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Article(id=1322506475793138222, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322506335908901129, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2024.11.533, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1732464000000, receivedDateStr=2024-11-25, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793239714811, onlineDateStr=2026-10-29, pubDate=1760457600000, pubDateStr=2025-10-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793239714811, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793239714811, creator=13701087609, updateTime=1793239714811, updator=13701087609, issue=Issue{id=1322506335908901129, tenantId=1146029695717560320, journalId=1281212831689347082, year='2025', volume='45', issue='10', pageStart='991', pageEnd='1100', issueExtLink='null', onlineDate='null', pubDate='1760457600000', pubDateStr='2025-10-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793239681460, creator='13701087609', updateTime=1793244369709, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext={EN=IssueExt(id=1322525999896576487, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322506335908901129, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1322525999896576488, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322506335908901129, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null, downloadFileDto=null}, startPage=1057, endPage=1064, ext={EN=ArticleExt(id=1322506475956716079, articleId=1322506475793138222, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Estimation of Grid Search Algorithm of Seamount Morphology Using Vertical Gravity Gradient Anomalies, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Seamounts are previously approximated as an axisymmetric Gaussian shape, and the vertical gravity gradient(VGG) anomalies combined with ship-track bathymetry data are used to evaluate seamount morphology. However, there are significant challenges in describing the complex morphology of seamounts. Therefore, we classify seamounts into two distinct types: conical and elliptical conical shapes, which can be determined by three and five parameters, respectively. We perform a grid search on seamount parameters, obtaining VGG anomalies corresponding to each parameter through forward calculation. The forward VGG anomalies are compared with the observed VGG anomalies, and the minimal root mean square(RMS) error between them is used as a search criterion for estimating. We validate the effectiveness of grid search algorithm through numerical simulations, and apply it to practical examples. The estimation results are compared with multi-beam bathymetry data, the RMS errors ranging from 150 m to 640 m, and the fitting of VGG anomalies is within 8 Eötvös.

, authors=Huan XU1, Yuwei TIAN1, Jinhai YU1, Zhongmiao SUN2, *, He TANG1, authorsList=Huan XU, Yuwei TIAN, Jinhai YU, Zhongmiao SUN, He TANG, authorCompany=null, correspAuthors=Zhongmiao SUN, authorNote=

XU Huan, PhD candidate, majors in gravity field refinement calculation and altimetry gravity application, E-mail:

, 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=1322506479295382080, articleId=1322506475793138222, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=利用垂直重力梯度异常评估海山形态的格网搜索算法, columnId=1322506242858271689, journalTitle=大地测量与地球动力学, columnName=地球物理解释, runingTitle=null, highlight=null, articleAbstract=

以往研究将海山近似为轴对称的高斯形，利用垂直重力梯度(VGG)异常结合船测深度数据来评估海山形态，但在描述复杂海山形态时存在精度损失。本文将海山近似成轴对称的圆锥形和椭圆锥形，分别采用3个和5个参数进行建模。通过对参数进行不同分辨率的格网搜索，并正演计算各组参数对应海山生成的VGG异常，将正演VGG异常与观测VGG异常进行比较，以两者RMS误差最小为搜索准则来评估海山形态。通过数值模拟验证格网搜索算法的有效性，并将算法应用于实际算例。评估结果与多波束测深数据相比，RMS误差在150~640 m之间，VGG异常的拟合RMS误差在8 Eötvös以内。

, authors=徐焕1, 田彧玮1, 于锦海1, 孙中苗2, *, 唐河1, authorsList=徐焕, 田彧玮, 于锦海, 孙中苗, 唐河, authorCompany=null, correspAuthors=孙中苗, authorNote=

徐焕，博士生，主要从事重力场精化计算和测高重力应用研究，E-mail:

, correspAuthorsNote=

孙中苗，博士，研究员，主要从事大地测量与导航定位研究，E-mail:

SUN Zhongmiao, PhD, professor, majors in geodesy and navigation and positioning, E-mail:

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XU Huan, PhD candidate, majors in gravity field refinement calculation and altimetry gravity application, E-mail:

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徐焕，博士生，主要从事重力场精化计算和测高重力应用研究，E-mail:

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Statistics of search parameters of conical and elliptical conical seamounts

, figureFileSmall=null, figureFileBig=null, tableContent=
h/km d/km R/km RMS/m RMSVGG/Eötvös
CS-1 3.3 2.0 14.4 330.1 4.1
CS-2 2.2 2.0 5.2 155.8 6.6
h/km d/km S/km L/km RMS/m RMSVGG /Eötvös
ES-1 4.0 2.0 15.2 18.1 542.9 6.6
ES-2 2.0 3.0 26.0 35.0 632.1 7.4
), ArticleFig(id=1322506485049967229, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506475793138222, language=CN, label=表1, caption=

圆锥形和椭圆锥形海山的搜索参数统计

, figureFileSmall=null, figureFileBig=null, tableContent=
h/km d/km R/km RMS/m RMSVGG/Eötvös
CS-1 3.3 2.0 14.4 330.1 4.1
CS-2 2.2 2.0 5.2 155.8 6.6
h/km d/km S/km L/km RMS/m RMSVGG /Eötvös
ES-1 4.0 2.0 15.2 18.1 542.9 6.6
ES-2 2.0 3.0 26.0 35.0 632.1 7.4
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利用垂直重力梯度异常评估海山形态的格网搜索算法
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徐焕 1 , 田彧玮 1 , 于锦海 1 , 孙中苗 2, * , 唐河 1
大地测量与地球动力学 | 地球物理解释 2025,45(10): 1057-1064
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大地测量与地球动力学 |地球物理解释 2025 , 45 (10) : 1057 -1064
利用垂直重力梯度异常评估海山形态的格网搜索算法
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徐焕1 , 田彧玮1, 于锦海1, 孙中苗2, * , 唐河1
作者信息
  • 1 中国科学院大学地球与行星科学学院，北京市玉泉路19号甲, 100049
  • 2 地理信息工程国家重点实验室，西安市雁塔路中段1号, 710054
通讯作者:

孙中苗，博士，研究员，主要从事大地测量与导航定位研究，E-mail:

SUN Zhongmiao, PhD, professor, majors in geodesy and navigation and positioning, E-mail:

作者简介:

徐焕，博士生，主要从事重力场精化计算和测高重力应用研究，E-mail:

Estimation of Grid Search Algorithm of Seamount Morphology Using Vertical Gravity Gradient Anomalies
Huan XU1 , Yuwei TIAN1, Jinhai YU1, Zhongmiao SUN2, * , He TANG1
Affiliations
  • 1 College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, A19 Yuquan Road, Beijing 100049, China
  • 2 State Key Laboratory of Geo-Information Engineering, 1 Mid-Yanta Road, Xi'an 710054, China

    • XU Huan, PhD candidate, majors in gravity field refinement calculation and altimetry gravity application, E-mail:

出版时间: 2025-10-15 doi: 10.14075/j.jgg.2024.11.533
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摘要
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以往研究将海山近似为轴对称的高斯形，利用垂直重力梯度(VGG)异常结合船测深度数据来评估海山形态，但在描述复杂海山形态时存在精度损失。本文将海山近似成轴对称的圆锥形和椭圆锥形，分别采用3个和5个参数进行建模。通过对参数进行不同分辨率的格网搜索，并正演计算各组参数对应海山生成的VGG异常，将正演VGG异常与观测VGG异常进行比较，以两者RMS误差最小为搜索准则来评估海山形态。通过数值模拟验证格网搜索算法的有效性，并将算法应用于实际算例。评估结果与多波束测深数据相比，RMS误差在150~640 m之间，VGG异常的拟合RMS误差在8 Eötvös以内。

