clock offset epoch difference results of average (AVG) and root mean square (RMS) at four time-frequency links for the DF, TF, QF and FF1 PPP models on DOYs 190 to 196 in 2020 (ps).

Items | Links | DF | TF | QF | FF1 |
---|---|---|---|---|---|

AVG | BRUX-CEBR | 7.38 | 7.38 | 7.38 | 7.38 |

BRUX-PTBB | 0.01 | 0.01 | 0.01 | 0.01 | |

BRUX-ROAG | 0.08 | 0.08 | 0.08 | 0.08 | |

BRUX-USN7 | −0.01 | −0.01 | −0.01 | −0.01 | |

RMS | BRUX-CEBR | 10.54 | 10.44 | 10.50 | 10.49 |

BRUX-PTBB | 7.02 | 6.88 | 6.97 | 6.87 | |

BRUX-ROAG | 7.22 | 7.09 | 7.17 | 7.14 | |

BRUX-USN7 | 7.98 | 7.88 | 8.02 | 8.14 |

**Table 5.**Clock offset epoch difference results at four time-frequency links for the FF1, FF2, FF3, FF4 and UC PPP models on DOYs 190 to 196 in 2020 (ps).

Item | Links | FF1 | FF2 | FF3 | FF4 | UC |
---|---|---|---|---|---|---|

AVG | BRUX-CEBR | 7.38 | 7.38 | 7.40 | 7.38 | 7.38 |

BRUX-PTBB | 0.01 | 0.01 | 0.01 | −0.01 | 0.01 | |

BRUX-ROAG | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | |

BRUX-USN7 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | |

RMS | BRUX-CEBR | 10.49 | 10.51 | 10.53 | 10.40 | 10.46 |

BRUX-PTBB | 6.87 | 6.92 | 6.95 | 6.84 | 6.92 | |

BRUX-ROAG | 7.14 | 7.15 | 7.16 | 7.09 | 7.03 | |

BRUX-USN7 | 8.14 | 8.04 | 8.04 | 8.00 | 8.23 |

**Table 6.**Clock offset epoch difference RMS at four time-frequency links by the WUM and GRG product PPP solutions on DOYs 190 to 196 in 2020 (ps).

Products | Links | DF | TF | QF | FF1 | FF2 | FF3 | FF4 | UC |
---|---|---|---|---|---|---|---|---|---|

WUM | BRUX-CEBR | 10.62 | 10.59 | 10.55 | 10.49 | 10.49 | 10.52 | 10.52 | 10.55 |

BRUX-PTBB | 7.1 | 7.02 | 7.04 | 6.96 | 7.02 | 7.02 | 6.95 | 6.98 | |

BRUX-ROAG | 7.27 | 7.11 | 7.16 | 7.12 | 7.16 | 7.16 | 7.04 | 7.13 | |

BRUX-USN7 | 8.08 | 9.02 | 8.05 | 8.15 | 8.11 | 8.08 | 8.34 | 8.06 | |

GRG | BRUX-CEBR | 10.65 | 10.53 | 10.54 | 10.51 | 10.55 | 10.51 | 10.76 | 10.52 |

BRUX-PTBB | 7.08 | 7.01 | 7.00 | 6.92 | 6.99 | 6.91 | 7.31 | 6.96 | |

BRUX-ROAG | 7.30 | 7.24 | 7.21 | 7.17 | 7.20 | 7.16 | 7.27 | 7.19 | |

BRUX-USN7 | 8.07 | 8.11 | 8.00 | 8.12 | 8.04 | 8.11 | 8.92 | 8.03 |

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## Share and Cite

**MDPI and ACS Style**

Xu, W.; Shen, W.-B.; Cai, C.-H.; Li, L.-H.; Wang, L.; Shen, Z.-Y.
Modeling and Performance Evaluation of Precise Positioning and Time-Frequency Transfer with Galileo Five-Frequency Observations. *Remote Sens.* **2021**, *13*, 2972.
https://doi.org/10.3390/rs13152972

**AMA Style**

Xu W, Shen W-B, Cai C-H, Li L-H, Wang L, Shen Z-Y.
Modeling and Performance Evaluation of Precise Positioning and Time-Frequency Transfer with Galileo Five-Frequency Observations. *Remote Sensing*. 2021; 13(15):2972.
https://doi.org/10.3390/rs13152972

**Chicago/Turabian Style**

Xu, Wei, Wen-Bin Shen, Cheng-Hui Cai, Li-Hong Li, Lei Wang, and Zi-Yu Shen.
2021. "Modeling and Performance Evaluation of Precise Positioning and Time-Frequency Transfer with Galileo Five-Frequency Observations" *Remote Sensing* 13, no. 15: 2972.
https://doi.org/10.3390/rs13152972