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Article(id=1322567054117417633, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322567020940481144, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.04.136, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1745337600000, receivedDateStr=2025-04-23, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793254157809, onlineDateStr=2026-10-29, pubDate=1776182400000, pubDateStr=2026-04-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793254157809, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793254157809, creator=13701087609, updateTime=1793254157809, updator=13701087609, issue=Issue{id=1322567020940481144, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='4', pageStart='392', pageEnd='521', issueExtLink='null', onlineDate='null', pubDate='1776182400000', pubDateStr='2026-04-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793254149899, creator='13701087609', updateTime=1793254149899, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=481, endPage=486, ext={EN=ArticleExt(id=1322567054297772706, articleId=1322567054117417633, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Geoid Innermost Area Effects Calculation Using Stokes Integral, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

To analyze the differences of different singular integral methods in geoid calculation using Stokes integral, based on the traditional methods using circular approximation and biquadratic polynomial interpolation, we derive a de-singularized method for Stokes integral within the integration radius. The results show that when the innermost area is selected as the single grid where the calculation point is located, the accuracy of traditional method and biquadratic polynomial interpolation method are similar, and the de-singularized method has been improved using the point kernel numerical integration. The mean kernel numerical integration can improve the calculation accuracy of traditional method and biquadratic polynomial interpolation method, while the de-singularized method is not significant.

, authors=Xiaoyan GENG1, Hui ZHAO1, *, Xinwei GUO1, Jingjuan MENG1, Wenchao WANG1, authorsList=Xiaoyan GENG, Hui ZHAO, Xinwei GUO, Jingjuan MENG, Wenchao WANG, authorCompany=null, correspAuthors=Hui ZHAO, 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=1322567057175065263, articleId=1322567054117417633, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=Stokes积分计算大地水准面中央区效应分析, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

为分析Stokes积分计算大地水准面中不同奇异积分处理方法的差异, 在研究基于圆形近似的传统方法与双二次多项式插值方法的基础上, 推导积分半径内Stokes积分的去奇异方法。选择实验区进行计算分析, 结果表明, 中央区取计算点所在的单个格网时, 使用点核数值积分, 传统方法与双二次多项式插值法结果相近, 去奇异法精度有所改进; 平均核数值积分能够提升传统方法与双二次多项式插值法计算精度, 而去奇异法变化不大。

, authors=耿晓燕1, 赵辉1, *, 郭鑫伟1, 孟静娟1, 王文超1, authorsList=耿晓燕, 赵辉, 郭鑫伟, 孟静娟, 王文超, authorCompany=null, correspAuthors=赵辉, authorNote=

耿晓燕, 高级工程师, 主要从事大地测量数据处理研究, E-mail:

, correspAuthorsNote=
赵辉, 高级工程师, 主要从事大地测量数据处理研究, E-mail:
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耿晓燕, 高级工程师, 主要从事大地测量数据处理研究, E-mail:

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Difference of point kernel numerical integration for three schemes with 1′×1′ grid

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数 方案 平均值/cm 标准差/cm 最小值/cm 最大值/cm
120 方案1 0.23 0.68 －1.52 2.20
方案2 0.24 0.72 －1.61 2.31
方案3 0.04 0.12 －0.48 0.30
2 160 方案1 －0.02 1.48 －4.63 3.45
方案2 －0.02 1.59 －5.02 3.54
方案3 0.07 1.14 －3.28 5.39
), ArticleFig(id=1322567063076451042, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567054117417633, language=CN, label=表1, caption=

1′×1′格网3种方案点核数值积分差值

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数 方案 平均值/cm 标准差/cm 最小值/cm 最大值/cm
120 方案1 0.23 0.68 －1.52 2.20
方案2 0.24 0.72 －1.61 2.31
方案3 0.04 0.12 －0.48 0.30
2 160 方案1 －0.02 1.48 －4.63 3.45
方案2 －0.02 1.59 －5.02 3.54
方案3 0.07 1.14 －3.28 5.39
), ArticleFig(id=1322567063172920035, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567054117417633, language=EN, label=Tab. 2, caption=

Difference of point kernel numerical integration for three schemes with 30″×30″ grid

