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Article(id=1322916916884538200, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322916858604696290, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.12.430, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1765123200000, receivedDateStr=2025-12-08, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793337571593, onlineDateStr=2026-10-30, pubDate=1786723200000, pubDateStr=2026-08-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793337571593, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793337571593, creator=13701087609, updateTime=1793337571593, 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=957, endPage=963, ext={EN=ArticleExt(id=1322916917157167961, articleId=1322916916884538200, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Weighting Schemes for Multi-Source Time-Variable Gravity Field Model Fusion, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Time-variable gravity field models provide critical data support for monitoring large-scale surface mass changes. However, discrepancies in accuracy exist among models released by different institutions. To fully utilize the advantages of multi-source time-variable gravity field models, this study focuses on weighted fusion methods, investigating various weighting strategies, including variance component estimation (VCE), ocean noise RMS-based weighting, and degree variance-based weighting (incorporating two reference models). Time-variable gravity field models released by eight institutions worldwide, including HUST, were selected for fusion experiments and accuracy assessment. The results indicate: 1) The choice of weighting schemes across different degrees/orders influences the VCE fusion outcomes. When truncated to degree 90, the optimal weighting orders are 70 for GRACE models and 60 for GRACE-FO models. 2) All four weighting fusion schemes enhance model accuracy. Compared to the Tongji model, the pre-filtering degree-variance fusion (using the GOCO06s reference model) reduces ocean noise RMS by up to 23%, while the post-filtering VCE fusion achieves a maximum reduction of 29%. 3) Over six major river basins, the terrestrial water storage trends derived from different time-variable models show good consistency, and the fusion models exhibit higher correlation coefficients with Mascon solutions.

, authors=Peng YAO1, Xingfu ZHANG1, *, Jianhao XUAN2, Xu YU1, authorsList=Peng YAO, Xingfu ZHANG, Jianhao XUAN, Xu YU, authorCompany=null, correspAuthors=Xingfu ZHANG, 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=1322916918885221222, articleId=1322916916884538200, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=多源时变重力场模型融合的不同定权方法研究, 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=

时变重力场模型为大尺度地表质量变化监测提供了关键数据支撑，但不同机构发布的模型精度存在差异。为综合利用多源时变重力场模型优势，本文围绕加权融合方法，重点研究VCE定权、海洋噪声RMS定权和阶方差定权（含2种参考模型）等不同定权方法，并选取HUST等8个国内外机构发布的时变重力场模型开展融合实验与精度评估。结果表明：1）不同阶次定权方案对VCE融合结果有影响，截断至90阶时，GRACE和GRACE-FO模型用于定权的最优球谐次数分别为70和60；2）4种定权融合方案均能提高模型精度，相比Tongji模型，滤波前阶方差定权融合（GOCO06s参考模型）最高可降低23%的海洋噪声RMS，滤波后VCE定权融合最高可降低29%的海洋噪声RMS；3）在6个流域，各时变模型的陆地水储量变化趋势具有较好的一致性，融合模型与Mascon结果相关系数更高。

, authors=姚鹏1, 张兴福1, *, 禤键豪2, 余旭1, authorsList=姚鹏, 张兴福, 禤键豪, 余旭, authorCompany=null, correspAuthors=张兴福, authorNote=

姚鹏，硕士生，主要从事卫星重力应用研究，E-mail：

, correspAuthorsNote=
张兴福, 教授, 主要从事卫星重力学理论、方法及应用研究, E-mail:
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Time-variable gravity field models

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机构全称 机构简称 GRACE版本 GRACE-FO版本
Center for Space Research at University of Texas, Austin CSR CSR Release 06 CSR Release 06.3 (GFO)
GFZ Helmholtz Centre for Geosciences GFZ GFZ Release 06 GFZ Release 06.3 (GFO)
Astronomical Institute of the University of Bern AIUB AIUB-G3P AIUB-GRACE-FO_RL02op
Institute of Geodesy at Graz University of Technology ITSG ITSG-Grace2018 ITSG-Grace_op
Jet Propulsion Laboratory JPL JPL Release 06 JPL Release 06.3 (GFO)
Centre National d'Études Spatiales CNES CNES_GRGS_RL05_CHOL
同济大学 Tongji Tongji-Grace2022
华中科技大学 HUST HUST-Grace2024
), ArticleFig(id=1322916925289923480, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916916884538200, language=CN, label=表1, caption=

