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Article(id=1322933587015762124, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322933542426116991, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2024.02.085, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1709136000000, receivedDateStr=2024-02-29, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793341546061, onlineDateStr=2026-10-30, pubDate=1736870400000, pubDateStr=2025-01-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793341546061, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793341546061, creator=13701087609, updateTime=1793341546061, updator=13701087609, issue=Issue{id=1322933542426116991, tenantId=1146029695717560320, journalId=1281212831689347082, year='2025', volume='45', issue='1', pageStart='1', pageEnd='110', issueExtLink='null', onlineDate='null', pubDate='1736870400000', pubDateStr='2025-01-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793341535431, creator='13701087609', updateTime=1793341535431, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=86, endPage=90, ext={EN=ArticleExt(id=1322933587246448845, articleId=1322933587015762124, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Detection of Earth's Free Oscillation Modes Based on NTFT, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Large earthquakes can simultaneously excite numerous normal modes of Earth's free oscillations. Due to the Earth's ellipticity, rotation, and internal anisotropy, these normal modes undergo splitting, with frequencies of splitting modes becoming very close(only a few μHz apart). This imposes higher demands on the detection of Earth's free oscillation modes. This article derives and verifies a new method for detecting free oscillation modes based on normal time-frequency transform(NTFT). Taking the detection of 3S1 splitting modes as an example, compared with the classical FT spectrum method and the latest OSE method, our method has higher frequency resolution for detecting free oscillation modes.

, authors=Yingqi ZHANG1, Gong XU1, Chenchen WANG1, Zheng MU1, Yaxin ZHAO1, authorsList=Yingqi ZHANG, Gong XU, Chenchen WANG, Zheng MU, Yaxin ZHAO, authorCompany=null, correspAuthors=null, authorNote=

ZHANG Yingqi, postgraduate, majors in principles and applications of geodesy,

, correspAuthorsNote=null, copyrightStatement=All rights reserved, without authorization, 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=1322933589876277466, articleId=1322933587015762124, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=基于NTFT的地球自由振荡模型检测, columnId=1322506242858271689, journalTitle=大地测量与地球动力学, columnName=地球物理解释, runingTitle=null, highlight=null, articleAbstract=

大地震能够同时激发出许多的地球自由振荡简正模, 且地球的椭率、自转和内部的各向异性也会引起简正模的分裂, 使各单线态之间的频率更接近(仅为几个μHz), 这对地球自由振荡模型的检测提出更高的要求。本文以标准时频变换为基础, 推导并验证一种自由振荡模型检测的新方法。以3S1模型的检测为例, 与经典的FT谱方法和最新的OSE方法相比, 该方法具有更高的频率分辨率。

, authors=张英琦1, 徐工1, 王辰晨1, 穆峥1, 赵亚鑫1, authorsList=张英琦, 徐工, 王辰晨, 穆峥, 赵亚鑫, authorCompany=null, correspAuthors=null, authorNote=

张英琦, 硕士生, 主要从事大地测量原理与应用研究, E-mail:

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ZHANG Yingqi, postgraduate, majors in principles and applications of geodesy,

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ZHANG Yingqi, postgraduate, majors in principles and applications of geodesy,

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张英琦, 硕士生, 主要从事大地测量原理与应用研究, E-mail:

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张英琦, 硕士生, 主要从事大地测量原理与应用研究, E-mail:

