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Article(id=1322566886542398110, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322566845215912204, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.03.067, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1741104000000, receivedDateStr=2025-03-05, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793254117856, onlineDateStr=2026-10-29, pubDate=1771084800000, pubDateStr=2026-02-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793254117856, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793254117856, creator=13701087609, updateTime=1793254117856, 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=256, endPage=261, ext={EN=ArticleExt(id=1322566886718558879, articleId=1322566886542398110, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Fast Parallel Computation and Marine Gravity Anomalies Using the Inverse Vening-Meinesz Formula, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

To address the low computational efficiency of serial programs in integral calculations for marine gravity anomaly inversion, this study takes the inverse Vening-Meinesz formula as a case study to optimize the spherical harmonic synthesis and integration processes. Two parallel computing methods, OpenMP and MPI, are compared, with an improved task allocation strategy proposed for MPI. MPI is ultimately selected to implement the rapid computation of marine gravity anomaly inversion using the remove-restore technique. The results demonstrate that parallel computing significantly enhances the efficiency of integral inversion calculations. OpenMP exhibits higher parallel efficiency, while MPI demonstrates stronger acceleration capabilities on cluster servers. By employing the MPI-based parallel method with optimized task allocation, the inversion process requires only 20.87 seconds under the following conditions: a cluster server with 80 processes (CPU model: Intel Xeon E5-2670 v2 @ 2.50 GHz), a computational region of 10°×10°, an inversion resolution of 1'×1', and an integration radius of 60'. The computational accuracy of this approach slightly surpasses that of the FFT method.

, authors=Qi FENG1, Taoyong JIN1, *, Jiancheng LI1, Heyang SUN1, authorsList=Qi FENG, Taoyong JIN, Jiancheng LI, Heyang SUN, authorCompany=null, correspAuthors=Taoyong JIN, 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=1322566887624528549, articleId=1322566886542398110, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=基于逆Vening-Meinesz积分公式的海洋重力异常快速并行计算分析, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

针对海洋重力异常积分计算中串行程序计算效率较低的问题, 以逆Vening-Meinesz公式为例, 优化球谐综合和积分计算过程, 比较OpenMP和MPI两种并行方法, 并改进MPI的任务分配模式, 选择MPI实现移去恢复法反演海洋重力异常的快速计算。结果表明, 并行计算可有效提升积分反演计算效率, OpenMP并行效率较高, MPI在集群服务器上加速能力较强。利用优化任务分配的MPI并行方法, 在CPU型号为Intel Xeon E5-2670 v2 @ 2.50 GHz的集群服务器80进程、10°×10°计算区域、1'×1'反演分辨率、60'积分半径条件下, 反演过程仅需20.87 s, 且计算精度略优于FFT方法。

, authors=冯崎1, 金涛勇1, *, 李建成1, 孙鹤洋1, authorsList=冯崎, 金涛勇, 李建成, 孙鹤洋, authorCompany=null, correspAuthors=金涛勇, authorNote=

冯崎, 硕士生, 主要研究方向为卫星大地测量, E-mail:

, correspAuthorsNote=
金涛勇, 博士, 教授, 主要研究方向为卫星大地测量, E-mail:
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冯崎, 硕士生, 主要研究方向为卫星大地测量, E-mail:

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Journal of Geodesy and Geodynamics, 2009, 29 (6): 126- 129, articleTitle=null, refAbstract=null)], funds=[Fund(id=1322566892980654810, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, awardId=42192533, language=CN, fundingSource=国家自然科学基金(42192533), fundOrder=null, country=null), Fund(id=1322566894704513756, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, awardId=42388102, language=CN, fundingSource=国家自然科学基金(42388102), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1322566887842632358, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, xref=null, ext=[AuthorCompanyExt(id=1322566887851020967, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, companyId=1322566887842632358, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1 School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China), AuthorCompanyExt(id=1322566887859409576, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, companyId=1322566887842632358, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1 武汉大学测绘学院, 武汉, 430079)])], figs=[ArticleFig(id=1322566891193881287, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=EN, label=Fig. 1, caption=Variation of OpenMP speedup, computing time, and parallel efficiency, figureFileSmall=9xrBYRnWAzujoz+QyCdFdQ==, figureFileBig=l2dVItqfF3lIoXSIkN32Mw==, tableContent=null), ArticleFig(id=1322566891273573064, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=CN, label=图1, caption=OpenMP加速比、计算时间、并行效率变化, figureFileSmall=9xrBYRnWAzujoz+QyCdFdQ==, figureFileBig=l2dVItqfF3lIoXSIkN32Mw==, tableContent=null), ArticleFig(id=1322566891638477513, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=EN, label=Fig. 2, caption=Variation of MPI speedup, computing time, and parallel efficiency, figureFileSmall=vkrelPCn1EGzWL6/UnkaKw==, figureFileBig=pO97rwdeW4K6+R9Oq6vVRg==, tableContent=null), ArticleFig(id=1322566891730752202, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=CN, label=图2, caption=MPI加速比、计算时间、并行效率变化, figureFileSmall=vkrelPCn1EGzWL6/UnkaKw==, figureFileBig=pO97rwdeW4K6+R9Oq6vVRg==, tableContent=null), ArticleFig(id=1322566891814638283, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=EN, label=Tab. 1, caption=

Comparison of cross-order recurrence optimization for associated Legendre functions in 180′×180′ computational domain

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计算阶次优化前时间/s优化后时间/s优化前/优化后
3601.300.168.13
7205.150.648.05
1 08011.221.398.07
1 80031.123.798.21
2 16044.875.418.29
), ArticleFig(id=1322566891890135756, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=CN, label=表1, caption=

180′×180′计算区域缔合勒让德函数跨阶次递推优化对比

, figureFileSmall=null, figureFileBig=null, tableContent=
计算阶次优化前时间/s优化后时间/s优化前/优化后
3601.300.168.13
7205.150.648.05
1 08011.221.398.07
1 80031.123.798.21
2 16044.875.418.29
), ArticleFig(id=1322566891969827533, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=EN, label=Tab. 2, caption=

Performance comparison of integral optimization for the inverse Vening-Meinesz formula in 180′×180′ computational domain

, figureFileSmall=null, figureFileBig=null, tableContent=
积分半径/(′)优化前时间/s优化后时间/s优化前/优化后
106.513.581.82
30165.4191.431.81
60567.91296.891.91
901 688.64792.732.13
), ArticleFig(id=1322566892045325006, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=CN, label=表2, caption=

180′×180′计算区域逆Vening-Meinesz公式积分优化对比

, figureFileSmall=null, figureFileBig=null, tableContent=
积分半径/(′)优化前时间/s优化后时间/s优化前/优化后
106.513.581.82
30165.4191.431.81
60567.91296.891.91
901 688.64792.732.13
), ArticleFig(id=1322566892145988303, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=EN, label=Tab. 3, caption=

OpenMP parallel computing time

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线程数60′×60′120′×120′180′×180′
时间/s加速比/效率时间/s加速比/效率时间/s加速比/效率
1672.831.00/1.002 803.931.00/1.006 326.411.00/1.00
2344.191.95/0.981 432.261.96/0.983 196.961.96/0.98
4193.833.47/0.87710.263.95/0.981 599.483.96/0.99
898.706.82/0.85372.867.52/0.94796.817.93/0.99
1265.8010.23/0.85227.7811.41/0.95533.4511.86/0.99
1649.1913.67/0.85183.3215.30/0.96405.8215.58/0.98
2039.2117.15/0.86140.4219.21/0.96318.1619.88/0.99
), ArticleFig(id=1322566892217291472, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=CN, label=表3, caption=

OpenMP并行计算时间

, figureFileSmall=null, figureFileBig=null, tableContent=
线程数60′×60′120′×120′180′×180′
时间/s加速比/效率时间/s加速比/效率时间/s加速比/效率
1672.831.00/1.002 803.931.00/1.006 326.411.00/1.00
2344.191.95/0.981 432.261.96/0.983 196.961.96/0.98
4193.833.47/0.87710.263.95/0.981 599.483.96/0.99
898.706.82/0.85372.867.52/0.94796.817.93/0.99
1265.8010.23/0.85227.7811.41/0.95533.4511.86/0.99
1649.1913.67/0.85183.3215.30/0.96405.8215.58/0.98
2039.2117.15/0.86140.4219.21/0.96318.1619.88/0.99
), ArticleFig(id=1322566892301177553, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=EN, label=Tab. 4, caption=

