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Article(id=1322941057054417227, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322940829001724376, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2024.06.289, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1718726400000, receivedDateStr=2024-06-19, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793343327058, onlineDateStr=2026-10-30, pubDate=1747238400000, pubDateStr=2025-05-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793343327058, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793343327057, creator=13701087609, updateTime=1793343327057, 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=506, endPage=511, ext={EN=ArticleExt(id=1322941057243160908, articleId=1322941057054417227, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Noise Reduction of GPS Coordinate Time Series Based on GAN, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

This paper applies generative adversarial networks (GAN) to the denoising of GPS coordinate time series. The generator produces time series data, which is then distinguished from clean time series data by the discriminator. During training, the generator is iteratively optimized until the discriminator can no longer differentiate between the generated data and the real clean data, thereby achieving the denoising effect. Simulation results show that, compared with traditional methods such as WNNM, BiLSTM, and SSA, the GAN method achieves the closest correlation with the original noise in key indicators such as residual amplitude, residual spectral index, and velocity uncertainty, demonstrating more efficient denoising performance. Experimental results using real data further validate the effectiveness of the proposed algorithm in denoising actual GPS coordinate time series.

, authors=Xinyu ZHOU1, Gang CHEN2, Jiawen BIAN1, *, authorsList=Xinyu ZHOU, Gang CHEN, Jiawen BIAN, authorCompany=null, correspAuthors=Jiawen BIAN, authorNote=

ZHOU Xinyu, postgraduate, majors in deep learning and GPS coordinate time series, 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=1322941061051588962, articleId=1322941057054417227, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=基于GAN的GPS坐标时间序列降噪, 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=

将生成对抗网络(generative adversarial networks, GAN)应用于GPS坐标时间序列的降噪处理中, 通过生成器生成时间序列数据, 并利用判别器对生成数据与干净的时间序列数据进行区分。在训练过程中, 不断迭代优化生成器, 直至判别器无法区分生成数据与真实干净数据, 从而实现降噪效果。仿真实验结果表明, 与WNNM、BiLSTM和SSA等传统方法相比, GAN方法在残差振幅、残差谱指数和速度不确定度等关键指标上与原始噪声的相关指标最接近, 展现出更高效的降噪性能。真实数据的实验结果进一步验证了本文算法在实际GPS坐标时间序列降噪中的有效性。

, authors=周新宇1, 陈刚2, 边家文1, *, authorsList=周新宇, 陈刚, 边家文, authorCompany=null, correspAuthors=边家文, authorNote=

周新宇, 硕士生, 主要研究方向为深度学习与GPS坐标时间序列, E-mail:

, correspAuthorsNote=

边家文, 博士, 副教授, 主要研究方向为时间序列分析与统计信号处理, E-mail:

BIAN Jiawen, PhD, associate professor, majors in time series analysis and statistical signal processing, E-mail:

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ZHOU Xinyu, postgraduate, majors in deep learning and GPS coordinate time series, E-mail:

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caption=Denoised GPS coordinate time series by various methods in U direction, figureFileSmall=ecSyPKRtxMaZLXlD92VUNw==, figureFileBig=Gv04q5kcxDzhefgm+mI2QA==, tableContent=null), ArticleFig(id=1322941070316806548, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322941057054417227, language=CN, label=图10, caption=U方向上各方法GPS时间序列降噪, figureFileSmall=ecSyPKRtxMaZLXlD92VUNw==, figureFileBig=Gv04q5kcxDzhefgm+mI2QA==, tableContent=null), ArticleFig(id=1322941070404886933, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322941057054417227, language=EN, label=Tab. 1, caption=

