# Helmert Variance Component Estimation for Multi-GNSS Relative Positioning

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## Abstract

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## 1. Introduction

## 2. Method

#### 2.1. Helmert Variance Component Estimation for Robust Kalman Filtering

#### 2.2. Flow Chart of Multi-GNSS HVCE for Robust Kalman Filtering Algorithm

## 3. Experiment Setup

#### 3.1. Station Selection

#### 3.2. Data Processing Strategy

## 4. Experimental Results and Discussion

#### 4.1. Weight Proportions of Multi-GNSS Phase and Code Observations

#### 4.2. Accuracy of HVCE Posterior Weighting-Based Multi-GNSS Positioning

#### 4.3. Frozen Variance of Unit Weight-Based Multi-GNSS Positioning

## 5. Summary and Conclusions

- Multi-GNSS observations and the HVCE method improve the positioning accuracy. Compared with the corresponding GPS-only strategies, the positioning ENU accuracy is improved 34.3%, 39.5% and 45.9% by ED Multi-GNSS, and 47.9% 49.0% and 52.4% by HVCE Multi-GNSS. With respect to ED method, the HVCE method improves positioning ENU accuracy by 7.4%, 6.4% and 5.9% in the GPS-only strategy, and 20.5%, 15.6% and 12.3% in the Multi-GNSS strategy.
- The quality of phase observations is almost equivalent among GPS, BDS, GLONASS and Galileo, as their variances of unit weight are all close to 1.0. In contrast, the quality of the code observations of different GNSS constellations differs to a great extent, presenting an average relationship as ${\mathsf{\sigma}}_{\mathrm{R},\mathrm{C}}^{2}>{\mathsf{\sigma}}_{\mathrm{C},\mathrm{C}}^{2}>{\mathsf{\sigma}}_{\mathrm{G},\mathrm{C}}^{2}>{\mathsf{\sigma}}_{\mathrm{E},\mathrm{C}}^{2}$. The ${\mathsf{\sigma}}_{\mathrm{E},\mathrm{C}}^{2}$ is the lowest in all baselines, which strongly indicates that Galileo has the best quality of code observations.
- The variances of unit weights of both phase and code were quite consistent in each baseline during the 30 experimental days, which allowed the freezing.
- Comparing with ED Multi-GNSS, the FVUW Multi-GNSS improves the positioning accuracy by 20.0%, 14.1% and 11.1% in ENU, similar to the corresponding improvements of 20.5%, 15.6% and 12.3% obtained by HVCE method. At the same time, the FVUW method saves 88% time consumption compared to the HVCE method.
- When the frozen variances of unit weight are extended to the positioning experiment for the next 30 days, the positioning accuracy can still be improved by 18.1%, 13.2% and 10.6% in ENU, indicating the effectiveness of the frozen variances of unit weight.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Locations of six independent baselines with co-located International GNSS Service (IGS) stations.

**Figure 3.**Time series of variances of unit weight of phase (

**top**) and code (

**bottom**) for baseline South Africa (SUT) from day of year (DoY) 183 to 186 in 2019.

**Figure 4.**Daily average variance of unit weight of Multi-GNSS phase observations in different baseline tests.

**Figure 5.**Daily average variance of unit weight of Multi-GNSS code observations in different baseline tests.

**Figure 6.**Time series of kinematic positioning errors of east (

**top**), north (

**middle**) and up (

**bottom**) components for baseline SUT from DoY 183 to 186 in 2019. The reset appearing in the start of each day is caused by the independent daily process.

**Figure 7.**RMSs of kinematic relative positioning using four weighting strategies, from top to bottom: ED GPS-only (

**first**), HVCE GPS-only (

**second**), ED Multi-GNSS (

**third**) and HVCE Multi-GNSS (

**last**) from DoY 171 to 200 in 2019.

**Figure 8.**Time series of Multi-GNSS kinematic positioning errors of east (

**top**), north (

**middle**) and up (

**bottom**) components for baseline SUT from DoY 183 to 186 in 2019. The reset appearing at the start of each day is caused by the independent daily process.

