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Article(id=1322566946248311467, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322566927927591451, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.03.090, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1742486400000, receivedDateStr=2025-03-21, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793254132090, onlineDateStr=2026-10-29, pubDate=1773504000000, pubDateStr=2026-03-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793254132090, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793254132090, creator=13701087609, updateTime=1793254132090, updator=13701087609, issue=Issue{id=1322566927927591451, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='3', pageStart='262', pageEnd='391', issueExtLink='null', onlineDate='null', pubDate='1773504000000', pubDateStr='2026-03-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793254127723, creator='13701087609', updateTime=1793254127723, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=273, endPage=280, ext={EN=ArticleExt(id=1322566946411889324, articleId=1322566946248311467, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Cycle Slip Detection Method of BDS Triple Frequency Data, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

In view of the problems that traditional pseudo-range phase combination method and STPIR method are greatly affected by observation noise and multi-path effect under low sampling rate, cycle slip is easily misjudged under strong noise environment, and detection performance is affected during periods of frequent or drastic changes in ionospheric activity. We introduce the time window method to improve the judgment conditions of pseudo-range phase combination method and STPIR method considering altitude angle, and the weighted recursive smoothing method is combined with time window to improve the algorithm. Experimental verification is conducted using observation data of IGS stations. The results show that the improved algorithm can detect and repair various types of cycle slips well.

, authors=Chenglin CAI1, Xuanwen QIN1, Wenhui GUAN2, *, authorsList=Chenglin CAI, Xuanwen QIN, Wenhui GUAN, authorCompany=null, correspAuthors=Wenhui GUAN, 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=1322566950220317379, articleId=1322566946248311467, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=BDS三频数据的周跳探测方法研究, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

针对在低采样率下, 传统的伪距相位组合方法和STPIR法受观测噪声和多路径效应的影响较大, 在强噪声环境下容易误判周跳, 以及在电离层活动频繁或剧烈变化时导致探测性能受到影响的问题, 提出结合高度角变化量, 引入时间窗口方法, 对伪距相位组合方法和STPIR法的判定条件进行改进, 并采用加权递推平滑与时间窗口相结合的方法改进算法。利用IGS测站观测数据进行实验验证, 结果表明, 改进的算法能够很好地探测与修复各种类型的周跳。

, authors=蔡成林1, 覃炫文1, 关文绘2, *, authorsList=蔡成林, 覃炫文, 关文绘, authorCompany=null, correspAuthors=关文绘, authorNote=

蔡成林, 博士, 教授, 主要研究方向为卫星导航与无线通信, E-mail:

, correspAuthorsNote=
关文绘, 博士生, 主要研究方向为实时精密单点定位, E-mail:
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蔡成林, 博士, 教授, 主要研究方向为卫星导航与无线通信, E-mail:

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蔡成林, 博士, 教授, 主要研究方向为卫星导航与无线通信, E-mail:

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figureFileSmall=1J4mV2LOyE98so+JcyhjlA==, figureFileBig=kQEgwxN2DYcW0yRwTQim1A==, tableContent=null), ArticleFig(id=1322566955492557558, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566946248311467, language=EN, label=Tab. 1, caption=

Selection of pseudo-range phase combination coefficients for BDS-2 and BDS-3

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卫星类型 采样率/s [i j k] λijk/m K 中误差/周
BDS-230 [1 4 －5] 6.370 7 0.321 4 0.099 4
30 [1 3 －4] 2.764 6 0.281 3 0.114 2
BDS-330 [－1 －4 5] 4.884 2 －0.475 8 0.104 5
30 [1 3 －4] 9.768 4 0.408 9 0.076 3
), ArticleFig(id=1322566955563860727, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566946248311467, language=CN, label=表1, caption=

BDS-2、BDS-3伪距相位组合系数选择

, figureFileSmall=null, figureFileBig=null, tableContent=
卫星类型 采样率/s [i j k] λijk/m K 中误差/周
BDS-230 [1 4 －5] 6.370 7 0.321 4 0.099 4
30 [1 3 －4] 2.764 6 0.281 3 0.114 2
BDS-330 [－1 －4 5] 4.884 2 －0.475 8 0.104 5
30 [1 3 －4] 9.768 4 0.408 9 0.076 3
), ArticleFig(id=1322566955639358200, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566946248311467, language=EN, label=Tab. 2, caption=

