Three-dimensional Magnetotelluric Forward Modeling And Inversion Based On Adaptive Vector Finite Element Method

Magnetotelluric(MT)is one of the most widely used electromagnetic methods.It has the advantages of convenient construction,large exploration depth and low cost.With the improvement of computer computing capabilities and advances in numerical simulation methods,The three-dimensional forward modeling and inversion technique of the magnetotelluric method has achieved considerable development and has been gradually applied in practice.However,there are still some problems with these methods.The first is that the widely used simulation methods(integral equation method,finite difference method,etc.)use orthogonal Cartesian grids.Accurate simulation of irregular interfaces formed by undulating terrain or complex anomalies cannot be performed.The second is the grid setting problem.Grids have a great influence on the accuracy of forward modeling and the resolution and results of inversion.In the process of meshing,the influence of many factors must be taken into consideration,requiring engineers and technicians to have a deeper understanding of the theoretical methods.This limits the application of the three-dimensional forward modeling and inversion method.The third issue is the speed of computation.The three-dimensional forward modeling and inversion of the magnetotelluric method requires a lot of computational time.How to speed up the forward modeling and inversion algorithm is the key to the practical application of the three-dimensional forward modeling and inversion method.Solving the above problems is of great significance for improving the application effect of the magnetotelluric method and advancing the practical application of the three-dimensional forward modeling and inversion method.In this work,we first studies the three-dimensional adaptive vector finite element forward modeling method of the magnetotelluric method.The control equation of the magnetotelluric method was deduced from the Maxwell equations.We use the vector basis function and based on the weighted residual method to obtain a weak form that equivalent to a partial differential equation.In order to solve the simulation problems of terrain and complex geologic bodies,we use unstructured hexahedral elements to discretize the computational domain.The posterior error estimation method is deduced based on the finite element theory,and the computation grid is adaptively refined using the posterior error estimator.As a result,the accuracy of solution is improved under the premise of saving computing resources,and the meshing problem of the forward modeling is solved.The numerical results show that the error convergence rate of adaptive grid refinement is much faster than that of global grid refinement.At the same time,we discuss the terrain effect of three-dimensional magnetotellurics.To solve the difficult problem of solving the linear system araising in the discretization of Maxwell's equations,we studied the solving methods of linear equations,including iterative method and efficient preconditions.We transform the linear system of complex coefficients into its equivalent real form,and construct the block diagonal preconditioner.FGMRES is used to solve the real linear system.Application of block diagonal preconditioner requires the solution of two smaller real-valued symmetric problems,we solve them by using either a direct solver or the conjugate gradient method preconditioned with the auxiliary space preconditioner.The numerical examples show that the general iterative methods and preconditioners are very unstable in solving linear system with complex coefficients.In most cases,they cannot converge.However,the number of iterations of the FGMRES combined with the direct solver or auxiliary space preconditioner is independent of frequencies and the number of degrees of freedom.At the same time,we analyzed the computational time and memory usage of different solvers.The computational time and memory usage of FGMRES combined with auxiliary space preconditioner increased the slowest with the number of degrees of freedom.Thus this method is very suitable for large-scale three-dimensional problems.In order to further improve the computational speed of forward modeling,we parallelized the forward modeling method based on distributed grid partitioning technology.By partitioning the grid,the computational task is decomposed into multiple sub-processes,and parallel linear equations storage and solving techniques are used.By using these methods,we achieved a very high speedup.Since the computation of each frequency in the forward modeling is independent,we also implement a parallel method based on the frequency division strategy.By decoupling the forward modeling mesh and inversion mesh,we apply the three-dimensional adaptive vector finite element forward modeling method to three-dimensional inversion and thus solve the problems of forward calculation accuracy and inversion parameterization.We first make a comparative analysis of commonly used inversion algorithms,according to the need we select the L-BFGS method with a small memory demand and a faster convergence rate.In order to reduce the non-uniqueness of the inversion,the upper and lower bound constraints of the resistivity parameters in the inversion were studied.The inversion of a simple 3D model shows that the L-BFGS method has great advantages in terms of convergence speed and time efficiency.For terrain problems,we have performed inversion of multiple terrain-influenced data.The results show that the terrain problems can be effectively solved by including terrain in the initial model of inversion.We believe that this is a fundamental solution to the terrain problem of magnetotelluric.