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Article(id=1322567021527683704, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322567020940481144, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.05.175, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1747843200000, receivedDateStr=2025-05-22, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793254150038, onlineDateStr=2026-10-29, pubDate=1776182400000, pubDateStr=2026-04-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793254150038, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793254150038, creator=13701087609, updateTime=1793254150038, updator=13701087609, issue=Issue{id=1322567020940481144, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='4', pageStart='392', pageEnd='521', issueExtLink='null', onlineDate='null', pubDate='1776182400000', pubDateStr='2026-04-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793254149899, creator='13701087609', updateTime=1793254149899, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=510, endPage=521, ext={EN=ArticleExt(id=1322567021716427385, articleId=1322567021527683704, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=A Method for Satellite Clock Offset Prediction Based on Singular Spectrum Analysis, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

A "decomposition-reconstruction-prediction" satellite clock offset prediction method based on singular spectrum analysis (SSA) is proposed. This method first decomposes the clock offset time series into trend, periodic, and random signals. Then, based on the significant difference in Hurst exponent among the three types of signals, the reconstructed clock offset signal is determined. Finally, the SSA iterative interpolation algorithm is used to extrapolate the reconstructed clock offset signal and obtain the final predicted clock offset time series. Taking GPS and BDS-3 satellite clock offsets as an example, the results show that SSA can accurately extract trend and periodic signals in clock offset time series. Compared to the quadratic polynomial model with periodic term, grey model and autoregressive integrated moving average model, the proposed method has a smaller root mean square error in prediction and more stable prediction results. For the 12-hour prediction, the average prediction accuracy of GPS satellite clock offset is improved by 38.81%, 59%, and 72.11% compared to the other three models, respectively, while the accuracy of BDS-3 satellite clock offset is improved by 71.54%, 74.82%, and 70.09%, respectively. For the 24-hour prediction, the improvements for GPS satellite clock offset are 51%, 70.66%, and 83.99%, respectively, while the improvements for BDS-3 satellite clock offset are 86.36%, 85.25%, and 81.05%, respectively. It is concluded that the proposed method is effective and practical for satellite clock offset prediction.

, authors=Chuanlei LÜ1, Yu LEI1, *, Danning ZHAO2, authorsList=Chuanlei LÜ, Yu LEI, Danning ZHAO, authorCompany=null, correspAuthors=Yu LEI, authorNote=null, correspAuthorsNote=null, copyrightStatement=All rights reserved. Unauthorized reproduction is prohibited., copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1322567025906537105, articleId=1322567021527683704, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=基于奇异谱分析的卫星钟差预报方法, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

提出一种基于奇异谱分析(SSA)的"分解-重构-预报"卫星钟差预报方法。该方法首先将钟差时间序列分解为趋势信号、周期信号与随机信号, 然后根据3类信号的Hurst指数存在明显差异这一特性确定钟差重构信号, 最后利用SSA迭代插补算法对钟差重构信号进行外推, 获得最终的钟差预报时间序列。以GPS和BDS-3卫星钟差为例进行分析与预报研究, 结果表明, SSA能够准确提取钟差的趋势信号与周期信号, 相比于附加周期项的二次多项式模型、灰色模型和差分整合移动平均自回归模型, 本文方法预报结果的均方根误差更小且更稳定。对于12 h预报, GPS卫星钟差平均预报精度较3种模型分别提高38.81%、59%和72.11%, BDS-3卫星钟差平均预报精度分别提高71.54%、74.82%和70.09%;对于24 h预报, GPS卫星钟差平均预报精度分别提高51%、70.66%和83.99%, BDS-3卫星钟差平均预报精度分别提高86.36%、85.25%和81.05%, 验证了本文方法的有效性与实用性。

, authors=吕传磊1, 雷雨1, *, 赵丹宁2, authorsList=吕传磊, 雷雨, 赵丹宁, authorCompany=null, correspAuthors=雷雨, authorNote=

吕传磊, 主要研究方向为GNSS数据处理, E-mail:

, correspAuthorsNote=
雷雨, 博士, 副教授, 主要研究方向为GNSS时间传递, E-mail:
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吕传磊, 主要研究方向为GNSS数据处理, E-mail:

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figureFileSmall=v0ZKNWc1ln18CL12jpUwEw==, figureFileBig=SxghaUvqC1ggf5O2VNsvFQ==, tableContent=null), ArticleFig(id=1322567031233303234, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567021527683704, language=CN, label=图10, caption=各模型和SSA方法的卫星钟差预报均方根误差, figureFileSmall=v0ZKNWc1ln18CL12jpUwEw==, figureFileBig=SxghaUvqC1ggf5O2VNsvFQ==, tableContent=null), ArticleFig(id=1322567031308800707, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567021527683704, language=EN, label=Fig. 11, caption=Range of satellite clock offset prediction from each model and SSA method, figureFileSmall=dC0uEmsN+G6qqQW7AGTlnQ==, figureFileBig=fAGSgj93/dslfGahmkx+tg==, tableContent=null), ArticleFig(id=1322567031380103876, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567021527683704, language=CN, label=图11, caption=各模型和SSA方法的卫星钟差预报极差, figureFileSmall=dC0uEmsN+G6qqQW7AGTlnQ==, figureFileBig=fAGSgj93/dslfGahmkx+tg==, tableContent=null), ArticleFig(id=1322567031438824133, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567021527683704, language=EN, label=Tab. 1, caption=