时变重力场模型

, figureFileSmall=null, figureFileBig=null, tableContent=
机构全称 机构简称 GRACE版本 GRACE-FO版本
Center for Space Research at University of Texas, Austin CSR CSR Release 06 CSR Release 06.3 (GFO)
GFZ Helmholtz Centre for Geosciences GFZ GFZ Release 06 GFZ Release 06.3 (GFO)
Astronomical Institute of the University of Bern AIUB AIUB-G3P AIUB-GRACE-FO_RL02op
Institute of Geodesy at Graz University of Technology ITSG ITSG-Grace2018 ITSG-Grace_op
Jet Propulsion Laboratory JPL JPL Release 06 JPL Release 06.3 (GFO)
Centre National d'Études Spatiales CNES CNES_GRGS_RL05_CHOL
同济大学 Tongji Tongji-Grace2022
华中科技大学 HUST HUST-Grace2024
), ArticleFig(id=1322916925365420953, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916916884538200, language=EN, label=Tab. 2, caption=

RMS of ocean noise of VCE weighting schemes for GRACE/GRACE-FO models (30 to 90 degrees and orders)

, figureFileSmall=null, figureFileBig=null, tableContent=
模型 阶 次
30 40 50 60 70 80 90
GRACE融合模型 30 2.38
40 4.94 4.86
50 10.87 10.85 10.84
60 21.83 21.89 21.91 22.13
70 50.58 50.65 51.17 52.08 51.47
80 91.85 91.81 92.19 93.03 93.85 94.44
90 145.00 145.30 145.19 144.56 144.18 145.07 144.99
GRACE-FO融合模型 30 2.24
40 3.44 3.40
50 6.27 6.14 6.02
60 13.10 12.82 12.59 12.56
70 30.38 29.61 29.25 29.24 28.92
80 106.02 102.60 100.54 101.47 103.28 102.18
90 173.00 165.92 160.01 158.96 159.68 162.83 162.66
), ArticleFig(id=1322916925436724122, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916916884538200, language=CN, label=表2, caption=

GRACE/GRACE-FO模型30 ~ 90阶次VCE定权方案的海洋噪声RMS

, figureFileSmall=null, figureFileBig=null, tableContent=
模型 阶 次
30 40 50 60 70 80 90
GRACE融合模型 30 2.38
40 4.94 4.86
50 10.87 10.85 10.84
60 21.83 21.89 21.91 22.13
70 50.58 50.65 51.17 52.08 51.47
80 91.85 91.81 92.19 93.03 93.85 94.44
90 145.00 145.30 145.19 144.56 144.18 145.07 144.99
GRACE-FO融合模型 30 2.24
40 3.44 3.40
50 6.27 6.14 6.02
60 13.10 12.82 12.59 12.56
70 30.38 29.61 29.25 29.24 28.92
80 106.02 102.60 100.54 101.47 103.28 102.18
90 173.00 165.92 160.01 158.96 159.68 162.83 162.66
), ArticleFig(id=1322916925499638683, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916916884538200, language=EN, label=Tab. 3, caption=

RMS/SNR of various fusion models and selected time-variable models before and after filtering

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滤波前后 海洋噪声定权融合 阶方差定权融合(GOCO06s) 阶方差定权融合(Mean) VCE融合 ITSG Tongji
无滤波 172.47 cm/0.21 139.30 cm/0.21 139.74 cm/0.21 151.49 cm/0.21 181.56 cm/0.21 180.97 cm/0.21
DDK3 2.11 cm/1.70 2.04 cm/1.77 2.03 cm/1.77 2.01 cm/1.79 2.37 cm/1.52 2.83 cm/1.35
), ArticleFig(id=1322916925570941852, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916916884538200, language=CN, label=表3, caption=

