| Frequency-domain electromagnetic(EM)method is an efficient tool for geophysical exploration.It can serve the exploration of mineral and hydrocarbon resources,solve environmental and engineering problems and therefore accelerate national infrastructure construction and economic development.Research on 3D forward modeling of frequency-domain EM method can promote the development of3 D inversion technology and provide theoretical support for field work.In order to improve the performance of numerical modeling algorithm,the wavelet finite-element method based on B-spline wavelet on the interval is introduced in this thesis to study the 3D forward modeling of airborne EM method(AEM)and marine controlled-source EM method(MCSEM)in frequency-domain.The wavelet finite-element method is a numerical algorithm that takes the scaling functions or wavelet functions in wavelet theory as the finite-element basis functions.Compared with conventional polynomial interpolation,the wavelet has more excellent numerical properties so that it is able to improve the performance of finite-element method.The wavelet finite-element method can approximate the unknown function in each element with basis functions at different scales to achieve multiresolution solutions.At the same time,due to the sparsity of the wavelets,the coefficient matrix of the wavelet finite-element equation is usually sparse,which is beneficial to accelerate the solution speed.The B-spline wavelet on the interval is defined on a sequence of nodes within the closed reference interval [0,1] and is composed of segmented B-spline polynomials between these nodes.The B-spline wavelets on the interval have explicit expressions.Meanwhile,different from the traditional wavelets defined on the real axis,the B-spline wavelet on the interval is defined on a closed interval.The scaling functions and wavelet functions at the boundary are specially designed to ensure the numerical accuracy at the boundary.Starting from the Maxwell's equations,the frequency-domain coupled-potential formulation with coulomb gauge is derived in this thesis based on the definition of the magnetic vector potential and electric scalar potential.In order to overcome the source singularity and to reduce the number of meshes,a field separation algorithm is utilized to construct the right-hand side based on the background electric field.Finally,the secondary coupled-potential formulation in frequency-domain is solved.In this thesis,the electrical field responses excited by a magnetic dipole in full air-space is used as the background for AEM.Its analytical solution can be obtained directly.For MCSEM,the response excited by an electric dipole in a layered medium is used as the background field.This response does not have any analytical solution.Instead,a 1D modeling code is introduced to calculate its semi-analytical solution.Furthermore,the finite-element basis functions are constructed by using the three-dimensional(3D)tensor product of B-spline wavelet on the interval to approximate the unknown potentials in the elements.The numerical models are discretized into structured hexahedral meshes and the governing equations are discretized by Galerkin method to form large linear systems.In this thesis,the direct solver MUMPS is used to efficiently solve the multi-source AEM forward problem.In order to reduce the memory consumption and to accelerate the forward modeling of large-scale MCSEM problems,we introduce the stabilized biconjugate gradient method and algebraic multigrid preconditioner to solve the linear equations system.At last,the EM field at receivers is obtained from potentials by moving least-square interpolation.For AEM modeling,to verify the accuracy and efficiency of the proposed algorithm,different mesh sizes and scales of basis functions are adopted in this thesis based on a half-space model.At the same time,the comparison with the conventional node-and edge-based finite-element methods shows that the proposed algorithm can obtain higher accuracy in less time.The modeling results of a conductive body show that the proposed algorithm agrees well with those of spectral-element method.Furthermore,the AEM responses for several typical models are simulated and analyzed.For MCSEM modeling,the response of a layered model with a high-resistivity imtermediate layer is simulated and the accuracy is verified by comparing with 1D semi-analytical solution.Then,the influence of mesh size and scale of basis functions is analyzed.The efficiency of algebraic multigrid preconditioner is verified by comparing with different preconditioners such as Jacobi preconditioer or SOR preconditioner.Finally,the influence of seawater depth and buried depth of reservoir on MCSEM responses is analyzed.Numerical examples show that the wavelet finiteelement method based on B-spline wavelet on the interval has high accuracy and can approximate the solutions at different scales.Meanwhile,the accuracy can be improved by two strategies,namely refining the mesh and increasing the scale of basis function. |
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