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Article(id=1322940875155845926, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322940829001724376, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2024.06.286, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1718553600000, receivedDateStr=2024-06-17, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793343283690, onlineDateStr=2026-10-30, pubDate=1747238400000, pubDateStr=2025-05-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793343283690, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793343283690, creator=13701087609, updateTime=1793343283690, updator=13701087609, issue=Issue{id=1322940829001724376, tenantId=1146029695717560320, journalId=1281212831689347082, year='2025', volume='45', issue='5', pageStart='441', pageEnd='550', issueExtLink='null', onlineDate='null', pubDate='1747238400000', pubDateStr='2025-05-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793343272686, creator='13701087609', updateTime=1793343272686, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=545, endPage=550, ext={EN=ArticleExt(id=1322940876883899175, articleId=1322940875155845926, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Improved IGGIII Robust Method with Membership Function for Crustal Movement Velocity Field Fitting and Analysis, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

To address the challenges of traditional hard-threshold methods in dealing with ambiguous boundaries when processing abnormal stations, this paper proposes a new method that treats station anomalies caused by tectonic movements and observational errors as gross errors and uses a membership function to construct an IGGIII equivalent weight function to mitigate the impact of abnormal stations. The new method incorporates fuzzy set theory and is an improved IGGIII robust estimation method based on the membership function of the fuzzy subset to which standardized residuals belong to gross errors(membership function IGGIII, MF-IGGIII). Firstly, the membership function of the fuzzy subset most affected by gross errors in standardized residuals is determined through fuzzy statistics; then, an equivalent weight function is constructed based on these membership function values, and a weight iteration process is performed. Simulation experiments and the fitting analysis of the GNSS velocity field in the Bohai rim area show that this method outperforms hypothesis testing, mean shift methods, and the IGGIII method in abnormal station detection, demonstrating strong capability in identifying abnormal stations.

, authors=Haifang JIAN1, Yuyan SHI1, Xinqiang LEI1, Yixu LI2, Jun ZHANG1, *, authorsList=Haifang JIAN, Yuyan SHI, Xinqiang LEI, Yixu LI, Jun ZHANG, authorCompany=null, correspAuthors=Jun ZHANG, authorNote=

JIAN Haifang, postgraduate, majors in plate movement and crust deformation, 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=1322940877844394798, articleId=1322940875155845926, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=用于地壳运动速度场拟合分析的隶属度函数改进IGGIII抗差方法, columnId=1322506197874356701, journalTitle=大地测量与地球动力学, columnName=地壳运动, runingTitle=null, highlight=null, articleAbstract=

为了解决传统硬阈值方法在处理异常测站时难以应对模糊边界的问题, 提出一种新方法, 将构造运动和观测误差引起的测站异常均视为粗差, 并利用隶属度函数构建IGGIII等价权函数, 以减小异常测站的影响。新方法引入模糊集理论, 是一种基于标准化残差属于粗差的模糊子集隶属度函数改进的IGGIII抗差估计方法(membership function IGGIII, MF-IGGIII)。首先, 通过模糊统计确定标准化残差受粗差污染程度最大的模糊子集的隶属度函数; 然后, 依据这些隶属度函数值构建等价权函数, 并进行选权迭代。模拟实验和对环渤海区域GNSS速度场的拟合分析结果表明, 该方法在异常测站检测方面的表现优于假设检验法、均值漂移法和IGGIII法, 显示出较强的异常测站识别能力。

, authors=简海方1, 石雨燕1, 雷新强1, 李屹旭2, 张俊1, *, authorsList=简海方, 石雨燕, 雷新强, 李屹旭, 张俊, authorCompany=null, correspAuthors=张俊, authorNote=

简海方, 硕士生, 主要从事板块运动及地壳形变研究, E-mail:

, correspAuthorsNote=

张俊, 博士, 副教授, 主要从事地壳运动及形变研究, E-mail:

ZHANG Jun, PhD, associate professor, majors in crustal movement and deformation, E-mail:

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JIAN Haifang, postgraduate, majors in plate movement and crust deformation, E-mail:

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Simulation data fitting results

