# The Selection of Basic Functions for a Time-Varying Model of Unmodeled Errors in Medium and Long GNSS Baselines

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methodology

#### 2.1. Unmodeled Error Data

#### 2.1.1. Baseline Data

#### 2.1.2. Inversion of Unmodeled Errors

#### 2.2. Methodology

#### 2.2.1. Alternative Basic Functions

#### 2.2.2. Model Testing and Evaluation

#### 2.2.3. Positioning Verification

## 3. Results

#### 3.1. Fitting Experiments

#### 3.1.1. Polynomials

#### 3.1.2. Sinusoidal Functions

#### 3.1.3. Combinatorial Functions

#### 3.2. Positioning Experiments

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Unmodeled errors of the G31−G10 satellite pair in No. 3 baseline in differential positioning modes.

**Figure 2.**Residuals of different polynomials for DD satellite pairs of baselines No. 1~4 in differential positionings. Columns (

**a**,

**b**) display “estimating atmospheric delays” and “using IF combination”, respectively.

**Figure 3.**R-squares and RMSEs of different polynomials for DD satellite pairs of baselines No. 1~4 in differential positionings. Columns (

**a**,

**b**) display “estimating atmospheric delays” and “using IF combination”, respectively.

**Figure 4.**Residuals of sinusoidal functions for the DD satellite pairs of baselines No. 1~4 in differential positionings. Columns (

**a**,

**b**) display “estimating atmospheric delays” and “using IF combination”, respectively.

**Figure 5.**R-square and RMSE values of the sinusoidal functions for DD satellite pairs of baselines No. 1~4 in differential positionings. Columns (

**a**,

**b**) display “estimating atmospheric delays” and “using IF combination”, respectively.

**Figure 6.**Residuals of combinatorial functions for DD satellite pairs of baselines No. 1~4 in differential positionings. Columns (

**a**,

**b**) display “estimating atmospheric delays” and “using IF combination”, respectively.

**Figure 7.**R-square and RMSE values of the combinatorial functions for the DD satellite pairs of baselines No. 1~4 in differential positionings. Columns (

**a**,

**b**) display “estimating atmospheric delays” and “using IF combination”, respectively.

**Figure 8.**Positioning deviations for uncorrected observations and corrected observations in the mode of estimating atmospheric delays.

**Figure 9.**Positioning deviations for uncorrected observations and corrected observations in the mode of IF combination.

Baseline Number | Base Station–Rover Station | Observation Type | Receiver Type at the Base Station | Antenna and Radome Types at the Base Station | Epoch (s) | Distance (km) |
---|---|---|---|---|---|---|

1 | TEHA–MRRY | L1C, L2W | TRIMBLE NETR9 | TRM57971.00 NONE | 10,800 | 30.7 |

2 | TEHA–BFSH | L1C, L2W | TRIMBLE NETR9 | TRM57971.00 NONE | 10,800 | 49.8 |

3 | SIMM–CHOW | L1C, L2W | TRIMBLE NETR9 | TRM57971.00 NONE | 10,800 | 193.3 |

4 | BFSH–CHOW | L1C, L2W | TRIMBLE NETR5 | TRM57971.00 NONE | 10,800 | 226.1 |

Categories | Settings | |
---|---|---|

Estimating Atmospheric Delays | Using IF Combination | |

Observations | Uncombined L1, L2 | $\frac{{f}_{1}^{2}{L}_{1}-{f}_{2}^{2}{L}_{2}}{{f}_{1}^{2}-{f}_{2}^{2}}$ |

Stochastic model | ${\sigma}^{2}={a}^{2}+\frac{{b}^{2}}{{\mathrm{sin}}^{2}(el)}$ | ${\sigma}^{2}={a}^{2}+\frac{{b}^{2}}{{\mathrm{sin}}^{2}(el)}$ |

Cut-off angle | 10 degrees | 10 degrees |

Parameter estimation | Extended Kalman filter | Extended Kalman filter |

Satellite orbit | Precise ephemeris | Precise ephemeris |

Clock bias | DD | DD |

Ionospheric delay | DD + Parameter estimation | DD + IF combination |

Tropospheric delay | DD + Parameter estimation | DD + Parameter estimation |

Relativistic effect | Model correction | Model correction |

Earth Solid Tide | IERS 2010 | IERS 2010 |

Ambiguity resolution | Continuous | Continuous |

**Table 3.**The convergence time of residuals in Figure 2.

Positioning Mode | Baseline Number | Convergence Time (s) | |||
---|---|---|---|---|---|

