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Article(id=1322916930243396541, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322916858604696290, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.11.402, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1763568000000, receivedDateStr=2025-11-20, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793337574778, onlineDateStr=2026-10-30, pubDate=1786723200000, pubDateStr=2026-08-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793337574778, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793337574778, creator=13701087609, updateTime=1793337574778, updator=13701087609, issue=Issue{id=1322916858604696290, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='8', pageStart='913', pageEnd='1038', issueExtLink='null', onlineDate='null', pubDate='1786723200000', pubDateStr='2026-08-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793337557698, creator='13701087609', updateTime=1793337557698, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=1013, endPage=1021, ext={EN=ArticleExt(id=1322916930474083262, articleId=1322916930243396541, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=High-Precision Normal Mode Detection Considering Phase Consistency, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

In this paper, high-precision normal mode detection is achieved by using the observation data of the superconducting gravimeter and combining the normal time-frequency transform (NTFT) and optimal sequence estimation (OSE) methods. Firstly, the instantaneous phases of each station were extracted through NTFT, and it was found that the instantaneous phases of each station were not the same. It was speculated that there was a non-synchronous oscillation phenomenon in the normal mode. When different quantities or different combinations of station data were used, the detection results of the OSE method showed significant differences, indicating that this phenomenon had an important impact on the detection accuracy of the OSE method. Subsequently, OSE experiments were conducted using stations with relatively consistent phases, and it was found that the detection frequency was close to the PREM theoretical value, indicating that the issue of phase consistency must be taken into account when applying the OSE method. Based on this, this paper proposes a high-precision normal mode detection method that integrates NTFT and OSE and takes into account phase consistency. Taking the split state detection of the 0S2 model as an example, the effectiveness of this method was verified, and it was also demonstrated that phase consistency is a key factor for the effective application of the date stacking method. This research provides a reliable data foundation and technical support for deeply revealing the deep structure of the Earth and optimizing the Earth model.

, authors=Yaxin ZHAO1, Gong XU1, *, Hanwei ZHANG1, authorsList=Yaxin ZHAO, Gong XU, Hanwei ZHANG, authorCompany=null, correspAuthors=Gong XU, 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=1322916932017587147, articleId=1322916930243396541, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=顾及相位一致性的高精度简正模检测, columnId=1281933709121138897, journalTitle=大地测量与地球动力学, columnName=地球动力学, runingTitle=null, highlight=null, articleAbstract=

利用超导重力仪观测数据，结合标准时频变换(NTFT)与最优序列估计(OSE)方法，实现对简正模的高精度检测。首先，通过NTFT提取各台站的瞬时相位，发现各台站瞬时相位不尽相同，推测简正模存在非同步振荡现象，使用不同数量或不同组合的台站数据时OSE方法的检测结果存在显著差异，表明该现象对OSE方法的检测精度产生了重要影响；然后，使用相位较一致的台站进行OSE实验，发现其检测频率同PREM理论值较接近，表明在应用OSE方法时必须考虑相位一致性问题。基于此，本文提出一种融合NTFT与OSE同时顾及相位一致性的高精度简正模检测方法，并以0S2模型的分裂态检测为例，验证了该方法的有效性，说明相位一致性是有效运用叠积方法的关键因素。本文研究为深入揭示地球深部结构及优化地球模型提供了可靠的数据基础与技术支撑。

, authors=赵亚鑫1, 徐工1, *, 张捍卫1, authorsList=赵亚鑫, 徐工, 张捍卫, authorCompany=null, correspAuthors=徐工, authorNote=

