Spectral Element Method For Electromagnetic Exploration

Low-frequency subsurface electromagnetic measurements are important tools for characterizing natural resources and environmental wastes.Rapid simulations of low-frequency subsurface electromagnetic measurements are still a challenge because of the large computational domain and low-frequency breakdown phenomenon,but they are important for large-scale inversion and processing of such data.In actual applications of geophysical prospecting,such as oil exploration and natural disasters prediction,the running speed needs to be high enough,because of the huge size of the research object.Finite-element method(FEM)is a robust and flexible simulation tool in geophysical prospecting,and has been developed and applied extensively with great success.However,the accuracy and efficiency need to be further improved.In this paper,we introduce the spectral element method(SEM),which is also known for its high degree of accuracy with a small number of unknowns to analyze 3-D electromagnetic problems.With the choice of nodal points and quadrature integration points based on Gauss-Lobatto-Legendre(GLL)points,this SEM can be considered as a special class of the general FEM but with high-order convergence rate.However,geophysical subsurface sensing often needs to use a low-frequency source for electromagnetic fields to penetrate the lossy ground to obtain deep geologic information.As the system matrix is highly singular at low frequencies,it is difficult to solve the whole system iteratively because of its slow convergence,while a direct solver costs huge memory and CPU time.We develop an effective method to simulate these extremely low-frequency subsurface electromagnetic measurements by using the SEM together with a domain decomposition method(DDM).The DDM can mitigate the low-frequency breakdown problem in part by directly solving the small subdomain matrices,and can reduce the computational costs by avoiding solving large matrix equations;the SEM has exponential convergence behaviors,i.e.,its error decreases exponentially with the order of basis functions.A specific mesh has been designed based on the traveling wave nature in the air and diffusion field nature in the underground space to greatly reduce the number of unknowns.The frequency-domain version of the Riemann solver(upwind flux)is used as an effective transmission condition to simulate the interactions between neighboring subdomains in DDM.With this method,we study the Grounded Electrical-source Airborne Transient EM(GREATEM)system.Several numerical examples of demonstrate the efficiency of the proposed approach in extremely low-frequency simulations.In order to fundamentally overcome the low-frequency breakdown phenomenon in 2.5-D and 3-D numerical computation,we propose a 2.5-D mixed spectral element method(2.5-D mixed SEM)and a 3-D mixed SEM,and apply these methods to solve the 2.5-D Marine Controlled-Source Electromagnetic(MCSEM)system and the 3-D Surface to Borehole Electromagnetic(SBEJM)system,respectively.Since the Gauss'law is now explicitly enforced in the mixed SEM to make the system matrix well-conditioned even at extremely low frequency,we can solve these linear systems from DC to high frequencies.Numerical examples show that the mixed SEM is accurate and efficient,and has significant advantages over conventional methods.The new contributions of this work are:(1)This is the first application of the SEM-DDM to overcome the low-frequency breakdown problem.The flexible meshing in the SEM significantly improves the efficiency of the ATEM modeling compared to the conventional FEM.(2)For the first time,we show the condition number of the 2.5-D marine CSEM systems,and discuss the singularity of the system matrix from the conventional method reported in the previous literatures.This is also the first application of the 2.5-D mixed SEM with divergence-free constraint to eliminate the low-frequency breakdown phenomenon.(3)The first application of the 3-D mixed SEM in Kikuchi's scheme for overcoming the low-frequency breakdown problem.Our scheme is concise and easy to apply.We also propose an effective normalization to improve the complex system matrix arising from the mixed spectral element discretization.