# Correction Method for the Observed Global Navigation Satellite System Ultra-Rapid Orbit Based on Dilution of Precision Values

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principle of Orbit Correction Method

_{i}-th epoch,

**V**(t

_{i}) the residuals of the equation,

**X**(t

_{i}) the orbit parameters (positions and velocities),

**L**(t

_{i}) the observations, and

**A**(t

_{i}) the design matrix expressed as [10]:

_{m}and station r

_{n}with reference to the satellite positions and velocities as in [26]:

_{r}is the signal reception time and ∆t the time taken for the signal to go from satellite to station.

**P**is the weight matrix of the observations, which is elevation-dependent for observations below 30° as in [26]; the observations below 10° are deleted when using Equation (1). Then, the DOP value of the k-th parameter is:

_{0}is related to the accuracy of observations.

_{i}

_{−1}epoch.

**X**(t

_{i}), which can be expressed as functions of [

**DOP**(t

_{i})]; hence:

**X**(t

_{i}) denotes the corrections for positions and velocities of a satellite as given in Equation (3). From the equations of the orbit determination, the DOP values of the orbit parameters of each epoch can be accurately acquired. Therefore, the function models between the orbit parameter corrections and the DOP values can be built based on different mathematical models, which should be selected before orbit correction. In this section, to clearly describe the correction method, the polynomial model chosen as an example to discuss is:

_{j}the epoch number. Stacking all available DOP values as a vector, Equation (11) becomes:

**G**represents the coefficient matrix, $d[{\overline{\mathit{D}\mathit{O}\mathit{P}}}_{k}]$ the residuals of the fitting models, and $\theta =[\begin{array}{ccccc}{\theta}_{0}& {\theta}_{1}& {\theta}_{2}& \cdots & {\theta}_{b}\end{array}]$

^{T}. Thus, the coefficient of the DOP function is:

## 3. Experiments Results of Ultra-Rapid Orbit Determination Correction

#### 3.1. Accuracy Analysis of the Observed Ultra-Rapid Orbit

#### 3.2. Experiments of Orbit Correction

- Step 1:
- Prepare the navigation files, list of stations, and station coordinates and merge daily observations (without the last 3 h of observations); in addition, all observations are preprocessed to refine the initial list of stations in the orbit determination;
- Step 2:
- Calculate epoch-wise the DOP values of each parameter, then accumulate and add them to the orbit correction equations;
- Step 3:
- Compare the determined and predicted ultra-rapid orbit with the multi-GNSS rapid precise orbit of GFZ to obtain the orbit residuals;
- Step 4:
- Establish the function models between the orbit state parameters and its corresponding accumulated DOP values;
- Step 5:
- Predict the DOP values of the last 3 h of the observed parts;
- Step 6:
- Incorporate the predicted DOP values into the orbit correction function to correct the observed part and obtain the predicted parts.

#### 3.3. Ultra-Rapid Orbit Determination

_{i}indicates the DOP values based on all stations except the i-th station, and DOP

_{0}represents the total DOP values before elimination. Note that, given the same distribution of stations, the more stations there are, the smaller are the DOP values. Moreover, in Equation (15), DOP

_{0}is one more station than DOP

_{i}. Therefore, DOP

_{i}is always than DOP

_{0}in Equation (15). Based on different amplification factors of the DOP values, the main steps to optimize the tracking stations distribution are the following:

- Step 1:
- Obtain the initial stations list, observation files, navigation files, and the corresponding stations coordinates;
- Step 2:
- Calculate the DOP
_{0}values of initial stations list after data preprocessing; - Step 3:
- Loop all stations to output the k
_{i}(amplification factors of i-th station) of every station; - Step 4:
- Compare k
_{i}with the given k; if k_{i}is greater than k, the corresponding station is stored in the list of stations; - Step 5:
- Assess whether timeliness can be meet with the requirements based on the selected list of stations; if not, continue to expand k;
- Step 6:
- Output the final list of stations for orbit determination.

