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Article(id=1251856526948909819, tenantId=1146029695717560320, journalId=1251234268282663017, issueId=1251856520619700745, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1003-3106.2025.11.008, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1756137600000, receivedDateStr=2025-08-26, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1776395454394, onlineDateStr=2026-04-17, pubDate=1762272000000, pubDateStr=2025-11-05, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1776395454394, onlineIssueDateStr=2026-04-17, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1776395454394, creator=13701087609, updateTime=1776395454394, updator=13701087609, issue=Issue{id=1251856520619700745, tenantId=1146029695717560320, journalId=1251234268282663017, year='2025', volume='55', issue='11', pageStart='2131', pageEnd='2324', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1776395452885, creator=13701087609, updateTime=1776395571911, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1251857019939013255, tenantId=1146029695717560320, journalId=1251234268282663017, issueId=1251856520619700745, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1251857019939013256, tenantId=1146029695717560320, journalId=1251234268282663017, issueId=1251856520619700745, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=2206, endPage=2217, ext={EN=ArticleExt(id=1251856528249144104, articleId=1251856526948909819, tenantId=1146029695717560320, journalId=1251234268282663017, language=EN, title=Ambiguity Resolution Method for BDS-3 Based on GIF Model, columnId=1251856525023719991, journalTitle=Radio Engineering, columnName=TT&C, Remote Sensing and Navigation & Positioning, runingTitle=null, highlight=null, articleAbstract=

The BeiDou-3 Global Navigation Satellite System ( BDS-3) can provide data of six frequencies at present, which provides more choices for Multi-frequency Carrier Ambiguity Resolution (MCAR). Focusing on BDS-3, the basic method of Geometry and Ionosphere Free (GIF) model in MCAR is comprehensively studied, including the application of three-frequency GIF model in ambiguity resolution. Based on the theory of three-frequency linear combinations, the basic mathematical model under three-frequency is given. The optimal frequency combination for ambiguity resolution using GIF model is discussed under the possibility of 20 combinations of any three of the six frequencies. Meanwhile, the optimal linear combination of each frequency combination is also systematically discussed. In addition, the high-quality linear combinations for single-epoch ambiguity resolution using GIF model are also analyzed. The experiment is carried out by using the real BDS-3 six-frequency data. Through theoretical analysis and practical demonstration, the results show that when using the GIF model for ambiguity resolution, the optimal frequency combination is (B1C, B3I, B2a). In this method, if the sum of the coefficients of two fixed Extra Wide Lane /Wide Lane (EWL/WL) combinations equals zero, the standard deviation of the ambiguity for the third Narrow Lane (NL) combination is theoretically dependent solely on the frequency characteristics. However, due to the influence of unmodeled errors, the actual results may deviate from theoretical expectations. The GIF model effectively eliminates ionospheric delay effects and avoids Geometry Base ( GB) errors, demonstrating significant advantages. The ambiguity resolution based on GIF model exhibits strong potential particularly in ionosphere-active environments and medium-to-long baseline scenarios.

, correspAuthors=Zhetao ZHANG, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, 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, authorCompany=null, fund=null, authors=null, authorsList=Fang WANG, Zhetao ZHANG), CN=ArticleExt(id=1251856545995243999, articleId=1251856526948909819, tenantId=1146029695717560320, journalId=1251234268282663017, language=CN, title=基于GIF模型的北斗三号模糊度固定方法, columnId=1251856525191492157, journalTitle=无线电工程, columnName=测控遥感与导航定位, runingTitle=null, highlight=null, articleAbstract=

北斗三号全球卫星导航系统（Global Beidou Navigation Satellite System，BDS-3）目前可提供6个频率的数据，为多频相位模糊度固定（Multi-frequency Carrier Ambiguity Resolution，MCAR）提供了更多选择。针对BDS-3，深入研究了MCAR中无几何无电离层（Geometry and Ionosphere Free，GIF）模型基本方法，主要包括三频GIF模型在模糊度固定中的应用。从三频线性组合理论出发，给出了三频下的基本数学模型。系统探讨了在六频中任选3个频率的20种组合可能下，GIF模型固定模糊度的最优频率组合以及各自的最优线性组合。此外，还分析了利用GIF模型单历元模糊度固定的高质量线性组合。利用实测BDS-3六频观测数据进行了试验，通过理论分析和实际论证，结果表明，采用GIF模型固定模糊度时，最优频率组合为（B1C，B3I，B2a）；在该方法中，若2个已固定的超宽巷/宽巷（Extra Wide Lane /Wide Lane，EWL/WL）组合的系数之和为0，则第3个窄巷（Narrow Lane，NL）组合模糊度的标准差理论上仅与频率相关。然而，受非模型化误差等影响，实际结果与理论可能存在偏差。GIF模型能有效消除电离层延迟影响，且不受几何相关（Geometry Based，GB）误差影响，基于GIF模型的模糊度固定在电离层活跃和中长基线场景下具有良好的潜力。

, correspAuthors=章浙涛, authorNote=null, correspAuthorsNote=
章浙涛 男，（1988—），博士，副教授。主要研究方向：卫星精密定位与多源融合导航。
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王芳 女，（2001—），硕士研究生。主要研究方向：GNSS导航定位。

