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Article(id=1322916947863666735, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322916945741348911, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.12.406, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1764604800000, receivedDateStr=2025-12-02, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793337578979, onlineDateStr=2026-10-30, pubDate=1789401600000, pubDateStr=2026-09-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793337578979, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793337578979, creator=13701087609, updateTime=1793337578979, updator=13701087609, issue=Issue{id=1322916945741348911, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='9', pageStart='1039', pageEnd='1170', issueExtLink='null', onlineDate='null', pubDate='1789401600000', pubDateStr='2026-09-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793337578473, creator='13701087609', updateTime=1793337578473, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=1156, endPage=1161, ext={EN=ArticleExt(id=1322916948060799024, articleId=1322916947863666735, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=A Real-Time Fusion Method for Beidou Static and RTK Deformation Monitoring Results Based on Multi-Scale Decomposition, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Aiming at the problem that the static relative positioning technology has a long calculation cycle and cannot timely reflect abrupt deformations when using Beidou/GNSS technology for infrastructure deformation monitoring currently, this paper proposes a real-time fusion method for monitoring results of static and RTK with different time scales and positioning accuracies, based on wavelet decomposition and dynamic weighted fusion of sliding time windows. Through experimental tests in the ultra-short baseline field, it is verified that this method can provide millimeter-level monitoring accuracy results when the monitored object is stable or deforms slowly, and can timely output the actual displacement when abrupt deformation occurs, with accuracy superior to that of RTK monitoring results. The method can be implemented based on general-purpose GNSS data processing software available on the market, and has a good prospect of engineering application.

, authors=Han HE1, Jifeng ZHANG1, Jian QIN1, Peifen PAN1, Xiaolei XU1, Yang LI2, authorsList=Han HE, Jifeng ZHANG, Jian QIN, Peifen PAN, Xiaolei XU, Yang LI, authorCompany=null, correspAuthors=null, 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=1322916949776269371, articleId=1322916947863666735, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=基于多尺度分解的北斗静态与RTK变形监测结果实时融合方法, columnId=1322566785874908034, journalTitle=大地测量与地球动力学, columnName=特色栏目: 地震大地测量学, runingTitle=null, highlight=null, articleAbstract=

针对目前基于北斗/GNSS技术进行基础设施变形监测时，采用静态相对定位技术解算周期长，不能及时反映突发变形的问题，基于滑动时间窗口的多尺度分解与动态加权融合，提出一种将静态与RTK两种不同时间尺度、不同定位精度的监测结果进行实时融合的方法。通过在超短基线场进行实验测试，验证该方法能在监测对象稳定或缓慢变形的时候，提供mm级监测精度结果，在突发变形时能及时输出实际位移情况，且精度优于RTK监测结果。本文方法基于市场上通用的GNSS数据处理软件即可实现，具有较好的工程应用前景。

, authors=贺晗1, 张吉峰1, 秦健1, 潘佩芬1, 徐晓磊1, 李洋2, authorsList=贺晗, 张吉峰, 秦健, 潘佩芬, 徐晓磊, 李洋, authorCompany=null, correspAuthors=null, authorNote=

贺晗，助理研究员，主要研究方向为GNSS在铁路行业中的变形监测应用，E-mail:

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贺晗，助理研究员，主要研究方向为GNSS在铁路行业中的变形监测应用，E-mail:

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On Analysis and Application Approach for Kalman Filter Parameters[J]. 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Experimental equipment and algorithm configuration parameters

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参数项 参数值
天线型号 华信HXCCSX601A
定位模块型号 和芯星通UM980
天线截止高度角 10°
导航系统 单北斗
RTK平面测量精度 8 mm
静态平面测量精度 1.5 mm
原始观测数据采集频率 1 Hz
RTK结果输出间隔 1 s
静态结果输出间隔 1 h
多尺度分解层数 4
滑动窗口大小 60 s
滑动窗口步长 3 s
), ArticleFig(id=1322916957024026741, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916947863666735, language=CN, label=表1, caption=

实验设备与算法配置参数

, figureFileSmall=null, figureFileBig=null, tableContent=
参数项 参数值
天线型号 华信HXCCSX601A
定位模块型号 和芯星通UM980
天线截止高度角 10°
导航系统 单北斗
RTK平面测量精度 8 mm
静态平面测量精度 1.5 mm
原始观测数据采集频率 1 Hz
RTK结果输出间隔 1 s
静态结果输出间隔 1 h
多尺度分解层数 4
滑动窗口大小 60 s
滑动窗口步长 3 s
), ArticleFig(id=1322916957095329910, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916947863666735, language=EN, label=Tab. 2, caption=