赵亚鑫，硕士生，主要从事现代大地测量原理与应用研究，E-mail：

, correspAuthorsNote=
徐工，博士，讲师，主要从事现代大地测量原理与应用研究，E-mail：
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pageEnd=549, url=null, language=null, rfNumber=31, rfOrder=45, authorNames=Dahlen F A, journalName=Geophysical Journal International, refType=null, unstructuredReference=Dahlen F A.The Effect of Data Windows on the Estimation of Free Oscillation Parameters[J].Geophysical Journal International,1982,69(2):537-549, articleTitle=The Effect of Data Windows on the Estimation of Free Oscillation Parameters, refAbstract=null)], funds=[Fund(id=1322916939303093245, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, awardId=12473068, language=CN, fundingSource=国家自然科学基金(12473068), fundOrder=null, country=null), Fund(id=1322916939374396414, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, awardId=23-2-1-217-zyyd-jch, language=CN, fundingSource=青岛市自然科学基金(23-2-1-217-zyyd-jch), fundOrder=null, country=null), Fund(id=1322916939487642623, tenantId=1146029695717560320, journalId=1281212831689347082, 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figs=[ArticleFig(id=1322916935754712040, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=EN, label=Fig. 1, caption=0S2 mode of Sumatra earthquake detected by NTFT, figureFileSmall=ud0Gj7+MzzcRxbx76m71JQ==, figureFileBig=hYWoQMTRcKUBUOhFQ4L41g==, tableContent=null), ArticleFig(id=1322916935846986729, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=CN, label=图1, caption=NTFT检测苏门答腊地震0S2模型, figureFileSmall=ud0Gj7+MzzcRxbx76m71JQ==, figureFileBig=hYWoQMTRcKUBUOhFQ4L41g==, tableContent=null), ArticleFig(id=1322916936094450666, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=EN, label=Fig. 2, caption=Instantaneous phases of each station extracted from the three major earthquakes by NTFT, figureFileSmall=Fs4M4EfuUyFbXfW9CZEEUQ==, figureFileBig=geDFwl89wrzef6v87j9GWQ==, tableContent=null), ArticleFig(id=1322916936182531051, 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journalId=1281212831689347082, articleId=1322916930243396541, language=EN, label=Fig. 4, caption=OSE detection results of the m=－2 multiplet for each station group and phase spectra of each station with m=－2 for the Japan earthquake, figureFileSmall=vDjzDEyAsk/Qw2/+8Bqd9Q==, figureFileBig=KioGo/E2WxAj1YdJ7VnKnA==, tableContent=null), ArticleFig(id=1322916936652293103, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=CN, label=图4, caption=日本地震各组台站m=－2分裂态OSE检测结果及m=－2各台站相位谱, figureFileSmall=vDjzDEyAsk/Qw2/+8Bqd9Q==, figureFileBig=KioGo/E2WxAj1YdJ7VnKnA==, tableContent=null), ArticleFig(id=1322916936723596272, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=EN, label=Fig. 5, caption=OSE detection results of the m=－2 multiplet for each station group and phase spectra of each station with m=－2 for the Chile earthquake, figureFileSmall=S8olPGK4cyLPHUM0VNDHcQ==, figureFileBig=hJDvJVzQ2P3excY0neJs6Q==, tableContent=null), ArticleFig(id=1322916936803288049, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=CN, label=图5, caption=智利地震各组台站m=－2分裂态OSE检测结果及m=－2各台站相位谱, figureFileSmall=S8olPGK4cyLPHUM0VNDHcQ==, figureFileBig=hJDvJVzQ2P3excY0neJs6Q==, tableContent=null), ArticleFig(id=1322916936882979826, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=EN, label=Fig. 6, caption=OSE detection results of the m=－2 multiplet for each station group and phase spectra of each station with m=－2 for the Sumatra earthquake, figureFileSmall=T3556mo6m8z6M2RV5QjFrQ==, figureFileBig=t9Wjdn2txBVzwvqDmvvOOQ==, tableContent=null), ArticleFig(id=1322916936971060211, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=CN, label=图6, caption=苏门答腊地震各组台站m=－2分裂态OSE检测结果及m=－2各台站相位谱, figureFileSmall=T3556mo6m8z6M2RV5QjFrQ==, figureFileBig=t9Wjdn2txBVzwvqDmvvOOQ==, tableContent=null), ArticleFig(id=1322916937054946292, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=EN, label=Tab. 1, caption=

Information of each station

, figureFileSmall=null, figureFileBig=null, tableContent=
台站名称编号仪器名称经度/(°)纬度/(°)高程/m
WettzellWE-1
WE-2		GWR CD029_U，GWR CD029_L12.87849.144613.700
WalferdangeWAOSG-CT406.152 849.664 7295.000
TIGO ConcepcionTCGWR RT038－73.025 65－36.843 76156.140
SutherlandSU-1
SU-2		D037_U，D037_L20.810 9－32.381 41 791.000
Ny-AlesundNYGWR C03911.867 1778.930 6143.000
MoxaMO-1
MO-2		CD034_U，CD034_L11.61650.645455.000
MetsahoviMEGWR T02024.395 860.217 255.600
StrasbourgSTGWR C0267.683 848.621 7180.000
MembachMBGWR C0216.006 650.609 3250.000
KamiokaKAGWR T016137.308 436.425 3358.000
CanberraCBGWR C031149.007 66－35.320 64762.749
), ArticleFig(id=1322916938694919158, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=CN, label=表1, caption=

各台站信息

, figureFileSmall=null, figureFileBig=null, tableContent=
台站名称编号仪器名称经度/(°)纬度/(°)高程/m
WettzellWE-1
WE-2		GWR CD029_U，GWR CD029_L12.87849.144613.700
WalferdangeWAOSG-CT406.152 849.664 7295.000
TIGO ConcepcionTCGWR RT038－73.025 65－36.843 76156.140
SutherlandSU-1
SU-2		D037_U，D037_L20.810 9－32.381 41 791.000
Ny-AlesundNYGWR C03911.867 1778.930 6143.000
MoxaMO-1
MO-2		CD034_U，CD034_L11.61650.645455.000
MetsahoviMEGWR T02024.395 860.217 255.600
StrasbourgSTGWR C0267.683 848.621 7180.000
MembachMBGWR C0216.006 650.609 3250.000
KamiokaKAGWR T016137.308 436.425 3358.000
CanberraCBGWR C031149.007 66－35.320 64762.749
), ArticleFig(id=1322916938774610935, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=EN, label=Tab. 2, caption=