Index values of various methods when noise amplitude is 10 mm/a0.25

, figureFileSmall=null, figureFileBig=null, tableContent=
方法 不带趋势项 带趋势项
k 残差振幅A/mm·a-0.25 速度不确定度/mm·a-1 misfit k 残差振幅A/mm·a-0.25 速度不确定度/mm·a-1 misfit
SSA -0.967 3 9.502 7 0.035 0 1.345 7 -0.970 7 9.531 7 0.034 6 1.237 6
MOLS -0.942 0 9.313 5 0.038 2 2.205 1 -0.946 6 9.421 0 0.037 9 2.193 6
WNNM -0.958 6 9.532 7 0.036 4 1.598 0 -0.961 5 9.553 9 0.036 0 1.461 7
BiLSTM -0.975 7 9.666 4 0.034 3 1.017 2 -0.978 1 9.721 7 0.034 2 0.972 6
GAN -0.980 3 9.720 4 0.033 8 0.871 4 -0.981 6 9.736 1 0.033 7 0.868 9
真实值 -0.988 5 9.879 9 0.033 2 -0.988 5 9.879 9 0.033 2
), ArticleFig(id=1322941070723654038, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322941057054417227, language=CN, label=表1, caption=

噪声振幅为10 mm/a0.25情形下各种方法的指标值

, figureFileSmall=null, figureFileBig=null, tableContent=
方法 不带趋势项 带趋势项
k 残差振幅A/mm·a-0.25 速度不确定度/mm·a-1 misfit k 残差振幅A/mm·a-0.25 速度不确定度/mm·a-1 misfit
SSA -0.967 3 9.502 7 0.035 0 1.345 7 -0.970 7 9.531 7 0.034 6 1.237 6
MOLS -0.942 0 9.313 5 0.038 2 2.205 1 -0.946 6 9.421 0 0.037 9 2.193 6
WNNM -0.958 6 9.532 7 0.036 4 1.598 0 -0.961 5 9.553 9 0.036 0 1.461 7
BiLSTM -0.975 7 9.666 4 0.034 3 1.017 2 -0.978 1 9.721 7 0.034 2 0.972 6
GAN -0.980 3 9.720 4 0.033 8 0.871 4 -0.981 6 9.736 1 0.033 7 0.868 9
真实值 -0.988 5 9.879 9 0.033 2 -0.988 5 9.879 9 0.033 2
), ArticleFig(id=1322941070794957207, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322941057054417227, language=EN, label=Tab. 2, caption=

Results of each index in three directions at BJFS station

, figureFileSmall=null, figureFileBig=null, tableContent=
方向 方法 k 残差振幅A/mm·a-0.25 速度不确定度/mm·a-1
N SSA -0.953 2 3.521 1 0.013 8
MOLS -0.947 2 3.214 1 0.012 9
WNNM -0.966 1 3.678 1 0.013 6
BiLSTM -0.962 3 3.651 3 0.013 7
GAN -0.958 9 3.432 7 0.013 1
E SSA -0.782 5 3.327 1 0.026 1
MOLS -0.722 1 3.059 7 0.030 2
WNNM -0.736 2 3.472 5 0.032 5
BiLSTM -0.765 9 3.502 1 0.029 3
GAN -0.771 4 3.274 6 0.026 8
U SSA -0.751 4 9.978 1 0.088 2
MOLS -0.609 2 9.421 4 0.141 3
WNNM -0.702 6 10.512 8 0.111 9
BiLSTM -0.721 5 10.172 6 0.100 8
GAN -0.772 3 9.872 1 0.080 5
), ArticleFig(id=1322941070874648984, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322941057054417227, language=CN, label=表2, caption=

BJFS站3个方向上各指标计算结果

, figureFileSmall=null, figureFileBig=null, tableContent=
方向 方法 k 残差振幅A/mm·a-0.25 速度不确定度/mm·a-1
N SSA -0.953 2 3.521 1 0.013 8
MOLS -0.947 2 3.214 1 0.012 9
WNNM -0.966 1 3.678 1 0.013 6
BiLSTM -0.962 3 3.651 3 0.013 7
GAN -0.958 9 3.432 7 0.013 1
E SSA -0.782 5 3.327 1 0.026 1
MOLS -0.722 1 3.059 7 0.030 2
WNNM -0.736 2 3.472 5 0.032 5
BiLSTM -0.765 9 3.502 1 0.029 3
GAN -0.771 4 3.274 6 0.026 8
U SSA -0.751 4 9.978 1 0.088 2
MOLS -0.609 2 9.421 4 0.141 3
WNNM -0.702 6 10.512 8 0.111 9
BiLSTM -0.721 5 10.172 6 0.100 8
GAN -0.772 3 9.872 1 0.080 5
), ArticleFig(id=1322941070945952153, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322941057054417227, language=EN, label=Tab. 3, caption=