各融合模型与部分时变模型滤波前后的RMS/SNR

, figureFileSmall=null, figureFileBig=null, tableContent=
滤波前后 海洋噪声定权融合 阶方差定权融合(GOCO06s) 阶方差定权融合(Mean) VCE融合 ITSG Tongji
无滤波 172.47 cm/0.21 139.30 cm/0.21 139.74 cm/0.21 151.49 cm/0.21 181.56 cm/0.21 180.97 cm/0.21
DDK3 2.11 cm/1.70 2.04 cm/1.77 2.03 cm/1.77 2.01 cm/1.79 2.37 cm/1.52 2.83 cm/1.35
), ArticleFig(id=1322916925646439325, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916916884538200, language=EN, label=Tab. 4, caption=

Correlation coefficients of selected time-variable models with Mascon models across various basins

, figureFileSmall=null, figureFileBig=null, tableContent=
AIUB Tongji VCE 阶方差融合(GOCO06s)
亚马孙河流域 0.996 0.997 0.999 0.999
鄂毕河流域 0.978 0.986 0.992 0.991
恒河流域 0.965 0.991 0.990 0.991
奥里诺科河流域 0.989 0.990 0.996 0.996
圣弗朗西斯科河流域 0.966 0.977 0.984 0.983
塔兹河流域 0.930 0.971 0.979 0.977
), ArticleFig(id=1322916925717742494, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916916884538200, language=CN, label=表4, caption=

部分时变模型在各流域与Mascon模型的相关系数

, figureFileSmall=null, figureFileBig=null, tableContent=
AIUB Tongji VCE 阶方差融合(GOCO06s)
亚马孙河流域 0.996 0.997 0.999 0.999
鄂毕河流域 0.978 0.986 0.992 0.991
恒河流域 0.965 0.991 0.990 0.991
奥里诺科河流域 0.989 0.990 0.996 0.996
圣弗朗西斯科河流域 0.966 0.977 0.984 0.983
塔兹河流域 0.930 0.971 0.979 0.977
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多源时变重力场模型融合的不同定权方法研究
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姚鹏 1 , 张兴福 1, * , 禤键豪 2 , 余旭 1
大地测量与地球动力学 | 特色栏目: 地震大地测量学 2026,46(8): 957-963
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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 (8) : 957 -963
多源时变重力场模型融合的不同定权方法研究
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姚鹏1 , 张兴福1, * , 禤键豪2, 余旭1
作者信息
  • 1 广东工业大学土木与交通工程学院, 广州, 510006
  • 2 同济大学测绘与地理信息学院, 上海, 200092
通讯作者:
张兴福, 教授, 主要从事卫星重力学理论、方法及应用研究, E-mail:
作者简介:

姚鹏，硕士生，主要从事卫星重力应用研究，E-mail：

Weighting Schemes for Multi-Source Time-Variable Gravity Field Model Fusion
Peng YAO1 , Xingfu ZHANG1, * , Jianhao XUAN2, Xu YU1
Affiliations
  • 1 School of Civil and Transportation Engineering, Guangdong University of Technology, Guangzhou 510006, China
  • 2 College of Surveying and Geo-Informatics, Tongji University, Shanghai 200092, China
出版时间: 2026-08-15 doi: 10.14075/j.jgg.2025.12.430
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摘要
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时变重力场模型为大尺度地表质量变化监测提供了关键数据支撑，但不同机构发布的模型精度存在差异。为综合利用多源时变重力场模型优势，本文围绕加权融合方法，重点研究VCE定权、海洋噪声RMS定权和阶方差定权（含2种参考模型）等不同定权方法，并选取HUST等8个国内外机构发布的时变重力场模型开展融合实验与精度评估。结果表明：1）不同阶次定权方案对VCE融合结果有影响，截断至90阶时，GRACE和GRACE-FO模型用于定权的最优球谐次数分别为70和60；2）4种定权融合方案均能提高模型精度，相比Tongji模型，滤波前阶方差定权融合（GOCO06s参考模型）最高可降低23%的海洋噪声RMS，滤波后VCE定权融合最高可降低29%的海洋噪声RMS；3）在6个流域，各时变模型的陆地水储量变化趋势具有较好的一致性，融合模型与Mascon结果相关系数更高。

