基于空气地球分离策略的大地电磁三维正演

An air-earth decomposition formula for three-dimensional magnetotellurics forward modeling

大地电磁正演常采用电场双旋度方程,由于工作频率低、空气电导率为零,电场双旋度方程含有空解空间,导致离散化的线性方程组严重病态、具有高的条件数.为降低大地电磁场双旋度正演方程的病态性,本文提出了一种基于空气与地球分离策略的大地电磁混合正演公式.首先,在空气层中将双旋度算子转换为Laplace算子,在地下利用电导率大于零的特征保留双旋度算子,建立空气与地下分离的混合公式系统,以便实现混合公式系统对应的离散化线性方程组条件数的大幅度降低.然后,在空气与地下分界面上,施加电磁场的切向与法向连续性条件实现空气与地下区域的耦合与信息交换,以保证电磁场分布的唯一性.再则,利用非结构化四面体网格离散技术来高精度模拟任意起伏地形和复杂地下三维地电结构,并结合节点有限元与棱边有限元法,高精度求解该混合公式系统.最后,利用国际标准大地电磁测试模型来验证该空气与地球分离算法的正确性,并展示其有效降低原有电场双旋度方程的病态性的能力.本文提出的空气与地球分离思想为寻求大地电磁最优化控制方程开启了新的求解思路,预计会产生更多、性能更佳的大地电磁混合公式,以便服务于大地电磁正反演技术的快速发展.

The electric field curl-curl equation is often used in the forward modeling of magnetotelluric.Due to the low operating frequency and zero air conductivity,the electric field curl-curl equation contains Null-space solution,resulting in a severe ill-conditioned discretized linear equation system with a high condition number.In order to reduce the ill-conditioned nature of the curl-curl forward equation of the magnetotelluric field,this paper proposes a hybrid magnetotelluric forward equation based on the air-earth decomposition strategy.First,the curl-curl operator is converted into the Laplace operator in the air layer,and the curl-curl operator is retained in the underground by using the characteristic of conductivity greater than zero to establish a hybrid formula system for the separation of air and underground,so as to realize a substantial reduction in the system matrixs′condition number of the discretized linear equations of the hybrid formula.Then,on the interface between the air and the underground,the tangential and normal continuity conditions of the electromagnetic field are applied to realize the coupling and information exchange between the air and the underground area,so as to ensure the uniqueness of the electromagnetic field distribution.Furthermore,the unstructured tetrahedral mesh discretization technique is used to simulate arbitrary undulating terrain and complex underground 3 D geoelectric structures with high precision,and the hybrid formula system is solved with high precision by combining node finite element and edge finite element methods.Finally,the correctness of the air-earth decomposition algorithm is verified by using the international standard magnetotelluric testing model,and its ability to effectively reduce the ill-conditioned nature of the original electric field curl-curl equation is demonstrated.The idea of separation of air and earth proposed in this paper opens up a new solution idea for seeking the optimal governing equation of magnetotelluric.It is expected that more and better mixed magnetotelluric formulas will be produced to serve the rapid development of magnetotelluric forward and inversion technology.