﻿ 有限圆柱体重力位三阶梯度张量正演计算公式
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 大地测量与地球动力学  2022, Vol. 42 Issue (1): 96-99  DOI: 10.14075/j.jgg.2022.01.018

### 引用本文

LENG Jie, QIU Feng. Forward Modeling Formulae for Third-Order Gradient Tensor of Gravitational Potential Caused by Finite Cylinder[J]. Journal of Geodesy and Geodynamics, 2022, 42(1): 96-99.

### 第一作者简介

LENG Jie, lecturer, majors in underwater acoustic engineering, underwater target detection and equipment application, E-mail: 2118573864@qq.com.

### 文章历史

1. 海军航空大学青岛校区，青岛市，266041;
2. 大连测控技术研究所，大连市滨海街16号，116013

1 公式推导

 $U=G\rho \iiint\limits_{V}{\frac{1}{r}\text{d}\xi }\text{d}\eta \text{d}\zeta =G\rho S\int_{{{y}_{0}}-L}^{{{y}_{0}}+L}{\frac{1}{r}}\text{d}\eta$ (1)
 图 1 圆柱体几何示意 Fig. 1 Cylinder geometry schematic

 \left\{ \begin{align} & \int{\frac{\text{d}u}{\sqrt{{{u}^{2}}+{{A}^{2}}}}}=\ln (u+\sqrt{{{u}^{2}}+{{A}^{2}}})+C \\ & \int{\frac{\text{d}u}{{{({{u}^{2}}+{{A}^{2}})}^{\frac{3}{2}}}}}=\frac{u}{{{A}^{2}}\sqrt{{{u}^{2}}+{{A}^{2}}}}+C \\ & \int{\frac{u\text{d}u}{{{({{u}^{2}}+{{A}^{2}})}^{\frac{3}{2}}}}}=-\frac{1}{\sqrt{{{u}^{2}}+{{A}^{2}}}}+C \\ \end{align} \right. (2)

ξ-x=aη-y=bζ-z=c

 $\left[ \begin{matrix} {{U}_{xx}} & {{U}_{xy}} & {{U}_{xz}} \\ {{U}_{yx}} & {{U}_{yy}} & {{U}_{yz}} \\ {{U}_{zx}} & {{U}_{zy}} & {{U}_{zz}} \\ \end{matrix} \right]=\left[ \begin{matrix} \frac{{{\partial }^{2}}U}{\partial {{x}^{2}}} & \frac{{{\partial }^{2}}U}{\partial x\partial y} & \frac{{{\partial }^{2}}U}{\partial x\partial z} \\ \frac{{{\partial }^{2}}U}{\partial y\partial x} & \frac{{{\partial }^{2}}U}{\partial {{y}^{2}}} & \frac{{{\partial }^{2}}U}{\partial y\partial z} \\ \frac{{{\partial }^{2}}U}{\partial z\partial x} & \frac{{{\partial }^{2}}U}{\partial z\partial y} & \frac{{{\partial }^{2}}U}{\partial {{z}^{2}}} \\ \end{matrix} \right]$ (3)

 \begin{align} & \ \ \ \ \ \ \ \ \ \ \ \ {{U}_{xx}}=G\rho S\cdot \\ & \frac{b[{{a}^{2}}({{a}^{2}}+{{c}^{2}})+{{r}^{2}}({{a}^{2}}-{{c}^{2}})]}{{{r}^{3}}{{({{a}^{2}}+{{c}^{2}})}^{2}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right. \\ \end{align} (4a)
 ${{U}_{xy}}=-G\rho S\frac{a}{{{r}^{3}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (4b)
 ${{U}_{xz}}=G\rho S\frac{abc(3{{r}^{2}}-{{b}^{2}})}{{{r}^{3}}{{[{{a}^{2}}+{{c}^{2}}]}^{2}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (4c)
 ${{U}_{yy}}=\frac{\partial {{V}_{y}}}{\partial y}=-G\rho S\frac{b}{{{r}^{3}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (4d)
 ${{U}_{yz}}=\frac{\partial {{V}_{y}}}{\partial z}=-G\rho S\frac{c}{{{r}^{3}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (4e)
 \begin{align} & \ \ \ \ \ \ \ \ \ \ \ \ \ {{U}_{zz}}=G\rho S\cdot \\ & \frac{b[{{c}^{2}}({{c}^{2}}+{{a}^{2}})+{{r}^{2}}({{c}^{2}}-{{a}^{2}})]}{{{r}^{3}}{{({{a}^{2}}+{{c}^{2}})}^{2}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right. \\ \end{align} (4f)