多源时变重力场  /  定权方法  /  模型融合  /  精度分析  /  加权融合方法

Time-variable gravity field models provide critical data support for monitoring large-scale surface mass changes. However, discrepancies in accuracy exist among models released by different institutions. To fully utilize the advantages of multi-source time-variable gravity field models, this study focuses on weighted fusion methods, investigating various weighting strategies, including variance component estimation (VCE), ocean noise RMS-based weighting, and degree variance-based weighting (incorporating two reference models). Time-variable gravity field models released by eight institutions worldwide, including HUST, were selected for fusion experiments and accuracy assessment. The results indicate: 1) The choice of weighting schemes across different degrees/orders influences the VCE fusion outcomes. When truncated to degree 90, the optimal weighting orders are 70 for GRACE models and 60 for GRACE-FO models. 2) All four weighting fusion schemes enhance model accuracy. Compared to the Tongji model, the pre-filtering degree-variance fusion (using the GOCO06s reference model) reduces ocean noise RMS by up to 23%, while the post-filtering VCE fusion achieves a maximum reduction of 29%. 3) Over six major river basins, the terrestrial water storage trends derived from different time-variable models show good consistency, and the fusion models exhibit higher correlation coefficients with Mascon solutions.

multi-source time-variable gravity field  /  weighting schemes  /  model fusion  /  accuracy analysis  /  weighted fusion method
姚鹏, 张兴福, 禤键豪, 余旭. 多源时变重力场模型融合的不同定权方法研究. 大地测量与地球动力学, 2026 , 46 (8) : 957 -963 . DOI: 10.14075/j.jgg.2025.12.430
Peng YAO, Xingfu ZHANG, Jianhao XUAN, Xu YU. Weighting Schemes for Multi-Source Time-Variable Gravity Field Model Fusion[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (8) : 957 -963 . DOI: 10.14075/j.jgg.2025.12.430
正文
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时变地球重力场可以监测地球系统质量迁移，揭示陆地水储量变化、冰川消融、海平面上升等水文气候过程[1-2]。国际重力场模型中心(ICGEM)已发布超过30个时变重力场模型，这些模型被广泛用于水储量变化等相关科学研究，促进了大地测量、海洋科学以及水文学等相关学科的交叉与融合[3]。