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数 方案 平均值/cm 标准差/cm 最小值/cm 最大值/cm
120 方案1 0.11 0.28 －0.62 0.86
方案2 0.12 0.30 －0.65 0.91
方案3 0.02 0.09 －0.35 0.24
2 160 方案1 0.02 1.01 －3.69 2.92
方案2 0.02 1.05 －3.84 2.97
方案3 0 0.87 －3.32 2.64
), ArticleFig(id=1322567063273583332, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567054117417633, language=CN, label=表2, caption=

30″×30″格网3种方案点核数值积分差值

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数 方案 平均值/cm 标准差/cm 最小值/cm 最大值/cm
120 方案1 0.11 0.28 －0.62 0.86
方案2 0.12 0.30 －0.65 0.91
方案3 0.02 0.09 －0.35 0.24
2 160 方案1 0.02 1.01 －3.69 2.92
方案2 0.02 1.05 －3.84 2.97
方案3 0 0.87 －3.32 2.64
), ArticleFig(id=1322567063542018789, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567054117417633, language=EN, label=Tab. 3, caption=

Difference of mean kernel numerical integration for three schemes with 1′×1′ grid

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数 方案 平均值/cm 标准差/cm 最小值/cm 最大值/cm
120 方案1 0.11 0.35 －0.82 1.23
方案2 0.12 0.39 －0.91 1.35
方案3 0.04 0.12 －0.48 0.30
2 160 方案1 －0.06 1.01 －3.22 3.51
方案2 －0.06 1.04 －3.46 3.42
方案3 0.06 1.11 －3.03 5.21
), ArticleFig(id=1322567063642682086, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567054117417633, language=CN, label=表3, caption=

1′×1′格网3种方案平均核数值积分差值

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数 方案 平均值/cm 标准差/cm 最小值/cm 最大值/cm
120 方案1 0.11 0.35 －0.82 1.23
方案2 0.12 0.39 －0.91 1.35
方案3 0.04 0.12 －0.48 0.30
2 160 方案1 －0.06 1.01 －3.22 3.51
方案2 －0.06 1.04 －3.46 3.42
方案3 0.06 1.11 －3.03 5.21
), ArticleFig(id=1322567063722373863, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567054117417633, language=EN, label=Tab. 4, caption=

Difference of mean kernel numerical integration for three schemes with 30″×30″ grid

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数 方案 平均值/cm 标准差/cm 最小值/cm 最大值/cm
120 方案1 0.05 0.13 －0.37 0.46
方案2 0.05 0.15 －0.40 0.51
方案3 0.02 0.09 －0.35 0.24
2 160 方案1 0.01 0.86 －3.24 2.69
方案2 0.01 0.87 －3.23 2.70
方案3 0 0.87 －3.28 2.67
), ArticleFig(id=1322567063944671976, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322567054117417633, language=CN, label=表4, caption=

30″×30″格网3种方案平均核数值积分差值

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数 方案 平均值/cm 标准差/cm 最小值/cm 最大值/cm
120 方案1 0.05 0.13 －0.37 0.46
方案2 0.05 0.15 －0.40 0.51
方案3 0.02 0.09 －0.35 0.24
2 160 方案1 0.01 0.86 －3.24 2.69
方案2 0.01 0.87 －3.23 2.70
方案3 0 0.87 －3.28 2.67
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Stokes积分计算大地水准面中央区效应分析
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耿晓燕 1 , 赵辉 1, * , 郭鑫伟 1 , 孟静娟 1 , 王文超 1
大地测量与地球动力学 | 大地测量学 2026,46(4): 481-486
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大地测量与地球动力学 |大地测量学 2026 , 46 (4) : 481 -486
Stokes积分计算大地水准面中央区效应分析
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耿晓燕1 , 赵辉1, * , 郭鑫伟1, 孟静娟1, 王文超1
作者信息
  • 1 自然资源部大地测量数据处理中心, 西安, 710054
通讯作者:
赵辉, 高级工程师, 主要从事大地测量数据处理研究, E-mail:
作者简介:

耿晓燕, 高级工程师, 主要从事大地测量数据处理研究, E-mail:

Geoid Innermost Area Effects Calculation Using Stokes Integral
Xiaoyan GENG1 , Hui ZHAO1, * , Xinwei GUO1, Jingjuan MENG1, Wenchao WANG1
Affiliations
  • 1 Geodetic Data Processing Center, MNR, Xi'an 710054, China
出版时间: 2026-04-15 doi: 10.14075/j.jgg.2025.04.136
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摘要
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为分析Stokes积分计算大地水准面中不同奇异积分处理方法的差异, 在研究基于圆形近似的传统方法与双二次多项式插值方法的基础上, 推导积分半径内Stokes积分的去奇异方法。选择实验区进行计算分析, 结果表明, 中央区取计算点所在的单个格网时, 使用点核数值积分, 传统方法与双二次多项式插值法结果相近, 去奇异法精度有所改进; 平均核数值积分能够提升传统方法与双二次多项式插值法计算精度, 而去奇异法变化不大。

Stokes积分  /  大地水准面  /  中央区效应  /  奇异积分

To analyze the differences of different singular integral methods in geoid calculation using Stokes integral, based on the traditional methods using circular approximation and biquadratic polynomial interpolation, we derive a de-singularized method for Stokes integral within the integration radius. The results show that when the innermost area is selected as the single grid where the calculation point is located, the accuracy of traditional method and biquadratic polynomial interpolation method are similar, and the de-singularized method has been improved using the point kernel numerical integration. The mean kernel numerical integration can improve the calculation accuracy of traditional method and biquadratic polynomial interpolation method, while the de-singularized method is not significant.