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王芳 女，（2001—），硕士研究生。主要研究方向：GNSS导航定位。

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王芳 女，（2001—），硕士研究生。主要研究方向：GNSS导航定位。

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city=null, postcode=null, companyName=null, departmentName=null, remark=School of Earth Sciences and Engineering, Hohai University, Nanjing 211100, China), AuthorCompanyExt(id=1251856546225930728, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, companyId=1251856546209153510, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=河海大学 地球科学与工程学院,江苏 南京 211100)])], figs=[ArticleFig(id=1251856547744268839, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Fig.1, caption=All three-frequency combinations in six frequencies, figureFileSmall=g9gIbV0EvG0jnDh+Vw1gSg==, figureFileBig=hzIUUFbXZIDmQMbV53ZEpA==, tableContent=null), ArticleFig(id=1251856547802989099, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=CN, label=图1, caption=六频中的所有三频组合, figureFileSmall=g9gIbV0EvG0jnDh+Vw1gSg==, figureFileBig=hzIUUFbXZIDmQMbV53ZEpA==, tableContent=null), ArticleFig(id=1251856549463933486, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Fig.2, caption=DD ionospheric delays for two baselines, figureFileSmall=WPaoVrLP55R4uV7ek8u2Nw==, figureFileBig=Ua0DSY5xll0nT11oFYdZhQ==, tableContent=null), ArticleFig(id=1251856549526848048, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=CN, label=图2, caption=2条基线双差电离层延迟量, figureFileSmall=WPaoVrLP55R4uV7ek8u2Nw==, figureFileBig=Ua0DSY5xll0nT11oFYdZhQ==, tableContent=null), ArticleFig(id=1251856549602345522, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Fig.3, caption=The errors of ambiguity resolution based on（B1C, B3I, B2a）on the day of year 124 in 2025, figureFileSmall=chzVSZNGyJXQskxqxxx1gg==, figureFileBig=pbmHJ/gUo9uNm7PaTy8anA==, tableContent=null), 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tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Fig.5, caption=The errors of NL ambiguity resolution with pots-leij in LC1 and LC2 on the day of year 225 in 2024, figureFileSmall=6NsJNyGAPviMCBaQLyGbSw==, figureFileBig=MjN7pNOpfcZeT55xHS0S3g==, tableContent=null), ArticleFig(id=1251856550042747451, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=CN, label=图5, caption=年积日225（2024）pots-leij组合1和组合2 NL模糊度误差, figureFileSmall=6NsJNyGAPviMCBaQLyGbSw==, figureFileBig=MjN7pNOpfcZeT55xHS0S3g==, tableContent=null), ArticleFig(id=1251856550155993662, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Fig.6, caption=The errors of NL ambiguity resolution with pots-leij in LC1 and LC2 on the day of year 124 in 2025, figureFileSmall=PUO5miswCL5o4jRv3W0Lzg==, figureFileBig=O+vvycFKU9Jo4r+IYiOASw==, tableContent=null), 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ArticleFig(id=1251856550575424068, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Tab.1, caption=

Characteristics of BDS-3 signals

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信号频率/MHz波长/m
B1C1575.4200.1903
B1I1561.0980.1920
B3I1268.5200.2363
B2b1207.1400.2483
B2a+b1191.7950.2515
B2a1176.4500.2548
), ArticleFig(id=1251856550663504454, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=CN, label=表1, caption=

BDS-3卫星信号频率

, figureFileSmall=null, figureFileBig=null, tableContent=
信号频率/MHz波长/m
B1C1575.4200.1903
B1I1561.0980.1920
B3I1268.5200.2363
B2b1207.1400.2483
B2a+b1191.7950.2515
B2a1176.4500.2548
), ArticleFig(id=1251856550772556360, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Tab.2, caption=

Optimal signals and related information in allthree-frequency scenarios of BDS-3

, figureFileSmall=null, figureFileBig=null, tableContent=
频点系数波长/mσTNLΦ/cycle
B1CB3IB2b1-544.8840.340
01-14.8840.281
B1CB3IB2a+b1-433.9070.222
01-13.9070.429
B1CB3IB2a1-322.4420.310
1-439.7680.338
B1IB3IB2b1-546.3710.292
1-432.7650.320
B1IB3IB2a+b1-434.8040.222
01-13.9070.421
B1IB3IB2a1-322.7650.266
1-4318.3160.372
B3IB2bB2a+b01-119.5370.081
1-329.7680.183
B3IB2bB2a01-19.7680.131
1-219.7680.142
B3IB2a+bB2a01-119.5370.082
1-326.5120.218
B2bB2a+bB2a1-1019.5370.078
01-119.5370.078
B1CB1IB3I1-1020.9320.089
B1CB1IB2b1-1020.9320.089
B1CB1IB2a+b1-1020.9320.089
B1CB1IB2a1-1020.9320.089
B1CB2bB2a+b01-119.5370.106
B1CB2bB2a01-19.7680.189
B1CB2a+bB2a01-119.5370.107
B1IB2bB2a+b01-119.5370.104
B1IB2bB2a01-19.7680.186
B1IB2a+bB2a01-119.5370.106
), ArticleFig(id=1251856550869025354, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=CN, label=表2, caption=