MPI parallel computing time

, figureFileSmall=null, figureFileBig=null, tableContent=
线程数60′×60′120′×120′180′×180′
时间/s加速比/效率时间/s加速比/效率时间/s加速比/效率
1672.831.00/1.002 803.931.00/1.006 326.411.00/1.00
1078.018.62/0.86302.889.26/0.93681.059.29/0.93
2039.4617.05/0.85153.4718.27/0.92335.0618.88/0.94
3027.2824.66/0.82102.0927.47/0.92216.7829.18/0.97
4026.7125.19/0.6379.4635.28/0.88190.6032.17/0.81
6013.4350.10/0.8450.2155.85/0.93112.9456.01/0.93
8013.4350.10/0.8450.1355.94/0.70110.4357.52/0.72
), ArticleFig(id=1322566892376675026, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566886542398110, language=CN, label=表4, caption=

MPI并行计算时间

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线程数60′×60′120′×120′180′×180′
时间/s加速比/效率时间/s加速比/效率时间/s加速比/效率
1672.831.00/1.002 803.931.00/1.006 326.411.00/1.00
1078.018.62/0.86302.889.26/0.93681.059.29/0.93
2039.4617.05/0.85153.4718.27/0.92335.0618.88/0.94
3027.2824.66/0.82102.0927.47/0.92216.7829.18/0.97
4026.7125.19/0.6379.4635.28/0.88190.6032.17/0.81
6013.4350.10/0.8450.2155.85/0.93112.9456.01/0.93
8013.4350.10/0.8450.1355.94/0.70110.4357.52/0.72
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Comparison of MPI task distribution optimization results

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计算区域进程数优化前优化后效率提升
时间/s加速比/效率时间/s加速比/效率
60′×60′4026.7125.19/0.6321.6931.02/0.780.15
5025.2026.70/0.5317.6338.16/0.760.23
120′×120′5074.7937.49/0.7567.2341.71/0.830.08
8050.1355.94/0.7044.3263.27/0.790.09
180′×180′40190.6032.17/0.81187.8733.67/0.840.03
80110.4357.52/0.72101.6262.26/0.780.06
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MPI任务分配优化对比

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时间/s加速比/效率时间/s加速比/效率
60′×60′4026.7125.19/0.6321.6931.02/0.780.15
5025.2026.70/0.5317.6338.16/0.760.23
120′×120′5074.7937.49/0.7567.2341.71/0.830.08
8050.1355.94/0.7044.3263.27/0.790.09
180′×180′40190.6032.17/0.81187.8733.67/0.840.03
80110.4357.52/0.72101.6262.26/0.780.06
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Statistical differences between parallel integration method recovery results and SIO model gravity anomalies

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积分半径/(′)移去阶次最大值/mGal最小值/mGal平均值/mGal标准差/mGal计算时间/s
1036030.86-21.690.284.251.88
1 0809.71-9.180.220.91
2 1908.04-9.060.220.81
303609.30-8.830.161.046.39
1 0807.77-7.610.180.56
2 1907.97-7.390.180.54
603607.66-7.890.130.5620.87
1 0808.04-7.190.130.48
2 1907.97-7.200.130.48
903608.08-7.410.080.5244.02
1 0807.90-7.380.090.47
2 1907.86-7.300.090.46
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并行积分法反演结果与SIO模型重力异常差值的统计

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1036030.86-21.690.284.251.88
1 0809.71-9.180.220.91
2 1908.04-9.060.220.81
303609.30-8.830.161.046.39
1 0807.77-7.610.180.56
2 1907.97-7.390.180.54
603607.66-7.890.130.5620.87
1 0808.04-7.190.130.48
2 1907.97-7.200.130.48
903608.08-7.410.080.5244.02
1 0807.90-7.380.090.47
2 1907.86-7.300.090.46
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Statistical differences between FFT method recovery results and SIO model gravity anomalies

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移去阶次最大值/mGal最小值/mGal平均值/mGal标准差/mGal
3608.90-8.100.160.98
1 0807.45-7.390.170.56
2 1907.45-7.220.170.54
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FFT方法反演结果与SIO模型重力异常差值的统计结果