Cycle slip detection results of two combined methods

, figureFileSmall=null, figureFileBig=null, tableContent=
卫星号 历元 周跳模拟值 伪距相位+STPIR 本文方法
C301 400 (0, 0, 1) √ √
1 500 (1, 0, 1) √ √
1 600 (2, 2, 2) √ √
1 700 (1, 1, 1) × √
1 800 (4, －3, 0) √ √
1 900 (1, 3, 2) √ √
2 000 (1, 2, 1) √ √
2 001 (3, 1, 2) √ √
C39100 (0, 0, 1) √ √
200 (1, 0, 1) √ √
300 (2, 2, 2) √ √
400 (1, 1, 1) × √
2 000 (4, －3, 0) √ √
2 100 (1, 3, 2) √ √
2 200 (1, 2, 1) √ √
2 201 (3, 1, 2) √ √
), ArticleFig(id=1322566955710661369, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566946248311467, language=CN, label=表2, caption=

2种联合方法的周跳探测结果

, figureFileSmall=null, figureFileBig=null, tableContent=
卫星号 历元 周跳模拟值 伪距相位+STPIR 本文方法
C301 400 (0, 0, 1) √ √
1 500 (1, 0, 1) √ √
1 600 (2, 2, 2) √ √
1 700 (1, 1, 1) × √
1 800 (4, －3, 0) √ √
1 900 (1, 3, 2) √ √
2 000 (1, 2, 1) √ √
2 001 (3, 1, 2) √ √
C39100 (0, 0, 1) √ √
200 (1, 0, 1) √ √
300 (2, 2, 2) √ √
400 (1, 1, 1) × √
2 000 (4, －3, 0) √ √
2 100 (1, 3, 2) √ √
2 200 (1, 2, 1) √ √
2 201 (3, 1, 2) √ √
), ArticleFig(id=1322566955773575930, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566946248311467, language=EN, label=Tab. 3, caption=

Cycle slip detection and repaired results of C40 satellite

, figureFileSmall=null, figureFileBig=null, tableContent=
历元 周跳模拟值周跳浮点解周跳修复值
[1 3 －4] [－1 －4 5] [1 1 －2]
1 800 (2, －1, 1) 2.172 3 －0.827 7 1.172 3 (2, －1, 1)
1 900 (0, 1, 2) －0.126 1 0.873 9 1.873 9 (0, 1, 2)
2 000 (1, 1, 1) 0.758 6 0.758 6 0.758 6 (1, 1, 1)
2 100 (2, 2, 2) 1.980 4 1.980 4 1.980 4 (2, 2, 2)
2 200 (0, 0, 1) 0.019 5 0.019 5 1.019 5 (0, 0, 1)
2 300 (2, 1, 0) 2.110 5 1.110 5 0.110 5 (2, 1, 0)
2 400 (1, 4, 2) 1.046 6 4.046 6 2.046 6 (1, 4, 2)
2 500 (1, －2, 3) 0.866 2 －2.133 8 2.866 2 (1, －2, 3)
2 600 (1, 3, 2) 1.343 0 3.343 0 2.343 0 (1, 3, 2)
2 700 (3, 2, 1) 2.823 6 1.823 6 0.823 6 (3, 2, 1)
2 701 (2, 1, 3) 2.219 7 1.219 7 3.219 7 (2, 1, 3)
), ArticleFig(id=1322566955857462011, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566946248311467, language=CN, label=表3, caption=

C40卫星周跳探测与修复结果

, figureFileSmall=null, figureFileBig=null, tableContent=
历元 周跳模拟值周跳浮点解周跳修复值
[1 3 －4] [－1 －4 5] [1 1 －2]
1 800 (2, －1, 1) 2.172 3 －0.827 7 1.172 3 (2, －1, 1)
1 900 (0, 1, 2) －0.126 1 0.873 9 1.873 9 (0, 1, 2)
2 000 (1, 1, 1) 0.758 6 0.758 6 0.758 6 (1, 1, 1)
2 100 (2, 2, 2) 1.980 4 1.980 4 1.980 4 (2, 2, 2)
2 200 (0, 0, 1) 0.019 5 0.019 5 1.019 5 (0, 0, 1)
2 300 (2, 1, 0) 2.110 5 1.110 5 0.110 5 (2, 1, 0)
2 400 (1, 4, 2) 1.046 6 4.046 6 2.046 6 (1, 4, 2)
2 500 (1, －2, 3) 0.866 2 －2.133 8 2.866 2 (1, －2, 3)
2 600 (1, 3, 2) 1.343 0 3.343 0 2.343 0 (1, 3, 2)
2 700 (3, 2, 1) 2.823 6 1.823 6 0.823 6 (3, 2, 1)
2 701 (2, 1, 3) 2.219 7 1.219 7 3.219 7 (2, 1, 3)
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BDS三频数据的周跳探测方法研究
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蔡成林 1 , 覃炫文 1 , 关文绘 2, *
大地测量与地球动力学 | 大地测量学 2026,46(3): 273-280
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大地测量与地球动力学 |大地测量学 2026 , 46 (3) : 273 -280
BDS三频数据的周跳探测方法研究
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蔡成林1 , 覃炫文1, 关文绘2, *
作者信息
  • 1 湘潭大学自动化与电子信息学院, 湘潭, 411105
  • 2 湘潭大学数学与计算科学学院, 湘潭, 411105
通讯作者:
关文绘, 博士生, 主要研究方向为实时精密单点定位, E-mail:
作者简介:

蔡成林, 博士, 教授, 主要研究方向为卫星导航与无线通信, E-mail:

Cycle Slip Detection Method of BDS Triple Frequency Data
Chenglin CAI1 , Xuanwen QIN1, Wenhui GUAN2, *
Affiliations
  • 1 School of Automation and Electronic Information, Xiangtan University, Xiangtan 411105, China
  • 2 School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
出版时间: 2026-03-15 doi: 10.14075/j.jgg.2025.03.090
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摘要
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针对在低采样率下, 传统的伪距相位组合方法和STPIR法受观测噪声和多路径效应的影响较大, 在强噪声环境下容易误判周跳, 以及在电离层活动频繁或剧烈变化时导致探测性能受到影响的问题, 提出结合高度角变化量, 引入时间窗口方法, 对伪距相位组合方法和STPIR法的判定条件进行改进, 并采用加权递推平滑与时间窗口相结合的方法改进算法。利用IGS测站观测数据进行实验验证, 结果表明, 改进的算法能够很好地探测与修复各种类型的周跳。

周跳探测与修复  /  北斗系统  /  三频数据  /  高度角

In view of the problems that traditional pseudo-range phase combination method and STPIR method are greatly affected by observation noise and multi-path effect under low sampling rate, cycle slip is easily misjudged under strong noise environment, and detection performance is affected during periods of frequent or drastic changes in ionospheric activity. We introduce the time window method to improve the judgment conditions of pseudo-range phase combination method and STPIR method considering altitude angle, and the weighted recursive smoothing method is combined with time window to improve the algorithm. Experimental verification is conducted using observation data of IGS stations. The results show that the improved algorithm can detect and repair various types of cycle slips well.

