# Pass-by-Pass Ambiguity Resolution in Single GPS Receiver PPP Using Observations for Two Sequential Days: An Exploratory Study

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Observation Models

#### 2.2. Ambiguity Resolution

#### 2.3. Least-Squares Estimation with Fixed Ambiguities

^{10}[6]. With the constraints adding to the Normal Equation (NEQ) in Equation (5), the new NEQ can read as

#### 2.4. Data Processing Scheme

- The first step includes data preprocessing mathematical modelling, parameter estimation and residual editing. In data preprocessing, the data gaps and cycle-slips are detected based on the Turboedit algorithm [27]. Then, the code and carrier-phase observations are modelled with the IF combinations, and the parameters are estimated with the sequential least squares method to obtain the real-valued solution. It should be noted that we do not eliminate the ambiguity parameter as [7] did, because the number of parameters is not large for a single station.
- The second step is on the ambiguity resolution. Firstly, the un-differenced estimates of wide-lane ambiguities are derived from HMW combinations. Then, for each satellite, search the corresponding WL ambiguity for the second sidereal day through the start and end epochs plus 2864, and form the SD WL ambiguity. Note that, even though the specific GPS satellite has slightly different repeat periods [28,29], observations with 30s sampling rate would not have to consider the specific repeat period. However, for the higher sampling rates, such as less than 10s, the exact repeat period needs to be considered for the specific satellite. Next, the DD ambiguity is formed between satellites and inserted to Equation (14) to check whether the ambiguity can be successfully resolved. If it is fixable, the un-differenced IF ambiguities are mapped as the WL ambiguities did to obtain DD IF ambiguity, and DD NL ambiguity is derived by the fixed WL ambiguity and the DD IF ambiguity as in Equation (16). Then, Equation (14) will be applied to fix the DD NL ambiguity.
- All the ambiguities are searched and fixed in step 2, however, only a group of independent DD ambiguities are added and constrained to the NEQ. Then, the normal equation will be resolved again based on the least squares estimation to obtain the ambiguity-fixed solution.

## 3. Results

#### 3.1. Data Collection and Processing Strategies

#### 3.2. Fractional Parts of DD WL and NL Ambiguities

#### 3.3. Ambiguity Fixing Rate

#### 3.4. Positioning Accuracy Assessment

#### 3.5. Deformation Monitoring Application

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Distribution of the fractional parts of all the WL and NL ambiguities for the PPP method on DOY 002-003.

**Figure 5.**Distribution of the fractional parts of all the WL and NL ambiguities for network resolution on DOY 002.

**Figure 6.**Distribution of the fractional parts of all the WL and NL ambiguities for BJFS and YELL on the whole year of 2018.

**Figure 9.**Averaged RMS statistics in east, north and up directions on each day during the experimental period.

**Figure 10.**RMS of the float and fixed position estimates for 1 h hourly data length with respect to the weekly estimates for 35 stations during 7days.

**Figure 11.**RMS of the float and fixed position estimates for 2 h hourly data length with respect to the weekly estimates for 35 stations during 7days.

Items | Strategies |
---|---|

Observations | Undifferenced IF combination of code and phase observations |

Parameter estimation | Least squares |

Reference Frame | ITRF2014 |

Cut-off elevation | 7º |

Sampling rate | 30s |

Session length | 48h |

Weight method | Elevation dependent weighting method |

Phase Center Offset | PCO/PCV corrections, igs14_2045.atx |

Phase wind-up | Corrected |

Tropospheric delays | Mapping function: GMF [32]; Zenith delay parameters for station with a 1 h interval; 48 h gradients for north and east horizontal delays [33] |

Receiver clocks | Solved at each epoch (white noise process) |

Tidal Corrections | FES2004 [34] |

**Table 2.**Mean position RMS in the float and fixed 1 h/2 h hourly solutions, and the accuracy improvements after fixing the ambiguities for the 35 stations.

Session Length | Float Solutions (cm) | Fixed Solutions (cm) | Improvement | ||||||
---|---|---|---|---|---|---|---|---|---|

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

1 h | 1.3 | 3.7 | 3.5 | 1.2 | 3.2 | 3.1 | 8.4% | 11.4% | 12.4% |

2 h | 0.8 | 2.0 | 2.1 | 0.7 | 1.7 | 1.8 | 10.8% | 15.8% | 11.8% |

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**MDPI and ACS Style**

Xi, R.; Chen, Q.; Meng, X.; Psimoulis, P.; Jiang, W.; Xu, C.
Pass-by-Pass Ambiguity Resolution in Single GPS Receiver PPP Using Observations for Two Sequential Days: An Exploratory Study. *Remote Sens.* **2021**, *13*, 3728.
https://doi.org/10.3390/rs13183728

**AMA Style**

Xi R, Chen Q, Meng X, Psimoulis P, Jiang W, Xu C.
Pass-by-Pass Ambiguity Resolution in Single GPS Receiver PPP Using Observations for Two Sequential Days: An Exploratory Study. *Remote Sensing*. 2021; 13(18):3728.
https://doi.org/10.3390/rs13183728

**Chicago/Turabian Style**

Xi, Ruijie, Qusen Chen, Xiaolin Meng, Panos Psimoulis, Weiping Jiang, and Caijun Xu.
2021. "Pass-by-Pass Ambiguity Resolution in Single GPS Receiver PPP Using Observations for Two Sequential Days: An Exploratory Study" *Remote Sensing* 13, no. 18: 3728.
https://doi.org/10.3390/rs13183728