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Article(id=1322566816598175823, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322566763112415546, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.02.034, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1739203200000, receivedDateStr=2025-02-11, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793254101179, onlineDateStr=2026-10-29, pubDate=1768406400000, pubDateStr=2026-01-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793254101179, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793254101179, creator=13701087609, updateTime=1793254101179, updator=13701087609, issue=Issue{id=1322566763112415546, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='1', pageStart='1', pageEnd='138', issueExtLink='null', onlineDate='null', pubDate='1768406400000', pubDateStr='2026-01-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793254088427, creator='13701087609', updateTime=1793254088427, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=48, endPage=54, ext={EN=ArticleExt(id=1322566816765947984, articleId=1322566816598175823, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Numerical Analysis of Gravimetric Quasi-Geoid Determination Based on SRBF Method, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

This paper employs the spherical radial basis function (SRBF) method to integrate terrestrial gravity and airborne gravity data for the computation of the gravity quasi-geoid. Experiments are conducted from aspects such as terrain effects, model order, and basis function parameters to investigate the impact and variation patterns of different parameters.The results show that: 1) The burial depth, effective distance, and expansion order of the basis function all affect the quality of the quasi-geoid refined by the SRBF method. The precision difference of the geoidal height anomalies corresponding to different parameter settings can reach up to 2.4 cm; 2) Considering the terrain effects when calculating residual gravity data can improve the final quasi-geoid precision by 21.6%; 3) Using two sets of GNSS/leveling points with different precisions and distributions for external validation of the quasi-geoid results, the best validation precision differs by 4.3 cm; 4) In mountainous areas, by optimizing parameters to integrate terrestrial and airborne data, the optimal precision of the quasi-geoid result can reach 2.7 cm.

, authors=Jingjuan MENG1, Hui ZHAO1, *, Wenchao WANG1, Xiaoyan GENG1, Junying CHEN1, authorsList=Jingjuan MENG, Hui ZHAO, Wenchao WANG, Xiaoyan GENG, Junying CHEN, 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=1322566818431086683, articleId=1322566816598175823, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=基于SRBF法确定似大地水准面的数值分析, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

基于球面径向基函数法(SRBF)融合地面重力、航空重力数据计算重力似大地水准面，从地形影响、模型阶次、基函数参数等方面进行实验，研究不同参数的影响程度与变化规律。结果表明：1)基函数的埋深、作用距离、展开阶数均对SRBF法精化的重力似大地水准面质量产生影响，不同参数设置对应的高程异常的精度差异可达2.4 cm; 2)在计算残差重力数据时考虑地形影响，可使最终似大地水准面精度提升21.6%;3)利用2组精度和分布不同的GNSS/水准点对似大地水准面成果进行外部检核，最佳检核精度相差4.3 cm; 4)在山区，通过优化参数完成地面与航空数据的融合，似大地水准面成果精度最优达2.7 cm。

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

孟静娟, 工程师, 主要从事大地测量数据处理, E-mail:

, correspAuthorsNote=
赵辉, 高级工程师, 主要从事大地测量数据处理, E-mail:
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Statistics of external validation results before and after removing topography effects

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地形影响最大值/cm最小值/cm平均值/cm标准差/cm
不考虑-70.3-129.5-103.88.8
考虑-82.9-128.7-102.26.9
), ArticleFig(id=1322566821648117902, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566816598175823, language=CN, label=表1, caption=

地形影响移去前后外部检核结果统计

, figureFileSmall=null, figureFileBig=null, tableContent=
地形影响最大值/cm最小值/cm平均值/cm标准差/cm
不考虑-70.3-129.5-103.88.8
考虑-82.9-128.7-102.26.9
), ArticleFig(id=1322566821727809679, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566816598175823, language=EN, label=Tab. 2, caption=