Correlation coefficients of the denoised time series by various methods in three directions at BJFS station

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方向 方法 SSA MOLS WNNM BiLSTM GAN
N SSA 1
MOLS 0.995 1
WNNM 0.997 0.993 1
BiLSTM 0.998 0.995 0.996 1
GAN 0.999 0.992 0.993 0.997 1
E SSA 1
MOLS 0.993 1
WNNM 0.996 0.996 1
BiLSTM 0.998 0.993 0.996 1
GAN 0.999 0.992 0.996 0.997 1
U SSA 1
MOLS 0.971 1
WNNM 0.973 0.974 1
BiLSTM 0.976 0.976 0.977 1
GAN 0.981 0.982 0.982 0.983 1
), ArticleFig(id=1322941071017255322, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322941057054417227, language=CN, label=表3, caption=

BJFS站3个方向上各方法去噪时间序列的相关系数

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方向 方法 SSA MOLS WNNM BiLSTM GAN
N SSA 1
MOLS 0.995 1
WNNM 0.997 0.993 1
BiLSTM 0.998 0.995 0.996 1
GAN 0.999 0.992 0.993 0.997 1
E SSA 1
MOLS 0.993 1
WNNM 0.996 0.996 1
BiLSTM 0.998 0.993 0.996 1
GAN 0.999 0.992 0.996 0.997 1
U SSA 1
MOLS 0.971 1
WNNM 0.973 0.974 1
BiLSTM 0.976 0.976 0.977 1
GAN 0.981 0.982 0.982 0.983 1
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基于GAN的GPS坐标时间序列降噪
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周新宇 1 , 陈刚 2 , 边家文 1, *
大地测量与地球动力学 | 现代大地测量 2025,45(5): 506-511
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大地测量与地球动力学 |现代大地测量 2025 , 45 (5) : 506 -511
基于GAN的GPS坐标时间序列降噪
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周新宇1 , 陈刚2, 边家文1, *
作者信息
  • 1 中国地质大学(武汉)数学与物理学院, 武汉市鲁磨路388号, 430074
  • 2 中国地质大学(武汉)海洋学院, 武汉市鲁磨路388号, 430074
通讯作者:

边家文, 博士, 副教授, 主要研究方向为时间序列分析与统计信号处理, E-mail:

BIAN Jiawen, PhD, associate professor, majors in time series analysis and statistical signal processing, E-mail:

作者简介:

周新宇, 硕士生, 主要研究方向为深度学习与GPS坐标时间序列, E-mail:

Noise Reduction of GPS Coordinate Time Series Based on GAN
Xinyu ZHOU1 , Gang CHEN2, Jiawen BIAN1, *
Affiliations
  • 1 School of Mathematics and Physics, China University of Geosciences, 388 Lumo Road, Wuhan 430074, China
  • 2 College of Marine Science and Technology, China University of Geosciences, 388 Lumo Road, Wuhan 430074, China

    • ZHOU Xinyu, postgraduate, majors in deep learning and GPS coordinate time series, E-mail:

出版时间: 2025-05-15 doi: 10.14075/j.jgg.2024.06.289
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摘要
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将生成对抗网络(generative adversarial networks, GAN)应用于GPS坐标时间序列的降噪处理中, 通过生成器生成时间序列数据, 并利用判别器对生成数据与干净的时间序列数据进行区分。在训练过程中, 不断迭代优化生成器, 直至判别器无法区分生成数据与真实干净数据, 从而实现降噪效果。仿真实验结果表明, 与WNNM、BiLSTM和SSA等传统方法相比, GAN方法在残差振幅、残差谱指数和速度不确定度等关键指标上与原始噪声的相关指标最接近, 展现出更高效的降噪性能。真实数据的实验结果进一步验证了本文算法在实际GPS坐标时间序列降噪中的有效性。

GPS坐标时间序列  /  深度学习  /  GAN  /  信号降噪

This paper applies generative adversarial networks (GAN) to the denoising of GPS coordinate time series. The generator produces time series data, which is then distinguished from clean time series data by the discriminator. During training, the generator is iteratively optimized until the discriminator can no longer differentiate between the generated data and the real clean data, thereby achieving the denoising effect. Simulation results show that, compared with traditional methods such as WNNM, BiLSTM, and SSA, the GAN method achieves the closest correlation with the original noise in key indicators such as residual amplitude, residual spectral index, and velocity uncertainty, demonstrating more efficient denoising performance. Experimental results using real data further validate the effectiveness of the proposed algorithm in denoising actual GPS coordinate time series.