## 4. Discussion

## 5. Conclusions and Prospects

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Teunissen, P.; Joosten, P.; Odijk, D. The Reliability of GPS Ambiguity Resolution. GPS Solut.
**1999**, 2, 63–69. [Google Scholar] [CrossRef] - Li, X.; Ge, M.; Douša, J.; Wickert, J. Real-time precise point positioning regional augmentation for large GPS reference networks. GPS Solut.
**2014**, 18, 61–71. [Google Scholar] [CrossRef] - International GNSS Monitoring & Assessment System. Available online: http://www.igmas.org/ (accessed on 12 December 2017).
- Choi, K.; Ray, J.; Griffiths, J.; Bae, T. Evaluation of GPS orbit prediction strategies for the IGS Ultra-rapid products. GPS Solut.
**2013**, 17, 403–412. [Google Scholar] [CrossRef] - Li, Y.; Gao, Y.; Li, B. An impact analysis of arc length on orbit prediction and clock estimation for PPP ambiguity resolution. GPS Solut.
**2015**, 19, 201–213. [Google Scholar] [CrossRef] - Stacey, P.; Ziebart, M. Long-term extended ephemeris prediction for mobile devices. In Proceedings of the International Technical Meeting of the Satellite Division of the Institute of Navigation, Salt Lake City, UT, USA, 20–23 September 2011; Volume 28, pp. 3235–3244. [Google Scholar]
- Wang, Q.; Hu, C.; Xu, T.; Chang, T.; Moraleda, A. Impacts of Earth rotation parameters on GNSS ultra-rapid orbit prediction: Derivation and real-time correction. Adv. Space Res.
**2017**, 60, 2855–2870. [Google Scholar] [CrossRef] - Yang, Y.; Li, J.; Xu, J.; Tang, J.; Guo, H.; He, H. Contribution of the Compass satellite navigation system to global PNT users. Chin. Sci. Bull.
**2011**, 56, 2813–2819. [Google Scholar] [CrossRef][Green Version] - Montenbruck, O.; Hauschild, A.; Steigenberger, P.; Hugentobler, U.; Teunissen, P.; Nakamura, P. Initial assessment of the COMPASS/BeiDou-2 regional navigation satellite system. GPS Solut.
**2013**, 17, 211–222. [Google Scholar] [CrossRef] - Hu, C.; Wang, Q.; Wang, Z.; Peng, X. An Optimal Stations Selected Model Based on the GDOP Value of Observation Equation. Geomat. Inf. Sci. Wuhan Univ.
**2017**, 42, 838–844. [Google Scholar] [CrossRef] - Wang, Q.; Zhang, K.; Wu, S.; Zou, Y.; Hu, C. A method for identification of optimal minimum number of multi-GNSS tracking stations for ultra-rapid orbit and ERP determination. Adv. Space Res.
**2017**. [Google Scholar] [CrossRef] - Ge, M.; Gendt, G.; Dick, G.; Zhang, F.; Rothacher, M. A new data processing strategy for huge GNSS global networks. J. Geod.
**2006**, 80, 199–203. [Google Scholar] [CrossRef] - Chen, H.; Jiang, W.; Ge, M.; Wickert, J.; Schuh, H. An enhanced strategy for GNSS data processing of massive networks. J. Geod.
**2014**, 88, 857–867. [Google Scholar] [CrossRef] - Blewitt, G. Fixed point theorems of GPS carrier phase ambiguity resolution and their application to massive network processing: Ambizap. J. Geophys. Res.
**2008**, 113, B12410. [Google Scholar] [CrossRef] - Chen, J.; Zhang, Y.; Xie, Y.; Zhou, X. Rapid data processing of huge networks and Multi-GNSS constellation. Geomat. Inf. Sci. Wuhan Univ.
**2014**, 39, 253–257. [Google Scholar] [CrossRef] - Zhang, R.; Zhang, Q.; Huang, G.; Wang, L.; Qu, W. Impact of tracking station distribution structure on BeiDou satellites orbit determination. Adv. Space Res.
**2015**, 56, 2177–2187. [Google Scholar] [CrossRef] - He, L.; Ge, M.; Wang, J.; Wickert, J.; Schuh, H. Experimental study on the precise orbit determination of the BeiDou navigation satellite system. Sensors
**2013**, 13, 2911–2928. [Google Scholar] [CrossRef] [PubMed] - Guo, J. The Impacts of Attitude, Solar Radiation and Function Model on Precise Orbit Determination for GNSS Satellites. Ph.D. Thesis, Wuhan University, Wuhan, China, 2014. [Google Scholar]
- Blanco-Delgado, N.; Nunes, F.; Seco-Granados, G. On the relation between GDOP and the volume described by the user-to-satellite unit vectors for GNSS positioning. GPS Solut.
**2017**, 21, 1139–1147. [Google Scholar] [CrossRef][Green Version] - Yarlagadda, R.; Ali, I.; Al-Dhahir, N.; Hershey, J. GPS GDOP metric. IEEE Proc.
**2000**, 147, 259–264. [Google Scholar] [CrossRef] - Li, J.; Li, Z.; Zhou, W.; Si, S. Study on the minimum of GDOP in satellite navigation and its application. Acta Geod. Cartogr. Sin.
**2011**, 40, 85–88. [Google Scholar] - Xue, S.; Yang, Y.; Dang, Y.; Chen, W. Dynamic positioning configuration and its first-order optimization. J. Geod.
**2014**, 88, 127–143. [Google Scholar] [CrossRef] - Wang, Q.; Dang, Y.; Xu, T. The method of Earth rotation parameter determination using GNSS observations and precision analysis. Lect. Notes Electr. Eng.
**2013**, 243, 247–256. [Google Scholar] [CrossRef] - Malkin, Z. On comparison of the Earth orientation parameters obtained from different VLBI networks and observing programs. J. Geod.
**2009**, 83, 547–556. [Google Scholar] [CrossRef] - Dvorkin, V.; Karutin, S. Optimization of the global network of tracking stations to provide GLONASS users with precision navigation and timing service. Gyrosc. Navig.
**2013**, 4, 181–187. [Google Scholar] [CrossRef] - Xu, G. GPS-Theory, Algorithms and Applications; Springer: New York, NY, USA, 2007. [Google Scholar]
- Akaike, H. Likelihood of a model and information criteria. J. Econ.
**1981**, 16, 3–14. [Google Scholar] [CrossRef] - Hu, C.; Wang, Q.; Wang, Z.; Moraleda, A. New-Generation BeiDou (BDS-3) Experimental Satellite Precise Orbit Determination with an Improved Cycle-Slip Detection and Repair Algorithm. Sensors
**2018**, 18, 1402. [Google Scholar] [CrossRef] [PubMed] - Dow, J.; Neilan, R.; Rizos, C. The international GNSS service (IGS) in a changing landscape of Global Navigation Satellite Systems. J. Geod.
**2009**, 83, 191–198. [Google Scholar] [CrossRef] - Wuhan University GNSS Center. Available online: http://igs.gnsswhu.cn/ (accessed on 14 June 2016).
- German Research Center for Geosciences. Available online: ftp://ftp.gfz-potsdam.de/ (accessed on 1 January 2013).
- Estey, L.; Meertens, C. TEQC: The Multi-Purpose Toolkit for GPS/GLONASS Data. GPS Solut.
**1999**, 3, 42–49. [Google Scholar] [CrossRef]