Statistical results of the calculation outcomes for the eight sets of parameters

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统计项目 参数组合
abcdefgh
最低阶数120120120120360360360360
埋深/km1010303010103030
作用距离/km50100501005010050100
成果标准差/cm3.53.85.05.73.43.74.24.5
A组残差标准差/cm7.27.28.39.47.07.57.67.7
B组残差标准差/cm3.52.73.13.84.64.14.24.6
), ArticleFig(id=1322566821799112848, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566816598175823, language=CN, label=表2, caption=

对8组参数计算成果的统计结果

, figureFileSmall=null, figureFileBig=null, tableContent=
统计项目 参数组合
abcdefgh
最低阶数120120120120360360360360
埋深/km1010303010103030
作用距离/km50100501005010050100
成果标准差/cm3.53.85.05.73.43.74.24.5
A组残差标准差/cm7.27.28.39.47.07.57.67.7
B组残差标准差/cm3.52.73.13.84.64.14.24.6
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基于SRBF法确定似大地水准面的数值分析
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孟静娟 1 , 赵辉 1, * , 王文超 1 , 耿晓燕 1 , 陈俊英 1
大地测量与地球动力学 | 大地测量学 2026,46(1): 48-54
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大地测量与地球动力学 |大地测量学 2026 , 46 (1) : 48 -54
基于SRBF法确定似大地水准面的数值分析
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孟静娟1 , 赵辉1, * , 王文超1, 耿晓燕1, 陈俊英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, E-mail:
作者简介:

孟静娟, 工程师, 主要从事大地测量数据处理, E-mail:

Numerical Analysis of Gravimetric Quasi-Geoid Determination Based on SRBF Method
Jingjuan MENG1 , Hui ZHAO1, * , Wenchao WANG1, Xiaoyan GENG1, Junying CHEN1
Affiliations
  • 1 Geodetic Data Processing Center, MNR, Xi'an 710054, China
出版时间: 2026-01-15 doi: 10.14075/j.jgg.2025.02.034
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摘要
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基于球面径向基函数法(SRBF)融合地面重力、航空重力数据计算重力似大地水准面，从地形影响、模型阶次、基函数参数等方面进行实验，研究不同参数的影响程度与变化规律。结果表明：1)基函数的埋深、作用距离、展开阶数均对SRBF法精化的重力似大地水准面质量产生影响，不同参数设置对应的高程异常的精度差异可达2.4 cm; 2)在计算残差重力数据时考虑地形影响，可使最终似大地水准面精度提升21.6%;3)利用2组精度和分布不同的GNSS/水准点对似大地水准面成果进行外部检核，最佳检核精度相差4.3 cm; 4)在山区，通过优化参数完成地面与航空数据的融合，似大地水准面成果精度最优达2.7 cm。

球面径向基函数  /  航空重力  /  地面重力  /  GNSS/水准  /  局部重力场

This paper employs the spherical radial basis function (SRBF) method to integrate terrestrial gravity and airborne gravity data for the computation of the gravity quasi-geoid. Experiments are conducted from aspects such as terrain effects, model order, and basis function parameters to investigate the impact and variation patterns of different parameters.The results show that: 1) The burial depth, effective distance, and expansion order of the basis function all affect the quality of the quasi-geoid refined by the SRBF method. The precision difference of the geoidal height anomalies corresponding to different parameter settings can reach up to 2.4 cm; 2) Considering the terrain effects when calculating residual gravity data can improve the final quasi-geoid precision by 21.6%; 3) Using two sets of GNSS/leveling points with different precisions and distributions for external validation of the quasi-geoid results, the best validation precision differs by 4.3 cm; 4) In mountainous areas, by optimizing parameters to integrate terrestrial and airborne data, the optimal precision of the quasi-geoid result can reach 2.7 cm.