BDS-3所有三频场景下最优信号及相关信息

, figureFileSmall=null, figureFileBig=null, tableContent=
频点系数波长/mσTNLΦ/cycle
B1CB3IB2b1-544.8840.340
01-14.8840.281
B1CB3IB2a+b1-433.9070.222
01-13.9070.429
B1CB3IB2a1-322.4420.310
1-439.7680.338
B1IB3IB2b1-546.3710.292
1-432.7650.320
B1IB3IB2a+b1-434.8040.222
01-13.9070.421
B1IB3IB2a1-322.7650.266
1-4318.3160.372
B3IB2bB2a+b01-119.5370.081
1-329.7680.183
B3IB2bB2a01-19.7680.131
1-219.7680.142
B3IB2a+bB2a01-119.5370.082
1-326.5120.218
B2bB2a+bB2a1-1019.5370.078
01-119.5370.078
B1CB1IB3I1-1020.9320.089
B1CB1IB2b1-1020.9320.089
B1CB1IB2a+b1-1020.9320.089
B1CB1IB2a1-1020.9320.089
B1CB2bB2a+b01-119.5370.106
B1CB2bB2a01-19.7680.189
B1CB2a+bB2a01-119.5370.107
B1IB2bB2a+b01-119.5370.104
B1IB2bB2a01-19.7680.186
B1IB2a+bB2a01-119.5370.106
), ArticleFig(id=1251856550965494347, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Tab.3, caption=

Error budgets under different baseline lengths

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误差项中长基线长基线
等价伪距观测值噪声
等价相位观测值噪声
一阶电离层延迟
二阶电离层延迟
对流层延迟
轨道误差
), ArticleFig(id=1251856551040991821, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=CN, label=表3, caption=

不同基线类型下的各误差预算

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误差项中长基线长基线
等价伪距观测值噪声
等价相位观测值噪声
一阶电离层延迟
二阶电离层延迟
对流层延迟
轨道误差
), ArticleFig(id=1251856551108100686, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Tab.4, caption=

Optimal phase combinations underdifferent baseline lengths

, figureFileSmall=null, figureFileBig=null, tableContent=
基线类型序号ijkλ（i,j,k）/mβ（i,j,k）μ（i,j,k）σTNLΦ/cycle
中长基线A1-322.442-0.61038.6400.187
B01-13.256-1.66318.7910.213
C1-439.7682.549207.8350.237
D1-211.395-1.06114.9410.323
长基线a1-322.442-0.61038.6400.310
b1-439.7682.549207.8350.338
c2-751.9540.02272.3850.388
d01-13.256-1.66318.7910.519
), ArticleFig(id=1251856551187792464, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=CN, label=表4, caption=

不同长度基线下的最优相位组合

, figureFileSmall=null, figureFileBig=null, tableContent=
基线类型序号ijkλ（i,j,k）/mβ（i,j,k）μ（i,j,k）σTNLΦ/cycle
中长基线A1-322.442-0.61038.6400.187
B01-13.256-1.66318.7910.213
C1-439.7682.549207.8350.237
D1-211.395-1.06114.9410.323
长基线a1-322.442-0.61038.6400.310
b1-439.7682.549207.8350.338
c2-751.9540.02272.3850.388
d01-13.256-1.66318.7910.519
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The value of κ in all three-frequencyscenarios of BDS-3

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频点κ
B1CB1IB3I1467.354
B1CB1IB2b1191.108
B1CB1IB2a+b1136.159
B1CB1IB2a1085.525
B1CB3IB2b243.651
B1CB3IB2a+b191.227
B1CB3IB2a156.678
B1CB2bB2a+b816.983
B1CB2bB2a398.134
B1CB2a+bB2a770.550
B1IB3IB2b253.665
B1IB3IB2a+b199.0695
B1IB3IB2a163.1094
B1IB2bB2a+b845.034
B1IB2bB2a411.580
B1IB2a+bB2a795.740
B3IB2bB2a+b4079.291
B3IB2bB2a1950.252
B3IB2a+bB2a3240.920
B2bB2a+bB2a14775.288
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BDS-3所有三频场景下κ值

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频点κ
B1CB1IB3I1467.354
B1CB1IB2b1191.108
B1CB1IB2a+b1136.159
B1CB1IB2a1085.525
B1CB3IB2b243.651
B1CB3IB2a+b191.227
B1CB3IB2a156.678
B1CB2bB2a+b816.983
B1CB2bB2a398.134
B1CB2a+bB2a770.550
B1IB3IB2b253.665
B1IB3IB2a+b199.0695
B1IB3IB2a163.1094
B1IB2bB2a+b845.034
B1IB2bB2a411.580
B1IB2a+bB2a795.740
B3IB2bB2a+b4079.291
B3IB2bB2a1950.252
B3IB2a+bB2a3240.920
B2bB2a+bB2a14775.288
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The information of baselines