Information of the 14 selected satellites

, figureFileSmall=null, figureFileBig=null, tableContent=
卫星编号 卫星类型 星钟类型 启用日期 卫星编号 卫星类型 星钟类型 启用日期
G02 Block ⅡR 铷钟 2024-11-12 G23 Block ⅢA 铷钟 2020-10-01
G16 Block ⅡR 铷钟 2003-02-18 C60 GEO 氢钟 2020-03-09
G17 Block ⅡR-M 铷钟 2005-11-13 C62 GEO 氢钟 2023-05-17
G29 Block ⅡR-M 铷钟 2008-01-02 C38 IGSO 氢钟 2019-04-20
G25 Block ⅡF 铷钟 2010-08-27 C39 IGSO 氢钟 2019-06-25
G30 Block ⅡF 铷钟 2014-05-30 C26 MEO 氢钟 2018-08-25
G11 Block ⅢA 铷钟 2022-05-25 C43 MEO 氢钟 2019-11-23
), ArticleFig(id=1322567031505932998, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567021527683704, language=CN, label=表1, caption=

选取的14颗卫星信息

, figureFileSmall=null, figureFileBig=null, tableContent=
卫星编号 卫星类型 星钟类型 启用日期 卫星编号 卫星类型 星钟类型 启用日期
G02 Block ⅡR 铷钟 2024-11-12 G23 Block ⅢA 铷钟 2020-10-01
G16 Block ⅡR 铷钟 2003-02-18 C60 GEO 氢钟 2020-03-09
G17 Block ⅡR-M 铷钟 2005-11-13 C62 GEO 氢钟 2023-05-17
G29 Block ⅡR-M 铷钟 2008-01-02 C38 IGSO 氢钟 2019-04-20
G25 Block ⅡF 铷钟 2010-08-27 C39 IGSO 氢钟 2019-06-25
G30 Block ⅡF 铷钟 2014-05-30 C26 MEO 氢钟 2018-08-25
G11 Block ⅢA 铷钟 2022-05-25 C43 MEO 氢钟 2019-11-23
), ArticleFig(id=1322567031568847559, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567021527683704, language=EN, label=Tab. 2, caption=

The dominant periodic terms of satellite clock offset

, figureFileSmall=null, figureFileBig=null, tableContent=
卫星编号 周期1/h 周期2/h 周期3/h 周期4/h
G02 6 12 24 4
G16 12 6 24 4
G17 12 6 24 4
G29 24 12 6 4
G25 12 6 24 4
G30 12 6 24 4
G11 6 24 12 4
G23 6 24 12 4
C60 12 24 8 6
C62 12 24 8 6
C38 24 12 8 4
C39 24 12 8 4
C26 12 6 24 4
C43 12 6 24 4
), ArticleFig(id=1322567031631762120, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567021527683704, language=CN, label=表2, caption=

卫星钟差的主周期项

, figureFileSmall=null, figureFileBig=null, tableContent=
卫星编号 周期1/h 周期2/h 周期3/h 周期4/h
G02 6 12 24 4
G16 12 6 24 4
G17 12 6 24 4
G29 24 12 6 4
G25 12 6 24 4
G30 12 6 24 4
G11 6 24 12 4
G23 6 24 12 4
C60 12 24 8 6
C62 12 24 8 6
C38 24 12 8 4
C39 24 12 8 4
C26 12 6 24 4
C43 12 6 24 4
), ArticleFig(id=1322567031698870985, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567021527683704, language=EN, label=Tab. 3, caption=

Mean RMSE and range of satellite clock offset prediction

, figureFileSmall=null, figureFileBig=null, tableContent=
模型GPS(12 h) GPS(24 h) BDS-3(12 h) BDS-3(24 h)
RMSE 极差 RMSE 极差 RMSE 极差 RMSE 极差
GM 1.00 1.73 1.67 2.89 1.39 1.74 2.44 3.48
ARIMA 1.47 2.64 3.06 5.59 1.17 1.88 1.90 3.40
QPP 0.67 0.93 1.00 1.63 1.23 2.09 2.64 4.89
SSA 0.41 0.84 0.49 1.17 0.35 0.54 0.36 1.38
), ArticleFig(id=1322567031791145674, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567021527683704, language=CN, label=表3, caption=