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Computers and Geosciences, 2005, 31 (5): 631- 640, articleTitle=Tsoft: Graphical and Interactive Software for the Analysis of Time Series and Earth Tides, refAbstract=null)], funds=[Fund(id=1322940452793615003, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, awardId=41704003, language=EN, fundingSource=National Natural Science Foundation of China(41704003), fundOrder=null, country=null), Fund(id=1322940452873306780, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, awardId=41704003, language=CN, fundingSource=国家自然科学基金(41704003), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1322940444363063901, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, xref=null, ext=[AuthorCompanyExt(id=1322940444371452510, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, companyId=1322940444363063901, language=EN, 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label=图1, caption=3S1模型的仿真信号, figureFileSmall=HIr15l4YkHZc98bCSgJi9g==, figureFileBig=IIu14pSEPywcgCbaQYGpTg==, tableContent=null), ArticleFig(id=1322940449241039494, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=EN, label=Fig. 2, caption=NTFT spectrum of the simulated 3S1 signal, figureFileSmall=00GUEIV7q3EesoUzIpjNoQ==, figureFileBig=tyv9tT+8/4SPEFjxFJBJfA==, tableContent=null), ArticleFig(id=1322940449475920519, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=CN, label=图2, caption=3S1仿真信号的NTFT谱, figureFileSmall=00GUEIV7q3EesoUzIpjNoQ==, figureFileBig=tyv9tT+8/4SPEFjxFJBJfA==, tableContent=null), ArticleFig(id=1322940449635304072, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=EN, label=Fig. 3, caption=Results of the 3S1 mode detection for each stations after the Sumatra earthquake based on FT spectrum, figureFileSmall=mYo4eY6F+i09P8doKWP84w==, figureFileBig=y67dQMHOOWwBi7v4PrJp3Q==, tableContent=null), ArticleFig(id=1322940449765327497, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=CN, label=图3, caption=基于FT谱的苏门答腊地震各台站3S1模型检测结果, figureFileSmall=mYo4eY6F+i09P8doKWP84w==, figureFileBig=y67dQMHOOWwBi7v4PrJp3Q==, tableContent=null), ArticleFig(id=1322940449845019274, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=EN, label=Fig. 4, caption=Results of the 3S1 mode detection for each station after the Sumatra earthquake based on OSE, figureFileSmall=bL7+IjD0BJhgxWCM5vc53w==, figureFileBig=xG7v3upuIbfCL7VsjT4Tuw==, tableContent=null), ArticleFig(id=1322940449916322443, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=CN, label=图4, caption=基于OSE的苏门答腊地震3S1模型检测结果, figureFileSmall=bL7+IjD0BJhgxWCM5vc53w==, 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figureFileBig=GUm0HGomWh6R1Hh4FBRmHg==, tableContent=null), ArticleFig(id=1322940451828925072, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=CN, label=图6, caption=采用不同方法检测到的3S1模型的频率及偏差, figureFileSmall=Wh032Yr991ovC2eCq3UKQA==, figureFileBig=GUm0HGomWh6R1Hh4FBRmHg==, tableContent=null), ArticleFig(id=1322940451896033937, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=EN, label=Tab. 1, caption=

Frequencies of 3S1 mode based on FT spectrum

, figureFileSmall=null, figureFileBig=null, tableContent=
m=-1 m=0 m=1
PREM 0.942 27 0.944 22 0.945 47
NTFT 0.942 07 0.944 11 0.945 68
), ArticleFig(id=1322940451967337106, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=CN, label=表1, caption=

基于FT谱的3S1模型频率

, figureFileSmall=null, figureFileBig=null, tableContent=
m=-1 m=0 m=1
PREM 0.942 27 0.944 22 0.945 47
NTFT 0.942 07 0.944 11 0.945 68
), ArticleFig(id=1322940452030251667, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=EN, label=Tab. 2, caption=

Information about superconducting gravimeters

, figureFileSmall=null, figureFileBig=null, tableContent=
台站名称(缩写) 纬度/(°) 经度/(°) 高程/m
Canberra(CB) -35.320 6 149.007 7 762.749 0
Sutherland(SU) -32.381 4 20.810 9 1 791.000 0
Strasbourg(ST) 48.621 7 7.683 8 180.000 0
Moxa(MO) 50.645 0 11.616 0 455.000 0
Kamioka(KA) 36.425 3 137.308 4 358.000 0
Membach(MB) 50.609 3 6.006 6 250.000 0
), ArticleFig(id=1322940452135109268, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=CN, label=表2, caption=

超导重力仪相关信息

, figureFileSmall=null, figureFileBig=null, tableContent=
台站名称(缩写) 纬度/(°) 经度/(°) 高程/m
Canberra(CB) -35.320 6 149.007 7 762.749 0
Sutherland(SU) -32.381 4 20.810 9 1 791.000 0
Strasbourg(ST) 48.621 7 7.683 8 180.000 0
Moxa(MO) 50.645 0 11.616 0 455.000 0
Kamioka(KA) 36.425 3 137.308 4 358.000 0
Membach(MB) 50.609 3 6.006 6 250.000 0
), ArticleFig(id=1322940452218995349, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=EN, label=Tab. 3, caption=

