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Article(id=1322566846339994038, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322566845215912204, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.03.060, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1740931200000, receivedDateStr=2025-03-03, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793254108271, onlineDateStr=2026-10-29, pubDate=1771084800000, pubDateStr=2026-02-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793254108271, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793254108271, creator=13701087609, updateTime=1793254108271, updator=13701087609, issue=Issue{id=1322566845215912204, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='2', pageStart='139', pageEnd='261', issueExtLink='null', onlineDate='null', pubDate='1771084800000', pubDateStr='2026-02-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793254108003, creator='13701087609', updateTime=1793254108003, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=234, endPage=243, ext={EN=ArticleExt(id=1322566846520349111, articleId=1322566846339994038, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Comparative Analysis of GRACE RL06.3 Time-Variable Gravity Field Models, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

We conduct a comparative analysis of RL06.3 time-variable gravity field models released by CSR, GFZ, and JPL, focusing on the degree one, C20 and C30 and their surface mass anomalies, mean degree variances, signal-to-noise ratios of global surface mass changes, and terrestrial water storage changes in typical regions. The results show that degree one, C20 and their surface mass anomalies of three models exhibit significant differences in trend but little differences in annual terms. For the C30, the trend shows considerable differences, while the annual terms are largely consistent. The C20 and C30 of RL06.3 models from the same institution with different truncation degrees have little impact on the trend and annual terms, but the differences between institutions are notable, particularly in the trend. The mean degree variances of three models are highly consistent in low-degree terms, while CSR RL06.3 shows the most gradual rise in noise fitting curves for high-degree terms. The terrestrial water storage inversion results from three models tend to be consistent, but the CSR and JPL models exhibit superior performance in terms of inversion accuracy and consistency. Conversely, the GFZ model generally presents a higher degree of uncertainty in its inversion results. When inverting terrestrial water storage changes, if the uncertainty of results is disregarded, it is recommended to use the GRACE time-variable gravity field models with higher truncation orders released by CSR or JPL. Otherwise, it is advisable to use the GRACE time-variable gravity field models with lower truncation orders released by CSR.

, authors=Jinhui ZHANG1, Shanshan LI1, *, Haopeng FAN1, Diao FAN1, authorsList=Jinhui ZHANG, Shanshan LI, Haopeng FAN, Diao FAN, authorCompany=null, correspAuthors=Shanshan LI, 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=1322566850156810696, articleId=1322566846339994038, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=GRACE RL06.3时变重力场模型比较分析, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

对CSR、GFZ和JPL发布的RL06.3时变重力场模型, 从一阶项、C20和C30及其地表质量异常、平均阶方差、全球地表质量变化的信噪比、典型区域陆地水储量变化等方面进行比较分析。结果表明, 3家机构模型的一阶项、C20及其地表质量异常在趋势项上差异显著, 但周年项差异较小; C30项在趋势项上差异较大, 而周年项基本一致。同一机构不同阶次的RL06.3时变重力场模型的C20和C30项的趋势项和周年项基本无差异, 但不同机构间的差异较为明显, 尤其是趋势项差异更为显著。3家机构模型的平均阶方差在低阶项的信号拟合曲线高度一致, 在高阶项CSR RL06.3模型的噪声拟合曲线上升最为平缓; 3家机构模型的陆地水储量反演结果趋于一致, 但CSR和JPL两家机构模型在反演精度和一致性方面表现更优, 而GFZ RL06.3模型反演结果的不确定度普遍较大。在反演陆地水储量变化时, 若忽略结果的不确定度, 建议使用CSR或JPL发布的截断阶数较高的GRACE时变重力场模型, 否则建议使用CSR发布的截断阶数较低的GRACE时变重力场模型。

, authors=张金辉1, 李姗姗1, *, 范昊鹏1, 范雕1, authorsList=张金辉, 李姗姗, 范昊鹏, 范雕, authorCompany=null, correspAuthors=李姗姗, authorNote=

张金辉, 副教授, 主要从事测量数据处理和海平面变化研究, E-mail:

, correspAuthorsNote=
李姗姗, 教授, 博士生导师, 主要从事物理大地测量学研究, E-mail:
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张金辉, 副教授, 主要从事测量数据处理和海平面变化研究, E-mail:

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张金辉, 副教授, 主要从事测量数据处理和海平面变化研究, E-mail:

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The main models used for solving GRACE RL06.3 time-varying gravity field models released by CSR, GFZ, and JPL

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解算模型CSR RL06.3GFZ RL06.3JPL RL06.3
背景静态重力场(阶次)GGM05C(360阶)EIGEN-6C4(200阶)GGM05C(360阶)
第三体扰动DE-430DE-430DE-421
固体潮IERS-2010IERS-2010IERS-2010
海潮GOT4.8FES2014FES2014b
极潮弹性地球模型+IERS-2010
(线性极移模型)		弹性地球模型+IERS-2010
(线性极移模型)		非弹性地球模型+IERS-2010
(三次极移模型)
大气和非潮汐变化的海洋AOD1B RL06
(ECMWF+MPIOM)		AOD1B RL06
(ECMWF+MPIOM)		AOD1B RL06
(ECMWF+OMCT)
), ArticleFig(id=1322566858981626360, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566846339994038, language=CN, label=表1, caption=