Results of error analysis

, figureFileSmall=null, figureFileBig=null, tableContent=
方法 突发变形前误差值/mm 开始突发变形至T+5时误差值/mm T+5时至T+7时误差值/mm
最大值 RMSE 最大值 RMSE 最大值 RMSE
静态监测 1.3 0.8 20.1 18.7 1 0.7
RTK监测 8.9 2 8.6 2.4 13.8 2.9
卡尔曼滤波 8.4 1.9 7.2 2.4 12.2 2.7
融合方法 1.3 0.7 11 2.3 1.2 0.8
), ArticleFig(id=1322916957367959671, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916947863666735, language=CN, label=表2, caption=

误差分析结果

, figureFileSmall=null, figureFileBig=null, tableContent=
方法 突发变形前误差值/mm 开始突发变形至T+5时误差值/mm T+5时至T+7时误差值/mm
最大值 RMSE 最大值 RMSE 最大值 RMSE
静态监测 1.3 0.8 20.1 18.7 1 0.7
RTK监测 8.9 2 8.6 2.4 13.8 2.9
卡尔曼滤波 8.4 1.9 7.2 2.4 12.2 2.7
融合方法 1.3 0.7 11 2.3 1.2 0.8
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基于多尺度分解的北斗静态与RTK变形监测结果实时融合方法
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贺晗 1 , 张吉峰 1 , 秦健 1 , 潘佩芬 1 , 徐晓磊 1 , 李洋 2
大地测量与地球动力学 | 特色栏目: 地震大地测量学 2026,46(9): 1156-1161
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大地测量与地球动力学 |特色栏目: 地震大地测量学 2026 , 46 (9) : 1156 -1161
基于多尺度分解的北斗静态与RTK变形监测结果实时融合方法
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贺晗，助理研究员，主要研究方向为GNSS在铁路行业中的变形监测应用，E-mail:

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贺晗，助理研究员，主要研究方向为GNSS在铁路行业中的变形监测应用，E-mail:

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贺晗1 , 张吉峰1, 秦健1, 潘佩芬1, 徐晓磊1, 李洋2
作者信息
  • 1 中国铁道科学研究院集团有限公司电子计算技术研究所, 北京, 100081
  • 2 北京经纬信息技术有限公司, 北京, 100081
作者简介:

贺晗，助理研究员，主要研究方向为GNSS在铁路行业中的变形监测应用，E-mail:

A Real-Time Fusion Method for Beidou Static and RTK Deformation Monitoring Results Based on Multi-Scale Decomposition
Han HE1 , Jifeng ZHANG1, Jian QIN1, Peifen PAN1, Xiaolei XU1, Yang LI2
Affiliations
  • 1 Institute of Computing Technology, China Academy of Railway Sciences Co Ltd, Beijing 100081, China
  • 2 Beijing Jingwei Information Technology Co Ltd, Beijing 100081, China
出版时间: 2026-09-15 doi: 10.14075/j.jgg.2025.12.406
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摘要
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针对目前基于北斗/GNSS技术进行基础设施变形监测时，采用静态相对定位技术解算周期长，不能及时反映突发变形的问题，基于滑动时间窗口的多尺度分解与动态加权融合，提出一种将静态与RTK两种不同时间尺度、不同定位精度的监测结果进行实时融合的方法。通过在超短基线场进行实验测试，验证该方法能在监测对象稳定或缓慢变形的时候，提供mm级监测精度结果，在突发变形时能及时输出实际位移情况，且精度优于RTK监测结果。本文方法基于市场上通用的GNSS数据处理软件即可实现，具有较好的工程应用前景。

北斗  /  变形监测  /  多尺度分解  /  滑动窗口  /  突发变形

Aiming at the problem that the static relative positioning technology has a long calculation cycle and cannot timely reflect abrupt deformations when using Beidou/GNSS technology for infrastructure deformation monitoring currently, this paper proposes a real-time fusion method for monitoring results of static and RTK with different time scales and positioning accuracies, based on wavelet decomposition and dynamic weighted fusion of sliding time windows. Through experimental tests in the ultra-short baseline field, it is verified that this method can provide millimeter-level monitoring accuracy results when the monitored object is stable or deforms slowly, and can timely output the actual displacement when abrupt deformation occurs, with accuracy superior to that of RTK monitoring results. The method can be implemented based on general-purpose GNSS data processing software available on the market, and has a good prospect of engineering application.