cycle slip detection and repair  /  BDS  /  triple frequency data  /  altitude angle
蔡成林, 覃炫文, 关文绘. BDS三频数据的周跳探测方法研究. 大地测量与地球动力学, 2026 , 46 (3) : 273 -280 . DOI: 10.14075/j.jgg.2025.03.090
Chenglin CAI, Xuanwen QIN, Wenhui GUAN. Cycle Slip Detection Method of BDS Triple Frequency Data[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (3) : 273 -280 . DOI: 10.14075/j.jgg.2025.03.090
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在卫星导航高精度定位中，周跳探测是系统数据预处理的重要组成部分，周跳的探测与修复可以提高定位结果精度。目前较为常用的周跳探测方法很多，对于单频数据，Li等[1]采用低阶多项式约束位置参数实现单频数据的周跳探测，但该算法复杂且会受到观测噪声影响。对于双频数据，Xu等[2]提出改进的TurboEdit法，结合MW组合和GF组合，能够高效探测周跳，适用于连续观测数据中的周跳检测，但存在对噪声敏感、受电离层干扰和不敏感周跳的问题。在此基础上，刘宁等[3]提出结合高度角因子的方法，可在一定程度上减少周跳的误探，但在观测数据出现多个周跳时可能会造成周跳漏探的问题。熊旭等[4]提出的伪距相位组合联合STPIR法能够在静态低采样率下探测各种类型周跳，但仅考虑了GEO卫星，并未对其他类型的北斗卫星进行分析。
上述方法多是在高采样率下对周跳探测进行研究，而实际应用中多数接收机的数据采集为30 s的低采样率。在低采样率下，这些方法受到伪距噪声以及多路径效应的影响更大，会严重影响探测效果。多频组合观测值具有波长更长、受电离层延迟和观测噪声影响更小等优势，有利于提高周跳探测性能。本文在熊旭等[4]的研究基础上提出针对低采样率观测数据，利用卫星高度角变化量，结合伪距相位组合和STPIR组合原理，构建顾及卫星高度角因子以及窗口因子的周跳检验量，以解决窗口期间发生周跳变化但未超过阈值而漏探的问题；最后采用最小二乘方法得到周跳修复结果，并利用IGS观测数据进行实验验证。
1 顾及高度角的三频伪距相位组合
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1.1 三频伪距相位组合法原理
构造三频组合伪距相位观测方程时，考虑到观测过程中存在多路径延迟和电离层延迟等误差，观测方程可表示为：
$P_{a b c}=\rho+\beta_{a b c} I_{1}+d_{a b c}+m_{a b c}+\varepsilon_{a b c}$
$\lambda_{i j k} \varphi_{i j k}=\rho-\beta_{i j k} I_{1}+\lambda_{i j k} N_{i j k}+d_{i j k}+m_{i j k}+\varepsilon_{i j k}$
式中，Pabc为三频伪距组合观测量，Pabc=aP1+bP2+cP3，a+b+c=1[5]，a, b, c∈R；φijk为三频载波相位组合观测量，φijk=iφ1+jφ2+kφ3，i, j, k∈Z；P1、P2、P3和φ1、φ2、φ3分别对应载波频率f1、f2、f3的伪距观测值和载波相位观测值。$\lambda_{i j k}=\frac{C}{i \cdot f_1+j \cdot f_2+k \cdot f_3} $为组合相位观测值的波长，其中C为光速；ρ为卫星至接收机的几何距离；I1为载波频率f1的电离层延迟量；Nijk=iN1+jN2+kN3为组合相位观测值的整周模糊度；βabc与βijk、dabc与dijk、mabc与mijk、εabc与εijk分别为组合伪距观测量和组合相位观测量的电离层延迟影响系数、硬件延迟、多路径误差、观测噪声。
将式(1)减去式(2)，整理后得到：
$\begin{gather*}N_{i j k}=\varphi_{i j k}-\frac{P_{a b c}}{\lambda_{i j k}}+\frac{\beta_{a b c}+\beta_{i j k}}{\lambda_{i j k}} I_{1}- \\\frac{d_{i j k}-d_{a b c}+m_{i j k}-m_{a b c}+\varepsilon_{i j k}-\varepsilon_{a b c}}{\lambda_{i j k}} \end{gather*}$
历元间作差后表达式为：
$\begin{gather*}\Delta N_{i j k}=\Delta \varphi_{i j k}-\frac{\Delta P_{a b c}}{\lambda_{i j k}}+\frac{\beta_{a b c}+\beta_{i j k}}{\lambda_{i j k}} \cdot \\\Delta I_{1}-\frac{\Delta \varepsilon_{i j k}-\Delta \varepsilon_{a b c}}{\lambda_{i j k}} \end{gather*}$