各研究机构在时变重力场模型的反演方法、数据处理策略等方面存在差异，导致各模型的噪声水平不同[4]，理论上融合多源时变重力场模型可以发挥各时变模型的优势，有利于减弱单个时变模型的噪声[5]，提升时变模型的信噪比和稳定性。部分学者采用谱组合法[6]、算术平均法[7]、区域加权平均法[8]、熵权法和变异系数法[9]、维纳最优评估法[10]等方法开展了多源重力场模型及其衍生产品的融合研究，融合模型精度较单个模型有所提高。为进一步推动多源时变重力场模型融合，国际上成立了COST-G(combination service for time-variable gravity fields)，深入开展了GRACE/GRACE-FO等时变重力场模型融合的方差分量估计(variance component estimation，VCE)法研究。VCE融合可降低模型噪声，融合模型的信噪比优于单个模型[11]。虽然COST-G开展了VCE融合方法研究，并发布了相应的融合模型，但由于不同机构发布的时变模型在不同阶次呈现不同的噪声水平，深入分析时变模型不同定权方案对融合结果的影响，仍然具有较高的实际应用价值。为此，本文开展基本各模型不同阶次的VCE定权融合、海洋噪声RMS定权、阶方差定权(含2种参考模型)等融合方法研究，并对融合结果进行分析。
1 原理与方法
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1.1 时变重力场模型精度评估方法
选取阶方差[12]、海洋噪声均方根(RMS)和信噪比(SNR)这3个指标来评估时变模型精度，并用于后续各时变模型权比计算。海洋噪声均方根计算公式为[13]：
$\mathrm{RMS}=\sqrt{\frac{\sum\limits_{i=1}^N \sin \left(\theta_i\right) \rho_i^2\left(\theta_i, \lambda_i\right)}{\sum\limits_{i=1}^N \sin \left(\theta_i\right)}}$
式中，N为格网数量；ρi为噪声，即第i个格网的水储量变化反演值与拟合信号值之差；θi和λi分别为第i个格网的余纬和经度。
考虑到海洋区域在全球水储量变化中具有变化幅度小、时空稳定性高的特点，因此选择海洋区域的SNR作为评估准则，计算公式为[14]：
$\mathrm{SNR}=\frac{\mathrm{RMS}_s}{\mathrm{RMS}_n}$
式中，RMSs为海洋区域水储量值信号的均方根；RMSn为海洋噪声的均方根。
1.2 融合方法
由于各时变模型的参考基准不同，在融合前需进行统一[15]。本文采用的多源时变模型融合方法，其本质是不同模型的加权融合，应用该方法的关键是通过不同方法确定各模型的合理权重，计算公式如下：
$\overline{X}_n^t=\frac{\sum\limits_{i=1}^N W_n^{i, t} X_n^{i, t}}{\sum\limits_{i=1}^N W_n^{i, t}}$
式中，i为第i个模型；t为对应月份；n为对应阶数；N为参与融合的模型数量；Xni, t与$ \overline{X}_n^t$分别为融合前后的球谐系数；Wni, t为权重，当单个模型的所有阶的权重都为同一值时，则下标n可以省略。
多源时变模型的加权融合可采用整体加权法与按阶(次)系数加权法两大类方法。整体加权法即对每个参与融合的模型分配一个权值，通过权值调控各模型的整体贡献度；按阶(次)系数加权法则是对各模型的相同阶(次)系数分别赋予不同权值。本文采用VCE定权和海洋噪声RMS定权2种整体定权方法，以及2种不同参考模型下的阶方差定权方法，各方法的具体实现如下：
1) VCE定权。VCE定权法是一种通过迭代优化权重来实现多源时变模型融合的方法，其基于各个模型与所有模型融合结果的残差信息，自适应估计各模型的方差，并以方差估值的倒数为权重进行融合，通过多次迭代，动态调整权重分配，使其逐渐逼近最优状态。计算公式为[16]：
$W^{i, t, k}=\left[\frac{\sum\limits_{n=2}^{n_{\max }} \sum\limits_{m=0}^n\left(X_{n, m}^{i, t}-\hat{X}_{n, m}^{t, k-1}\right)^2}{N_{\text {coef }} \cdot\left(1-\frac{W^{i, t, k-1}}{\sum\limits_{i=1}^{N_{\mathrm{sol}}} W^{i, t, k-1}}\right)}\right]^{-1}$
式中，Ncoef和Nsol分别为模型参与定权的系数数量与模型数量；X为时变模型的球谐系数；$ \hat{X}_{n, m}^{k-1}$为第k-1次迭代时系数的加权平均值。
由于不同阶次的球谐系数噪声水平存在差异，如果仅选取时变模型的最高阶次进行定权，可能会受高阶次噪声影响，导致最终融合结果并非最优。因此，本文在进行VCE融合定权时，通过选取不同阶次的球谐系数参与定权计算，从而分析不同定权方案对融合结果的影响。
