Kalman filter-based satellite clock bias prediction algorithm with frequency difference estimation correction

doi:10.11947/j.AGCS.2025.20250055

Abstract

Abstract:

Satellite clock bias prediction is of great significance for time synchronization, real-time positioning, and autonomous navigation, and its accuracy directly affects the service quality of navigation systems. The traditional Kalman filter model (KFM) is widely used in clock bias prediction because of its minimum variance estimation characteristics, which enable it to obtain optimal estimates of time difference, frequency difference, and frequency drift. However, KFM does not explicitly model the periodic terms in clock bias, resulting in periodic fluctuations in the estimated values of time difference, frequency difference, and frequency drift. This periodic estimation bias increases the prediction error of KFM and further amplifies it over time. To address this issue, this paper proposes an improved Kalman filter model (IKFM) based on frequency difference estimation correction. This model first identifies the periodic terms in the frequency difference estimates through spectral analysis and fits their parameters using least squares. Then, periodic fluctuations are subtracted from the estimated values to eliminate the interference of periodic terms on the state estimation. Finally, the time difference is extrapolated based on the corrected frequency differences. The experimental results based on GPS clock bias data show that, compared with KFM, IKFM reduced the error in 1~24 h predictions by up to 32.14%; and compared with the gray model, quadratic polynomial model, and spectral analysis model, IKFM showed the best accuracy and stability for all prediction durations. By effectively suppressing periodic term interference, IKFM provides a reliable solution for high-precision satellite clock bias prediction, especially for spaceborne atomic clocks with significant periodic fluctuations.

Key words: satellite clock bias prediction, Kalman filter, prediction accuracy, atomic clock state estimation, periodic terms

CLC Number:

P228

Cong SHEN, Guocheng WANG, Lintao LIU, Huiwen HU, Zhiwu CAI. Kalman filter-based satellite clock bias prediction algorithm with frequency difference estimation correction[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(9): 1596-1607.