垂直重力梯度  /  海山形态  /  格网搜索  /  数值模拟

Seamounts are previously approximated as an axisymmetric Gaussian shape, and the vertical gravity gradient(VGG) anomalies combined with ship-track bathymetry data are used to evaluate seamount morphology. However, there are significant challenges in describing the complex morphology of seamounts. Therefore, we classify seamounts into two distinct types: conical and elliptical conical shapes, which can be determined by three and five parameters, respectively. We perform a grid search on seamount parameters, obtaining VGG anomalies corresponding to each parameter through forward calculation. The forward VGG anomalies are compared with the observed VGG anomalies, and the minimal root mean square(RMS) error between them is used as a search criterion for estimating. We validate the effectiveness of grid search algorithm through numerical simulations, and apply it to practical examples. The estimation results are compared with multi-beam bathymetry data, the RMS errors ranging from 150 m to 640 m, and the fitting of VGG anomalies is within 8 Eötvös.

vertical gravity gradient  /  seamount morphology  /  grid search  /  numerical simulation
徐焕, 田彧玮, 于锦海, 孙中苗, 唐河. 利用垂直重力梯度异常评估海山形态的格网搜索算法. 大地测量与地球动力学, 2025 , 45 (10) : 1057 -1064 . DOI: 10.14075/j.jgg.2024.11.533
Huan XU, Yuwei TIAN, Jinhai YU, Zhongmiao SUN, He TANG. Estimation of Grid Search Algorithm of Seamount Morphology Using Vertical Gravity Gradient Anomalies[J]. Journal of Geodesy and Geodynamics, 2025 , 45 (10) : 1057 -1064 . DOI: 10.14075/j.jgg.2024.11.533
正文
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通过精化海洋重力异常和丰富船测深度数据可以改进海底地形模型，但在特定海域，现有地形预测算法效果不佳。如在近岸和岛礁等浅水海域，由于测高卫星的工作原理，这些海域的重力异常精度偏低，导致地形预测结果精度不足。利用光学遥感数据预测岛礁地形可有效克服该局限[1-2]，进一步联合光学遥感和测高重力数据来精化岛礁海域地形[3]。但对于地形起伏较大，特别是存在海山的海域[4]，地形预测精度仍存在较大偏差。