式中，ΔNijk为伪距相位组合周跳检验量，Δ为历元间差分算子。伪距相位组合周跳检验量的电离层延迟系数K可表示为：
$\begin{equation*}K=\frac{\beta_{a b c}+\beta_{i j k}}{\lambda_{i j k}} \end{equation*}$
式中，$\beta_{a b c}=a+b\left(\frac{\lambda_2}{\lambda_1}\right)^2+c\left(\frac{\lambda_3}{\lambda_1}\right)^2 ; \beta_{i j k}=\frac{\lambda_{i j k}}{\lambda_1}(i+ \left.j \frac{\lambda_2}{\lambda_1}+k \frac{\lambda_3}{\lambda_1}\right)$。
在周跳探测过程中，很难对多路径效应和观测噪声进行建模并将其分离出来，但经过历元间差分后变化很小，同时在电离层活动变化不大时，组合周跳检验量可表示为：
$\begin{equation*}\Delta \hat{N}_{i j k}=\Delta \varphi_{i j k}-\frac{\Delta P_{a b c}}{\lambda_{i j k}} \end{equation*}$
式中，$\Delta \hat{N}_{i j k} $为忽略观测噪声和电离层残差后的周跳检验量。
假设各频率载波相位和伪距观测值的噪声项在时间上互不相关，同时方差保持一致，伪距相位组合的周跳检验量中误差为：
$\begin{equation*}\sigma_{\Delta \dot{N}}=\sqrt{2} \sqrt{\left(i^{2}+j^{2}+k^{2}\right) \sigma_{\varphi}^{2}+\frac{\left(a^{2}+b^{2}+c^{2}\right) \sigma_{P}^{2}}{\lambda_{i j k}^{2}}} \end{equation*}$
式中，σφ为载波相位观测噪声标准差，一般取值为0.01周；σP为伪距观测噪声标准差，一般取值为0.3 m[6]；采用等权模型[7]，即a=b=c=1/3。由式(7)可以看出，中误差值与组合系数i、j、k和组合波长有关。因此，选择组合系数时，应使组合波长较长，电离层延迟系数较小，组合周跳检验量中误差较小[8]。
针对BDS-2频点B1I(1 561.098 MHz)、B2I(1 207.140 MHz)、B3I(1 268.520 MHz)和BDS-3频点B1C(1 575.420 MHz)、B2a(1 176.450 MHz)、B3I，通过比较各种组合的波长和中误差，同时考虑到条件数过大会造成矩阵病态，导致解算结果偏离真实值。因此，对于BDS-2卫星系数组合选择[1 4-5]、[1 3-4]，对于BDS-3卫星系数组合选择[-1-4 5]、[1 3-4]。具体组合系数见表 1。
1.2 结合高度角的伪距相位组合法的改进模型
传统的伪距相位组合周跳探测判定表达式为：
$\begin{equation*}\left|\Delta \hat{N}_{i j k}\right|=\left|\Delta \varphi_{i j k}-\frac{\Delta P_{a b c}}{\lambda_{i j k}}\right| \geqslant 4 \sigma_{\Delta \dot{N}} \end{equation*}$
式中，取4倍中误差作为判定阈值检验是否发生周跳。但在30 s的低采样率下，受多路径效应、观测噪声、电离层变化等影响，周跳探测与修复的准确性可能会降低。此外，卫星位置时刻在变化，卫星信号受到大气层延迟、多路径效应等的干扰也与高度角变化有关，会导致复杂情况下的周跳误判。
以武汉站2025-02-22实测GNSS数据(采样率为30 s的“干净”数据)为例，当天电离层活动稳定，选取BDS-2中GEO卫星C01、IGSO卫星C07、MEO卫星C11和BDS-3中MEO卫星C30、IGSO卫星C39卫星，在未发生周跳时伪距相位组合得到的周跳检验结果见图 1。
由图 1可知，在低采样率下，BDS GEO卫星的高度角变化不大，伪距相位组合的周跳检验量未发生明显波动，且在阈值范围内。而IGSO和MEO卫星的高度角变化较大，伪距相位组合检验量随着卫星高度角的下降而有明显波动，并且在卫星到达某些低高度角时，波动会超出阈值。原因可能是低高度角处存在较大的观测噪声，从而导致部分检验量超出经验阈值，使得周跳探测时发生误判。因此，本文结合高度角变化量，引入时间窗口因子，构建附加高度角约束的周跳检验量和中误差，进行加权递推求解[3]。表达式为：
$\begin{gathered}\Delta \bar{N}_{i j k}(n)=\Delta \bar{N}_{i j k}(n-1)+\frac{\sin E(n)}{\sum\limits_{l=n}^{n+m-1} \sin E(l)} \cdot \\{\left[\Delta \hat{N}_{i j k}(n)-\Delta \bar{N}_{i j k}(n-1)\right]}\end{gathered}$