Frequency differences of OSE simulation results of the 0S2 mode with m=－2 for the Japan earthquake

, figureFileSmall=null, figureFileBig=null, tableContent=
同基准信号的相位差/(°)同基准台站组合的OSE结果频率差/μHz
00.042 3
200.087 5
400.102 6
600.107 9
800.121 1
1000.116 9
1200.119 8
1400.096 1
1600.072 0
1800.061 5
), ArticleFig(id=1322916938854302712, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=CN, label=表2, caption=

日本地震0S2模型m=－2单线态OSE仿真结果频率差

, figureFileSmall=null, figureFileBig=null, tableContent=
同基准信号的相位差/(°)同基准台站组合的OSE结果频率差/μHz
00.042 3
200.087 5
400.102 6
600.107 9
800.121 1
1000.116 9
1200.119 8
1400.096 1
1600.072 0
1800.061 5
), ArticleFig(id=1322916938921411577, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=EN, label=Tab. 3, caption=

Stations used in each combination

, figureFileSmall=null, figureFileBig=null, tableContent=
台站组合苏门答腊地震智利地震日本地震
组合1SU-1, SU-2, MO-2, WE-1, WE-2MB, MO-1, WA, TC, WE-2MO-1, ST, SU-2, WE-1, WE-2
组合2SU-1, SU-2, MO-2, WE-1, WE-2, CBMB, MO-1, WA, TC, WE-2, KAMO-1, ST, SU-2, WE-1, WE-2, ME
组合3SU-1, SU-2, MO-2, WE-1, WE-2, MBMB, MO-1, WA, TC, WE-2, STMO-1, ST, SU-2, WE-1, WE-2, NY
组合4SU-1, SU-2, MO-2, WE-1, WE-2, CB, MBMB, MO-1, WA, TC, WE-2, KA, STMO-1, ST, SU-2, WE-1, WE-2, ME, NY
组合5SU-1, SU-2, MO-2, WE-1, WE-2, KAMB, MO-1, WA, TC, WE-2, NYMO-1, ST, SU-2, WE-1, WE-2, CB
组合6SU-1, SU-2, MO-2, WE-1, WE-2, TCMB, MO-1, WA, TC, WE-2, SU-2MO-1, ST, SU-2, WE-1, WE-2, MB
组合7SU-1, SU-2, MO-2, WE-1, WE-2, KA, TCMB, MO-1, WA, TC, WE-2, NY, SU-2MO-1, ST, SU-2, WE-1, WE-2, CB, MB
), ArticleFig(id=1322916938992714746, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=CN, label=表3, caption=

各组合所用台站

, figureFileSmall=null, figureFileBig=null, tableContent=
台站组合苏门答腊地震智利地震日本地震
组合1SU-1, SU-2, MO-2, WE-1, WE-2MB, MO-1, WA, TC, WE-2MO-1, ST, SU-2, WE-1, WE-2
组合2SU-1, SU-2, MO-2, WE-1, WE-2, CBMB, MO-1, WA, TC, WE-2, KAMO-1, ST, SU-2, WE-1, WE-2, ME
组合3SU-1, SU-2, MO-2, WE-1, WE-2, MBMB, MO-1, WA, TC, WE-2, STMO-1, ST, SU-2, WE-1, WE-2, NY
组合4SU-1, SU-2, MO-2, WE-1, WE-2, CB, MBMB, MO-1, WA, TC, WE-2, KA, STMO-1, ST, SU-2, WE-1, WE-2, ME, NY
组合5SU-1, SU-2, MO-2, WE-1, WE-2, KAMB, MO-1, WA, TC, WE-2, NYMO-1, ST, SU-2, WE-1, WE-2, CB
组合6SU-1, SU-2, MO-2, WE-1, WE-2, TCMB, MO-1, WA, TC, WE-2, SU-2MO-1, ST, SU-2, WE-1, WE-2, MB
组合7SU-1, SU-2, MO-2, WE-1, WE-2, KA, TCMB, MO-1, WA, TC, WE-2, NY, SU-2MO-1, ST, SU-2, WE-1, WE-2, CB, MB
), ArticleFig(id=1322916939114349563, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=EN, label=Tab. 4, caption=

