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Article(id=1322506567619031798, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322506567065383670, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2024.07.346, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1721059200000, receivedDateStr=2024-07-16, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793239736703, onlineDateStr=2026-10-29, pubDate=1752508800000, pubDateStr=2025-07-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793239736703, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793239736703, creator=13701087609, updateTime=1793239736703, updator=13701087609, issue=Issue{id=1322506567065383670, tenantId=1146029695717560320, journalId=1281212831689347082, year='2025', volume='45', issue='7', pageStart='661', pageEnd='770', issueExtLink='null', onlineDate='null', pubDate='1752508800000', pubDateStr='2025-07-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793239736572, creator='13701087609', updateTime=1793239736572, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=705, endPage=711, ext={EN=ArticleExt(id=1322506569376445176, articleId=1322506567619031798, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Adaptive Carrier Phase Smoothing Pseudorange and Single Point Positioning Accuracy, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

We propose an adaptive carrier phase smoothing pseudorange method based on multi frequency observations, which allows for adaptive selection of effective carriers to continue smoothing the pseudorange of all frequencies in the case of partial frequency signal loss of lock or cycle slip. The results show that the adaptive smoothing L1/L2/L5 pseudorange noise of GPS is reduced by about 62%, 50%, and 33% respectively compared to traditional single frequency smoothing, and is reduced by about 60%, 54%, and 61% respectively compared to traditional dual frequency smoothing. In addition, the B1I/B2I/B3I pseudorange noise of BDS-2 is reduced by about 64%, 67%, 51% respectively compared to traditional single frequency smoothing, and is reduced by about 55%, 59%, 50% respectively compared to traditional dual frequency smoothing. Similar effects are also observed in other GNSS frequencies. Conducting dynamic positioning experiments on road overshadowed by trees and tall buildings, the results indicate that standard deviation(STD) of adaptive smoothing pseudorange positioning is reduced by about 45%, 76%, 47% in E, N, U directions respectively compared to single frequency smoothing and about 41%, 69%, 49% compared to dual frequency smoothing. Adaptive smoothing method can improve position accuracy effectively.

, authors=Xianzhong WANG1, Jinghui LIU1, Liyuan LI1, Liangliang WANG1, Lifang WEN1, authorsList=Xianzhong WANG, Jinghui LIU, Liyuan LI, Liangliang WANG, Lifang WEN, authorCompany=null, correspAuthors=null, authorNote=

WANG Xianzhong, senior engineer, majors in GNSS positioning algorithm, GNSS/INS combined positioning algorithm and product development, E-mail:

, 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=1322506570873811717, articleId=1322506567619031798, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=自适应载波相位平滑伪距及单点定位精度分析, columnId=1322506169105633839, journalTitle=大地测量与地球动力学, columnName=现代大地测量, runingTitle=null, highlight=null, articleAbstract=

提出基于多频观测值的自适应载波相位平滑伪距方法，在部分频点信号失锁或载波周跳时可以自适应选择有效载波继续平滑所有频点的伪距。测试结果表明，自适应平滑的GPS L1/L2/L5频点伪距噪声相比传统单频平滑分别减小约62%、50%、33%，相比传统双频平滑分别减小约60%、54%、61%；自适应平滑的BDS-2 B1I/B2I/B3I频点伪距噪声相比传统单频平滑分别减小约64%、67%、51%，相比传统双频平滑分别减小约55%、59%、50%；其他GNSS频点也有类似的效果。在有树荫和高楼遮挡的路段进行动态定位实验，结果表明，自适应平滑伪距定位在E、N、U方向的标准差相比传统单频平滑分别减小约45%、76%、47%，相比传统双频平滑分别减小约41%、69%、49%。自适应平滑方法能明显提升伪距定位精度。

, authors=王献中1, 刘京慧1, 李丽媛1, 王亮亮1, 温丽芳1, authorsList=王献中, 刘京慧, 李丽媛, 王亮亮, 温丽芳, authorCompany=null, correspAuthors=null, authorNote=

王献中，高级工程师，主要研究方向为GNSS定位算法、GNSS/INS组合定位算法开发及产品研发，E-mail:

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WANG Xianzhong, senior engineer, majors in GNSS positioning algorithm, GNSS/INS combined positioning algorithm and product development, E-mail:

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WANG Xianzhong, senior engineer, majors in GNSS positioning algorithm, GNSS/INS combined positioning algorithm and product development, E-mail:

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王献中，高级工程师，主要研究方向为GNSS定位算法、GNSS/INS组合定位算法开发及产品研发，E-mail:

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王献中，高级工程师，主要研究方向为GNSS定位算法、GNSS/INS组合定位算法开发及产品研发，E-mail:

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Experimental schemes of carrier phase smoothing pseudorange

, figureFileSmall=null, figureFileBig=null, tableContent=
方案具体方法
单频平滑GPS：L1载波平滑L1伪距，L2载波平滑L2伪距，L5载波平滑L5伪距
BDS-2：B1I载波平滑B1I伪距，B2I载波平滑B2I伪距，B3I载波平滑B3I伪距
双频平滑GPS：L1+L2载波平滑L1和L2伪距，L1+L5载波平滑L1和L5伪距
BDS-2：B1I+B2I载波平滑B1I和B2I伪距，B1I+B3I载波平滑B1I和B3I伪距
自适应平滑BDS-2、GPS：同§1.2.3
), ArticleFig(id=1322506575852450617, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506567619031798, language=CN, label=表1, caption=

载波相位平滑伪距实验方案

, figureFileSmall=null, figureFileBig=null, tableContent=
方案具体方法
单频平滑GPS：L1载波平滑L1伪距，L2载波平滑L2伪距，L5载波平滑L5伪距
BDS-2：B1I载波平滑B1I伪距，B2I载波平滑B2I伪距，B3I载波平滑B3I伪距
双频平滑GPS：L1+L2载波平滑L1和L2伪距，L1+L5载波平滑L1和L5伪距
BDS-2：B1I+B2I载波平滑B1I和B2I伪距，B1I+B3I载波平滑B1I和B3I伪距
自适应平滑BDS-2、GPS：同§1.2.3
), ArticleFig(id=1322506575927948090, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506567619031798, language=EN, label=Tab. 2, caption=

GPS pseudorange noise of different smoothing schemes

, figureFileSmall=null, figureFileBig=null, tableContent=
频点平滑模式PRN01PRN06PRN08PRN14PRN24PRN27PRN30均值
L1单频平滑6.05.84.66.613.44.27.66.9
双频平滑5.47.87.21.312.89.22.36.6
自适应平滑1.71.61.81.36.93.60.82.5
L2单频平滑14.55.19.16.98.18.94.08.1
双频平滑14.210.99.06.97.310.23.78.9
自适应平滑2.11.97.16.85.63.21.34.0
L5单频平滑1.84.32.62.76.62.91.53.2
双频平滑2.610.76.81.16.37.74.45.7
自适应平滑1.21.32.50.66.22.20.82.1
), ArticleFig(id=1322506576003445563, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506567619031798, language=CN, label=表2, caption=

不同方案平滑的GPS伪距噪声

, figureFileSmall=null, figureFileBig=null, tableContent=
频点平滑模式PRN01PRN06PRN08PRN14PRN24PRN27PRN30均值
L1单频平滑6.05.84.66.613.44.27.66.9
双频平滑5.47.87.21.312.89.22.36.6
自适应平滑1.71.61.81.36.93.60.82.5
L2单频平滑14.55.19.16.98.18.94.08.1
双频平滑14.210.99.06.97.310.23.78.9
自适应平滑2.11.97.16.85.63.21.34.0
L5单频平滑1.84.32.62.76.62.91.53.2
双频平滑2.610.76.81.16.37.74.45.7
自适应平滑1.21.32.50.66.22.20.82.1
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BDS-2 pseudorange noise of different smoothing schemes

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频点平滑模式PRN06PRN07PRN08PRN09PRN10PRN13PRN16均值
B1I单频平滑3.72.38.86.92.34.52.24.4
双频平滑2.41.17.25.80.96.21.33.5
自适应平滑1.00.91.92.40.82.51.21.5
B2I单频平滑3.02.16.83.41.78.12.64.0
双频平滑2.71.05.34.50.86.71.43.2
自适应平滑1.20.92.30.80.61.81.31.3
B3I单频平滑2.71.93.63.51.43.92.92.8
双频平滑1.40.56.45.40.63.21.92.8
自适应平滑0.90.41.32.80.52.01.51.3
), ArticleFig(id=1322506576158634813, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506567619031798, language=CN, label=表3, caption=