, figureFileSmall=null, figureFileBig=null, tableContent=
拟合方案 欧拉运动参数/10-9 rad·a-1 应变参数/10-9 ·a-1 RMSE/mm·a-1
ωx ωy ωz εee εen εnn
模拟数据 5.001 2 -10.962 1 -8.266 0 -3.52 -2.78 2.12
方案1 4.764 2 -10.386 0 -7.721 8 1.456 3
方案2 4.772 2 -10.417 1 -7.754 3 1.398 9
方案3 4.811 0 -10.497 4 -7.840 2 1.091 0
方案4 4.841 7 -10.572 4 -7.913 9 1.048 0
方案5 4.982 2 -10.833 0 -8.108 2 -3.56 -2.76 2.02 0.951 8
方案6 4.965 2 -10.831 8 -8.108 6 -3.66 -2.89 1.84 0.836 3
方案7 4.951 5 -10.834 0 -8.125 5 -3.56 -2.68 2.08 0.659 8
方案8 4.929 6 -10.801 7 -8.103 4 -3.59 -2.58 2.24 0.606 7
), ArticleFig(id=1322940882592346975, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940875155845926, language=CN, label=表1, caption=

模拟数据拟合结果

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拟合方案 欧拉运动参数/10-9 rad·a-1 应变参数/10-9 ·a-1 RMSE/mm·a-1
ωx ωy ωz εee εen εnn
模拟数据 5.001 2 -10.962 1 -8.266 0 -3.52 -2.78 2.12
方案1 4.764 2 -10.386 0 -7.721 8 1.456 3
方案2 4.772 2 -10.417 1 -7.754 3 1.398 9
方案3 4.811 0 -10.497 4 -7.840 2 1.091 0
方案4 4.841 7 -10.572 4 -7.913 9 1.048 0
方案5 4.982 2 -10.833 0 -8.108 2 -3.56 -2.76 2.02 0.951 8
方案6 4.965 2 -10.831 8 -8.108 6 -3.66 -2.89 1.84 0.836 3
方案7 4.951 5 -10.834 0 -8.125 5 -3.56 -2.68 2.08 0.659 8
方案8 4.929 6 -10.801 7 -8.103 4 -3.59 -2.58 2.24 0.606 7
), ArticleFig(id=1322940882672038752, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940875155845926, language=EN, label=Tab. 2, caption=

Statistics of abnormal station detection in simulated experiments

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异常测站 模拟数据 方案1 方案2 方案3 方案4 方案5 方案6 方案7 方案8
1 11 533 11 533 11 533
2 11 134 11 134 11 134 11 134 11 134
3 11 434 11 434 11 434
4 11 834 11 834
5 11 235 11 235 11 235
6 11 336 11 336 11 336
7 11 836 11 836
8 11 838 11 838 11 838 11 838 11 838
9 11 339 11 339 11 339 11 339 11 339 11 339 11 339
10 11 841 11 841 11 841 11 841 11 841
共计 10 2 2 17 3 3 10
与模拟重合点 2 2 8 3 3 10
与模拟非重合点 9
), ArticleFig(id=1322940882739147617, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940875155845926, language=CN, label=表2, caption=

模拟实验异常测站检测情况统计

, figureFileSmall=null, figureFileBig=null, tableContent=
异常测站 模拟数据 方案1 方案2 方案3 方案4 方案5 方案6 方案7 方案8
1 11 533 11 533 11 533
2 11 134 11 134 11 134 11 134 11 134
3 11 434 11 434 11 434
4 11 834 11 834
5 11 235 11 235 11 235
6 11 336 11 336 11 336
7 11 836 11 836
8 11 838 11 838 11 838 11 838 11 838
9 11 339 11 339 11 339 11 339 11 339 11 339 11 339
10 11 841 11 841 11 841 11 841 11 841
共计 10 2 2 17 3 3 10
与模拟重合点 2 2 8 3 3 10
与模拟非重合点 9
), ArticleFig(id=1322940882844005218, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940875155845926, language=EN, label=Tab. 3, caption=

Fitting of velocity field in the Bohai Rim region

, figureFileSmall=null, figureFileBig=null, tableContent=
拟合方案 欧拉运动参数/10-10 rad·a-1 应变参数/10-10·a-1 σΔv/mm·a-1
ωx ωy ωz εee εen εnn
方案1 8.434 2 -1.580 4 -6.817 4 1.877 6
方案2 9.058 6 -3.322 2 -8.174 4 1.438 6
方案3 8.716 1 -2.825 1 -7.686 4 1.114 9
方案4 8.498 3 -2.627 5 -7.463 5 1.054 6
方案5 7.242 2 0.754 3 -4.765 9 -4.96 0.39 -27.42 1.782 2
方案6 8.623 5 -2.388 9 -7.375 2 -1.01 -1.79 -28.71 1.362 0
方案7 8.406 3 -2.234 5 -7.183 2 -2.10 -1.55 -28.50 1.066 9
方案8 8.551 3 -2.650 9 -7.524 1 -2.04 -2.26 -28.61 1.015 9
), ArticleFig(id=1322940882927891299, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940875155845926, language=CN, label=表3, caption=