Beidou  /  deformation monitoring  /  multi-scale decomposition  /  sliding window  /  abrupt deformation
贺晗, 张吉峰, 秦健, 潘佩芬, 徐晓磊, 李洋. 基于多尺度分解的北斗静态与RTK变形监测结果实时融合方法. 大地测量与地球动力学, 2026 , 46 (9) : 1156 -1161 . DOI: 10.14075/j.jgg.2025.12.406
Han HE, Jifeng ZHANG, Jian QIN, Peifen PAN, Xiaolei XU, Yang LI. A Real-Time Fusion Method for Beidou Static and RTK Deformation Monitoring Results Based on Multi-Scale Decomposition[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (9) : 1156 -1161 . DOI: 10.14075/j.jgg.2025.12.406
正文
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地表及工程构筑物受自然因素或人为活动影响，会不可避免地发生位移与变形，为确保结构安全，同时预防灾害，开展变形监测至关重要。传统监测主要采用全站仪、水准仪等非自动化设备进行人工测量，存在投入人力多、成本高、实时性差等问题[1]。随着北斗三号卫星系统的全面建成，基于北斗/全球导航卫星系统(GNSS)的变形监测技术凭借其在监测上的实时性与连续性优势，被广泛应用于高精度变形监测领域[2-4]。
使用北斗/GNSS技术进行连续高精度变形监测主要采用2种数据处理技术，分别为静态相对定位技术与实时动态载波相位差分(RTK)技术[5]，这2种数据处理技术各具优缺点。由于静态相对定位技术需要累积一定周期的观测数据进行解算，以获得mm级监测结果，一个周期一般为数十分钟至数小时[6]，导致其监测结果时效性较差，无法及时对监测对象的突发性变形进行报警。RTK技术可在1 s至数秒内得到一个监测结果，时效性高但监测精度不足，只能达到cm级[7]，无法反映大坝、铁路路基等监测对象缓慢、微小的变形。为兼顾监测精度与时效性，有学者通过融合加速度计等传感器数据，在抑制低精度解、提高监测频率方面开展了研究[8-10]，也有学者利用监测体的变化特征在粗差和周跳实时处理、模糊度短时中断后的快速恢复等方面对RTK解算的精度和稳定性进行了改进[11-13]。这些研究方法在实验阶段表现出了较好的效果，但未形成成熟的软硬件产品。而常见通用的北斗/GNSS监测设备与数据处理软件无法直接支持这些方法，需要增加硬件设备并对数据处理软件的核心解算部分进行定制开发，应用难度与成本较高。
本文旨在不改变设备传输的数据内容及不增加额外硬件的前提下，将通用北斗/GNSS数据处理软件可以解算得到的静态与RTK监测结果作为输入值，利用滑动窗口的多尺度分解提取局部趋势, 结合系数加权融合与小波重构，对2种监测结果进行融合处理，使最终输出的结果具备较高的监测精度，并能及时识别突发变形。
1 离散小波变换
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北斗/GNSS技术变形监测结果为监测站相对于参考站的三维位置变化量。大部分的监测对象，特别是固定基础设施，在多数情况下处于稳定状态，位移速率极为缓慢。然而，受技术本身特点、观测设备精度及观测环境等因素影响，监测得到的位移值会出现上下波动。这种情况下非常适合采用小波变换来分离实际位移趋势和上下波动产生的随机噪声，且小波变换的时频局部性使其在处理非平稳信号时效果显著，能快速响应实际位移趋势的突变。
小波分析的概念最早由法国工程师Morlet在1984年处理地震信号时提出，发展至今已成为应用数学、信号处理等领域的重要工具，但连续小波变换运算量大，且连续积分无法直接通过计算机进行计算。20世纪80年代末期，多分辨率分析框架、小波系数快速计算等方法的提出奠定了离散小波的理论与应用基础，其高计算效率和实现简便的特点让小波分析领域迎来关键突破[14]。
设L2(R)为实数轴上的可积函数空间，对于连续信号x(t)∈L2(R)的小波变换，其公式为：
$\operatorname{CWT}_{x}(a, b)=\frac{1}{\sqrt{a}} \int_{R}\left[x(t) \bar{\psi}\left(\frac{t-b}{a}\right)\right] \mathrm{d} t $
