# Introduction of the Double-Differenced Ambiguity Resolution into Precise Point Positioning

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Models

_{1}and f

_{2}are the frequencies of L1 and L2 observations; ${N}_{n,u}^{k,i}$ and ${N}_{w,u}^{k,i}$ are SD NL and WL integer ambiguities at user u and their corresponding wavelengths are n and w. ${b}_{n}^{k,i}$ and ${b}_{w}^{k,i}$ are SD NL and WL satellite FCB; and $c$ is the light speed. Equation (2) shows that the SD PPP ambiguity still has no integer property and there are satellite biases. When a high-quality SD real-value IFC ambiguity is introduced, the SD IFC FCB can be removed and Equation (2) can be rewritten as:

#### 2.1. DD AR

_{w}

^{k,i}is SD WL satellite FCB. Equation (4) shows that the SD WL FCB is removed and a DD WL ambiguity is generated, when a high-quality SD WL ambiguity from reference station or other user is introduced. It could be written as:

#### 2.2. PPP AR and Positioning

## 3. Data and Experiments

_{j}is the satellite elevation at epoch j. The corrections for the Earth rotation, Earth tides, relativistic effects, phase center variation (PCV), and differential code bias (DCB) are implemented [37]. The tropospheric delay was corrected using the Saastamoinen model and the rest wet part was estimated by setting up a piece wise constant (PWC) at an interval of 1 h. The settings for the PPP processing are shown in Table 1.

## 4. Discussion

#### 4.1. PPP Solutions

#### 4.2. Application in Displacement Monitoring of the Reference Station

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AR | Ambiguity resolution |

DCB | Differential code bias |

DD | Double-differenced |

FCB | Fractional cycle bias |

GNSS | Global navigation satellite system |

IFC | Ionosphere-free combination |

NL | Narrow lane |

PCV | Phase center variation |

PPP | Precise point positioning |

RMS | Root mean square |

WL | Wide lane |

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**Figure 2.**Stations distribution and inter-station distance (km) of the local (red) and regional (blue) reference stations, circle is user station.

**Figure 3.**Twenty-four hour skyplot of visible GPS satellites observed from Shanghai on 20 August 2014.

**Figure 4.**Ten days (11 to 20 August 2014) positioning errors of strategies #1 (

**top**), #2 (

**middle**), and #3 (

**bottom**) for 1-h sessions.

**Figure 6.**PPP results (cm) for the three coordinate components of north, east, and up, and the ratio values of the selected DD ambiguity processing.

**Figure 7.**Residuals of the ionosphere-free phase combination of the PPP computation for the satellites of G16, G23, G27, and G31 of the stations of CMMZ (

**top**), LGXC (

**middle**), and SHJD (

**bottom**).

**Figure 8.**Results for the baseline components estimated with the DD algorithm for north (

**top**), east (

**middle**) and up (

**bottom**) directions.

Model | Settings | |
---|---|---|

Measurements | Ionosphere-free code and phase combination | |

Adjustment | Least square | |

Weighting | Elevation-dependent function | |

Corrections | DCB(P1-C1) | Products provided by CODE |

Tides corrections | Solid tide and ocean tide correction | |

Phase center variation(PCV) | Absolute IGS 08 correction mode | |

Relativity | Corrected | |

Parameters | Station coordinates | Estimated |

Troposphere | Correction: Saastamoinen model Residual: Estimate as piece wise mode | |

Receiver clock error | Solved for at each epoch as white noise | |

Phase ambiguity | Float and fixing results |

**Table 2.**PPP solution Root mean square (RMS) (cm) of the three strategies using data from the local reference stations.