OSE detection results of each combination of the0S2 (m=－2) multiplet

, figureFileSmall=null, figureFileBig=null, tableContent=
台站组合苏门答腊地震/mHz智利地震/mHz日本地震/mHz
PREM0.30.30.3
组合10.299 770.302 120.300 15
组合2—0.301 30—
组合30.298 58——
组合4—0.293 38—
组合50.298 230.304 62—
组合60.302 870.292 550.297 38
组合70.299 42——
), ArticleFig(id=1322916939202429948, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916930243396541, language=CN, label=表4, caption=

0S2(m=－2)分裂态各组合OSE检测结果

, figureFileSmall=null, figureFileBig=null, tableContent=
台站组合苏门答腊地震/mHz智利地震/mHz日本地震/mHz
PREM0.30.30.3
组合10.299 770.302 120.300 15
组合2—0.301 30—
组合30.298 58——
组合4—0.293 38—
组合50.298 230.304 62—
组合60.302 870.292 550.297 38
组合70.299 42——
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顾及相位一致性的高精度简正模检测
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赵亚鑫 1 , 徐工 1, * , 张捍卫 1
大地测量与地球动力学 | 地球动力学 2026,46(8): 1013-1021
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大地测量与地球动力学 |地球动力学 2026 , 46 (8) : 1013 -1021
顾及相位一致性的高精度简正模检测
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赵亚鑫1 , 徐工1, * , 张捍卫1
作者信息
  • 1 山东理工大学建筑工程与空间信息学院，淄博，255000
通讯作者:
徐工，博士，讲师，主要从事现代大地测量原理与应用研究，E-mail：
作者简介:

赵亚鑫，硕士生，主要从事现代大地测量原理与应用研究，E-mail：

High-Precision Normal Mode Detection Considering Phase Consistency
Yaxin ZHAO1 , Gong XU1, * , Hanwei ZHANG1
Affiliations
  • 1 School of Civil Engineering and Geomatics, Shandong University of Technology, Zibo 255000, China
出版时间: 2026-08-15 doi: 10.14075/j.jgg.2025.11.402
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摘要
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利用超导重力仪观测数据，结合标准时频变换(NTFT)与最优序列估计(OSE)方法，实现对简正模的高精度检测。首先，通过NTFT提取各台站的瞬时相位，发现各台站瞬时相位不尽相同，推测简正模存在非同步振荡现象，使用不同数量或不同组合的台站数据时OSE方法的检测结果存在显著差异，表明该现象对OSE方法的检测精度产生了重要影响；然后，使用相位较一致的台站进行OSE实验，发现其检测频率同PREM理论值较接近，表明在应用OSE方法时必须考虑相位一致性问题。基于此，本文提出一种融合NTFT与OSE同时顾及相位一致性的高精度简正模检测方法，并以0S2模型的分裂态检测为例，验证了该方法的有效性，说明相位一致性是有效运用叠积方法的关键因素。本文研究为深入揭示地球深部结构及优化地球模型提供了可靠的数据基础与技术支撑。

NTFT  /  OSE  /  相位一致性  /  非同步振荡  /  简正模检测

In this paper, high-precision normal mode detection is achieved by using the observation data of the superconducting gravimeter and combining the normal time-frequency transform (NTFT) and optimal sequence estimation (OSE) methods. Firstly, the instantaneous phases of each station were extracted through NTFT, and it was found that the instantaneous phases of each station were not the same. It was speculated that there was a non-synchronous oscillation phenomenon in the normal mode. When different quantities or different combinations of station data were used, the detection results of the OSE method showed significant differences, indicating that this phenomenon had an important impact on the detection accuracy of the OSE method. Subsequently, OSE experiments were conducted using stations with relatively consistent phases, and it was found that the detection frequency was close to the PREM theoretical value, indicating that the issue of phase consistency must be taken into account when applying the OSE method. Based on this, this paper proposes a high-precision normal mode detection method that integrates NTFT and OSE and takes into account phase consistency. Taking the split state detection of the 0S2 model as an example, the effectiveness of this method was verified, and it was also demonstrated that phase consistency is a key factor for the effective application of the date stacking method. This research provides a reliable data foundation and technical support for deeply revealing the deep structure of the Earth and optimizing the Earth model.