2) 海洋噪声RMS定权。通过计算各时变模型未滤波水储量变化的海洋噪声RMS值来确定权重。计算公式为：
$W^{i, t}=\frac{\sigma_0^2}{\left(\mathrm{RMS}_n^{i, t}\right)^2}$
式中，σ0为单位权中误差，可取所有时变模型RMS的平均值；RMSni, t为第i个模型在t月对应的未滤波海洋噪声RMS。
3) 阶方差定权。阶方差定权的本质是选定一个参考重力场模型，分别计算各时变模型的阶方差作为定权依据。计算公式为：
$W_n^{i, t}=\frac{\sigma_0^2}{\left(\sigma_n^{i, t}\right)^2}$
式中，Wni, t为n阶系数权重；σ0为单位权中误差；σni, t为相对于参考重力场模型的阶方差。
2 实验结果与分析
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2.1 数据源
选取ICGEM网站公开发布的时变重力场模型作为实验数据，实验模型的选取准则为模型最大阶数不低于90，同时需包含GRACE和GRACE-FO两个观测时间段的数据。最终选定8个机构的时变模型用于分析，各模型信息见表 1。为确保在统一的时段内所有时变模型均可参与融合，将GRACE和GRACE-FO模型时间覆盖范围分别统一选取为2002-04—2017-06和2018-06—2024-08。
2.2 各时变模型精度分析
图 1为以GOCO06s静态重力场为参考模型计算的2005-08和2019-10各模型的阶方差。结果表明，在前20阶，各模型之间的阶方差差异较小，说明各模型在此频段内精度相当；20阶之后，各模型之间的阶方差呈现出较大差异，说明各模型不同阶之间的噪声水平存在差异，因此探讨多源时变模型融合具有研究价值。Tongji模型与ITSG模型的阶方差较低，精度优于其他模型，后续将选取这2个模型与融合模型进行相关精度比较，为融合有效性提供参考依据。
2.3 不同定权方案下VCE融合结果分析
为分析VCE融合过程中采用不同阶次定权对融合结果的影响，设计多组对比实验。由于球谐系数的噪声在高阶部分较大[17]，故忽略其低阶部分，选择阶数范围为30~90阶、次数范围为30~90次，其中定权次数不大于定权阶数，共构建28种实验方案。为保证融合的合理性，各模型仅截断至对应定权方案的阶数进行融合，并采用海洋噪声RMS与SNR对融合模型精度进行分析，RMS评估结果见表 2(单位cm)。
表 2结果表明，采用同阶不同次数定权方案对VCE融合结果存在影响，对于GRACE模型的30~50阶融合结果，最优次均为最高次；对于60~90阶融合结果，最优次分别为30、30、40和70次。对于GRACE-FO模型的30~70阶融合结果，最优次均为最高次，80和90阶融合结果的最优次分别为50和60次。
GRACE融合模型在30~90阶融合结果的SNR分别为0.98、0.50~0.51、0.28、0.22、0.21、0.20~0.21、0.21；GRACE-FO融合模型在30~90阶融合结果的SNR分别为0.71、0.52、0.37~0.38、0.33、0.32~0.33、0.29、0.30。可以看出，由于融合模型未滤波，噪声较大，所以融合模型的SNR较小，且随阶次升高而降低，同阶不同次的SNR结果差异基本可以忽略。
综上可知，在90阶条件下，对于本文的GRACE模型融合，最优定权次数为70次，而对于GRACE-FO模型融合，最优定权次数为60次。由于GRACE和GRACE-FO卫星的定轨和加速度计等观测数据精度存在差异，且不同机构采用的重力场反演方法、加速度计校正方法以及数据处理策略等不同，导致两类卫星时变模型的精度和不同阶次的噪声水平存在差异，进而使融合最优次不同[3]。
图 2为不同VCE定权方案下各时变模型权重占比图，结果表明，在本文选取的最优定权方案中，精度较高的ITSG、Tongji模型的权重占比高于其他模型[18]，能发挥各模型的精度优势，且对于不同月份各时变模型的贡献也存在差异。
2.4 阶方差定权融合分析
本文选用2种参考模型进行阶方差定权计算：1)选用高精度静态重力场模型GOCO06s作为参考模型；2)以输入的所有时变模型系数的平均值作为参考模型(简称为Mean模型)。图 3为不同方案中各模型的权重，结果表明，当阶数小于20时，权值分布较为接近，说明各模型在此频段内精度较为一致；当阶数大于20时，各模型权值差异较大，精度较高的Tongji、ITSG等模型的权值较高；2种定权方案下的权值分布一致性较高，仅部分模型的权值在中高阶(30~50阶)存在差异，说明该定权方法对参考模型的选择依赖性不高。
2.5 海洋噪声RMS定权融合及各定权融合方法对比分析