卫星钟差预报的平均均方根误差和极差

, figureFileSmall=null, figureFileBig=null, tableContent=
模型GPS(12 h) GPS(24 h) BDS-3(12 h) BDS-3(24 h)
RMSE 极差 RMSE 极差 RMSE 极差 RMSE 极差
GM 1.00 1.73 1.67 2.89 1.39 1.74 2.44 3.48
ARIMA 1.47 2.64 3.06 5.59 1.17 1.88 1.90 3.40
QPP 0.67 0.93 1.00 1.63 1.23 2.09 2.64 4.89
SSA 0.41 0.84 0.49 1.17 0.35 0.54 0.36 1.38
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基于奇异谱分析的卫星钟差预报方法
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吕传磊 1 , 雷雨 1, * , 赵丹宁 2
大地测量与地球动力学 | 大地测量学 2026,46(4): 510-521
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大地测量与地球动力学 |大地测量学 2026 , 46 (4) : 510 -521
基于奇异谱分析的卫星钟差预报方法
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吕传磊1 , 雷雨1, * , 赵丹宁2
作者信息
  • 1 西安邮电大学计算机学院, 西安, 710121
  • 2 宝鸡文理学院电子电气工程学院, 宝鸡, 721016
通讯作者:
雷雨, 博士, 副教授, 主要研究方向为GNSS时间传递, E-mail:
作者简介:

吕传磊, 主要研究方向为GNSS数据处理, E-mail:

A Method for Satellite Clock Offset Prediction Based on Singular Spectrum Analysis
Chuanlei LÜ1 , Yu LEI1, * , Danning ZHAO2
Affiliations
  • 1 School of Computer Science and Technology, Xi'an University of Posts and Telecommunications, Xi'an 710121, China
  • 2 School of Electronic and Electrical Engineering, Baoji University of Arts and Sciences, Baoji 721016, China
出版时间: 2026-04-15 doi: 10.14075/j.jgg.2025.05.175
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摘要
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提出一种基于奇异谱分析(SSA)的"分解-重构-预报"卫星钟差预报方法。该方法首先将钟差时间序列分解为趋势信号、周期信号与随机信号, 然后根据3类信号的Hurst指数存在明显差异这一特性确定钟差重构信号, 最后利用SSA迭代插补算法对钟差重构信号进行外推, 获得最终的钟差预报时间序列。以GPS和BDS-3卫星钟差为例进行分析与预报研究, 结果表明, SSA能够准确提取钟差的趋势信号与周期信号, 相比于附加周期项的二次多项式模型、灰色模型和差分整合移动平均自回归模型, 本文方法预报结果的均方根误差更小且更稳定。对于12 h预报, GPS卫星钟差平均预报精度较3种模型分别提高38.81%、59%和72.11%, BDS-3卫星钟差平均预报精度分别提高71.54%、74.82%和70.09%;对于24 h预报, GPS卫星钟差平均预报精度分别提高51%、70.66%和83.99%, BDS-3卫星钟差平均预报精度分别提高86.36%、85.25%和81.05%, 验证了本文方法的有效性与实用性。

卫星钟差  /  钟差预报  /  奇异谱分析  /  Hurst指数

A "decomposition-reconstruction-prediction" satellite clock offset prediction method based on singular spectrum analysis (SSA) is proposed. This method first decomposes the clock offset time series into trend, periodic, and random signals. Then, based on the significant difference in Hurst exponent among the three types of signals, the reconstructed clock offset signal is determined. Finally, the SSA iterative interpolation algorithm is used to extrapolate the reconstructed clock offset signal and obtain the final predicted clock offset time series. Taking GPS and BDS-3 satellite clock offsets as an example, the results show that SSA can accurately extract trend and periodic signals in clock offset time series. Compared to the quadratic polynomial model with periodic term, grey model and autoregressive integrated moving average model, the proposed method has a smaller root mean square error in prediction and more stable prediction results. For the 12-hour prediction, the average prediction accuracy of GPS satellite clock offset is improved by 38.81%, 59%, and 72.11% compared to the other three models, respectively, while the accuracy of BDS-3 satellite clock offset is improved by 71.54%, 74.82%, and 70.09%, respectively. For the 24-hour prediction, the improvements for GPS satellite clock offset are 51%, 70.66%, and 83.99%, respectively, while the improvements for BDS-3 satellite clock offset are 86.36%, 85.25%, and 81.05%, respectively. It is concluded that the proposed method is effective and practical for satellite clock offset prediction.