式中，a为连续尺度参数；b为连续平移参数；ψ为满足容许条件的小波母函数。
通过对尺度参数a和平移参数b进行离散化，得到:
$\left\{\begin{array}{l}a=a_{0}^{j} \\b=k \cdot b_{0} \cdot a_{0}^{j}\end{array}\right.$
式中，j、k为整数。由此可得离散小波的定义为：
$\operatorname{DWT}_{x}(j, k)=\frac{1}{\sqrt{a_{0}^{j}}} \int_{R}\left[x(t) \bar{\psi}\left(a_{0}^{-j} t-k b_{0}\right)\right] \mathrm{d} t $
式中，a0一般取值为2；b0一般取值为1。
在工程应用时，常用Mallat算法[15]将采样后的离散时域数据转化为多层的低通、高通滤波与下采样的迭代计算，使时域数据分解为表征整体趋势的低频近似系数及表征局部突变特征的高频细节系数。
对于离散数据x(n)∈L2(Z)，将其进行逐级分解，第j层的近似系数Aj(k)和细节系数Dj(k)分别为：
$\left\{\begin{array}{l}A_{j}(k)=\sum\nolimits_{n} A_{j-1}(n) h(n-2 k) \\D_{j}(k)=\sum\nolimits_{n} A_{j-1}(n) g(n-2 k)\end{array}\right.$
式中，A0(n)=x(n)为初始近似系数；n为离散信号采样点数；k为其中第k个点；h和g分别为低通和高通滤波系数且满足正交关系。
离散小波变换的重构与分解相反，为上采样与滤波迭代计算，最终多层叠加，其公式为：
$\begin{gather*}A_{j-1}(n)=\sum\nolimits_{k} A_{j}(k) h(n-2 k)+ \\\sum\nolimits_{k} D_{j}(k) g(n-2 k) \end{gather*}$
2 融合方法
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2.1 时间滑动窗口建立
监测结果序列随时间不断更新，为实现融合结果的持续输出，需要通过时间滑动窗口将持续更新的监测结果序列划分为固定窗口长度的局部数据段进行处理。滑动窗口按照一定的滑动步长持续向后滑动，用最新的监测结果替换窗口中部分陈旧数据，从而实现逐段处理、持续输出。假设L(n)为最新原始监测结果序列，则当前窗口数据集Xi及上一滑动窗口数据集Xi-1分别为：
$\left\{\begin{array}{l}X_{i}=\left\{L^{\prime}(j) \mid j=n-N, \cdots, j=n\right\} \\X_{i-1}=\{L(j) \mid j=n-2 N-S, \cdots, j=n-N-S\}\end{array}\right.$
式中，N为设定的滑动窗口长度；S为滑动窗口滑动步长。
每个滑动窗口作为后续融合方法的一个计算单元，通过对其数据集进行小波分解及系数加权融合处理，获得最终结果序列中对应的输出值，前后滑动窗口内的数据重叠率高，最终输出结果的平滑性较好。
2.2 多尺度分解
分别对每个滑动窗口的静态与RTK位移监测结果序列进行离散小波多尺度分解，分离窗口内监测结果的位移波动，提取真实位移趋势。考虑到位移监测结果可能发生突变及滑动窗口数据处理的实时性需求，本文选择计算效率高、对突变信号敏感的Haar小波函数。
Haar小波是最简单的正交紧支撑小波，支撑长度仅为1，其低通滤波器系数h(n)和高通滤波器系数g(n)取值仅在n为0或1时非零：
$\left\{\begin{array}{l}h(0)=h(1)=\frac{1}{\sqrt{2}} \\g(0)=\frac{1}{\sqrt{2}}, g(1)=-\frac{1}{\sqrt{2}}\end{array}\right.$
结合式(4)可得到第j层的近似与细节系数公式为：
$\left\{\begin{array}{l}A_{j}(k)=\frac{1}{\sqrt{2}}\left(A_{j-1}(2 k)+A_{j-1}(2 k+1)\right) \\D_{j}(k)=\frac{1}{\sqrt{2}}\left(A_{j-1}(2 k)-A_{j-1}(2 k+1)\right)\end{array}\right.$