图 4为各时变模型、RMS定权融合模型、VCE定权融合模型、阶方差定权融合模型的海洋噪声RMS值。结果表明，不同模型的RMS存在差异，因此可将RMS作为定权依据。从融合结果来看，各融合模型的RMS值比AIUB、GFZ等单个时变模型更低，表明各融合方案可起到降低噪声的效果。
图 5为4种融合方法所得模型在2013-06和2022-04的阶方差，表 3为各融合模型与单个模型在滤波前后的RMS及SNR值。图 5和表 3结果表明，在低阶范围内，4种融合方案的阶方差较为接近；当阶数大于50时，阶方差融合方案的阶方差略低于其他2种方案，且2种阶方差融合方案的结果较为接近。未滤波时，4种融合方案精度较单个模型均有所提升，其中阶方差融合方案(GOCO06s)降噪效果最优，较Tongji模型RMS降低约23%，VCE融合方案效果次之，RMS降低约16%；滤波后，所有融合方案结果均优于单个模型，其中VCE融合方案RMS降低约29%，信噪比提升32%。表明4种融合方案均可以降低噪声，具有实际应用价值。
综上可知，无论滤波前后，本文方法均可以降低各时变模型噪声。如果仅需要未滤波融合模型，可以首选阶方差定权方法；如果需要融合模型的滤波结果，阶方差定权和VCE方法均可。从各融合方法的适用性来看，VCE方法需要3个及以上时变模型才能有效融合，海洋噪声RMS定权融合和阶方差定权融合适用于2个及以上时变模型融合。海洋噪声RMS定权融合和VCE均是模型整体定权，而阶方差定权融合可以体现不同阶位系数差异。
2.6 流域水储量分析
为进一步评估融合模型精度，本文选取6个流域进行水储量反演，并将Mascon结果作为参考基准进行趋势变化分析。时变模型的后处理方法见参考文献[19]，其中滤波采用DDK3，并以2004—2009年的平均水储量值为参考基准，扣除后得到水储量变化值，计算各流域2002—2024年水储量变化的趋势项。
图 6为6个流域的陆地水储量变化时间序列，图中实线表示水储量变化值, 虚线表示趋势。表 4为各融合模型与Mascon模型的相关系数。图 6和表 4结果表明，融合模型与单个模型的水储量时间序列变化趋势具有较好的一致性，且融合模型与Mascon结果具有更高的相关性。
3 结语
收起
本研究采用基于VCE、海洋噪声RMS以及2种参考模型的阶方差定权融合方法，对多源时变重力场模型进行融合，并开展实验验证与精度分析，主要结论如下：
1) 在VCE不同定权方案中，在同一阶数条件下，使用不同次数进行定权对融合结果精度有影响，且不同模型可能会对应不同的最优定权方案。进行90阶模型融合时，对于GRACE模型，70次为最佳定权方案；对于GRACE-FO模型，60次为最佳定权方案。
2) 4种融合方案结果表明，滤波前后融合模型的精度均有提升，其中在滤波前，2种阶方差定权融合方案效果较优；滤波后则为VCE定权融合方案较优。基于全球6个流域的陆地水储量变化分析结果表明，融合模型与单个时变模型的变化趋势具有较好的一致性，且融合模型与参考模型Mascon具有更高的相关性。
基金
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  • 国家自然科学基金(42374100)
文献
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参考文献 引证文献
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2026年第46卷第8期
32
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引用本文
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doi: 10.14075/j.jgg.2025.12.430
  • 接收时间：2025-12-08
  • 首发时间：2026-10-30
  • 出版时间：2026-08-15
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  • 收稿日期：2025-12-08
基金
国家自然科学基金(42374100)
作者信息
    1 广东工业大学土木与交通工程学院, 广州, 510006
    2 同济大学测绘与地理信息学院, 上海, 200092

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张兴福, 教授, 主要从事卫星重力学理论、方法及应用研究, E-mail:
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2种不同金属材料的力学参数
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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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