# LEO-Constellation-Augmented BDS Precise Orbit Determination Considering Spaceborne Observational Errors

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## Abstract

**:**

## 1. Introduction

## 2. Constellation Design and Data Simulation

#### 2.1. BDS/LEO Combined Constellation Configurations

#### 2.2. Simulation of Observations

^{−13}s) and the GeoForschungsZentrum (GFZ) 5-min interval multi-GNSS final clock offset products were used to simulate the BDS-3 satellite clock errors. The Earth orientation parameters (EOP) were taken from the IGS values.

## 3. LEO-Constellation-Augmented BDS POD Strategies

- (1)
- First, the designed BDS and LEO ephemerides are converted to a broadcast ephemeris and used as the initial values for the POD. An 18-parameter model [31] is used for the BDS broadcast ephemeris design.
- (2)
- The initial BDS positions are obtained from the generated broadcast ephemeris and then the BDS initial dynamics information is derived via orbit dynamics fitting. The positions of the LEO satellites are derived from the kinematic POD using the BDS broadcast ephemeris and LEO spaceborne data, and then the LEO orbit initial dynamics information is obtained via orbit dynamics fitting.
- (3)
- Orbit integration is applied for the BDS and LEO satellite dynamics to obtain the initial reference orbit and state transition matrix.
- (4)
- The unknown parameters are ranked, combined, and added with the constraints listed in Table 3, and the POD equations are linearized at the reference initial orbit. The observations at all epochs are stacked on a combined adjustment for parameter estimation.
- (5)
- The adjusted BDS and LEO reference orbits are obtained via orbit integration after having resolved the normal equations.
- (6)
- Considering several random cycles of cycle slips are added to the observations, posterior residual editing is conducted until new cycle slips are detected and calibrated.
- (7)
- Steps 3–5 are repeated until the residuals are smaller than the given threshold values. In terms of the carrier-phase measurements, the threshold is usually defined as three times the post-fit STD value. The carrier-phase measurement can be regarded as “bad” and abandoned if the corresponding residual is larger than the threshold value. An ambiguity parameter is introduced and involved in calculation when the residual is larger than one and a half times the STD value. Otherwise, the residuals are regarded as white noises with normal distribution. This step is necessary because the actual-measured code and carrier-phase residuals are included in the observation simulation.
- (8)
- Finally, the BDS and LEO satellite orbit dynamics and clock products are calculated and the precise orbits can be derived via orbit integration with the orbit dynamics information. The estimated final orbits are compared with the “true” orbits, which are described in Section 2.1. A flowchart of the above procedures is given in Figure 3.

## 4. Experimental Analysis

#### 4.1. Analysis of the Spaceborne Measurement Noise

#### 4.2. BDS/LEO Combined POD during a Quiet Period of Solar Activity

#### 4.3. GNSS/LEO Combined POD during a Solar Storm

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Designed BDS-3/LEO constellation: (

**a**) BDS-3, (

**b**) Walker 2/1/0, (

**c**) Walker 6/2/1, (

**d**) Walker 12/3/1, (

**e**) Walker 24/4/2, and (

**f**) Walker 60/6/4.

**Figure 4.**BDS orbit solutions calculated with different levels of spaceborne measurement noise. From left to right: 0.001 m, 0.005 m, 0.01 m, and 0.05 m.

**Figure 5.**Elevation-dependent GF pseudo-range and carrier-phase measurement residuals for Swarm-A (

**top**), GRACE-A (

**middle**), and Sentinel-3A (

**below**).

**Figure 7.**The ODOP values of G05/G10/C02/C03/C06/C11 satellites with respect to the regional ground stations; the regional network and W2/1/0 LEOs; the regional network and W6/2/1 LEOs; the regional network and W12/3/1 LEOs; the regional network and W24/4/2 LEOs; and the regional network and W60/6/4 LEOs, which are shown from subplots left to right. The epoch time interval is 10 s.

**Figure 8.**IF carrier-phase measurement residuals for the Swarm-A and GRACE-A spaceborne GPS observations.

**Figure 9.**The number of tracked BDS C02 (left three subplots)/C06 (middle three subplots)/C11 (right three subplots) satellites by LEO satellites. The green, red, and blue dots represent the W2/1/0, W12/3/1, and W60/6/4 LEO constellation, respectively.

**Figure 10.**Orbit differences between the estimated and the “references” of C02/C06/C11 satellites for scheme S2, S4, and S6, respectively.