Stokes integral  /  geoid  /  innermost area effects  /  singular integral
耿晓燕, 赵辉, 郭鑫伟, 孟静娟, 王文超. Stokes积分计算大地水准面中央区效应分析. 大地测量与地球动力学, 2026 , 46 (4) : 481 -486 . DOI: 10.14075/j.jgg.2025.04.136
Xiaoyan GENG, Hui ZHAO, Xinwei GUO, Jingjuan MENG, Wenchao WANG. Geoid Innermost Area Effects Calculation Using Stokes Integral[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (4) : 481 -486 . DOI: 10.14075/j.jgg.2025.04.136
正文
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研究地球形状是大地测量学的主要任务之一。大地水准面定义为与全球无潮平均海水面密合的重力等位面，用于表征地球形状，是测定正高的基准面。高精度大地水准面对于区域高程基准维持具有重要作用[1]。
基于Stokes理论确定大地水准面是目前使用最广、最重要的方法之一[2]，即使利用Molodensky理论确定似大地水准面，一阶项也是Stokes积分形式。在实际计算中，Stokes积分通常采用数值积分方式实现，而计算点邻近的中央区距离趋近于0时，积分值变得无穷大，会产生奇异积分效应。为确保计算精度，需要单独处理中央区积分。一般中央区取计算点周围的圆形区域，在平面近似下圆内的重力异常可通过计算点处的泰勒级数展开表示，以推导Stokes积分的中央区影响[3]。但重力异常数据通常以经差和纬差分割的标准格网形式存储，圆形处理方法与格网数据的实际分布存在差异[4]。为解决中央区为近似矩形的奇异问题，将中央区格网的重力异常表示为双二次多项式插值，在平面近似下引入非奇异变换，可在中央区进行数值积分[5-9]。如果将积分核函数改进为积分核函数减计算点核函数值，那么当积分点趋于计算点时，新的积分核函数为0，奇异点被中和，这种思路已应用于谐函数径向导数计算[3]和泊松积分迭代法[10]。
本文基于圆形近似的传统方法和双二次多项式插值方法，推导Stokes积分的去奇异方法。为分析不同方法计算Stokes积分中央区效应的差异，采用地球重力场模型计算实验区重力异常，进而通过级数展开计算近区积分的严密性，用于结果检验。
1 Stokes积分计算大地水准面
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1.1 Stokes积分
Stokes公式是位理论第三边值问题的解，通过调整大地水准面外部地形质量，对重力异常进行全球积分来计算大地水准面。但由于通常仅能获取一定范围内的实测重力数据，Stokes积分所要求的全球积分难以满足。实际计算过程中将积分区域分为2个部分，以半径为ψ0的球冠区域σ0为近区，球面剩余部分σ－σ0为远区，远区影响由截断系数表示为级数形式。
大地水准面N的计算公式[11]为：
$N=\frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} \Delta g S(\psi) \mathrm{d} \sigma+\frac{R}{2 \gamma} \sum\limits_{n=2}^{\infty} Q_n \Delta g_n$
式中，S(ψ)为Stokes函数；Qn为截断系数；Δgn为重力异常Δg的n阶拉普拉斯谐函数；R为地球平均半径；γ为正常椭球面正常重力；ψ为计算点到积分面元的角距；dσ为单位球积分面元。
Stokes函数表达式为：
$\begin{gathered} S(\psi)=\frac{1}{\sin \frac{\psi}{2}}-6 \sin \frac{\psi}{2}+1-5 \cos \psi- \\3 \cos \psi \ln \left(\sin \frac{\psi}{2}+\sin ^2 \frac{\psi}{2}\right) \end{gathered}$
由扰动位和重力异常谐函数关系Tn= $\frac{R}{n-1} \Delta g_n$，将大地水准面表示为级数形式：
$N=\frac{R}{2 \gamma} \sum\limits_{n=2}^{\infty} \frac{2}{n-1} \Delta g_n$
结合式(1)和式(3)可得检验近区Stokes积分的标准，表达式为：
$\frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} \Delta g S(\psi) \mathrm{d} \sigma=\frac{R}{2 \gamma} \sum\limits_{n=2}^{\infty}\left(\frac{2}{n-1}-Q_n\right) \Delta g_n$
1.2 平均核数值积分
近区积分基于数值积分法进行计算，对于积分半径内的格网qk，重力异常以平均值$\Delta \bar{g}_k$表示，则[3]：
$\frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} \Delta g S(\psi) \mathrm{d} \sigma \approx \frac{R}{4 \mathsf{π} \gamma} \sum\limits_k c_k \Delta \bar{g}_k$
式中，系数ck可表示为：
$c_k=\iint\limits_{q_k} S\left(\psi_k\right) \mathrm{d} \sigma \approx \bar{S}_k \cos \varphi_Q \Delta \varphi \Delta \lambda$
式中，$\bar{S}_k$为格网积分核函数平均值；φQ为积分流动点纬度；Δφ和Δλ分别为格网纬差和经差。
