Techniques For Improving The Accuracy Of Finite Element Solution Of MT Data

The Electromagnetic(EM)method due to its broad frequency spectrum is the widely used method in geophysical exploration.The method is used to study the shallow subsurface in engineering,the environment application using the radio-magnetotelluric(RMT)method or deep structure of the Earth utilize the magnetotelluris(MT)approaches.The goal of this dissertation is to develop a numerical method in C plus-plus(Cpp)using high order finite elements(FE)method,which will be utilized to model Magnetotelluric(MT)data on unstructured grids in isotropic media,and a linear finite element method for MT data modeling scheme in the anisotropic media using also unstructured grids.Modules from an existing code(MT2D),a freely available online code was extended to take into account both high order spatial and high order discretization.High order methods have capacity to yield to far more accurate numerical results for certain problems when compared to corresponding mesh refinement of the first order methods,and often has a significant reduction in total computational cost.The governing equations to be solved are electromagnetic(EM)partial differential equations either in primary or secondary fields for electric or magnetic fields from two-dimensional electromagnetic fields induction in the isotropic and anisotropic media.Unstructured mesh is used to construct the high order finite element based on triangular element and the basis function are defined for linear element(3 vertices),quadratic element(6 vertices),cubic element(10 vertices)and fourth element(15 vertices).We also developed an algorithm for a 2D anisotropic medium for MT data modelling.The first part of this dissertation presents a summary of the available modeling strategies for the Magnetotelluric method in both isotropic and anisotropic media.The assumptions of the MT method and the fundamental equations of Maxwell’s equations used as well as the necessary mathematics required for the MT boundary value problem.For the finite element analysis,the Galerkin’s method of the weak formulation to approximate the unknown field on the vertices of the computational mesh is used.The derived system of equation from the transformation is resolved using the direct method.The implementation of the high order scheme is designed using Object oriented programming features and translated in Cpp language.Even though the finite element method was successfully used in different Magnetotelluric forward modeling,it was not successful here because of lack of efficiency in the framework that was used and the limited amount of computational memory.The second half addresses the implementation of the method and its validation using benchmark problems.The validation of the computational algorithm is performed using a homogeneous Earth,where the analytic solution of the MT problem is known.The convergence of the solution is analyzed for different mesh degree of freedom(DOF)and different finite element polynomial order.The obtainedresult is presented in terms of maximum error percentages and average relative error percentages of the apparent resistivity and the phase computed on the Earth surface.The result for each finite element order is quite satisfactory.Finally,we studied horizontal anisotropy in two different cases namely: 2D isotropic slab in a horizontal anisotropic half-space and 2D anisotropic slab in a horizontal isotropic half-space.In both situations the code was able to detect the anomaly.In the case of the anisotropy we discussed how MT responses behave for some proposed synthetic examples.These computational algorithms could be performed and extended with the use of adaptative method and can also be implemented for 2D inversion of MT data.