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Article(id=1322570989779669581, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322570898712937423, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.07.249, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1752508800000, receivedDateStr=2025-07-15, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793255096143, onlineDateStr=2026-10-29, pubDate=1778774400000, pubDateStr=2026-05-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793255096143, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793255096143, creator=13701087609, updateTime=1793255096143, updator=13701087609, issue=Issue{id=1322570898712937423, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='5', pageStart='522', pageEnd='661', issueExtLink='null', onlineDate='null', pubDate='1778774400000', pubDateStr='2026-05-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793255074432, creator='13701087609', updateTime=1793255074432, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=587, endPage=598, ext={EN=ArticleExt(id=1322570989985190478, articleId=1322570989779669581, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=A Satellite Clock Bias Estimation Method Considering the Noise Component of GNSS Spaceborne Atomic Clocks, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Given the issue that multi-system satellite clock bias estimation is susceptible to interference from various noise types, an estimation model that fully considers the characteristics of satellite clock noise components is proposed. This model employs two strategies, namely sliding window and overall modeling, and utilizes the Hadamard variance to solve for noise parameters. Based on this, clock bias estimation and accuracy assessment are conducted. In the experimental section, multiple representative satellites from the four major satellite navigation systems—GPS, BDS-3, Galileo, and GLONASS are selected as research subjects. A systematic comparative analysis is performed on the performance differences of the two modeling strategies across different systems and satellites. The experimental results reveal that the BDS-3 exhibits significant inter-satellite variability. Specifically, most satellites in MEO demonstrate more pronounced time-varying noise characteristics, making them more suitable for the sliding modeling strategy. Conversely, for some satellites equipped with MEO rubidium clocks and stable IGSO satellites, the overall modeling strategy offers greater advantages. In the case of the Galileo, the global modeling strategy performs better. For the GPS and GLONASS, after adopting the sliding window strategy, the positioning accuracy in E, N, U directions improves by 10.34%, 17.07%, and 4.30%, respectively, and by 6.27%, 10.34%, and 3.76%, respectively, compared to the overall modeling strategy.

, authors=Linfeng ZHU1, Zhongyuan WANG1, *, Chao HU2, Jiaxing YANG1, authorsList=Linfeng ZHU, Zhongyuan WANG, Chao HU, Jiaxing YANG, authorCompany=null, correspAuthors=Zhongyuan WANG, 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=1322570993705538145, articleId=1322570989779669581, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=顾及GNSS星载原子钟噪声分量的卫星钟差估计方法, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

鉴于多系统卫星钟差估计易受多种噪声类型干扰的问题, 提出一种充分考虑卫星钟噪声分量特性的估计模型。该模型采用滑动窗口与整体建模2种策略, 借助哈达玛方差来解算噪声参数, 并在此基础上开展钟差估计与精度评定工作。在实验环节, 选取GPS、BDS-3、Galileo、GLONASS四大卫星导航系统中的多个具有代表性的卫星作为研究对象, 对2类建模策略在不同系统以及不同卫星上的表现差异进行系统性对比分析。实验结果显示, BDS-3呈现出显著的星间差异特性。其中, 大部分处于MEO轨道的卫星, 其噪声时变特性更为突出, 更适合采用滑动窗口建模策略; 而对于部分采用MEO铷钟的卫星以及性能稳定的IGSO卫星, 整体建模策略则更具优势。在Galileo方面, 整体建模策略表现更优。对于GPS和GLONASS, 采用滑动窗口建模策略后, 在E、N、U三个方向上的定位精度相较于整体建模策略分别提升10.34%、17.07%、4.30%和6.27%、10.34%、3.76%。

, authors=朱麟丰1, 王中元1, *, 胡超2, 杨嘉兴1, authorsList=朱麟丰, 王中元, 胡超, 杨嘉兴, authorCompany=null, correspAuthors=王中元, authorNote=

朱麟丰, 硕士生, 主要研究方向为星载原子钟性能评估与实时钟差估计, E-mail:

, correspAuthorsNote=
王中元, 博士, 副教授, 主要研究方向为GNSS空间大地测量技术, E-mail:
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朱麟丰, 硕士生, 主要研究方向为星载原子钟性能评估与实时钟差估计, E-mail:

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Journal of Geodesy, 2023, 97 (2), articleTitle=BDS-3 Precise Orbit and Clock Solution at Wuhan University: Status and Improvement, refAbstract=null)], funds=[Fund(id=1322571002589074080, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570989779669581, awardId=42404015, language=CN, fundingSource=国家自然科学基金(42404015), fundOrder=null, country=null), Fund(id=1322571002668765857, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570989779669581, awardId=42274048, language=CN, fundingSource=国家自然科学基金(42274048), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1322570994833805922, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570989779669581, xref=null, ext=[AuthorCompanyExt(id=1322570994842194531, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570989779669581, companyId=1322570994833805922, language=EN, country=null, province=null, city=null, postcode=null, 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C为BDS-3; E为Galileo; G为GPS; R为GLONASS

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C为BDS-3; E为Galileo; G为GPS; R为GLONASS

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The influence of different window lengths on the sliding window modeling results

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系统窗口长度/minQ1/10-24Q2/10-34Q3/10-44
BDS-3151.7192.0762.091
301.9372.3612.619
GPS152.07125.77526.242
302.57427.36930.868
Galileo151.5981.9241.928
301.8732.0172.197
GLONASS1554.70345.31143.949
3068.53456.86550.169
), ArticleFig(id=1322571000374481561, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570989779669581, language=CN, label=表1, caption=