通常直接利用格网中心点的核函数值进行数值积分，但由于核函数在计算点附近变化迅速，使用点核函数值进行计算可能精度不足，因此计算点附近需要足够密的格网才能满足要求。将高斯-勒让德积分法用于计算平均核，仅需少量高斯点，就可快速收敛到平均值[12]。
对于范围为φ∈[φ1, φ2]、λ∈[λ1, λ2]的格网，使用二维高斯-勒让德积分法计算权重系数wi和wj及流动点φQj 和λQi ，高斯-勒让德平均核可表示为：
$\begin{aligned}& \int_{\lambda_1}^{\lambda_2} \int_{\varphi_1}^{\varphi_2} S(\psi) \cos \varphi \mathrm{d} \varphi \mathrm{~d} \lambda \approx \frac{1}{\sum w_i w_j} . \\& \sum\limits_{i=1}^n \sum\limits_{j=1}^m w_i w_j S\left(\psi\left(\varphi_P, \lambda_P, \varphi_{Q_j}, \lambda_{Q_i}\right)\right)\end{aligned}$
式中，n=m表示高斯-勒让德阶数。
2 中央区效应计算方法
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2.1 传统方法
假设中央区为计算点P周围半径为s0的圆形区域，当ψ→0时，对积分核函数进行近似，Stokes公式可写为[3]：
$\mathsf{δ} N=\frac{1}{2 \mathsf{π} \gamma} \int_{\alpha=0}^{2 \mathsf{π}} \int_{s=0}^{s_0} \frac{\Delta g}{s} s \mathrm{~d} s \mathrm{~d} \alpha$
在P点将重力异常按泰勒级数展开：
$\Delta g=\Delta g_P+s\left(g_x \cos \alpha+g_y \sin \alpha\right)+\cdots$
将式(9)代入式(8)，对α和s进行积分，并舍去高阶项，得到：
$\mathsf{δ} N=\frac{s_0}{\gamma} \Delta g_P$
由球冠和格网的等面积关系，根据格网纬差Δφ和经差Δλ，计算s0后代入式(10)得：
$\mathsf{δ} N=\frac{R}{\gamma} \sqrt{\frac{\cos \varphi_P \Delta \varphi \Delta \lambda}{\mathsf{π}}} \Delta g_P$
2.2 双二次多项式插值法
利用计算点P及周围9个节点构造重力异常插值函数，用于表达中央区重力异常变化。以计算点为原点建立切平面直角坐标系，x轴沿子午线方向指北，y轴沿平行圈方向指东。
中央区平面近似下Stokes公式可写为[5]：
$\mathsf{δ} N=\frac{1}{2 \mathsf{π} \gamma} \iint \frac{\Delta g}{l_0} \mathrm{~d} x \mathrm{~d} y$
式中，$l_0=\sqrt{\left(x-x_P\right)^2+\left(y-y_P\right)^2}$为计算点与流动点的平面距离。
设x方向格网间隔a=RΔφ，y方向格网间隔b=RΔλcosφ，引入新的变量u=x/a和v=y/b，则中央区范围为$\left[-\frac{1}{2}<u<\frac{1}{2},-\frac{1}{2}<\right.$$\left.v<\frac{1}{2}\right]$。
将中央区任意点的重力异常用双二次多项式插值表示：
$\Delta g=\left[\begin{array}{lll}1 & u & u^2\end{array}\right]\left[\begin{array}{lll}c_{00} & c_{01} & c_{02} \\c_{10} & c_{11} & c_{12} \\c_{20} & c_{21} & c_{22}\end{array}\right]\left[\begin{array}{c}1 \\v \\v^2\end{array}\right]$
式中，cij为多项式系数。利用9个节点处已知重力异常为条件，解线性方程得：
$\left\{\begin{aligned}c_{00}= & \Delta g_{0, 0} \\c_{02}= & \frac{1}{2}\left(\Delta g_{0, 1}+\Delta g_{0, -1}-2 \Delta g_{0, 0}\right) \\c_{20}= & \frac{1}{2}\left(\Delta g_{1, 0}+\Delta g_{-1, 0}-2 \Delta g_{0, 0}\right) \\c_{22}= & \frac{1}{4}\left(\Delta g_{1, -1}-2 \Delta g_{1, 0}+\Delta g_{1, 1}-2 \Delta g_{0, -1}+\right. \\& 4 \Delta g_{0, 0}-2 \Delta g_{0, 1}+\Delta g_{-1, -1}-2 \Delta g_{-1, 0}+ \\& \left.\Delta g_{-1, 1}\right)\end{aligned}\right.$
将式(13)代入式(12)，并顾及奇偶性质，将4个象限的和由第一象限表示为：
$\mathsf{δ} N=\frac{4 a b}{2 \mathsf{π} \gamma} \int_0^{\frac{1}{2}} \int_0^{\frac{1}{2}} \frac{c_{00}+c_{20} u^2+c_{02} v^2+c_{22} u^2 v^2}{\sqrt{a^2 u^2+b^2 v^2}} \mathrm{~d} u \mathrm{~d} v$