3S1 mode frequencies for each station after the Sumatra earthquake based on FT spectrum

, figureFileSmall=null, figureFileBig=null, tableContent=
m=-1 m=0 m=1
PREM 0.942 27 0.944 22 0.945 47
CB 0.942 61 - 0.945 79
SU 0.942 77 - 0.945 58
ST 0.942 73 - 0.945 79
MO 0.942 73 - 0.945 79
KA 0.942 65 - 0.945 70
MB 0.942 77 - 0.945 83
均值 0.942 71 - 0.945 75
), ArticleFig(id=1322940452294492822, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=CN, label=表3, caption=

基于FT谱的苏门答腊地震各台站的3S1模型频率

, figureFileSmall=null, figureFileBig=null, tableContent=
m=-1 m=0 m=1
PREM 0.942 27 0.944 22 0.945 47
CB 0.942 61 - 0.945 79
SU 0.942 77 - 0.945 58
ST 0.942 73 - 0.945 79
MO 0.942 73 - 0.945 79
KA 0.942 65 - 0.945 70
MB 0.942 77 - 0.945 83
均值 0.942 71 - 0.945 75
), ArticleFig(id=1322940452374184599, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322933587015762124, language=EN, label=Tab. 4, caption=

3S1 mode frequencies after the Sumatra earthquake based on OSE

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m=-1 m=0 m=1
PREM 0.942 27 0.944 22 0.945 47
OSE 0.942 73 0.945 27 0.945 92
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基于OSE的苏门答腊地震的3S1模型频率

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m=-1 m=0 m=1
PREM 0.942 27 0.944 22 0.945 47
OSE 0.942 73 0.945 27 0.945 92
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3S1 mode frequencies for each station after the Sumatra earthquake based on NTFT

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m=-1 m=0 m=1
PREM 0.942 27 0.944 22 0.945 47
CB 0.942 58 - 0.945 76
SU 0.942 48 - 0.945 66
ST 0.942 62 - 0.945 83
MO 0.942 55 - 0.945 76
KA 0.942 50 - 0.945 75
MB 0.942 51 - 0.945 80
均值 0.942 54 - 0.945 76
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基于NTFT的苏门答腊地震各台站的3S1模型频率

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m=-1 m=0 m=1
PREM 0.942 27 0.944 22 0.945 47
CB 0.942 58 - 0.945 76
SU 0.942 48 - 0.945 66
ST 0.942 62 - 0.945 83
MO 0.942 55 - 0.945 76
KA 0.942 50 - 0.945 75
MB 0.942 51 - 0.945 80
均值 0.942 54 - 0.945 76
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基于NTFT的地球自由振荡模型检测
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张英琦 1 , 徐工 1 , 王辰晨 1 , 穆峥 1 , 赵亚鑫 1
大地测量与地球动力学 | 地球物理解释 2025,45(1): 86-90
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大地测量与地球动力学 |地球物理解释 2025 , 45 (1) : 86 -90
基于NTFT的地球自由振荡模型检测
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张英琦1 , 徐工1, 王辰晨1, 穆峥1, 赵亚鑫1
作者信息
  • 1 山东理工大学建筑工程与空间信息学院, 山东省淄博市新村西路266号, 255000
作者简介:

张英琦, 硕士生, 主要从事大地测量原理与应用研究, E-mail:

Detection of Earth's Free Oscillation Modes Based on NTFT
Yingqi ZHANG1 , Gong XU1, Chenchen WANG1, Zheng MU1, Yaxin ZHAO1
Affiliations
  • 1 School of Civil Engineering and Geomatics, Shandong University of Technology, 266 West-Xincun Road, Zibo 255000, China

    • ZHANG Yingqi, postgraduate, majors in principles and applications of geodesy,

出版时间: 2025-01-15 doi: 10.14075/j.jgg.2024.02.085
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摘要
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大地震能够同时激发出许多的地球自由振荡简正模, 且地球的椭率、自转和内部的各向异性也会引起简正模的分裂, 使各单线态之间的频率更接近(仅为几个μHz), 这对地球自由振荡模型的检测提出更高的要求。本文以标准时频变换为基础, 推导并验证一种自由振荡模型检测的新方法。以3S1模型的检测为例, 与经典的FT谱方法和最新的OSE方法相比, 该方法具有更高的频率分辨率。