System | BDS-3 | LEO | ||
---|---|---|---|---|

Orbit Type | GEO | IGSO | MEO | Polar |

Satellite number | 3 | 3 | 24 | 2, 6, 12, 24, and 60, respectively |

Constellation | Placed at east longitude 80°, 110.5°, and 140° | Placed at geographical longitude 0°, 120°, and 240° | Walker 24/3/1 | 1, 2, 3, 4, and 6 planes, respectively |

Inclination | 0 | 55° | 55° | 86.4° |

altitude [km] | 35,786 | 35,786 | 21,528 | 1200 |

Dynamic Model | BDS | LEO |
---|---|---|

Earth gravity | EIGEN-6C (12 × 12) [20] | EIGEN-6C (120 × 120) |

N-body | JPL DE405 | JPL DE405 |

Solid tide and pole tide | IERS 2010 [21] | IERS 2010 |

Ocean tide | FES 2004 [22] | FES 2004 |

Relatively | IERS 2010 | IERS 2010 |

Solar radiation pressure | ECOM2 7-parameter model [3] | Box-Wing model [23] |

Atmospheric drag | None | DTM94 [24] |

Item | Description | A Priori Constraints |
---|---|---|

Observations | Undifferenced dual-frequency IF LC and PC | Ground pseudo-range observations: 1 m; carrier-phase: 0.02 cycles; spaceborne observations: calibrated with real-calculated observations |

Cut-off elevation | Spaceborne observations: 1°; ground observations: 7° | |

Arc length | 72 h | |

Sampling interval | Ground and spaceborne BDS observations: 10 s | |

Satellite position and velocity | Initial value from the broadcast ephemeris | Position: 10 m; velocity: 0.1 m/s |

Station coordinates | PPP static solutions | 2 mm for horizontal and 5 mm for vertical component |

Tropospheric delay | LEO spaceborne: none; ground stations: Saastamoinen model/GMF, the zenith wet delay estimated every 2 h | 0.2 + 0.2/sqrt (hours) |

BDS and LEO satellite clock offsets | Broadcast ephemeris and white noise (estimated with a BDS clock as reference) | 5000 m |

Spaceborne receiver clock offset | Estimated as white noise | 1000 m |

Ground receiver clock offset | Estimated as white noise | 9000 m |

EOP | Fixed | IGS final products |

Ambiguity | Float solution |

LEO Constellation | Swarm | GRACE | Sentinel-3 |
---|---|---|---|

Satellite | Swarm-A/B/C | GRACE-A/B | Sentinel-3A/B |

Spaceborne observations | GPS | GPS | GPS |

Altitude | A/C: ~460 km, B: ~510 km | ~500 km | ~814.5 km |

Inclination | A/C: 87.35°, B: 87.75° | 89.5° | 98.65° |

Orbit type | Circular near-polar orbits | Circular near-polar orbits | Repeating frozen sun-synchronous orbit [33] |

Repeat cycle | 7–10 months | A sparse repeat track of 61 revolutions every four days [34] | 27-day repeat cycle, 14 + 7/27 orbits per day [34] |

Goal | Geomagnetic observation | Detection of the Earth’s gravity variations | Earth monitoring and operational oceanography |

Satellite | S2 (W2/1/0) | S4 (W/12/3/1) | S6 (60/6/4) | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Item | C02 | C06 | C11 | C02 | C06 | C11 | C02 | C06 | C11 | |

Mean ODOP | 297.5 | 237.5 | 203.1 | 84.0 | 96.4 | 72.6 | 40.2 | 38.8 | 18.7 | |

${\sigma}_{0}^{2}$ [m^{2}] | ${1.12}^{2}=1.25$ | ${1.83}^{2}=3.35$ | ${0.88}^{2}=0.77$ | |||||||

${\sigma}_{0}^{2}\cdot ODO{P}_{mean}$ | 371.9 | 296.9 | 253.9 | 281.4 | 322.9 | 243.2 | 31.0 | 29.9 | 14.4 | |

Orbital 3D RMS [m] | 1.18 | 1.07 | 0.75 | 1.10 | 1.15 | 0.69 | 0.60 | 0.43 | 0.44 |

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**MDPI and ACS Style**

Li, M.; Xu, T.; Ge, H.; Guan, M.; Yang, H.; Fang, Z.; Gao, F.
LEO-Constellation-Augmented BDS Precise Orbit Determination Considering Spaceborne Observational Errors. *Remote Sens.* **2021**, *13*, 3189.
https://doi.org/10.3390/rs13163189

**AMA Style**

Li M, Xu T, Ge H, Guan M, Yang H, Fang Z, Gao F.
LEO-Constellation-Augmented BDS Precise Orbit Determination Considering Spaceborne Observational Errors. *Remote Sensing*. 2021; 13(16):3189.
https://doi.org/10.3390/rs13163189

**Chicago/Turabian Style**

Li, Min, Tianhe Xu, Haibo Ge, Meiqian Guan, Honglei Yang, Zhenlong Fang, and Fan Gao.
2021. "LEO-Constellation-Augmented BDS Precise Orbit Determination Considering Spaceborne Observational Errors" *Remote Sensing* 13, no. 16: 3189.
https://doi.org/10.3390/rs13163189