GPS coordinate time series  /  deep learning  /  GAN  /  signal denoise
周新宇, 陈刚, 边家文. 基于GAN的GPS坐标时间序列降噪. 大地测量与地球动力学, 2025 , 45 (5) : 506 -511 . DOI: 10.14075/j.jgg.2024.06.289
Xinyu ZHOU, Gang CHEN, Jiawen BIAN. Noise Reduction of GPS Coordinate Time Series Based on GAN[J]. Journal of Geodesy and Geodynamics, 2025 , 45 (5) : 506 -511 . DOI: 10.14075/j.jgg.2024.06.289
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GPS坐标时间序列分析面临线性趋势、周期性信号和多源噪声的挑战[1]，这些噪声包括白噪声、闪烁噪声和随机漫步噪声等[2]。为了准确提取构造信号，研究者们采用了多种降噪方法，如SSA[3]、MOLS[4]、WNNM[5]和TNNR[6]等。深度学习方法凭借在特征学习和解决复杂问题方面的优势，也被应用于GPS时间序列的建模和预测，包括LSTM[7-8]、VMD-LSTM[9]、BiLSTM[10]等。生成对抗网络(GAN)的核心思想源于博弈论中的纳什均衡[11]，它由生成网络和判别网络组成。生成器的主要目的是通过样本学习真实数据的分布，它以随机噪声作为输入，生成虚假数据样本，使这些虚假样本尽可能接近真实样本的分布。生成器的目标是生成能够欺骗判别器的合成数据，使其无法区分生成数据和真实数据。判别器也是一个神经网络，实质是个二分类器，它接收来自生成器的合成数据和真实数据的输入，将生成网络生成的虚假样本判定为假，将输入的真实样本数据判定为真，生成器和判别器相互竞争，从而使整个模型的性能不断提升。Yin等[11]将GAN用于脑电图(electroencephalography, EEG)信号中，提出一个由CNN和Transformer构成的生成器用于捕获局部和全局时间依赖性，判别器则纠正干净和去噪EEG信号之间的整体不一致性。实验结果表明，相较于LSTM等方法，该方法的RRMSE平均降低11.15%，SNR提高9.81%。
基于上述研究，本文提出一种基于GAN的GPS坐标时间序列降噪方法。生成器采用卷积自编码架构，以提取全局特征。由于经典GAN训练过程中容易出现梯度消失问题，本文用最小二乘损失替代交叉熵损失。为了更好地适应降噪任务，引入距离函数和局部最大差函数，以约束整体与局部特征，从而获得更精确的结果，提升模型的整体降噪能力。
1 基于GAN的GPS坐标时间序列降噪
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1.1 时变振幅GPS坐标时间序列模型
GPS坐标时间序列一般由趋势项、周期项和噪声项组成，带有时变振幅的GPS坐标时间序列可表示为[12]：
$\begin{gathered}y(t)=y_0+\nu t+\sum\limits_{i=1}^2 a_i(t) \cos \left(2 \pi f_i t\right)+ \\b_i(t) \sin \left(2 \pi f_i t\right)+\varepsilon(t)\end{gathered}$
式中，ai(t)和bi(t)(i=1, 2)分别为年周期和半年周期对应的季节振幅；f1和f2分别为年周期和半年周期的频率；ε(t)为观测噪声。ai(t)和bi(t)可以表达为[13]：
$\begin{aligned}& a_i(t)=a_i+\mathrm{e}^{p \sin t} \\& b_i(t)=b_i+\mathrm{e}^{q \sin t}\end{aligned}$
式中，ai(t)(i=1, 2, …, c)和bi(t)(i=1, 2, …, c)为季节信号的常数项振幅；epsint和eqsint用于描述GPS坐标时间序列中时变的季节性变化，p和q为波动参数。
1.2 生成对抗网络
将生成对抗网络的生成网络记为G，判别网络记为D，噪声数据分布记为pz(z)，将随机初始化的噪声作为G的输入得到输出G(z)，由D判断G(z)是来自初始数据分布或者由G生成。模型G和D同时训练，固定模型D，调整G的参数使得lg[1－D(G(z))]的期望最小化，使判别器尽量犯错；固定模型G，调整D的参数使lgD(x)+lg[1－D(G(z))]的期望最大化，因此目标函数为：