NTFT  /  OSE  /  phase consistency  /  non-synchronous oscillation  /  normal mode detection
赵亚鑫, 徐工, 张捍卫. 顾及相位一致性的高精度简正模检测. 大地测量与地球动力学, 2026 , 46 (8) : 1013 -1021 . DOI: 10.14075/j.jgg.2025.11.402
Yaxin ZHAO, Gong XU, Hanwei ZHANG. High-Precision Normal Mode Detection Considering Phase Consistency[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (8) : 1013 -1021 . DOI: 10.14075/j.jgg.2025.11.402
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地球自由振荡(又称简正模)是地球在经历强烈地震等巨大震动后产生的整体性持续振动，其振荡频率和振幅中携带有地球内部结构及震源机制的关键信息。精确检测振荡频率对反演地球深部圈层构造具有重要意义，因此分析简正模的模态特征与分裂特征，是理解相关地球物理过程、揭示地球内部结构特征的核心[1-6]，为揭示地球内部分层结构提供关键信息[7-12]。在地震学研究中，简正模频率是反演地震矩张量、推断震源机制及确定破裂过程所必需的关键参数之一[7]，也被广泛应用于检验现有地球模型(如PREM模型)的准确性[8]。此外，简正模的分裂模型频率能够反映地球深部物质成分、温度或压力的横向不均匀性，如地幔过渡带异常和内核各向异性。因此，实现简正模频率的高精度探测，对深入理解地球深部结构具有重要意义[13-15]。
单个台站难以检测出振幅微弱的简正模信号，通过多个台站信息的叠加，可提高信噪比，进而提取出目标信号[16-24]。目前，基于叠积方法的简正模检测已较为成熟，其前提是假设全球同步振荡[18-25]，即在受到冲击后，地球上任一地点的简正模同时被激发并持续振荡，采用多个台站共同检测起到了信号放大的作用。
本文结合标准时频变换(NTFT)和最优序列估计(OSE)方法，首先分析各台站的瞬时相位，在此基础上提出一种顾及相位一致性的高精度简正模检测方法。该方法在充分利用OSE弱信号检测能力的同时，能有效兼顾相位一致性的影响，提升简正模检测精度。
1 简正模频率与相位检测方法
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1.1 叠积方法
常见的叠积方法包括线性叠积、加权叠积、球谐叠积(SHS)、多台站实验技术(MSE)及最优序列估计(OSE)等。在众多研究中，丁浩等[18]证实OSE方法的检测效果优于SHS与MSE等方法，因此本文选用OSE方法进行简正模频率的检测。
仅考虑滞弹性地球模型表面位移的径向分量，则地球表面台站为Ω=(θ, φ)，其中θ、φ表示余纬和经度，径向位移uR可表示为[21]：
$u_R(\varOmega, t)=\sum\limits_m{ }_n U_l Y_l^m(\varOmega) s_m\left(r_0\right) \mathrm{e}^{\mathrm{i} \sigma_m t}$
式中，t为时间；nUl为地表的径向本征函数；Ylm(Ω)为球谐函数；sm(r0)为每个分裂态的径向震源激发项，由地震矩张量和应变张量决定；r0为震源处的位置；σm为分裂态的复本征频率。
引入符号εm=nUlsm(r0)和Am=εmeiσmt，则式(1)可简化为：
$u_R(\varOmega, t)=\sum\limits_m A_m Y_l^m(\varOmega)$
对于简正模模型，方程组的矩阵形式为：
$\left[\begin{array}{l}\;\;\;u_R\left(\varOmega_1, t\right)=A_{-l} Y_l^{-l}\left(\varOmega_1\right)+A_{-l+1} Y_l^{-l+1}\left(\varOmega_1\right)+\cdots+A_l Y_l^l\left(\varOmega_1\right) \\\;\;\;u_R\left(\varOmega_2, t\right)=A_{-l} Y_l^{-l}\left(\varOmega_2\right)+A_{-l+1} Y_l^{-l+1}\left(\varOmega_2\right)+\cdots+A_l Y_l^l\left(\varOmega_2\right) \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vdots\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vdots \\u_R\left(\varOmega_N, t\right)=A_{-l} Y_l^{-l}\left(\varOmega_N\right)+A_{-l+1} Y_l^{-l+1}\left(\varOmega_N\right)+\cdots+A_l Y_l^l\left(\varOmega_N\right)\end{array}\right]$
用Ωj(j=1, 2, ⋯, N)表示第j个超导重力仪台站的位置，则N个台站的叠积可表示为：
$\boldsymbol{U}=\boldsymbol{Y A}$
$\boldsymbol{U}=\left[\begin{array}{c}u_R\left(\varOmega_1, t\right) \\u_R\left(\varOmega_2, t\right) \\\vdots \\u_R\left(\varOmega_N, t\right)\end{array}\right]$