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移去阶次最大值/mGal最小值/mGal平均值/mGal标准差/mGal
3608.90-8.100.160.98
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2 1907.45-7.220.170.54
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基于逆Vening-Meinesz积分公式的海洋重力异常快速并行计算分析
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冯崎 1 , 金涛勇 1, * , 李建成 1 , 孙鹤洋 1
大地测量与地球动力学 | 大地测量学 2026,46(2): 256-261
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大地测量与地球动力学 |大地测量学 2026 , 46 (2) : 256 -261
基于逆Vening-Meinesz积分公式的海洋重力异常快速并行计算分析
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冯崎1 , 金涛勇1, * , 李建成1, 孙鹤洋1
作者信息
  • 1 武汉大学测绘学院, 武汉, 430079
通讯作者:
金涛勇, 博士, 教授, 主要研究方向为卫星大地测量, E-mail:
作者简介:

冯崎, 硕士生, 主要研究方向为卫星大地测量, E-mail:

Fast Parallel Computation and Marine Gravity Anomalies Using the Inverse Vening-Meinesz Formula
Qi FENG1 , Taoyong JIN1, * , Jiancheng LI1, Heyang SUN1
Affiliations
  • 1 School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China
出版时间: 2026-02-15 doi: 10.14075/j.jgg.2025.03.067
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摘要
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针对海洋重力异常积分计算中串行程序计算效率较低的问题, 以逆Vening-Meinesz公式为例, 优化球谐综合和积分计算过程, 比较OpenMP和MPI两种并行方法, 并改进MPI的任务分配模式, 选择MPI实现移去恢复法反演海洋重力异常的快速计算。结果表明, 并行计算可有效提升积分反演计算效率, OpenMP并行效率较高, MPI在集群服务器上加速能力较强。利用优化任务分配的MPI并行方法, 在CPU型号为Intel Xeon E5-2670 v2 @ 2.50 GHz的集群服务器80进程、10°×10°计算区域、1'×1'反演分辨率、60'积分半径条件下, 反演过程仅需20.87 s, 且计算精度略优于FFT方法。

海洋重力异常反演  /  垂线偏差  /  移去恢复  /  OpenMP  /  MPI

To address the low computational efficiency of serial programs in integral calculations for marine gravity anomaly inversion, this study takes the inverse Vening-Meinesz formula as a case study to optimize the spherical harmonic synthesis and integration processes. Two parallel computing methods, OpenMP and MPI, are compared, with an improved task allocation strategy proposed for MPI. MPI is ultimately selected to implement the rapid computation of marine gravity anomaly inversion using the remove-restore technique. The results demonstrate that parallel computing significantly enhances the efficiency of integral inversion calculations. OpenMP exhibits higher parallel efficiency, while MPI demonstrates stronger acceleration capabilities on cluster servers. By employing the MPI-based parallel method with optimized task allocation, the inversion process requires only 20.87 seconds under the following conditions: a cluster server with 80 processes (CPU model: Intel Xeon E5-2670 v2 @ 2.50 GHz), a computational region of 10°×10°, an inversion resolution of 1'×1', and an integration radius of 60'. The computational accuracy of this approach slightly surpasses that of the FFT method.