$\begin{gathered}\min _G \max _D V(D, G)= \\E_{x \sim p_{\text {data }}(x)}[\lg D(x)]+ \\E_{z \sim p_z(z)}[\lg (1-D(G(z)))]\end{gathered}$
式中，x为真实数据；z为随机噪声变量；G(z)为G生成的数据，该数据服从真实数据的分布Pdata(x)；Pz(z)为输入变量为z的先验分布；E为数学期望。
生成对抗网络的缺点之一是没有指定生成数据的方向，当输入数据复杂度高、数量多时，生成器总是生成同样的样本点，无法训练出期望的数据。为解决GAN训练结果不稳定、生成目标不明确的问题，Masci等[14]提出条件生成对抗网络(conditional generative adversarial networks, CGAN)，即在生成网络和判别网络中加入额外信息y，让额外信息作为条件，指导数据生成，其架构图如图1~3所示。
经典生成对抗网络的损失函数是交叉熵函数，这种损失函数的缺点是非常容易过拟合，并且函数的输出并不是零均值的，对于非常小的数据点x，交叉熵损失函数容易忽略x到决策边界的距离，这就可能会导致学习过程中存在梯度消失问题。为克服这个问题，本文使用最小二乘生成对抗网络(least squares GAN, LSGAN)中的最小二乘损失函数来代替交叉熵损失函数[14]，其中G和D的目标函数如下：
$\left.\min _G V_{\mathrm{LSGAN}}(G)=\frac{1}{2} E_{z \sim p_z(z)}[D(G(z))-1)^2\right]$
$\begin{gathered}\min _G V_{\mathrm{LSGAN}}(D)=\frac{1}{2} E_{x \sim p_{\text {data }}(x)}\left[(D(x)-1)^2\right]+ \\\frac{1}{2} E_{z \sim p_z(z)}\left[(D(G(z)))^2\right]\end{gathered}$
1.3 生成器与判别器
本文采用卷积自编码网络构建生成器，卷积层能够在潜在的高级特征表示中保持空间局部性和输入的邻域关系[15]。GAN本质上是学习从输入变量z到实际数据的映射，而CGAN则是学习从条件y与输入变量z到实际数据的映射。鉴于本文的主要任务是针对GPS坐标时间序列进行降噪，因此需要通过条件y生成期望的数据；而对于变量z的取舍，研究发现，删除输入变量z的情况下，模型降噪性能得到提高[16]，并且保留了更多细节。借鉴图像处理领域的经验，本文在后续实验中不再添加输入变量z。
本文使用的编码器由7个卷积层组成，尺寸分别为512×1、512×16、512×32、512×64、256×128、128×256和32×1 024。解码器是编码器的镜像，每个卷积之间设置PReLU作为激活函数。为便于梯度反传，设置跳跃连接。由于池化层会丢失信息，因此不设置池化层[14]。
对于LSGAN目标函数，本文通过剔除变量z，并且考虑添加距离函数以及局部最大差函数来完善LSGAN的目标函数，以达到降噪目的。2个函数定义如下：
$l_{\mathrm{dist}}=\sqrt{\sum\limits_{n=1}^N\left|\hat{x}_n-x_n\right|}$
$l_{\max }=\max \left(\left|\hat{x}_1-x_1\right|, \cdots, \left|\hat{x}_N-x_N\right|\right)$
式中，N为样本数；$ \hat{x}_n$为降噪后的信号；xN为干净信号。得到生成器的损失函数为：
$\begin{gathered}\min _G V(G)= \\\left.E_{\tilde{x} \sim p_{\text {noisy }}(\tilde{x})}[(D(G(\tilde{\boldsymbol{x}}), \tilde{\boldsymbol{x}}))-1)^2\right]+ \\\lambda_1 l_{\text {dist }}+\lambda_2 l_{\max }\end{gathered}$
式中，$ P_{\text {noisy }}(\tilde{x})$表示噪声信号$\tilde{\boldsymbol{x}}$的分布；λ1和λ2为用于调节ldist和lmax的权重系数。