$\begin{gathered}\Delta \sigma_{\Delta \bar{N}}^{2}(n)=\Delta \sigma_{\Delta \bar{N}}^{2}(n-1)+\frac{\sin E(n)}{\sum\limits_{l=n}^{n+m-1} \sin E(l)} \cdot\\\left\{\left[\Delta \hat{N}_{i j k}(n)-\Delta \bar{N}_{i j k}(n-1)\right]^{2}-\Delta \sigma_{\Delta \bar{N}}^{2}(n-1)\right\} \end{gathered}$
式中，$\Delta \bar{N}_{i j k}, \sigma_{\Delta \bar{N}} $分别为伪距相位组合检验量利用滑动窗口内部分历元的高度角加权平均值及其均方根中误差；n为历元数；m为滑动窗口因子；E为当时历元卫星的高度角。基于式(9)和式(10)，周跳判定阈值可表示为：
$\begin{equation*}\left|\Delta \hat{N}_{i j k}(n)-\Delta \bar{N}_{i j k}(n-1)\right| \geqslant 4 \sigma_{\Delta \bar{N}}(n-1) \end{equation*}$
滑动窗口的选取会影响$\Delta \bar{N}_{i j k}、\sigma_{\Delta \bar{N}} $值，基于大量实验，本文取m=20。
对选取的卫星进行周跳探测，此时结合高度角的伪距相位组合周跳探测方法可以很好地避免在低采样率和低高度角情况下对周跳的探测误判，结果见图 2。
利用式(11)探测周跳，实验数据分析表明，当多个周跳发生时，周跳探测可能会出现漏判和偏零值较大的情况。以BDS-3的C30、C39卫星为例，在C30卫星第1 400历元、1 430历元、1 800历元分别添加周跳(0, 1, 0)、(1, 0, 0)、(2, 0, －1)，在C39卫星第500历元、800历元、830历元分别添加周跳(2, 0, －1)、(0, 1, 0)、(1, 0, 0)。周跳探测结果见图 3。
由图 3可知，以C30卫星为例，第1 400历元探测成功，但后续第1 430历元则被漏判。第1 800历元能够探测到周跳，但后续部分周跳检验量偏离零点，探测到的周跳值越大，偏零值就越严重。原因是递推算法的后值会受到前值影响，在发生周跳后，$\Delta \bar{N}_{i j k}(n)、 \sigma_{\Delta \bar{N}}(n) $会因为$\Delta \hat{N}_{i j k}(n) $在后续的递推中造成误差影响，从而导致周跳探测的漏判和偏零值较大。为解决该问题，对其阈值判定算法进行改进，表达式为：
$\Delta \bar{N}_{i j k}(n) =\Delta \bar{N}_{i j k}(n-1)$
$\sigma_{\Delta \bar{N}}(n) =\sigma_{\Delta \bar{N}}(n-1) $
当式(11)阈值判定出周跳发生后，利用式(12)、(13)对$\Delta \bar{N}_{i j k}(n)、\sigma_{\Delta \bar{N}}(n) $进行修正，可以有效解决$\Delta \hat{N}_{i j k}(n) $对$\Delta \bar{N}_{i j k}(n)、\sigma_{\Delta \bar{N}}(n) $的误差影响。对比结果见图 4，可以看出, 改进后的算法结果中不再出现漏判和偏零值的问题。
2 顾及高度角的三频STPIR法
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2.1 三频STPIR法原理
当观测数据为低采样率时(如30 s)，相邻的电离层变化较大，为削弱电离层误差、对流层误差以及钟差影响[9-10]，选择合适的组合系数对电离层残差再进行一次历元间作差。
研究表明[4, 6]，选择组合系数[1 1 －2]对BDS-2和BDS-3卫星进行数据分析并构建三频电离残差组合量时，该组合系数对于BDS-2的电离层延迟系数为－0.068 5，中误差为0.008 0；对于BDS-3的电离层延迟系数为－0.055 5，中误差为0.008 1。两者结果均很小，因此可以将[1 1 －2]作为STPIR法的组合系数。具体表达式为：
$\begin{gather*}\varphi_{\mathrm{PIR} 3}=\varphi_{1}+\frac{\lambda_{2}}{\lambda_{1}} \varphi_{2}-2 \frac{\lambda_{3}}{\lambda_{1}} \varphi_{3}=N_{1}+ \\\frac{\lambda_{2}}{\lambda_{1}} N_{2}-2 \frac{\lambda_{3}}{\lambda_{1}} N_{3}+I_{123} \end{gather*}$
式中，φPIR3为三频电离层残差构造量；$I_{123}=-\frac{I_1}{\lambda_1} \left(1+\frac{\lambda_2}{\lambda_1}-2 \frac{\lambda_3}{\lambda_1}\right)$为三频组合电离层延迟。对式(14)历元间作差可得：
$\begin{gather*}\Delta \varphi_{\mathrm{PIR} 3}=\varphi_{\mathrm{PIR} 3}(n)-\varphi_{\mathrm{PIR} 3}(n-1)= \\{\left[\Delta N_{1}+\frac{\lambda_{2}}{\lambda_{1}} \Delta N_{2}-2 \frac{\lambda_{3}}{\lambda_{1}} \Delta N_{3}\right](n)+\Delta I_{123}(n)} \end{gather*}$