 $\frac{\partial r}{\partial x}=-\frac{\xi-x}{r}, \frac{\partial r}{\partial y}=-\frac{\eta-y}{r}, \frac{\partial r}{\partial z}=-\frac{\zeta-z}{r}$ (5)

 \begin{align} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{U}_{xxx}}=\frac{\partial {{U}_{xx}}}{\partial x} \\ & =G\rho S\cdot ab\frac{[{{a}^{2}}({{a}^{2}}+{{c}^{2}})+{{r}^{2}}({{a}^{2}}-{{c}^{2}})][3({{a}^{2}}+{{c}^{2}})+4{{r}^{2}}]-{{r}^{2}}(6{{a}^{2}}+2{{r}^{2}})({{a}^{2}}+{{c}^{2}})}{{{r}^{5}}{{({{a}^{2}}+{{c}^{2}})}^{3}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right. \\ \end{align} (6a)
 ${{U}_{xxy}}=\frac{\partial {{U}_{xx}}}{\partial y}=G\rho S\frac{{{r}^{2}}-3{{a}^{2}}}{{{r}^{5}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (6b)
 ${{U}_{xxz}}=\frac{\partial {{U}_{xx}}}{\partial z}=G\rho S\cdot bc\frac{(-3{{r}^{2}}+{{b}^{2}}-6{{a}^{2}}){{r}^{2}}({{a}^{2}}+{{c}^{2}})+{{a}^{2}}(3{{r}^{2}}-{{b}^{2}})[3({{a}^{2}}+{{c}^{2}})+4{{r}^{2}}]}{{{r}^{5}}{{[{{a}^{2}}+{{c}^{2}}]}^{3}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (6c)
 ${{U}_{xyy}}=\frac{\partial {{U}_{xy}}}{\partial y}=-G\rho S\frac{3ab}{{{r}^{5}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (6d)
 ${{U}_{xyz}}=\frac{\partial {{U}_{xy}}}{\partial z}=-G\rho S\frac{3ac}{{{r}^{5}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (6e)
 ${{U}_{xzz}}=\frac{\partial {{U}_{xz}}}{\partial z}=G\rho S\cdot ab\frac{(-3{{r}^{2}}+{{b}^{2}}-6{{c}^{2}}){{r}^{2}}({{a}^{2}}+{{c}^{2}})+{{c}^{2}}(3{{r}^{2}}-{{b}^{2}})[3({{a}^{2}}+{{c}^{2}})+4{{r}^{2}}]}{{{r}^{5}}{{[{{a}^{2}}+{{c}^{2}}]}^{3}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (6f)
 ${{U}_{yyy}}=\frac{\partial {{U}_{yy}}}{\partial y}=G\rho S\frac{{{r}^{2}}-3{{b}^{2}}}{{{r}^{5}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (6g)
 ${{U}_{yyz}}=\frac{\partial {{U}_{yy}}}{\partial z}=-G\rho S\frac{3bc}{{{r}^{5}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (6h)
 ${{U}_{yzz}}=\frac{\partial {{U}_{yz}}}{\partial z}=G\rho S\frac{{{r}^{2}}-3{{c}^{2}}}{{{r}^{5}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right.$ (6i)
 \begin{align} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{U}_{zzz}}=\frac{\partial {{U}_{zz}}}{\partial z}= \\ & G\rho S\cdot bc\frac{[{{c}^{2}}({{a}^{2}}+{{c}^{2}})+{{r}^{2}}({{c}^{2}}-{{a}^{2}})][3({{a}^{2}}+{{c}^{2}})+4{{r}^{2}}]-{{r}^{2}}(6{{c}^{2}}+2{{r}^{2}})({{a}^{2}}+{{c}^{2}})}{{{r}^{5}}{{({{a}^{2}}+{{c}^{2}})}^{3}}}\left| \begin{matrix} {{y}_{0}}+L \\ {{y}_{0}}-L \\ \end{matrix} \right. \\ \end{align} (6j)
2 正演计算公式的正确性检验