判别器使用4个卷积层和1个全连接层，每个卷积层后面是虚拟批处理规范化层和LeakyReLU激活函数层。在最终的全连接层之后，sigmoid函数输出0~1之间的概率值，其结构如图4所示。图4左侧为GPS坐标时间序列降噪过程，右侧为模型的训练过程。本文采用卷积自编码网络构建包含编码和解码阶段的生成器，其输入是带噪声的信号，输出是降噪后的信号。判别器接受一对信号作为输入，该对信号是降噪后的信号和干净信号，并对生成器生成的降噪信号与干净信号进行区分。通过生成器和判别器之间的对抗性训练，生成器不断学习输入信号和原始干净信号之间的差异，判别器通过其自身输出形成的判别器损失函数来更新D的参数。生成器通过由其自身输出和判别器输出组成的损失函数来更新G的参数。模型训练完成后，保存生成器模型，之后将带噪信号输入到保存的模型中即可输出降噪信号。
2 实验结果与分析
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2.1 实验数据及网络参数设置
从全球671个IGS站点的分析来看[13]，90%以上GNSS坐标时间序列U方向的年周期项振幅在0~8 mm之间，半年周期项振幅在0~2 mm波动；而N和E方向的年周期项振幅在0~2 mm之间，半年周期项振幅小于1 mm；噪声类型为闪烁噪声，振幅在7~20 mm之间。基于此，模拟产生10 000条仿真数据，年周期振幅在0~8 mm之间，半年周期振幅在0~2 mm之间，时变振幅的p和q设置为0.3，以模拟实际GPS坐标时间序列的时序变化；同时加入0~20 mm/a0.25的闪烁噪声，每条数据的闪烁噪声振幅递增1 mm，依次循环，以便能够自适应地完成不同振幅的GPS坐标时间序列降噪任务。
选取misfit、残差振幅、残差谱指数、速度不确定度等4个指标定量分析GAN的降噪效果，并且与WNNM、MOLS、SSA、BiLSTM方法进行对比。MOLS和SSA的参数设置与前文保持一致，BiLSTM的隐藏层数为2，隐藏层节点数为256，然后由全连接将输出结果转换为与输入数据相同的维度，选择Adam优化器，学习率α设置为0.001，损失函数选用均方误差，批次大小设置为64，训练1 000轮。GAN优化器选用Adam，学习率α设置为0.001，批次大小设置为64，训练1 000轮。
2.2 仿真实验结果分析
GAN、WNNM、SSA、BiLSTM和MOLS在噪声幅度为10 mm/a0.25时的降噪信号分别如图5、6所示，图5为不带趋势项的结果，图6为带趋势项的结果。结合图5、6可知，无论是否带有趋势项，在噪声振幅为10 mm/a0.25时，GAN方法降噪后的信号更贴合干净信号。从定量分析来看，GAN方法的misfit最小，同时残差振幅、残差谱指数和速度不确定度都是所有方法中最接近真实值的(表1)，说明该方法降噪效果较好，能够较为充分地去除观测信号中的噪声。从图7功率谱密度(power spectral density, PSD)来看，在闪烁噪声振幅为10 mm/a0.25的条件下，PSD曲线在1 cpy和2 cpy处都有明显的波谷，每种方法距离真实噪声的PSD曲线均有一定差距，说明在估计年和半年周期信号时都或多或少地吸收了一部分噪声；在低频区域，也就是1 cpy之前，GAN方法比其他方法的PSD曲线更接近噪声的PSD曲线，这表明GAN方法能更加准确地提取GPS坐标时间序列中的季节信号。
2.3 实际GPS坐标时间序列分析
实际数据为北京房山站(BJFS)2001-09-09—2020-01-17的N、E、U三个方向带有趋势项的真实观测数据，该观测数据可以通过SOPAC下载。实验前使用3σ准则剔除缺失值与异常值[17]，使用基于KKF插值的方法填补缺失数据[18]，再使用BiLSTM、WNNM、MOLS、SSA和GAN方法对数据进行降噪处理(图8~10)。由于实际站点的无噪声序列未知，因此不计算misfit。
从图8~10可以看出，在N和E方向上，线性趋势占主导地位，季节性变化并不十分明显；U方向上可以看到明显的季节性信号。从降噪效果来分析，N和E分量上的噪声振幅不是很大，各方法降噪效果差异不是很明显，各方法对GPS坐标时间序列的趋势项信号都能够体现出来。U方向上的噪声振幅明显比其他方向的噪声振幅大，各方法降噪效果有所差异：MOLS更容易受到噪声的干扰，当时间序列长度较短时，U方向上的周期信号基本能够体现出来，但是随时间序列长度增加，MOLS不能很好地体现时间序列中的周期信号，同时MOLS方法降噪的时间序列在U方向存在不光滑的问题，说明该方法可能拟合了一部分噪声；其他方法在均能较好地拟合周期信号，并且GAN方法拟合的曲线更为平滑。
为了更好地体现各个方法的差异性，计算每个方法获得的速度不确定度、谱指数、残差振幅和每个方法之间的相关系数，各个指标及相关系数分别如表2、3所示。可以看出，GAN方法与其他4种方法的相关系数接近1，其速度不确定度、谱指数和残差振幅的结果与其他4种方法都很接近，进一步印证了GAN用于GPS坐标时间序列降噪的有效性。