**Figure 9.**RMSs of kinematic relative positioning using FVUW Multi-GNSS positioning from DoY 171 to 200 in 2019.

Baseline | Length(m) | Station | Receiver Type | Antenna Type |
---|---|---|---|---|

CHP | 1850 | CHPG | TRIMBLE NETR9 | TRM59800.00 NONE |

CHPI | SEPT POLARX5 | TPSCR.G3 NONE | ||

DAE | 0 | DAEJ | TRIMBLE NETR9 | TRM59800.00 SCIS |

DAE2 | TRIMBLE NETR9 | TRM59800.00 SCIS | ||

GOD | 65 | GODE | SEPT POLARX5TR | AOAD/M_T JPLA |

GODN | JAVAD TRE_3 DELTA | TPSCR.G3 SCIS | ||

STR | 70 | STR1 | SEPT POLARX5 | ASH701945C_M NONE |

STR2 | TRIMBLE NETR9 | LEIAR25.R3 NONE | ||

SUT | 142 | SUTH | SEPT POLARX5 | ASH701945G_M NONE |

SUTM | JAVAD TRE_3 | JAVRINGANT_G5T NONE | ||

TLS | 1265 | TLSE | TRIMBLE NETR9 | TRM59800.00 NONE |

TLSG | SEPT POLARX5TR | TRM59800.00 NONE |

ED GPS-Only | HVCE GPS-Only | ED Multi-GNSS | HVCE Multi-GNSS | FVUW Multi-GNSS |
---|---|---|---|---|

Convergence Time (minutes) | ||||

18.4 | 16.6 | 6.1 | 5.1 | 5.2 |

Number of Available Positions Per Day | ||||

459.8 | 462.1 | 475.8 | 477.1 | 476.8 |

**Table 3.**Average common satellite number and positioning dilution of precision (PDOP) values of different baselines.

Baseline | G | C | R | E | G + C + R + E |
---|---|---|---|---|---|

Average Available Satellites Number | |||||

CHP | 7.87 | 4.18 | 5.66 | 5.86 | 22.92 |

DAE | 7.73 | 11.42 | 5.82 | 4.42 | 29.31 |

GOD | 7.56 | 5.26 | 6.04 | 5.23 | 24.01 |

STR | 7.76 | 10.95 | 5.92 | 5.27 | 29.87 |

SUT | 7.67 | 7.74 | 5.69 | 5.79 | 26.85 |

TLS | 7.66 | 5.13 | 6.13 | 5.68 | 24.51 |

Average PDOP Value | |||||

CHP | 1.17 | 2.85 | 1.62 | 1.58 | 0.67 |

DAE | 1.21 | 1.08 | 1.52 | 2.47 | 0.60 |

GOD | 1.25 | 2.04 | 1.49 | 1.82 | 0.66 |

STR | 1.21 | 1.11 | 1.48 | 1.78 | 0.59 |

SUT | 1.22 | 1.25 | 1.62 | 1.62 | 0.62 |

TLS | 1.25 | 2.49 | 1.45 | 1.64 | 0.66 |

**Table 4.**Average phase and code variances of unit weight calculated by HVCE Multi-GNSS strategy and their corresponding root mean square (RMS) values from DoY 171 to 200 in 2019.

Baseline | $\mathbf{Phase}\text{}{\mathsf{\sigma}}_{\mathbf{L}}^{2}\pm \mathbf{R}\mathbf{M}{\mathbf{S}}_{\mathbf{L}}$ | $\mathbf{Code}\text{}{\mathsf{\sigma}}_{\mathbf{C}}^{2}\pm \mathbf{R}\mathbf{M}{\mathbf{S}}_{\mathbf{C}}$ | ||||||
---|---|---|---|---|---|---|---|---|

G | C | R | E | G | C | R | E | |

CHP | 1.00 | 1.47 ± 0.57 | 1.47 ± 0.61 | 1.83 ± 0.74 | 3.18 ± 1.30 | 2.02 ± 0.87 | 2.68 ± 0.91 | 1.13 ± 0.42 |

DAE | 1.00 | 2.20 ± 0.77 | 2.11 ± 0.92 | 1.53 ± 0.51 | 4.06 ± 1.55 | 4.65 ± 1.74 | 4.51 ± 1.61 | 1.33 ± 0.60 |

GOD | 1.00 | 1.44 ± 0.56 | 1.51 ± 0.64 | 1.50 ± 0.51 | 2.66 ± 0.62 | 5.68 ± 2.37 | 8.10 ± 2.59 | 1.92 ± 0.76 |

STR | 1.00 | 1.20 ± 0.51 | 1.38 ± 0.61 | 1.30 ± 0.54 | 8.28 ± 2.81 | 10.30 + 4.28 | 10.86 + 4.40 | 2.55 ± 0.93 |

SUT | 1.00 | 1.36 ± 0.40 | 1.36 ± 0.47 | 0.88 ± 0.43 | 6.55 ± 3.29 | 12.05 ± 4.08 | 17.77 ± 3.75 | 1.61 ± 0.79 |

TLS | 1.00 | 1.60 ± 0.54 | 1.67 ± 0.51 | 1.57 ± 0.67 | 3.34 ± 1.23 | 2.47 ± 1.29 | 3.61 ± 1.61 | 1.02 ± 0.40 |

Average | 1.00 | 1.55 ± 0.32 | 1.58 ± 0.26 | 1.44 ± 0.29 | 4.68 ± 2.04 | 6.20 ± 3.77 | 7.92 ± 5.22 | 1.59 ± 0.52 |

**Table 5.**Accuracy improvement percentages of Multi-GNSS strategies based on ED and HVCE compared with the corresponding GPS-only strategies from DoY 171 to 200 in 2019.

