一种新的三维大地电磁积分方程正演方法

A new integral equation approach for 3D magnetotelluric modeling

采用规则六面体单元和并矢Green函数奇异积分等效积分技术,已有的大地电磁积分正演方法具有不能有效模拟地下复杂地质体和计算精度偏低的缺点.本文提出了一种新的三维大地电磁积分方程正演技术,即采用四面体单元、解析的并矢Green函数奇异积分表达式,达到既能模拟地下复杂异常体,又能有效提高已有积分方程法计算精度的目的.首先,采用四面体网格技术离散地下复杂异常体,获得四面体单元上的大地电磁积分方程.然后,利用针对四面体单元开发的新的奇异值积分的解析表达式,准确计算线性方程中的并矢Green函数的奇异积分,从而获得精确的线性方程.借助于PARDISO高性能并行直接求解器,实现了三维大地电磁问题的高精度求解.最后,基于国际标准3D-1模型和六棱柱模型,通过与其他方法结果的对比分析,验证了本文方法的正确性、处理高电导率对比度的能力(1000:1)和处理复杂模型的能力.

Magnetotelluric method is a technique for imaging the electrical conductivity and structure of the Earth. It is widely used in mining exploration, the detection of deep geologic structure and the study of earth dynamics. As a key point of inversion and interpretation, forward is always focused on by many authors. Compared to other forwarding methods, for instance, finite difference method, finite element method and finite volume method, integral equation method has a better precision and it only need to discrete the anomalous bodies in the earth, which deserves to be implemented. Traditional integral equation approaches for magnetotelluric problems generally use regularly structured grids which are difficult to approximate geometrically complicated models. In addition, the traditional integral equation algorithms replace the cubic cell by a spherical cell of equal volume to approximately calculate singular integral in the dyadic Green function, which would reduce the accuracy of numerical solutions. In this paper, we propose a new approach for three dimensional magnetotelluric modeling. It uses the tetrahedral grids and the closed-form solutions for the singular dyadic Green volume integrals, to simulate the complicated anomalous bodies, as well as improve the accuracy of numerical solutions. Firstly, integral equation on anomalous body for magnetotelluric problem is derived from Maxwell＇s equations. The expression for linear equation is obtained by using tetrahedral grids to discrete the anomaly and assuming that the electrical field in each element is a constant. Then, the closed-form solutions for the singular dyadic Green volume integrals on tetrahedron are applied to compute the singular integral in the dyadic Green function. PARDISO, a high-performance parallel solver, is chosen to achieve accurate results for three dimensional magnetotelluric problems. Finally, the 3D-1 model and the hexagonal prism model are tested to verify the accuracy of the new integral equation approach and validate its abilities for modeling high conductivity contrast problem up to 1000：1, and simulating complicated models.