Haar小波将式(4)中原本的卷积计算变成了简单的加减法计算，使其能够快速完成信号的分解与重构，占用的计算资源较少。
随着多尺度分解层数的增加，可逐步降低位移监测结果数据中的高频波动，使近似系数的趋势更平稳。但过度分解会导致计算效率降低，且层数越多将导致边界效应越明显，使分解后的边界点与真实值偏差愈发显著，因此需要选择合适的分解层数。
由于Mallat算法每层数据的长度减半，因此最大理论层数为：
$K \leqslant \log _{2} N $
式中，N为滑动窗口的数据长度。在处理坐标时间序列时，综合考虑效果与效率，一般分解层数选为3~4层[16-17]。
2.3 加权重构
分别对滑动窗口的静态和RTK位移监测结果序列进行多尺度分解后，得到各自表征真实位移趋势的近似系数A1和A2及表征位移波动的细节系数。在融合过程中，舍弃细节系数，对两者的近似系数进行加权融合。采用中误差平方的倒数作为权重比例，即
$\left\{\begin{array}{l}w_{1}=\frac{1 / \sigma_{1}^{2}}{1 / \sigma_{1}^{2}+1 / \sigma_{2}^{2}} \\w_{2}=\frac{1 / \sigma_{2}^{2}}{1 / \sigma_{1}^{2}+1 / \sigma_{2}^{2}}\end{array}\right.$
式中，σ1和σ2分别为监测设备标称的静态与RTK测量精度，也可根据滑动窗口静态与RTK位移监测结果序列的中误差确定。融合后的新小波近似系数为：
$A_{\text {fuse }}=w_{1} A_{1}+w_{2} A_{2}$
基于近似系数Afuse进行小波重构，即可得到该滑动窗口融合了静态与RTK位移监测结果的新位移序列。为避免小波变换导致的边界效应，选择融合后的新位移序列滑动窗口中点对应的位移值作为该滑动窗口的输出结果，计算滑动窗口的输出结果与对应时刻的RTK位移监测结果的差值：
$\varDelta=\left|T_{\mathrm{RTK}}-T_{\text {fuse }}\right|$
当Δ>ξ时, 认为发生了突发位移，此时静态位移监测结果已失去意义，将其权重设为0，使用表征RTK位移监测结果趋势的近似系数进行小波重构。阈值ξ可基于式(10)中σ2取值。
2.4 数据融合流程
基于北斗/GNSS技术开展变形监测时，为顾及监测精度，一般采用短基线方式布设参考基准站与变形监测站。设备实时采集原始观测数据并回传至数据处理中心，由数据处理软件解算得到相应的监测结果。
本文方法要求数据处理软件同步解算并输出静态与RTK监测结果，作为融合算法的输入项。在RTK监测结果输出时刻，检查静态监测结果是否更新，如未更新，则取上一解算周期结果，分别加入到RTK与静态监测序列中。由监测序列构建时间滑动窗口，对窗口内的RTK与静态监测数据进行融合处理，输出当前窗口结果并按照设定的步长等待监测数据更新，然后滑动到下一时间窗口进行处理。融合方法的流程见图 1。
3 实验验证
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3.1 实验数据
为验证融合方法的有效性，在GNSS超短基线场布置模拟环境进行测试实验(图 2)。在GBW1号观测墩布设参考站，在GBW2号观测墩布设监测站。将监测站对天观测天线布置于精密位移平台上，该平台可通过滑轨带动观测天线进行平面位移操作，水平方向最大10 cm、高程方向最大1 cm，通过滑轨标尺可得知准确位移量。
测试环境布置完成后，将精密位移平台的位移量设置为0并采集1 d的原始观测数据用于对参考站与监测站相对位置进行标定，设定为初始状态。由于采用的精密位移平台在高程方向的最大位移量小于监测设备RTK测量的标称精度，因此本实验通过水平方向的位移开展融合方法验证。
实验共采集了7 h观测数据。首先在初始状态下，从T时前开始采集参考站与监测站观测数据，在T+4时整点时刻前2 min开始匀速调整位移平台滑轨，使其在平面滑轨方向位移20 mm后保持静止，连续观测3 h。实验使用的参考站与监测站设备型号一致，设备详细参数及算法配置参数见表 1。
3.2 结果分析
分别将实验期间静态、RTK及融合方法输出的坐标结果转换为以参考站为原点的站心ENU坐标系变化值，通过与真实位移量进行对比可以验证融合方法对于突发形变识别的有效性与监测精度。