对电离层残差组合进行二次历元间差分[4]，表达式为：
$\begin{gathered}\Delta \varphi_{\mathrm{STPIR} 3}(n)=\varphi_{\mathrm{PIR} 3}(n)-2 \varphi_{\mathrm{PIR} 3}(n-1)+ \\\varphi_{\mathrm{PIR} 3}(n-2)=\left[\Delta N_1+\frac{\lambda_2}{\lambda_1} \Delta N_2-2 \frac{\lambda_3}{\lambda_1} \Delta N_3\right]. \\(n)-\left[\Delta N_1+\frac{\lambda_2}{\lambda_1} \Delta N_2-2 \frac{\lambda_3}{\lambda_1} \Delta N_3\right] .\\(n-1)+\Delta I_{123}(n)-\Delta I_{123}(n-1)\end{gathered}$
相较于电离层残差，二次差分使得历元间的电离层变化对周跳检验量的影响很小，并使周跳检验序列始终在零值附近波动，更有利于周跳探测。
三频STPIR法周跳探测检验量的中误差为：
$\begin{equation*}\sigma_{\mathrm{STPIR} 3}=2 \sigma_{\varphi} \sqrt{1+\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{2}+\left(-2 \frac{\lambda_{3}}{\lambda_{1}}\right)^{2}} \end{equation*}$
式中，σSTPIR3为三频STPIR法周跳检验量中误差。虽然STPIR法可以很好地削弱电离层残差，但随着卫星高度角的降低，多路径效应增强，周跳检验量数值波动也会变大，此时取3倍或4倍中误差会导致在低高度角时发生误判现象。以武汉站2025-02-22实测GNSS数据为例，采样率为30 s，选取BDS-2卫星C01、C07、C11和BDS-3卫星C30、C39，选择3倍中误差，所得结果见图 5。
由图 5可知，GEO卫星的高度角变化幅度不大，周跳检验量仍然可以使用经验阈值来检测；而对于MEO和IGSO卫星，随着高度角的下降，周跳检验量波动剧烈，在到达某些角度时，周跳检验量超过原本阈值，导致周跳误判。
2.2 结合高度角的三频STPIR法的改进模型
结合高度角变化量，引入时间窗口因子，构建附加高度角约束的加权递推公式对中误差进行求解，表达式为：
$\begin{gather*}\Delta \sigma_{n}^{2}=\Delta \sigma_{n-1}^{2}+\frac{\sin E(n)}{\sum\limits_{l=n}^{n+m-1} \sin E(l)}\cdot \\{\left[\Delta \varphi_{\text {STPIR3 }}(n)^{2}-\Delta \sigma_{n-1}^{2}\right]} \end{gather*}$
式中，Δσn为利用滑动窗口内部分历元的高度角加权均方根中误差。利用式(16)计算出周跳检验量后，以|ΔφSTPIR3(n) |≥3Δσn－1为阈值判定周跳。当周跳发生后，用Δσ(n)=Δσ(n－1)来修正中误差，避免ΔφSTPIR3对Δσ造成影响，从而导致发生周跳时漏判。窗口大小m同样取20，选择3倍中误差进行周跳检验，对比结果见图 6。可以看出，改进判定阈值后的STPIR法可以有效避免周跳误判的问题。
改进的伪距相位法依然无法探测到等值周跳，改进的STPIR法也依然存在敏感周跳，2种探测方法结合起来可以互补。本文选择2组改进后的伪距相位组合系数和1组改进后的STPIR组合系数构建组合方程组，表达式为：
$\left[\begin{array}{c}\Delta \hat{N}_{i_{1} j_{1} k_{1}} \\\Delta \hat{N}_{i_{2} j_{2} k_{2}} \\\Delta \varphi_{\text {STPIR3 }}\end{array}\right]=\left[\begin{array}{ccc}i_{1} & j_{1} & k_{1} \\i_{2} & j_{2} & k_{1} \\1 & \frac{\lambda_{2}}{\lambda_{1}} & -2 \frac{\lambda_{3}}{\lambda_{1}}\end{array}\right]\left[\begin{array}{l}\Delta N_{1} \\\Delta N_{2} \\\Delta N_{3}\end{array}\right]$
本文在系数选择时已考虑条件数过多造成方程组病态的问题，最终BDS-2卫星选择周跳探测组合[1 4 －5]、[1 3 －4]、[1 1 －2]，BDS-3卫星选择周跳探测组合[－1 －4 5]、[1 3 －4]、[1 1 －2]。采用最小二乘方法求解周跳浮点解，四舍五入得到周跳整数值。
3 实验结果与分析
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以武汉站2025-02-22实测GNSS数据(采样率为30 s的“干净”数据)为例，当天电离层活动稳定，选取5颗BDS卫星(C01、C07、C11、C30、C39)作为实验对象，设计2个实验来验证改进的联合算法的实用性与有效性。