**Figure 1.**3D RMSs between the observed ultra-rapid orbit of WHU and the rapid orbit of GFZ during the last three hours: (

**a**) GPS, (

**b**) GLONASS, (

**c**) BeiDou, (

**d**) Galileo.

**Figure 2.**3D RMSs between the predicted ultra-rapid orbit based on WHU and the rapid orbit of GFZ during 24 h: (

**a**) GPS, (

**b**) GLONASS, (

**c**) BeiDou, (

**d**) Galileo.

**Figure 3.**DOP values and orbit accuracy of different schemes with different number of stations of last 3 h for each epoch (30 s interval: (

**a**,

**b**) G09, (

**c**,

**d**) C13, (

**e**,

**f**) E19, (

**g**,

**h**) R11).

**Figure 6.**10-day (DOY 141–150, 2016) 3D RMSs of the GNSS observed orbit for the last 3 h based on improvement and not improvement method.

**Figure 7.**Orbit accuracy before and after correction of the last three hours and the corresponding DOP values on DOY 141: (

**a**) DOP values with and without prediction, (

**b**) orbit accuracy based on observed, predicted and correction, respectively.

**Figure 9.**Distributions of stations for the different schemes: (

**a**) all stations, (

**b**) 5%, (

**c**) 10%, (

**d**) 15%, (

**e**) 20%, (

**f**) GFZ_sites.

**Figure 10.**Orbit 1D RMSs of different orbit determination schemes: (

**a**) GPS, (

**b**) GLONASS, (

**c**) Galileo, (

**d**) BeiDou.

Systems | 1–20 h | 21 h | 22 h | 23 h | 24 h |
---|---|---|---|---|---|

GPS | 3.6 | 3.5 | 4.6 | 6.3 | 7.5 |

GLONASS | 6.1 | 5.6 | 6.1 | 7.5 | 10.2 |

BeiDou | 12.9 | 12.1 | 12.4 | 15.6 | 21.8 |

Galileo | 8.2 | 13.2 | 13.5 | 14.4 | 17.9 |

Systems | 2 h | 4 h | 6 h | 1–12 h | 1–24 h |
---|---|---|---|---|---|

GPS | 7.8 | 9.8 | 10.6 | 10.8 | 16.1 |

GLONASS | 12.5 | 13.4 | 13.6 | 14.3 | 21.9 |

BeiDou | 23.2 | 39.9 | 61.1 | 70.1 | 139.9 |

Galileo | 16.9 | 26.6 | 32.5 | 34.4 | 49.9 |

Effective (%) | MP1 (m) | MP2 (m) | CSR | ||
---|---|---|---|---|---|

1–21 h | Minimum | 96.8 | 0.04 | 0.03 | 0.02 |

Maximum | 100.0 | 0.28 | 0.31 | 4.22 | |

Average | 95.1 | 0.22 | 0.26 | 1.49 | |

21–24 h | Minimum | 94.2 | 0.06 | 0.09 | 0.02 |

Maximum | 100.0 | 0.28 | 0.41 | 4.81 | |

Average | 96.3 | 0.21 | 0.38 | 2.48 |

Station Numbers | Correlation Factors |
---|---|

409 | 0.9283 |

200 | 0.8846 |

150 | 0.8635 |

100 | 0.8134 |

0 | - |

**Table 5.**10-day (DOY 141–150, 2016) results (mm) of orbit accuracy for the last three hours based on improvement and no improvement method and its improvement rate.