不同窗口长度对滑动窗口建模结果的影响

, figureFileSmall=null, figureFileBig=null, tableContent=
系统窗口长度/minQ1/10-24Q2/10-34Q3/10-44
BDS-3151.7192.0762.091
301.9372.3612.619
GPS152.07125.77526.242
302.57427.36930.868
Galileo151.5981.9241.928
301.8732.0172.197
GLONASS1554.70345.31143.949
3068.53456.86550.169
), ArticleFig(id=1322571000437396122, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570989779669581, language=EN, label=Tab. 2, caption=

Specific parameter values of different GNSS

, figureFileSmall=null, figureFileBig=null, tableContent=
系统轨道类型星钟类型PRNQ1/10-24Q2/10-34Q3/10-44
整体建模滑动窗口建模整体建模滑动窗口建模整体建模滑动窗口建模
BDS-3MEORBC368.0921.1418.5911.2127.4511.141
PHMC268.0871.1618.5561.2407.5191.172
IGSOPHC388.1251.0908.603 151.1577.7521.083
GPSMEOⅡFG1075.7968.654104.48277.45169.40375.370
ⅢAG1172.8458.640103.31677.32266.54975.154
ⅡRG2873.7018.286106.52773.92168.22071.683
ⅡR-MG2977.68753.744142.404103.39185.77481.060
GalileoMEOPHME257.5601.0858.6661.1541.65091.080
GLONASSMEOCSR0884.58238.593209.701118.0781.285105.701
), ArticleFig(id=1322571000521282203, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570989779669581, language=CN, label=表2, caption=

不同GNSS具体参数数值

, figureFileSmall=null, figureFileBig=null, tableContent=
系统轨道类型星钟类型PRNQ1/10-24Q2/10-34Q3/10-44
整体建模滑动窗口建模整体建模滑动窗口建模整体建模滑动窗口建模
BDS-3MEORBC368.0921.1418.5911.2127.4511.141
PHMC268.0871.1618.5561.2407.5191.172
IGSOPHC388.1251.0908.603 151.1577.7521.083
GPSMEOⅡFG1075.7968.654104.48277.45169.40375.370
ⅢAG1172.8458.640103.31677.32266.54975.154
ⅡRG2873.7018.286106.52773.92168.22071.683
ⅡR-MG2977.68753.744142.404103.39185.77481.060
GalileoMEOPHME257.5601.0858.6661.1541.65091.080
GLONASSMEOCSR0884.58238.593209.701118.0781.285105.701
), ArticleFig(id=1322571002186420892, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570989779669581, language=EN, label=Tab. 3, caption=

Secondary difference values of different satellites for different GNSS

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系统轨道类型星钟类型二次差差值均值/10-12s
BDS-3MEORB-0.619
PHM-1.113
IGSOPH0.309
GPSMEOⅡF-4.383
ⅢA-3.837
ⅡR-2.865
ⅡR-M-3.290
GalileoMEOPHM27.470
GLONASSMEOCS-9 933.11
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不同GNSS不同卫星二次差差值

, figureFileSmall=null, figureFileBig=null, tableContent=
系统轨道类型星钟类型二次差差值均值/10-12s
BDS-3MEORB-0.619
PHM-1.113
IGSOPH0.309
GPSMEOⅡF-4.383
ⅢA-3.837
ⅡR-2.865
ⅡR-M-3.290
GalileoMEOPHM27.470
GLONASSMEOCS-9 933.11
), ArticleFig(id=1322571002341610142, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322570989779669581, language=EN, label=Tab. 4, caption=

Statistics of PPP positioning accuracy and convergence time

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卫星系统钟差产品RMS/cm收敛时间/min
E方向N方向U方向E方向N方向U方向
BDS-3cal(all)4.873.136.7323.413.323.46
cal(hd)4.763.216.3624.2814.5725.55
pre2.482.743.8920.0312.5823.37
Galileocal(all)3.392.455.1810.098.0417.03
cal(hd)3.312.384.5911.599.6319.94
pre1.541.852.648.957.4016.92
GPScal(all)3.872.465.3518.849.9122.64
cal(hd)3.472.045.1217.219.2521.09
pre1.491.902.7815.559.1220.39
GLONASScal(all)7.825.619.5838.4833.5347.68
cal(hd)7.335.039.2237.8232.8945.72
pre6.644.377.7734.9628.1441.95
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PPP定位精度与收敛时间统计

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卫星系统钟差产品RMS/cm收敛时间/min
E方向N方向U方向E方向N方向U方向
BDS-3cal(all)4.873.136.7323.413.323.46
cal(hd)4.763.216.3624.2814.5725.55
pre2.482.743.8920.0312.5823.37
Galileocal(all)3.392.455.1810.098.0417.03
cal(hd)3.312.384.5911.599.6319.94
pre1.541.852.648.957.4016.92
GPScal(all)3.872.465.3518.849.9122.64
cal(hd)3.472.045.1217.219.2521.09
pre1.491.902.7815.559.1220.39
GLONASScal(all)7.825.619.5838.4833.5347.68
cal(hd)7.335.039.2237.8232.8945.72
pre6.644.377.7734.9628.1441.95
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顾及GNSS星载原子钟噪声分量的卫星钟差估计方法
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朱麟丰 1 , 王中元 1, * , 胡超 2 , 杨嘉兴 1
大地测量与地球动力学 | 大地测量学 2026,46(5): 587-598
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大地测量与地球动力学 |大地测量学 2026 , 46 (5) : 587 -598
顾及GNSS星载原子钟噪声分量的卫星钟差估计方法
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朱麟丰1 , 王中元1, * , 胡超2, 杨嘉兴1
作者信息
  • 1 中国矿业大学环境与测绘学院, 徐州, 221116
  • 2 安徽理工大学空间信息与测绘工程学院, 淮南, 232001
通讯作者:
王中元, 博士, 副教授, 主要研究方向为GNSS空间大地测量技术, E-mail:
作者简介:

朱麟丰, 硕士生, 主要研究方向为星载原子钟性能评估与实时钟差估计, E-mail:

A Satellite Clock Bias Estimation Method Considering the Noise Component of GNSS Spaceborne Atomic Clocks
Linfeng ZHU1 , Zhongyuan WANG1, * , Chao HU2, Jiaxing YANG1
Affiliations
  • 1 School of Environment and Surveying, China University of Mining and Technology, Xuzhou 221116, China
  • 2 School of Spatial Information and Surveying Engineering, Anhui University of Science and Technology, Huainan 232001, China
出版时间: 2026-05-15 doi: 10.14075/j.jgg.2025.07.249
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摘要
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鉴于多系统卫星钟差估计易受多种噪声类型干扰的问题, 提出一种充分考虑卫星钟噪声分量特性的估计模型。该模型采用滑动窗口与整体建模2种策略, 借助哈达玛方差来解算噪声参数, 并在此基础上开展钟差估计与精度评定工作。在实验环节, 选取GPS、BDS-3、Galileo、GLONASS四大卫星导航系统中的多个具有代表性的卫星作为研究对象, 对2类建模策略在不同系统以及不同卫星上的表现差异进行系统性对比分析。实验结果显示, BDS-3呈现出显著的星间差异特性。其中, 大部分处于MEO轨道的卫星, 其噪声时变特性更为突出, 更适合采用滑动窗口建模策略; 而对于部分采用MEO铷钟的卫星以及性能稳定的IGSO卫星, 整体建模策略则更具优势。在Galileo方面, 整体建模策略表现更优。对于GPS和GLONASS, 采用滑动窗口建模策略后, 在E、N、U三个方向上的定位精度相较于整体建模策略分别提升10.34%、17.07%、4.30%和6.27%、10.34%、3.76%。

GNSS  /  噪声分量  /  哈达玛方差  /  钟差估计  /  精度评定

Given the issue that multi-system satellite clock bias estimation is susceptible to interference from various noise types, an estimation model that fully considers the characteristics of satellite clock noise components is proposed. This model employs two strategies, namely sliding window and overall modeling, and utilizes the Hadamard variance to solve for noise parameters. Based on this, clock bias estimation and accuracy assessment are conducted. In the experimental section, multiple representative satellites from the four major satellite navigation systems—GPS, BDS-3, Galileo, and GLONASS are selected as research subjects. A systematic comparative analysis is performed on the performance differences of the two modeling strategies across different systems and satellites. The experimental results reveal that the BDS-3 exhibits significant inter-satellite variability. Specifically, most satellites in MEO demonstrate more pronounced time-varying noise characteristics, making them more suitable for the sliding modeling strategy. Conversely, for some satellites equipped with MEO rubidium clocks and stable IGSO satellites, the overall modeling strategy offers greater advantages. In the case of the Galileo, the global modeling strategy performs better. For the GPS and GLONASS, after adopting the sliding window strategy, the positioning accuracy in E, N, U directions improves by 10.34%, 17.07%, and 4.30%, respectively, and by 6.27%, 10.34%, and 3.76%, respectively, compared to the overall modeling strategy.