不同方案平滑的BDS-2伪距噪声

, figureFileSmall=null, figureFileBig=null, tableContent=
频点平滑模式PRN06PRN07PRN08PRN09PRN10PRN13PRN16均值
B1I单频平滑3.72.38.86.92.34.52.24.4
双频平滑2.41.17.25.80.96.21.33.5
自适应平滑1.00.91.92.40.82.51.21.5
B2I单频平滑3.02.16.83.41.78.12.64.0
双频平滑2.71.05.34.50.86.71.43.2
自适应平滑1.20.92.30.80.61.81.31.3
B3I单频平滑2.71.93.63.51.43.92.92.8
双频平滑1.40.56.45.40.63.21.92.8
自适应平滑0.90.41.32.80.52.01.51.3
), ArticleFig(id=1322506576246715198, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506567619031798, language=EN, label=Tab. 4, caption=

Positioning errors of four smoothing schemes

, figureFileSmall=null, figureFileBig=null, tableContent=
伪距模式RMSSTD
ENUENU
原始伪距4.373.4011.501.250.803.08
单频平滑4.063.2410.640.980.662.36
双频平滑4.043.1710.650.910.532.44
自适应平滑3.343.588.800.530.161.23
), ArticleFig(id=1322506576322212671, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506567619031798, language=CN, label=表4, caption=

4种平滑伪距方案的定位误差

, figureFileSmall=null, figureFileBig=null, tableContent=
伪距模式RMSSTD
ENUENU
原始伪距4.373.4011.501.250.803.08
单频平滑4.063.2410.640.980.662.36
双频平滑4.043.1710.650.910.532.44
自适应平滑3.343.588.800.530.161.23
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自适应载波相位平滑伪距及单点定位精度分析
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王献中 1 , 刘京慧 1 , 李丽媛 1 , 王亮亮 1 , 温丽芳 1
大地测量与地球动力学 | 现代大地测量 2025,45(7): 705-711
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大地测量与地球动力学 |现代大地测量 2025 , 45 (7) : 705 -711
自适应载波相位平滑伪距及单点定位精度分析
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王献中1 , 刘京慧1, 李丽媛1, 王亮亮1, 温丽芳1
作者信息
  • 1 和芯星通科技(北京)有限公司, 北京市丰贤东路7号, 100094
作者简介:

王献中，高级工程师，主要研究方向为GNSS定位算法、GNSS/INS组合定位算法开发及产品研发，E-mail:

Adaptive Carrier Phase Smoothing Pseudorange and Single Point Positioning Accuracy
Xianzhong WANG1 , Jinghui LIU1, Liyuan LI1, Liangliang WANG1, Lifang WEN1
Affiliations
  • 1 Unicore Communications Co Ltd, 7 East-Fengxian Road, Beijing 100094, China

    • WANG Xianzhong, senior engineer, majors in GNSS positioning algorithm, GNSS/INS combined positioning algorithm and product development, E-mail:

出版时间: 2025-07-15 doi: 10.14075/j.jgg.2024.07.346
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摘要
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提出基于多频观测值的自适应载波相位平滑伪距方法，在部分频点信号失锁或载波周跳时可以自适应选择有效载波继续平滑所有频点的伪距。测试结果表明，自适应平滑的GPS L1/L2/L5频点伪距噪声相比传统单频平滑分别减小约62%、50%、33%，相比传统双频平滑分别减小约60%、54%、61%；自适应平滑的BDS-2 B1I/B2I/B3I频点伪距噪声相比传统单频平滑分别减小约64%、67%、51%，相比传统双频平滑分别减小约55%、59%、50%；其他GNSS频点也有类似的效果。在有树荫和高楼遮挡的路段进行动态定位实验，结果表明，自适应平滑伪距定位在E、N、U方向的标准差相比传统单频平滑分别减小约45%、76%、47%，相比传统双频平滑分别减小约41%、69%、49%。自适应平滑方法能明显提升伪距定位精度。

