Three-Dimensional Vector Finite-Element Forward Modeling For Controlled-Source Electromagnetics Using Rational Krylov And Algebraic Multigrid Methods

With the urgent need for resources exploration in the deep earth or deep sea,the ground electromagnetics(EM),the airborne and marine electromagnetics carried out on moving platforms have gone through a rapid development during the past decade.The EM methods are playing a more and more important role in mineral exploration,hydrocarbon detection,hydrogeophysical exploration,environmental and engineering geophysics.Therefore,reliable and effective three-dimensional(3D)forward and inverse techniques are crucial for detailed interpretations of geophysical data.The existing numerical simulation techniques usually develop an algorithm for a specific EM exploration method,lacking generalizability.The forward modeling methods using structured hexahedral grids can't accurately delineate the real topography,the irregular interface and the complicated abnormal bodies.The forward modeling approaches adopted direct solvers cost plenty of memories,therefore limiting the practical inversions for large-scale models.Therefore,resolving the above theoretical bottlenecks and technical problems will provide theoretical support for 3D EM data processing,and lay a solid foundation for 3D EM data inversion.To solve the above mentioned issues,I focus on investigating accurate and efficient forward modeling methodologies for both frequency-and time-domain controlled-source EM methods by combining the unstructured tetrahedral grids and vector finite elements.Starting from the Maxwell's equations,I firstly derive the governing equations and boundary conditions for the total electric field.The tetrahedral grids are applied to handle the complex geoelectrical models.Nédélec H(curl)-conforming vector basis functions are adopted to approximate the electric field,such that the spatial discretization is complete.The Nédélec elements meet the continuity condition of tangential electric field and automatically satisfy the condition that the divergence of the electric field vanishes.For any complicated-shaped electric or magnetic transmitting sources,I simply approximate them by a series of short electric dipoles,and consequently establish a unified forward modeling framework for any controlled-source EM methods.Furthermore,I formulate the scalar finite element equation for the electric potential,to calculate the initial electric field for the time-domain controlled-source EM problems with electric transmitting sources.For either vector or scalar finite element methods,the Galerkin weighted-residual method is adopted to obtain the final linear system for finite-element discretization.To verify the accuracy and reliability of the total electric-field approach,I perform the forward modelings using direct solvers.For frequency-domain EM modeling,I solve the complex linear equations system related to the electric field and the frequency,while for time-domain EM modeling,the partial differential equation is discretized in time using the unconditionally stable backward Euler scheme,where the linear system include the electric field and the time step size is solved at each time step.I calculate the electric field by LU decomposition with the sparse direct solver MUMPS,and evaluate the EM fields at the receivers by interpolations.The numerical experiments for four kinds of dipole sourcs are carried out to demonstrate the versatility of the total-field-based finite-element method.To improve the efficiency of time-domain EM modeling,I propose to solve the time-domain problems using the block rational Krylov method.The explicit solutions to the initial value problem of electric field are obtained directly as the product of a matrix exponential function with a vector,without any time stepping.I put forward a weighted residual optimization algorithm to find the optimal shift parameters based on the estimated errors of rational Arnoldi approximations,which significantly reduces the number of rational Arnoldi iterations and further improves the speed.In addition,by constructing the rational Krylov basis with the block rational Arnoldi algorithm,the electric field solutions at arbitrary times are calculated with the orthogonal basis.I simulate several typical time-domain airborne EM and marine controlled-source EM models to validate the correctness of the block rational Krylov method and to compare its efficiency with the backward Euler scheme.Considering that the direct solvers are not scalable and require a large amount of memory,they are not suitable for large-scale EM modeling.To mitigate this problem,I use the conjugate gradient method to iteratively solve the linear system for time-domain EM problems.At each time step,the linear system obtained by the second-order backward Euler scheme is indeed the finite-element discretization for the Maxwell's equation with real coefficients.Based on the Hiptmair-Xu decomposition,the primary H(curl)space is split into three auxiliary spaces,and I utilize the efficient algebraic multigrid preconditioner,to iteratively solve the linear system at each step.I study the influence of air conductivity and time step sizes on the robustness of the conjugate gradient solver,and evaluate how the initial value optimization could improve the computational efficiency.The last section of this thesis is devoted to iteratively solving the complex linear equations system arising from the curl-curl equations for frequency-domain EM.I formulate the complex equation to its equivalent real-value form and introduce a block diagonal preconditioner.This results in a preconditioned system solving the real-coefficient Maxwell's equation.The condition number of the preconditioned linear system is proved to be less than or equal to square root of two.In the outer loop,I apply the flexible general minimal residual solver to solve the original linear system with 2N unknowns.While in the inner loop,I use the preconditioned conjugate gradient solver with an anlgebraic multigrid preconditioner to solve the preconditioned system with N unknowns.The numerical examples for a magnetic dipole in the full space and half space,and an electric dipole in the half space are provided to demonstrate the feasibility of the solver for Maxwell's equations with constant and variable coefficients,respectlvely.For marine controlled-source EM forward modeling,I study the influence of air conductivity and frequency on the robustness of the solver,and summarize the memory footprint of iterative solvers for different number of unknowns.Based on the above work,I further prove via a lot of numerical experiments that the total electric field approach is accurate,reliable and versatile.Using the weighted shift parameter optimization algorithm,both the single-vector and block rational Krylov method have high accuracy.Benefitted from better cache utilization and higher floating-point operations per communication ratio,the block rational technique results in 1.26 to 1.48 fold speedup for moderate number of sources compared to the single-vector method.The two rational Krylov methods are 7 to 13 times faster than the backward Euler schemes.The iterative solutions for time-domain EM show that,the air conductivity and time steps merely affect the convergence of conjugate gradient solver,where the initial value optimization technology could bring a speedup of 17% to 34%.For frequency-domain EM,the auxiliary-space preconditioned flexible general minimum residual solver takes only dozens of iterations to convergence to the preset tolerance.In addition,the air conductivity and the frequency for marine controlled-source EM have no effect on the robustness of the solver.Benefitted from the excellent memory performance,I successfully solve a frequency-domain EM problem with 25 million of real unknowns on a common personal workstation,which indicates the great potential of the algebraic multigrid preconditioned iterative solvers.