Baseline | ED Method | HVCE Method | ||||
---|---|---|---|---|---|---|

E | N | U | E | N | U | |

CHP | 23.5% | 33.2% | 44.4% | 40.9% | 39.3% | 45.2% |

DAE | 37.7% | 45.2% | 39.0% | 43.6% | 49.5% | 41.3% |

GOD | 21.4% | 52.1% | 49.3% | 31.8% | 58.6% | 53.1% |

STR | 53.9% | 52.4% | 55.7% | 57.0% | 59.1% | 59.9% |

SUT | 45.5% | 20.8% | 52.8% | 53.1% | 29.8% | 57.4% |

TLS | 23.8% | 33.0% | 34.5% | 37.6% | 36.4% | 40.9% |

Average | 34.3% | 39.5% | 45.9% | 44.0% | 45.4% | 49.6% |

**Table 6.**Accuracy improvement percentages of HVCE GPS-only and HVCE Multi-GNSS strategies compared with the corresponding ED methods from DoY 171 to 200 in 2019.

Baseline | GPS-Only | Multi-GNSS | ||||
---|---|---|---|---|---|---|

E | N | U | E | N | U | |

CHP | 2.7% | 4.6% | 5.6% | 24.8% | 13.3% | 7.0% |

DAE | 5.7% | 5.6% | 2.2% | 14.7% | 13.1% | 5.9% |

GOD | 6.5% | 1.6% | 2.2% | 18.9% | 14.9% | 9.5% |

STR | 13.7% | 8.2% | 10.5% | 19.4% | 21.0% | 18.9% |

SUT | 10.7% | 12.7% | 9.6% | 23.2% | 22.6% | 18.4% |

TLS | 5.0% | 3.9% | 5.0% | 22.2% | 8.9% | 14.3% |

Average | 7.4% | 6.1% | 5.9% | 20.5% | 15.6% | 12.3% |

**Table 7.**Averaged time consumption at adjustment process per epoch for five strategies in the unit of millisecond.

ED GPS-Only | HVCE GPS-Only | ED Multi-GNSS | HVCE Multi-GNSS | FVUW Multi-GNSS |
---|---|---|---|---|

1 | 2 | 5 | 41 | 5 |

**Table 8.**Accuracy improvement percentages of FVUW method compared with ED prior weighting strategy in Multi-GNSS positioning from DoY 171 to 200 in 2019.

Components | CHP | DAE | GOD | STR | SUT | TLS | Average |
---|---|---|---|---|---|---|---|

East | 24.4% | 10.2% | 18.3% | 17.0% | 25.5% | 24.6% | 20.0% |

North | 5.5% | 10.3% | 13.4% | 23.8% | 21.1% | 10.4% | 14.1% |

Up | 4.7% | 4.5% | 8.9% | 20.0% | 21.2% | 7.2% | 11.1% |

**Table 9.**Accuracy improvement percentages of frozen variances of the unit weight strategy compared with the ED prior weighting strategy in Multi-GNSS positioning from DoY 201 to 230 in 2019.

Components | CHP | DAE | GOD | STR | SUT | TLS | Average |
---|---|---|---|---|---|---|---|

East | 16.8% | 8.1% | 24.3% | 21.5% | 20.8% | 17.4% | 18.1% |

North | 6.1% | 10.0% | 13.9% | 25.0% | 18.2% | 6.0% | 13.2% |

Up | 3.0% | 7.3% | 5.5% | 22.8% | 18.4% | 6.9% | 10.6% |

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**MDPI and ACS Style**

Li, M.; Nie, W.; Xu, T.; Rovira-Garcia, A.; Fang, Z.; Xu, G.
Helmert Variance Component Estimation for Multi-GNSS Relative Positioning. *Sensors* **2020**, *20*, 669.
https://doi.org/10.3390/s20030669

**AMA Style**

Li M, Nie W, Xu T, Rovira-Garcia A, Fang Z, Xu G.
Helmert Variance Component Estimation for Multi-GNSS Relative Positioning. *Sensors*. 2020; 20(3):669.
https://doi.org/10.3390/s20030669

**Chicago/Turabian Style**

Li, Mowen, Wenfeng Nie, Tianhe Xu, Adria Rovira-Garcia, Zhenlong Fang, and Guochang Xu.
2020. "Helmert Variance Component Estimation for Multi-GNSS Relative Positioning" *Sensors* 20, no. 3: 669.
https://doi.org/10.3390/s20030669