Strategy | 1 h | 2 h | 4 h | ||||||
---|---|---|---|---|---|---|---|---|---|

North | East | Up | North | East | Up | North | East | Up | |

#1 | 2.30 | 2.87 | 3.12 | 1.92 | 2.01 | 2.27 | 0.80 | 1.01 | 1.23 |

#2 | 2.03 | 2.41 | 2.75 | 1.43 | 1.55 | 2.09 | 0.62 | 0.86 | 0.91 |

#3 | 1.76 | 2.22 | 2.56 | 1.41 | 1.52 | 1.83 | 0.53 | 0.58 | 0.77 |

Improvement (%) | 12 | 16 | 12 | 26 | 23 | 8 | 23 | 15 | 26 |

23 | 23 | 18 | 27 | 24 | 19 | 34 | 43 | 37 |

**Table 3.**PPP solution RMS (cm) of the three strategies using data from the regional reference stations.

Strategy | 1 h | 2 h | 4 h | ||||||
---|---|---|---|---|---|---|---|---|---|

North | East | Up | North | East | Up | North | East | Up | |

#1 | 2.28 | 2.89 | 3.15 | 1.93 | 2.03 | 2.25 | 0.80 | 1.01 | 1.23 |

#2 | 2.09 | 2.44 | 2.79 | 1.47 | 1.58 | 2.10 | 0.62 | 0.86 | 0.91 |

#3 | 1.71 | 2.19 | 2.53 | 1.38 | 1.49 | 1.83 | 0.53 | 0.58 | 0.77 |

Improvement (%) | 8 | 16 | 11 | 24 | 22 | 7 | 23 | 15 | 26 |

25 | 24 | 20 | 28 | 27 | 19 | 34 | 43 | 37 |

Strategy | 1 h | 2 h | 4 h | ||||||
---|---|---|---|---|---|---|---|---|---|

North | East | Up | North | East | Up | North | East | Up | |

#1 | 0.93 | 0.95 | 0.99 | 0.85 | 0.92 | 0.97 | 0.77 | 0.65 | 0.25 |

#2 | 0. 89 | 0.90 | 0.91 | 0.80 | 0.88 | 0.91 | 0.76 | 0.64 | 0.22 |

#3 | 0.84 | 0.85 | 0.89 | 0.78 | 0.82 | 0.88 | 0.72 | 0.60 | 0.20 |

Improvement (%) | 4 | 5 | 8 | 6 | 4 | 6 | 1 | 2 | 12 |

10 | 11 | 10 | 8 | 11 | 9 | 6 | 8 | 20 |

Strategy | 1 h | 2 h | 4 h | ||||||
---|---|---|---|---|---|---|---|---|---|

North | East | Up | North | East | Up | North | East | Up | |

#1 | 0.94 | 0.96 | 0.98 | 0.86 | 0.91 | 0.98 | 0.78 | 0.64 | 0.26 |

#2 | 0.90 | 0.90 | 0.89 | 0.81 | 0.87 | 0.93 | 0.76 | 0.63 | 0.23 |

#3 | 0.85 | 0.87 | 0.89 | 0.77 | 0.82 | 0.89 | 0.71 | 0.62 | 0.19 |

Improvement (%) | 4 | 6 | 9 | 6 | 4 | 5 | 3 | 2 | 12 |

10 | 9 | 9 | 10 | 10 | 9 | 9 | 3 | 27 |

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**MDPI and ACS Style**

Li, H.; Xiao, J.; Zhang, S.; Zhou, J.; Wang, J.
Introduction of the Double-Differenced Ambiguity Resolution into Precise Point Positioning. *Remote Sens.* **2018**, *10*, 1779.
https://doi.org/10.3390/rs10111779

**AMA Style**

Li H, Xiao J, Zhang S, Zhou J, Wang J.
Introduction of the Double-Differenced Ambiguity Resolution into Precise Point Positioning. *Remote Sensing*. 2018; 10(11):1779.
https://doi.org/10.3390/rs10111779

**Chicago/Turabian Style**

Li, Haojun, Jingxin Xiao, Shoujian Zhang, Jin Zhou, and Jiexian Wang.
2018. "Introduction of the Double-Differenced Ambiguity Resolution into Precise Point Positioning" *Remote Sensing* 10, no. 11: 1779.
https://doi.org/10.3390/rs10111779