3-D GDS Anisotropic Modeling Based On Unstructured Edge-based Finite-element Method

Geomagnetic depth sounding(GDS)is a geophysical electromagnetic(EM)method that considers the current in the magnetosphere and ionosphere as the source and study the deep structure and composition of the earth by using the long-period EM signal received by geomagnetic observatories and satellites.The study of the deep structure and the law of material circulation has been a difficult and hot issue in the scientific community for many years.Since the conductivity is very sensitive to temperature,phase state and water content,the study of global-scale GDS has become an important tool to explore the deep structure of the Earth.With several studies revealing the significant lateral inhomogeneity of the Earth's interior,the one-dimensional(1-D)model in GDS can no longer meet the demand and the three-dimensional(3-D)GDS modeling and inversion become a key direction in related fields.However,whether for 1-D or 3-D modeling,the existing research is mainly based on isotropic medium models,and the research on anisotropic models is still a blank.The complex tectonic movement and material transformation in the deep earth cannot ensure that the medium is isotropic,and even there is a link between the material composition and geological evolution process and the anisotropic conductivity.Studies have proved that there is an obvious anisotropy in the deep earth.Therefore,the study on 3-D anisotropic modeling for GDS is of great significance to further understand the plate movement,orogenic activities,seismic and volcanic activities,deep structure and evolution,the nature of the soft flow circle,and the mantle thermochemical state,etc.In this paper,I carried out 3-D GDS anisotropic modeling based on the unstructured tetrahedral mesh and edge-based finite-element(FE)method.The unstructured tetrahedral mesh can describe irregular anomalies in the earth and complex ocean-land boundaries at high precision and flexibility.At the same time,the unstructured tetrahedra can be locally refined.The edge-based FE method uses vector basis functions defined on the edges,which guarantees the tangential continuity and normal discontinuity of the electric field at the electrical interface,while satisfying the scattering condition to ensure that the modeling does not generate spurious mode.In this paper,the equatorial ring current of the magnetosphere is used as the excitation source,and the primary magnetic field generated by the ring current is added to the equation as the boundary condition.Since the global-scale EM modeling is very complex,the degrees of freedom of the equation may reach tens of millions.In this situation,it is difficult to solve the equation using direct solvers or traditional iterative solvers.Therefore,in this paper,a preconditioned iterative solver is used for solving super large linear equations.In order to reduce the condition number of the curl-curl governing equation for the electric field and to achieve fast convergence of iterative solvers,I transform the complex equation into an equivalent real equations system and introduce a block diagonal preconditioner to preprocess the linear equation system,then I use the flexible generalized minimum residual method to solve the real equations.In the inner iteration,I solve the preconditioned system iteratively based on the subsidiary space preconditioning and conjugate gradient method until the relative residual falls below 10^-2,which can ensure the overall convergence.I verify the accuracy of the proposed algorithm by comparing my results with previous researches.The algorithm in this paper has higher computational efficiency and less memory requirements while achieving high-precision modeling.The effect of anisotropy on GDS is analyzed based on theoretical examples.Then,I calculate the global-scale EM responses based on the global near-surface conductivity distribution model and analyze the influence of ocean effect.Finally,I design a subduction zone model to simulate the response of the subduction zone under isotropic and anisotropic conditions.It can be seen from the results that the GDS responses can reveal the deep electrical anomaly and has a greater exploration depth than the magnetotelluric method.GDS has great potential in the study of deep structures such as subduction zone,mantle column,mantle transition zone,etc.The research in this paper provides theoretical and algorithmic support for high-accuracy and efficient anisotropic inversion for large-scale GDS,and it will also lay the foundation for geodynamic studies based on anisotropic inversion and analysis of deep geological problems.