载波相位平滑伪距  /  自适应  /  多频  /  伪距噪声  /  单点定位

We propose an adaptive carrier phase smoothing pseudorange method based on multi frequency observations, which allows for adaptive selection of effective carriers to continue smoothing the pseudorange of all frequencies in the case of partial frequency signal loss of lock or cycle slip. The results show that the adaptive smoothing L1/L2/L5 pseudorange noise of GPS is reduced by about 62%, 50%, and 33% respectively compared to traditional single frequency smoothing, and is reduced by about 60%, 54%, and 61% respectively compared to traditional dual frequency smoothing. In addition, the B1I/B2I/B3I pseudorange noise of BDS-2 is reduced by about 64%, 67%, 51% respectively compared to traditional single frequency smoothing, and is reduced by about 55%, 59%, 50% respectively compared to traditional dual frequency smoothing. Similar effects are also observed in other GNSS frequencies. Conducting dynamic positioning experiments on road overshadowed by trees and tall buildings, the results indicate that standard deviation(STD) of adaptive smoothing pseudorange positioning is reduced by about 45%, 76%, 47% in E, N, U directions respectively compared to single frequency smoothing and about 41%, 69%, 49% compared to dual frequency smoothing. Adaptive smoothing method can improve position accuracy effectively.

carrier phase smoothing pseudorange  /  adaptive  /  multi frequency  /  pseudorange noise  /  single point positioning
王献中, 刘京慧, 李丽媛, 王亮亮, 温丽芳. 自适应载波相位平滑伪距及单点定位精度分析. 大地测量与地球动力学, 2025 , 45 (7) : 705 -711 . DOI: 10.14075/j.jgg.2024.07.346
Xianzhong WANG, Jinghui LIU, Liyuan LI, Liangliang WANG, Lifang WEN. Adaptive Carrier Phase Smoothing Pseudorange and Single Point Positioning Accuracy[J]. Journal of Geodesy and Geodynamics, 2025 , 45 (7) : 705 -711 . DOI: 10.14075/j.jgg.2024.07.346
正文