GNSS  /  noise component  /  Hadamard variance  /  clock bias estimation  /  accuracy assessment
朱麟丰, 王中元, 胡超, 杨嘉兴. 顾及GNSS星载原子钟噪声分量的卫星钟差估计方法. 大地测量与地球动力学, 2026 , 46 (5) : 587 -598 . DOI: 10.14075/j.jgg.2025.07.249
Linfeng ZHU, Zhongyuan WANG, Chao HU, Jiaxing YANG. A Satellite Clock Bias Estimation Method Considering the Noise Component of GNSS Spaceborne Atomic Clocks[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (5) : 587 -598 . DOI: 10.14075/j.jgg.2025.07.249
正文
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在全球导航卫星系统(GNSS)中，高精度的轨道与钟差产品是保障导航定位及授时服务精准性与可靠性的基石。卫星钟差建模作为GNSS高精度定位的核心要素，其研究进展对于国家空间信息基础设施的建设与完善具有举足轻重的作用[1-3]。目前，精密轨道产品通常由各大分析中心通过事后处理的方式生成，这使得产品存在一定的时间滞后性。尽管当前导航卫星所配备的原子钟性能已达到较高水准，但钟差的细微变化仍可能对最终的定位结果产生显著影响。因此，对钟差进行精密建模与有效改正，在提升定位精度方面仍具有不可忽视的重要意义[4-5]。
在现有的钟差估计研究中，研究人员往往聚焦于单一GNSS系统的特性，针对性地进行模型优化。例如，王浩浩等[6]基于非差法，实现了对BDS实时钟差的精准估计；谭粤等[7]、武美芳等[8]分别提出创新方法，前者利用轨道标准差优化随机模型并剔除异常卫星，以此提升实时钟差估计的准确性，后者则基于小时观测文件拼接技术，设计出一种实时钟差算法。这些方法在各自针对的特定系统中均取得了良好的应用效果，然而，其普适性存在一定的局限性，难以广泛适用于其他系统。为构建能够兼顾不同GNSS系统的通用钟差模型，后续研究逐渐聚焦于卫星钟的物理特性。研究发现，深入挖掘卫星钟的物理特性，如钟速、频漂等，并构建精细化的模型，是提升钟差估计精度的有效途径。例如，胡超等[9]设计了一种充分考虑BDS-3/GNSS星钟状态特征的超快速估算方案；周圣淇等[10]从钟差的短时相关性角度出发，构建了钟速与频漂的高效实时估计模型。这些研究成果充分表明，依据卫星钟的物理特性构建适宜的钟差估计模型，是优化钟差估计效果的关键手段。陈雄川等[11]基于平方根信息滤波(SRIF)框架构建实时估计模型，实现了GPS钟差精度达到0.27 ns，BDS的GEO、IGSO、MEO卫星钟差精度分别达到0.39 ns、0.47 ns、1.37 ns，定位精度达到cm级，能够满足实时精密单点定位(PPP)的需求。然而，这种性能差异也揭示出一个问题：将统一的钟差模型应用于不同的GNSS系统时，可能无法充分适配各系统自身的随机特性，进而导致性能表现不均衡，对最终的定位结果产生不利影响。
在实际应用中，针对不同系统、不同原子钟类型进行卫星钟差估计时，往往会受到不同噪声类型的干扰，统一的钟差模型难以满足多样化的需求。同时，原子钟本身的物理特性以及噪声随机分量也是影响钟差估计精度的重要因素。综合考虑这些因素，本文选取整体建模和滑动窗口建模2种策略对钟差数据进行处理。在卡尔曼滤波框架下，通过优化过程噪声，使模型能够更好地适应不同卫星的噪声特性。其中，双频无电离层组合模型作为观测模型，承担着建立原始观测值与待估钟差等参数之间联系的重要任务。通过哈达玛方差分析确定钟差协方差参数，整体建模法基于全时段数据统一求解协方差参数；滑动窗口建模法则通过在时间轴上滑动取样的方式，动态地估计卫星钟差的噪声分量参数。这2种方法均以双频无电离层组合观测模型为基础，并结合钟速与频漂构建实时钟差估计模型。最终，通过钟差二次差精度评估和PPP定位实验，系统比较2种参数估计方法对钟差解算及定位精度的影响差异，为GNSS卫星钟差估计提供更科学、更精准的方法参考。
1 基本原理与方法
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1.1 方差计算
鉴于低频、甚低频噪声的影响以及铷钟存在显著的线性漂移特性，本文选用基于三次差分的重叠哈达玛方差(Hadamard variance)进行方差计算[12]。在时域内，设有一组原始相位数据{yi，i=1，2，3, …，N}，N为数据总个数，采样间隔为τ0。具体公式表示如下：
$\begin{gathered}H \sigma_y^2(\tau)=\frac{1}{6 \tau^2(N-3 m)} \sum\limits_{i=1}^{N-3 m}\left[x_{i+3 m}-\right. \\\left.3 x_{i+2 m}+3 x_{i+m}-x_i\right]^2\end{gathered}$
式中，τ=mτ0为平滑时间；m为平滑因子，一般取1≤m≤int((N－1)/3);Hσy2(τ)为哈达玛方差。
1.2 钟差估计方法
双频无电离层组合观测模型如下：