CSR、GFZ和JPL GRACE RL06.3时变重力场模型解算所采用的主要模型

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解算模型CSR RL06.3GFZ RL06.3JPL RL06.3
背景静态重力场(阶次)GGM05C(360阶)EIGEN-6C4(200阶)GGM05C(360阶)
第三体扰动DE-430DE-430DE-421
固体潮IERS-2010IERS-2010IERS-2010
海潮GOT4.8FES2014FES2014b
极潮弹性地球模型+IERS-2010
(线性极移模型)		弹性地球模型+IERS-2010
(线性极移模型)		非弹性地球模型+IERS-2010
(三次极移模型)
大气和非潮汐变化的海洋AOD1B RL06
(ECMWF+MPIOM)		AOD1B RL06
(ECMWF+MPIOM)		AOD1B RL06
(ECMWF+OMCT)
), ArticleFig(id=1322566859258450425, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566846339994038, language=EN, label=Tab. 2, caption=

Trend, annual amplitude and annual phase of degree one of GRCAE RL06.3 time-variable gravity models released by CSR、GFZ and JPL

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模型一阶项系数趋势/10-12·a-1周年振幅/10-10周年相位/(°)
CSR RL06.3C10-49.391 8±0.147 22.391 3±0.168 53.96±9.27
GFZ RL06.3-44.199 2±0.156 22.399 6±0.178 3-18.80±8.50
JPL RL06.3-50.520 2±0.139 12.395 2±0.160 74.43±7.84
CSR RL06.3C11-9.717 6±0.179 31.378 6±0.146 223.53±12.81
GFZ RL06.3-12.020 1±0.173 51.377 5±0.197 524.22±11.90
JPL RL06.3-10.294 9±0.183 21.364 8±0.209 424.30±13.11
CSR RL06.3S118.815 9±0.202 41.544 5±0.231 2-138.85±3.43
GFZ RL06.311.752 3±0.349 41.434 5±0.397 2-136.67±2.41
JPL RL06.38.016 8±0.483 21.528 1±0.321 7-137.88±3.16
), ArticleFig(id=1322566859463971322, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566846339994038, language=CN, label=表2, caption=

CSR、GFZ和JPL GRACE RL06.3时变重力场模型一阶项的趋势、周年振幅与周年相位

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模型一阶项系数趋势/10-12·a-1周年振幅/10-10周年相位/(°)
CSR RL06.3C10-49.391 8±0.147 22.391 3±0.168 53.96±9.27
GFZ RL06.3-44.199 2±0.156 22.399 6±0.178 3-18.80±8.50
JPL RL06.3-50.520 2±0.139 12.395 2±0.160 74.43±7.84
CSR RL06.3C11-9.717 6±0.179 31.378 6±0.146 223.53±12.81
GFZ RL06.3-12.020 1±0.173 51.377 5±0.197 524.22±11.90
JPL RL06.3-10.294 9±0.183 21.364 8±0.209 424.30±13.11
CSR RL06.3S118.815 9±0.202 41.544 5±0.231 2-138.85±3.43
GFZ RL06.311.752 3±0.349 41.434 5±0.397 2-136.67±2.41
JPL RL06.38.016 8±0.483 21.528 1±0.321 7-137.88±3.16
), ArticleFig(id=1322566859740795388, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566846339994038, language=EN, label=Tab. 3, caption=

Trend, annual amplitude and annual phase of C20 and C30 terms of different models

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模型系数趋势/10-11·a-1周年振幅/10-11周年相位/(°)
CSR RL06.3/60C20-2.402 5±0.002 15.343 7±0.024 223.60±0.56
CSR RL06.3/96-2.402 7±0.002 25.334 7±0.023 623.74±0.66
GFZ RL06.3/60-2.961 1±0.002 45.917 9±0.026 811.67±0.57
GFZ RL06.3/96-2.839 1±0.002 46.741 4±0.027 513.36±0.41
JPL RL06.3/60-2.266 2±0.002 46.620 2±0.027 418.26±0.30
JPL RL06.3/96-2.190 4±0.002 46.725 5±0.027 516.88±0.65
SLR TN14-5.179 0±0.000 84.123 0±0.009 0-50.74±0.48
CSR RL06.3/60C30-10.587 1±0.058 65.613 2±0.066 97.86±0.10
CSR RL06.3/96-10.532 8±0.064 25.614 6±0.073 37.81±0.09
GFZ RL06.3/60-3.068 3±0.066 55.049 3±0.075 912.63±0.13
GFZ RL06.3/96-3.168 1±0.063 34.914 0±0.072 413.01±0.12
JPL RL06.3/60-4.406 1±0.056 07.578 2±0.064 117.39±0.10
JPL RL06.3/96-4.559 9±0.090 97.502 7±0.104 816.61±0.17
SLR TN14-6.563 1±0.170 67.734 2±0.110 416.66±0.13
), ArticleFig(id=1322566859912761853, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566846339994038, language=CN, label=表3, caption=