环渤海区域速度场拟合情况

, figureFileSmall=null, figureFileBig=null, tableContent=
拟合方案 欧拉运动参数/10-10 rad·a-1 应变参数/10-10·a-1 σΔv/mm·a-1
ωx ωy ωz εee εen εnn
方案1 8.434 2 -1.580 4 -6.817 4 1.877 6
方案2 9.058 6 -3.322 2 -8.174 4 1.438 6
方案3 8.716 1 -2.825 1 -7.686 4 1.114 9
方案4 8.498 3 -2.627 5 -7.463 5 1.054 6
方案5 7.242 2 0.754 3 -4.765 9 -4.96 0.39 -27.42 1.782 2
方案6 8.623 5 -2.388 9 -7.375 2 -1.01 -1.79 -28.71 1.362 0
方案7 8.406 3 -2.234 5 -7.183 2 -2.10 -1.55 -28.50 1.066 9
方案8 8.551 3 -2.650 9 -7.524 1 -2.04 -2.26 -28.61 1.015 9
), ArticleFig(id=1322940883003388772, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940875155845926, language=EN, label=Tab. 4, caption=

Statistics of abnormal station detection in the Bohai Rim region

, figureFileSmall=null, figureFileBig=null, tableContent=
异常测站 方案1 方案2 方案3 方案4 方案5 方案6 方案7 方案8
共计 4 30 30 103 4 24 24 92
与方案1重合点 4 4 4 4 4 4 4 4
与方案2重合点 4 30 30 30 4 17 17 27
与方案3重合点 4 30 30 30 4 17 17 27
与方案4重合点 4 30 30 103 4 24 24 74
与方案5重合点 4 4 4 4 4 4 4 4
与方案6重合点 4 17 17 24 4 24 24 16
与方案7重合点 4 17 17 24 4 24 24 16
与方案8重合点 4 20 20 53 4 16 16 92
构造带附近点 4 22 22 78 4 17 17 69
), ArticleFig(id=1322940883083080549, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940875155845926, language=CN, label=表4, caption=

环渤海区域异常测站检测情况统计

, figureFileSmall=null, figureFileBig=null, tableContent=
异常测站 方案1 方案2 方案3 方案4 方案5 方案6 方案7 方案8
共计 4 30 30 103 4 24 24 92
与方案1重合点 4 4 4 4 4 4 4 4
与方案2重合点 4 30 30 30 4 17 17 27
与方案3重合点 4 30 30 30 4 17 17 27
与方案4重合点 4 30 30 103 4 24 24 74
与方案5重合点 4 4 4 4 4 4 4 4
与方案6重合点 4 17 17 24 4 24 24 16
与方案7重合点 4 17 17 24 4 24 24 16
与方案8重合点 4 20 20 53 4 16 16 92
构造带附近点 4 22 22 78 4 17 17 69
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用于地壳运动速度场拟合分析的隶属度函数改进IGGIII抗差方法
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简海方 1 , 石雨燕 1 , 雷新强 1 , 李屹旭 2 , 张俊 1, *
大地测量与地球动力学 | 地壳运动 2025,45(5): 545-550
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大地测量与地球动力学 |地壳运动 2025 , 45 (5) : 545 -550
用于地壳运动速度场拟合分析的隶属度函数改进IGGIII抗差方法
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简海方, 硕士生, 主要从事板块运动及地壳形变研究, E-mail:

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简海方1 , 石雨燕1, 雷新强1, 李屹旭2, 张俊1, *
作者信息
  • 1 贵州大学矿业学院, 贵阳市花溪大道南段2708号, 550025
  • 2 贵州大学农学院, 贵阳市花溪大道南段2708号, 550025
通讯作者:

张俊, 博士, 副教授, 主要从事地壳运动及形变研究, E-mail:

ZHANG Jun, PhD, associate professor, majors in crustal movement and deformation, E-mail:

作者简介:

简海方, 硕士生, 主要从事板块运动及地壳形变研究, E-mail:

Improved IGGIII Robust Method with Membership Function for Crustal Movement Velocity Field Fitting and Analysis
Haifang JIAN1 , Yuyan SHI1, Xinqiang LEI1, Yixu LI2, Jun ZHANG1, *
Affiliations
  • 1 College of Mining, Guizhou University, 2708 South Segment of Huaxi Road, Guiyang 550025, China
  • 2 College of Agriculture, Guizhou University, 2708 South Segment of Huaxi Road, Guiyang 550025, China

    • JIAN Haifang, postgraduate, majors in plate movement and crust deformation, E-mail:

出版时间: 2025-05-15 doi: 10.14075/j.jgg.2024.06.286
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摘要
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为了解决传统硬阈值方法在处理异常测站时难以应对模糊边界的问题, 提出一种新方法, 将构造运动和观测误差引起的测站异常均视为粗差, 并利用隶属度函数构建IGGIII等价权函数, 以减小异常测站的影响。新方法引入模糊集理论, 是一种基于标准化残差属于粗差的模糊子集隶属度函数改进的IGGIII抗差估计方法(membership function IGGIII, MF-IGGIII)。首先, 通过模糊统计确定标准化残差受粗差污染程度最大的模糊子集的隶属度函数; 然后, 依据这些隶属度函数值构建等价权函数, 并进行选权迭代。模拟实验和对环渤海区域GNSS速度场的拟合分析结果表明, 该方法在异常测站检测方面的表现优于假设检验法、均值漂移法和IGGIII法, 显示出较强的异常测站识别能力。

地壳形变分析  /  隶属度函数  /  MF-IGGIII  /  等价权函数  /  异常测站

To address the challenges of traditional hard-threshold methods in dealing with ambiguous boundaries when processing abnormal stations, this paper proposes a new method that treats station anomalies caused by tectonic movements and observational errors as gross errors and uses a membership function to construct an IGGIII equivalent weight function to mitigate the impact of abnormal stations. The new method incorporates fuzzy set theory and is an improved IGGIII robust estimation method based on the membership function of the fuzzy subset to which standardized residuals belong to gross errors(membership function IGGIII, MF-IGGIII). Firstly, the membership function of the fuzzy subset most affected by gross errors in standardized residuals is determined through fuzzy statistics; then, an equivalent weight function is constructed based on these membership function values, and a weight iteration process is performed. Simulation experiments and the fitting analysis of the GNSS velocity field in the Bohai rim area show that this method outperforms hypothesis testing, mean shift methods, and the IGGIII method in abnormal station detection, demonstrating strong capability in identifying abnormal stations.