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GNSS伪距定位应用广泛，但伪距观测值的多路径误差和观测噪声较大，定位精度较低。载波相位观测值精度为mm级[1]，使用载波相位平滑伪距可以降低伪距噪声，进而提升伪距定位精度[2]。载波相位平滑伪距方法包括单频平滑和双频平滑[3]。其中，Hatch滤波递推模型[4]应用最为广泛，但采用固定平滑常数的Hatch滤波算法在电离层延迟变化较快时可能引入严重的发散误差[5]。Park等[6]建立电离层模型并基于电离层延迟估计值确定平滑因子，采用不固定的平滑因子进行最优Hatch滤波。Zhang等[7]使用SBAS电离层网格模型计算电离层延迟变化，基于电离层延迟变化和卫星高度角自适应改变平滑窗口大小，其结果优于传统Hatch滤波方法。韩澎涛等[8]利用Klobuchar模型改正相邻历元间电离层延迟变化，验证了单频平滑方法可以在不同程度上提高静动态伪距定位精度。Hwang等[9]提出双频平滑方法，该方法可以消除电离层延迟误差的影响，不受平滑时间限制，能有效提升单点及差分定位精度，对静态和动态场景都适用[10]。
载波平滑伪距方法通常使用该频点的载波平滑对应频点的伪距，当载波失锁或周跳时中断平滑，导致对多路径和白噪声的抑制效果下降。在某频点信号受干扰或信噪比较低时，载波失锁无效，而伪距依然有效，此时的伪距误差较大，需通过平滑来减少噪声。鉴于当前多频接收机使用较为广泛，本文推导得到多频自适应载波相位平滑伪距模型，即根据频点跟踪及载波周跳情况自适应选择有效频点的载波来平滑所有频点的伪距，使得在部分频点的载波失锁或周跳时，该频点的伪距依然能被连续平滑。最后通过实验验证本文方法的有效性。
1 数学模型
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1.1 载波相位平滑伪距原理
GNSS原始伪距和载波相位观测方程可表示为[11]：
$\left\{\begin{array}{l}P_i=\rho+t_{\mathrm{r}}-t^{\mathrm{s}}+\frac{\mathit{λ}_i^2}{\mathit{λ}_1^2} \cdot I_1+Z+ \\\quad b_{\mathrm{r}, i}-b_i^{\mathrm{s}}+\varepsilon_P \\L_i=\rho+t_{\mathrm{r}}-t^{\mathrm{s}}-\frac{\mathit{λ}_i^2}{\mathit{λ}_1^2} \cdot I_1+Z+ \\\quad \mathit{λ}_i\left(N_i+B_{\mathrm{r}, i}-B_i^{\mathrm{s}}\right)+\varepsilon_L\end{array}\right.$
式中，Pi和Li分别为i频点的伪距观测量和载波相位观测量等效距离；ρ为卫星至测站的几何距离；tr为接收机钟差；ts为卫星钟差；λi为波长；I1为第一频点的电离层延迟；Z为对流层延迟；Ni为整周模糊度；br, i和bis分别为接收机端和卫星端的伪距硬件延迟；Br, i和Bis分别为接收机端和卫星端的相位延迟；εP和εL分别为伪距和载波相位的白噪声与多路径总和。
由于伪距噪声通常较大，载波相位噪声较小，根据载波可以得到高精度的历元间伪距增量，通常利用该增量来平滑伪距[12]，平滑公式为：
$\left\{\begin{array}{l}\bar{P}_i^{(k)}=\frac{1}{k} P_i^{(k)}+\frac{k-1}{k}\left(\bar{P}_i^{(k-1)}+\Delta P_{i, k, k-1}\right) \\\bar{P}_i^{(1)}=P_i^{(1)}\end{array}\right.$
式中，$\Delta P_{i, k, k-1}$为i频点的k和k－1历元间伪距增量；Pi(k)为原始伪距；$\bar{P}_i^{(k)}$和$\bar{P}_i^{(k-1)}$分别为k和k－1历元的平滑后伪距。对于跟踪的某颗卫星，分别对原始伪距和载波相位在k和k－1历元间作差，可得到单差伪距和载波相位观测方程为：
$\left\{\begin{array}{l}\Delta P_{i, k, k-1}=\Delta \rho_{k, k-1}+\Delta t_{\mathrm{r}, k, k-1}-\Delta t_{k, k-1}^{\mathrm{s}}+ \\\quad \frac{\mathit{λ}_i^2}{\mathit{λ}_1^2} \cdot \Delta I_{1, k, k-1}+\Delta Z_{k, k-1}+\varepsilon_{\Delta P} \\\Delta L_{i, k, k-1}=\Delta \rho_{k, k-1}+\Delta t_{\mathrm{r}, k, k-1}-\Delta t_{k, k-1}^{\mathrm{s}}- \\\quad \frac{\mathit{λ}_i^2}{\mathit{λ}_1^2} \cdot \Delta I_{1, k, k-1}+\Delta Z_{k, k-1}+\varepsilon_{\Delta L}\end{array}\right.$