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基线长度/km年积日观测时间/h采样间隔/s
pots-leij123.64124（2025）2430
225（2024）2430
pots-warn209.39124（2025）2430
225（2024）2430
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基线信息

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基线长度/km年积日观测时间/h采样间隔/s
pots-leij123.64124（2025）2430
225（2024）2430
pots-warn209.39124（2025）2430
225（2024）2430
), ArticleFig(id=1251856551598834266, tenantId=1146029695717560320, journalId=1251234268282663017, articleId=1251856526948909819, language=EN, label=Tab.7, caption=

The EWL/WL combinations used in the experiment

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基线序号被选组合a1a2
pots-leij1ABE3.282586-2.282586
2CDE0.7907540.209246
pots-warn3abe0.2391380.760862
4cde2.051616-1.051616
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实验所用EWL/WL组合

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基线序号被选组合a1a2
pots-leij1ABE3.282586-2.282586
2CDE0.7907540.209246
pots-warn3abe0.2391380.760862
4cde2.051616-1.051616
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基于GIF模型的北斗三号模糊度固定方法
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王芳 , 章浙涛 *
无线电工程 | 测控遥感与导航定位 2025,55(11): 2206-2217
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无线电工程 | 测控遥感与导航定位 2025, 55(11): 2206-2217
基于GIF模型的北斗三号模糊度固定方法
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王芳, 章浙涛*
作者信息
  • 河海大学 地球科学与工程学院,江苏 南京 211100

    • 王芳 女，（2001—），硕士研究生。主要研究方向：GNSS导航定位。

通讯作者:

章浙涛 男，（1988—），博士，副教授。主要研究方向：卫星精密定位与多源融合导航。
Ambiguity Resolution Method for BDS-3 Based on GIF Model
Fang WANG, Zhetao ZHANG*
Affiliations
  • School of Earth Sciences and Engineering, Hohai University, Nanjing 211100, China
出版时间: 2025-11-05 doi: 10.3969/j.issn.1003-3106.2025.11.008
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摘要
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北斗三号全球卫星导航系统（Global Beidou Navigation Satellite System，BDS-3）目前可提供6个频率的数据，为多频相位模糊度固定（Multi-frequency Carrier Ambiguity Resolution，MCAR）提供了更多选择。针对BDS-3，深入研究了MCAR中无几何无电离层（Geometry and Ionosphere Free，GIF）模型基本方法，主要包括三频GIF模型在模糊度固定中的应用。从三频线性组合理论出发，给出了三频下的基本数学模型。系统探讨了在六频中任选3个频率的20种组合可能下，GIF模型固定模糊度的最优频率组合以及各自的最优线性组合。此外，还分析了利用GIF模型单历元模糊度固定的高质量线性组合。利用实测BDS-3六频观测数据进行了试验，通过理论分析和实际论证，结果表明，采用GIF模型固定模糊度时，最优频率组合为（B1C，B3I，B2a）；在该方法中，若2个已固定的超宽巷/宽巷（Extra Wide Lane /Wide Lane，EWL/WL）组合的系数之和为0，则第3个窄巷（Narrow Lane，NL）组合模糊度的标准差理论上仅与频率相关。然而，受非模型化误差等影响，实际结果与理论可能存在偏差。GIF模型能有效消除电离层延迟影响，且不受几何相关（Geometry Based，GB）误差影响，基于GIF模型的模糊度固定在电离层活跃和中长基线场景下具有良好的潜力。

北斗三号全球卫星导航系统  /  多频  /  模糊度固定  /  无几何无电离层模型  /  电离层延迟

The BeiDou-3 Global Navigation Satellite System ( BDS-3) can provide data of six frequencies at present, which provides more choices for Multi-frequency Carrier Ambiguity Resolution (MCAR). Focusing on BDS-3, the basic method of Geometry and Ionosphere Free (GIF) model in MCAR is comprehensively studied, including the application of three-frequency GIF model in ambiguity resolution. Based on the theory of three-frequency linear combinations, the basic mathematical model under three-frequency is given. The optimal frequency combination for ambiguity resolution using GIF model is discussed under the possibility of 20 combinations of any three of the six frequencies. Meanwhile, the optimal linear combination of each frequency combination is also systematically discussed. In addition, the high-quality linear combinations for single-epoch ambiguity resolution using GIF model are also analyzed. The experiment is carried out by using the real BDS-3 six-frequency data. Through theoretical analysis and practical demonstration, the results show that when using the GIF model for ambiguity resolution, the optimal frequency combination is (B1C, B3I, B2a). In this method, if the sum of the coefficients of two fixed Extra Wide Lane /Wide Lane (EWL/WL) combinations equals zero, the standard deviation of the ambiguity for the third Narrow Lane (NL) combination is theoretically dependent solely on the frequency characteristics. However, due to the influence of unmodeled errors, the actual results may deviate from theoretical expectations. The GIF model effectively eliminates ionospheric delay effects and avoids Geometry Base ( GB) errors, demonstrating significant advantages. The ambiguity resolution based on GIF model exhibits strong potential particularly in ionosphere-active environments and medium-to-long baseline scenarios.