satellite clock offset  /  clock offset prediction  /  singular spectrum analysis  /  Hurst exponent
吕传磊, 雷雨, 赵丹宁. 基于奇异谱分析的卫星钟差预报方法. 大地测量与地球动力学, 2026 , 46 (4) : 510 -521 . DOI: 10.14075/j.jgg.2025.05.175
Chuanlei LÜ, Yu LEI, Danning ZHAO. A Method for Satellite Clock Offset Prediction Based on Singular Spectrum Analysis[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (4) : 510 -521 . DOI: 10.14075/j.jgg.2025.05.175
正文
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卫星钟是导航卫星的最关键载荷之一，可为卫星系统提供高稳定性的时频基准。尽管导航卫星均搭载高精度的铯钟、铷钟或氢钟，但与系统时间之间仍存在偏差或漂移，因此卫星钟差是全球导航卫星系统(GNSS)定位中一项非常重要的误差源。目前IGS及其分析中心提供的24 h超快速卫星钟差产品仍然是获取钟差预报值的主要途径，这就要求建立精确的卫星钟差模型进行短期预报。
钟差预报模型如二次多项式模型[1]、灰色模型(GM)[2]、差分整合移动平均自回归模型(ARIMA)[3]、指数平滑模型[4]和卡尔曼滤波模型[5]等，受卫星钟自身因素及温度和磁场等空间环境的影响，多星定轨条件下的卫星钟差不仅具有频偏和频漂等趋势性变化，还会出现显著的周期性和随机性波动[6-7]，使得模型精度难以取得较大改进。为提高钟差预报精度，部分学者提出基于神经网络与深度学习的钟差预报方法，如小波神经网络[8]、非线性自回归神经网络[9]和循环神经网络[10]等，这些方法虽可提高钟差预报精度，但仍存在3个问题：1)神经元数量和学习率等网络参数难以确定，通过经验进行调节使得网络性能不够稳定；2)网络训练速度较慢，会影响预报效率；3)网络过拟合或欠拟合可能导致网络泛化能力较弱。这些因素限制了神经网络与深度学习在钟差预报中的应用。
奇异谱分析(SSA)是一种非参数、无模型的非线性和非平稳时间序列分析方法，通过将时间序列数据转化为矩阵形式，并利用奇异值分解来提取时间序列中的趋势信号、周期信号与噪声，从而实现对时间序列的消噪、重构与预报[11]。该方法提取信号无需先验信息，不受正弦波假定的约束，能较好地从含噪时间序列中提取时变信号，近年来在大地测量领域得到广泛应用[12-15]。
针对卫星钟差复杂的时频特性，本文提出一种基于SSA的卫星钟差预报方法。该方法不仅采用SSA重构钟差趋势项和周期项，还利用SSA迭代插补算法对重构信号进行直接外推预报。首先，使用SSA对卫星钟差时间序列进行分解，得到钟差的多个信号分量；然后，根据趋势信号、周期信号与随机噪声的Hurst指数存在明显差异这一特性，确定钟差重构趋势信号分量和周期信号分量；最后，利用迭代插补算法对重构信号进行外推，获得最终的钟差预报序列。利用IGS提供的GPS精密卫星钟差数据和德国地学研究中心(GFZ)提供的BDS-3精密卫星钟差数据分别进行1 h、6 h、12 h和24 h的预报对比分析，验证本文算法的有效性。
1 卫星钟差物理模型
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多星定轨条件下的卫星钟差不仅包含由频偏、频漂等产生的误差，也包含受轨道和钟差耦合影响而产生的周期性波动。卫星钟差的这种物理特性通常可采用附加周期项的二次多项式(QPP)模型表示[16]：
$\begin{aligned}& x(t)=x_0+y_0 t+\frac{1}{2} d t^2+ \\& \sum\limits_{i=1}^n A_i \sin \left(2 \pi f_i t+\varphi_i\right)+v(t)\end{aligned}$
式中，x(t)为t时刻的卫星钟差；x0为卫星钟初始相位；y0为卫星钟钟速；d为卫星钟频漂率；A、f和φ分别为周期性波动的振幅、频率和初始相位；n为钟差周期项数目；υ(t)为随机性误差。
2 预报方法
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基于SSA的卫星钟差预报主要包括钟差时间序列分解、长期趋势项和周期项主成分识别及其外推预报3个步骤[11]。
2.1 钟差时间序列分解与主成分识别
卫星钟差序列包含长期趋势项、周期项和随机项非线性成分，首先利用SSA对非线性、非平稳序列的分析与处理能力，对钟差序列y1, y2, …, yN进行分解，其中N为钟差序列长度，SSA分解包含以下步骤。