$\boldsymbol{Y}=\left[\begin{array}{cccc}Y_l^{-l}\left(\varOmega_1\right) & Y_l^{-l+1}\left(\varOmega_1\right) & \cdots & Y_l^l\left(\varOmega_1\right) \\Y_l^{-l}\left(\varOmega_2\right) & Y_l^{-l+1}\left(\varOmega_2\right) & \cdots & Y_l^l\left(\varOmega_2\right) \\\vdots & & \vdots & \\Y_l^{-l}\left(\varOmega_N\right) & Y_l^{-l+1}\left(\varOmega_N\right) & \cdots & Y_l^l\left(\varOmega_N\right)\end{array}\right]$
$\boldsymbol{A}=\left[\begin{array}{c}\varepsilon_{-l} \mathrm{e}^{\mathrm{i} \sigma}-l^t \\\varepsilon_{-l+1} \mathrm{e}^{\mathrm{i} \sigma}-l+1^t \\\vdots \\\varepsilon_l \mathrm{e}^{\mathrm{i} \sigma_l t}\end{array}\right]$
式中，t为从1到x的时间样本；U为N×x阶超导重力仪的径向分量观测矩阵；Y为N×(2l+1)阶球谐函数矩阵；A为需要反演的矩阵，其每一行表示简正模的一个分裂态。当N＞2l+1时，通过最小二乘法可得：
$\boldsymbol{A}=\left(\boldsymbol{Y}^{\mathrm{T}} \boldsymbol{P} \boldsymbol{Y}\right)^{-1} \boldsymbol{Y}^{\mathrm{T}} \boldsymbol{P U}$
为兼顾各台站的观测质量，式(8)中 P表示权重矩阵，与台站的信噪比成反比。
1.2 基于NTFT的无偏相位检测
标准时频变换能够无偏估计非平稳周期信号的瞬时振幅、周期与相位，在处理非线性信号方面展现出独特优势。本文基于该理论论证NTFT可无偏获取简正模的相位信息，从而为将其应用于振荡模型的相位检测奠定理论基础。
1.2.1 NTFT理论
函数f(t)∈L1(R)的标准时频变换定义为[25-28]：
$\varPsi f(\tau, \tilde{\omega})=\int_R f(t) \overline{\psi(t-\tau, \tilde{\omega})} \mathrm{d} t \quad \tau, \tilde{\omega} \in R$
式中，τ为时间因子；$\tilde{\omega} $为频率因子；上划线表示共轭算子；$\psi(t, \tilde{\omega}) $为时频变换的核函数，满足：
$\hat{\psi}(\omega, \tilde{\omega})=\int_R \psi(t, \tilde{\omega}) \exp (-\mathrm{i} \omega t) \mathrm{d} t \;\dot{\neq}\; 0$
式中，“·”表示几乎处处；ω表示信号真实频率。
一类典型的NTFT核函数为：
$\begin{aligned}\psi(t, \tilde{\omega}) & =|\mu(\tilde{\omega})| w(\mu(\tilde{\omega}) t) \exp (\mathrm{i} \tilde{\omega} t), \\w(t) & \in \varOmega(R), (\mu(\tilde{\omega}) \in R) \;\dot{\neq} \;0\end{aligned}$
式中，w(t)表示窗函数。
1.2.2 无偏相位检测
简正模理论模型可表达衰减振荡余弦信号之和，每个信号可表示为：
$u(t)=\sum\limits_k\left|A_k\right| \exp \left(-\alpha_k t\right) \cos \left(\beta_k t+\varphi_k\right)$
式中，|Ak|为振幅；αk为衰减因子；βk为角频率；φk为初相。由欧拉定理可得：
$u(t)=\frac{1}{2}(O(t)+\overline{O(t)})$
式中，O(t)=|A|exp(－αt)exp(i(βt+φ))为u(t)的复信号，上划线表示共轭算子。
NTFT核函数为$\psi(t, \tilde{\omega})=w(t) \exp (\mathrm{i} \tilde{\omega} t) $，其中$w(t)=\frac{1}{\sqrt{2 \mathtt{π}} \sigma} \exp \left(-\frac{t^2}{2 \sigma^2}\right) $, σ>0为标准高斯窗函数，则O(t)和$\overline{O(t)} $的NTFT为：
$\begin{aligned}\varPsi O(\tau, \tilde{\omega})= & O(\tau) \exp \left(\frac{\sigma^2 \alpha^2}{2}-\frac{\sigma^2(\tilde{\omega}-\beta)^2}{2}\right) \cdot \\& \exp \left(\mathrm{i} \sigma^2 \alpha(\tilde{\omega}-\beta)\right)\end{aligned}$
