The correction method of relativistic effects for GNSS and LEO satellites

doi:10.11947/j.AGCS.2025.20250226

Abstract

Abstract:

Relativistic effects in satellite navigation stem from the differential motion states between satellites and terrestrial users, manifesting as gravitational frequency shifts and time dilation. These effects are more pronounced for low earth orbit (LEO) satellites due to stronger perturbations from Earth's non-spherical gravity, raising questions about the applicability of existing correction methods. This study reviews the rigorous relativistic correction formula and two approximations used by IGS Analysis Centers and global navigation satellite systems: the traditional formula, which assumes a small orbital eccentricity and only considers Earth's central gravity, and a modified formula that additionally accounts for gravitational perturbations from Earth's higher-order terms. We first evaluate these formulas for GPS, GLONASS, Galileo, and BDS-3 satellites. Subsequently, we analyze the relationship between their correction accuracy and the orbital inclination, semi-major axis, and eccentricity of LEO satellites using simulated and measured data. Results indicate that for MEO satellites (excluding E14/E18), the modified formula reduces the periodic error amplitude from 0.11 ns to 0.05 ns. For BDS-3 IGSO satellites, however, the traditional formula yields a better accuracy of 0.05 ns compared to 0.06 ns from the modified one. For LEO satellites, the accuracy of both formulas decreases significantly and is strongly influenced by orbital parameters. Specifically, correction accuracy decreases with greater orbital inclination, lower orbital altitude, and larger eccentricity, with periodic errors for near-polar LEO satellites reaching up to 1 ns.

Key words: relativistic effect, GNSS, low-orbit satellite, atomic clock, satellite clock offset

CLC Number:

P228

Tao GENG, Qiang LI, Lingyue CHENG, Jingnan LIU. The correction method of relativistic effects for GNSS and LEO satellites[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(12): 2129-2141.