式中，$\Delta P_{i, k, k-1}$为i频点的历元间伪距增量；$\Delta L_{i, k, k-1}$为i频点的历元间载波相位增量；$\Delta \rho_{k, k-1}$为历元间卫地距增量；$\Delta t_{\mathrm{r}, k, k-1}$为历元间接收机钟差增量；$\Delta t_{k, k-1}^{\mathrm{s}}$为历元间卫星钟差增量；$\Delta I_{1, k, k-1}$为第一频点的历元间电离层延迟增量；$\Delta Z_{k, k-1}$为历元间对流层延迟增量；$\varepsilon_{\Delta P}$为历元间单差伪距噪声；$\varepsilon_{\Delta L}$为历元间单差相位噪声。
根据式(3)对i频点的历元间伪距增量$\Delta P_{i, k, k-1}$和j频点的历元间载波相位增量$\Delta L_{j, k, k-1}$作差，忽略观测噪声，可以得到：
$\Delta P_{i, k, k-1}=\Delta L_{j, k, k-1}+\frac{\mathit{λ}_i^2+\mathit{λ}_j^2}{\mathit{λ}_1^2} \cdot \Delta I_{1, k, k-1}$
式中，i和j表示任意频点，且i和j可以相等，即表示同一频点。利用式(2)和式(4)即可实现载波相位平滑伪距，根据$\Delta I_{1, k, k-1}$的获取方法可分为单频和双频载波相位平滑伪距。
1.2 自适应载波相位平滑伪距模型
1.2.1 自适应单频相位平滑伪距
使用单频相位平滑伪距方法的前提是历元间电离层延迟变化量较小[13]，实际应用中可以通过电离层模型值或SBAS电离层格网参数判断电离层变化并确定单频平滑的适用性[7-8]。在适用场景下，历元间电离层延迟约为0，即：
$\Delta I_{1, k, k-1} \approx 0$
将式(5)代入式(4)可得：
$\Delta P_{i, b, b-1} \approx \Delta L_{i, b, b-1}$
由于连续进行单频平滑会导致电离层误差累积发散，为提升单频平滑的适用性，通过设置合理的平滑时长(比如200 s)，利用双路滤波器切换来抑制电离层误差的影响[14]。将式(6)代入式(2)可得单频平滑模型：
$\left\{\begin{array}{l}\bar{P}_i^{(k)}=\frac{1}{k} P_i^{(k)}+\frac{k-1}{k}\left(\bar{P}_i^{(k-1)}+\Delta L_{j, k, k-1}\right) \\\bar{P}_i^{(1)}=P_i^{(1)}\end{array}\right.$
式中，i和j表示任意频点，只要能稳定跟踪频点j的载波，就可以计算得到载波相位历元间增量$\Delta L_{j, k, k-1}$，根据式(7)可以平滑所有频点的原始伪距$P_i^{(k)}$，得到平滑后的伪距$\bar{P}_i^{(k)}$。
传统单频伪距平滑方法是i频点的载波只平滑i频点的伪距，如果i频点的载波无效，则不平滑i频点的伪距，会导致i频点的伪距误差增大。而单频自适应平滑方法是当i频点的载波出现周跳或失锁时，可自适应地选择未发生周跳的j频点载波计算的$\Delta L_{j, k, k-1}$来平滑伪距，减小重置滤波器的概率，增加伪距平滑的连续性。
1.2.2 自适应双频相位平滑伪距
同一颗星的j和m频点的载波相位历元间增量分别为$\Delta L_{j, k, k-1}$和$\Delta L_{m, k, k-1}$，对二者作差，忽略观测噪声，根据式(3)可以得到历元间电离层延迟变化量为：
$\begin{gathered}\Delta I_{1, k, k-1}=\frac{\mathit{λ}_1^2}{\mathit{λ}_m^2-\mathit{λ}_j^2} \cdot \\\left(\Delta L_{j, k, k-1}-\Delta L_{m, k, k-1}\right), j \neq m\end{gathered}$
将式(8)代入式(4)，整理可得历元间的伪距变化量为：
$\begin{gathered}\Delta P_{i, k, k-1}=\frac{\mathit{λ}_i^2+\mathit{λ}_m^2}{\mathit{λ}_m^2-\mathit{λ}_j^2} \cdot \Delta L_{j, k, k-1}- \\\frac{\mathit{λ}_i^2+\mathit{λ}_j^2}{\mathit{λ}_m^2-\mathit{λ}_j^2} \cdot \Delta L_{m, k, k-1}, j \neq m\end{gathered}$