3 结语
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本文引入深度学习领域中的生成对抗网络，结合卷积自编码网络的生成器以及最小二乘生成对抗网络的损失函数，保持高级特征的空间局部性和输入的邻域关系，并使用该模型组成的生成对抗网络对GPS坐标时间序列含有的非平稳噪声进行降噪。在模拟实验中，分别进行带有趋势项和不带有趋势项下的不同噪声振幅实验，从拟合曲线、指标分析和功率谱密度分析来看，本文提出的基于GAN的降噪方法相比于其他方法能够更好地拟合GPS坐标时间序列，同时各项指标相对于WNNM、BiLSTM、SSA等方法都是最接近真实值的，说明该方法的降噪效果更加优秀。同时选取BJFS站N、E和U三个方向的真实数据进行实验。结果表明，GAN与WNNM、BiLSTM、SSA的评价指标值均比较接近，并表现出较高的相关性，证实了GAN用于GPS坐标时间序列降噪的有效性。
基金
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  • 国家自然科学基金(42274012)
文献
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参考文献 引证文献
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1
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Williams S D P , Bock Y , Fang P , et al. Error Analysis of Continuous GPS Position Time Series[J]. Journal of Geophysical Research: Solid Earth, 2004, 109 (B3)
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Chen Q , Dam T , Sneeuw N , et al. Singular Spectrum Analysis for Modeling Seasonal Signals from GPS Time Series[J]. Journal of Geodynamics, 2013, 72, 25- 35
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Klos A , Bos M S , Bogusz J . Detecting Time-Varying Seasonal Signal in GPS Position Time Series with Different Noise Levels[J]. GPS Solutions, 2017, 22 (1)
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2025年第45卷第5期
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文章信息
doi: 10.14075/j.jgg.2024.06.289
  • 接收时间：2024-06-19
  • 首发时间：2026-10-30
  • 出版时间：2025-05-15
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  • 收稿日期：2024-06-19
基金
National Natural Science Foundation of China(42274012)
国家自然科学基金(42274012)
作者信息
    1 中国地质大学(武汉)数学与物理学院, 武汉市鲁磨路388号, 430074
    2 中国地质大学(武汉)海洋学院, 武汉市鲁磨路388号, 430074

通讯作者:

边家文, 博士, 副教授, 主要研究方向为时间序列分析与统计信号处理, E-mail:

BIAN Jiawen, PhD, associate professor, majors in time series analysis and statistical signal processing, E-mail:

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