由于精密位移平台的滑轨方向基本指向北，因此从图 3(a)可以看出，在模拟的突然变形阶段，水平E方向上仅有微小位移，融合结果整体与静态监测结果贴合。由图 3(b)可以看出，水平N方向在未发生形变时，本文方法静态监测结果多尺度分解出的近似系数权重大，使得融合结果整体贴合静态监测结果并遵循RTK监测结果的波动趋势，围绕静态监测结果产生微小幅度波动，而RTK监测结果以5 mm幅度上下振荡。当T+4时刻突然发生20 mm位移时，本文方法在加权重构阶段舍弃静态监测结果，采用RTK监测结果经多尺度分解出的近似系数进行小波重构，使得融合结果能够快速反映突发变形且波动幅度相对较低。静态监测结果需要在变形发生后的下一整点解算时刻(T+5)跳变至位移真值附近，此时融合结果重新贴合静态监测结果。
3种结果的水平位移量与对应真实位移量的误差结果见图 4。
为进一步验证本文融合方法的有效性，选取常用的卡尔曼滤波方法对RTK监测结果进行处理，并将其结果与本文融合方法所得的结果进行对比分析。由于监测对象一般处于稳定或缓慢沉降状态，因此卡尔曼滤波的动力学模型选择为常速度模型，将监测点的位移值和速度作为状态向量。测量噪声通过T时至T+1时未发生位移的RTK监测数据进行计算，由于实验过程中位移变化通过精密位移平台匀速调整，因此过程噪声设为固定值且取值越小越好[18]，本文设为测量噪声的0.01倍。
RTK监测结果的卡尔曼滤波值与真实位移量的误差见图 5，可以看出，相较于原始RTK监测结果，卡尔曼滤波结果的精度虽有一定提升，但与本文融合方法相比仍存在明显差距。
突发变形前、开始突然变形至静态解算结果更新前(T+5时)、静态解算结果更新后(T+5时至T+7时)3个阶段不同监测方法结果的误差最大值与均方根误差见表 2。
可以看出，突发变形前，本文融合方法的精度与静态监测结果相近；突发变形后，静态监测结果不能及时更新导致误差较大，最大值为平台位移完成后静态监测结果(T+3时刻值)与实际位移真值的差值；本文融合方法在平台开始位移但未达到设定的突变检测阈值时，静态监测结果的权重较大，导致此时误差大于RTK监测结果，但在之后能够快速跟随RTK监测结果及时反映突发变形，且由于经过滑动窗口的趋势提取，整体精度优于RTK监测结果。当T+5时刻静态监测结果完成更新后，本文融合方法的输出精度可与静态监测结果水平维持一致。
4 结语
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本文基于离散小波的多尺度分解理论，通过建立时间滑动窗口，将静态与RTK监测结果划分为局部数据段，分解提取局部趋势，经过系数加权、小波重构与残差判断，将2种监测结果进行处理并输出融合结果，得到以下结论。
1) 相较于单独采用静态或RTK解算，本文方法在提高监测结果输出频率的同时，能够在监测对象稳定或缓慢变形时提供静态mm级监测精度结果，在监测点位移发生时及时反映突发形变，为后续的报警发布提供支撑。
2) 本文方法基于目前市场上通用的GNSS数据处理软件即可实现，具有较好的工程应用前景。
3) 本文实验在超短基线场中进行，观测条件较好，后续还需要在环境复杂的现场开展相关测试。
4) 本文方法的时间滑动窗口长度越大，得到的结果序列越平滑，但由于每个滑动窗口的输出结果为窗口中点的融合结果，导致输出结果存在时间窗口大小一半的延迟，需要根据实际需求进行选择。
基金
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  • 中国国家铁路集团有限公司科技研究开发计划(Q2025X011)
文献
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2026年第46卷第9期
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doi: 10.14075/j.jgg.2025.12.406
  • 接收时间：2025-12-02
  • 首发时间：2026-10-30
  • 出版时间：2026-09-15
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  • 收稿日期：2025-12-02
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中国国家铁路集团有限公司科技研究开发计划(Q2025X011)
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    1 中国铁道科学研究院集团有限公司电子计算技术研究所, 北京, 100081
    2 北京经纬信息技术有限公司, 北京, 100081
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