GNSS与低轨卫星相对论效应改正方法

doi:10.11947/j.AGCS.2025.20250226

测绘学报 ›› 2025, Vol. 54 ›› Issue (12): 2129-2141.doi: 10.11947/j.AGCS.2025.20250226

GNSS与低轨卫星相对论效应改正方法

耿涛¹(), 李强¹^,²(), 程凌岳², 刘经南¹

^1.武汉大学卫星导航定位技术研究中心，湖北　武汉　430079
^2.武汉大学测绘学院，湖北　武汉　430079

收稿日期:2025-06-04 修回日期:2025-12-25 出版日期:2026-01-15 发布日期:2026-01-15
通讯作者: 李强 E-mail:gt_gengtao@whu.edu.cn;lq_liqiang@whu.edu.cn
作者简介:耿涛（1982—），男，博士，教授，研究方向为卫星大地测量及卫星精密定轨。E-mail：gt_gengtao@whu.edu.cn
基金资助:
国家自然科学基金(42374030);广东省重点领域研发计划(2023B1111050013)

The correction method of relativistic effects for GNSS and LEO satellites

Tao GENG¹(), Qiang LI¹^,²(), Lingyue CHENG², Jingnan LIU¹

^1.GNSS Research Center, Wuhan University, Wuhan 430079, China
^2.School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China

Received:2025-06-04 Revised:2025-12-25 Online:2026-01-15 Published:2026-01-15
Contact: Qiang LI E-mail:gt_gengtao@whu.edu.cn;lq_liqiang@whu.edu.cn
About author:GENG Tao (1982—), male, PhD, professor, majors in satellite geodesy and procise orbit determination. E-mail: gt_gengtao@whu.edu.cn
Supported by:
The National Natural Science Foundation of China(42374030);The Key-Area Research and Development Program of Guangdong Province(2023B1111050013)

摘要/Abstract

摘要：

全球导航卫星系统的相对论效应误差来源于卫星与用户接收机间的运动状态差异，涉及引力频移和时间膨胀效应。相较于中高轨卫星，低轨导航卫星受地球非球形引力摄动和时间膨胀效应更为显著，已有相对论效应改正方法的适用性需要进一步探讨。本文回顾了相对论效应改正严密公式和目前IGS分析中心与各导航系统使用的两种近似公式（传统公式和改进公式），其中传统公式仅考虑了卫星受到的地球中心引力并针对小偏心率情况进行了简化，改进公式则进一步考虑了地球J₂项引力影响。分析了两种近似公式对GPS、GLONASS、Galileo和BDS-3系统的相对论效应改正精度，并通过仿真与实测数据重点研究了两种近似公式的改正精度与低轨卫星轨道高度、倾角和偏心率等参数之间的关系。结果表明，相对论效应改正传统公式对中圆轨道卫星（E14、E18除外）的周期性改正误差振幅为0.11 ns，改进公式减小至0.05 ns；传统公式对倾斜地球同步轨道卫星的周期性改正误差振幅为0.05 ns，优于改进公式的0.06 ns；两种近似公式对低轨卫星相对论效应的改正精度明显下降，且受轨道参数影响显著，轨道倾角越大、轨道高度越低和偏心率越大时，改正精度越低，近极轨卫星的周期性改正误差振幅可达1 ns。

关键词: 相对论效应, GNSS, 低轨卫星, 星载原子钟, 卫星钟差

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

中图分类号:

P228

耿涛, 李强, 程凌岳, 刘经南. GNSS与低轨卫星相对论效应改正方法[J]. 测绘学报, 2025, 54(12): 2129-2141.

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.

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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

GNSS与低轨卫星相对论效应改正方法

The correction method of relativistic effects for GNSS and LEO satellites

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