标准时频变换  /  傅里叶变换  /  最优序列估计  /  谱峰分裂  /  模型检测

Large earthquakes can simultaneously excite numerous normal modes of Earth's free oscillations. Due to the Earth's ellipticity, rotation, and internal anisotropy, these normal modes undergo splitting, with frequencies of splitting modes becoming very close(only a few μHz apart). This imposes higher demands on the detection of Earth's free oscillation modes. This article derives and verifies a new method for detecting free oscillation modes based on normal time-frequency transform(NTFT). Taking the detection of 3S1 splitting modes as an example, compared with the classical FT spectrum method and the latest OSE method, our method has higher frequency resolution for detecting free oscillation modes.

normal time-frequency transform  /  Fourier transform  /  optimal sequence estimation  /  peak splitting  /  mode detection
张英琦, 徐工, 王辰晨, 穆峥, 赵亚鑫. 基于NTFT的地球自由振荡模型检测. 大地测量与地球动力学, 2025 , 45 (1) : 86 -90 . DOI: 10.14075/j.jgg.2024.02.085
Yingqi ZHANG, Gong XU, Chenchen WANG, Zheng MU, Yaxin ZHAO. Detection of Earth's Free Oscillation Modes Based on NTFT[J]. Journal of Geodesy and Geodynamics, 2025 , 45 (1) : 86 -90 . DOI: 10.14075/j.jgg.2024.02.085
正文
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地球自由振荡蕴含丰富的地球物理学信息，其特性与地球自转、地球的椭率、地球内部的横向非均匀性以及各向异性紧密相关[1-2]。对地球自由振荡的模态分裂和耦合现象进行研究，有助于揭示地球的非均匀性特征，并可用于确定地球的密度结构。因此，地球自由振荡已成为探索地球内部结构，尤其是深层结构的关键工具之一。
地球自由振荡一般采用振荡衰减模型来表征，其有4个重要参数：初始振幅、频率、衰减因子(Q值)和相位。如1066A地球模型是由1 066个简正模反演得到[3]；PREM地球模型参考约1 000个地球自由振荡模态的简并频率及约500个模态的Q值[4]；最新的三维地球模型S40RTS[5]、SAVANI[6]、SP12RTS[7]都采用大量的简正模及其谱峰分裂模型的频率和Q值。精确地检测地球自由振荡简正模及其谱峰分裂，可以有效提高地球大尺度密度结构研究的精度[8-9]。地球自由振荡的振幅和相位与震源的破裂方式和破裂程度密切相关[10-11]，这是目前唯一可对震源机制解进行总体检验和约束的地震学方法[12]。
傅里叶变换可以将时域上的信号转换为频域上的信号，便于对信号的频率特征进行分析和处理。因此，目前大多数的自由振荡检测方法都是基于傅里叶变换发展起来的。例如积谱分析(product spectrum analysis，PSA)[13]、球谐叠加技术(spherical harmonic stacking, SHS)[14]、多台站实验技术(multi-station experiment, MSE)[15]、最优序列估计方法(optimal sequence estimation，OSE)[16-17]等。PSA利用多个观测数据进行叠积操作以增强信号，提高信噪比。SHS和MSE是基于目标模态的特性而提出的方法，二者均旨在将多重态自由振荡信号的单线态逐个剥离，以达到识别信号的目的。OSE与MSE方法原理类似，优点是具有比MSE方法更高的SNR。但这些方法都是在频域上分析信号，无法反映信号的强弱和信号随时间的变化。
本文将时频分析引入自由振荡模型检测中，推导标准时频变换(normal time-frequency transform, NTFT)无偏检测和确定自由振荡模型频率的公式，并通过仿真实例验证该方法的有效性。以3S1模型为例，通过检测苏门答腊地震震后的自由振荡模型，证实新方法在自由振荡模型检测中更具优势，可为进一步研究地球密度结构、精化地球模型提供依据。
1 理论方法