spherical radial basis function  /  airborne gravity  /  terrestrial gravity  /  GNSS/leveling  /  regional gravity field
孟静娟, 赵辉, 王文超, 耿晓燕, 陈俊英. 基于SRBF法确定似大地水准面的数值分析. 大地测量与地球动力学, 2026 , 46 (1) : 48 -54 . DOI: 10.14075/j.jgg.2025.02.034
Jingjuan MENG, Hui ZHAO, Wenchao WANG, Xiaoyan GENG, Junying CHEN. Numerical Analysis of Gravimetric Quasi-Geoid Determination Based on SRBF Method[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (1) : 48 -54 . DOI: 10.14075/j.jgg.2025.02.034
正文
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对地球局部重力场的探索和构建一直是大地测量领域的热点。随着重力测量技术的不断进步，重力数据日益丰富，测量精度也不断提高，为高精度重力场建模提供了良好的数据基础。与此同时，各种新理论和新方法的涌现，为提升重力场确定的精度开辟了新的途径。对不同特性的多源重力数据进行有效融合，实现不同类型重力数据的优势互补，是为局部重力场建模提供可靠重力信息的关键。
通常，重力似大地水准面的确定主要依赖于数值积分法，包括基于Molodensky边值理论的级数解、将第二类Helmert凝聚引入Stokes边值问题的Stokes-Helmert法以及基于Bjerhammar边值理论的解析延拓法等。这些方法大多以重力异常为输入数据，需要提前对不同类型的重力数据进行融合处理。最小二乘配置法在多源重力数据融合方面具有天然优势，是常用的多源数据融合计算方法，但在求解法方程时，协方差函数的确定是一个较大的挑战。球面径向基函数(SRBF)具有良好的局部化特性，能够在求解过程中通过基函数融合多源重力信息，其计算效率明显优于传统的最小二乘配置法[1]，近年来在局部重力场逼近中得到了广泛应用。吴怿昊等[2]采用Poisson小波径向基函数融合船载、航空、地面等多源重力数据，构建了欧洲局部区域陆海统一的似大地水准面模型，其中荷兰似大地水准面的检核精度达到1.12 cm；Bentel等[3]同时采用Poisson多极函数、Blackman函数、Cubic函数和Abel-Poisson函数等4种基函数对喜马拉雅地区的局部重力场进行建模，从多角度比较了不同基函数的特性，并分析了不同类型基函数对建模精度的影响。
尽管SRBF理论在局部重力场逼近中具有显著优势，但其逼近效果极易受到基函数选择和参数设置的影响。如果基函数及其参数设置合理，将获得较优的逼近效果；反之，若选择不当，则很难达到满意的重力场逼近效果。Poisson小波核函数在空域和频域均具有较好的局部化特性，对于融合不同高度的重力数据具有显著优势[4-5]。因此，本文选择Poisson小波核函数作为基函数，结合实验区域的实测重力数据，对不同参数组合下SRBF逼近局部重力场的表现进行详细分析，并利用实测GNSS/水准点对计算的局部重力场进行精度检核。以GNSS/水准点高程异常残差标准差为评价依据，分析不同参数设置下SRBF计算的重力场的逼近效果，最终总结影响重力场逼近效果的原因，以实现优化重力场逼近方案的目标。
1 理论基础
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1.1 球面径向基函数表达重力场
依据重力场的“可叠加性”，地球外部任意空间点x的扰动位T(x)均可分解为2个部分：模型部分和残余部分：
$T(x)=T_{\mathrm{mdl}}(x)+T_{\mathrm{res}}(x)$
式中，Tmdl(x)表示扰动位的模型部分; Tres(x)表示移除扰动位的模型部分后的残余扰动位，该方法称为“两步法”。若考虑地形影响，则Tres(x)可进一步细分为Tdem(x)和Tres(x)，其中Tdem(x)表示地形影响；Tres(x)表示移除模型部分和地形影响后的残余扰动位，这部分可由K个球面径向基函数Φk(x)的线性组合表示：
