# Multipath Error Fusion Modeling Methods for Multi-GNSS

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sidereal Filtering for Multi-GNSS Multipath Correction

#### 2.2. Multi-Point Hemispherical Grid Model for Multi-GNSS Multipath Correction

## 3. Results

#### 3.1. Use the Observations from Days of the Previous Orbit Repeat Period for Testing

#### 3.2. Use the Observations from Multi-Day for Testing

## 4. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Scatter plots of the daily time shifts for GPS, BDS, and Galileo in 2018 and 2020 based on the broadcast ephemeris data. (

**a**) Time shifts for the MEO satellite of GPS. (

**b**) Time shifts for the GEO and IGSO satellites of BDS2 and BDS3. (

**c**) Time shifts for the MEO satellite of BDS2 and BDS3. (

**d**) Time shifts for the MEO satellite of Galileo.

**Figure 3.**Location relationship and surrounding observation environment of the stations. A for station A which a metal baffle is mounted in the west, B, C for stations B and C which is normal.

**Figure 4.**Distribution histogram and normal distribution curve of double-differenced residuals before and after the correction of different models.

**Figure 5.**Double-differenced residual sequences and signal power spectrum before and after model correction. (

**a**) The double differenced residual series of G19 and G22 on station B and station C and the power spectrum analysis of the residual series. (

**b**) The double differenced residual series of C04 and C08 on station A and station B and the power spectrum analysis of the residual series. (

**c**) The double differenced residual series of E01 and E31 on station A and station C and the power spectrum analysis of the residual series.

**Figure 6.**Mean RMSs of the double-differenced residuals in the ambiguity-fixed period after multipath error correction using the observations from the days of the previous orbit repeat period and multi-day.

**Figure 7.**Distribution histogram and normal distribution curve of average double-differenced residuals after M (10) strategy correction.

Constellation Type | $\mathit{n}$ | $\mathit{k}$ | |
---|---|---|---|

GPS | MEO | 1 | 2 |

BDS | GEO | 1 | 1 |

IGSO | 1 | 1 | |

MEO | 7 | 13 | |

Galileo | MEO | 10 | 17 |

Parameter | Model |
---|---|

Satellites | GPS+BDS+Galileo |

Observations | Phase and code observations |

Signal | GPS L1+BDS B1+Galileo E1 |

Sampling rate | 1s |

Observation weight | Satellite elevation |

Cutoff elevation | 7 degree |

Estimator | Least square method |

Phase wind-up | Corrected |

Phase center pattern | igs14.atx |

Tropospheric delay | GPT2+Saastamonient+GMF |

Satellite clock | Broadcast + Process |

Receiver clock | Estimated+white noise |

Station displacement | Solid earth tide+Pole tide+Ocean tide loading |

Station coordinate | Fixed; Estimated for static positioning |

Terrestrial frame | ITRF2014 |

**Table 3.**Statistical results of the average root mean square (RMS) of the double-differenced residuals in an ambiguity-fixed period under different processing strategies.

DOY in 2018 | Mean Value of RMS of Double-Differenced Residuals/mm | ||||||||
---|---|---|---|---|---|---|---|---|---|

GPS | BDS | Galileo | |||||||

N | S | M(1) | N | S | M(1) | N | S | M(1) | |

225 | 8.54 | 3.48 | 3.47 | 5.99 | 2.66 | 2.57 | 8.67 | 3.74 | 3.01 |

226 | 8.38 | 5.29 | 5.60 | 5.84 | 2.96 | 2.73 | 8.45 | 3.78 | 3.26 |

227 | 6.55 | 3.10 | 3.48 | 5.98 | 3.45 | 2.95 | 8.70 | 2.81 | 3.04 |

228 | 8.98 | 7.47 | 6.98 | 6.42 | 3.41 | 2.68 | 8.33 | 4.28 | 4.30 |

229 | 7.86 | 4.92 | 4.77 | 7.09 | 5.06 | 3.20 | 8.14 | 5.04 | 4.33 |

230 | 8.87 | 6.62 | 6.92 | 6.21 | 3.67 | 3.48 | 9.10 | 5.19 | 4.34 |

231 | 9.71 | 4.63 | 3.72 | 5.87 | 3.10 | 3.03 | 9.09 | 4.26 | 3.90 |

232 | 9.60 | 3.12 | 3.21 | 6.07 | 3.04 | 2.99 | 9.59 | 3.97 | 3.16 |

233 | 9.68 | 2.96 | 2.94 | 5.82 | 2.90 | 2.81 | 8.41 | 3.25 | 2.91 |

234 | 9.29 | 3.00 | 2.97 | 5.88 | 3.36 | 2.88 | 11.25 | 3.91 | 3.17 |

Average improvement | / | 48.5% | 48.9% | / | 45.4% | 52.0% | / | 54.8% | 60.1% |

**Table 4.**Multi-GNSS static positioning statistics for one hour with different processing strategies.

Observation Environment | A (Simulated) | B (Normal) | |||||
---|---|---|---|---|---|---|---|

Processing Strategies | N | S | M(1) | N | S | M(1) | |

Positioning accuracy RMS (mm) | N | 2.80 | 0.96 | 0.89 | 2.43 | 0.95 | 1.11 |

E | 4.31 | 1.84 | 1.40 | 3.33 | 1.68 | 1.92 | |

U | 9.47 | 3.10 | 1.83 | 5.84 | 2.21 | 1.54 | |

3D improvement | / | 65.3% | 77.1% | / | 59.0% | 62.3% |

DOY for Modeling in 2018 | DOY for Verification in 2018 | |||
---|---|---|---|---|

GPS | BDS (IGSO+GEO) | BDS (MEO) | Galileo | |

224 | 224 | 218 | 215 | 225 |

225 | 225 | 219 | 216 | 226 |

226 | 226 | 220 | 217 | 227 |

227 | 227 | 221 | 218 | 228 |

228 | 228 | 222 | 219 | 229 |

229 | 229 | 223 | 220 | 230 |

230 | 230 | 224 | 221 | 231 |

231 | 231 | 225 | 222 | 232 |

232 | 232 | 226 | 223 | 233 |

233 | 233 | 227 | 224 | 224 |

DOYs for Modeling in 2018 | DOY for Verification in 2018 |
---|---|

215~224 | 225 |

216~225 | 226 |

217~226 | 227 |

218~227 | 228 |

219~228 | 229 |

220~229 | 230 |

221~230 | 231 |

222~231 | 232 |

223~232 | 233 |

224~233 | 234 |

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**MDPI and ACS Style**

Zou, X.; Li, Z.; Wang, Y.; Deng, C.; Li, Y.; Tang, W.; Fu, R.; Cui, J.; Liu, J.
Multipath Error Fusion Modeling Methods for Multi-GNSS. *Remote Sens.* **2021**, *13*, 2925.
https://doi.org/10.3390/rs13152925

**AMA Style**

Zou X, Li Z, Wang Y, Deng C, Li Y, Tang W, Fu R, Cui J, Liu J.
Multipath Error Fusion Modeling Methods for Multi-GNSS. *Remote Sensing*. 2021; 13(15):2925.
https://doi.org/10.3390/rs13152925

**Chicago/Turabian Style**

Zou, Xuan, Zhiyuan Li, Yawei Wang, Chenlong Deng, Yangyang Li, Weiming Tang, Ruinan Fu, Jianhui Cui, and Jingnan Liu.
2021. "Multipath Error Fusion Modeling Methods for Multi-GNSS" *Remote Sensing* 13, no. 15: 2925.
https://doi.org/10.3390/rs13152925