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1.1 NTFT
NTFT由Liu等[18-19]提出。自由振荡模型为$u(t)=|A| \exp (-\alpha t) \cos (\beta t+\varphi)$，根据欧拉定理，自由振荡的模型可写为[20]：
$u(t)=\frac{1}{2}(O(t)+\overline{O(t)})$
式中，$O(t)=|A| \exp (-\alpha t) \exp (\mathrm{i}(\beta t+\varphi)) $，上划线表示共轭算子。
选用NTFT核函数$\psi(t, \widetilde{\omega})=w(t) \exp (\mathrm{i} \widetilde{\omega} t)$，其中$ w(t)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{t^2}{2 \sigma^2}\right), \sigma>0$是标准高斯窗函数。$O(t)$和$\overline{O(t)}$的NTFT分别为：
$\begin{aligned}\Psi O(\tau, \widetilde{\omega})= & O(\tau) \exp \left(\frac{\sigma^2 \alpha^2}{2}-\frac{\sigma^2(\tilde{\omega}-\beta)^2}{2}\right) \cdot \\& \exp \left(\mathrm{i} \sigma^2 \alpha(\widetilde{\omega}-\beta)\right)\end{aligned}$
$\begin{aligned}\Psi \overline{O(\tau, \tilde{\omega})}= & \overline{O(\tau)} \exp \left(\frac{\sigma^2 \alpha^2}{2}-\frac{\sigma^2(\tilde{\omega}+\beta)^2}{2}\right) \cdot \\& \exp \left(\mathrm{i} \sigma^2 \alpha(\tilde{\omega}+\beta)\right)\end{aligned}$
根据标准窗函数的性质，在$\widetilde{\omega}=\beta$附近，$\overline{O(\tau)}$的Ψ变换$(\tilde{\omega}>0)$几乎为0，因而u(t)的NTFT可近似为：
$\begin{gathered}\Psi u(\tau, \tilde{\omega})=\frac{1}{2} \Psi O(\tau)=\frac{1}{2} O(\tau) \cdot \\\exp \left(\frac{\sigma^2 \alpha^2}{2}-\frac{\sigma^2(\tilde{\omega}-\beta)^2}{2}\right) \exp \left(\mathrm{i} \sigma^2 \alpha(\tilde{\omega}-\beta)\right)\end{gathered}$
其对应的NTFT谱为：
$\begin{gathered}|\Psi u(\tau, \widetilde{\omega})|=\frac{1}{2}|A| \cdot \\\exp (-\alpha \tau) \exp \left(\frac{\sigma^2 \alpha^2}{2}-\frac{\sigma^2(\tilde{\omega}-\beta)^2}{2}\right)\end{gathered}$
可以看出，当$\widetilde{\omega}=\beta$时，式(5)取得最大值，从理论上保证NTFT能够无偏地确定简正模及其单线态的频率。
将式(5)确定的频率β代入式(4)中，可以得到NTFT系数：
$\Psi u(\tau, \beta)=\frac{1}{2} O(\tau) \exp \left(\frac{\sigma^2 \alpha^2}{2}\right)$
从式(6)看出，NTFT系数的实部是自由振荡模型本身乘以一个实数$\frac{1}{2} \exp \left(\frac{\sigma^2 \alpha^2}{2}\right)$，因而，利用NTFT的系数能够表征自由振荡模型的相位。
1.2 仿真实例
采用PREM模型频率和衰减的理论值构造3S1的3个单线态信号，根据频率从低到高的顺序，振幅依次设为1 nm/s2、1.4 nm/s2、1.2 nm/s2，再加入-0.2 dB的噪声后构成3S1仿真信号，见图1。该信号去除边缘效应后的NTFT谱见图2，频率结果见表1(单位mHz)。
从图2可以看出，NTFT谱不仅可以清晰地检测出3S1的3个单线态，而且能够揭示信号的强弱和信号随时间的衰减变化。从表1可以看出，NTFT方法能检测到自由振荡模型，平均偏差为0.17 μHz，偏差值很小，仅为理论频率的0.018%，基本能够无偏检测自由振荡模型。
2 数据准备
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选取IGETS数据库中的6个超导重力仪台站在苏门答腊地震中的观测数据[21]，台站信息见表2，采样频率为1 min。