不同模型C20和C30项系数的趋势、周年振幅与周年相位

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模型系数趋势/10-11·a-1周年振幅/10-11周年相位/(°)
CSR RL06.3/60C20-2.402 5±0.002 15.343 7±0.024 223.60±0.56
CSR RL06.3/96-2.402 7±0.002 25.334 7±0.023 623.74±0.66
GFZ RL06.3/60-2.961 1±0.002 45.917 9±0.026 811.67±0.57
GFZ RL06.3/96-2.839 1±0.002 46.741 4±0.027 513.36±0.41
JPL RL06.3/60-2.266 2±0.002 46.620 2±0.027 418.26±0.30
JPL RL06.3/96-2.190 4±0.002 46.725 5±0.027 516.88±0.65
SLR TN14-5.179 0±0.000 84.123 0±0.009 0-50.74±0.48
CSR RL06.3/60C30-10.587 1±0.058 65.613 2±0.066 97.86±0.10
CSR RL06.3/96-10.532 8±0.064 25.614 6±0.073 37.81±0.09
GFZ RL06.3/60-3.068 3±0.066 55.049 3±0.075 912.63±0.13
GFZ RL06.3/96-3.168 1±0.063 34.914 0±0.072 413.01±0.12
JPL RL06.3/60-4.406 1±0.056 07.578 2±0.064 117.39±0.10
JPL RL06.3/96-4.559 9±0.090 97.502 7±0.104 816.61±0.17
SLR TN14-6.563 1±0.170 67.734 2±0.110 416.66±0.13
), ArticleFig(id=1322566860101505534, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566846339994038, language=EN, label=Tab. 4, caption=

SNR statistics of global surface mass variation form April 2002 to August 2024

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模型CSR RL06.3/60CSR RL06.3/96GFZ RL06.3/60GFZ RL06.3/96JPL RL06.3/60JPL RL06.3/96
最大值3.201 23.041 43.182 43.091 73.194 83.152 6
最小值1.487 51.353 51.456 41.186 11.507 41.409 2
平均值2.199 22.169 22.161 32.133 02.182 22.180 3
标准差0.238 80.239 80.250 80.251 80.248 30.252 8
), ArticleFig(id=1322566860294443520, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566846339994038, language=CN, label=表4, caption=

2002-04—2024-08全球地表质量变化的信噪比统计

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模型CSR RL06.3/60CSR RL06.3/96GFZ RL06.3/60GFZ RL06.3/96JPL RL06.3/60JPL RL06.3/96
最大值3.201 23.041 43.182 43.091 73.194 83.152 6
最小值1.487 51.353 51.456 41.186 11.507 41.409 2
平均值2.199 22.169 22.161 32.133 02.182 22.180 3
标准差0.238 80.239 80.250 80.251 80.248 30.252 8
), ArticleFig(id=1322566860596433409, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566846339994038, language=EN, label=Tab. 5, caption=

Trends and uncertainties of terrestrial water storage in typical river basins

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典型流域CSR RL06.3/60CSR RL06.3/96GFZ RL06.3/60GFZ RL06.3/96JPL RL06.3/60JPL RL06.3/96
趋势不确定度趋势不确定度趋势不确定度趋势不确定度趋势不确定度趋势不确定度
亚马逊0.070.280.070.290.081.150.070.820.080.550.070.73
密西西比-0.370.14-0.370.17-0.512.48-0.502.64-0.370.37-0.370.67
伏尔加-0.320.35-0.320.42-0.301.19-0.291.27-0.350.29-0.350.57
刚果0.610.420.600.410.551.200.551.240.590.400.580.91
长江0.480.290.470.490.510.900.501.100.470.400.460.89
默里河0.311.150.321.120.373.140.387.090.301.280.301.71
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典型流域陆地水储量的趋势与不确定度

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典型流域CSR RL06.3/60CSR RL06.3/96GFZ RL06.3/60GFZ RL06.3/96JPL RL06.3/60JPL RL06.3/96
趋势不确定度趋势不确定度趋势不确定度趋势不确定度趋势不确定度趋势不确定度
亚马逊0.070.280.070.290.081.150.070.820.080.550.070.73
密西西比-0.370.14-0.370.17-0.512.48-0.502.64-0.370.37-0.370.67
伏尔加-0.320.35-0.320.42-0.301.19-0.291.27-0.350.29-0.350.57
刚果0.610.420.600.410.551.200.551.240.590.400.580.91
长江0.480.290.470.490.510.900.501.100.470.400.460.89
默里河0.311.150.321.120.373.140.387.090.301.280.301.71
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GRACE RL06.3时变重力场模型比较分析
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张金辉 1 , 李姗姗 1, * , 范昊鹏 1 , 范雕 1
大地测量与地球动力学 | 大地测量学 2026,46(2): 234-243
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大地测量与地球动力学 |大地测量学 2026 , 46 (2) : 234 -243
GRACE RL06.3时变重力场模型比较分析
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张金辉1 , 李姗姗1, * , 范昊鹏1, 范雕1
作者信息
  • 1 信息工程大学地理空间信息学院, 郑州, 450001
通讯作者:
李姗姗, 教授, 博士生导师, 主要从事物理大地测量学研究, E-mail:
作者简介:

张金辉, 副教授, 主要从事测量数据处理和海平面变化研究, E-mail:

Comparative Analysis of GRACE RL06.3 Time-Variable Gravity Field Models
Jinhui ZHANG1 , Shanshan LI1, * , Haopeng FAN1, Diao FAN1
Affiliations
  • 1 School of Surveying and Mapping, Information Engineering University, Zhengzhou 450001, China
出版时间: 2026-02-15 doi: 10.14075/j.jgg.2025.03.060
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对CSR、GFZ和JPL发布的RL06.3时变重力场模型, 从一阶项、C20和C30及其地表质量异常、平均阶方差、全球地表质量变化的信噪比、典型区域陆地水储量变化等方面进行比较分析。结果表明, 3家机构模型的一阶项、C20及其地表质量异常在趋势项上差异显著, 但周年项差异较小; C30项在趋势项上差异较大, 而周年项基本一致。同一机构不同阶次的RL06.3时变重力场模型的C20和C30项的趋势项和周年项基本无差异, 但不同机构间的差异较为明显, 尤其是趋势项差异更为显著。3家机构模型的平均阶方差在低阶项的信号拟合曲线高度一致, 在高阶项CSR RL06.3模型的噪声拟合曲线上升最为平缓; 3家机构模型的陆地水储量反演结果趋于一致, 但CSR和JPL两家机构模型在反演精度和一致性方面表现更优, 而GFZ RL06.3模型反演结果的不确定度普遍较大。在反演陆地水储量变化时, 若忽略结果的不确定度, 建议使用CSR或JPL发布的截断阶数较高的GRACE时变重力场模型, 否则建议使用CSR发布的截断阶数较低的GRACE时变重力场模型。

GRACE  /  时变重力场模型  /  RL06.3模型

We conduct a comparative analysis of RL06.3 time-variable gravity field models released by CSR, GFZ, and JPL, focusing on the degree one, C20 and C30 and their surface mass anomalies, mean degree variances, signal-to-noise ratios of global surface mass changes, and terrestrial water storage changes in typical regions. The results show that degree one, C20 and their surface mass anomalies of three models exhibit significant differences in trend but little differences in annual terms. For the C30, the trend shows considerable differences, while the annual terms are largely consistent. The C20 and C30 of RL06.3 models from the same institution with different truncation degrees have little impact on the trend and annual terms, but the differences between institutions are notable, particularly in the trend. The mean degree variances of three models are highly consistent in low-degree terms, while CSR RL06.3 shows the most gradual rise in noise fitting curves for high-degree terms. The terrestrial water storage inversion results from three models tend to be consistent, but the CSR and JPL models exhibit superior performance in terms of inversion accuracy and consistency. Conversely, the GFZ model generally presents a higher degree of uncertainty in its inversion results. When inverting terrestrial water storage changes, if the uncertainty of results is disregarded, it is recommended to use the GRACE time-variable gravity field models with higher truncation orders released by CSR or JPL. Otherwise, it is advisable to use the GRACE time-variable gravity field models with lower truncation orders released by CSR.