marine gravity anomalies recovery  /  vertical deflection  /  remove-restore  /  OpenMP  /  MPI
冯崎, 金涛勇, 李建成, 孙鹤洋. 基于逆Vening-Meinesz积分公式的海洋重力异常快速并行计算分析. 大地测量与地球动力学, 2026 , 46 (2) : 256 -261 . DOI: 10.14075/j.jgg.2025.03.067
Qi FENG, Taoyong JIN, Jiancheng LI, Heyang SUN. Fast Parallel Computation and Marine Gravity Anomalies Using the Inverse Vening-Meinesz Formula[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (2) : 256 -261 . DOI: 10.14075/j.jgg.2025.03.067
正文
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海洋重力场是地球重力场的重要组成部分，在高程基准确定、海底地形反演和水下匹配导航中有重要作用[1-3]。当前，海洋重力场主要利用多源卫星测高数据进行反演[4-5]，常用的方法包括逆Vening-Meinesz公式、逆Stokes法和基于Laplace方程的解析方法，其中逆Vening-Meinesz和逆Stokes法都涉及积分过程，难以高效计算，一般采用其卷积式由FFT算法进行快速计算[6]。
随着卫星测高数据的长期积累，特别是多种新型雷达高度计的出现，包括合成孔径雷达高度计、合成孔径雷达干涉高度计等[7-9]，海面观测数据呈海量增加，且SWOT卫星的成功发射首次获取了同步的二维海面高观测数据[10]，有助于提升测高数据解算海洋重力场的空间分辨率和精度[11-13]。尽管高分辨率格网可以提供积分半径内更丰富的有效信息，但受限于计算机运行速度，积分格网点的增加将显著加重串行程序的计算负担。当今计算机算力发展迅速，在此情况下，利用积分过程结合海量观测数据和并行计算使更加严密地确定海洋重力场成为可能，具有实用价值。并行计算通过将任务分配到多个计算核心以减少计算耗时，目前基于OpenMP和MPI的CPU并行方法已被广泛应用于重力场参量求解[14-15]、卫星重力数据处理[16-17]、重力场模型构建[18-19]等计算领域。利用逆Vening-Meinesz公式和逆Stokes法由积分过程反演海洋重力场，需要采用移去恢复方法引入先验地球重力场模型，反演残余重力异常后进行恢复。全过程主要有两个步骤涉及大量计算，包括利用超高阶地球重力场模型由球谐综合计算参考垂线偏差、重力异常和大地水准面及利用残余垂线偏差或残余大地水准面积分计算残余重力异常。
由于球谐综合中格网点不存在相关性，适合分块并行计算，且海洋重力异常反演的格网计算也具有天然的并行性。本文针对逆Vening-Meinesz公式由垂线偏差积分反演海洋重力异常，以球谐综合计算重力异常为例比较OpenMP和MPI两种并行方法，由于MPI在集群服务器的提速能力更强，最后基于MPI实现了包括球谐综合和逆Vening-Meinesz积分反演在内的海洋重力异常快速并行计算，并进一步分析其计算效率、参考模型阶次和积分半径对反演结果的影响。
1 海洋重力异常反演方法
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逆Vening-Meinesz公式以垂线偏差为输入，是目前反演海洋重力场广泛采用的方法。由于本文旨在实现海洋重力异常反演积分过程的快速计算，故仅采用SIO(Scripps Institution of Oceanography)团队发布的V31版本1′×1′格网垂线偏差模型模拟卫星测高格网垂线偏差，不考虑实测数据计算垂线偏差及其格网化过程，并选择EGM2008地球重力场位系数模型作为移去和恢复的参考模型。基于逆Vening-Meinesz公式使用移去恢复法反演海洋重力异常的具体步骤如下：
1) 根据球谐综合计算出截断阶次下EGM2008模型参考重力异常Δgmod和参考垂线偏差分量ξmod、ηmod。
2) 将SIO模型垂线偏差分量ξSIO、ηSIO与ξmod、ηmod作差，得到残余垂线偏差分量ξres、ηres，作为反演输入值。
3) 利用逆Vening-Meinesz公式，选择积分半径反演得到残余重力异常Δgres，并计算内圈带效应值Δgi。
4) 将参考重力异常Δgmod、残余重力异常Δgres和内圈带效应值Δgi相加，得到海洋重力异常反演结果Δg。
1.1 球谐综合计算参考重力异常和垂线偏差
在球近似条件下，重力异常Δg、垂线偏差子午圈分量ξ和卯酉圈分量η的球谐展开式为：