BDS-3  /  multi-frequency  /  ambiguity resolution  /  GIF model  /  ionospheric delay
王芳, 章浙涛. 基于GIF模型的北斗三号模糊度固定方法. 无线电工程, 2025 , 55 (11) : 2206 -2217 . DOI: 10.3969/j.issn.1003-3106.2025.11.008
Fang WANG, Zhetao ZHANG. Ambiguity Resolution Method for BDS-3 Based on GIF Model[J]. Radio Engineering, 2025 , 55 (11) : 2206 -2217 . DOI: 10.3969/j.issn.1003-3106.2025.11.008
正文
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0　引言
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北斗卫星导航系统（Beidou Navigation Satellite System,BDS）是中国着眼于国家安全和经济社会发展需要,自主建设运行的全球卫星导航系统,是为全球用户提供全天候、全天时、高精度的定位、导航和授时服务的国家重要时空基础设施[1-2]。目前,随着BDS-3卫星全球组网成功,已公开提供6个频率的数据。如何充分利用这些多频信号资源,进而提升导航定位的精度和效率是研究热点之一。
整周模糊度固定是实现高精度全球卫星导航系统（Global Navigation Satellite System,GNSS）定位的重要基础[3-5]。然而在实际应用中,非模型化误差[6-7]（如对流层延迟、电离层延迟以及多路径效应等）会导致模糊度参数产生偏差,进而降低浮点解的精度。特别是在大范围测量或复杂环境下的定位场景中[8-9],如何实现模糊度的快速解算乃至单历元固定,成为当前GNSS精密定位技术面临的关键挑战。为此,诸多学者展开了研究。Forssell等[10]首次提出了三频相位模糊度解算（Three-Frequency Carrier Ambiguity Resolution, TCAR）方法。Enge等[11]提出了级联模糊度解算（Cascading Integer Resolution,CIR）方法,其本质与TCAR类似,都是基于无几何（Geometry-Free,GF）模型,按照模糊度容易固定的EWL组合到模糊度最难固定的NL组合的顺序,通过四舍五入的方法逐级固定相应组合的模糊度。由于伽利略和北斗提供多频观测值,章浙涛等[12]系统研究了MCAR的基本理论和方法,包括TCAR、四频相位模糊度固定（Four-Frequency Carrier Ambiguity Resolution,FCAR）和五频相位模糊度固定（Five-Frequency Carrier Ambiguity Resolution,FiCAR）,结果表明MCAR能有效进行单历元模糊度固定,频率数增加可显著提高模糊度固定成功率。之后,相关研究进一步拓展到了GB模型。Feng[13]对三频GB模型的最优电离层削弱（Ionosphere Reduced,IR）信号进行了研究,基本思想是基于线性组合的总噪声水平（Total Noise Level,TNL）确定3个最优的线性组合来实现更可靠的模糊度解算。
然而,上述方法本质上采用的是IR模型,因此在电离层活跃和长基线等场景下,其性能会受到影响。基于此,文献[14-15]首次采用2个模糊度已固定的EWL观测值与一个模糊度待固定的NL观测值组成同时满足GF和IF的GIF线性组合来实现不受基线距离限制的模糊度解算。基于参考站间基线已知的特点,高旺等[16]提出了一种分步解算的IF方法,实验结果表明所提方法能够有效消除长基线情况下电离层延迟的影响,同时受观测噪声的影响较小。此后,高扬骏等[17]提出一种顾及电离层延迟影响并具有良好自适应抗差特性的改进TCAR算法,提高NL模糊度固定正确率。针对BDS-3三频观测值,艾力·库尔班等[18]采用GF模型和GIF模型分别对实测BDS-3卫星数据进行单历元TCAR解算,证明GIF模型在中长基线解算时相比于GF模型具有更好的模糊度固定能力。然而,这些研究并未从实际角度深入论证GIF方法在中长基线乃至长基线场景下的有效性,并且缺乏对北斗全频率信号优势的挖掘。
基于此,本文首先针对BDS-3全频率观测值,深入探讨了不同频率组合下采用GIF方法的效果；其次通过中长基线和长基线数据,从理论和实际2个角度详细分析了GIF方法在不同条件下模糊度固定的性能,论证了GIF方法进行模糊度固定的有效性。
1　BDS-3三频线性组合理论
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2020年7月31日, BDS-3正式开通[19]。现BDS-3可播发B1I、B3I、B1C、B2a、B2b及B2a+b信号,为多频模糊度固定提供了更多优势。BDS-3的6种信号具体信息如表1所示。鉴于目前市面上大多数多频接收机主要支持接收3个频率的信号,因此本文重点探讨三频线性组合。
设3个频率分别为f1、f2、f3,且满足f1＞f2＞f3。则三频观测值线性组合的频率、波长和双差（Double Differenced,DD）整周模糊度分别为：