$\left\{\begin{aligned}P_{\mathrm{r}}^{\mathrm{s}}= & \rho_{\mathrm{r}}^{\mathrm{s}}-c \cdot \Delta t^{\mathrm{s}}+c \cdot \Delta t_{\mathrm{r}}+d^{\mathrm{s}}- \\& d_{\mathrm{r}}+Z_{\mathrm{r}}+\varepsilon_{\mathrm{r}}^{\mathrm{s}} \\L_{\mathrm{r}}^{\mathrm{s}}= & \rho_{\mathrm{r}}^{\mathrm{s}}-c \cdot \Delta t^{\mathrm{s}}+c \cdot \Delta t_{\mathrm{r}}+b^{\mathrm{s}}- \\& b_{\mathrm{r}}+Z_{\mathrm{r}}+\lambda_{\mathrm{IF}} N_{\mathrm{IF}}^{\mathrm{s}}+\varepsilon_{\mathrm{r}}^{\mathrm{s}}\end{aligned}\right.$
式中，s、r分别为卫星与接收机；Prs、Lrs分别为伪距和相位观测值；ρrs为接收机与卫星的几何距离；c为光速；Δtr、Δts为接收机钟差、卫星钟差参数；dr、ds和br、bs分别为接收机及卫星方面的伪距、相位硬件延迟；Zr为对流层延迟参数；λIF为组合波长；NsIF为整周模糊度。
依据原子钟类型的不同，对其进行建模处理的方法亦不相同。本文采用多项式模型来描述钟差状态随时间的演变。模型的阶数依据原子钟的物理特性选取：对于频率漂移不显著的铯钟，可采用包含钟差和钟速的一阶模型；对于具有显著线性频率漂移的铷钟，采用包含钟差、钟速与频漂的二阶模型[13]。基于上述原则，原子钟的滤波状态方程可表示为：
$\left[\begin{array}{c}x(t+\tau) \\y(t+\tau) \\z(t+\tau)\end{array}\right]=\left[\begin{array}{ccc}1 & \tau & \tau^2 / 2 \\0 & 1 & \tau \\0 & 0 & 1\end{array}\right] \cdot\left[\begin{array}{c}x(t) \\y(t) \\z(t)\end{array}\right]+\left[\begin{array}{c}\varepsilon_x \\\varepsilon_y \\\varepsilon_z\end{array}\right]$
式中，x(t)、y(t)、z(t)分别为t时刻卫星钟的钟差、钟速及钟漂；τ为采样时间；εx、εy、εz分别为随机变化分量。
当前，一般采用Kalman滤波或最小二乘法进行钟差的实时估计。本文采用Kalman滤波进行估计，参数估计时，保留前一历元时刻的状态向量及其相应的方差-协方差矩阵，同时综合考虑先验预测信息与此刻的观测值[14-15]。Kalman滤波一般包括观测方程、状态方程2个部分，前者为观测值与状态参数之间的函数关系，后者为状态参数随时间变化的函数关系。具体观测方程为：
$\boldsymbol{L}_k=\boldsymbol{H}_k \cdot \boldsymbol{X}_k+\boldsymbol{v}_k, \boldsymbol{v}_k \sim N(0, \boldsymbol{R})$
式中，k为第k个历元；Hk为观测方程的系数矩阵；Xk为待估参数矩阵；vk为观测噪声；R为观测方程中噪声方差矩阵。钟差实时估计的状态方程如下：
$\boldsymbol{X}_k=\boldsymbol{\varphi}_{k, k-1} \boldsymbol{X}_{k-1}+\boldsymbol{\omega}_{k-1}, \boldsymbol{\omega}_{k-1} \sim N(0, \boldsymbol{Q})$
式中，φk, k－1为状态转移方程的系数矩阵；ωk－1为状态噪声；Q为状态噪声的方差矩阵。当状态转移矩阵中速度、加速度对应的系数均设为0时，将其与其他相关参数进行结合，进而可以推导出矩阵 φk, k－1的表达式，以及与之对应的方差-协方差矩阵$\sum w_k$[16]。
同时考虑观测噪声中调相白噪声的影响，联合上述3种噪声，可推导出如下方差表达式：
$\begin{gathered}H \sigma_y^2(\tau)=(10 / 3) Q_0 \tau^{-2}+Q_1 \tau^{-1}+ \\ (1 / 6) Q_2 \tau+(11 / 120) Q_3 \tau^3\end{gathered}$
式中，Q0为调相白噪声参数；Q1为钟差过程噪声参数，表现为调相随机游走噪声；Q2为钟速过程噪声参数，表现为调频随机游走噪声；Q3为钟漂过程噪声参数，表现为调频随机奔跑噪声；Hσy2(τ)为哈达玛方差。基于该式，选取不同采样间隔计算对应的哈达玛方差，并利用最小二乘拟合估计噪声参数Q0、Q1、Q2、Q3的值，从而实现原子钟噪声特性的具体数值表述。
Kalman滤波的具体步骤为：基于状态方程，利用前一历元时刻的参数估值Xk-1+及协方差矩阵Pk-1+，计算当前时刻的状态参数的预测值Xk-及协方差矩阵Pk-；再利用当前时刻的观测值，可得当前时刻的最佳估计Xk+、协方差矩阵Pk+；然后利用Kalman滤波进行迭代计算，获取瞬时的各类钟差数据；最后利用式(3)的滤波模型获取任意时刻的钟差、钟速，再进行下一步的各类计算、评估[17]。
1.3 精度评估