$\left\{\begin{array}{l}\Delta g=\frac{G M}{r^2} \sum _{n=2}^N(n-1)\left(\frac{a}{r}\right)^n \sum _{m=0}^n \\\left(\bar{C}_{n m}^* \cos m \lambda+\bar{S}_{n m} \sin m \lambda\right) \bar{P}_{n m}(\cos \theta) \\\xi=\frac{G M}{r^2 \gamma} \sum _{n=2}^N\left(\frac{a}{r}\right)^n \sum _{m=0}^n\left(\bar{C}_{n m}^* \cos m \lambda+\right. \\\left.\bar{S}_{n m} \sin m \lambda\right) \frac{\mathrm{d} \bar{P}_{n m}(\cos \theta)}{\mathrm{d} \theta} \\\eta=\frac{G M}{r^2 \gamma \sin \theta} \sum _{n=2}^N\left(\frac{a}{r}\right)^n \sum _{m=0}^n m \\\left(\bar{C}_{n m}^* \sin m \lambda-\bar{S}_{n m} \cos m \lambda\right) \bar{P}_{n m}(\cos \theta)\end{array}\right.$
式中，GM为地心引力常数；a为椭球长半球；r为椭球表面点至地心的距离；γ为椭球表面点的正常重力；${\bar C}$nm*和${\bar S}$nm为扰动位的球谐系数；λ为地心经度；θ为地心余纬。
采用跨阶次递推法计算缔合勒让德函数，递推公式为：
$\begin{gathered}\bar{P}_{n m}(\cos \theta)=\alpha_{n m} \bar{P}_{n-2, m}(\cos \theta)+ \\\beta_{n m} \bar{P}_{n-2, m-2}(\cos \theta)-\gamma_{n m} \bar{P}_{n, m-2}(\cos \theta)\end{gathered}$
式中，αnm、βnm、γnm为递推系数，具体公式见参考文献[14]。
1.2 逆Vening-Meinesz公式
使用Molodenskii推导的逆Vening-Meinesz公式[20]：
$\left\{\begin{array}{l}\Delta g(p)=\frac{\gamma_0}{4 \pi} \iint M(\psi)\left(\xi \cos \alpha_{q p}+\eta \sin \alpha_{q p}\right) \mathrm{d} \sigma \\M(\psi)=3 \csc \psi_{p q}-\csc \psi_{p q} \csc \frac{\psi_{p q}}{2}-\tan \frac{\psi_{p q}}{2}\end{array}\right.$
式中，ξ和η为垂线偏差子午圈和卯酉圈分量；γ0为平均正常重力；M(ψ)为核函数；ψpq和αqp分别为计算点p与流动点q间的球面角距和方位角，可根据球面三角函数公式求出：
$\left\{\begin{array}{l}\cos \psi_{p q}=\sin \varphi_p \sin \varphi_q+\cos \varphi_p \cos \varphi_q \cos \left(\lambda_q-\lambda_p\right) \\\tan \alpha_{q p}=\frac{-\cos \varphi_p \sin \left(\lambda_q-\lambda_p\right)}{\cos \varphi_q \sin \varphi_p-\sin \varphi_q \cos \varphi_p \cos \left(\lambda_q-\lambda_p\right)}\end{array}\right.$
式中，φ为地心纬度。当ψ=0时，核函数M(ψ)发生奇异，无法使用式(3)进行计算，本文使用Hwang[21]推导的内圈带效应模型进行计算。
2 串行程序计算优化
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为提升整体效率，首先优化串行程序计算过程，所有计算均在集群服务器上完成，CPU型号为Intel Xeon E5-2670 v2 @ 2.50 GHz，每节点配备20个CPU核心。
2.1 缔合勒让德函数递推优化
由式(2)可知，勒让德函数${\bar P}$nm(cosθ)的取值与地心余纬θ有关，所以采用逐纬度计算的方法，将纬圈循环放在外侧，同一纬度仅递推一次$\bar{P}_{n m}(\cos \theta)$，以避免重复计算。其次，递推系数αnm、βnm、γnm、anm和bnm仅与计算阶次有关，故预先计算并存入数组，后续递推时可直接调用。
