3D Frequency-domain Electromagnetic Modeling Using Spectral Element Method With Gauss-Lobatto-Chebyshev Polynomials

Electromagnetic(EM)exploration field data interpretation is mainly based on one-dimensional(1D)and two-dimensional(2D)inversion.However,for the practical explorations,the earth is always very complicated and three-dimensional(3D)modeling and inversion are necessary.Fast and accurate 3D electromagnetic(EM)modeling has become a research hotpot.Conventional 3D EM modeling methods have problems of low accuracy,slow speed and instability which seriously limited the development of 3D inversion in industry.In my thesis,I introduce the spectral element method(SEM)with Gauss-Lobatto-Chebyshev(GLC)Polynomials into 3D frequency-domain EM modeling and apply this method to frequency-domain airborne electromagnetic(AEM)modeling and marine controlled-source electromagnetic(MCSEM)modeling.The SEM is an accurate and efficient numerical method for EM modelling.It uses high-order complete orthogonal polynomials for interpolation that have spectral accuracy and exponential convergence.It combines the flexibility of the finite-element method and the high accuracy of the spectral method by a simple application of spectral method to each element,which means that,like in the finite-element method,SEM discretize the modeling area into a set of elements,and like in the spectral method,the field inside each element is expressed via a series of orthogonal polynomials(the spectral approximation).Starting from the Maxwell's equations,we obtain a vector Helmholtz equation for the scattered electric field with the primary field of a 1D model as the source to avoid the sharp changes of EM field near the transmitting source.Galerkin method is used to discretize the Helmholtz equation,in which the curl-conforming Gauss-Lobatto-Chebyshev(GLC)polynomials are used as basis functions.The GLC polynomials help to derive analytical expressions of entries in the system matrix and thus SEM with GLC polynomial guarantees higher modeling accuracy.AEM and MCSEM modeling is a problem of multiple frequencies and multiple sources.I use the direct solver MUMPS to solve the large linear systems,which is very efficient as it only needs to do the factorization once for each individual frequency,and all fields of different sources are solved just by replacing the source term.Finally,the secondary electric fields at the receiver locations are obtained by interpolation,and the magnetic field is calculated by Faraday's law.To verify the effectiveness of the SEM algorithm and codes,I model and compare responses for 1D models with the semi-analytical solutions for AEM method and with the solutions from open-source software for MCSEM method.Then,I analyze the characteristics of SEM with different GLC interpolation polynomials orders.After that,I analyze efficiency of SEM by comparing with the adaptive finite-element method,the finite-difference method and the integer equation method for 3D models.The results show that SEM with GLC polynomials is an efficient and effective method for EM modeling,it can deliver very accurate results and has a significant advantage of less sensitive to mesh quality than the finite-element method and the finite-difference method.Finally,I simulate the AEM responses and MCSEM responses for typical 3D models and analyze the responses characteristics.