1) 构建轨迹矩阵。根据窗口长度M将钟差序列构建成M×(N-M+1)阶轨迹矩阵X =[Y1, Y2, …, YN－M+1]，矩阵X的每行和每列均为钟差序列的子序列，即Yi=[yi, yi+1, …, yi+M－1]T，其中1≤i≤N-M+1。
2) 奇异值分解。对轨迹矩阵X的协方差矩阵W=XXT进行奇异值分解，获得特征值λ1≥λ2≥…≥λM>0和特征向量U1, U2, …, UM。令$\boldsymbol{V}_i=\sqrt{\lambda_i} \boldsymbol{X}^{\mathrm{T}} \boldsymbol{U}_i(i=1, 2, \cdots, M)$，则X=X1+ X2+…+ XM*，其中M*=min(M, N－M+1)，初等矩阵$\boldsymbol{X}_i=\sqrt{\lambda_i} \boldsymbol{U}_i \boldsymbol{V}_i^{\mathrm{T}}$。
3) 分组。将初等矩阵Xi的下标{1, 2, …, M*}分成p个不相交的子集I1, I2, …, Ip，称为分组，则合成矩阵X=X1+X2+…+XM*可写为X=XI1+ XI2+… +XIp。
4) 重构。利用对角平均化将矩阵XIj(j=1, 2, …, p)重构为长度为N的时间序列，每个重构分量(RC)代表钟差序列的不同特征。
SSA分解得到的RC分量可表征原始钟差序列中的长期趋势项、周期项和随机项成分，其中长期趋势项和周期项存在长期记忆性，具有长时间相关效应，其Hurst指数大于0.5；随机项的未来值与历史值相关性不强，长期记忆性较弱，其Hurst指数小于0.5。Hurst指数可通过重标极差(R/S)分析方法计算得到，具体计算方法参考文献[17]。本文将Hurst指数大于0.5的RC分量识别为趋势项或周期项，Hurst指数小于0.5的RC分量识别为随机项。
2.2 SSA预报
SSA预报算法是由SSA迭代插值算法发展而来，利用SSA对钟差序列进行预报实质上是SSA迭代插值，其核心思想为：首先将钟差序列中需要外推预报时刻的预报值用零值填补；然后利用SSA对填补后的时间序列进行分解，将零值用RC分量代替；最后选取适当的迭代次数重复上述过程，直至获得预期预报效果。具体步骤如下：
1) 设原始时间序列长度为N，预报长度为P，并将P个时刻的预报值用零值填补。
2) 对长度为N+P的延长序列进行SSA分解，P个时刻的预报值用第一阶RC分量(记为RC1，下同)的最后P个时刻的数值代替，再对新的延长序列进行分解，重复该过程，直至2次预报值的均方根误差(RMSE)小于阈值。
3) 在步骤2)基础上添加第二阶RC分量(RC2)并代替预报值，即预报值为RC1+RC2，重复步骤2)。选取适当数量的RC分量(RC重构阶次r)进行SSA迭代预报，直至得到预期预报结果。
2.3 参数选取
钟差预报过程需要确定窗口长度M和RC重构阶次r两个参数。对于SSA分解，若窗口长度太小，钟差趋势项、周期项和随机项无法完全分解，RC分量会叠加不同成分；反之，窗口长度过大，信号易出现频率混叠效应，会产生虚假钟差成分。目前SSA窗口长度M基本上是根据时间序列特点主观选取，缺乏统一标准，通常取1 < M≤N/2。由于受轨道与钟差耦合的影响，多星定轨条件下的卫星钟差存在明显的周期性变化，其周期约为卫星轨道周期的1/2倍、1倍或2倍[18]。基于此，本文选取卫星钟差时间序列中周期项的最小公倍数作为SSA窗口长度。
钟差序列SSA分解获得的前r阶RC分量的方差贡献率较大，反映了信号的主要能量，故前r阶RC分量是长期趋势项和周期项主成分[15]，需要进行SSA迭代预报，若重构阶次太小，钟差的主成分被视为随机噪声而忽略，会降低钟差预报精度；反之，若重构阶次过大，随机噪声会参与SSA迭代预报，也会影响预报质量。本文根据长期趋势项和周期项主成分与随机噪声的Hurst指数来确定RC重构阶次，核心思想为：给定窗口长度M，令r=1，计算扣除RC1分量后剩余RC分量之和RCr+1+RCr+2+…+RCM的Hurst指数，若H＞0.5，则令r=r+1，重复上述过程，直至RCr+1+RCr+2+…+RCM的Hurst指数H≤0.5，选取此时的r作为RC重构阶次，即前r阶RC分量为钟差长期趋势项和周期项主成分，需要进行SSA迭代预报。
3 实验结果与分析
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以GPS卫星钟差和BDS-3卫星钟差作为研究对象，验证本文方法的有效性。GPS卫星钟差数据来自IGS，BDS-3卫星钟差数据来自GFZ，钟差数据时间段为2024-04-14—17，数据采样间隔为5 min。GPS星座由中圆地球轨道(MEO)卫星构成，包含Block Ⅱ R、Block Ⅱ R-M和现代化的Block Ⅱ F和Block Ⅲ A卫星类型，BDS-3星座由地球同步轨道(GEO)卫星、倾斜地球同步轨道(IGSO)卫星和MEO卫星构成，从各类型卫星中随机选取2颗卫星进行分析与预报，选取的14颗卫星的具体信息见表 1。