为验证递推系数预存策略的效果，以180′×180′区域为例，计算并统计阶次为360、720、1 080、1 800、2 160时跨阶次递推法的计算时间，如表 1所示。不同阶次下优化前后的时间比值均大于8，显著提升了勒让德函数的计算效率，该方法同样适用其一阶导数的计算。重力场元计算中sinmλ和cosmλ仅与经度与截断阶次有关，也可预存至数组以加快计算。
2.2 逆Vening-Meinesz公式积分优化
由式(3)可知，求解计算点的残余重力异常通过累加求和方式进行，每个计算点处都需要计算整个区域内的ψpq和积分半径内的αqp。由于积分区域在均匀格网平面上可视为圆域，因此将ψpq的计算范围限定在圆域所内切的正方形区域内，以减少时间消耗。
式(4)表明，ψpq与αqp仅与φp、φq及经差λq－λp有关。在同一纬度带上，计算点与流动点的相对位置保持不变，故可预先计算一次ψpq与αqp并存入数组，后续直接调用。以180′×180′计算区域为例，统计积分半径为10′、30′、60′、90′时逆Vening-Meinesz公式积分计算的时间，如表 2所示。结果显示，计算时间随积分半径的增大而显著增加，优化前后的时间比值在2左右，表明优化策略有效提升了积分计算的效率。
3 并行方案设计与实现
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3.1 并行方案设计
并行方案的设计关键在于降低并行开销和提升程序的并行度。本文采用数据并行的任务划分策略，结合串行程序优化中逐纬度计算的思路，在并行实现中亦将格网数据按纬度条带分解计算任务，以行为基本单元分配给各线程或进程，实现并行计算。
在OpenMP并行中，并行域内所有线程均可访问共享内存区域，不同线程对同一变量的写入值不同会引发数据竞争，因此需合理划分变量的私有性和共享性。如${\bar P}$(cosθ)、cosmλ、sinmλ等中间变量在线程间取值不同，应在并行域内使用private()子句定义为私有变量；Δgmod、ξmod、ηmod和Δg可在并行域外以数组形式定义为共享变量，由各线程分块写入，完成数据传输。
MPI基于分布式储存系统，进程间数据互不共享，无需考虑内存冲突的问题，但需使用通信函数传递数据。因此在球谐综合计算结束后，使用MPI_ALLgatherv()函数共享各进程Δgmod、ξmod和ηmod，最后使用MPI_Gatherv()函数汇总Δg至根进程。通过以下步骤实现MPI任务划分：
1) 创建维数与总进程数相同的一维数组，数组每个元素即为每个进程分配纬度带的个数，并可通过进程编号调用；
2) 用纬度带条数除以总进程数，将可整除部分均分给各个进程，余数部分自0号进程按进程编号再进行均分。
3.2 OpenMP和MPI并行效率对比
基于上述优化和并行方案设计，取2 190阶次的EGM2008模型，以球谐综合计算参考重力异常为例，分别在60′×60′、120′×120′、180′×180′三种分辨率的实验区域上对比OpenMP和MPI的计算效率。各实验条件下不同线程(进程)数对应的计算时间如表 3和4所示，结果对比如图 1和2所示。
由表 3可知，OpenMP在180′×180′实验区域、20线程的条件下，加速比最大可达19.88，显著提升了计算效率。在相同线程数条件下，计算区域越大，OpenMP并行效率越高，60′×60′计算区域下并行效率为0.85~0.98，180′×180′计算区域则提升至0.98~0.99。随着计算范围的扩大，计算时间相应延长，使得OpenMP在线程创建和调度上的开销占比逐渐减小。在图 1(a)中表现为加速比趋于线性递增，表明并行任务负载越重，其并行效率越高。
由表 4和图 2可知，MPI的并行效率同样随实验区域的增大而提高，但整体效率相较于OpenMP略低，在60′×60′区域下并行效率仅为0.63~0.86，180′×180′区域下才提升至0.72~0.97。这主要由于MPI需通过通信函数收集各进程计算结果，通信开销拉低了整体并行效率。相比之下，OpenMP基于共享内存系统，线程间可通过读写共享变量实现数据传递，通信开销较小。然而，由于MPI支持跨节点并行，其加速能力远超OpenMP，加速比最大可达57.52。
此外，由图 2可知，当计算区域纬度带条数无法被进程数整除时，MPI的并行效率显著下降。以图 2中60′×60′区域和40进程的条件为例，其中20个进程负责计算1条纬度带，另外20个进程负责计算2条纬度带，导致任务负载差异达到2倍，显著影响了并行效率。
3.3 MPI并行任务分配优化
为保证进程间任务负载均衡，本文将任务分配模式调整为以格网点为单位，并将二维循环转换为一维循环，以尽量减少进程间任务分配的差异。只有当格网点为进程起始点或位于每条纬度带的起始位置时，才计算勒让德函数。选取计算区域中纬度带条数无法整除进程数的情况进行实验，其他参数与上节相同，改进前后方法对比如表 5所示。结果表明，改进后的任务分配模式能显著提高纬度带条数无法整除时的并行效率；且在相同范围格网下，进程数越多，效率提升越大。
4 海洋重力异常反演