将式(9)代入式(2)，可得使用j、m双频载波平滑伪距表达式为：
$\left\{\begin{array}{l}\bar{P}_i^{(k)}=\frac{1}{k} P_i^{(k)}+\frac{k-1}{k}\left(\bar{P}_i^{(k-1)}+\right. \\\quad \frac{\mathit{λ}_i^2+\mathit{λ}_m^2}{\mathit{λ}_m^2-\mathit{λ}_j^2} \cdot \Delta L_{j, k, k-1}- \\\left.\quad \frac{\mathit{λ}_i^2+\mathit{λ}_j^2}{\mathit{λ}_m^2-\mathit{λ}_j^2} \cdot \Delta L_{m, k, k-1}\right), j \neq m \\\bar{P}_i^{(1)}=P_i^{(1)}\end{array}\right.$
式中，i为任意频点，j和m为不同频点。
根据式(10)可知，只要存在任意2个能稳定跟踪且无周跳的频点j和m，即可对所有频点的伪距进行平滑，当j或m失锁或周跳时，自适应选择其他能稳定跟踪的2个频点继续进行相位平滑伪距；当有效频点达到3个及以上时，优先选择跟踪时间较长、$\left|\mathit{λ}_m^2-\mathit{λ}_j^2\right|$较大的2个频点，以减少测量噪声。
1.2.3 自适应单双频相位平滑伪距
对于多频接收机，由于信号干扰或环境遮挡等原因，如果只能稳定跟踪一个频点的载波，而能跟踪多个频点的伪距，则启用§1.2.1中的单频自适应平滑方法；如果能跟踪2个及以上频点的载波，则启用§1.2.2中的双频自适应平滑方法。在复杂环境下，这种自适应平滑方法对提升平滑的连续性、降低伪距噪声、提升定位精度都有帮助。
2 实验与精度分析
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和芯星通UM980接收机支持GNSS多系统多频点信号跟踪和输出，使用该接收机分别采集1 Hz静态和动态数据，其中静态数据采集于北京有遮挡场景，采集时间为2024-03-09，时长约1 h；动态数据采集于深圳市区树荫和高楼遮挡场景，采集时间为2024-02-28，时长约1 h。通过实时动态差分定位(real-time kinematic，RTK)获得采集点的参考坐标。
对单频平滑、双频平滑和自适应平滑3种方案的平滑连续性、伪距噪声和定位精度进行对比分析，具体平滑方案见表 1。
2.1 平滑伪距连续性分析
采用表 1中3种方案对静态数据的伪距进行平滑，图 1为GPS伪随机噪声码(pseudo random noise code，PRN)01号卫星的平滑连续性结果，使用载波锁定时间表示载波跟踪连续性，锁定时间持续增长表示载波跟踪连续，锁定时间为0表示载波失锁。由图可知，L1和L2频点的载波跟踪不连续，失锁时刻载波相位无效，L5频点跟踪连续，载波相位一直有效，单频平滑方法在L1、L2频点信号失锁时无法继续平滑对应频点的伪距；双频平滑方法在L1和L2任一频点信号失锁时无法继续平滑L1和L2频点的伪距；自适应平滑方法只要有一个能稳定跟踪的频点就可以实现对所有频点伪距的连续平滑。
2.2 平滑伪距残差分析
为分析自适应平滑方法在抑制伪距噪声方面的性能，利用2块零基线板卡采集静态数据。在所采集的数据中，GPS L1和L2频点受干扰较为严重，信号失锁较为频繁，L5频点信号基本未发生失锁；BDS-2频点整体上跟踪较为连续，个别卫星B1I或B2I频点信号有失锁现象。采用表 1中3种平滑方案分别对2块板卡所采集数据的伪距进行平滑，然后通过双差方法得到不同平滑模式下的伪距噪声[15]。
以GPS PRN06号卫星为例，伪距噪声和信号跟踪情况如图 2所示。由图可知，L1和L2频点信号失锁时，单频平滑的L1、L2频点伪距噪声均增大，L5频点伪距噪声正常；双频平滑的L1、L2、L5频点伪距噪声均增大；自适应平滑的3个频点的伪距噪声均正常。
统计多颗GPS卫星的伪距噪声，结果如图 3和表 2(单位cm)所示。可以看出，单频平滑模式下L1、L2和L5频点的伪距噪声均值分别为6.9 cm、8.1 cm和3.2 cm，由于L1和L2频点载波失锁较为频繁，会影响伪距平滑连续性，因此L1和L2频点的伪距噪声较L5频点更大。双频平滑模式下L1、L2和L5频点的伪距噪声均值分别为6.6 cm、8.9 cm和5.7 cm，其中L1和L2频点的伪距噪声和单频平滑基本相当，但L5频点伪距噪声相比单频平滑增大，这是因为受L1频点载波失锁的影响，L1和L5频点多个历元无法进行双频平滑，影响L5频点平滑效果。自适应平滑模式下L1、L2和L5频点的伪距噪声均值分别为2.5 cm、4.0 cm和2.1 cm，相比单频平滑分别减小约62%、50%、33%，相比双频平滑分别减小约60%、54%、61%。可见自适应平滑方法能够在部分频点受干扰时维持平滑连续性，比单频和双频平滑更能提升伪距观测值精度。