海山分布对海底航行、矿产资源分布和火山活动等具有重要影响[5-6]。随着海洋观测数据的持续改善，全球海山分布也在不断精化和更新[7-13]。早期主要依靠卫星测高获取的大地水准面高数据来评估海山，对海山进行定位并分析其高度[13]。随着数据的丰富，Baudry等[14]利用高密度测高数据来评估海山形态。此后，部分学者利用船测深度数据研究海山特性，将海山形状近似为截断的直角圆锥，并统计分析海山各参数间的关系[15-16]。Smith[17]设定海山为高斯形，利用单颗测高卫星获取的剖线来评估海山特征。在高斯形海山设定的基础上，部分学者结合船测深度和海洋重力数据定量分析海山高度、底部半径与距海面距离的对应关系[5, 18]。然而，真实的海山存在更强的复杂性，高斯形海山无法准确捕捉其真实形态。
基于此，本文进一步将海山近似成轴对称的圆锥形和椭圆锥形，2种海山分别可用3个和5个参数进行描述。对参数进行不同分辨率的格网搜索，并正演计算各组参数对应海山生成的VGG异常，统计正演VGG异常与观测VGG异常，以两者RMS误差最小为搜索准则来评估海山形态。最后将算法应用于实际算例，分别选取2个类圆锥形和2个类椭圆锥形海山进行评估，并与多波束测深数据进行比较以检验算法的有效性。
1 理论与方法
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1.1 海山形态建模
本文将海山简化成图 1中2种形态，图 1(a)为圆锥形海山示意图，其由海山高度h、海山顶部到海面距离d及海山半径R三个参数组{h, d, R}确定；图 1(b)为椭圆锥形海山示意图，其由海山高度h、海山顶部到海面距离d、海山方位角θ、底面半长轴长S和半短轴长L五个参数组{h, d, θ, S, L}确定。确定圆锥形和椭圆锥形海山对应的参数组，即完成对海山形态的评估。本文利用海面观测的VGG异常对海山进行评估，需要计算海山Ω在海面上任意一点P(ζ, ξ, ψ)生成的VGG异常。
建立图 1所示的局部坐标系，则海山Ω在点P生成的重力位V(P)可表示为：
$G \rho \iiint_{\varOmega} \frac{1}{\sqrt{(x-\zeta)+(y-\xi)^2+(z-\varPsi)^2}} \mathrm{~d} \varOmega$
求V(P)沿Z轴方向的二阶导数，并转化到极坐标下：
$V_{z z}(P)=\frac{\partial^2 V(P)}{\partial z^2}=G \rho \int_0^{2 \mathsf{π}} \int_0^h \int_0^{R_1} \frac{r_{\varOmega}\left[r_{\varOmega}^2+r_p^2-2(h+d-z)^2-2 r_{\varOmega} r_p \cos \left(\theta_{r_{\varOmega}}-\theta_{r_p}\right)\right]}{\left[r_{\varOmega}^2+r_p^2+(h+d-z)^2-2 r_{\varOmega} r_p \cos \left(\theta_{r_{\varOmega}}-\theta_{r_p}\right)\right]^{5 / 2}} \mathrm{~d} r_{\varOmega} \mathrm{d} z \mathrm{~d} \theta_{r_{\varOmega}}$
当海山为圆锥形时，式(2)积分上限$R_1=R\left(1-\frac{z}{h}\right)$；当为椭圆锥形时，$R_1=\frac{(h-z)}{h}\cdot\sqrt{L^2 \cos ^2 \theta_{r_{\varOmega}}+S^2 \sin ^2 \theta_{r_{\varOmega}}}$。若将式(2)中密度ρ改成海底基岩与海水的密度差Δρ=ρs－ρw，即可算出海山在海平面上生成的VGG异常。本文将基岩和海水密度分别设置为2.8 g/cm3和1.03 g/cm3。
1.2 格网搜索算法
格网搜索是通过对VGG异常观测值进行搜索来评估海山形态，搜索标准为海山在海面上正演生成的VGG异常与观测VGG异常间的RMS误差最小，计算公式为：
$\mathrm{RMS}_{\mathrm{VGG}}=\sqrt{\frac{\sum\limits_{i=1}^n\left(\mathrm{VGG}_i-\mathrm{VGG}_i^0\right)^2}{n}}$
