基于非结构化有限元的三维井地电阻率法约束反演

doi:10.11720/wtyht.2022.0181

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摘要

电磁探测反演是典型的不适定问题,易造成反演结果的多解性,不适定性是反演自身固有的特征,没有求解的附加信息这一本质困难是很难克服的,解决该问题的有效方法是研究约束反演。本文采用目前较为主流的高斯牛顿—共轭梯度法(GN-CG),在反演目标函数中直接施加约束条件,将介质电阻率的取值范围作为先验信息和约束条件以外点罚函数法的方式引入到反演目标函数中,与常规三维电阻率反演目标函数相比,增加了不等式约束项的目标函数,理论上可以压制反演的多解性。通过多种理论模型的测试结果表明,本文基于不等式约束的三维井地电阻率反演算法有效地改善了反演结果的精度,以惩罚函数法施加不等式约束条件的方式是现实可行及有效的。

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	王智
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	方思南

关键词 ：井地电阻率法, 反演, 高斯牛顿—共轭梯度法, 不等式约束, 惩罚函数法

Abstract：

The inversion of electromagnetic detection data is a typical ill-posed problem and is prone to cause a multiplicity of solutions of the inversion results. The ill-posedness is an inherent characteristic of inversion and is difficult to overcome without additional information. An effective way to solve this problem is constrained inversion. In this study, the Gauss-Newton - conjugate gradient (GN-CG) method was used to directly impose constraints on the inversion objective function. Specifically, the dielectric resistivity range was introduced into the inversion objective function as the prior information and constraints using the exterior penalty function method. Compared with the conventional three-dimensional resistivity inversion objective function, the objective function with inequality constraints can suppress the multiplicity of solutions in theory. As revealed by the testing results of various theoretical models, the three-dimensional borehole-to-surface resistivity inversion algorithm based on inequality constraints effectively improves the precision of inversion results, and the way of imposing inequality constraints using the penalty function method is feasible and effective.

Key words： borehole-to-surface resistivity method inversion GN-CG inequality constraint penalty function method

收稿日期: 2022-04-13 修回日期: 2022-07-09 出版日期: 2022-12-20

ZTFLH:

P631

基金资助:国家自然科学基金项目“起伏地形下的井中激电井—地方式并行反演研究”(41604093);天地科技股份有限公司科技创新创业资金专项(2020-TD-QN11);中国博士后科学基金(2017M622382)

作者简介: 王智(1985-),男,湖北省武汉市人,博士,副教授,主要研究方向为电磁法数值模拟。Email:1324385898@qq.com

引用本文:

王智, 王程, 方思南. 基于非结构化有限元的三维井地电阻率法约束反演[J]. 物探与化探, 2022, 46(6): 1431-1443.
WANG Zhi, WANG Cheng, FANG Si-Nan. Constraint inversion of three-dimensional borehole-to-surface resistivity based on unstructured finite element. Geophysical and Geochemical Exploration, 2022, 46(6): 1431-1443.

链接本文:

https://www.wutanyuhuatan.com/CN/10.11720/wtyht.2022.0181 或 https://www.wutanyuhuatan.com/CN/Y2022/V46/I6/1431

Fig.1 球体模型结构及网格剖分示意
a—球体模型示意;b—球体模型网格剖分放大效果;c—点与测点网格加密效果;d—网格剖分示意

Fig.2 地下球体的解析解与数值解对比

Fig.3 解析解与数值解的相对误差

Fig.4 平坦地形下长方体模型
a—xoy水平截面; b—xoz垂直断面;c—三维异常体模型;d—测点加密示意;e—正演网格;f——反演网格

Fig.5 迭代次数与RMS的关系

Fig.6 三维反演结果
a—传统电阻率反演结果z=-7 m处xoy切片;b—传统电阻率反演结果y=25 m处xoz切片;c—施加不等式约束的反演结果z=-7 m处xoy切片;d—施加不等式约束的反演结果y=25 m处xoz切片;e—传统电阻率反演结果的三维剖面(电阻率小于50

Ω · m

);f—施加不等式约束反演结果的三维剖面(电阻率小于50

Ω · m

)

长方体编号	电阻率/ $(Ω · m)$	尺寸/(m×m×m)
S1	200	$100 × 100 × 40$
S2	100	$100 × 100 × 40$
S3	2000	$350 × 50 × 40$
B1	2000	$350 × 100 × 200$
B2	100	$100 × 150 × 180$

Table 1 长方体参数

Fig.7 异常体模型位置与网格剖分
a—三维异常体模型位置; b—地表数据观测系统;c—垂直剖面yoz; d—非结构化网格加密示意

Fig.8 井—地二极装置传统正则化反演与约束反演结果
a—传统正则化反演x=400 m处yoz剖面; b—传统正则化反演x=600 m处yoz剖面;c—约束反演x=400 m处yoz剖面; d—约束反演x=600 m处yoz剖面

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