Figures/Tables 10

Tab. 1

Fig. 1

Tab. 2

Fig. 2

Fig. 3

Fig. 4

Tab. 3

Tab. 4

Fig. 5

Tab. 5

References 30

[1]	伍贻威, 杨斌, 肖胜红, 等. 原子钟模型和频率稳定度分析方法[J]. 武汉大学学报(信息科学版), 2019, 44(8): 1226-1232.
	WU Yiwei, YANG Bin, XIAO Shenghong, et al. Atomic clock models and frequency stability analyses[J]. Geomatics and Information Science of Wuhan University, 2019, 44(8): 1226-1232.
[2]	樊礼谦, 焦文海, 孟轶男. 基于Kalman滤波的导航星座集中式守时算法研究[J]. 时间频率学报, 2023, 46(1): 21-31.
	FAN Liqian, JIAO Wenhai, MENG Yinan. Research on centralized time-keeping algorithm for navigation constellations based on Kalman filter[J]. Journal of Time and Frequency, 2023, 46(1): 21-31.
[3]	FORMICHELLA V, GALLEANI L, SIGNORILE G, et al. Time-frequency analysis of the Galileo satellite clocks: looking for the J2 relativistic effect and other periodic variations[J]. GPS Solutions, 2021, 25(2): 56.
[4]	LIANG Yifeng, XU Jiangning, LI Fangneng, et al. A VMD-PE-SG denoising method based on K-L divergence for satellite atomic clock[J]. Measurement Science and Technology, 2023, 34(5): 55012.
[5]	胡超, 王潜心et al.. 顾及BDS-3/GNSS星钟状态的超快速钟差参数估计模型[J]. 测绘学报, 2024, 53(12): 2268-2281. DOI: . doi: 10.11947/j.AGCS.2024.20240112
	HU Chao, WANG Qianxinet al.. BDS-3/GNSS satellite ultra-rapid clock offsets estimation model with the aid of onboard clock states solution[J]. Acta Geodaetica et Cartographica Sinica, 2024, 53(12): 2268-2281. DOI: . doi: 10.11947/j.AGCS.2024.20240112
[6]	JIANG Weiping, ZHAO Qile, LI Min, et al. The progress of IGS analysis center at Wuhan university[J]. Journal of Geodesy and Geoinformation Science, 2023, 6(3): 46-57.
[7]	潘雄, 赵万卓, 张龙杰, 等. 联合半参数与优化BiLSTM的BDS-3钟差超短期预报[J]. 中国惯性技术学报, 2024, 32(10): 985-993.
	PAN Xiong, ZHAO Wanzhuo, ZHANG Longjie, et al. Ultra-short-term prediction method for BDS-3 clock offset by combined semi-parametric and improved BiLSTM models[J]. Journal of Chinese Inertial Technology, 2024, 32(10): 985-993.
[8]	蒋春华, 朱美珍, 薛慧杰, 等. 基于长短时记忆神经网络的Multi-GNSS卫星钟差建模预报[J]. 大地测量与地球动力学, 2024, 44(3): 257-262.
	JIANG Chunhua, ZHU Meizhen, XUE Huijie, et al. Multi-GNSS satellite clock offset prediction model based on long short-term memory neural network[J]. Journal of Geodesy and Geodynamics, 2024, 44(3): 257-262.
[9]	TAN Xiaorong, XU Jiangning, HE Hongyang, et al. Short-term satellite clock bias forecast based on complementary ensemble empirical mode decomposition and quadratic polynomial[J]. Survey Review, 2023, 55(389): 127-136.
[10]	雷雨, 赵丹宁. 附加周期项与随机项补偿的卫星钟差预报算法[J]. 中国惯性技术学报, 2024, 32(10): 994-1000.
	LEI Yu, ZHAO Danning. Satellite clock offset prediction algorithm with additional periodic and stochastic terms compensated[J]. Journal of Chinese Inertial Technology, 2024, 32(10): 994-1000.
[11]	SHEN Cong, WANG Guocheng, LIU Lintao, et al. Extraction of periodic terms in satellite clock bias based on Fourier basis pursuit bandpass filter[J]. Remote Sensing, 2025, 17(5): 827.
[12]	XUE Huijie, XU Tianhe, NIE Wenfeng, et al. An enhanced prediction model for BDS ultra-rapid clock offset that combines singular spectrum analysis, robust estimation and gray model[J]. Measurement Science and Technology, 2021, 32(10): 105002.
[13]	朱祥维, 肖华, 雍少为, 等. 卫星钟差预报的Kalman算法及其性能分析[J]. 宇航学报, 2008, 29(3): 966-970, 1052.
	ZHU Xiangwei, XIAO Hua, YONG Shaowei, et al. The Kalman algorithm used for satellite clock offset prediction and its performance analysis[J]. Journal of Astronautics, 2008, 29(3): 966-970, 1052.
[14]	杨旭, 王潜心, 吕伟才, 等. 顾及卫星间相关性的Kalman短期钟差预报[J]. 导航定位学报, 2022, 10(3): 59-68.
	YANG Xu, WANG Qianxin, LÜ Weicai, et al. A method of Kalman filter multi-satellite clock offset short-term prediction based on inter-satellite correlation[J]. Journal of Navigation and Positioning, 2022, 10(3): 59-68.
[15]	郭海荣, 杨元喜, 何海波, 等. 导航卫星原子钟Kalman滤波中噪声方差-协方差的确定[J]. 测绘学报, 2010, 39(2): 146-150.
	GUO Hairong, YANG Yuanxi, HE Haibo, et al. Determination of covariance matrix of Kalman filter used for time prediction of atomic clocks of navigation satellites[J]. Acta Geodaetica et Cartographica Sinica, 2010, 39(2): 146-150.
[16]	宋会杰, 董绍武, 屈俐俐, 等. 基于Sage窗的自适应Kalman滤波用于钟差预报研究[J]. 仪器仪表学报, 2017, 38(7): 1809-1816.
	SONG Huijie, DONG Shaowu, QU Lili, et al. Research on clock difference prediction using adaptive Kalman filter based on Sage window[J]. Chinese Journal of Scientific Instrument, 2017, 38(7): 1809-1816.
[17]	林旭, 罗志才. 一种新的卫星钟差Kalman滤波噪声协方差估计方法[J]. 物理学报, 2015, 64(8): 18-23.
	LIN Xu, LUO Zhicai. A new noise covariance matrix estimation method of Kalman filter for satellite clock errors[J]. Acta Physica Sinica, 2015, 64(8): 18-23.
[18]	谭小容, 许江宁, 何泓洋, 等. GM(1, 1)模型初始条件优化及其在GNSS钟差预报中的应用[J]. 大地测量与地球动力学, 2022, 42(9): 919-924.
	TAN Xiaorong, XU Jiangning, HE Hongyang, et al. Optimized initial condition of GM(1, 1) model and its application in GNSS clock offset prediction[J]. Journal of Geodesy and Geodynamics, 2022, 42(9): 919-924.
[19]	袁德宝, 张建, 张振超, 等. 基于SAFA-FDGM(1, 1)模型的BDS钟差预报[J]. 大地测量与地球动力学, 2021, 41(7): 672-675.
	YUAN Debao, ZHANG Jian, ZHANG Zhenchao, et al. BDS clock error prediction based on SAFA-FDGM(1, 1) model[J]. Journal of Geodesy and Geodynamics, 2021, 41(7): 672-675.
[20]	WU Yiwei. Determination of theoretical Kalman filter performance for atomic clock estimation through equivalent linear time-invariant systems[J]. IEEE Transactions on Aerospace and Electronic Systems, 2023, 59(6): 7908-7922.
[21]	王宇谱, 薛申辉, 王威, 等. 基于一次差分预报原理的常用钟差预报模型效果分析[J]. 大地测量与地球动力学, 2021, 41(4): 336-341.
	WANG Yupu, XUE Shenhui, WANG Wei, et al. Effect analysis of common prediction models specific to satellite clock bias based on the principle of single difference prediction[J]. Journal of Geodesy and Geodynamics, 2021, 41(4): 336-341.
[22]	WANG Guocheng, LIU Lintao, XU Aigong, et al. On the capabilities of the inaction method for extracting the periodic components from GPS clock data[J]. GPS Solutions, 2018, 22(4): 92.
[23]	WU Yiwei, GONG Hang, ZHU Xiangwei, et al. A clock steering method: using a third-order type 3 DPLL equivalent to a Kalman filter with a delay[J]. Metrologia, 2015, 52(6): 864.
[24]	WANG Jian, HAN Houzeng, LIU Fei, et al. Performance analysis of GNSS/MIMU tight fusion positioning model with complex scene feature constraints[J]. Journal of Geodesy and Geoinformation Science, 2021, 4(2): 1-13.
[25]	伍贻威, 龚航, 朱祥维, 等. 原子钟两级驾驭算法及在建立GNSS时间基准中的应用[J]. 电子学报, 2016, 44(7): 1742-1750.
	WU Yiwei, GONG Hang, ZHU Xiangwei, et al. Twice atomic clock steering algorithm and its application in forming a GNSS time reference[J]. Acta Electronica Sinica, 2016, 44(7): 1742-1750.
[26]	WU Yiwei, GONG Hang, ZHU Xiangwei, et al. A DPLL method applied to clock steering[J]. IEEE Transactions on Instrumentation and Measurement, 2016, 65(6): 1331-1342.
[27]	LI Fangneng, LIANG Yifeng, XU Jiangning, et al. Advances in satellite atomic clock technologies for the GNSS[J]. Measurement Science and Technology, 2024, 35(1): 015027.
[28]	XIE Xin, GENG Tao, ZHAO Qile, et al. Orbit and clock analysis of BDS-3 satellites using inter-satellite link observations[J]. Journal of Geodesy, 2020, 94(7): 64.
[29]	WANG Xu, CHAI Hongzhou. Developing an innovative high-precision approach to predict medium-term and long-term satellite clock bias[J]. Journal of Geodesy and Geoinformation Science, 2023, 6(1): 47-58.
[30]	GLANER M F, WEBER R. An open-source software package for precise point positioning: raPPPid[J]. GPS Solutions, 2023, 27(4): 174.