式中，VGGi0为沿剖线采样的VGG异常观测值；VGGi为搜索的海山正演生成的VGG异常。格网搜索步骤如下：
1) 当为圆锥形海山时，确定VGG异常观测值最大点的坐标(xLon, yLat)。将纬度yLat固定并沿经度方向以1 km分辨率对VGG异常观测值进行采样，记为VGGc0(图 1(a))。
2) 当为椭圆锥形海山时，先采样VGG异常观测值的等高线，并进行椭圆拟合，计算得到椭圆方位角θ，然后以VGG异常观测值最大点为起点，沿长轴方向进行1 km分辨率采样得到采样剖线，记为VGGe0(图 1(b))。
3) 根据2个特殊值(VGG异常为0时距坐标原点的距离和VGG异常的最大值)，结合先验数据来缩小海山参数组{h, d, R}的搜索范围。对圆锥形海山的参数组{h, d, R}进行1 km格网搜索，即每个参数以1 km递增，正演得到其沿采样剖线生成的VGG异常，记为VGGc。
4) 利用式(3)计算各组参数{h, d, R}对应的圆锥形海山正演生成的VGGc与观测VGGc0之间的RMSVGG，并对RMSVGG进行排序，选取前30个最小RMSVGG对应的圆锥形海山的参数组{h, d, R}1。
5) 对步骤4)中得到的参数组{h, d, R}1进行0.1 km格网搜索，并正演其沿采样剖线生成的VGG异常；重复步骤4)，将最小RMSVGG对应的参数组{h, d, R}0.1作为最终海山形态结果。对于椭圆锥形海山，由于对等高线进行拟合可确定海山方位角θ，故只需搜索其他4个参数组成的参数组{h, d, S, L}，按照步骤3)~5)操作即可。
2 数值模拟
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2.1 圆锥形海山评估
模拟3个不同大小的圆锥形海山(图 2(a)~2(c))，模拟海山的精度分别为1 km、0.1 km和0.01 km。利用式(2)正演计算3个海山在海面上方生成的VGG异常，结果如图 2(d)~2(f)所示，图中紫色虚线表示VGG异常的1 km采样剖线，对应图 3(b)~3(d)中红圈点线。结果表明，3个海山对应的特殊值组分别为9.0 km/242.6 Eötvös、10.9 km/156.2 Eötvös和13.1 km/184.6 Eötvös，以此来缩小参数组的搜索范围。
通过正演计算获取先验数据，设定不同圆锥形海山的参数h、d和R，其中h和d范围都为1~5 km，并以1 km步长递增；R范围为2~18 km，并以2 km步长递增，利用式(2)计算并建立不同海山参数对应的先验数据。以海山S-1的特殊值组9.0 km/242.6 Eötvös为例，要满足VGG异常最大值为242.6 Eötvös，对应的R应在6~10 km之间；VGG异常为0时距离中心约9.0 km，其对应的h在2~5 km之间，d在1~2 km之间。据此可缩小海山S-1参数组{h, d, R}的变化范围，从而大大节省搜索时间。
以海山S-1正演生成的VGG异常为例(图 2(d))，对任意一条等高线进行采样(图 3(a)红圈)并进行圆形拟合(图 3(a)蓝线)。可以看出，采样点和拟合曲线完全重合，且拟合圆心即为VGG异常的最大值点。以VGG异常的最大值点为起点，沿Y=0进行1 km分辨率采样，将采样值作为搜索的VGG异常观测值(图 3(b)红圈点线)。由于海山S-1的参数组{h, d, R}精度为1 km，故采用1 km分辨率进行格网搜索即可确定参数组{h, d, R}，拟合的VGG异常如图 3(b)蓝色点线所示。拟合的VGG异常与VGG异常观测值完全重合，搜索的参数组也与模拟的真实参数组完全一致，即搜索结果无误差。
海山S-2和S-3分别进行3次和4次格网搜索，对应的拟合VGG异常曲线如图 3(c)~3(d)所示。对于海山S-3，分别对参数组进行1 km、0.5 km、0.1 km和0.01 km精度的格网搜索。前3次格网搜索对应VGG异常的RMS误差分别为2.83 Eötvös、2.33 Eötvös和0.24 Eötvös，0.1 km精度格网搜索的参数组{h, d, R}为{2.7 km, 2.1 km, 14.3 km}。搜索结果与真值仅在0.01 km精度上存在差异，该参数精度误差造成VGG异常的RMS误差仅为0.24 Eötvös。由于目前10-1 Eötvös量级的VGG异常误差是难以实现的，故在实际算例中将格网搜索的精度设定为0.1 km。
2.2 椭圆锥形海山形态评估