$T_{\mathrm{res}}(x)=\frac{G M}{r} \sum\limits_{k=1}^K d_k \varPhi_k\left(x, x_k\right)$
式中，GM为地球引力常数；r为地心向径；dk为基函数系数；K为基函数系数的个数；x和xk分别为计算点和径向基函数节点。
由式(2)可知，只要确定基函数Φk(x, xk)，利用重力观测数据，列观测方程解未知数(即系数dk)，即可获得区域重力场的残余部分，随后恢复扰动位的模型部分和地形影响，即可计算空间任意点的高程异常。
1.2 球面径向基函数
根据重力场大地边值理论，用于求解重力场的径向基函数必须满足拉普拉斯方程，即具备调和性质。目前常见的径向基函数包括点质量核函数、Poisson核函数、Shannon核函数[6]、Blackman核函数、Cubic核函数、径向多极核函数以及Poisson小波核函数。其中，点质量理论[7]是将离散分布于地表的实测数据通过解析延拓至某一虚拟球面，但在计算范围较大或分辨率较高时，其在求解法方程的过程中容易失去稳定性。Blackman核函数[8]则相当于对球面径向基函数(SRBF)进行加权滤波处理。Poisson核函数、Poisson小波核函数和径向多极核函数均具有明显的带通滤波特性，这种特性使其在提升局部重力场建模精度方面更具优势。相较于Poisson核函数和径向多极核函数，Poisson小波核函数不仅具备时域和频域的局部性，还具备多尺度分析能力，使用起来更加灵活高效。因此，本文最终选择Poisson小波核函数，为提高计算效率，采用其解析形式：
$\varPhi_k^m(x)=2\left(\chi_{m+1}-\chi_m\right), \chi_m=\left(r_k \frac{\partial}{\partial r_k}\right)^m / L$
式中，m为基函数的次，本文取m=1；L为地球外部计算点与Bjerhammer球面上球面径向基函数节点的空间距离；rk为球面径向基函数节点的地心向径。
1.3 多源重力场融合的参数估计
扰动位T与重力异常、重力扰动、高程异常存在以下关系：
$\left\{\begin{array}{l}\Delta g(x)=-\frac{\partial T}{\partial r}-\frac{2 T}{r} \\\delta g(x)=-\frac{\partial T}{\partial r} \\\zeta(x)=\frac{T}{\gamma}\end{array}\right.$
将式(2)代入式(4)，分别得到重力异常、重力扰动、高程异常的基函数表达形式。以M个实测数据yi作为观测量，建立误差方程，求解基函数系数dk：
$\left[\begin{array}{c}v_1 \\v_2 \\\vdots \\v_M\end{array}\right]=\left[\begin{array}{cccc}A_{11} & A_{12} & \cdots & A_{1 K} \\A_{21} & A_{22} & \cdots & A_{2 K} \\\vdots & \vdots & \ddots & \vdots \\A_{M 1} & A_{M 2} & \cdots & A_{M K}\end{array}\right]\left[\begin{array}{c}d_1 \\d_2 \\\vdots \\d_K\end{array}\right]-\left[\begin{array}{c}y_1 \\y_2 \\\vdots \\y_M\end{array}\right]$
式中，A为M×K设计矩阵；vi表示第i个观测值的残差。
1.4 大地水准面高到高程异常的转换
本文是在正常高系统下进行似大地水准面的计算，而实测的GNSS/水准点的水准数据为正高，因此采用似大地水准面与大地水准面严密转换模型[9]，完成实测数据从大地水准面高到高程异常的转换，模型为：
$N-\zeta \approx \frac{1}{\bar{\gamma}}\left(\Delta g_B h+V_0^t-V^t-\frac{h^2}{2} \frac{\partial \Delta g_B}{\partial h}\right)$
式中，Vt和V0t分别为地面点及其在大地水准面上的投影点的地形引力位；γ为全球正常重力平均值；ΔgB和$ \frac{\partial \Delta g_B}{\partial h}$分别表示计算点的布格重力异常及其对应的布格重力异常梯度。
