基于多尺度有限元法的频率域航空电磁三维正演研究

Research on 3D forward modeling of AEM in frequency domain based on multiscale finite element method

正演模拟是电磁数据反演的基础,其计算速度与精度一直是制约电磁反演的两大核心问题.在三维电磁正反演中,传统方法通过加密网格或增加插值基函数阶数提高计算精度,但由此也降低了计算效率,制约了三维电磁反演的实用化.因此,如何实现大尺度模型高精度快速正演是目前电磁三维正反演中亟需解决的问题.本文将多尺度有限元法应用到麦克斯韦方程求解中.我们首先在粗网格尺度上构建满足局部特性微分算子的多尺度基函数,进而在粗网格尺度上对原问题进行求解,通过建立粗细两套网格间场的映射关系,在未知数较少的粗网格上实现电磁问题求解之后,利用粗细两套网格间场的映射关系获取细网格上电磁场响应,由此可以在保证计算精度前提下快速获取不同尺度电磁场正演响应,计算速度得到很大提高.此外,本文还基于八叉树思想进行网格优化,进一步改善三维正演效率.我们通过对典型地电结构进行多尺度有限元正演模拟并与传统有限元结果对比验证算法的有效性.最后,我们通过模拟加拿大Voisey's Bay卵形体镍铜硫化矿区航空电磁响应以检验本文算法模拟地下复杂异常体的能力.

Forward modeling is the basis of Electromagnetic (EM) data inversion, and its computational speed and accuracy have been the two core problems constraining EM inversion. In the 3D EM forward and inversion, the traditional method improves the calculation accuracy by refining the grid or increasing the order of the interpolation basis function, but this also reduces the computational efficiency and restricts the practical application of the 3D EM inversion. Therefore, how to achieve high-precision and fast forward modeling of large-scale models is an urgent problem to be solved in the current 3D EM forward and inversion. In this paper, the multiscale finite element method is applied to the solve Maxwell's equations. We first construct the multiscale basis function that satisfies the local characteristic differential operator on the coarse grid scale, and then solve the original problem on the coarse grid scale. By establishing the field mapping relationship between the two sets of coarse and fine grids, after solving the EM problem on a small number of coarse grids, the EM field response on the fine grid can be obtained by using the field mapping relationship between the two sets of coarse and fine grids, Therefore, the forward modeling response of EM fields at different scales can be quickly obtained under the premise of ensuring the calculation accuracy, and the calculation speed is greatly improved. In addition, this paper further improves the efficiency of 3D forward modeling based on the idea of octree. Finally, we verify the effectiveness of the algorithm by performing multiscale finite element forward modeling on typical geoelectric structures and comparing with the traditional finite element results, we also calculate the EM response for a realistic model—the ovoid zone ore body located at Voisey's Bay, Labrador, Canada, to verify the effectiveness of our algorithm.