# A Long Baseline Three Carrier Ambiguity Resolution with a New Ionospheric Constraint

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{5}signal, in addition to the current L

_{1}and L

_{2}signals. Chinese Beidou satellites can transmit L

_{2}, L

_{7}and L

_{6}signals, and Galileo was designed to provide signals centred at L

_{1}, E6, E5B and E5A. The third frequency is close to the second frequency, which creates favourable conditions for directly fixing extra wide-lane (EWL) ambiguity and further achieving wide-lane (WL) and narrow-lane (NL) ambiguity resolutions (AR). Thus, using triple-frequency signals to improve the efficiency and reliability of AR in long baselines has become an active research topic.

## 2. On the Fundamental Combinations Concerning TCAR

_{1}, f

_{2}, f

_{3}are the frequencies of the three carriers, and they satisfy f

_{1}> f

_{2}> f

_{3}; N

_{(1,0,0)}, N

_{(0,1,0)}and N

_{(0,0,1)}are DD ambiguities on the three frequencies; the subscribe represents different frequency for the corresponding DD phase observables. The DD non-dispersive delay mainly consists of geometric distances and tropospheric delay. I

_{1}denotes the DD ionospheric delay with respect to the first frequency. The parameters at the end of the equations represent the pseudorange and phase measurement noise, respectively. In the following sections, the standard deviations of pseudorange and phased observables are assumed to be 0.2 m and 0.003 m, respectively [2]. Assuming that the combination coefficients i, j and k are arbitrary integers, the linear combined DD pseudorange observation can be modelled as follows:

## 3. The Current TCAR Method

#### 3.1. EWL and WL Resolutions

#### 3.2. NL Resolution

_{EWL}, V

_{WL}and V

_{1}are the residual vector of phase observables, respectively; and I

_{EWL}, I

_{WL}and I

_{1}are the corresponding OMC vectors. In the filtering process, the LAMBDA algorithm is used to search and fix narrow ambiguity [18,19,20,21]. Equation (6) can be used to estimate the ionospheric delay. However, the precision of its estimate is approximately 0.3 m; thus, the ionospheric estimations only minimally contribute to the ambiguity resolution even if it is treated as a constraint (Equation (5)) [8,17]. For comparison with the new ionospheric model, Equation (6) is defined as the “ionospheric combination method” in the following sections.

## 4. New Ionosphere-Weighted Model

#### 4.1. New Model of Estimating DD Ionospheric Delay

- Geometry-free condition:${x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}+{x}_{6}=0$
- Retain the remaining DD ionosphere delay:$-{x}_{1}\cdot {\eta}_{(1,0,0)}-{x}_{2}\cdot {\eta}_{(0,1,0)}-{x}_{3}\cdot {\eta}_{(0,0,1)}+{x}_{4}\cdot {\eta}_{(1,0,0)}+{x}_{5}\cdot {\eta}_{(0,1,0)}+{x}_{6}\cdot {\eta}_{(0,0,1)}=1$
- Eliminate N
_{(1,0,0)}:$-{x}_{1}\cdot {\lambda}_{(1,0,0)}+{x}_{7}+{x}_{8}=0$ - Eliminate N
_{(0,1,0)}:$-{x}_{2}\cdot {\lambda}_{(0,1,0)}-{x}_{7}=0$ - Eliminate N
_{(0,0,1)}:$-{x}_{3}\cdot {\lambda}_{(0,0,1)}-{x}_{8}=0$ - Minimize the noise condition:$({x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2})\cdot {\sigma}_{p}^{2}+({x}_{4}^{2}+{x}_{5}^{2}+{x}_{6}^{2})\cdot {\sigma}_{\varphi}^{2}=\mathrm{min}$

#### 4.2. Smoothed Ionospheric Delay Estimates

#### 4.3. NL Ambiguity Resolution

## 5. Experiments and Analysis

#### 5.1. Evaluate the Precision of the Ionospheric Estimates

#### 5.2. Evaluate the Performance of NL Resolution

_{1}and T

_{2}, respectively, represent the TFFS for the old and the new method, I represents the improvement, and the calculation process is as follows:

## 6. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The theoretical standard deviations (std) of the DD ionospheric delay estimates versus the std of the pseudorange observations; the std of the phase observation is 2 mm (

**top**); 3 mm (

**middle**) and 4 mm (

**bottom**). “Adaptive” represents the estimation where the ratio satisfies the real value, and the other lines with different colours represent estimations when the ratios are constants (75, 100, 125, 150, and 175).

**Figure 2.**The differences between theoretical std of the DD ionospheric delay estimates and the adaptive estimates; the std of the phase observation is, respectively, 2 mm (

**top**) and 3 mm (

**bottom**). The “adaptive” represents the estimation with the ratio is satisfied with the real one, and the other lines with different colours represent the estimations when the ratios are constants (75, 100, 125, 150, and 175).

**Figure 3.**The theoretical standard deviation (std) of ionospheric versus the smoothing length. Blue indicates the results when the std of ionospheric is 0.02 m, red indicate the results when the std of ionospheric is 0.01 m, solid lines represent the results from the new ionospheric model, and dashed lines represent the estimates of ionospheric delay derived from the combination method.