椭圆锥形海山模拟结果如图 4(a)所示，对应的4个参数{h, d, S, L}为{3.3 km, 1.4 km, 5.2 km, 10.1 km}，方位角设为45°(椭圆长轴与X轴的夹角)。利用式(2)正演计算其在海面上方生成的VGG异常，结果如图 4(b)所示，图中紫色虚线为沿着长半轴和短半轴采样的VGG异常的1 km分辨率剖线。对比沿短半轴的采样VGG异常可知，沿长半轴采样的VGG异常具有更大量级且衰减速度较慢，表明其具有更优的信噪比，因此在模拟实验和实算中使用长半轴采样的VGG异常。选取椭圆锥形海山正演生成的VGG异常(图 4(c)红圈)，并进行椭圆拟合(图 4(c)蓝线)，图中紫色空星点和实星点分别为拟合的椭圆中心点和VGG异常最大值点，橙色和绿色星线分别为长轴和短轴端点，利用长轴端点即可计算椭圆锥形方位角从而确定海山方位角。对椭圆锥形海山的参数组{h, d, S, L}进行1 km、0.5 km、0.1 km精度格网搜索，搜索结果对应的VGG异常值如图 4(d)所示。由于海山参数的模拟精度为0.1 km，采用0.1 km精度格网搜索即可确定海山参数组的真值。
3 算例分析
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3.1 圆锥形海山
选取2个类圆锥形海山来验证格网搜索算法的有效性。VGG异常来自SIO公布的1′×1′格网数据，该异常模型结合了最新的卫星测高数据，可有效降低噪声影响[19]。选取的2个类圆锥形海山(CS-1和CS-2)的VGG异常分布如图 5(a)和5(b)所示，以VGG异常最大值为起点，沿纬度进行1 km采样，采样值如图 5(c)和5(d)中红圈所示。搜索的参数组统计见表 1，海山CS-1和CS-2参数组{h, d, R}的搜索结果分别为{3.3 km, 2.0 km, 14.4 km}和{2.2 km, 2.0 km, 5.2 km}，二维平面图如图 6(a)和6(b)所示。
多波束测深数据来自NOAA，具体见图 6(c)和6(d)。由表 1可知，2个海山评估的最大深度分别为-5 300 m和-4 200 m，而对应的多波束测深数据的最大深度分别为-5 393 m和-4 359 m，表明评估的最大深度与多波束测深数据基本一致。两者最浅深度分别相差1 071 m和764 m，即评估结果的最浅深度偏大，这也导致低估了海山高度。由于当前1′×1′分辨率重力场模型是插值且平滑后的结果，即实际分辨率更低。插值后的重力场模型会削弱对海山的识别能力，同时平滑效果使得重力场的宽度大于海山真实宽度，从而高估海山底部尺寸和低估海山真实高度[12]。
海山评估结果与多波束测深数据的误差(多波束测深减去评估结果)分布如图 6(e)和6(f)所示。可以看出，评估结果的最大误差均出现在海山顶部附近，其中最大正误差分别为1 320 m和899 m；而最大负误差出现在远离海山中心处，即评估结果低估了海山最大深度。2个海山评估结果的RMS误差分别为330.1 m和155.8 m，但拟合VGG异常的RMS误差分别为4.1 Eötvös和6.6 Eötvös，表明VGG异常拟合效果无法直接反映最终的评估结果。从测深数据来看，海山CS-2的等高线相较于海山CS-1更加平滑且近似为圆形，表明其评估结果相对更优。
3.2 椭圆锥形海山
选取2个类椭圆锥形海山对应的VGG异常模型(图 7(a)和7(b))。由于VGG异常模型并非标准椭圆，即每条等高线的长轴方向存在偏差，这会影响海山方位角的确定。因此，选取2个海山VGG异常观测值的4条等高线进行椭圆拟合，计算拟合椭圆的4条长轴方位角并求平均值，分别将4.7°和22.8°作为海山的方位角。沿长轴方向对VGG异常观测值进行1 km分辨率采样(图 7(c)和7(d)中红圈)，搜索结果对应的VGG异常如图 7(c)和7(d)中蓝色点线所示。搜索结果统计见表 1，海山ES-2的长轴和短轴明显大于海山ES-1，这与2个海山VGG异常观测值的尺寸相对应。
利用搜索的参数组及确定的海山方位角绘制二维平面图(图 8(a)和8(b))，对应海域的多波束测深数据如图 8(c)和8(d)所示，两者的差异分布如图 8(e)和8(f)所示。可以看出，海山ES-1的最大和最小深度与多波束测深数据基本一致，但高估海山ES-2的最浅深度约1 000 m。海山ES-1的较大负误差主要为图 8(e)中蓝点误差，最大负误差为-1 657 m。相比于图 8(c)多波束测深数据，其明显深于附近的其他多波束测深结果，故有理由怀疑此处的测深数据存在一定误差。而海山ES-2的误差分布表现为山顶附近正误差、远离山顶逐渐变为负误差。这表明高估了海山的最浅深度而低估了最大深度。
