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A Direct Approach to Determine the External Disturbing Gravity Field by Applying Green Integral with the Ground Boundary Value
TIAN Jialei , WU Xiaoping, LI Shanshan
Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, ChinaAbstract
First author: TIAN Jialei(1988—), male, PhD candidate, majors in physical geodesy. E-mail： tianjialei3037@163.com
Abstract: By using the ground as the boundary, Molodensky problem usually gets the solution in form of series. Higher order terms reflect the correction between a smooth surface and the ground boundary. Application difficulties arise from not only computational complexity and stability maintenance, but also data-intensiveness. Therefore, in this paper, starting from the application of external gravity disturbance, Green formula is used on digital terrain surface. In the case of ignoring the influence of horizontal component of the integral, the expression formula of external disturbance potential determined by boundary value consisted of ground gravity anomalies and height anomaly difference are obtained, whose kernel function is reciprocal of distance and Poisson core respectively. With this method, there is no need of continuation of ground data. And kernel function is concise, and suitable for the stochastic computation of external disturbing gravity field.
Key words: Green formula     Stokes formula     disturbing gravity     gravity anomaly     boundary value problem

﻿ 1 扰动重力位的地面边值问题

2 地面边值问题的格林公式表示

 图 1 内法线方向示意图 Fig. 1 Schematic diagram of inner normal direction

 图 2 球近似下各元素的几何关系 Fig. 2 Geometric relations among the elements under the condition of the ball approximation

3 基于格网数字地形面的扰动重力位边值问题解式

 图 3 格网数字地形面示意图 Fig. 3 Schematic diagram of gridded digital terrain surface

4 由地面边值确定扰动重力场元

5 试验计算分析

5.1 改进的格林公式与Stokes公式的等效性验证

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 13.429 8 -8.732 1 0.030 3 2.123 1 经度方向分量 0.579 1 0.498 5 0.000 2 0.098 8 纬度方向分量 0.991 6 -1.007 6 0.002 0.208 4

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 0.420 3 -0.532 6 -0.001 3 0.084 2 经度方向分量 0.579 1 -0.501 1 0.000 2 0.098 8 纬度方向分量 0.991 4 -1.007 6 0.002 0 0.208 4

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 1.336 0 -0.761 5 0.002 5 0.210 1 经度方向分量 0.154 8 -0.158 8 0.001 8 0.049 4 纬度方向分量 0.355 2 -0.314 2 0.002 3 0.112 3

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 4.376 2 -8.767 4 -0.008 9 1.311 4 经度方向分量 6.118 2 -6.109 9 -0.002 1.074 5 纬度方向分量 6.243 9 -5.711 4 -0.003 3 1.049 8

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 3.771 4 -7.612 3 -0.005 1 1.109 9 经度方向分量 5.334 9 -5.274 2 0.001 0.937 4 纬度方向分量 5.727 3 -5.283 2 -0.001 7 0.966 5
5.2 实际地形面为边界面下格林积分公式优越性验证

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 18.311 6 -16.160 1 0.032 3 4.178 7 经度方向分量 9.292 7 -10.799 3 0.028 4 2.243 8 纬度方向分量 9.067 -7.655 2 0.019 3 2.312 1

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 4.750 8 -5.586 5 0.005 4 1.333 9 经度方向分量 7.590 3 -8.691 1 0.021 6 1.839 6 纬度方向分量 9.943 1 -11.142 3 0.042 6 2.506 4

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 5.843 2 -7.222 6 0.022 8 1.783 6 经度方向分量 0.918 4 -1.349 4 0.000 7 0.291 6 纬度方向分量 1.298 8 -1.289 7 0.041 5 0.274 1

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 3.514 9 -4.626 1 0.022 7 1.104 2 经度方向分量 1.104 1 -1.056 2 0.006 4 0.212 1 纬度方向分量 1.678 2 -1.384 2 0.037 2 0.334 0

 mGal 扰动重力分量 最大差值 最小差值 平均值 标准差 径向分量 1.739 2 -2.215 7 0.036 5 0.511 7 经度方向分量 0.694 5 -0.762 9 0.001 0.133 3 纬度方向分量 1.673 2 -1.413 4 -0.029 1 0.363 6

6 结 论

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http://dx.doi.org/10.11947/j.AGCS.2015.20140516

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#### 文章信息

TIAN Jialei, WU Xiaoping, LI Shanshan

A Direct Approach to Determine the External Disturbing Gravity Field by Applying Green Integral with the Ground Boundary Value

Acta Geodaeticaet Cartographica Sinica, 2015, 44(11): 1189-1195.
http://dx.doi.org/10.11947/j.AGCS.2015.20140516