Figures/Tables 16

Fig. 1

Tab. 1

Fig. 2

Fig. 3

Fig. 4

Tab. 2

Tab. 3

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Tab. 4

Tab. 5

Fig. 11

References 33

[1]	ASHBY N. Relativity and the global positioning system[J]. Physics Today, 2002, 55(5): 41-47.
[2]	SPILKER J J, AXELRAD P, PARKINSON B W, et al. Global positioning system: theory and applications, volume I[M]. Washington, D. C.: American Institute of Aeronautics and Astronautics, 1996: 623-625.
[3]	KOUBA J. Improved relativistic transformations in GPS[J]. GPS Solutions, 2004, 8(3): 170-180.
[4]	GAO Weiguang, XIE Xin, MENG Yinan, et al. Performance analysis of LEO augmented GNSS precise point positioning from in-orbitCENTISPACE^TM satellites[J]. Measurement Science and Technology, 2025, 36(1): 016338.
[5]	REID T G R. Orbital diversity for global navigation satellite systems[D]. Stanford: Stanford University, 2017.
[6]	XIE Xin, GENG Tao, ZHAO Qile, et al. Precise orbit determination for BDS-3 satellites using satellite-ground and inter-satellite link observations[J]. GPS Solutions, 2019, 23(2): 40.
[7]	LÜ Yifei, GENG Tao, ZHAO Qile, et al. Initial assessment of BDS-3 preliminary system signal-in-space range error[J]. GPS Solutions, 2019, 24(1): 16.
[8]	KOUBA J. Relativistic time transformations in GPS[J]. GPS Solutions, 2002, 5(4): 1-9.
[9]	HAN Chunhao, CAI Zhiwu. Relativistic effects to the onboard BeiDou satellite clocks[J]. NAVIGATION, 2019, 66(1): 49-53.
[10]	FORMICHELLA V. The J2 relativistic periodic component of GNSS satellite clocks[C]//Proceedings of 2018 IEEE International Frequency Control Symposium (IFCS). Olympic Valley: IEEE, 2018: 1-7.
[11]	王棣星, 李江伟, 陶清瑞. 部分相对论效应对北斗原子钟性能影响分析[J]. 测绘科学, 2022, 47(3): 29-36, 64.
	WANG Dixing, LI Jiangwei, TAO Qingrui. Analysis of the influence of partial relativistic effect on the performance of BDS atomic clock[J]. Science of Surveying and Mapping, 2022, 47(3): 29-36, 64.
[12]	WANG Dixing, LI Min, XUE Huijie, et al. Analysis of the J₂ relativistic effect on the performance of on-board atomic clocks[J]. GPS Solutions, 2023, 27(3): 114.
[13]	FANG Shanchuan, DU Lan, GAO Yunpeng, et al. Orbital elements ephemerides and interfaces design of LEO satellites[J]. Journal of Geodesy and Geoinformation Science, 2019, 2(4): 44-52.
[14]	REN Xia, YANG Yuanxi. Development of comprehensive PNT and resilient PNT[J]. Journal of Geodesy and Geoinformation Science, 2023, 6(3): 1-8.
[15]	YANG Yuanxi. Resilient PNT concept frame[J]. Journal of Geodesy and Geoinformation Science, 2019, 2(3): 1-7.
[16]	GE Haibo, WU Tianhao, LI Bofeng. Characteristics analysis and prediction of low Earth orbit (LEO) satellite clock corrections by using least-squares harmonic estimation[J]. GPS Solutions, 2022, 27(1): 38.
[17]	LI Wenwen, YANG Qiangwen, DU Xiaodong, et al. LEO augmented precise point positioning using real observations from two CENTISPACE^TM experimental satellites[J]. GPS Solutions, 2023, 28(1): 44.
[18]	ARSON K M, ASHBY N, HACKMAN C, et al. An assessment of relativistic effects for low Earth orbiters: the GRACE satellites[J]. Metrologia, 2007, 44(6): 484-490.
[19]	WU Meifang, WANG Kan, LIU Jiawei, et al. Relativistic effects of LEO satellite and its impact on clock prediction[J]. Measurement Science and Technology, 2023, 34(9): 095005.
[20]	IAU. IAU transactions. Vol. 11B[M]. Dordrecht: Kluwer Academic Publishers, 1991.
[21]	IERS. IERS conventions (2003): IERS technical note 32[R]. Frankfurt am Main: International Earth Rotation and Reference Systems Service, 2003.
[22]	PETIT G. Importance of a common framework for the realization of space-time reference systems[M]//Towards an integrated global geodetic observing system (IGGOS). Berlin: Springer, 2000: 3-7.
[23]	Global Positioning System Joint Program Office. Interface control document: NAVSTAR GPS space segment/navigation user interface: ICD-GPS-200[S]. [S. l.]: Global Positioning System Joint Program Office, 1993.
[24]	ASHBY N, SPILKER J J. Introduction to relativistic effects on the global positioning system[M]//Global positioning system: theory and applications, Vol. I. Washington: American Institute of Aeronautics and Astronautics, 1996: 623-697.
[25]	KOUBA J. Testing of general relativity with two Galileo satellites in eccentric orbits[J]. GPS Solutions, 2021, 25(4): 139.
[26]	KOUBA J. Relativity effects of Galileo passive hydrogen maser satellite clocks[J]. GPS Solutions, 2019, 23(4): 117.
[27]	张小红, 马福建. 低轨导航增强GNSS发展综述[J]. 测绘学报, 2019, 48(9): 1073-1087. DOI: . doi: 10.11947/j.AGCS.2019.20190176
	ZHANG Xiaohong, MA Fujian. Review of the development of LEO navigation-augmented GNSS[J]. Acta Geodaetica et Cartographica Sinica, 2019, 48(9): 1073-1087. DOI: . doi: 10.11947/j.AGCS.2019.20190176
[28]	YANG Yuanxi, MAO Yue, REN Xia, et al. Demand and key technology for a LEO constellation as augmentation of satellite navigation systems[J]. Satellite Navigation, 2024, 5(1): 11.
[29]	袁俊军, 李凯, 唐成盼, 等. 面向精密位置服务的低轨卫星轨道预报精度分析[J]. 测绘学报, 2022, 51(5): 640-647. DOI: . doi: 10.11947/j.AGCS.2022.20210473
	YUAN Junjun, LI Kai, TANG Chengpan, et al. Accuracy analysis of LEO satellites orbit prediction for precise position service[J]. Acta Geodaetica et Cartographica Sinica, 2022, 51(5): 640-647. DOI: . doi: 10.11947/j.AGCS.2022.20210473
[30]	刘林, 胡松杰, 王歆. 航天动力学引论[M]. 南京: 南京大学出版社, 2006.
	LIU Lin, HU Songjie, WANG Xin. An introduction of astrodynamics[M]. Nanjing: Nanjing University Press, 2006.
[31]	FOLCIK Z J, CEFOLA P J. A general solution to the second order J₂ contribution in a mean equinoctial element semianalytic satellite theory[C]//Proceedings of 2012 Advanced Maui Optical and Space Surveillance Technologies Conference. Wailea: The Maui Economic Development Board, 2012: 45.
[32]	VASHKOV'YAK M A. Constructive-analytical solution of the problem of the secular evolution of polar satellite orbits[J]. Solar System Research, 2017, 51(4): 315-326.
[33]	WANG Yuan, SUN Xiucong, HE Lixuan, et al. An accurate and efficient second-order J₂ model for the draper semianalytic satellite theory[J]. Acta Astronautica, 2024, 225: 169-185.