1.5 残差地形影响
选用ETOPO20地形数据，其格网分辨率为30″，用于精确计算局部地形质量对重力点重力值以及GNSS/水准点高程异常的局部影响。在处理地形数据时，采用残差地形模型(RTM)方法。该模型仅考虑地形面与平滑参考面之间的残差质量：参考面上方的多余质量被移除，下方的缺失质量被填充。通过这种方式，RTM能够有效逼近由地形起伏引起的高频信号，从而为重力场建模提供更精准的地形改正信息。以计算点为原点，基于棱柱法计算地形扰动位：
$T=G \rho \int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} \frac{1}{r} \mathrm{~d} x \mathrm{~d} y \mathrm{~d} z$
式中，(x1，y1，z1)、(x2，y2，z2)为四棱柱顶点的坐标。
积分后，扰动位T的计算公式表示为：
$\begin{gathered}T=G \rho| | \mid x y \ln (z+r)+y z \ln (x+r)+ \\z x \ln (y+r)-\frac{x^2}{2} \arctan \frac{y z}{x r}- \\\frac{{{y^2}}}{2}\arctan \frac{{zx}}{{yr}} - \frac{{{z^2}}}{2}\arctan \frac{{xy}}{{zr}}\left. {} \right|_{{x_1}}^{{x_2}}\left. {} \right|_{{y_1}}^{{y_2}}\left| {_{{z_1}}^{{z_2}}} \right.\end{gathered}$
2 采用数据
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2.1 研究区域与数据
实验区选择美国西部科罗拉多州，区域内多山地，地形分布较复杂。中西部是密集的森林山区，海拔最高达4 220 m，最低约940 m，区域内高程变化较剧烈，平均值约2 015 m(图 1)。
从美国国家大地测量局(National Geodetic Survey，NGS)收集到该区域已有的地面重力数据59 303点[10]、航空重力数据28 371点[11]，2组分别为509点和221点的GNSS/水准数据，数据覆盖范围为110°~102°W、35°~40°N，分布如图 2所示。图中绿色点位表示地面重力数据，黑色点位表示航空重力数据，蓝色点和红色点分别表示来自于2个不同项目的GNSS/水准点。
由图 2可见，地面重力数据已基本覆盖整个研究区域，但其分布密度存在明显差异：在平坦地区，地面重力点的平均间距约为3 km；而在山区，平均间距则扩大到约10 km。由于地面重力点的高程为正高，而在计算模型值时需要点位的大地高信息，因此本文通过Geoid 2018模型进行插值，获取点位的大地水准面高，进而将正高转换为椭球高，用于计算地面重力点的空间异常。
航空重力数据来源于美国垂直基准重定义计划(GRAVD)，测量时间从2015-11持续至2017-09。其主测线呈东西走向，线距约为10 km；检测线则为南北走向，线距约为80 km。这些数据整体分布于实验区域的中南部，主要用于计算测线上的扰动重力。
此外，还收集了2组GNSS/水准点数据。其中一组在研究区内分布较为广泛，记为A组，共包含509个点(图 2中标注为蓝色点位)，这些点位由多年、多期的测量成果组合而成。不同点的测量时间跨度较大，导致各成果之间的精度差异显著。为了减少边缘效应对最终检测精度的影响，A组数据的范围相较于重力数据覆盖范围向内缩进了0.5°(图 2中紫色矩形框所示)。经过筛选，最终参与重力场精度检核的A组点位共计309个GNSS/水准点。另一组GNSS/水准点数据来自GSVS17项目[12]，包含221个实测点(图 2中标注为红色三角点)。GSVS项目共进行3次测量活动，旨在更新和改善美国的国家高程基准。GSVS17是该项目的第3次也是最后一次大地水准面坡度调查测量，由美国国家大地测量局于2017年夏季在地形复杂的南科罗拉多地区完成。此次测量以严格的测点间距和高规格的观测精度，实测了GPS数据、二等水准和天文垂线偏差，最终实现了优于2 cm精度的大地水准面检核目标。因此，本文选用GSVS17项目的GNSS水准成果作为B组数据。