Systems | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 | Improvement Rate |
---|---|---|---|---|---|---|---|---|---|---|---|

GPS (no improvement) | 42 | 43 | 45 | 48 | 39 | 48 | 36 | 42 | 48 | 42 | - |

GPS (improvement) | 34 | 31 | 39 | 36 | 34 | 38 | 28 | 37 | 34 | 37 | 20% |

GLONASS (no improvement) | 59 | 56 | 67 | 61 | 66 | 66 | 72 | 61 | 77 | 78 | - |

GLONASS (improvement) | 44 | 50 | 65 | 58 | 58 | 62 | 64 | 55 | 26 | 35 | 22% |

Galileo (no improvement) | 81 | 88 | 76 | 74 | 88 | 83 | 74 | 83 | 90 | 88 | - |

Galileo (improvement) | 73 | 78 | 68 | 63 | 79 | 75 | 63 | 66 | 79 | 76 | 13% |

BDS_MEO (no improvement) | 80 | 75 | 88 | 67 | 86 | 68 | 66 | 79 | 88 | 88 | - |

BDS_MEO (improvement) | 65 | 66 | 80 | 56 | 72 | 59 | 60 | 74 | 71 | 85 | 12% |

BDS_IGSO (no improvement) | 85 | 78 | 88 | 70 | 88 | 88 | 78 | 79 | 89 | 74 | - |

BDS_IGSO (improvement) | 66 | 64 | 68 | 62 | 72 | 74 | 71 | 71 | 76 | 63 | 16% |

Satellites | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 |
---|---|---|---|---|---|---|---|---|---|---|

G09 | 94% | 90% | 92% | 88% | 89% | 92% | 93% | 94% | 88% | 90% |

R11 | 79% | 76% | 76% | 82% | 86% | 82% | 76% | 76% | 68% | 86% |

C13 | 78% | 68% | 68% | 84% | 72% | 63% | 67% | 78% | 79% | 70% |

E19 | 76% | 66% | 62% | 62% | 64% | 56% | 66% | 67% | 60% | 76% |

**Table 7.**Orbits 1D RMS (cm) of the last 3 h before and after correction and the corresponding station numbers for different schemes.

Schemes | GPS | GLONASS | Galileo | BeiDou | ||||
---|---|---|---|---|---|---|---|---|

ORB (cm) 1D RMSs | Number of Stations | ORB (cm) 1D RMSs | Number of Stations | ORB (cm) 1D RMSs | Number of Stations | ORB (cm) 1D RMSs | Number of Stations | |

All stations | - | 409 | - | 281 | - | 118 | - | 38 |

5% | 1.27 | 171 | 3.31 | 165 | 3.53 | 101 | 3.38 | 38 |

10% | 1.61 | 131 | 4.14 | 129 | 4.61 | 92 | 4.33 | 38 |

15% | 2.48 | 101 | 4.95 | 100 | 5.63 | 76 | 7.63 | 35 |

20% | 3.30 | 91 | 5.72 | 90 | 6.93 | 70 | 9.72 | 34 |

GFZ_site | 2.97 | 111 | 5.25 | 106 | 6.42 | 76 | 8.62 | 20 |

90 stations | 3.99 | 90 | 6.53 | 90 | 7.89 | 66 | 11.95 | 26 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, Q.; Hu, C.; Mao, Y. Correction Method for the Observed Global Navigation Satellite System Ultra-Rapid Orbit Based on Dilution of Precision Values. *Sensors* **2018**, *18*, 3900.
https://doi.org/10.3390/s18113900

**AMA Style**

Wang Q, Hu C, Mao Y. Correction Method for the Observed Global Navigation Satellite System Ultra-Rapid Orbit Based on Dilution of Precision Values. *Sensors*. 2018; 18(11):3900.
https://doi.org/10.3390/s18113900

**Chicago/Turabian Style**

Wang, Qianxin, Chao Hu, and Ya Mao. 2018. "Correction Method for the Observed Global Navigation Satellite System Ultra-Rapid Orbit Based on Dilution of Precision Values" *Sensors* 18, no. 11: 3900.
https://doi.org/10.3390/s18113900