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由上述结果可知，计算区域较大时，MPI能充分发挥服务器集群的优势，获得更好的加速效果。本文使用MPI实现了移去恢复法反演海洋重力异常的整体并行计算，实验选取我国南海(10°~20°N，110°~120°E)作为计算区域，使用1′×1′的SIO模型垂线偏差作为逆Vening-Meinesz公式的输入，EGM2008模型作为移去恢复参考模型。为评估反演精度，将海洋重力异常反演结果与SIO模型的重力异常进行比较。统计在80个进程下，不同积分半径和移去阶次下反演结果的精度和时间，结果如表 6所示；同时，给出基于公式卷积形式并采用FFT加速计算的反演结果精度，如表 7所示。
由表 6可知，当积分半径大于等于60′时，海洋重力异常反演结果与SIO模型的差值标准差最大为0.56 mGal，平均值最大为0.13 mGal；60′积分半径下反演时间仅为20.87 s，验证了本文重力异常反演的并行计算在精度和效率上的可靠性。在相同积分半径条件下，所用参考模型的阶次越高，反演精度越高。在10′积分半径下，移去360阶和2 190阶参考模型的标准差差异为3.44 mGal；而在30′、60′和90′积分半径下，标准差差异依次降低至0.50 mGal、0.08 mGal和0.06 mGal，表明随着积分半径的增加，模型阶次对结果精度的影响逐渐减小。
此外，由表 7可知，当积分方法采用30′积分半径时，其反演精度已接近FFT方法，采用60′积分半径时则优于FFT结果。随着参考模型阶次的升高，球谐综合中勒让德函数的计算量显著增加，导致高阶参考重力场的计算耗时显著增加。同时，勒让德函数的计算误差会随着递推累积，影响高阶重力场元的计算精度。综合考虑计算效率与精度，在实际反演操作时可仅移去参考模型的中低阶项，选择60′积分半径，该策略与已有研究结论相符[22]。此外，实际应用中亦可单独实现参考重力场的球谐综合过程，仅考虑不同区域的积分计算并行，从而在满足精度要求的前提下提高海洋重力异常的反演效率。
5 结语
收起
本文针对海洋重力异常积分计算中串行程序效率较低的问题，以逆Vening-Meinesz积分公式为例，实现了基于MPI的移去恢复法反演重力异常的快速计算。通过逐纬度计算和预存系数的方法，优化了勒让德函数递推与逆Vening-Meinesz公式的串行计算，并设计了OpenMP和MPI的并行方案，以球谐综合为例比较了二者的效率和加速能力，优化了MPI任务分配方案。最后选择加速能力更强的MPI实现了移去恢复法积分计算重力异常的并行，并研究了移去参考模型阶次和积分半径对于精度的影响。
结果表明，本文使用的串行程序优化方法可以提升勒让德函数递推与逆Vening-Meinesz公式的计算效率。OpenMP和MPI均能有效实现程序并行，其中OpenMP并行效率高于MPI，但加速比受限于节点核数；由于进程间通信开销较大，MPI并行效率低于OpenMP，但能充分利用集群服务器计算资源，具备更强的加速能力。在CPU型号为Intel Xeon E5-2670 v2 @ 2.50 GHz的集群服务器80进程、10°×10°计算区域、1′×1′反演分辨率、60′积分半径条件下, 反演过程仅需20.87 s。重力异常反演结果表明，在90′积分半径内，参考模型阶次越高，积分半径越大，反演精度越高；但积分半径大于60′后，模型阶次对精度的影响相对较小。为兼顾反演精度和计算效率，可仅移去参考模型的中低阶项，选择60′的积分半径。
基金
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  • 国家自然科学基金(42192533)
  • 国家自然科学基金(42388102)
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2026年第46卷第2期
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doi: 10.14075/j.jgg.2025.03.067
  • 接收时间：2025-03-05
  • 首发时间：2026-10-29
  • 出版时间：2026-02-15
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  • 收稿日期：2025-03-05
基金
国家自然科学基金(42192533)
国家自然科学基金(42388102)
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    1 武汉大学测绘学院, 武汉, 430079

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金涛勇, 博士, 教授, 主要研究方向为卫星大地测量, E-mail:
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