偏心率e的数值范围	GNSS卫星
0.000<e<0.002	G01、G11
	R01、R04、R05、R09、R11、R12、R14、R15、R16、R17、R18、R19、R20、R21、R22、R24
	E02、E03、E04、E05、E06、E07、E08、E09、E10、E11、E12、E13、E15、E16、E19、E21、E23、E24、E25、E26、E27、E29、E30、E31、E33、E34、E36
	C19、C20、C21、C22、C23、C24、C25、C26、C27、C28、C29、C30、C32、C33、C34、C35、C36、C37、C41、C42、C43、C44、C45、C48
0.002≤e<0.005	G04、G06、G09、G18、G20、G29
	R02、R03、R07、R08
	C38、C39、C40
0.005≤e<0.010	G03、G05、G12、G13、G14、G23、G26、G30、G32
0.010≤e<0.100	G02、G07、G08、G10、G15、G16、G17、G19、G21、G22、G24、G25、G27、G31
0.100≤e<0.200	E14、E18

卫星	传统公式	改进公式
GPS	0.09	0.04
GLONASS	0.11	0.05
Galileo（除E14、E18外）	0.08	0.04
BDS-3 MEO	0.08	0.04
Galileo E14、E18	0.12	0.07
BDS-3 IGSO	0.05	0.06

轨道参数	仿真数值
高度/km	500、750、1000、1250、1500
偏心率	0.001、0.01、0.02
轨道倾角/（°）	20、40、60、89

偏心率	轨道高度/km	轨道倾角
		精度提高/ns				精度提高比例/（%）
		20°	40°	60°	89°	20°	40°	60°	89°
0.001	500	0.07	0.28	0.40	0.02	33	63	60	2
	750	0.09	0.27	0.38	0.03	58	73	62	3
	1000	0.08	0.26	0.36	0.02	62	75	61	3
	1250	0.08	0.25	0.34	0.02	64	76	60	3
	1500	0.07	0.24	0.32	0.02	65	77	61	3
0.01	500	0.08	0.27	0.39	0.03	32	57	56	3
	750	0.08	0.27	0.39	0.03	50	71	62	3
	1000	0.08	0.26	0.36	0.02	53	72	61	3
	1250	0.07	0.24	0.34	0.02	52	73	60	3
	1500	0.06	0.24	0.33	0.02	53	74	61	3
0.02	500	0.08	0.27	0.38	－0.05	22	46	47	－5
	750	0.07	0.27	0.39	0.03	39	68	63	3
	1000	0.07	0.26	0.36	0.02	42	70	61	3
	1250	0.06	0.24	0.34	0.02	40	70	60	3
	1500	0.06	0.24	0.33	0.02	43	71	61	3

卫星	轨道高度/km	轨道倾角/（°）	偏心率
COSMIC-Ⅱ-1	700	24	~0.001
Sentinel-6A	1336	66	<0.001
GRACE-FO-1	500	89	<0.005