3.1 实验1
单一算法都会存在不敏感周跳，改进的伪距相位法无法探测到等值周跳，改进的STPIR法无法探测到$1+\frac{\lambda_2}{\lambda_1}-2 \frac{\lambda_3}{\lambda_1}=0 $的周跳。为研究上述组合方法的周跳探测能力及修复效果，在“干净”的原始数据中每隔100个历元添加不同大小共计8组三频周跳，分别包括普通周跳、不敏感周跳、连续周跳。利用本文改进的方法进行周跳探测，并与原始伪距相位组合STPIR法探测结果进行对比。限于篇幅，本文只展示BDS-3 C30、C39卫星的周跳检查序列，BDS-3卫星周跳探测系数组合依次为[1 3 －4]、[－1 －4 5]、[1 1 －2]。探测结果见图 7。
从图 7可以看出，伪距相位组合法和STPIR法都能探测出普通周跳(0, 0, 1)、(1, 0, 1)；(2, 2, 2)与(1, 3, 2)、(4, －3, 0)分别为伪距相位组合与STPIR法的敏感周跳组合，因而各自未探测成功，两者互补可解决该问题。对C30卫星第2 000历元、第2 001历元，C39卫星第2 200历元、第2 201历元添加连续周跳，可以看到, 组合法也能很好地探测出周跳的发生。无论是C30还是C39卫星，原始伪距相位组合STPIR方法都会出现误探并且无法探测到周跳(1, 1, 1)。由表 2可知，本文的改进方法可以探测出添加的模拟周跳。
为验证本文改进算法对高度角变化幅度不大的GEO卫星数据的适用性，同时充分考虑各种可能出现的周跳组合，对“干净”的BDS-2 C01卫星实测观测数据每隔5个历元人为随机添加周跳组合，具体如(0, 0, 1), (0, 0, 2), …, (7, 7, 7)，共计511个周跳组合，周跳组合中同样包括普通周跳和不敏感周跳，BDS-2卫星周跳探测系数组合依次为[1 4 －5]、[1 3 －4]、[1 1 －2]，探测结果见图 8。通过实验统计分析可知，3个系数组合的探测成功率依次为97.5%、97.1%、97.3%，联合探测成功率为100%。人为随机加入的511个周跳全部探测成功，本文的改进算法对高度角变化幅度不大的GEO卫星仍能很好地实现周跳探测。
3.2 实验2
为进一步验证改进的算法在电离层活跃期的可行性，选取武汉站2024-10-10实时观测数据和广播星历，数据采样间隔为30 s，原观测数据未发生周跳。图 9为2024-10-10—11期间Kp和Dst指数时间序列，可以看出10-10 15:00—24:00 Dst指数均小于－50 nT，Kp指数都高达8及以上，表明当天地球磁场活动强烈并伴随有磁暴现象发生。
选取当天电离层最活跃的9 h原始观测数据，为验证该方法的正确性和有效性，每隔100个历元加入周跳组合。选择C40卫星实测数据进行实验，探测结果见图 10。
由图 10可知，所有加入的周跳组合均被探测出。进一步对周跳探测结果进行修复(表 3)，可以看出，周跳修复值与对应的模拟周跳值完全一致，表明所有加入的周跳均被探测与修复。因此，在电离层活动频繁期，本文方法依然可准确进行周跳探测与修复。
4 结语
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本文提出伪距相位组合STPIR结合高度角来修正判定阈值的算法，并采用IGS测站实测数据进行算法验证。结果表明，相比于传统伪距相位组合STPIR法在低采样率和低高度角情况下会出现周跳误探的问题, 本文结合高度角，引入时间窗口因子，通过改进加权递推平滑算法修正阈值，可克服经验阈值的局限性，即使在电离层活动频繁期也能避免周跳误探和漏探的情况。本文方法能够准确实现BDS中不同类型的周跳探测。
基金
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  • 国家重点研发计划(2020YFA0713501)
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2026年第46卷第3期
39
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文章信息
doi: 10.14075/j.jgg.2025.03.090
  • 接收时间：2025-03-21
  • 首发时间：2026-10-29
  • 出版时间：2026-03-15
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  • 收稿日期：2025-03-21
基金
国家重点研发计划(2020YFA0713501)
作者信息
    1 湘潭大学自动化与电子信息学院, 湘潭, 411105
    2 湘潭大学数学与计算科学学院, 湘潭, 411105

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2种不同金属材料的力学参数
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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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