# An Observation Density Based Method for Independent Baseline Searching in GNSS Network Solution

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## Abstract

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## 1. Introduction

## 2. Data and Method

#### 2.1. Data

#### 2.2. MST

#### 2.3. The Criteria—Distance, Observation, and Others

_{SHORTEST}(e) in MST is proportional to the baseline length, i.e., the smaller the baseline length, the smaller w_

_{SHORTEST}(e) is. This is based on the assumption that the closer two stations, the more thoroughly common errors can be eliminated or reduced by DD, thus the higher overall accuracies can be achieved for baseline solutions. For the OBS-MAX method, on the other hand, only the number of DD observations is used as the criterion, i.e., w_

_{OBS-MAX}(e) is considered to be inversely proportional to the number of observations. The more the DD observations, the smaller w_

_{OBS-MAX}(e) is. This strategy is based on the fact that more observations can bring higher redundancy in parameter estimation. WEIGHT is a synthesis of these two strategies. In this process, a normalization is introduced since the dimensions of OBS-MAX (number) and SHORTEST (meter) are not consistent [14]. Unfortunately, WEIGHT is still an empirical operation lacking theoretical support.

_{obs}denotes the number of co-viewing satellites observed by every two stations, m

_{sho}denotes the geodetic distance of each two stations; x

_{1}and x

_{2}denote the weights applied to the m

_{obs}and m

_{sho}factors, respectively, which can be obtained empirically or based on the a posteriori accuracies of the solutions of the two methods. Note that in this paper, the number of observations is counted on a daily basis.

#### 2.4. The Calculation Process of the Independent Baseline

**M**is formed, and each element m

_{obs}_{obs}represents the number of common observations between every station pair. At the same time, the geodetic distance between every station pair is calculated to form the distance matrix

**M**. The unit of element m

_{sho}_{sho}is meter.

**M**and

_{obs}**M**are shown in Equation (2).

_{sho}**M**, which chooses the solution that lets the sum of m

_{sho}_{sho}be the smallest. The main diagonal elements of the

**M**matrix represent all available observations of individual stations or 0 distances, which are not involved in the MST generation.

**M**, the reciprocal of each element or the maximum spanning tree should be used, since the largest observation needs to be chosen instead of the smallest. Correspondingly, the WEIGHT matrix

_{obs}**M**and the OBS-DEN matrix

_{wei}**M**can be computed as follows:

_{den}_{1}and x

_{2}of the WEIGHT were set to 0.5. In order to apply the above methods with Bernese, the generated baseline file can be used to replace the baseline file generated by Bernese’s default scheme. Except for the independent baseline option, all other processing sessions and parameter settings follow Bernese’s default options [8].

#### 2.5. Parallel Computation

## 3. Results

#### 3.1. Single-Day Solution

#### 3.2. One-Year Statistical Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Flow chart for independent baseline selection, starting from reading the RINEX files of all stations, to generate different independent baseline files according to different baseline selection strategies.

**Figure 4.**Histograms of single-day solutions. The x-axis of each subplot is the final station coordinate accuracy in millimeters, and the interval of each bin of North and East is 1 mm (3 mm for Up). The y-axis of each subplot represents the number of stations accommodated in each bin. The columns from left to right denote the East (E), North (N), and Up (U) component, respectively. The four methods from top to bottom are SHORTEST, OBS-MAX, WEIGHT, and OBS-DEN, respectively.

**Figure 6.**Variation of the number of DD observations between every two stations with distance. This is based on a single-day solution. Some of the stations have only GPS observations while others have both GPS and GLONASS observations, leading to two linear patterns in the plot.

**Figure 7.**Histograms of one-year solutions. The x-axis of each subplot is the final station coordinate accuracy in millimeters, and the interval of each bin is 1 mm. The y-axis of each subplot, which is on a logarithmic scale, represents the quotient of the number of stations accommodated in each bin and the total number. The columns from left to right denote the East (E), North (N), and Up (U) component, respectively. The four methods from top to bottom are SHORTEST, OBS-MAX, WEIGHT, and OBS-DEN, respectively.

**Table 1.**Accuracy comparison of single-day solutions of different methods. The statistics of station accuracies are calculated in the local coordinate system. The three axes of the local coordinate frame are East (E), North (N), and Up (U). The left, middle and right columns show the mean, the standard deviation (STD), and the root mean square (RMS) of the station coordinate errors of each method, respectively.

MEAN (mm) | STD (mm) | RMS (mm) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

E | N | U | E | N | U | E | N | U | 3D | |

SHORTEST | −0.99 | 0.89 | −0.53 | 3.66 | 3.15 | 5.97 | 3.79 | 3.28 | 6.00 | 7.81 |

OBS-MAX | −0.70 | −1.09 | 0.13 | 2.83 | 3.53 | 7.28 | 2.92 | 3.69 | 7.28 | 8.67 |

WEIGHT | −1.45 | 0.29 | 0.00 | 2.96 | 3.07 | 6.95 | 3.30 | 3.08 | 6.95 | 8.29 |

OBS-DEN | −0.70 | 0.38 | 0.19 | 2.78 | 2.41 | 6.25 | 2.86 | 2.44 | 6.25 | 7.30 |

**Table 2.**Statistics of one-year solutions. The left side represents the RMS, and the right side represents the probability that the 3D errors for each method exceed certain thresholds. The threshold ε is set as 9.67 mm, which is the average 3D RMS value of the four methods.

RMS | Probability | ||||||
---|---|---|---|---|---|---|---|

E (mm) | N (mm) | U (mm) | 3D | <ε | <2ε | <3ε | |

SHORTEST | 4.38 | 4.21 | 7.63 | 9.75 | 71.89% | 96.17% | 99.38% |

OBS-MAX | 3.92 | 3.94 | 7.79 | 9.57 | 71.96% | 96.54% | 99.16% |

WEIGHT | 4.14 | 3.92 | 7.78 | 9.64 | 71.82% | 96.77% | 99.41% |

OBS-DEN | 4.31 | 4.15 | 7.68 | 9.73 | 72.49% | 96.45% | 99.33% |

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**MDPI and ACS Style**

Liu, T.; Du, Y.; Nie, W.; Liu, J.; Ma, Y.; Xu, G.
An Observation Density Based Method for Independent Baseline Searching in GNSS Network Solution. *Remote Sens.* **2022**, *14*, 4717.
https://doi.org/10.3390/rs14194717

**AMA Style**

Liu T, Du Y, Nie W, Liu J, Ma Y, Xu G.
An Observation Density Based Method for Independent Baseline Searching in GNSS Network Solution. *Remote Sensing*. 2022; 14(19):4717.
https://doi.org/10.3390/rs14194717

**Chicago/Turabian Style**

Liu, Tong, Yujun Du, Wenfeng Nie, Jian Liu, Yongchao Ma, and Guochang Xu.
2022. "An Observation Density Based Method for Independent Baseline Searching in GNSS Network Solution" *Remote Sensing* 14, no. 19: 4717.
https://doi.org/10.3390/rs14194717