计算多颗BDS-2卫星的伪距噪声，统计结果见表 3(单位cm)和图 4。可以看出，自适应平滑的B1I、B2I和B3I频点的伪距噪声均值分别为1.5 cm、1.3 cm和1.3 cm，相比单频平滑分别减小约64%、67%、51%，相比双频平滑分别减小约55%、59%、50%，即自适应平滑方法相比于单频和双频平滑方法能有效减小BDS-2各频点的伪距噪声。
分析其他GNSS系统频点的伪距噪声，均能得到类似的结论。
2.3 定位精度分析
单频伪距定位时，电离层误差采用Klobuchar模型修改，对流层误差采用Hopfield模型修正，卫星截止高度角设为8°，采用广播星历计算卫星位置和钟差，根据信噪比和卫星仰角计算伪距噪声，采用加权最小二乘算法解算位置，然后与位置基准作差，统计分析定位误差的RMS和STD。
2.3.1 静态数据定位结果分析
在街边树荫环境下，采集1 Hz静态数据，车来车往等外部动态环境会干扰GNSS信号，有时甚至会导致部分频点的载波失锁。
基于GPS L1和BDS B1I频点的伪距，分别使用原始伪距、单频平滑伪距、双频平滑伪距和自适应平滑伪距进行定位，不同伪距模式下的定位结果在E、N、U方向的误差如图 5所示。可以看出，原始伪距定位存在dm级毛刺点抖动现象，这是因为该时段内卫星信号受到干扰导致跟踪失锁，原始伪距误差较大；单频和双频平滑伪距相比原始伪距定位误差毛刺点有所减少，但由于信号失锁时无法平滑伪距，因此仍存在不同程度的定位误差抖动情况；自适应平滑方法能对伪距进行连续平滑，可有效减小伪距噪声，消除误差毛刺点。可见自适应平滑方法能够最大程度地使用载波平滑伪距，减少伪距误差，提升定位精度。
2.3.2 动态数据定位结果分析
对动态数据进行分析，定位误差见图 6和表 4(单位m)。可以看出，高程误差约为10 m，主要原因是该时段内电离层延迟较大，使用广播星历中的Klobuchar参数进行电离层延迟改正后仍存在m级系统性误差，而且存在楼宇遮挡等，多路径噪声也较大。
由表 4可知，自适应平滑伪距定位在E、N、U方向的RMS分别为3.34 m、3.58 m、8.80 m，较原始伪距定位分别改善约23%、-5%、23%，较单频平滑分别改善约17%、-10%、17%，较双频平滑分别改善约17%、-12%、17%。由于该组实验数据的定位误差主要由电离层及多路径等系统性误差引起，载波平滑伪距无法消除这些系统性误差，所以对RMS的改善效果有限。自适应平滑伪距定位在E、N、U方向的STD分别为0.53 m、0.16 m、1.23 m，相比原始伪距定位分别减小约57%、80%、59%，相比单频平滑分别减小约45%、76%、47%，相比双频平滑分别减小约41%、69%、49%。自适应平滑能够最大程度地平滑伪距，使伪距多路径误差和测量噪声最低，定位精度改善明显。
3 结语
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本文推导得到基于多频观测值的自适应载波相位平滑伪距算法，在GNSS载波稳定跟踪时，该方法的性能和传统方法一致；在部分频点受干扰等导致载波失锁时，本文算法能自适应地选择有效频点的载波，对所有频点的伪距进行平滑，最大程度地保证伪距平滑的连续性，有利于减小伪距误差，提升定位精度。
通过实际数据分析，自适应平滑的GPS L1/L2/L5频点伪距噪声相比传统单频平滑分别减小约62%、50%、33%，相比传统双频平滑分别减小约60%、54%、61%；自适应平滑的BDS-2 B1I/B2I/B3I频点伪距噪声相比传统单频平滑分别减小约64%、67%、51%，相比传统双频平滑分别减小约55%、59%、50%；其他GNSS频点也有类似的效果。在有树荫和高楼遮挡的路段进行动态数据分析，自适应平滑伪距定位的STD相比原始伪距定位在E、N、U方向分别减小约57%、80%、59%，相比传统单频平滑分别减小约45%、76%、47%，相比传统双频平滑分别减小约41%、69%、49%。

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doi: 10.14075/j.jgg.2024.07.346
  • 接收时间：2024-07-16
  • 首发时间：2026-10-29
  • 出版时间：2025-07-15
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