3.1 钟差时间序列分解与重构
利用SSA对钟差时间序列进行分解，重点是选择合适的窗口长度M，一般1 < M≤N/2且为周期的最小公倍数。以G16卫星为例，先利用二次多项式模型对2024-04-14—16时段5 min的钟差序列进行拟合，去除钟差趋势项，再使用快速傅里叶变换对去除趋势项的残差序列进行频谱分析，原始钟差序列、拟合残差序列及其频谱分析结果见图 1。可以看出，原始钟差序列呈递增趋势，周期性表现不明显；去除趋势项的残差序列表现出显著的周期性和随机性波动，通过频谱分析可知，钟差序列包含4个主周期项，按照显著程度依次为12 h、6 h、24 h和4 h，分别对应GPS卫星轨道周期的1倍、1/2倍、2倍和1/3倍(GPS卫星轨道周期为11 h 58 min)，表明卫星钟差不仅与轨道周期耦合，还受到外界昼夜环境变化的影响，可能与卫星光压模型有关。因此，选择窗口长度M=24 h(288个历元)对G16卫星进行SSA分解，获得288个RC分量，限于篇幅，本文仅给出前20阶RC分量，结果如图 2所示。
从图 2可以看出，RC1分量与G16卫星原始钟差变化趋势非常接近，对应钟差长期趋势项；RC2分量也应属于钟差趋势变化的一部分，但时间序列两端可能存在一定程度的畸变和失真(在数据处理中称为端部效应)；RC3和RC4分量属于同一信号，这是因为根据SSA分解原理，针对周期信号，SSA分解可以获得一对近似相等的特征值，由此可以得到一对相似的RC分量，对RC3+RC4分量进行频谱分析(图 3(a))可知，RC3和RC4分量对应钟差序列中的12 h周期信号；类似地，RC5和RC6分量也属于同一信号，频谱分析(图 3(b))显示，这2个分量对应钟差序列中的6 h周期信号；RC7分量的周期性波动频率和振幅均小于RC2~RC5分量，对应钟差序列中的24 h周期信号(图 3(c))；RC8和RC9分量同样属于同一周期信号，对应钟差序列中的振幅较小的4 h周期信号(图 3(d))；RC9分量是上述不同频率混叠的周期信号；RC11~RC13分量对应钟差序列中2.4 h和3 h的微弱周期信号；其他RC分量表现为准周期性变化甚至非周期性变化(图 3(e))，且幅度较小。从图 2还可以发现，钟差时间序列中的主周期项不稳定，其波动幅度具有一定的时变特性，这与卫星钟受温度和磁场等空间环境的影响有关[7]。
为分析RC分量的波动特性，使用R/S方法计算RC分量的Hurst指数，结果如图 4所示。由图可知，RC1和RC2分量的Hurst指数为0.99，与原始钟差的Hurst指数0.98近乎相等，表明RC1和RC2分量波动较小，代表钟差长期趋势项；RC3~RC10分量的Hurst指数介于0.55~0.95之间，表现为持久性变化特征，其长期变化行为具有可预报性；其他RC分量的Hurst指数均小于0.5，且大部分不超过0.2，说明10阶以后的RC分量波动明显，时间序列变化特性复杂，表现为反持久性变化特征，其长期变化行为难以预报甚至不可预报。为进一步确定RC重构阶次r以提取钟差长期趋势项和周期项主成分，使用§2.3中方法确定RC重构阶次，得到RC重构阶次r=10，图 5为钟差重构信号和剩余信号。对比图 1(a)和5(a)可以看出，重构低频钟差信号与原始钟差变化趋势和幅值非常接近，而重构高频钟差信号则表现为随机性变化。重构低频钟差信号的频谱结果(图 3(f))显示，重构信号包含与原始钟差相同的12 h、6 h、24 h和4 h四个主周期项。实验结果表明，利用SSA提取钟差长期趋势项和周期项是可行、有效的。
3.2 钟差预报实验
3.2.1 SSA预报方法窗口长度影响
SSA预报方法是以钟差序列中周期项的最小公倍数作为SSA窗口长度，为验证窗口长度选取的合理性，首先利用频谱分析方法确定卫星钟差的主周期项，分析结果见表 2(周期按照显著程度依次给出)，然后选取主周期项的最小公倍数作为窗口长度，分析SSA方法的钟差重构效果。分别利用QPP模型和SSA方法对2024-04-14—16钟差序列进行拟合和重构，得到QPP模型和SSA方法的钟差残差(拟合或重构序列与钟差序列之差)，结果见图 6。可以看出，QPP模型虽然在钟差拟合时吸收了主周期项，但钟差拟合残差中仍存在部分周期项，Block Ⅱ R和Block Ⅱ R-M卫星钟差序列的拟合残差基本在1 ns以内，Block Ⅱ F和Block Ⅲ A卫星钟差序列的拟合残差在0.3 ns以内。对于BDS-3卫星，除C60卫星钟差序列的拟合残差较大外，其他5颗BDS-3卫星钟差序列的拟合残差在1.2 ns以内，这是由于卫星钟在轨运行的复杂性导致钟差周期性变化具有时变特性，QPP模型仅考虑了钟差周期项的确定性部分，而时变性部分残留在钟差残差项中，因此QPP模型无法完全提取钟差主周期项，具有一定的局限性。