2个椭圆锥形海山的RMS误差分别为542.9 m和632.1 m，明显大于2个圆锥形海山。图 7(c)和7(d)中红圈和蓝色点线分别为VGG异常观测值采样点和评估海山正演生成的VGG异常。可以看出，在距离海山顶部5 km范围内，评估海山正演生成的VGG异常结果明显大于VGG异常观测值，且最大差异分别为21.6 Eötvös和22.3 Eötvös。统计的RMS误差分别为6.6 Eötvös和7.4 Eötvös，相较于圆锥形海山的RMS误差4.1 Eötvös和6.6 Eötvös，两者仅相差1~3 Eötvös，而评估海山的RMS误差却在300~400 m之间。这表明VGG异常的RMS误差仅在一定程度上反映了评估结果精度，更重要的是海山形状对结果精度的影响。
4 结语
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本文将海山形态进行参数化，并利用VGG异常来搜索海山形态。通过数值模拟和实际算例得到以下认识：
1) VGG异常对纵向的质量分布更加敏感[12]，可更好地反映地形起伏，但会在一定程度上放大高频误差。从数值模拟实验可知，VGG异常拟合的RMS误差在4 Eötvös以内，对应海山参数组评估精度可达100 m；而本文实例的RMS误差都超过4 Eötvös，最终海山评估的RMS误差也大于100 m，即VGG数据的革新将进一步提高海山评估精度。
2) 除海山ES-1的高度被高估了166 m外，其余3个海山的高度都被低估超过1 000 m，出现低估的原因主要为重力异常的宽度远大于海山的实际宽度。4个海山评估结果的RMS误差受海山形状的影响较大，对于具有其他不规则形状的海山，特别是当存在多个不同大小的重叠海山时，会出现更大的估计误差。因此，海山参数的设置和优化是未来研究的重点。
3) 一般的地形评估算法获取的是格网化地形产品，因此网格分辨率将影响最终结果精度[20]。海山评估方法是通过估计海山的参数来确定海山的形状或形态的，可有效减小网格造成的误差。后续考虑使用“去除-恢复”方法来有效提高海山海域地形模型的准确性。
基金
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  • 国家自然科学基金(42274010)
  • 海军研究院项目(2022000573)
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2025年第45卷第10期
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doi: 10.14075/j.jgg.2024.11.533
  • 接收时间：2024-11-25
  • 首发时间：2026-10-29
  • 出版时间：2025-10-15
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  • 收稿日期：2024-11-25
基金
National Natural Science Foundation of China(42274010)
国家自然科学基金(42274010)
Project of Naval Research Institute(2022000573)
海军研究院项目(2022000573)
作者信息
    1 中国科学院大学地球与行星科学学院，北京市玉泉路19号甲, 100049
    2 地理信息工程国家重点实验室，西安市雁塔路中段1号, 710054

通讯作者:

孙中苗，博士，研究员，主要从事大地测量与导航定位研究，E-mail:

SUN Zhongmiao, PhD, professor, majors in geodesy and navigation and positioning, 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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