crustal deformation analysis  /  membership function  /  MF-IGGIII  /  equivalent weight function  /  abnormal station
简海方, 石雨燕, 雷新强, 李屹旭, 张俊. 用于地壳运动速度场拟合分析的隶属度函数改进IGGIII抗差方法. 大地测量与地球动力学, 2025 , 45 (5) : 545 -550 . DOI: 10.14075/j.jgg.2024.06.286
Haifang JIAN, Yuyan SHI, Xinqiang LEI, Yixu LI, Jun ZHANG. Improved IGGIII Robust Method with Membership Function for Crustal Movement Velocity Field Fitting and Analysis[J]. Journal of Geodesy and Geodynamics, 2025 , 45 (5) : 545 -550 . DOI: 10.14075/j.jgg.2024.06.286
正文
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在地壳形变分析时，异常测站误差影响较大[1]，只有排除异常测站的影响，才能获得区域地壳的整体运动及变形的空间分布特征。由于构造运动引起的测站异常与观测引起的测站异常均可视为粗差处理[2]，故本文采用粗差处理的有关理论对异常测站进行处理。粗差处理大致可分为两种途径：将粗差影响归入函数模型的均值漂移模型和将粗差影响归入随机模型的方差膨胀模型[3-7]。前者通过假设检验法检测、定位和删除粗差，后者通过残差“表现”采用分段定权的方式抑制粗差对数据处理结果的影响[8-9]。然而，这些方法的共同点是都采用了“硬阈值”法判别残差是否属于粗差，即以残差绝对值是否满足事先设定的阈值范围为依据判定粗差，这样的粗差识别过程忽略了粗差边界的模糊性。如，标准化残差wi=1.999 999，wj=2.000 001，二者较差仅为0.000 002，若同等对待，一般不会影响实际应用，但当阈值取2时，就会得出绝然不同的两个结论，这显然是不合理的[10]。地壳运动及形变分析时，处在断裂带附近的测站由于构造变形的影响，其变形通常具有相对接近的“离群”特征，但采用传统“硬阈值”法判定异常测站时，往往被区别对待。如何正确处理这些模糊异常测站是值得研究的课题。本文采用模糊数学思想，引入模糊集隶属度函数，提出以受粗差污染程度最大的标准化残差的模糊子集的隶属度值代替IGGIII抗差方案中直接以标准化残差绝对值为依据的等价权确定方法，即利用标准化残差的隶属度函数构建IGGIII分段等价权函数，进行地壳运动异常测站分析。本文将改进的方法记为MF-IGGIII(membership function IGGIII)。
1 刚性地壳运动分析模型与弹性地壳运动分析模型
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1.1 刚性地壳运动分析模型
欧拉矢量法是定量描述板块运动的基本模型，其本质上是块体旋转矢量的一种表述形式。欧拉矢量有3个参数，记为(φ，λ，ω)，其中φ、λ为纬度和经度，ω为旋转角速度。基于欧拉矢量法求板块刚性运动速度的公式如下[11]：
$\begin{gathered}\boldsymbol{V}_{\mathrm{g}}=\left[\begin{array}{l}\boldsymbol{V}_{\mathrm{e}} \\\boldsymbol{V}_{\mathrm{n}}\end{array}\right]_{\mathrm{g}}=\boldsymbol{A} \boldsymbol{\omega}= \\{\left[\begin{array}{ccc}-r \sin \varphi \cos \lambda & -r \sin \varphi \sin \lambda & r \cos \varphi \\r \sin \lambda & -r \cos \lambda & 0\end{array}\right]\left[\begin{array}{c}\omega_x \\\omega_y \\\omega_z\end{array}\right]}\end{gathered}$
式中，Ve指纬向(东向)速度；Vn指经向(北向)速度；λ为测站经度；φ为测站纬度；r为地球平均半径；ωx，ωy，ωz为欧拉旋转矢量在三维坐标轴上的分量。
1.2 弹性地壳运动分析模型
当地壳被看成具有弹塑性质的物理介质时，板块水平运动可以被描述为板块的整体刚性运动与内部变形的叠加，以此建立地壳的复合运动方程，并反演形变参数[12]。李延兴等[13]基于中国大陆及周边地区相对于全球框架和欧亚板块的速度场数据，研究块体的刚性运动模型和弹塑性运动模型，建立并检验了块体的刚性-弹塑性运动-均匀应变模型(REHSM)：
$\begin{gathered}\boldsymbol{V}_{\mathrm{h}}=\left[\begin{array}{l}\boldsymbol{V}_{\mathrm{e}} \\\boldsymbol{V}_{\mathrm{n}}\end{array}\right]_{\mathrm{h}}= \\{\left[\begin{array}{ccc}-r \sin \varphi \cos \lambda & -r \sin \varphi \sin \lambda & r \cos \varphi \\r \sin \lambda & -r \cos \lambda & 0\end{array}\right]\left[\begin{array}{c}\omega_x \\\omega_y \\\omega_z\end{array}\right]+} \\{\left[\begin{array}{ll}\varepsilon_{\mathrm{ee}} & \varepsilon_{\mathrm{en}} \\\varepsilon_{\mathrm{ne}} & \varepsilon_{\mathrm{nn}}\end{array}\right]\left[\begin{array}{c}r\left(\lambda-\lambda_0\right) \cos \varphi \\r\left(\varphi-\varphi_0\right)\end{array}\right]}\end{gathered}$