相比于QPP模型拟合残差，SSA重构残差变化较平稳，无显著的残余周期性信号，Block Ⅱ R和Block Ⅱ R-M卫星钟差序列的拟合差残基本在0.5 ns以内，Block Ⅱ F和Block Ⅲ A卫星钟差序列的拟合残差在0.1 ns以内，BDS-3卫星钟差序列的拟合残差在0.4 ns以内，钟差重构精度高于QPP模型，这是因为SSA属于非参数性的信号分析方法，本质上是对钟差进行非线性滤波，即使钟差序列中的周期项呈现时变特性，SSA方法也能够准确识别和分离时变性周期信号，从而提升钟差周期项提取效果，比QPP模型具有更好的适用性。此外，从图 6还可以发现，SSA方法对不同BDS-3卫星钟差序列的重构效果并无太大差异，但对不同GPS卫星钟差序列的重构效果存在一定差异，Block Ⅱ F和Block Ⅲ A卫星钟差序列的重构效果优于Block Ⅱ R和Block Ⅱ R-M卫星，这与卫星钟的噪声特性密切相关，Block Ⅱ F和Block Ⅲ A星载铷钟比Block Ⅱ R和Block Ⅱ R-M星载铷钟具有更为优良的噪声特性[19-20]，而BDS-3星载氢钟的噪声特性并无太大差异[21]。
为进一步验证SSA方法窗口长度选取的合理性，分析窗口长度M对钟差预报精度的影响，采用2024-04-14—16钟差数据，分别选取12 h、24 h和36 h窗口长度进行12 h和24 h预报实验，考虑到事后卫星钟差产品的精度约为0.01 ns[22]，预报过程中设置SSA迭代预报停止阈值为0.01 ns(下同)，并以最终钟差数据为真值，计算预报结果的RMSE，结果见图 7。可以看出，窗口长度对SSA方法的钟差预报精度有一定影响，但因钟差周期特性不一致和卫星类型存在差异，窗口长度对不同卫星钟差预报精度的影响不尽相同。以G23和G29卫星为例，窗口长度为12 h、24 h和36 h时，G23卫星钟差的24 h预报精度分别为1.88 ns、0.05 ns和0.89 ns，G29卫星钟差的24 h预报精度分别为1.53 ns、1.59 ns和1.96 ns。总体而言，窗口长度为12 h时预报精度较低，窗口长度为24 h时预报精度较高，再增大窗口长度预报精度无明显改善，甚至会降低预报精度。通过分析发现，窗口长度选取不合适，钟差序列中的周期分量无法被充分提取或存在过度分解，会对SSA迭代预报结果产生影响。本文以钟差主周期项的最小公倍数作为窗口长度时预报精度相对更高，因此基于“最小公倍数”原则选取的窗口长度是合理可行的，能够避免窗口长度选取的盲目性给钟差预报带来的不利影响。
3.2.2 多模型预报结果对比
分析SSA方法的短期预报效果，并将预报结果与其他模型进行对比。以2024-04-14—16钟差数据作为建模数据，运用QPP、GM、ARIMA模型和SSA方法对选取的14颗卫星进行12 h和24 h时长的钟差预报，各模型和SSA方法的钟差预报残差(预报值与真实值之差)见图 8和图 9。
从图 8和图 9可以看出，QPP、GM和ARIMA模型的预报残差总体上比较分散，而SSA方法的预报残差分布相对集中；随着预报时长的增加，各模型的预报残差变化较大，特别是GM和ARIMA模型的12 h和24 h预报残差表现出明显的发散趋势，而QPP模型和SSA方法的预报残差发散较小，表现出一定的稳定性，这是由于GM模型要求钟差呈指数变化规律，ARIMA模型要求钟差变化是平稳的，但钟差变化的复杂性使其难以满足这2种模型的要求。QPP模型和SSA方法不仅能考虑钟差的趋势性变化，还可顾及钟差的周期性变化，因此钟差预报残差相对更小，且变化较为稳定。此外，从图中还可以发现，SSA方法对Block Ⅱ F和Block Ⅲ A卫星钟差的预报残差比Block Ⅱ R和Block Ⅱ R-M卫星钟差的预报残差更小且变化更平稳，这是因为前者卫星钟具有更为优良的时频特性。为进一步反映各模型和SSA方法的钟差预报效果，图 10和图 11分别为14颗卫星钟差12 h和24 h预报时长的RMSE和极差，其中极差为钟差预报残差的最大值和最小值之差，用以衡量钟差预报结果的稳定性。表 3为各模型和SSA方法对GPS和BDS-3卫星钟差预报结果的平均RMSE和极差，分析结果如下。
1) 从图 10可以看出，除G04卫星外，无论对于12 h还是24 h预报时长，SSA方法对其他卫星钟差的预报RMSE均小于1 ns，预报精度未随预报时长的增加而降低，表明SSA方法的预报结果比较稳定。SSA方法对Block Ⅱ F和Block Ⅲ A卫星钟差的预报RMSE比Block Ⅱ R和Block Ⅱ R-M卫星钟差更小，表明SSA方法对新型卫星钟钟差的预报效果更好，这与新型卫星钟优良的时频特性密切相关。对比SSA方法，QPP、GM和ARIMA模型的预报RMSE较大，且预报精度随预报时长的增加均有不同程度的降低。此外，3种模型对个别卫星钟差的预报精度较低，如QPP模型对C60卫星钟差12 h和24 h的预报RMSE分别接近4 ns和7 ns，ARIMA模型对G25卫星钟差12 h和24 h的预报RMSE分别达到2 ns和5 ns，GM模型对G16卫星钟差12 h和24 h的预报RMSE分别超过2.5 ns和4 ns。这是因为QPP模型的趋势项参数的预报初始时刻偏差可能导致预报值偏离真实值[23]，ARIMA模型要求钟差序列满足平稳变化，GM模型要求钟差序列呈指数规律变化，但钟差长时序难以满足指数规律变化或平稳变化，表明QPP、GM和ARIMA模型对钟差预报的适用性不强。SSA方法少数情况下预报精度略低于其他模型，如G25卫星钟差的24 h预报精度，这可能是由于受卫星钟差系统性误差的影响，SSA方法在迭代预报过程存在一定的收敛性误差。