GRACE  /  time-variable gravity field model  /  RL06.3 model
张金辉, 李姗姗, 范昊鹏, 范雕. GRACE RL06.3时变重力场模型比较分析. 大地测量与地球动力学, 2026 , 46 (2) : 234 -243 . DOI: 10.14075/j.jgg.2025.03.060
Jinhui ZHANG, Shanshan LI, Haopeng FAN, Diao FAN. Comparative Analysis of GRACE RL06.3 Time-Variable Gravity Field Models[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (2) : 234 -243 . DOI: 10.14075/j.jgg.2025.03.060
正文
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为精确监测地球重力场变化，美国国家航空航天局与德国航空中心合作先后发射了重力场恢复与气候实验卫星GRACE及后续实验卫星GRACE-FO。该卫星能够全天候探测地球重力场中长波分量及其随时间的变化[1]，利用其观测解算的时变重力场模型已被广泛应用于地球质量迁移及全球气候变化响应研究中[2]，取得的研究成果不仅深化了对地球系统各圈层运动规律的理解，还显著提升了对地球系统物质迁移过程的认识，可为应对全球气候变化提供重要的科学依据。
自GRACE资料向全球公开以来，随着观测精度的不断提升和数据处理方法的持续改进，美国得克萨斯大学空间研究中心(CSR)、德国波茨坦地学研究中心(GFZ)和美国国家航空航天局喷气推进实验室(JPL)等国际主要科研机构陆续发布并不断更新GRACE产品。部分学者[3-4]从时变重力场模型阶方差、C20项时间序列及全球地表质量变化等方面，对早期不同版本截断至60阶的GRACE产品进行比较分析，指出新版本的时变重力场模型在数据质量和精度上较前一版本均有显著提升。然而，以往研究在利用时变重力场模型数据分析极地冰盖和冰川融化[5-6]、陆地水储量变化[7-8]、质量海平面变化[9-10]、地震重力变化[11-12]和冰后回弹[13]时，主要采用CSR发布的截断至60阶的时变重力场模型，而关于不同机构发布的更高阶(如截断至96阶)时变重力场模型与截断至60阶模型的差异及更高阶模型能否适用于上述研究目前尚无明确结论。因此，本文针对CSR、GFZ和JPL三家机构发布的GRACE RL06.3时变重力场模型进行较为全面的比较分析，旨在评估其数据质量和精度，为相关研究中选取合适的时变重力场模型提供参考依据。
1 时变重力场模型与全球地表质量异常球谐展开
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1.1 时变重力场模型
JPL、CSR和GFZ于2024年相继发布了各自的GRACE RL06.3时变重力场模型(以下简称RL06.3)，均以截断至60阶和96阶的球谐系数形式发布。表 1为上述3家机构在解算RL06.3时变重力场模型时所采用的主要模型，可以看出，除固体潮模型外，3家机构在其他模型的选择上均存在一定差异。
与RL06.1相比，RL06.3对处于WDB(wide dead band)模式的月份使用改进的GF-2加速计数据进行重新处理，但3家机构在WDB模式的具体月份上并不完全一致[14-16]，其他月份的数据则与各自机构发布的RL06.1时变重力场模型保持一致。值得一提的是，2024-04仅CSR发布了RL06.2时变重力场模型。该模型采用最优权重策略调整KBR和GPS之间的相对权重，使得2022年以后的低阶项系数有所改善[17]。
1.2 全球地表质量异常球谐展开
研究表明[18]，地表质量异常通常用地表面密度Δσ表示，其球谐展开形式为：
$\begin{gathered}\Delta \sigma(\theta, \lambda)=\sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^m\left[\Delta C_{m n}^{\prime} \cos (n \lambda)+\right. \\\left.\Delta S_{m n}^{\prime} \sin (n \lambda)\right] \widetilde{P}_{m n}(\cos \theta)\end{gathered}$
式中，θ为地心余纬；λ为地心经度；$\widetilde{P}_{m n}$为m阶n次完全规格化的缔合勒让德多项式。面密度变化球谐系数(ΔC′mn，ΔS′mn)可由地球重力场球谐系数转化求得：
$\left[\begin{array}{c}\Delta C_{m n}^{\prime} \\\Delta S_{m n}^{\prime}\end{array}\right]=\frac{a_{\mathrm{E}} \rho_{\mathrm{ave}}}{3} \frac{2 m+1}{1+k_m}\left[\begin{array}{l}\Delta C_{m n} \\\Delta S_{m n}\end{array}\right]$
式中，ae为地球赤道平均半径；ρave为地球平均密度；km为m阶负荷勒夫数。将式(1)除以水密度ρwater并顾及式(2)，得到以等效水高Δh表示的地球表面质量变化：
$\begin{gathered}\Delta h=\frac{a_{\mathrm{E}} \rho_{\text {ave }}}{3 \rho_{\text {water }}} \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^m \frac{2 m+1}{1+k_m}\left[\Delta C_{m n} \cos (n \lambda)+\right. \\\left.\Delta S_{m n} \sin (n \lambda)\right] \widetilde{P}_{m n}(\cos \theta)\end{gathered}$