2.1 理论模型的重力三阶梯度张量分布

 黑色矩形为有限圆柱体模型的水平位置;白色实线为图 3剖面的水平位置;1 pMKS=10-12/ms2 图 2 有限圆柱体的重力三阶梯度张量分布 Fig. 2 Distribution of third-order gradient tensor of the gravitational potential caused by a cylinde
2.2 解析解的正确性检验

 图 3 解析表达式与中心差分的计算结果对比 Fig. 3 Comparison between the calculating results by analytic expression and central difference

 图 4 解析表达式与中心差分的计算结果误差 Fig. 4 The calculating error results by analytic expression and central difference

 图 5 重力位三阶梯度张量部分元素相加结果 Fig. 5 The partial element addition results of third-order gradient tensor
3 结语

 [1] Li Y G, Oldenburg D W. 3-D Inversion of Gravity Data[J]. Geophysics, 1998, 63(1): 109-119 DOI:10.1190/1.1444302 (0) [2] Rim H, Li Y G. Gravity Gradient Tensor due to a Cylinder[J]. Geophysics, 2016, 81(4): G59-G66 DOI:10.1190/geo2015-0699.1 (0) [3] Na S H, Rim H, Shin Y H, et al. Calculation of Gravity due to a Vertical Cylinder Using a Spherical Harmonic Series and Numerical Integration[J]. Exploration Geophysics, 2015, 46(4): 381-386 DOI:10.1071/EG14123 (0) [4] Su Y J, Cheng L Z, Chouteau M, et al. New Improved Formulas for Calculating Gravity Anomalies Based on a Cylinder Model[J]. Journal of Applied Geophysics, 2012, 86: 36-43 DOI:10.1016/j.jappgeo.2012.07.001 (0) [5] Rao V B, Murty B V S. Gravity Anomalies over Long Semi-Circular Cylinders[J]. Geoexploration, 1971, 9(4): 207-212 DOI:10.1016/0016-7142(71)90038-X (0) [6] Murthy I V R, Swamy K V. Gravity Anomalies of a Vertical Cylinder of Polygonal Cross-Section and Their Inversion[J]. Computers and Geosciences, 1996, 22(6): 625-630 DOI:10.1016/0098-3004(95)00126-3 (0) [7] 曾华霖. 重力场与重力勘探[M]. 北京: 地质出版社, 2005 (Zeng Hualin. Gravity Field and Gravity Exploration[M]. Beijing: Geological Publishing House, 2005) (0) [8] 邱峰, 杜劲松, 陈超. 矩形棱柱体重力位三阶梯度张量正演计算公式[J]. 大地测量与地球动力学, 2019, 39(3): 313-316 (Qiu Feng, Du Jinsong, Chen Chao. Forward Modeling Formulae for Third-Order Gradient Tensor of Gravitational Potential Caused by the Right Rectangular Prism[J]. Journal of Geodesy and Geodynamics, 2019, 39(3): 313-316) (0)
Forward Modeling Formulae for Third-Order Gradient Tensor of Gravitational Potential Caused by Finite Cylinder
LENG Jie1     QIU Feng2
1. Naval Aviation University Qingdao Campus, Qingdao 266041, China;
2. Dalian Scientific Test and Control Technology Institute, 16 Binhai Street, Dalian116013, China
Abstract: Firstly, we derive the forward modeling formulae for third-order gradient tensor of gravitational potential caused by the finite cylinder. Then, the model experiment is carried out based on the theoretical formula, and two different approaches are utilized to verify the obtained analytic expressions. The first compares the results calculated by our proposed formulae and the central difference of gradient tensor of the gravitational potential. The second determines whether the sum of the components is zero acording to the Laplace equation.
Key words: third-order gradient tensor; finite cylinder; forward modeling