三维陆地可控源电磁法有限元模型降阶快速正演

三维陆地可控源电磁法有限元快速正演的主要瓶颈在于多频率大型稀疏方程组求解问题.本文引入一种基于模型降阶的Krylov子空间投影算法，推导了有限元刚度矩阵的模型降阶形式，构建了频率域传递函数；采用标准正交向量序列，构建一个远远小于有限元刚度矩阵维度的矩阵，该矩阵与频率无关，通过一次模型降阶即可实现多频点有限元方程快速求解.采用基于电场的变分方程，加入散度校正条件，以消除伪解；引入伪δ函数，消除了源点的奇异性，可适用于复杂背景模型三维有限元数值模拟，并为多源的求解奠定了基础；以层状介质模型解析解为标准，通过和基于Pardiso直接求解器的有限元算法（3DFEM）进行比较，模型降阶法计算时间小于前者的1/10，平均相对误差在1.72%，在满足精度要求下，实现了高效率三维有限元数值求解；分别设计了横向高低阻模型和纵向高低阻模型，分析了从近区到远区电场和卡尼亚视电阻率的变化规律，假极值的表现特征，阴影效应的影响等，从而也验证了该算法的正确性.最后，建立了一个地层陷落柱模型，通过模型降阶有限元正演模拟，发现视电阻率断面图在陷落柱上方出现"凹陷"，与模型设计吻合，表明该算法对复杂地层模拟具有同样的适用性.

Fast forward modeling of the 3D land controlled-source electromagnetic method based on model reduction

The main bottleneck of fast forward modeling of 3-D land controlled-source electromagnetics based on the finite element method lies in solving multi-frequency large-scale sparse equations. In this paper, a Krylov subspace projection algorithm based on model reduction is introduced, a model reduction form of the finite element stiffness matrix is derived, and a frequency domain transfer function is constructed. A standard orthogonal vector sequence is used to construct a matrix that is much smaller than the finite element stiffness matrix, which is unrelated to frequency. Through model reduction, it is possible to quickly solve these multiple frequency finite element equations. The variational equation based on the electric field is used, and divergence correction conditions are added to eliminate spurious solutions. The pseudo-delta function is introduced to eliminate the singularity of the source point, which can be applied to 3D finite element numerical simulation of complex models. Such an algorithm may be regarded as the basic principle for a solution to the problem of multiple sources. Using the analytical solution of the layered medium model as the standard, comparison shows that the averaged percentage computing error of the proposed method is 1.72%. Compared with the 3DFEM algorithm using the Pardiso direct solver, the CPU time consumption of the proposed method is reduced by 1/10 of the original one. Therefore, with a required accuracy, it is possible to achieve highly efficient 3-D finite element numerical solutions. In the numerical simulation, with the horizontal and vertical high-low resistance models, we analyze the changes of the electric field and the Cagniard apparent resistivity from the near to the far region, the performance characteristics of false extremes, and the effects of shadow effects, which verify the correctness of the proposed algorithm. Finally, we apply the proposed algorithm to simulation of a stratum collapse column model. The results show that the apparent resistivity cross section looks like a "sag" above the collapse column, which is consistent with the model design, and further indicates that the proposed algorithm can be applied to real complex geoelectric structure.