$\begin{aligned}\varPsi \overline{O(\tau, \tilde{\omega})}= & \overline{O(\tau)} \exp \left(\frac{\sigma^2 \alpha^2}{2}-\frac{\sigma^2(\tilde{\omega}+\beta)^2}{2}\right) \cdot \\& \exp \left(\mathrm{i} \sigma^2 \alpha(\tilde{\omega}+\beta)\right)\end{aligned}$
由标准高斯窗函数的性质可知，在$\tilde{\omega}=\beta $且$\tilde{\omega} $>0时，$\overline{O(t)} $的NTFT几乎为0，则简正模位移u(t)的NTFT可表示为：
$\begin{gathered}\varPsi u(\tau, \tilde{\omega})=\frac{1}{2} \varPsi O(\tau)=\frac{1}{2} O(\tau) \cdot \\\exp \left(\frac{\sigma^2 \alpha^2}{2}-\frac{\sigma^2(\tilde{\omega}-\beta)^2}{2}\right) \exp \left(\mathrm{i} \sigma^2 \alpha(\tilde{\omega}-\beta)\right)\end{gathered}$
其对应的NTFT谱为：
$\begin{gathered}|\varPsi u(\tau, \tilde{\omega})|=\frac{1}{2}|A| \exp (-\alpha \tau) \cdot \\\exp \left(\frac{\sigma^2 \alpha^2}{2}-\frac{\sigma^2(\tilde{\omega}-\beta)^2}{2}\right)\end{gathered}$
从式(17)可以看出，当$\tilde{\omega} $=β时，NTFT谱取得最大值，这从理论上保证了NTFT谱能够无偏揭示简正模的分裂态频率。
将β代入式(16)可得：
$\varPsi u(\tau, \beta)=\frac{1}{2} O(\tau) \exp \left(\frac{\sigma^2 \alpha^2}{2}\right)$
式(18)表明，NTFT系数本质上是简正模模型乘以一个实数$\frac{1}{2} \exp \left(\frac{\sigma^2 \alpha^2}{2}\right) $，由于实数不会改变简正模模型的相位，因而NTFT系数可以无偏揭示信号的即时相位。
2 非同步振荡验证
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2.1 数据准备
本文选取矩震级较高的2004年苏门答腊地震、2010年智利地震及2011年日本地震激发的0S2简正模微弱信号，从国际地球动力学与固体潮服务组织(IGETS)[29]中获取14个台站的超导重力仪观测数据，采样间隔1 min，选取震后5~300 h时间段数据[19, 30-31]，台站信息见表 1。
本文采用IGETS获取的Level 3数据[29]，该数据已经过固体潮、海潮负荷、大气负荷、地球自转及仪器漂移等改正，达到μGal级精度。
2.2 NTFT无偏获取相位信息
式(17)和式(18)表明，NTFT能够无偏地确定简正模的瞬时相位。本文利用NTFT提取全部台站的瞬时相位信息，采用震后5~300 h超导重力数据，由于持续时间较长，仅呈现震后30~32 h的瞬时相位信息。本文以苏门答腊地震MB台站为例，NTFT检测0S2模型的频率见图 1。
对检测到的分裂态信号进行提取，结果见图 2，三大地震中14个台站的瞬时相位并不完全一致，部分台站之间差异显著。在苏门答腊地震的m=－2分裂态中，KA站和TC站几乎与其他台站反相，而CB站与MB站也表现出较大的相位偏差；在智利地震与日本地震的m=－2分裂态中，出现反相或相位显著偏离的台站各不相同，表明该现象并非由观测误差引起，从而验证了地球的非同步振荡现象。具体而言，智利地震中NY站与SU-2站几乎完全反相，KA站与ST站也呈现较大相位偏差；而日本地震中CB站与MB站基本反相，NY站与ME站的相位偏差亦较为明显。智利地震中SU-2站尽管同其他台站几乎反相，但在日本地震和苏门答腊地震中，该台站同多数台站相位较一致。该结果充分表明，相关台站的相位差异并非由仪器固有偏差或观测环境因素所致。不同地震出现反相和相位偏差较大的台站各不相同，说明不是数据观测导致的相位不一致现象。
2.3 顾及相位一致性的简正模检测
OSE方法能够有效检测简正模的微弱信号，其前提假设是全球同步振荡。图 2结果验证了各台站之间存在相位差异，这种差异会影响检测精度和灵敏度。为保证微弱信号的检测精度，本文融合OSE和NTFT，提出一种顾及相位一致性的高精度简正模检测方法。该方法通过逐台站提取瞬时相位谱，筛选出瞬时相位基本一致的台站，再利用筛选出的台站进行OSE检测，通过剔除相位不一致的台站，最大程度保证叠积共振效果，进一步提升OSE方法的微弱信号检测能力。
以日本地震检测到的0S2模型m=－2单线态为研究对象，选取MO-1、ST、SU-2、WE-1、WE-2共5个台站作为基准台站进行OSE实验，对获取的实验结果进行NTFT变换，从中提取m=－2单线态信号分量，计算得到该信号的初始相位为－85.675°。结合PREM模型给出的理论频率0.3 mHz，通过设置不同初始相位参数构建系列仿真信号。将各仿真信号分别与基准台站组合开展OSE实验，得到的对比结果见图 3，不同初始相位仿真信号与基准台站组合对应的频率偏差数据如表 2所示。
