﻿ 利用大地水准面模型计算垂线偏差的方法及精度分析
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 大地测量与地球动力学  2019, Vol. 39 Issue (8): 843-848  DOI: 10.14075/j.jgg.2019.08.015

引用本文

LI Weichao, ZHANG Xingfu, CHEN Zhiwei, et al. Method and Accuracy Analysis of Calculating Deflection of the Vertical Based on Geoid Model[J]. Journal of Geodesy and Geodynamics, 2019, 39(8): 843-848.

Foundation support

National Natural Science Foundation of China, No.41674006, 41731069.

Corresponding author

ZHANG Xingfu, PhD, professor, majors in theory, method and application of satellite gravity, E-mail:xfzhang77@163.com.

第一作者简介

LI Weichao, postgraduate, majors in measurement data processing, E-mail:super724085917@163.com.

文章历史

1. 广东工业大学土木与交通工程学院，广州市外环西路100号，510006

1 原理与方法 1.1 由大地水准面模型计算垂线偏差的简便数学模型

 $u = - \frac{{{\rm{d}}N}}{{{\rm{d}}s}}$ (1)

 $u = \xi {\rm{cos}}\alpha + \eta {\rm{sin}}\alpha$ (2)

 $- \frac{{{\rm{d}}N}}{{{\rm{d}}s}} = \xi {\rm{cos}}\alpha + \eta {\rm{sin}}\alpha$ (3)

 $- \frac{{\Delta {N_{AB}}}}{{\Delta {s_{AB}}}} \approx \xi {\rm{cos}}{\alpha _{AB}} + \eta {\rm{sin}}{\alpha _{AB}} = u$ (4)

 $- \frac{{\Delta {h_{AB}} - \Delta {H_{AB}}}}{{\Delta {s_{AB}}}} \approx \xi {\rm{cos}}{\alpha _{AB}} + \eta {\rm{sin}}{\alpha _{AB}}$ (5)

 图 1 利用大地水准面模型计算垂线偏差原理 Fig. 1 The principle of calculating deflection of the vertical by geoid model

 $\left\{ \begin{array}{l} {\xi _1} = - \frac{{\Delta {N_{OA}}}}{{\Delta {s_{OA}}}}\\ {\eta _1} = - \frac{{\Delta {N_{OB}}}}{{\Delta {s_{OB}}}}\\ {\xi _2} = \frac{{\Delta {N_{OC}}}}{{\Delta {s_{OC}}}}\\ {\eta _2} = \frac{{\Delta {N_{OD}}}}{{\Delta {s_{OD}}}} \end{array} \right.$ (6)

 $\left\{ \begin{array}{l} {\xi _{O1}} = \frac{{{\xi _1} + ( - {\xi _2})}}{2}\\ {\eta _{O1}} = \frac{{{\eta _1} + ( - {\eta _2})}}{2} \end{array} \right.$ (7)

 $\left\{ \begin{array}{l} {\xi _3} = \frac{{{u_{OE}}\sin{\alpha _{OF}} - {u_{OF}}\sin{\alpha _{OE}}}}{{\sin\left( {{\alpha _{OF}} - {\alpha _{OE}}} \right)}}\\ {\eta _3} = \frac{{{u_{OE}}\cos{\alpha _{OF}} - {u_{OF}}\cos{\alpha _{OE}}}}{{\sin({\alpha _{OE}} - {\alpha _{OF}})}}\\ {\xi _4} = \frac{{{u_{OG}}\sin{\alpha _{OH}} - {u_{OH}}\sin{\alpha _{OG}}}}{{\sin({\alpha _{OH}} - {\alpha _{OG}})}}\\ {\eta _4} = \frac{{{u_{OG}}\cos{\alpha _{OH}} - {u_{OH}}\cos{\alpha _{OG}}}}{{\sin({\alpha _{OG}} - {\alpha _{OH}})}} \end{array} \right.$ (8)

 $\left\{ \begin{array}{l} {\xi _{O2}} = \frac{{{\xi _3} + {\xi _4}}}{2}\\ {\eta _{O2}} = \frac{{{\eta _3} + {\eta _4}}}{2} \end{array} \right.$ (9)

1.2 精度估计

 $\sigma _u^2 = \frac{1}{{\Delta s_{AB}^2}}(\sigma _{\Delta {N_{AB}}}^2) + {\left( {\frac{{\Delta {N_{AB}}}}{{\Delta s_{AB}^2}}} \right)^2}\sigma _{\Delta {s_{AB}}}^2$ (10)

 $\sigma _u^2 = \frac{1}{{\Delta s_{AB}^2}}(\sigma _{\Delta {N_{AB}}}^2)$ (11)

2 模拟计算

 图 2 模拟计算流程 Fig. 2 Simulation flow chart

3 算例分析 3.1 算例1

3.2 算例2

 图 3 垂线偏差计算值与模型值比较 Fig. 3 Comparison of the deflection of vertical between calculated values and the true values

4 结语

1) 模拟计算结果表明，若想获取优于±1 ″精度的垂线偏差，建议采取方案1，且取点间距为1′，大地水准面模型的相对精度为±10 mm。

2) 基于GEOID12B大地水准面模型，利用方案1计算GSVS2011和GSVS2014两个项目测站点垂线偏差，并与实测结果进行比较。结果表明，2个算例中垂线偏差的精度均优于±0.5″，说明利用相对精度为cm甚至亚cm级的大地水准面模型可获取较高精度的垂线偏差。在实际计算中，取点间距可与大地水准面模型的分辨率一致。

3) 本文模拟计算与实测数据处理结果稍有差异，主要原因可能是模拟计算点的垂线偏差较大，且估计的是单个点的垂线偏差精度；而实测数据处理中测区垂线偏差较小，且评估的是所有测站点垂线偏差的整体精度。

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Method and Accuracy Analysis of Calculating Deflection of the Vertical Based on Geoid Model
LI Weichao1     ZHANG Xingfu1     CHEN Zhiwei1     CUI Jiawu1
1. Faculty of Civil and Transportation Engineering, Guangdong University of Technology, 100 West-Waihuan Road, Guangzhou 510006, China
Abstract: According to the relationship between geoid and deflection of the vertical, a simplified formula for calculating deflection of the vertical using geoid model is given by designing a reasonable calculation scheme. The influence of the relative accuracy of geoid model, the distance between two points, the selection of known points and the number of known points on the calculation results, are discussed by simulating calculation. Using the GEOID12B model, the deflection of the vertical of the stations in the GSVS2011 and GSVS2014 projects and the western region of the United States (40°~45°N, 100°~105°W, resolution is 1') are calculated respectively. The calculated results are compared with the measured values of the GSVS projects and the DEFLEC12B model values. The results show that the calculation accuracy of the north-south component and the east-west component of the deflection of the vertical is better than ±0.5″. It is shown that the geoid models with relative accuracy of cm level or even sub cm level can obtain higher accuracy deflection of the vertical.
Key words: geoid model; deflection of the vertical; simplified formula; accuracy analysis