式中：i、j、k表示任意整数,c0表示真空中的光速,▽ΔN1,▽ΔN2,▽ΔN3分别表示3个频率的双差整周模糊度。
相应的三频双差伪距和载波相位观测值线性组合可表示为：
式中：▽ΔP1、▽ΔP2、▽ΔP3分别表示频率f1、f2和f3对应的双差伪距观测值,▽ΔΦ1,▽ΔΦ2,▽ΔΦ3分别表示频率f1、f2、f3对应的双差载波相位观测值。将式（4）和式（5）展开得双差伪距和载波相位组合观测值（单位m）可表示为：
式中：▽Δρ表示双差卫地距,▽Δδorb表示双差轨道误差,▽Δδtro表示双差对流层延迟,▽Δδion1表示频率f1对应的双差载波相位一阶电离层延迟,▽Δδion2表示频率f1对应的双差载波相位二阶电离层延迟, 和分别表示▽ΔP（i,j,k）和▽ΔΦ（i,j,k）的噪声,β（i,j,k）和θ（i,j,k）分别表示一阶和二阶电离层延迟对应的尺度因子,即：
不失一般性,将各频率的噪声视为相等且独立,即满足,,其中ε▽ΔP和ε▽ΔΦ分别表示等价双差伪距和载波相位观测值噪声。相应的三频双差伪距和载波相位线性组合观测值噪声精度满足：
式中：μ（i,j,k）表示噪声因子。
需要注意的是,当不同频率之间的伪距观测值精度差距很大且不可忽略时,此时不可认为,而是需要根据实际,为各个频率的观测值设定合适的权重。
2　GIF模型模糊度固定方法
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目前,在MCAR中固定模糊度的方法主要包括GB模型、GF模型和GIF模型3种。其中,GB模型是通过对基线三维坐标分量进行参数化的模型,其为非线性模型但不容易秩亏[20-21]；GF模型是以卫地距的形式进行参数化的模型,其为线性模型但通常秩亏[12]；GIF模型是在构造GF模型时,还考虑了IF的条件[15,22]。
2.1　高质量三频线性组合
为实现模糊度快速固定,首先需要选择用于固定模糊度的虚拟组合。在TCAR中,相应地需要选择3个独立的虚拟组合。所谓独立,是指这3个虚拟组合的系数所构成的矩阵,其秩应为满秩,即为3。目的是在固定好虚拟组合观测值的模糊度后,能够通过一定的线性关系,求得原始频率的整周模糊度。显然,若没有任何限制,则有无穷多种选择。
事实上,适合用于模糊度固定的虚拟组合应满足一定的条件,如：较长的波长、较小的电离层延迟尺度因子以及较小的观测值噪声因子等。为探索BDS-3播发的6个频率中,高质量三频线性组合,本文将同时满足系数在[-10,10],λ≥0.75 m,|β（i,j,k）|＜1.5以及|μ（i,j,k）|＜200的作为高质量信号[12]。其中在BDS-3播发的6种频率中,任意3种频率组合得到的所有排列组合如图1所示,其中红色矩形表示选中的频点,蓝色矩形则表示未选中的频点,一共得到20种组合。
TNL是评估GNSS多频模糊度固定中整体噪声影响的一项重要指标,其数值越小,表明观测值受各类误差干扰越小,质量越高,越有利于模糊度的精确固定。在高精度定位应用中,即使采用高质量信号,有时仍难以实现最佳的模糊度固定效果,主要原因在于虚拟组合观测值放大了原始观测值中各误差项的影响,总误差相对于自身波长并非最小。为此, Feng[13]和Zhang等[20]以及其他学者引入了TNL的概念。通过综合考虑轨道误差,电离层延迟、对流层延迟和观测噪声等的影响,根据式（6）和式（7）可以分别得到基于伪距（单位m）和载波相位观测值（单位cycle）的TNL：
式中：σTNLP和σTNLΦ分别表示伪距和载波相位观测值的TNL,表示轨道误差精度,表示对流层延迟精度；表示一阶电离层延迟精度, 表示二阶电离层延迟精度；此时,用于固定模糊度的最优组合应满足σTNLP=min和σTNLΦ=min的条件。
通过式（13）和式（14）计算得到BDS-3所有三频场景下最优信号及相关信息,如表2所示。可以看到,不同频率组成的三频组合的最优组合不尽相同。其中频率组合为（B1C, B1I, B3I）,（B1C, B1I, B2b）、（B1C, B1I, B2a+b）和（B1C, B1I,B2a）的最优组合一致,均是由第3种频率的系数取0,第一种频率的系数取1,而第二种频率的系数取-1得到；其组合波长达到了20.932 m,而其σTNLΦ仅有0.089周,利于模糊度固定。频率组合（B2b,B2a+b,B2a）的最优组合（1,-1,0）波长为19.537 m,σTNLΦ仅0.078周,其作为模糊度固定的组合潜能巨大。
由于GIF模型适用于长基线场景,因此讨论中长基线（100 km＜l＜200 km）和长基线（l≥200 km）场景的模糊度固定问题,此时误差预算设置如表3所示[15]。
由于在长基线中EWL/WL信号采用GF方法具有较好的固定率且易于实施[20],因此本文采用GF方法固定EWL/WL信号的模糊度。
根据表3给出的各误差预算,基于频率组合（B1C,B3I,B2a）可得到不同长度基线下的最优相位组合如表4所示。类似地,也可以得到不同长度基线下的最优伪距组合。其中以频率组合（B1C,B3I, B2a）为例,中长基线下的最优伪距组合为（10,9, 8）；长基线下的最优伪距组合为（10,4,-1）。需要特别注意的是,若不同频率伪距观测值的精度差异显著且不可忽略,将导致最优伪距组合的选择发生变化,因此在实际应用中必须考虑这一影响因素。
2.2　GIF模型
为了清晰地看到电离层延迟对模糊度固定的影响,可以利用模糊度已固定的EWL和WL信号求出双差电离层延迟量。由GF模型固定完模糊度的EWL/WL组合观测值可表示为：