**Figure 5.**Differences between ionospheric estimates and actual values as estimated by the new ionospheric model (

**top**) and the combination method (

**bottom**).

**Figure 6.**Differences between smoothed ionospheric delay estimates and the actual values when the smoothing length is 10 (

**top**) and 16 (

**bottom**) epochs. Detailed distributions are depicted on the right.

**Figure 7.**Ionospheric residuals derived by the old method (

**top**) and the new method (

**bottom**) for the first portion.

**Figure 8.**Positioning results with the old method (

**top**) and the new method (

**bottom**) for the first portion; the reciprocals of their ratio values are described on the right.

**Figure 9.**Average TFFS on the baselines of HOFN-MYVA (

**top**); OBE4-WTZ3 (

**middle**) and LLAG-MASL (

**bottom**).

**Figure 10.**Ionospheric residuals estimated by the old (

**top**) and the new (

**bottom**) methods during the first 60 epochs for the 12 cases.

**Figure 11.**Reciprocals of the ratios of the 12 portions; they are, respectively derived by the old (

**top**) and the new (

**bottom**) methods. The annotation 1 indicates that ratio = 1, the annotation 0 corresponds to fixed epochs, and the dash indicates that the ratio is 3.

**Table 1.**The wavelength, ionospheric scalar factor and noise amplitude factor for the NL/WL/EWL combinations.

Class | ${\mathit{\varphi}}_{(\mathit{i},\mathit{j},\mathit{k})}$ | ${\mathit{\lambda}}_{(\mathit{i},\mathit{j},\mathit{k})}$ | ${\mathit{\eta}}_{(\mathit{i},\mathit{j},\mathit{k})}$ | ${\mathit{\mu}}_{(\mathit{i},\mathit{j},\mathit{k})}$ |
---|---|---|---|---|

NL | ${\varphi}_{(1,0,0)}$ | 0.190 | 1 | 1 |

${\varphi}_{(0,1,0)}$ | 0.244 | 1.647 | 1 | |

${\varphi}_{(0,0,1)}$ | 0.255 | 1.793 | 1 | |

WL | ${\varphi}_{(1,0,-1)}$ | 0.751 | −1.339 | 4.930 |

${\varphi}_{(1,-1,0)}$ | 0.862 | −1.283 | 5.740 | |

EWL | ${\varphi}_{(1,-6,5)}$ | 3.256 | −0.0744 | 103.80 |

${\varphi}_{(0,1,-1)}$ | 5.861 | −1.719 | 33.24 |

x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} |
---|---|---|---|---|---|---|---|

−12.46 | 68.29 | −56.07 | −0.14 | 0.15 | 0.21 | −16.66 | 14.30 |

Observation Time Span | T_{1} (Epoch) | T_{2} (Epoch) | I (%) |
---|---|---|---|

1 | 39 | 32 | 17.95 |

2 | 36 | 30 | 16.67 |

3 | 50 | 39 | 22.00 |

4 | 34 | 25 | 26.47 |

5 | 16 | 15 | 6.25 |

6 | 61 | 51 | 16.39 |

7 | 27 | 22 | 18.52 |

8 | 71 | 53 | 25.35 |

9 | 39 | 29 | 25.64 |

10 | 48 | 37 | 22.92 |

11 | 40 | 30 | 25.00 |

12 | 65 | 50 | 23.08 |

Mean | 43.76 | 34.34 | 21.53 |

Baseline Name | Baseline Length (km) | T1 (min) | T2 (min) | I (%) |
---|---|---|---|---|

BJF1-BJXT | 78 | 20.67 | 17.01 | 17.71 |

LLAG-MASL | 104 | 21.42 | 17.30 | 19.23 |

OBE4-MTZ3 | 166 | 25.71 | 18.99 | 26.14 |

HOFN-MYVA | 173 | 23.39 | 19.61 | 16.16 |

SHA1-ZJKD | 207 | 28.50 | 22.42 | 21.33 |

WUH1-HBF2 | 244 | 27.82 | 23.64 | 15.03 |

CHA1-CHSW | 258 | 28.95 | 25.07 | 13.40 |

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**MDPI and ACS Style**

Ning, Y.; Yuan, Y.; Huang, Z.; Chai, Y.; Tan, B.
A Long Baseline Three Carrier Ambiguity Resolution with a New Ionospheric Constraint. *ISPRS Int. J. Geo-Inf.* **2016**, *5*, 198.
https://doi.org/10.3390/ijgi5110198

**AMA Style**

Ning Y, Yuan Y, Huang Z, Chai Y, Tan B.
A Long Baseline Three Carrier Ambiguity Resolution with a New Ionospheric Constraint. *ISPRS International Journal of Geo-Information*. 2016; 5(11):198.
https://doi.org/10.3390/ijgi5110198

**Chicago/Turabian Style**

Ning, Yafei, Yunbin Yuan, Zhen Huang, Yanju Chai, and Bingfeng Tan.
2016. "A Long Baseline Three Carrier Ambiguity Resolution with a New Ionospheric Constraint" *ISPRS International Journal of Geo-Information* 5, no. 11: 198.
https://doi.org/10.3390/ijgi5110198