2D And 3D Forward Modeling Of MT Using Unstructured Finite Element Method

Magnetotellurics whose sources are the natural electromagnetic field is a very popular electromagnetic geophysical method of imaging the electrical resistivity structure. The method are widely utilized in mining exploration, earth deep structure research and the study of earthquake prediction because of its advantages of low cost, wide frequencies band and great exploration depth. The forward modelling is the difficulty of geophysics and the premise of inversion and interpretation. This paper mainly study the two and three dimensional magnetelluircs finite element method forward, aiming at improving the accuracy and efficiency.Existing finite element method commonly used for the numerical simulation of two dimensional magnetotellurics employ linear interpolation shape functions, which may be insufficient for dramatic change of field. In this paper, quadratic interpolation shape functions are used to approximate 2-D magnetotelluric (MT) field at any point in a specified element. Unstructured grids which compose of irregular triangles are adopted in our finite element modelling, which can be refined locally and adaptively according to the complex geometry of computational domain or subsurface structures. The results show that the accuracy of solutions for quadratic interpolation have a considerable improvement compared with linear interpolation at the case of similar node numbers, especially for high frequencies. The relative errors were increased about two order of magnitude for some high frequencies, which persuasively indicate that improving the order of shape functions is a feasible and effective approach to obtain more desired solution. In addition, for sake of more accurate solutions, secondary fields rather than total fields are numerically computed, which make the errors to be limited to relatively small secondary fields. Heterogeneous permeability is taken into considerations in our algorithm, as a consequence, which can deal with heterogeneous permeability model. The accuracies of our approach are verified by comparisons with analytical solutions for 1-D layered media and previously published numerical simulations of a COMMEMI-2D1 model. The advantages of our approach are also illustrated by the forward modelling of models with arbitrary topography and complicated structures model, at the same time, the effect of topography and heterogeneous permeability for two dimensional magnetotellurics modelling results are discussed in detail.Traditional node-based finite element methods can effectively cope with the continuous variation of scalar field, such as direct current potential and two dimensional magnetotellurics field. However, problems are encountered for node-based finite element methods when dealing with the vector electromagnetic field. The node-based finite element methods do not satisfy the conditions that the normal electrical fields are discontinuous at the interface of electric variations and the electrical current density are divergence-free throughout the regions without source, which obviously violate Maxwell equations and result in spurious solutions. Some scholars employ the divergence correction to suppress spurious solutions, however, which can’t eliminate fundamentally. In order to overcoming these drawbacks of node-based finite element, vector finite element method is adopted for three dimensional magnetotellurics modelling in this paper. Unstructured grids which compose of irregular tetrahedrons are used in discretizing the computational domain, which can be more accurately approximated to the complex three dimensional subsurface structures. The heterogeneous permeability model is available for our algorithm, because the electrical and magnetic parameters are took into account comprehensively. We verify the accuracy of our approach by comparisons with analytical solutions for one dimensional layered media and previously published numerical simulations of a COMMEMI-3D1 model. The ideal solutions are obtained by the numerical simulations of both arbitrary topography model and complicated anomalous body model. In addition, the effect of three dimensional topography and heterogeneous permeability for modelling results are discussed detailed.In general, the large, sparse and symmetric linear equations will be obtained for the finite element method simulations, which need a large computer memory especially for three dimensional magnetotellurics modelling. In order to save the computer memory, the up-triangular elements of coefficient matrix are stored in CSR format. We use BICG algorithm and BICGSTAB algorithm, which is aimed specially at the symmetric systems, to solve finite element equations whose coefficient matrix is symmetric matrix. In addition, an appropriate preconditioner plays a significant role in coping with ill-condition problems and improving convergence rate. The results show that our algorithm is efficient and robust.