针对各GNSS系统，分别选取1颗卫星作为参考，先对该卫星的估计钟差与精密钟差进行一次差分，以消除时间基准引起的偏移，再将其余卫星的估计钟差同此参考精密钟差作一次差分。通过计算所得二次差分时间序列的标准差(STD)评估钟差偏离的离散程度[18]：
$\mathrm{CLK}_{\mathrm{STD}}=\sqrt{\frac{\sum\limits_{i=1}^n\left(\varDelta_t-\bar{\varDelta}\right)}{n-1}}$
2 实验与分析
收起
为比较不同系统中各类卫星钟差建模方法的性能差异，本文基于IGS提供的2025-03(共31 d)的近实时钟差数据(300 s间隔)，采用整体建模与滑动窗口建模2种方式求解噪声参数，重建钟差序列。通过与事后精密钟差二次作差，系统地评估2种建模策略的精度差异，进而剖析不同GNSS系统及卫星个体在噪声特性上的差异性。具体实验流程如图 1所示。
本文在传统钟差估计框架的基础上，针对整体建模与滑动窗口建模2种策略进行系统性对比研究，以分析不同建模策略在基于重叠哈达玛方差进行卫星钟差估计与精度评定中的性能差异及其对后续精密单点定位(PPP)结果的影响。其中，整体建模策略采用全时段联合估计方式，以整体时间序列为对象建立统一的钟差模型，具有较强的平滑性与全局一致性；滑动窗口建模策略则通过设定固定长度的时间窗口并进行滚动更新，实现实时性与局部敏感性的平衡。
2种策略分别输出钟差估计产品，并经由重叠哈达玛方差方法对钟差稳定性进行评定，最终通过PPP对比分析验证其在定位精度与稳定性方面的差异。
2.1 整体参数建模
选取31 d所有的钟差数据进行整体解算，以获取每颗卫星对应的稳定噪声模型参数Q。具体方法为：1)对每颗卫星的完整日间钟差数据进行预处理；2)采用哈达玛方差法求解不同τ尺度下对应的方差值；3)通过最小二乘拟合方法估计出最优的参数；4)将得到的最优参数输入至钟差估计模型中，进行整日钟差的估计；5)将估计钟差结果与IGS发布的精密钟差进行比较。
由图 2可见，BDS-3采用C19卫星作为参考卫星，与原始钟差二次作差后，总体精度在(0~4)×10-10 s之间；GPS采用G05作为参考卫星，总体精度在(0~1.6)×10-10 s之间；Galileo采用E19作为参考卫星，大部分卫星二次差精度在(0~1.8)×10-10 s之间；GLONASS采用R01作为参考卫星，大部分卫星二次差精度在(0~1.7)×10-7 s之间，而R02、R03、R04和R05四颗卫星的精度在(0~2)×10-10 s之间。经分析，GLONASS存在显著性差异与实验期间卫星观测数据的连续性强相关。R02、R03、R04和R05四颗卫星钟差估计序列平滑且稳定，相比之下，其他大部分卫星存在数据中断，导致钟差估计模型状态重置，从而在钟差序列中引入微小的不连续点或跳变，致使数据偏差较大。
2.2 滑动窗口建模
窗口长度是滑动窗口建模中的关键参数，由式(1)可知，过短的窗口(如5 min以下)因数据量不足，难以获得稳健的哈达玛方差估计；过长的窗口(如30 min以上)则会平滑掉钟差的短期特性，降低模型的敏感性，趋近于整体建模。综合考虑，本文选取15 min作为基准窗口长度，并与30 min窗口的结果进行对比分析。
为此，分别在15 min和30 min窗口下进行滑动窗口建模，对比2种窗口长度下估算出的过程噪声参数(表 1)。
由表 1可知，BDS-3与Galileo的高稳定性卫星，其随机特性本身就非常平稳，因此15 min和30 min窗口的结果接近，符合物理预期。对于GPS，30 min估算出的Q1/Q2/Q3值均高于15 min。这揭示了在更长的时间尺度上，GPS星钟的随机游走等长期噪声分量开始显现并累积。GLONASS星钟的不稳定性最为显著，采用较长时间窗口，会强制将特性迥异的短期数据段进行平均，导致模型钝化，无法准确反映其真实的、快速变化的噪声水平，从而估算出偏大且失真的噪声参数。因此，本文选取15 min的时间窗口长度。
为进一步探究部分卫星噪声特性存在时间变化的问题，利用滑动窗口方法进行局部建模。其步骤为：首先将整日钟差数据划分为多个重叠或不重叠的时间窗口(本实验中采用15 min)；再对每个窗口内的钟差数据分别计算哈达玛方差，并独立估计局部Q参数；然后针对每个不同窗口内计算出来的Q值取均值，利用此均值进行钟差估计；最后将估计钟差结果与IGS发布的精密钟差进行比较。
图 3展示了滑动窗口建模的钟差序列与事后精密钟差二次差的对比结果。分析显示，除Galileo外，其余系统在该方法下解算的精度与整体建模基本无异，多数卫星的二次差精度与整体建模相近。总体来看，2种建模策略解算的精度基本相当，且变化趋势高度一致。这是因为2种策略的核心区别在于对状态方程中过程噪声协方差(Q矩阵)的动态或静态设定。然而，对于高稳定性的星钟，其随机特性的过程噪声参数本身就处于较小的数值量级，因此，通过滑动窗口法得到钟差序列，在整体数值上与整体建模方法并无显著差异，需借助PPP定位有效检验其中细微但关键的模型差异。
2.3 不同建模策略参数值对比
从上述二次差结果对比(图 2、3)中较难看出2种方法的具体性能差异。为直接对比整体建模与滑动窗口建模各个噪声参数的具体数值差异，采用重叠哈达玛方差对原始钟差数据进行分析，计算出不同建模策略下的各个噪声参数的具体量化表征。