2 实验分析
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2.1 不同模型的一阶项、C20和C30比较
卫星重力观测采用的是地球系统质心坐标[1]，因此由其解算的地球重力场模型的一阶项(C10、C11、S11)均为0。而地表水文与海洋观测则是相对于地球自然表面构形中心坐标框架下的结果[1]，其地球重力场模型的一阶项并不为0。此外，由于可能受潮汐混频影响，导致地球重力场模型的C20项具有很大的不确定性[1]。为此，基于CSR、GFZ和JPL各自的海洋和大气模型，结合GRACE和GRACE-FO RL06.3的二阶项以上系数(C20除外)月解，利用改进的GRACE-OBP算法[19-20]，分别计算一阶项和C20项的月解数据。对上述3家机构RL06.3模型的一阶项、C20和C30项月解数据的时间序列进行最小二乘回归分析，得到相应的趋势和周年信息，结果见表 2和3。
表 2为CSR、GFZ和JPL三家机构RL06.3模型一阶项的趋势、周年振幅与周年相位结果。通过比较可知，对于趋势项，JPL RL06.3模型C10项下降速率最大，为-50.520 2×10-12 /a，与CSR结果大致相当，但比GFZ结果大14.30%；GFZ RL06.3模型C11项下降速率最大，为-12.020 1× 10-12 /a，比JPL结果大16.76%，比CSR结果大23.69%；GFZ RL06.3模型S11项上升速率最大，为11.752 3×10-12 /a，比CSR结果大33.31%，比JPL结果大46.60%。对于周年振幅，3家机构RL06.3模型一阶项的周年振幅均基本一致。对于周年相位，CSR和JPL RL06.3模型C10项周年相位基本相同(相差约0.47°)，但与GFZ结果反相，且相差约23°。3家机构RL06.3模型C11和S11项周年振幅与周年相位都基本一致。
表 3为CSR、GFZ和JPL三家机构分别截断至60阶和96阶的RL06.3模型与SLR TN14的C20和C30项的趋势、周年振幅与周年相位结果。可以看出，同一机构RL06.3模型截断阶数由60阶增加至96阶对C20和C30项的趋势和周年信号的影响均非常有限，但不同机构模型间的差异较为明显。对于趋势项，GFZ RL06.3模型C20项下降速率最大，约为-2.90×10-11 /a，比CSR和JPL结果分别大约20.71%和30.15%；CSR RL06.3模型C30项下降速率最大，约为-10.56×10-12 /a，约是GFZ和JPL结果的3.5倍和2.5倍。对于周年振幅，GFZ与JPL RL06.3模型C20项结果大体相当，但明显大于CSR结果；CSR与GFZ RL06.3模型C30项结果基本相当，但显著小于JPL结果。对于周年相位，CSR RL06.3模型C20项周年相位最大(约24°)，约为GFZ和JPL结果的4倍和1.5倍；JPL RL06.3模型C30项周年相位最大(约17°)，约为GFZ和CSR结果的1.5倍和2倍。
与SLR TN14相比，CSR、GFZ和JPL RL06.3模型C20项下降速率分别减小53.61%、44.01%和56.97%，周年振幅分别增加29.50%、53.52%和61.84%，周年相位与之反相，分别相差74.41°、63.26°和68.31°。JPL RL06.3模型C30项下降速率减小约31.69%，周年信号与SLR TN14基本一致；GFZ RL06.3模型C30项下降速率减小约52.49%，周年振幅减小约35.59%，周年相位减小约4°；CSR RL06.3模型C30项下降速率增加约60.90%，周年振幅减小约27.41%，周年相位减小约10°。
2.2 不同模型的平均阶方差比较
研究表明[18]，地球时变重力场模型的低阶项(2≤m＜21)主要反映了地幔以下的时变重力场信号，而高阶项(m≥30)包含更多噪声信息，两者之间则为信号和噪声的混合信息。对于信号和噪声的阶方差，可通过重力场模型阶方差进行估计，一般采用大地水准面起伏$\left(\delta_m=\right. \left.a_{\mathrm{E}} \sqrt{\sum \limits_{n=0}^m \Delta C_{m n}^2+\sum\limits_{n=1}^m \Delta S_{m n}^2}\right)$表示。
图 1为2002-04—2024-08期间CSR、GFZ和JPL三家机构RL06.3模型的平均阶方差分布, 图中红色点和蓝色点分别表示截断至60阶和96阶的RL06.3模型平均阶方差，黑色线表示RL06.3模型平均阶方差信号部分的拟合曲线，红色线和蓝色线分别表示截断至60阶和96阶的RL06.3模型平均阶方差噪声部分的拟合曲线。可以看出，3家机构RL06.3模型在低阶项的平均阶方差均随阶数增加迅速减小，且信号拟合曲线高度一致，拟合参数间的差异非常小；在高阶项，随着阶数增加，3家机构RL06.3模型的平均阶方差均逐渐增大，但噪声拟合曲线的差异显著。具体而言，在截断至60阶的RL06.3模型中，GFZ噪声拟合曲线上升速度最为缓慢，JPL噪声拟合曲线上升速度最快；在截断至96阶的RL06.3模型中，CSR噪声拟合曲线上升速度最为缓慢，JPL噪声拟合曲线上升速度最快。与同机构RL06.3/60噪声拟合曲线相比，RL06.3/96噪声拟合曲线上升速度略快，表明3家机构的RL06.3/96模型在高阶项的精度略低于RL06.3/60模型。
2.3 不同模型的一阶项、C20和C30质量异常比较
基于CSR、GFZ和JPL RL06.3模型提供的一阶项、C20和C30月解数据，利用式(3)分别计算一阶项、C20和C30项地表质量异常时间序列，并利用最小二乘回归分析获得相应的趋势、周年振幅和周年相位，结果如图 2~4所示，C20和C30项结果如图 5所示。