基于图 3的实验对比结果与表 2的频率偏差数据，对基准台站的相位偏差阈值进行科学界定：在一个信号周期内，若目标频率精度需达到0.01 μHz，则初始相位偏差需控制在20°以内，此类台站界定为相位相近台站；若目标频率精度需达到0.1 μHz，则初始相位偏差可放宽至40°以内，此类台站同样界定为相位相近台站。相位偏差阈值的选取需结合具体研究的精度需求确定，本文统一选取相位偏差≤20°的台站作为相位一致性较好的台站，用于后续分析。由式(8)可以得出，在一致性振荡的前提下，多余观测量越多，检测精度越高。为探究OSE方法与所用台站数量和组合之间的联系，分别选取三大地震中相位较一致的台站进行OSE实验，并逐个加入相位偏差较大的台站或几乎反相的台站。实验各组合所用台站见表 3，其中组合1为相位较一致的台站，组合2~4为引入相位偏差较大的台站，组合5~7为引入几乎反相的台站，实验结果见图 4~6。
图 4~6左图中的红色虚线对应PREM模型预测的0S2(m=－2)分裂态的理论频率，即0.3 mHz，可以看出，OSE方法的检测精度与台站分布之间存在显著的关联性，对台站组合的依赖性较强，使得相位不一致台站信息的加入会抵消或削弱叠积共振效应。各组合叠积实验结果如表 4所示。
图 4~6结果显示，OSE方法的检测精度并非与台站数量成正比。在三大地震中，当所用台站的瞬时相位较为接近时(如组合1中的5个台站)，检测到的0S2(m=－2)分裂态频率与PREM理论值高度吻合；然而，随着组合2~7中逐步引入相位不一致的台站，OSE检测结果与理论值的偏差显著增大。在日本地震和智利地震中引入近乎反相的台站，在苏门答腊地震中引入相位偏差较大的台站，均导致频域信号能量衰减，削弱了相位一致台站原有的共振放大效应，从而降低了对微弱信号的识别能力。这一现象与式(8)所预期的影响相矛盾。此外，在台站数量相同但组合不同时，OSE的检测结果也存在显著差异。这些结果表明，地球处于非同步振荡状态，各台站的相位存在差异。在叠加过程中，部分台站信号之间会相互抵消，从而导致不同台站组合下的OSE检测结果出现较大偏差。因此，为提高OSE方法的检测精度与可靠性，必须在数据处理中充分考虑各台站的相位信息。
综合本文OSE实验(图 3~6及表 3、4)的相位谱信息可以看出，不同台站数量和组合情况导致OSE检测结果相差较大，不同台站观测数据的瞬时相位不尽相同，采用相位较一致的台站组合进行OSE实验的检测结果最优，与PREM理论值高度吻合，相位不一致台站的加入会对高精度简正模检测带来影响，这些都证实了相位不一致现象的存在。使用OSE方法检测高精度简正模信号时，台站相位一致性是关键因素，这意味着地球不同区域的简正模在时间上存在差异，这种非同步性为观察地球内部不同区域的物理性质差异提供了一个窗口，这些实验结果也说明了叠积方法需要考虑相位不一致性问题。
3 结语
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1) 使用NTFT提取三大地震各台站的瞬时相位，发现台站之间的相位存在偏差甚至反相，表明简正模存在相位不一致现象(或称非同步振荡现象)。
2) OSE实验表明，非同步振荡台站的加入一方面会降低OSE的频率检测精度，另一方面会削弱OSE微弱信号检测能力。因此，为提高叠积方法的检测精度和灵敏度，必须考虑相位不一致性。
3) 本文提出顾及相位一致性的高精度简正模检测方法，该方法可兼顾NTFT无偏揭示相位和OSE微弱信号检测的能力。实验结果表明，新方法在微弱信号高精度检测上具备一定的优势，可为进一步约束和完善地球模型提供高精度的基础数据。
本文提出的非同步性振荡现象为观察地球内部不同区域的物理性质差异提供了一个窗口，后续工作将利用全球分布的宽频带地震仪观测数据揭示非同步振荡的空间分布特征，为研究地球内部，尤其是深内部的小尺度结构提供新的思路和方法。
基金
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  • 国家自然科学基金(12473068)
  • 青岛市自然科学基金(23-2-1-217-zyyd-jch)
  • 山东省研究生教育质量提升计划(SDYKC2023108)
文献
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2026年第46卷第8期
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doi: 10.14075/j.jgg.2025.11.402
  • 接收时间：2025-11-20
  • 首发时间：2026-10-30
  • 出版时间：2026-08-15
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  • 收稿日期：2025-11-20
基金
国家自然科学基金(12473068)
青岛市自然科学基金(23-2-1-217-zyyd-jch)
山东省研究生教育质量提升计划(SDYKC2023108)
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    1 山东理工大学建筑工程与空间信息学院，淄博，255000

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徐工，博士，讲师，主要从事现代大地测量原理与应用研究，E-mail：
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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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