$\mathit{m}=1$ | $\mathit{m}=2$ | $\mathit{m}=3$ | $\mathit{m}=4$ | ||

Estimating atmospheric delays | 1 | 319 | 25 | 9 | 5 |

2 | 68 | 11 | 5 | 1 | |

3 | 12 | 3 | 3 | 3 | |

4 | 146 | 30 | 11 | 6 | |

Using the IF combination | 1 | 720 | 39 | 14 | 8 |

2 | 243 | 4 | 1 | 1 | |

3 | 223 | 12 | 6 | 1 | |

4 | 597 | 35 | 11 | 6 |

Positioning Mode | Processing Time (s) | |||
---|---|---|---|---|

$\mathit{m}=1$ | $\mathit{m}=2$ | $\mathit{m}=3$ | $\mathit{m}=4$ | |

Estimating atmospheric delays | 0.467 | 0.592 | 0.739 | 0.875 |

Using the IF combination | 0.430 | 0.586 | 0.729 | 0.855 |

**Table 5.**Convergence times of residuals in Figure 6.

Positioning Mode | Baseline Number | Convergence Time (s) | |||||
---|---|---|---|---|---|---|---|

$\mathit{m}=2,\mathit{n}=0$ | $\mathit{m}=2,\mathit{n}=4$ | $\mathit{m}=2,\mathit{n}=6$ | $\mathit{m}=2,\mathit{n}=8$ | $\mathit{m}=2,\mathit{n}=10$ | $\mathit{m}=2,\mathit{n}=12$ | ||

Estimating atmospheric delays | 1 | 25 | 21 | 19 | 18 | 18 | 18 |

2 | 11 | 7 | 7 | 7 | 7 | 7 | |

3 | 3 | 2 | 2 | 2 | 2 | 2 | |

4 | 30 | 25 | 23 | 23 | 23 | 20 | |

uUsing IF combination | 1 | 39 | 39 | 35 | 35 | 35 | 35 |

2 | 4 | 1 | 1 | 1 | 1 | 1 | |

3 | 12 | 8 | 8 | 8 | 8 | 8 | |

4 | 35 | 31 | 26 | 26 | 26 | 26 |

**Table 6.**Processing time to fit the unmodeled errors of one satellite pair for second-order polynomial and combinatorial functions.

Positioning Mode | Processing Time (s) | |||||
---|---|---|---|---|---|---|

$\mathit{m}=2,\mathit{n}=0$ | $\mathit{m}=2,\mathit{n}=4$ | $\mathit{m}=2,\mathit{n}=6$ | $\mathit{m}=2,\mathit{n}=8$ | $\mathit{m}=2,\mathit{n}=10$ | $\mathit{m}=2,\mathit{n}=12$ | |

Estimating atmospheric delays | 0.592 | 1.306 | 1.595 | 1.909 | 2.068 | 2.531 |

Using IF combination | 0.586 | 1.284 | 1.598 | 1.905 | 2.214 | 2.533 |

Positioning Mode | Directions | RMSEs of Uncorrected Positioning (m) | RMSEs of Corrected Positioning (m) | Improvement (%) |
---|---|---|---|---|

estimating atmospheric delays | E | 0.0224 | 0.0080 | 64.29 |

N | 0.0051 | 0.0031 | 39.22 | |

U | 0.0165 | 0.0033 | 80.00 | |

using IF combination | E | 0.0115 | 0.0109 | 5.22 |

N | 0.0088 | 0.0053 | 39.77 | |

U | 0.0184 | 0.0066 | 64.13 |

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**MDPI and ACS Style**

Wang, J.; Yu, X.; Aragon-Angel, A.; Rovira-Garcia, A.; Wang, H.
The Selection of Basic Functions for a Time-Varying Model of Unmodeled Errors in Medium and Long GNSS Baselines. *Remote Sens.* **2023**, *15*, 5022.
https://doi.org/10.3390/rs15205022

**AMA Style**

Wang J, Yu X, Aragon-Angel A, Rovira-Garcia A, Wang H.
The Selection of Basic Functions for a Time-Varying Model of Unmodeled Errors in Medium and Long GNSS Baselines. *Remote Sensing*. 2023; 15(20):5022.
https://doi.org/10.3390/rs15205022

**Chicago/Turabian Style**

Wang, Jiafu, Xianwen Yu, Angela Aragon-Angel, Adria Rovira-Garcia, and Hao Wang.
2023. "The Selection of Basic Functions for a Time-Varying Model of Unmodeled Errors in Medium and Long GNSS Baselines" *Remote Sensing* 15, no. 20: 5022.
https://doi.org/10.3390/rs15205022