Evaluation of strategies for the ultra-rapid orbit prediction of BDS GEO satellites

Figures & data

Figure 1. Illustration of DYB (Sun-fixed) and XYZ (body-fixed) orthogonal frames (Rodriguez-Solano, Hugentobler, and Steigenberger 2012).

Table 1. Values of parameters in the a priori SRP model for BDS GEO satellites (Wang et al. 2019; units: nm/s²; $\beta$ angle in degrees).

Parameter	Value nm/s^2
$D_{0P}$	0.0136 ${\beta }^2-$ 112.1
$D_{1P}$	0.32
$D_{2P}$	−10.4
$D_{4P}$	−1.6
$Y_{1P}$	−0.416 $\beta$
$B_{0P}$	1.4
$B_{1P}$	−0.022 ${\beta }^2-$ 6.53
$B_{3P}$	−1.51

Figure 2. Distributions of MGEX (blue) and iGMAS (red) stations used in ultra-rapid orbit prediction experiment.

Table 2. Orbit dynamic models.

Models	Description
Geopotential (static)	EGM 08 up to 12 × 12 (Pavlis et al. 2012)
N-body	Sun, Moon, Jupiter, Venus, Mars Mercury, Uranus, Neptune, Saturn, Pluto JPL DE405 ephemeris (Standish et al. 1998)
Solid earth tides	International Earth rotation and Reference systems Service (IERS) conventions 2010 (Petit and Luzum 2010)
Ocean tides	IERS conventions 2010
Solid earth pole tides	IERS conventions 2010
Relativistic effects	IERS conventions 2010
Solar radiation pressure	Five-parameter ECOM with and without an a priori model (Wang et al. 2019; Springer, Beutler, and Rothacher 1999)
Earth radiation pressure	None
Orbit integration	Runge–Kutta–Fehlberg 7(8), Adams–Bashforth, and Adams–Moulton methods (Montenbruck and Gill 2012)

Figure 3. Correlations between empirical along-track acceleration and SRP parameters with 24-h, 48-h, and 72-h orbit arc lengths. Other BDS GEO satellites show similar performances.

Figure 4. Workflow of orbit prediction and comparison.

Figure 5. Daily OBDs time series of observed orbits for strategies 1, 2, and 3 in the radial (R), cross-track (C), and along-track (A) directions for BDS C01 satellite (unit: m). The gray blocks indicate the satellite eclipse seasons. The black lines indicate the elevation angle of the Sun above the satellite orbital plane. Other BDS GEO satellites showed similar performance.

Table 3. The daily OBDs RMS values of BDS GEO satellites.

Satellite	Strategy	R	C	A	3D
C01	1	0.345	0.133	0.490	0.614
	2	0.218	0.124	0.499	0.559
	3	0.258	0.053	0.490	0.563
C02	1	0.304	0.151	0.487	0.594
	2	0.228	0.143	0.486	0.556
	3	0.174	0.069	0.469	0.505
C03	1	0.415	0.147	0.664	0.797
	2	0.402	0.138	0.622	0.753
	3	0.324	0.056	0.620	0.702
C04	1	0.419	0.149	0.560	0.715
	2	0.385	0.131	0.570	0.701
	3	0.397	0.055	0.560	0.689
C05	1	0.410	0.138	0.582	0.725
	2	0.370	0.130	0.581	0.703
	3	0.379	0.068	0.563	0.682

Figure 6. SLR residual time series of observed orbits for strategies 1–3 for BDS C01 satellite (unit: m).

Table 4. Mean and STD of SLR residuals for observed orbits for BDS C01 satellite (unit: m), as determined by strategies 1–3.

Strategy	Mean	STD
1	−0.303	0.210
2	−0.026	0.103
3	−0.019	0.104

Figure 7. Daily RMS of BDS C01 orbit overlap differences in the radial, cross-track, and along-track directions for the fitting arc lengths of 24-h, 48-h, and 72-h with different strategies. Other BDS GEO satellites show similar performances.

Figure 8. Correlations between SRP parameters and the PX, PY, PZ (satellite position parameter in the X, Y and Z direction of the celestial reference frame) parameters, VX, VY VZ (satellite velocity parameter in the X, Y and Z direction of the celestial reference frame) parameters of 48-h orbit arc length for BDS C01 satellite when β is equal to 21 degrees, parameter correlations of 24-h and 72-h orbit arc length show similar characteristics.