图 2为CSR、GFZ和JPL RL06.3模型一阶项地表质量异常趋势，可以看出，3家机构RL06.3模型一阶项地表质量异常趋势的空间分布高度一致，大致从赤道向两极逐渐增加。但趋势极值存在一定差异：CSR趋势正负极值最小，为±3.04 mm/a，分别位于(76°S, 137°E)和(76°N, 43°W)；GFZ趋势正负极值最大，为±3.29 mm/a，分别位于(73°S, 136°E)和(73°N, 44°W)；JPL趋势正负极值为±3.10 mm/a，分别位于(76°S, 143°E)和(76°N, 37°W)。CSR与GFZ的趋势差异如图 2(d)所示，最大差异位于北大西洋和印度洋，幅值达0.30 mm/a；CSR与JPL的趋势差异如图 2(e)所示，最大差异位于南太平洋和亚欧非大陆，幅值达0.09 mm/a；GFZ与JPL的趋势差异如图 2(f)所示，最大差异位于北大西洋和印度洋，幅值达0.28 mm/a。
图 3为CSR、GFZ和JPL RL06.3模型一阶项地表质量异常的周年振幅，可以看出，3家机构RL06.3模型一阶项地表质量异常的周年振幅在空间分布和极值位置上高度一致。其中，GFZ周年振幅极值最大，为18.47 mm，位于(52°S, 136°E)，比CSR结果大1.94%，比JPL结果大2.27%。CSR与GFZ周年振幅差异如图 3(d)所示，最大差异出现在靠近两极区域，幅值达0.83 mm；CSR与JPL周年振幅差异如图 3(e)所示，最大差异集中在大西洋赤道附近以及印度洋与太平洋交界区域，幅值达0.13 mm；GFZ与JPL周年振幅差异如图 3(f)所示，最大差异同样出现在靠近两极区域，幅值达0.79 mm。
图 4为CSR、GFZ和JPL RL06.3模型一阶项地表质量异常的周年相位，可以看出，3家机构RL06.3模型一阶项地表质量异常的周年相位在空间分布上具有较好的一致性，即北半球以正相位为主，南半球以负相位为主，且南半球的正相位呈扁“U”形结构。但与CSR和JPL相比，GFZ周年正相位在南半球的分布范围明显偏小。3家机构的周年相位差异如图 4(d)~4(f)所示，主要差异集中在太平洋-南极-非洲南端的扁“U”形窄条带区域。
图 5为CSR、GFZ和JPL RL06.3模型与SLR TN14的C20与C30项地表质量异常的趋势、周年振幅与周年相位。可以看出，同一机构截断至60阶和96阶的RL06.3模型的C20项地表质量异常结果(图 5(a)~5(c))基本一致。C20项地表质量异常的趋势和周年振幅在赤道和两极区域取得极值，与SLR TN14相比，CSR、GFZ和JPL三家机构的趋势极值分别大36.94%、68.79%和29.17%，周年振幅极值分别大1.30%、12.19%和25.50%，周年相位分别大12°、2°和5°。此外，C20项地表质量异常的趋势和周年振幅在南北纬35.5°附近为0，而周年相位则在此处发生反相，且正反相位相差180°。
对于C30项地表质量异常(图 5(d)~5(f))，3家机构的结果基本重合。C30项地表质量异常的趋势和周年振幅在南北纬26.5°与两极处取得极值。3家机构的趋势极值比SLR TN14小34.01%，但其周年振幅和周年相位与SLR TN14基本一致。此外，C30项地表质量异常的趋势和周年振幅在南北纬51.5°与赤道处为0，而周年相位则在此处发生反相，且正反相位相差180°。
2.4 全球与典型区域地表质量变化结果比较
为获得全球地表质量变化的精确结果，对CSR、GFZ和JPL三家机构RL06.3模型数据补充一阶项球谐系数，使用SLR TN14模型中的C20和C30项替换RL06.3模型中低精度的C20和C30项；此外，采用Duan等[21]提出的去相关滤波与300 km高斯滤波的组合滤波方案，以有效抑制南北条带误差和高频噪声影响，并利用全球水文模型GLDAS通过多尺度因子法对泄漏误差进行改正，最后采用ICE6G-D模型扣除地壳均衡调整(GIA)的影响。
由于各月的反演结果较为相似，本文以2015-03全球地表质量变化为例进行分析。从图 6可以看出，未经任何滤波处理时，相较于截断至60阶的RL06.3模型反演结果(图 6(a)~6(c))，截断至96阶的RL06.3模型反演结果(图 6(d)~6(f))表现出更为严重的南北条带噪声，尤其是GFZ和JPL RL06.3模型。GFZ RL06.3/96模型的南北条带噪声主要集中在南北纬40°、赤道和南极半岛附近，而JPL RL06.3/96模型的南北条带噪声则主要分布在南北纬40°之间。经上述组合滤波处理后，全球地表质量变化信号得以清晰呈现(图 6(g)~6(i))，尽管JPL与CSR和GFZ反演结果在南美洲中西部、澳大利亚北部和非洲中西部等区域存在一定差异，但3家机构截断至60阶与96阶的RL06.3模型反演结果却表现出较高的一致性。
表 4(单位dB)为CSR、GFZ和JPL RL06.3模型经上述组合滤波方案处理后，2002-04—2024-08期间全球地表质量变化的信噪比统计结果。可以看出，3家机构RL06.3/60模型反演结果的信噪比最大值、最小值、平均值均略大于各自机构的RL06.3/96模型，而标准差则略小于各自机构的RL06.3/96模型。在3家机构RL06.3模型中，CSR反演结果的信噪比最大。结合图 6可知，尽管3家机构截断至96阶的RL06.3模型在未经滤波时表现出更为严重的南北条带噪声，但经滤波处理后，其地表质量变化的信噪比与截断至60阶时基本处于同一水平。这表明滤波后截断至96阶与60阶的RL06.3模型在真实信号的提取上具有较高的一致性，与图 6(g)~6(i)滤波后结果的一致性相吻合。
为进一步比较分析3家机构RL06.3模型的反演结果，选取时空分辨率更高的CSR RL06.3 Mascon模型(以下简称Mascon模型)作为参考(采用其他机构发布的Mascon模型，所得结果与之相似)，计算其与上述3家机构RL06.3模型反演的地表质量趋势差异与相关系数。本文以CSR RL06.3为例进行说明，结果如图 7所示，图中仅显示相关系数大于0.4的区域。可以看出，CSR不同阶次RL06.3模型与Mascon模型反演的地表质量变化除在两极附近(如格陵兰岛、南极半岛)和加拿大东部等区域存在较大差异(约为6 cm/a)和相关性较低外，其他绝大部分区域的趋势基本一致，且相关性也非常高，特别是在南美洲中北部、北美洲西南部、澳大利亚东部、非洲中部和南部、亚洲大陆南端、欧洲西北部等区域，相关系数均大于0.9。