利用Tsoft软件计算合成潮汐[22]，并从原始重力观测序列中扣除相应的重力潮汐信号。采用2004-10的超导重力仪数据计算各个台站所在地的气压导纳值，并对扣除重力潮汐信号的原始重力观测序列进行气压改正，得到重力残差序列。
3 3S1模型检测
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3.1 基于FT谱的模型检测
采用苏门答腊地震震后50~280 h的重力残差数据，加窗(hanning)后计算其傅里叶功率谱(power spectral density, PSD)，结果见图3。为揭示检测频率与理论频率的偏差，在图3中以红色虚线表示对应的PREM模型的理论值，蓝色虚线表示检测频率，m=-1、0、1时对应的理论频率分别为0.942 27 mHz、0.944 22 mHz、0.945 47 mHz。
根据PSD统计各台站在苏门答腊地震期间检测到的3S1模型各单线态的频率，见表3(单位mHz)。如图3和表3所示，各个台站所检测到的3S1模型单线态的数量相同，只能检测到m=±1的两个单线态，m=0的单线态难以检测到；不同台站检测到的频率存在一定差异，也与理论值存在偏差，最大偏差达0.5 μHz。由于m=±1的单线态的频率非常接近，FT谱方法难以完全分离两个信号，所以检测到的频率与理论值存在明显的偏差。
3.2 基于OSE的模型检测
采用苏门答腊地震震后50~280 h的重力残差数据，应用OSE检测3S1各单线态的结果，见图4(红色虚线和蓝色虚线分别代表理论频率和检测频率)。统计苏门答腊地震检测到的频率，见表4(单位mHz)。
可以看出，苏门答腊地震能够检测到3个单线态，但m=0的单线态非常弱，与理论值偏差较大。OSE方法检测到的单线态的频率与理论值也存在明显偏差，最大偏离达1.05 μHz。OSE在m=0的单线态检测到较大偏差的原因可能是6个台站很难检测到该信号，所以即使OSE方法综合了6个台站的信号，检测到m=0的单线态也存在较大偏差。
3.3 基于NTFT的模型检测
从上节可以看出，FT谱分析和OSE在自由振荡模型检测方面存在以下缺陷：1)单线态数量越多，各单线态的频率越接近，频率检测的偏差越大；2)这两种方法仅能检测自由振荡模型的频率，无法反映信号随时间的变化。
本文采用的新方法具有更高的时频聚集性，能够完全剥离3S1的单线态，为自由振荡模型检测提供更高的精度。采用苏门答腊地震震后20 d的重力残差数据，计算其NTFT谱，见图5。根据式(5)确定各台站单线态的频率，见表5(单位mHz)。
与FT谱相比，NTFT对频率更加敏感，NTFT检测到的频率更接近理论值，检测精度更高，最大偏差为0.36 μHz，仅占理论值的0.038%。NTFT能够清晰地揭示各单线态的强弱，以SU台站为例，m=1的单线态比m=-1的单线态的强度高，这可能是因为SU台站检测到的m=1的单线态比m=-1的单线态精度更高。
4 模型检测精度分析
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为了比较FT谱、OSE和NTFT方法检测单线态的精度，采用柱状图对3种方法检测到的3S1各单线态的频率进行统计分析，见图6。
FT谱、OSE和NTFT方法检测频率与理论频率偏差的均值分别为0.36 μHz、0.65 μHz和0.28 μHz，NTFT的精度与FT谱和OSE相比分别提高22%和57%。FT谱和NTFT方法都只检测到m=±1的单线态，没有检测到m=0的单线态，OSE方法检测到3个单线态，但检测到的m=0的单线态与理论值偏差较大，结果的精度有待进一步提高。综上所述，NTFT方法具有更高的频率分辨率，能更准确地检测自由振荡模型。
5 结语
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本文将时频分析引入自由振荡模型检测中，推导并验证了一种自由振荡模型检测的新方法。实验表明，本文方法检测精度比FT谱和OSE方法更高。
基金
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  • 国家自然科学基金(41704003)
文献
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doi: 10.14075/j.jgg.2024.02.085
  • 接收时间：2024-02-29
  • 首发时间：2026-10-30
  • 出版时间：2025-01-15
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  • 收稿日期：2024-02-29
基金
National Natural Science Foundation of China(41704003)
国家自然科学基金(41704003)
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    1 山东理工大学建筑工程与空间信息学院, 山东省淄博市新村西路266号, 255000
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
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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