Table 5. Average RMS values of all BDS GEO satellites in the three directions and the 3D component for strategies 1–3 with fitting arc lengths of 24-h, 48-h, and 72-h for eclipse (E) and non-eclipse (NE) seasons (unit: m).

Strategy	Fitting arc length (h)	R		C		A		3D
		E	NE	E	NE	E	NE	E	NE
1	24	1.220	0.951	0.117	0.197	5.630	3.145	5.775	3.287
	48	0.510	0.192	0.857	0.248	2.689	0.765	2.946	0.851
	72	0.499	0.272	1.515	0.361	2.927	1.031	3.514	1.168
2	24	1.432	1.325	0.210	0.464	5.624	5.478	5.842	5.683
	48	0.564	0.242	0.763	0.282	2.703	0.845	2.964	0.959
	72	0.550	0.264	1.504	0.365	2.978	0.985	3.558	1.130
3	24	1.432	1.330	0.209	0.462	5.627	5.528	5.845	5.727
	48	0.268	0.227	0.185	0.215	1.007	0.836	1.080	0.916
	72	0.287	0.265	0.117	0.169	1.042	0.984	1.105	1.050

Figure 9. Time series of SLR residuals for predicted orbits with 24-h, 48-h, and 72-h fitting arc lengths of BDS C01 satellites (unit: m). Other BDS GEO satellites show similar performances.

Table 6. Means and STDs of SLR residuals of orbit prediction for the three strategies (unit: m).

Strategy	Fit length (h)		Eclipse	Non-eclipse
1	24	Mean	−0.556	−0.964
		STD	0.726	0.872
	48	Mean	−0.248	−0.333
		STD	0.491	0.291
	72	Mean	−0.288	−0.249
		STD	0.528	0.328
2	24	Mean	−0.399	−0.467
		STD	0.869	0.797
	48	Mean	−0.024	−0.041
		STD	0.419	0.213
	72	Mean	−0.039	0.002
		STD	0.497	0.257
3	24	Mean	−0.395	−0.453
		STD	0.869	0.801
	48	Mean	0.038	−0.088
		STD	0.221	0.226
	72	Mean	−0.094	0.018
		STD	0.233	0.242

Figure 10. Estimated SRP parameters (D₀,Y₀,B₀,B_C, and B_S) time series with fitting arc lengths of 24-h, 48-h, and 72-h for strategies 1–3 for BDS C01. Other BDS GEO satellites show similar performances.

Figure 11. Correlations coefficients between the Y₀ and B_S parameters of BDS C01. The black lines indicate the elevation angle of the Sun.

Rodriguez-Solano, C.J., U. Hugentobler, and P. Steigenberger. 2012. “Adjustable Box-wing Model for Solar Radiation Pressure Impacting GPS Satellites.” Advances in Space Research 49 (7): 1113–1128. doi:10.1016/j.asr.2012.01.016.

 Google Scholar

Wang, C., J. Guo, Q. Zhao, and J. Liu. 2019. “Empirically Derived Model of Solar Radiation Pressure for BeiDou GEO Satellites.” Journal of Geodesy 93 (6): 791–807. doi:10.1007/s00190-018-1199-y.

 Google Scholar

Pavlis, N.K., S.A. Holmes, S. Kenyon, and J.K. Factor. 2012. “The Development and Evaluation of the Earth Gravitational Model 2008 (EGM2008).” Journal of Geophysical Research: Solid Earth 117 (B4): 4406. doi:10.1029/2011JB008916.

 Web of Science ®Google Scholar

Standish, E.M. “JPL Planetary and Lunar Ephemerides, DE405/LE405.” JPL IOM 1998, 312.F-98-048. http://iau-comm4.jpl.nasa.gov/de405iom/

 Google Scholar

Petit, G., and B. Luzum. 2010. IERS Conventions 2010 (IERS Technical Note, 179. Vol. 36. Frankfurt am Main: Verlag des Bundesamts für Kartographie und Geodäsie.

 Google Scholar

Springer, T.A., G. Beutler, and M. Rothacher. 1999. “A New Solar Radiation Pressure Model for GPS Satellites.” GPS Solutions 2 (3): 50–62. doi:10.1007/PL00012757.

 Google Scholar

Montenbruck, O., and E. Gill. 2012. Satellite Orbits: Models, Methods and Applications. Berlin, Germany: Springer Science & Business Media.

 Google Scholar