式中：和分别表示已经固定好模糊度的双差载波相位虚拟观测值,下标LCi（i=1,2, 3）表示线性组合i。通过式（15）和式（16）相减得到双差电离层延迟量：
GIF模型如下所示：
式中：a1和a2分别表示其前相应的系数,▽ΔΦLC3表示模糊度待固定的双差载波相位虚拟观测值, ▽ΔNLC3、λLC3和分别表示组合LC3的双差模糊度、波长和噪声。
为了满足GF的条件,要求：
同时为了满足IF的条件,须满足：
式中：βi（i=1,2,3）表示组合i的一阶电离层延迟对应的尺度因子。可以发现,在构造线性组合时, GIF模型不仅考虑到了几何无关,而且考虑到了IF的条件,最大程度地减少各种误差的影响,以达到获得最佳模糊度固定效果的目的。
若采用GIF方法固定模糊度,当模糊度已固定好的2个EWL/WL虚拟观测值各自的系数之和为0,则第3个NL虚拟观测值的模糊度标准差与其系数无关,而仅与所采用的频率相关[15,22],其三频模糊度估计的标准差（单位cycle）为：
将前面讨论的20种三频组合代入式（22）计算得到κ值,如表5所示。可以看到,不同频率组合的κ值大小差异明显,频率组合（B1C, B3I, B2a）的κ值最小,为156.678；频率组合（B2b,B2a+b,B2a）的κ值最大,为14775.288。因此根据实际需要,在可获取的频率范围内,有必要选择合适的频率组合。
由于总噪声较大,因此可通过平均取整确定NL组合的整周模糊度：
式中：t表示历元,n表示总历元数。事实上,由于GIF模型在构建组合时,已消除了一阶电离层延迟和GB误差影响,因此由GIF模型求解得到的模糊度系统误差较小,但由于在构建GIF模型时,噪声会急剧放大,理论上也可通过滤波的方法确定其模糊度固定解[23-25]。
3　实验与结果分析
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3.1　实验数据
为验证GIF方法固定模糊度的优势,选取potsleij这条长度为123.64 km的中长基线和pots-warn这条长度为209.39 km的长基线作为实验基线。为了比较不同电离层条件下,GIF方法的有效性,实验选取了2024年的年积日225和2025年的年积日124这2 d的数据。地磁报告显示,2024年的年积日225这1 d的全球地磁活动指数（Kp）均超过5,其中在12：00—15：00的Kp甚至达到8,电离层较为活跃；2025年的年积日124这1 d的Kp均不超过5,属于电离层不活跃时段。
同时计算了这2 d的pots-leij和pots-warn这2条基线的双差电离层延迟量,如图2所示。每一种颜色代表一对卫星对。时间系统采用的是全球定位系统时（Global Positioning System Time,GPST）。可以看到2024年的年积日225的电离层波动较2025年的年积日124更剧烈,符合实际情况。
选取pots-leij这条长度为123.64 km的中长基线和pots-warn这条长度为209.39 km的长基线在2024年的年积日225和2025年的年积日124这2 d的数据进行单历元模糊度固定实验,基线信息如表6所示。
根据前面的计算和分析,可发现频率组合为（B1C,B3I,B2a）的κ值最小,为156.678；频率组合（B2b,B2a+b,B2a）的κ值最大,为14775.288。因此本文首先针对这2种频率组合选用pots-warn这条基线进行实验,其中对于频率组合（B1C, B3I, B2a）,依据前面的计算,其实验所用的伪距组合为（10,4,-1）,EWL组合1（1,-3,2）,EWL组合2（1,-4,3）和NL组合（0,0,1）；对于频率组合（B2b,B2a +b,B2a）,依据前面的计算,其实验所用的伪距组合为（10,10,9）,EWL组合1（1,-1,0）,EWL组合2（0,1,-1）和NL组合（0,0,1）。
为了进一步分析GIF模型的有效性,根据前面的分析,采用表7所列组合进行下一步实验。根据前面的计算,其中pots-leij基线实验所用的最优伪距组合为（10,9,8）,pots-warn基线实验所用的最优伪距组合为（10,4,-1）。其中组合E：（0,0,1）,组合e：（0,0,1）。
3.2　实验分析
图3是2025年的年积日为124这天采用频率组合（B1C, B3I, B2a）进行实验得到的模糊度误差图,其中EWL组合1和EWL组合2采用GF方法固定,对于第3个最难固定的NL组合采用GIF方法进行固定。由图3可以看到,EWL组合1和EWL组合2的误差集中分布在-0.5～0.5周,与表2所述一致,保证了单历元模糊度固定的有效性。NL组合模糊度误差集中分布在-7.5～7.5周。
图4是年积日同样为2025年的年积日124这天采用频率组合（B2b,B2a+b,B2a）进行实验得到的模糊度误差图,其中EWL组合1和EWL组合2采用GF方法固定,对于第3个最难固定的NL组合采用GIF方法进行固定。
由图4可以看到,EWL组合1和EWL组合2的误差全部集中在-0.05～0.05周,与表2所得结论一致,可以较好地满足单历元模糊度固定需求。与此同时,NL组合的模糊度误差主要集中在-75.00～75.00周。与图3对比可以清晰地看到,二者在固定NL组合模糊度上,图3更有优势。同时再次证明了表5结论的正确性。因此,采用GIF方法固定NL组合模糊度,应选择频率组合（B1C, B3I,B2a）。