参考卫星的选择旨在全面揭示不同策略下的主要变化特征，为此，依据轨道类型、星钟类型、卫星批次等指标各选取1个在实验期间健康状态稳定且性能、数据变化幅度具有代表性的卫星。具体而言，针对BDS-3，C26、C36和C38分别代表了MEO氢钟、MEO铷钟及IGSO氢钟；同理，G10、G11、G28及G29分别代表GPS中Block Ⅱ F、Ⅲ A、Ⅱ R及Ⅱ R-M等不同批次的卫星；E25代表Galileo；R08代表GLONASS。图 4、5为不同GNSS的整体建模参数值与滑动窗口建模参数值的对比，从左往右依次为噪声分量Q1、Q2、Q3的结果。
由图 4可见，BDS-3中采用滑动窗口建模时，C26(MEO氢钟)、C38(IGSO氢钟)与C36(MEO铷钟)3颗卫星的噪声参数波动幅度相似，表明该方法对不同类型噪声具有较好的稳健性。相比之下，整体建模结果显示, MEO轨道的C26与IGSO轨道的C38两颗卫星性能相当，且参数值表现均普遍优于C36，反映出氢钟在中长期时间尺度上具备更优越的稳定性，符合其固有的高精度特性，适合中长期频率稳定性要求较高的应用，而铷钟则更适合强调短期稳定性的场景。
在Galileo中，滑动窗口建模得到的噪声参数波动幅度与BDS-3相当，而整体建模参数值较BDS-3的氢钟更优，体现其PHM氢钟具有更强的稳定性。
GLONASS的参数值结果则差异明显。整体建模参数值普遍比BDS-3及Galileo高出一个数量级，反映其频率噪声较大，稳定性较差。滑动窗口建模方法下，参数散点值跳跃明显，呈现出强烈的随机波动特征，表明其星载原子钟在短中期时间范围内均存在较大的不稳定性，这一现象与GLONASS的固有特性相符。已有研究指出，部分GLONASS卫星搭载的老一代铯钟性能相对落后，且系统在星间时差校正和轨道控制精度方面也与其他现代GNSS系统存在差距，这些因素均会导致其钟差波动性较大。
图 5展示了GPS在2种建模策略下的参数值，其结果优于GLONASS，但稍差于BDS-3、Galileo。同时，GPS卫星中不同原子钟类型间也存在显著差异。滑动窗口策略下，除ⅡR-M (G29)代卫星外，Ⅱ R、Ⅱ F、Ⅲ A代卫星的参数数值虽较BDS-3、Galileo高出一个数量级，但波动范围大致相近，反映其具备一定的短期稳定性。其中，Ⅲ A代卫星的表现最佳，其次为Ⅱ R与Ⅱ F代卫星; Ⅱ R-M代卫星波动幅度较大，类似GLONASS, 可能与Ⅱ R-M代卫星原子钟老化或运行状况相关。
从整体建模结果来看，GPS各代卫星在Q2和Q3参数上的数值与滑动窗口建模均值相近，原因是中长期频率噪声的影响较小。与之相比，Q1参数值存在较大差异，是因其对白噪声敏感，易受短期异常波动干扰。
总体来看，GPS频率稳定性良好，Ⅲ A代卫星尤为突出，适合高精度导航、授时；其余代卫星需结合运行状态进一步评估。
上述不同系统的具体参数数值如表 2所示。可以看出，2种策略下BDS-3和Galileo的噪声参数均处于最低水平，反映了其搭载的现代化高性能原子钟具有极高的内在稳定性。相比之下，GPS因其星座包含不同代次卫星而表现出更大的内部差异，GLONASS的噪声参数整体最高，与其星钟特性一致。因此，GPS与GLONASS更适合采用滑动窗口处理，揭示了捕捉短时钟差的局部波动特征，可提高建模的精度。
2.4 不同建模策略钟差估计性能对比
利用二次差差值分析法定量分析整体建模策略与滑动窗口建模策略在钟差估计中的相对优劣。具体步骤为：首先，以IGS精密钟差产品为参考，分别计算2种方法估计的钟差序列，并与IGS参考序列作差，得到各自的钟差残差序列；随后，将这2个残差序列再次作差，通过此差值来揭示不同建模策略对钟差估计结果的影响。具体差值结果如图 6所示。
由图 6可知，BDS-3中，存在不同轨道类型、原子钟类型的差异，对于MEO轨道，其所有氢钟与大部分铷钟的卫星适合滑动窗口建模，能更好地捕捉短时钟差的局部波动特征，提高建模的精度；而少部分铷钟频率噪声变化相对稳定，整体建模较优。对于IGSO轨道类型，其轨道相对固定，运行环境稳定，长期趋势可能更稳定、更具规律性，因此整体建模策略略优。因此，可依据原子钟的不同类型选取最优的估计策略，以提高精度。GPS中，滑动窗口处理得到的参数整体上更优，除Ⅱ R-M代卫星外，其余卫星均表现出较为稳定的波动，其中Ⅲ A代卫星效果最为显著。Galileo几乎均采用被动型氢钟(PHM)，噪声特性良好，适合整体建模; 对于E02、E03、E07、E08、E18、E25等卫星，采用滑动窗口建模结果更优，可能是因为卫星部署时间较早，存在稳定性下降或运行状态频繁切换的问题。GLONASS使用的卫星钟为铯钟，且频繁进行星务调整，整体波动范围较大，可选取滑动窗口建模进行钟差估计; 对于R05、R15、R16、R18、R20等卫星，采用整体建模结果更优，可能由于当时卫星运行状态稳定、维护频次较低，噪声变化较为稳定。2种方法作差后的具体数值如表 3所示。
由表 3可见，对于BDS-3的MEO卫星，滑动窗口建模展现出优势，特别是搭载被动型氢钟(PHM)的卫星，其优化效果比RB卫星更显著。IGSO的氢钟卫星计算结果为正值，这是因为其轨道相对固定，运行环境稳定，长期趋势可能更稳定、更具规律性，因此整体建模策略略优。这揭示了在BDS-3中，建模策略的优越性可能与轨道类型和星钟类型均相关。GPS卫星的二次差差值均为显著的负值, 表明对于搭载不同代次铷钟(RB)的GPS卫星，滑动窗口建模更为有效。Galileo与GLONASS呈现出截然不同的结果。Galileo搭载的高性能被动型氢钟(PHM)钟差值为较大的正数，说明整体建模对其适用性远超滑动窗口法。GLONASS的铯钟(CS)卫星得到了一个极大的负值，说明滑动窗口建模策略对于GLONASS卫星的适用性极强。
2.5 PPP定位精度分析