式中，εee、εen、εne、εnn称之为应变参数，其余参数含义同式(1)。
2 IGGIII抗差方案
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IGG方案通过设置多个阈值来判断数据，按照不同阈值将数据分为3段：正常段，即不包含粗差或异常值的数据，采用最小二乘估计求解最优解；可疑段，即超过一定范围，可能存在粗差或异常值的数据，通过降权处理，从而降低可疑数据的权重影响；淘汰段，即明显存在粗差或异常值的数据，采用淘汰或剔除的手段，即零权估计。杨元喜等[14]基于IGG方案，提出3段函数模型和等价权函数思想，设计了IGGIII抗差方案，具体如下：
$\omega\left(\nu_i\right)=\left\{\begin{array}{l}1, \left|\nu_i\right|<k_0 \sigma \\\frac{k_0 \sigma_{\nu_i}}{\left|\nu_i\right|}\left(\frac{k_1-\frac{\left|\nu_i\right|}{\sigma_{\nu_i}}}{k_1-k_0}\right)^2, k_0 \sigma \leqslant\left|\nu_i\right|<k_1 \sigma \\0, \left|\nu_i\right| \geqslant k_1 \sigma\end{array}\right.$
式中，ω(νi)为权函数；νi为残差；σνi为中误差；k0和k1为相应的调和系数。
令$ \left|w_i\right|=\left|\nu_i\right| / \sigma_{\nu_i}$，表示观测值残差向量标准化残差的绝对值，则式(3)可表示为：
$\omega\left(\nu_i\right)=\left\{\begin{array}{l}1, \left|w_i\right|<k_0 \\\frac{k_0}{\left|w_i\right|}\left(\frac{k_1-\left|w_i\right|}{k_1-k_0}\right)^2, k_0 \leqslant\left|w_i\right|<k_1 \\0, \left|w_i\right| \geqslant k_1\end{array}\right.$
式中，k0=1.0~1.5、k1=2.5~3.0[2]，本文取k0=1.5、k1=2.5。
3 隶属度函数
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在传统的集合论思想中，一个元素要么属于某个特定集合(隶属度为1)，要么不属于该集合(隶属度为0)；而“模糊集合”则允许一定程度的“隶属”存在，当隶属程度相差不大时，可以被统一归属为“属于”或“不属于”，非常适合于处理具有“模糊性”特征的问题。
经典集A可由其特征函数χA(x)唯一确定，映射表达为[15]：
$\begin{aligned}& \chi_A: X \rightarrow\{0, 1\} \\& x \mapsto \chi_A(x)=\left\{\begin{array}{l}1, x \in A \\0, x \notin A\end{array}\right.\end{aligned}$
式中，确定了X上的经典模糊子集A。χA(x)的值表示x对A的隶属程度，不过只有两种：一个元素x属于A(隶属度为1)或不属于A(隶属度为0)。
设U为论域，$\underset{\sim}{A}$为U上一个模糊子集，映射$ \mu_{\underset{\sim}{A}}$称为$ \underset{\sim}{A}$的隶属函数，$ \mu_{\underset{\sim}{A}}(x)$称为x对$ \underset{\sim}{A}$的隶属程度。模糊子集\underset{\sim}{A}$ $由隶属度函数$ \mu_{\underset{\sim}{A}}$唯一确定，映射表达为[15]：
$\begin{aligned}& \mu_{\underset{\sim}{A}}: U \rightarrow[0, 1] \\& x \mapsto \mu_{\underset{\sim}{A}}(x) \in[0, 1]\end{aligned}$
由映射定义可知，$ \mu_{\underset{\sim}{A}}$(x)=1时，x完全属于$ \underset{\sim}{A}$；$ \mu_{\underset{\sim}{A}}(x)$=0时，x完全不属于$ \underset{\sim}{A}$；随着$ \mu_{\underset{\sim}{A}}(x)$值的增大，x属于$ \underset{\sim}{A}$的程度也逐渐提高。由此可知，最大程度受粗差污染的标准化残差的模糊子集及其隶属度函数、映射如下[10]：
$\begin{aligned}& \mu_{\underset{\sim}{A}}: U \rightarrow[0, 1] \\& w \mapsto \mu_{\underset{\sim}{A}}(w) \in[0, 1]\end{aligned}$
