基于矢量有限元的带地形大地电磁三维反演研究

Three-dimensional magnetotelluric inversion with surface topography based on the vector finite element method

研究了基于矢量有限元方法的大地电磁带地形三维反演算法并开发了三维反演计算程序代码.在大地电磁场正演数值模拟方面,采用并行直接稀疏求解器PARDISO且无需进行散度校正的快速正演方案,对典型地形模型,在中等规模计算条件下,与双共轭梯度法(BICG)计算结果比较,发现PARDISO比BICG快10倍以上;通过理论模型试算,并与前人的有限元法计算结果对比,验证了带地形三维正演计算程序的正确性.在反演方面,本研究基于共轭梯度方法编写了大地电磁带地形三维反演代码,为了避免直接求取雅可比矩阵,将反演中的雅可比矩阵计算问题转为求解两次"拟正演"问题,进而将PARDISO的快速正演方案应用于"拟正演"问题的求解,以提高反演计算效率.利用开发的反演算法对多个带地形地电模型的合成数据进行了三维反演,反演结果能很好地重现理论模型的电性结构,验证了本文开发的三维反演算法的正确性和可靠性.最后,利用该算法反演了某矿区大地电磁实测数据,反演得到的三维电性结构清晰地反映了研究区的地电特征,将反演结果与该区已有地质资料结合进行解释,应用效果明显,进一步验证了本文算法的有效性.

In this paper,the three-dimension(3 D)Magnetotelluric(MT)inversion algorithm with topography based on the vector finite element method is studied and the 3 Dinversion program code is developed.In the 3 Dforward numerical modeling of the MT field,a fast forward modeling scheme based on Parallel Direct Sparse Solver(PARDISO)without divergence correction is adopted.For typical models with topography,under the conditions of medium-scale calculations,comparison shows that the PARDISO method is more than 10 times faster in calculation speed than Bi-Conjugate Gradient(BICG).The correctness of the 3 Dforward modeling program with topography is also confirmed by tests on a theoretical model and the comparison with the results in previous studies using the finite element method.In 3 Dinversion,an inversion code of MT with topography based on the Conjugate Gradient(CG)method is compiled in this study to avoid directly calculation of the Jacobian matrix,thus the problem existing in calculation of the Jacobian matrix is turned into solving two "quasi-forward"problems,and the fast forward solution of PARDISO is further applied to the solution of the "quasi-forward"modeling to improve the efficiency of inversion calculation.The developed algorithm is used to carry out 3 Dinversion of synthetic data of a series of geo-electric models with topography.The inversion results can well reproduce the electrical structure of the theoretical model,which verifies the correctness and reliability of the 3 Dinversion algorithm developed in this paper.Finally,the algorithm is used to invert the MT measurement data in a mine area.The resultant 3 Delectrical structure from the inversion clearly reveals geoelectric characteristics of the study area.Data interpretation is conducted successfully by combining the inversion results with geological data available,further verifying the effectiveness of the algorithm proposed in this work.