卫星类型	原子钟类型	卫星钟编号
ⅡR	Rb	G02、G13、G16、G19、G20、G21、G22
ⅡRM	Rb	G05、G07、G12、G15、G17、G29、G31
IIF	Rb	G01、G03、G06、G09、G10、G25、G26、G27、G30、G32
IIF	Cs	G08、G24
ⅢA	Rb	G04、G11、G14、G18、G23、G28

f=1/86 400 Hz	f=1/43 200 Hz	f=1/21 600 Hz
0.004 5	0.005 0	0.004 9
－90.760 7	－90.234 0	－90.091 9
－218.925 5	－218.398 2	－218.258 7
－0.001 2	－0.002 7	－0.005 5
0.399 6	0.206 0	0.101 9
0.404 9	0.208 7	0.103 3

模型	ⅡR	ⅡR-M	ⅡF-Rb	ⅡF-Cs	ⅢA	均值
GM	0.604 6	0.837 3	1.744 7	2.702 7	1.501 7	1.311 1
QPM	0.953 4	0.995 3	1.110 4	3.693 3	0.399 0	1.078 9
SAM	0.953 3	0.996 0	1.099 9	3.633 6	0.393 7	1.071 0
KFM	0.670 8	0.740 2	0.804 3	3.594 6	0.277 2	0.836 6
IKFM	0.575 3	0.639 0	0.543 1	1.836 2	0.236 2	0.594 4
均值	0.751 5	0.841 6	1.060 5	3.092 1	0.561 6

统计指标	模型	3 h预报	6 h预报	12 h预报	24 h预报
RMS/ns	GM	0.644 1	0.783 3	1.185 4	2.391 1
	QPM	0.541 2	0.695 3	1.027 5	1.957 6
	SAM	0.528 6	0.681 3	1.004 2	1.936 1
	KFM	0.557 1	0.724 8	1.033 3	1.687 1
	IKFM	0.509 6	0.611 8	0.783 6	1.144 8
预报提升率/（%）	IKFM vs. GM	20.87	21.89	33.89	52.12
	IKFM vs. QPM	5.82	12.01	23.73	41.51
	IKFM vs. SAM	3.57	10.20	21.97	40.87
	IKFM vs. KFM	8.51	15.59	24.16	32.14

测站	E方向RMS			N方向RMS			U方向RMS
测站	IGUP	KFM	IKFM	IGUP	KFM	IKFM	IGUP	KFM	IKFM
FALK	9.5	5.7	2.9	9.7	7.0	7.3	7.4	6.9	5.8
LROC	12.9	12.9	6.2	5.4	8.7	6.0	11.3	16.9	11.0
MAR6	9.4	8.5	7.9	4.5	9.1	4.4	8.6	9.6	5.2