在测量过程中，为了减少外部因素对成果精度的影响，B组数据的水准测量历时73 d完成，获得了精度为±0.7 mm/km的NAVD 88正高成果；GPS观测任务历时38 d完成，获得了平面精度优于1 cm的GPS大地高成果(IGS 08椭球高)。由于这些点位同时实测了椭球高和正高信息，因此该组数据反映的是大地水准面起伏。为了保持检测数据与成果的一致性，本文将点位的实测大地水准面高通过严密模型转换为高程异常，作为SRBF计算的重力似大地水准面精度的检测数据。
2.2 参考重力场模型
EGM2008模型是由美国国家地理情报局(National Geospatial-Intelligence Agency，NGA)开发的超高阶次地球重力场模型，它综合了地面重力、航空重力、卫星测高数据等多种数据源。凭借其极高的分辨率，该模型能够提供全球范围内的重力场数据。在美国境内，EGM2008模型采用了丰富的地面重力数据和高质量的卫星数据，使得其在美国境内的重力异常精度达到±(1~2) mGal的高水平。
为了最大限度地降低参考模型对重力场逼近结果的影响，同时综合考虑研究区的位置以及模型的整体适用性，本文选择EGM2008模型作为参考重力场模型，用于计算重力场的长波部分。此外，还利用EGM2008模型计算了GNSS/水准点高程异常的模型值，以确保数据的一致性和可靠性。
3 数值分析
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3.1 移去模型阶数对逼近效果的影响
为研究移去模型阶数对基函数性能的影响，采用EGM2008模型，分别移去180阶、360阶、540阶、810阶、1 080阶、1 350阶、1 620阶、1 890阶和2 160阶的模型重力值。随后，利用SRBF方法对残余重力进行处理，分别计算重力似大地水准面。计算完成后，使用A组GNSS/水准点对得到的重力似大地水准面进行精度检核，并统计实测高程异常与计算重力似大地水准面之间的差值标准差，结果如图 3所示。
由图可见，不同的移去模型阶数对最终似大地水准面的检核精度产生了显著影响。总体而言，随着移去模型阶数的增加，外部检核精度逐步提高。具体而言，当移去180阶模型重力值时，残余重力计算结果的外部检核精度为23.5 cm；而当移去2 160阶模型重力值时，计算结果的外部检核精度显著提升至9.1 cm。由此可见，较高的移去模型阶数能够更有效地减少模型误差对结果精度的影响。因此，为了最大限度地降低模型对结果精度的干扰，后续研究均以2 160阶作为重力场模型值的移去阶数。
3.2 地形影响对结果精度的影响
为了分析地形对重力似大地水准面计算精度的影响，本文采用残差地形模型(RTM)来计算各类数据点位的地形影响。在移去2 160阶模型值的基础上，设定最低展开阶数为120阶，最高展开阶数为2 160阶。分别使用未考虑地形影响和已考虑地形影响的残余重力文件计算重力似大地水准面，结果如图 4所示。随后，利用A组GNSS/水准点对考虑地形影响前后的重力似大地水准面进行精度检核，统计结果详见表 1。
由表 1可见，考虑与不考虑地形影响时，GNSS点的检验结果存在显著差异。在不考虑地形影响的情况下，使用A组309个点对计算得到的似大地水准面进行外部精度检核，结果显示, 高程异常差异的最小值为-129.5 cm，最大值为-70.3 cm，标准差为8.8 cm。而移去地形影响后的统计结果显示，高程异常差异的最小值为-128.7 cm，最大值为-8.9 cm，检核标准差从8.8 cm降至6.9 cm，整体精度提升了21.6%。
图 4表明，加入地形影响后，计算得到的似大地水准面精度得到显著提升。研究区全域的残差标准差从6.9 cm降至3.8 cm。特别是研究区西北角区域(110°~105°W，38°~40°N)，其残差标准差从12.7 cm降至5.5 cm，降低了56.7%。结合图 1的地形分布和图 2的数据分布情况分析可知，研究区西北角为复杂的高山地形，地形扰动对计算结果的影响较大。同时，该区域的数据分布密度相对较低，进一步影响了计算精度。而残差地形模型的应用有效平滑了局部重力场的高频扰动，从而显著改善了计算结果。
3.3 基函数参数对逼近效果的影响
基函数的频谱特性极易受到参数设置的影响，不同参数下的计算结果存在显著差异。基函数的性能主要受基函数阶数、埋藏深度、作用距离和密度分布等因素的影响。基函数埋藏深度越浅，对重力场高频信号的敏感度越高，尤其是在山区，重力场信号与地形起伏密切相关，需要较多的浅层基函数来捕捉这些高频信号。因此，本文选择基函数埋藏深度分别为10 km和30 km。
