﻿ 重力三维密度光滑反演软件设计与实现
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 大地测量与地球动力学  2024, Vol. 44 Issue (3): 283-286  DOI: 10.14075/j.jgg.2023.07.117

### 引用本文

PENG Yanwu, CHEN Wenjin, TAN Xiaolong, et al. Design and Implementation of Gravity Three-Dimensional Density Smooth Inversion Software[J]. Journal of Geodesy and Geodynamics, 2024, 44(3): 283-286.

### Foundation support

National Natural Science Foundation of China, No. 42264001; Science Foundation for Doctors of Jiangxi University of Science and Technology, No. 205200100588.

### Corresponding author

CHEN Wenjin, PhD, lecturer, majors in geodesy and surveying engineering, E-mail: cwjwhu@whu.edu.cn.

### 第一作者简介

PENG Yanwu, postgraduate, majors in physical geodesy and gravity inversion, E-mail: ywpeng@mail.jxust.edu.cn.

### 文章历史

1. 江西理工大学土木与测绘工程学院，江西省赣州市客家大道1958号，341000

1 方法原理 1.1 重力场正演

 $V(P)=G \iiint\limits_V \frac{\rho}{r} \mathrm{~d} V$ (1)

 $V\left(\boldsymbol{r}_i\right)=G \sum\limits_{j=1}^M \rho_j \int\limits_{\Delta V_j} \frac{1}{\boldsymbol{r}-\boldsymbol{r}_i} \mathrm{~d} V$ (2)

1.2 三维密度反演

 $S=\varphi_d+\mu \varphi_m$ (3)

 $\varphi_d=\left\|\boldsymbol{W}_d\left(\boldsymbol{d}^{\text {obs }}-\boldsymbol{d}^{\text {pre }}\right)\right\|_2^2$ (4)

 $\varphi_m(\boldsymbol{\rho})=\left(\boldsymbol{\rho}-\boldsymbol{\rho}_0\right)^{\mathrm{T}}\left(\sum\limits_{i=s, x, y, z} \boldsymbol{W}_i^{\mathrm{T}} \boldsymbol{W}_i\right)\left(\boldsymbol{\rho}-\boldsymbol{\rho}_0\right)$ (5)

 $\boldsymbol{w}(z)=\frac{1}{\left(z_0+z\right)^{\beta / 2}}$ (6)

 $\varphi(\boldsymbol{\rho})=\varphi(\boldsymbol{p})_d+\mu \varphi_m(\boldsymbol{p})$ (7)

 $\begin{gathered} \boldsymbol{\rho}=\boldsymbol{\rho}_0+\left(\boldsymbol{G}^{\mathrm{T}} \boldsymbol{W}_d^{\mathrm{T}} \boldsymbol{W}_d \boldsymbol{G}+\mu \boldsymbol{W}_m^{\mathrm{T}} \boldsymbol{W}_m\right)^{-1} \\ \boldsymbol{G}^{\mathrm{T}} \boldsymbol{W}_d^{\mathrm{T}} \boldsymbol{W}_d\left(\boldsymbol{d}_{\mathrm{obs}}-\boldsymbol{G} \boldsymbol{\rho}_0\right) \end{gathered}$ (8)
2 软件设计与功能

 图 1 系统设计框架与操作流程 Fig. 1 System design framework and operation process

 图 2 三维密度反演主界面与子界面 Fig. 2 The main interface and sub-interface of three-dimensional density inversion
3 数据测试 3.1 单长方体模型数据测试

 图 3 模型切面与重力观测数据 Fig. 3 Model profile and gravity observation data

 图 4 反演模型在x=500 m、z=200 m处密度切面 Fig. 4 Density profiles of inversion model at x=500 m and z=200 m

3.2 实测数据测试

 图 5 实测重力梯度数据与密度异常体预测数据 Fig. 5 Measured gravity gradient data and predicted density anomaly data

 图 6 反演模型在y=2 000 m、z=300 m和z=700 m处密度切面 Fig. 6 Density profiles of inversion model at y=2 000 m, z=300 m and z=700 m
4 结语

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Design and Implementation of Gravity Three-Dimensional Density Smooth Inversion Software
PENG Yanwu1     CHEN Wenjin1     TAN Xiaolong1     HONG Guoqing1
1. School of Civil and Surveying and Mapping Engineering, Jiangxi University of Science and Technology, 1958 Kejia Road, Ganzhou 341000, China
Abstract: Using the scientific computing language MATLAB, we develop a gravity three-dimensional density inversion software based on smooth constraints, including functions such as three-dimensional density inversion, gravity field forward modeling, data analysis, and visualization. The test results show that the software can invert reasonable density distribution of geological body, which is reliable and practical.
Key words: smooth constraint; 3D density inversion; MATLAB; software design