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

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 3D 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.