式中，$ \underset{\sim}{A}$为最大程度受粗差污染的标准化残差W的模糊子集，标准化残差绝对值|W|为论域U，$ \mu_{\underset{\sim}{A}}(w)$称为w对$ \underset{\sim}{A}$的隶属程度。当$ \mu_{\underset{\sim}{A}}(w) = 1$时，w完全属于$ \underset{\sim}{A}$，观测值在最大程度上受粗差污染；$ \mu_{\underset{\sim}{A}}(w) = 0$时，w完全不属于$ \underset{\sim}{A}$，观测值没有受粗差污染影响；随着$ \mu_{\underset{\sim}{A}}(w)$值的增大，w属于$ \underset{\sim}{A}$的程度也逐渐提高，观测值受粗差污染影响程度也增加。
4 MF-IGGIII抗差估计原理
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4.1 隶属度函数改进的IGGIII抗差方案中的等价权函数
考虑到粗差本身具有模糊性，本文在王新洲[10]方法的基础上进行简化，采用受粗差污染程度最大的标准化残差的模糊子集的隶属度函数值代替式(4)中的标准化残差，以此构建等价权函数并进行选权迭代。则相应的等价权函数定义如下：
$\bar{P}_{i+1}=\left\{\begin{array}{l}P_i, \mu\left(w_i\right) \leqslant \mu\left(k_0\right) \\P_i \frac{\mu\left(k_0\right)}{\mu\left(w_i\right)}\left(\frac{\mu\left(k_1\right)-\mu\left(w_i\right)}{\mu\left(k_1\right)-\mu\left(k_0\right)}\right), \\\mu\left(k_0\right)<\mu\left(w_i\right) \leqslant \mu\left(k_1\right) \\10^{-10}, \mu\left(w_i\right)>\mu\left(k_1\right)\end{array}\right.$
式中，${\bar P}$i+1为等价权函数；Pi表示第i次迭代的权；μ(·)为标准化残差最大程度属于粗差模糊子集的隶属度函数；i表示迭代次数；10－10代替淘汰段的零权估计，避免计算出现奇异矩阵的情形。
4.2 隶属度函数的确定方法
本文利用式(8)，即采用MF-IGGIII进行抗差估计时，必须给出残差标准差最大程度属于粗差模糊子集的隶属度函数。此处直接采用文献[10]构建的隶属度函数：
$\mu_{\underset{\sim}{A}}\left(w_i\right)=\left\{\begin{array}{l}0, w_i<1 \\\frac{1}{1+\frac{20+w_i}{w_i^{c^2}}}, w_i \geqslant 1\end{array}\right.$
式中，$ \mu_{\underset{\sim}{A}}\left(w_i\right)$表示wi(wi为论域|W|的元素)关于$ \underset{\sim}{A}$的隶属程度, 即wi属于的程度，$ \underset{\sim}{A}$其中
$c=\left\{\begin{array}{l}w_i, w_i<3 \\3, w_i \geqslant 3\end{array}\right.$
根据式(8)~式(10)，利用MF-IGGII抗差估计法进行异常测站处理步骤(图1)如下：
1) 根据地壳运动速度场确定观测值初始权阵，若精度相同，则直接用单位阵代替初始权阵，然后利用最小二乘估计获取初始残差V并计算获取标准化残差wi；
2) 根据式(9)计算各标准化残差的隶属函数值，代入式(8)计算等价权；
3) 利用步骤2)获取的等价权，根据选权迭代停止条件$ \left\|\boldsymbol{X}^{i+1}-\boldsymbol{X}^i\right\|$ < ε(本文取ε=10－10)，重复步骤1)，更新参数和残差估值；
4) 结束迭代过程，输出参数估值；
5) 以RRM和REHSM为主模型进行解算；
6) 根据解算结果进行形变分析。
5 算例及分析
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5.1 模拟实验及分析
以1°×1°的格网密度，在经度范围为111°~120°E，纬度范围为33°~42°N范围内，利用式(2)模型模拟了100个测站点的速率，模型具体参数见表1。在模拟数据中加入服从正态分布N(0，0.52)的偶然误差，得到速度场观测值。为验证本文方法的可行性，同时随机选取10个测站点，并加入5倍模拟数据中误差的粗差。现设计8种方案(方案1：RRM刚体运动地壳形变分析模型，假设检验法检测异常测站；方案2：RRM，均值漂移法；方案3：RRM，IGGIII；方案4：RRM，MF-IGGIII；方案5：REHSM，假设检验法；方案6：REHSM，均值漂移法；方案7：REHSM，IGGIII；方案8：REHSM，MF-IGGIII)对模拟速度场观测数据进行拟合分析，相关拟合结果列于表1，其中RMSE表示速率估值的均方根误差。
由表1可知，方案5、6、7和8的均方根误差均小于方案1、2、3和4，表明REHSM模型考虑块体局部变形后比RRM模型不考虑块体局部变形结果更合理；方案4相比于方案1、2和3，方案8相比于方案5、6、7的拟合精度均有改善，说明MF-IGGIII抗差方案具有更优的异常测站检测效果。由表2可知，方案1和5未能识别出模拟的异常测站，方案2、3、6和7仅识别出2~3个异常测站，说明假设检验法和传统分段阈值法判定粗差的做法过于僵硬，不能解决粗差模糊边界的问题。方案4和8分别识别出8个模拟异常测站和10个异常测站，与其他方案相比，优势明显，对模拟异常测站识别成功率高。尤其是方案8成功识别出全部模拟的异常测站，且没有出现错误识别的情况，说明MF-IGGIII新方法因考虑了粗差模糊边界问题，比传统IGGIII抗差方案稳健性更强、更合理。