Bentel等[3]指出，当格网点均匀分布且数量足够时，格网类型对局部重力场建模结果的影响较小。基于这一观点，本文采用Reuter格网来构造SRBF节点。Reuter格网的等级K与数据点之间的平均距离n之间存在一定的换算关系：
$K=\frac{180^{\circ}}{n}$
格网等级K与基函数的最大展开阶次存在关联，而数据点平均距离为8 km，对应分辨率约为5′，因此后文取最高展开阶数和格网等级K均设为2 160。
为分析参数对计算结果的影响，从最低阶数、埋深、作用距离3个方面组合形成8个参数方案，依次计算残差似大地水准面，结果如图 5所示，精度情况统计结果见表 2。结合表 2中成果标准差和图 5可以看出，作用距离方面，方案a与b、c与d、e与f、g与h，反映了基函数作用距离从50 km调整为100 km，高程异常残差标准差均小幅度增加；埋深方面，方案a与c、b与d、e与g、f与h，反映了埋深从10 km调整为30 km，高程异常残差标准差均有较大增加；最低展开阶数方面，方案a与e、b与f、c与g、d与h，反映了最低阶从120调整为360，高程异常残差标准差均有不同程度的减小。
为客观评价成果精度，分别采用A组和B组GNSS/水准点对计算结果进行精度检核，结果见表 2。如前文所述，这2组GNSS/水准数据在精度和分布方面差异较大，可直观表现检核点对计算结果精度的影响。
从表 2中A组GNSS/水准点的外部检核结果可以看出，当最低展开阶数为120阶时，外检精度整体较差，均在7 cm以上，最大为d组的9.4 cm，精度差异达2.4 cm。随埋深和作用距离的增大，外部检核精度逐渐变差；而当最低展开阶数调整为360阶时，随着埋深和作用距离的增大，检核精度依然保持逐步变差的规律。考虑A组检核点的分布较为广泛且整体精度不佳，检核结果与成果精度的变化基本保持一致。而从B组的检核结果可以看出，相较于A组，B组的外部检核结果均大幅下降，标准差最大值为h组(4.6 cm)，最低为b组(2.7 cm)。该结果不仅与B组的较高的数据精度有关，同时还与B组分布有一定关系。如前所述，B组点位全部位于研究区中部，整体呈线状分布，不同于A组的面状分布。该结果亦说明检核点精度状况影响参数设置的选择，高精度检核点才能准确反映参数规律。
4 结语
收起
本文通过调整移去重力场模型的阶数、径向基函数的埋藏深度、作用距离以及基函数密度等参数，利用2组GNSS/水准点对计算得到的重力似大地水准面进行精度分析，并进一步探讨了地形影响以及GNSS/水准点选择对重力似大地水准面精度的影响。基于本次实验数据，主要结论如下：
1) 移去模型的不同阶数对最终成果精度存在一定影响，最大差异可达14 cm。
2) 考虑地形影响的重力似大地水准面比不考虑地形影响的精度提升了1.9 cm。
3) 通过选择合适的参数，球面径向基函数法能够有效融合多源重力数据来计算重力似大地水准面，同时采用精度较高的GNSS/水准点对成果进行检核，其最优精度可达2.7 cm。
4) GNSS/水准点的精度和分布情况对重力似大地水准面的拟合精度有一定影响。文中2组不同精度的GNSS/水准点的最优检核精度相差4.3 cm，因此在进行检核之前，应先对GNSS/水准点的精度和覆盖情况进行综合评估。

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doi: 10.14075/j.jgg.2025.02.034
  • 接收时间：2025-02-11
  • 首发时间：2026-10-29
  • 出版时间：2026-01-15
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  • 收稿日期：2025-02-11
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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, 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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