根据图2可知,2024年的年积日225为电离层活跃时间段,2025年的年积日124为电离层不活跃时间段。因此选择这2 d对pots-leij这条基线利用表7所列的组合1和组合2进行了模糊度固定。经过计算发现,不同卫星对解算结果大致类似,因此选择其中具有代表性的3颗卫星C26、C40和C50进行分析,C42、C25和C21分别是其在各自时间段内的参考卫星,结果如图5和图6所示。从图5和图6可以看到,选择不同EWL/WL组合固定模糊度,对于NL组合（0,0,1）固定的效果基本一致。在不同电离层条件下,可以看到二者固定效果也大体一致,说明GIF方法有效消除了电离层延迟的影响。一方面,从理论上,在构建GIF组合时,通过特定组合方式消除了GB项和电离层延迟项,从原理上减少了电离层延迟对模糊度解算的影响。其次,在构建组合观测值时,考虑到了电离层尺度因子较小这一条件,使得组合后的信号对电离层变化不敏感,即使在电离层活跃期,也能保证模糊度解算的相对稳定性,从而有助于固定模糊度。另一方面,从实际上,在电离层活跃和不活跃2种情况下,同一条基线的NL组合模糊度误差基本保持一致,由于存在非模型化误差等的影响,实际与理论存在一些细小的差异。由此证明,在电离层活跃期间,GIF方法固定模糊度具有良好潜力。
为进一步说明GIF方法的优势,分析了在不同基线长度下GIF方法的性能。对2025年的年积日124的基线pots-leij和基线pots-warn进行了实验,结果如图7所示。同样,选择了3对具有代表性的卫星对进行分析,为便于说明,相同卫星对的颜色保持一致,从上到下依次是基线pots-leij和基线pots-warn的卫星对结果。从图7可以看到,不同基线长度下,其NL组合模糊度都在-10～10周波动,其中C40-C23和C50-C42卫星对的NL组合模糊度误差主要集中在-5～5周之间波动。但无论基线长度如何,其NL组合模糊度固定效果大体一致,由于非模型化误差等的存在,但不完全相同。这也再次证明了基于GIF方法固定模糊度与基线长度无关,因此在长基线场景下,GIF模型具有较好的应用前景。
4　结束语
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本文基于GIF模型的模糊度解算问题,深入研究了基于GIF模型的BDS-3模糊度固定方法。以BDS-3为例,探讨了全频率下,利用三频观测值构造线性组合进行模糊度固定效果的优劣,以及不同电离层条件和不同基线长度条件下使用GIF模型固定模糊度的性能。通过理论分析和实际论证,主要结论如下：
① 通过对6个频率任意选择3个频率进行组合,共有20种频率组合。各种频率组合下的最优EWL/WL组合不尽相同,其中波长最长可达到20.932 m,σTNLΦ最小可达到0.078周。通常情况下,频率组合（B1C,B3I,B2a）用作GIF模型固定模糊度效果最佳。
② 在使用GIF方法固定模糊度时,若模糊度已固定的2个EWL/WL虚拟观测值各自系数之和为0,则第3个NL虚拟观测值模糊度标准差与其系数无关,而仅与所采用的频率相关。然而受非模型化误差等的影响,实际结果与理论预期会存在一些差异,并不完全一致。
③ GIF模型有效消除了电离层延迟的影响,因此在电离层活跃的情况下,使用GIF模型固定NL组合模糊度具有一定优势；同时,GIF模型不受基线距离的影响,消除了GB的误差项,因此特别适合用于中长基线场景下固定模糊度。
综上,在中长基线场景下,GIF模型可以很好地解决模糊度固定问题,表现出优越的性能。未来可考虑将GIF模型应用于快速精密定位中,尤其是在大尺度条件及复杂环境下具有较好的应用前景。
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2025年第55卷第11期
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doi: 10.3969/j.issn.1003-3106.2025.11.008
  • 接收时间：2025-08-26
  • 首发时间：2026-04-17
  • 出版时间：2025-11-05
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  • 收稿日期：2025-08-26
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    河海大学 地球科学与工程学院,江苏 南京 211100

通讯作者:

章浙涛 男，（1988—），博士，副教授。主要研究方向：卫星精密定位与多源融合导航。
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2种不同金属材料的力学参数
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多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
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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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