5.2 利用MF-IGGIII抗差方案进行环渤海地区地壳形变分析
为验证本文方法，采用文献[11]给出的ITRF2008框架下的环渤海区域482个监测站，经度范围为111.48°~128.11°E，纬度范围为30.96°~42.54°N，共包括32个CORS基准站和450个区域站。GPS站速率通过st_filter软件处理获得, 相对于ITRF2008其精度优于2 mm/a。图2为环渤海区域主要地震分布图[2]。采用与§5.1模拟实验相同的数据处理方案，相关拟合结果列于表3，异常测站检测情况见表4，检测出的异常测站见图3。
由表3可知，方案1、2、3和4拟合结果比方案5、6、7和8差，说明考虑块体变形的REHSM比RRM更合理；方案1和5拟合结果最差，方案2和6次之，方案3和7较好，方案4和8最好，说明基于假设检验的地壳形变分析异常测站检测能力最弱，均值漂移模型对异常测站检测能力比假设检验法稍强，IGGIII对异常测站的检测能力较好，MF-IGGIII在本文中表现最好。其原因在于，均值漂移法通过迭代过程不断对测站速率进行修正，但其本质上与假设检验法原理是类似的，也是通过硬阈值来判定粗差的；IGGIII抗差方案采用分段定权的选权迭代方式进行两阈值法粗差判定，其比采用单一阈值法判定粗差的方法更加合理，但对于粗差的模糊边界问题仍然没有解决；MF-IGGIII考虑粗差模糊边界问题，以IGGIII为基础采用隶属度函数，得到更好的拟合结果，表明是一种适合的方法。
从表4统计的各方案检测出的异常测站数目来看，相对于其他方案，方案4和8检测出了更多的异常测站，且几乎包含了其他方案检测出的所有异常测站点。且从图2和图3可以看出，检测出的异常测站点大部分位于蓬莱断裂带、郯庐断裂带和20 a以来地震震中附近，其原因在于构造活动会引起局部点的异常变形，而MF-IGIII考虑了边界模糊问题，比采用硬阈值更适合断裂带及其附近异常测站的检测。故可以推测，MF-IGGIII检测出了其他方案难以判别的“模糊异常测站”，理论上是一种更合理的异常判别方法。
6 结语
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本文引入模糊数学思想，利用隶属度函数代替标准化残差构建MF-IGIII分段等价权函数，进行GNSS地壳运动速度场拟合分析。该方法与采用“硬阈值”法来判定粗差相比，可以使判定对象在某一个弹性区间内被“公正”地对待，避免了简单利用某一特定阈值僵硬淘汰或接受的弊端，较为符合实际情况，对于粗差处理是一种较好的方法。对环渤海地区实测速度场的拟合分析结果表明，MF-IGIII拟合结果更好，尤其是该方法能够识别出“模糊异常测站”，其识别出的异常测站不仅包含了IGGIII识别出的全部异常测站，还成功识别出了其他位于环渤海地区构造带及其附近和本区域20 a来地震震中附近的异常测站。这充分表明，利用MF-IGGIII法进行地壳形变分析，其结果更合理、更符合构造带实际、适用性更强。由于研究区域单一，未进行不同隶属度函数的讨论，今后将进行相关研究，构建更加精细和符合实际的MF-IGGIII模型。
基金
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  • 贵州省科学技术基础研究计划(黔科合基础[2017]1054)
  • 贵州省科技厅科技支撑计划(黔科合支撑[2022]一般204)
文献
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2025年第45卷第5期
27
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doi: 10.14075/j.jgg.2024.06.286
  • 接收时间：2024-06-17
  • 首发时间：2026-10-30
  • 出版时间：2025-05-15
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  • 收稿日期：2024-06-17
基金
Basic Research Project of Science and Technology of Guizhou Province(黔科合基础[2017]1054)
贵州省科学技术基础研究计划(黔科合基础[2017]1054)
Science and Technology Support Program ofthe Department of Science and Technology of Guizhou Province(黔科合支撑[2022]一般204)
贵州省科技厅科技支撑计划(黔科合支撑[2022]一般204)
作者信息
    1 贵州大学矿业学院, 贵阳市花溪大道南段2708号, 550025
    2 贵州大学农学院, 贵阳市花溪大道南段2708号, 550025

通讯作者:

张俊, 博士, 副教授, 主要从事地壳运动及形变研究, E-mail:

ZHANG Jun, PhD, associate professor, majors in crustal movement and deformation, E-mail:

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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
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红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
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