为检验2种模型在精密单点定位(PPP)中的精度表现，选用DOY074、DOY075、DOY076三天观测数据，选取全球范围内分布的25个MGEX测站，对事后静态PPP的定位精度与收敛时间进行统计分析，测站分布如图 7所示。设定当东方向(E)、北方向(N)与天方向(U)的偏差均小于10 cm时卡尔曼滤波开始收敛，统计收敛后3 h内定位偏差作为精度评估指标。
采用BDS-3、Galileo、GPS及GLONASS分别进行单系统定位实验，对上述25个测站进行为期3 d的PPP数据分析。从每天的11：50开始解算，直至23：50结束，对不同GNSS不同钟差产品3 d的PPP定位精度与收敛时间求取平均值，结果如表 4所示。
由表 4可知，BDS-3、Galileo中的整体建模估计钟差与滑动窗口建模估计钟差在进行PPP定位后，定位精度相差不大，只存在mm级数值差距，这可能与二者均采用了高精度的氢钟有关。对于GPS而言，与整体建模估计钟差相比，滑动窗口建模估计钟差在E、N、U三个方向的定位精度分别提升10.34%、17.07%和4.30%；GLONASS的整体建模估计钟差相比滑动窗口建模估计钟差在E、N、U三个方向的定位精度分别提升6.27%、10.34%和3.76%。图 8为ALIC、KIRU、NYA2三个测站的PPP定位误差序列，图 9为相应的定位精度柱状统计图。
由图 8、9可知，2种处理方式下, BDS-3与Galileo在各个测站3 d的PPP定位误差数值几乎相同，这一结果与图 6中描述有所差异，原因是：1)BDS-3和Galileo星载原子钟本身具有极高稳定性; 2)PPP是利用所有可见卫星进行综合解算的结果，解算结果会被其他性能差异不大的卫星进行平均化，从而稀释了单颗卫星的性能增益。对于GPS及GLONASS，滑动窗口建模估计钟差的定位精度要稍优于整体建模估计钟差。综上所述，GPS与GLONASS更适于采用滑动窗口建模估计钟差，BDS-3与Galileo两种方式估计钟差结果大致相当。
3 结语
收起
本文基于四大GNSS系统卫星钟差数据，分别采用整体建模与滑动窗口建模，利用哈达玛方差对噪声参数进行估计，并依据解算的参数值构建钟差估计模型。最终与IGS发布的事后精密钟差进行二次作差对比分析，同时也将整体建模与滑动窗口建模的估计钟差进行二次作差，进而评估2种建模策略在钟差估计中的性能差异。结果表明，不同系统、不同卫星所适用的建模策略存在明显差异，主要受限于钟类型特性、噪声稳定性以及星务状态变化等多种因素。具体结论如下：
1) BDS-3中，对于MEO卫星，滑动窗口建模能更好地捕捉其局部噪声波动特征，从而提高建模精度；少部分IGSO及采用铷钟的MEO卫星，其频率噪声特性相对稳定，适合采用整体建模策略。因此，BDS-3建议按卫星钟型进行差异化建模策略处理，以提升整体建模效果。Galileo中，多数卫星采用被动型氢钟(PHM)，其稳定性极优，适合整体建模。但对于部分长期服役、状态变更频繁的卫星(如E02、E03、E07、E08、E18、E25)，滑动窗口策略更佳，体现出即使高稳定的氢钟也需关注其实时运行状况。
2) GPS中，采用滑动窗口建模时，均表现出较低的波动幅度，表明其频率噪声具有非平稳性，整体建模解算难以全面反映不同时间尺度下的变化情况。所以，优先采用滑动窗口建模策略。GLONASS中，因普遍使用铯原子钟，同时受控调整频繁、噪声非平稳性高，因此滑动窗口建模更能捕获短周期的变化波动。
3) BDS-3与Galileo的整体建模和滑动窗口建模钟差在PPP定位中精度差异较小，可能与高精度的卫星钟类型有关。相比之下，GPS采用滑动窗口建模在E、N、U方向定位精度较整体建模分别提升10.34%、17.07%和4.30%；GLONASS利用滑动窗口建模在E、N、U方向精度较整体建模分别提升6.27%、10.34%和3.76%。
综上所述，解算钟差协方差参数时，可根据不同系统、卫星钟类型及运行状态选取合适的参数估计策略，从而得到更高精度的钟差数据，提高定位精度。
基金
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  • 国家自然科学基金(42404015)
  • 国家自然科学基金(42274048)
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2026年第46卷第5期
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doi: 10.14075/j.jgg.2025.07.249
  • 接收时间：2025-07-15
  • 首发时间：2026-10-29
  • 出版时间：2026-05-15
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  • 收稿日期：2025-07-15
基金
国家自然科学基金(42404015)
国家自然科学基金(42274048)
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    1 中国矿业大学环境与测绘学院, 徐州, 221116
    2 安徽理工大学空间信息与测绘工程学院, 淮南, 232001

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王中元, 博士, 副教授, 主要研究方向为GNSS空间大地测量技术, E-mail:
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