统计结果表明，同一机构不同阶次RL06.3模型与Mascon模型反演的地表质量趋势和相关性大体相同，但不同机构RL06.3模型与Mascon模型的平均相关性略有差异：CSR平均相关性为0.73，GFZ为0.70，JPL为0.71。综上可知，3家机构RL06.3模型采用独立的解算系统和不同的背景模型，经相同的数据处理后，其反演结果趋于一致，表明3家机构RL06.3模型反演结果具有稳健性，进一步增强了结果的可靠性。
为深入分析CSR、GFZ和JPL三家机构RL06.3模型的反演结果，选取南美洲亚马孙河流域、北美洲密西西比河流域、欧洲伏尔加河流域、非洲刚果河流域、亚洲长江流域和澳大利亚默里河流域等6个典型流域的陆地水储量变化时间序列进行研究，并采用广义三角帽法[18]评估其陆地水储量变化的不确定度，结果如图 8和表 5(趋势单位cm/a，不确定度单位cm)所示。可以看出，基于3家机构不同阶次RL06.3模型反演的上述6个典型流域陆地水储量变化时间序列具有较好的一致性，但2015-02长江流域(图 8(b)中阴影部分)、2004-09和2015-02默里河流域(图 8(e)中阴影部分)，GFZ RL06.3/96模型反演结果异常偏大。此外，在密西西比河流域(表 5)，GFZ反演的陆地水储量趋势明显偏大，比CSR和JPL结果约大38%，而在其他流域，3家机构的陆地水储量变化趋势结果基本一致。
从陆地水储量变化结果的不确定度来看，在亚马孙河、密西西比河和长江流域，CSR RL06.3/60模型反演结果的不确定度最小；在伏尔加河和刚果河流域，JPL RL06.3/60模型反演结果的不确定度最小；而在默里河流域，CRS RL06.3/96模型反演结果的不确定度最小。相比之下，GFZ RL06.3模型在上述6个典型流域反演结果的不确定度均最大。尽管3家机构RL06.3模型在绝大多数流域的反演结果表现出较好的一致性，但在某些流域，各模型反演结果的不确定度仍存在显著差异。整体而言，CSR和JPL RL06.3模型在大多流域表现出更高的精度和一致性，而GFZ RL06.3模型的不确定度则普遍较大。
3 结语
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从时变重力场模型的一阶项、C20和C30及其地表质量异常、平均阶方差、全球地表质量变化的信噪比、典型流域陆地水储量变化等多个方面对CSR、GFZ和JPL最新发布的RL06.3时变重力场模型进行较为全面的比较分析，得到以下结论：
1) 3家机构RL06.3时变重力场模型一阶项的趋势差异显著，周年振幅和周年相位大体一致；模型截断阶数由60阶增加到96阶对C20和C30项的趋势和周年信息影响较小，一般不超过5%。
2) 3家机构RL06.3时变重力场模型的平均阶方差在低阶项的噪声信号拟合曲线高度一致，而在高阶项中，JPL RL06.3模型的噪声拟合曲线上升速度最快，CSR RL06.3模型的噪声拟合曲线上升速度最慢。
3) 3家机构RL06.3时变重力场模型一阶项地表质量异常的趋势和周年信息差异显著；C20项地表质量异常的趋势和周年信息差异明显，且与SLR TN14的差异较大；C30项地表质量异常的趋势基本一致，比SLR TN14小34.01%，但其周年振幅和周年相位与SLR TN14基本相当。
4) 3家机构截断至96阶的RL06.3时变重力场模型表现出更为严重的南北条带噪声，但经滤波处理后其反演的全球地表质量变化与截断至60阶的模型反演结果趋于一致。整体而言，CSR和JPL RL06.3模型在反演陆地水储量变化的精度和一致性方面表现更优，而GFZ RL06.3模型的不确定度普遍较大，这为后续研究中选择合适的时变重力场模型提供了重要参考。
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  • 国家自然科学基金(42174007)
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2026年第46卷第2期
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doi: 10.14075/j.jgg.2025.03.060
  • 接收时间：2025-03-03
  • 首发时间：2026-10-29
  • 出版时间：2026-02-15
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  • 收稿日期：2025-03-03
基金
国家自然科学基金(42204009)
国家自然科学基金(42174007)
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    1 信息工程大学地理空间信息学院, 郑州, 450001

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李姗姗, 教授, 博士生导师, 主要从事物理大地测量学研究, E-mail:
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total species (%)		 属
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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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