基于非结构化有限元的三维井地电阻率法约束反演

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。

Constraint inversion of three-dimensional borehole-to-surface resistivity based on unstructured finite element

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.