将第一象限分为2个三角形区域σ1∈ $\left[0 <u<\frac{1}{2}, 0<v<u\right]$和$\sigma_2 \in\left[0<v<\frac{1}{2}, 0<u<v\right]$。对σ1引入新变量r=v/u，对σ2引入新变量t=u/v，将v=ru和u=tv分别代入2个区域，完成对u和v的积分，简化后可得：
$\begin{gathered}\mathsf{δ} N=\frac{a b}{\mathsf{π} \gamma} \int_0^1 \frac{c_{00}+\frac{1}{12} c_{20}+\frac{1}{12} c_{02} r^2+\frac{1}{80} c_{22} r^2}{\sqrt{a^2+b^2 r^2}} \mathrm{~d} r+ \\\frac{a b}{\mathsf{π} \gamma} \int_0^1 \frac{c_{00}+\frac{1}{12} c_{02}+\frac{1}{12} c_{20} t^2+\frac{1}{80} c_{22} t^2}{\sqrt{b^2+a^2 t^2}} \mathrm{~d} t\end{gathered}$
积分后得：
$\begin{gathered}\mathsf{δ} N=\frac{1}{480 a^2 b^2 \mathsf{π} \gamma}\left[\left(a b^3 \sqrt{a^2+b^2}-\right.\right. \\\left.b^5 \operatorname{arcsinh}\left(\frac{a}{b}\right)\right)\left(20 c_{20}+3 c_{22}\right)+ \\\left(a^3 b \sqrt{a^2+b^2}-a^5 \operatorname{arcsinh}\left(\frac{b}{a}\right)\right)\left(20 c_{02}+\right. \\\left.3 c_{22}\right)+40 a^2 b^3 \operatorname{arcsinh}\left(\frac{a}{b}\right)\left(12 c_{00}+c_{02}\right)+ \\\left.40 a^3 b^2 \operatorname{arcsinh}\left(\frac{b}{a}\right)\left(12 c_{00}+c_{20}\right)\right]\end{gathered}$
2.3 去奇异法
首先将积分半径内所有格网的重力异常减去计算点重力异常，则计算点所在格网重力异常为0，即不存在积分奇异问题，然后将积分结果加上计算点重力异常的影响。计算过程可分解为：
$\begin{gathered}\frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} S(\psi) \Delta g \mathrm{~d} \sigma=\frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} S(\psi) \cdot \\\left(\Delta g-\Delta g_P\right) \mathrm{d} \sigma+\Delta g_P \frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} S(\psi) \mathrm{d} \sigma\end{gathered}$
对于式(18)等号右端第2项，在以P点为原点的球面极坐标系下进行积分：
$\frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} S(\psi) \mathrm{d} \sigma=\frac{R}{4 \mathsf{π} \gamma} \int_{\alpha=0}^{2 \mathsf{π}} \int_{\psi=0}^{\psi_0} S(\psi) \sin \psi \mathrm{d} \psi \mathrm{~d} \alpha$
首先对ψ进行积分，根据定积分性质可得：
$\begin{gathered}\int_0^{\psi_0} S(\psi) \sin \psi \mathrm{d} \psi=\int_0^\mathsf{π} S(\psi) \sin \psi \mathrm{d} \psi- \\\int_{\psi_0}^\mathsf{π} S(\psi) \sin \psi \mathrm{d} \psi\end{gathered}$
上式等号右端第1项可看作ψ0=0时的截断系数Q0，此时为0；第2项为任意ψ0时的截断系数Q0，则：
$\int_0^{\psi_0} S(\psi) \sin \psi \mathrm{d} \psi=-Q_0$
然后对α进行积分，式(19)可转化为：
$\frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} S(\psi) \mathrm{d} \sigma=-\frac{R}{2 \gamma} Q_0$
将式(22)代入式(18)，近区Stokes积分的去奇异公式可表示为：
$\begin{gathered}\frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} S(\psi) \Delta g \mathrm{~d} \sigma=\frac{R}{4 \mathsf{π} \gamma} \iint\limits_{\sigma_0} S(\psi) \cdot \\\quad\left(\Delta g-\Delta g_P\right) \mathrm{d} \sigma-\frac{R}{2 \gamma} Q_0 \Delta g_P\end{gathered}$
由于计算点重力异常差为0，因此数值积分时无需考虑奇异问题。
3 数值实验
