3D Frequency-Domain/Time-Domain Airborne EM Forward Modeling Based On Spectral-Element Method With Arbitrary Hexahedral Grids

Recently,airborne electromagnetic(EM)survey is widely implemented and playing an increasing role in the fields of near surface,hydrology,engineering and environmental exploration.The high-resolution of airborne EM and huge dataset,which requires a high-efficiency and high-precision algorithm for the forward modeling and inversion,bring a great challenge to the practical airborne EM data processing and inversion.Now,the airborne EM forward modeling and inversion for 1D model is quite mature,however,the 1D algorithms fail to model and explain the data of the complex practical geological areas.Besides,the existing 3D forward modeling methods are limited by the accuracy and efficiency that restrict the progress of inversion and interpretation for the 3D survey data.To overcome the restriction on the efficiency and accuracy of the algorithms,I put forward 3D frequency-domain and time-domain airborne EM forward modeling method based on spectral-element method for the complex 3D geoelectric models.I apply the Guass-Lobatto-Legendre polynomials as the basis function for the spectral-element method and implement the arbitrary hexahedral elements for the mesh subdivisions.In this thesis,I will investigate how subdivisions of the the physical meshes,the order of the basis functions,and the improvement of the modeling efficiency and accuracy.I will also analyze the modeling ability of the spectral-element method for the earth models with electrical anisotropy,complex anomalous bodies,and 3D topography.In order to achieve the forward modeling for the regular shaped geological bodies,the calculating space is discretized by the regular hexahedral elements.Starting from the vector Helmholtz equation and electromagnetic vector wave equation,and by introducing the full conductivity tensor,the boundary value problem for the frequency-domain and time-domain airborne EM forward modeling can be derived,respectively.Then,with the full conductivity tensor simplified as a scalar conductivity,the boundary value problem for the electrical isotropy earth model can be obtained.To avoid the singularity of the airborne electromagnetic field around the transmitting source,I adopt in this thesis the field separation algorithm.For the frequency-domain and time-domain forward problems,the analytical solution for the full air-space and the semi-analytical solution for the half-earth space are adopted as the primary field,respectively.For the frequency-domain case,the calculating space is discretized by the basis function formed by the Gauss-Lobatto-Legendre polynomials.Based on the Galerkin weighted residual method,the spectral-element governing equation is obtained,and then with the mapping relationship between the physical domain and the reference domain,the element matrix is formed.Considering the property of ? function for the basis function at the collocation points,the reduced-order integration technique is applied to increase the sparsity of the total matrix.To efficiently handle the multi-source airborne EM problem,a direct solver is used.For the time-domain case,a time-domain spectral-element method that combines the unconditionally stable backward Euler scheme and spectral-element method is proposed,which can contribute to a direct solution for the time-domain forward modeling with no time step limitation.Based on the Galerkin weighted residual method and the backward Euler scheme,the time-domain spectral-element governing equation is obtained.Due to the diagonal mass matrix calculated by the reduced-order integration,the calculation of the total matrix and the right-side vector can be optimized.Besides,I implemente a direct solver to solve the equations system,thus for the same time step I can do only in once factorization but multiple back substitutions for different time channels.This can avoid time-consuming multiple factorization and realize a fast time-domain solution.To check the validity of the algorithm,the results for 1D electrical isotropy and anisotropy earth model is compared with the semi-analytical solution,and to further study the high efficiency of this method,I do a comparison of the present spectral-element method with finite-element and finite-difference methods,test the modeling ability of spectral-element method for the “nonlinear” airborne EM response within one cell.In this thesis,in order to synthetically analyze the modeling ability of the spectral-element method for airborne EM data,to study the influence on the modeling accuracy and efficiency caused by the order of the basis function and the physical mesh subdivision,and to discuss the “quadratic” optimizing strategy(the appropriate combination between the physical mesh subdivision and the order of basis function)for the airborne EM forward modeling,the frequency-domain forward modeling is tested.For the typical electrical anisotropic earth model,the characteristic of the EM response is analyzed,and an identification method for the anisotropic EM response is proposed.To check the accuracy of the time-domain spectral-element method,the validity of the four combinations of different orders of basis function and mesh refinements for a layered earth model is tested,and the influence of the mesh refinements and basis function orders on the model is analyzed,I also show the modeling capability of time-domain spectral-element for the multi-source airborne EM responses for 3D earth model at different time channels is demonstrated.To model the electromagnetic response for complex geological bodies and study the flexibility modeling of SE method,in this thesis,I introduce the discretization using deformed hexahedral cells.Then,I establish the frequency-domain and time-domain forward modeling theory based on spectral-element method using the deformed hexahedral elements.To calculate the stiffness and mass matrix for the deformed hexahedral cells,I build the relationship between the physical domain for the geological bodies and the reference domain for the basis function via the Jacobian matrix calculated by the function format.The combination of regular orthogonal and deformed hexahedral cells discretization for the complex geological bodies is adopted,the deformed cells are used to fit the complex boundaries with different physical properties,while the regular cells are applied in the area with the constant physical property.This is to speed up the formation of the matrix.The flexibility of the SE method is proved by the accuracy check of the 1D earth model and multiple 3D numerical experiments,and simultaneously,the analysis of the electromagnetic response characteristic for the complex geological bodies and 3D topography models is presented.To study the feasibility that a simple mesh subdivision can satisfy the modeling for a geophysical earth model,I propose a transformation of the conventional SE method to the idea of the Element-free Galerkin method,in which the conductivity is no longer assumed constant within a cell but instead can be handled by numerical integration that is similar to Element free Galerkin method.The basic theory for the new approach is presented.Via the 1D and 3D earth models,the efficiency and accuracy of the new approach is verified,the rules on how the order of basis function and the order of the numerical integration influence the modeling results is summarized.Further to show the flexibility of the approach,the forward modeling for a complex practical earth model is conducted.In this thesis,I establish an efficient and accurate frequency-domain and time-domain airborne EM forward modeling theory using spectral-element method,and develop the related algorithm,realize the 3D airborne EM forward modeling for anisotropic and complex earth model.This can provide foundation for the fast and efficient inversion of huge dataset for frequency-domain and time-domain airborne EM survey.