2) 从图 11可以看出，对于12 h预报时长，SSA方法对G17和G29卫星钟差的预报极差分别约为1.5 ns和2 ns，对其他卫星钟差的预报极差小于1 ns，QPP、GM和ARIMA模型对个别卫星钟差的预报极差达到4 ns，大部分卫星钟差的预报极差超过1 ns。对于24 h预报时长，SSA方法对G29和C60卫星钟差的预报极差分别约为2.5 ns和2 ns，对其他卫星钟差的预报极差小于1.5 ns，QPP、GM和ARIMA模型对个别卫星钟差的预报极差达到8 ns，大部分卫星钟差的预报极差超过2 ns。主要原因在于，钟差高频噪声项会干扰QPP、GM和ARIMA模型拟合，导致预报波动较大，而SSA方法仅对钟差趋势项和周期项进行建模预报，预报结果会更加稳定。
3) 由表 3(单位ns)可知，在各模型中，QPP模型对GPS卫星钟差的预报精度仅次于SSA方法，ARIMA模型精度最低，对于BDS-3卫星钟差，ARIMA模型的预报精度仅次于SSA方法，QPP和GM模型预报精度较低，这可能与BDS-3卫星钟差“日界不连续”有关[24]，ARIMA模型根据钟差差分数据进行建模，能够在一定程度上消除钟差不连续的不利影响。就平均预报RMSE而言，SSA方法对GPS和BDS-3卫星钟差12 h和24 h的预报精度均达到亚ns级，当预报时长为12 h时，与QPP、GM和ARIMA模型相比，SSA方法对GPS卫星钟差的平均预报精度分别提高38.81%、59%和72.11%，对BDS-3卫星钟差的平均预报精度分别提高71.54%、74.82%和70.09%；当预报时长为24 h时，SSA方法对GPS卫星钟差的平均预报精度分别提高51%、70.66%和83.99%，对BDS-3卫星钟差的平均预报精度分别提高86.36%、85.25%和81.05%，SSA方法预报精度提高显著。
4 结语
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针对卫星钟差时间序列的非线性和非平稳特性以及趋势信号、周期信号和随机信号相互干扰可能会影响预报精度的问题，提出一种以SSA为基础的“分解-重构-预报”钟差预报方法。该方法能充分发挥SSA对于趋势信号和周期信号良好的提取效果，同时根据趋势信号、周期信号和随机信号的Hurst指数存在明显差异的特性，对钟差趋势项和周期项进行重构。结果表明，SSA能够有效地提取钟差时间序列中的趋势项和周期项，比多项式拟合模型的提取效果更好。利用本文方法对GPS和BDS-3卫星钟差进行12 h和24 h预报实验，结果表明，本文方法的预报精度相较于常用的QPP、GM和ARIMA模型改进显著，预报精度达到亚ns级，12 h和24 h平均预报精度提高38%和51%以上，最大达74.82%和86.36%。随着预报时长的增加，SSA方法仍能保持较高的预报精度，表明SSA方法具有更好的预报稳定性。本文结论可为卫星钟差预报提供一种新方法，具有一定的应用价值。
基金
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  • 陕西省自然科学基础研究计划(2025JC-YBMS-049)
文献
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2026年第46卷第4期
48
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doi: 10.14075/j.jgg.2025.05.175
  • 接收时间：2025-05-22
  • 首发时间：2026-10-29
  • 出版时间：2026-04-15
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  • 收稿日期：2025-05-22
基金
陕西省自然科学基础研究计划(2025JC-YBMS-049)
作者信息
    1 西安邮电大学计算机学院, 西安, 710121
    2 宝鸡文理学院电子电气工程学院, 宝鸡, 721016

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雷雨, 博士, 副教授, 主要研究方向为GNSS时间传递, E-mail:
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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