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为分析采用不同方法计算Stokes积分中央区效应的差异，选择40°~45°N、84°~90°E为研究区，该区域位于天山山脉，重力异常相对变化较大，利用地球重力场模型生成数据进行实验。武汉大学发布的SGG-UGM-2模型融合了GOCE、GRACE卫星重力数据和多代卫星测高数据以及EGM2008模型重力异常数据，基于椭球谐分析而建立，在中国区域相较于其他模型精度有所提升[13]。选用该模型计算1′×1′和30″×30″两种分辨率的重力异常，模型阶数分别为120阶和2 160阶，中央区取计算点所在的单个格网，分别以传统方法(方案1)、双二次多项式插值法(方案2)和去奇异法(方案3)处理中央区效应，积分半径为0.5°，将式(3)等号右端级数求和结果作为真值进行验证分析。为减小边缘效应的影响，统计范围由研究区边界向内缩小1°。
图 1为利用重力场模型计算的120阶重力异常，可以看出，由于仅包含低频信息，重力异常变化平缓，在－162.373 9~82.975 6 mGal之间，标准差为52.363 3 mGal，能尽可能减少数值积分对中央区效应的影响。图 2为利用重力场模型计算的2 160阶重力异常，可以看出，由于阶数较高，包含高频信息，重力异常波动更大，在－234.510 8~398.959 0 mGal之间，标准差为104.956 7 mGal。
在使用1′×1′分辨率的重力异常格网进行计算的情况下，统计格网数共43 200个。由表 1可知，基于点核数值积分，对于120阶和2 160阶，方案1和方案2标准差基本一致，方案3标准差最小。由于2 160阶重力异常变化更为剧烈，相对于120阶，2 160阶差值的标准差、最小值、最大值均更大。
在使用30″×30″分辨率的重力异常格网进行计算的情况下，统计格网数共172 800个。对比表 2与表 1可知，基于点核数值积分，随着格网分辨率的提高，2个阶数的统计结果精度均有所提升；120阶差值的绝对值小于1 cm，2 160阶差值的绝对值小于4 cm。
以上计算均取格网中心点核函数进行点核数值积分，点核数值积分计算方便，但计算点周围核函数变化剧烈，仅用中心点核函数值进行计算，存在不同程度的近似。平均核数值积分基于高斯-勒让德数值积分计算格网核函数平均值，虽然计算速度会受影响，但精度有所提升。
为进一步分析平均核对数值积分结果的影响，实验中固定计算点所在格网，其余格网核函数平均值根据高斯-勒让德数值积分法进行计算。图 3、4为1′×1′格网2 160阶重力异常，方案1为点核数值积分、平均核数值积分结果分别与级数求和结果作差，可以看出，平均核差值相对降低。图 5、6为1′×1′格网2 160阶重力异常，方案3为点核数值积分、平均核数值积分结果分别与级数求和结果作差，二者分布接近，均与图 2所示重力异常变化具有较高相关性。
对比表 3、4和表 1、2可知，方案3采用平均核数值积分与点核数值积分结果基本一致，而方案1和方案2采用平均核数值积分比点核数值积分结果精度均有不同程度提升。与点核数值积分相比，采用平均核数值积分后，2种格网分辨率120阶统计结果方案3最优，2 160阶统计结果3种方案相差不大。此外，方案3在同等分辨率和阶数条件下，采用点核数值积分和平均核数值积分结果变化较小，说明去奇异法在数值积分中使用相对于计算点重力异常的变化，数值相对较小，积分结果较为稳定。
4 结语
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为解决Stokes积分中央区奇异的问题，本文研究中央区单个格网的双二次多项式插值法，推导去奇异方法，并利用实验数据对3种中央区处理方法在不同阶数和格网分辨率条件下进行计算分析。结果表明，基于点核数值积分，去奇异法在数值积分中可移去积分半径内的全部流动点所受到的计算点重力异常的影响，相较于传统方法和双二次多项式插值法，计算精度有所提升；基于平均核数值积分，相比于点核数值积分，传统方法和双二次多项式插值法计算精度均有所提升，去奇异法变化不大。对于重力异常变化剧烈的地区，需要提升格网分辨率才能确保精度满足需求；通过对中央区格网进行细化，可以更精细化地处理奇异问题。
基金
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  • 陕西测绘地理信息局科技创新项目(SCK2025-04)
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2026年第46卷第4期
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doi: 10.14075/j.jgg.2025.04.136
  • 接收时间：2025-04-23
  • 首发时间：2026-10-29
  • 出版时间：2026-04-15
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  • 收稿日期：2025-04-23
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陕西测绘地理信息局科技创新项目(SCK2025-04)
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    1 自然资源部大地测量数据处理中心, 西安, 710054

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赵辉, 高级工程师, 主要从事大地测量数据处理研究, E-mail:
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2种不同金属材料的力学参数
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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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