Forward Modeling and Inversion of 3D Electromagnetic Scattering Problems in Complicated Backgrounds

In this dissertation, four topics will be presented: (1) The volume integral eqaution method with the domain decomposition method (VIE-DDM) and the inversion using VIE-DDM as forward solver; (2)The numerical mode matching method with surface current boundary condition (NMM-SCBC); (3) NMM-VIE-DDM.VIE-DDM and the inversion: In many applications, electromagnetic scattering from inhomogeneous objects embedded in multiple layers needs to be simulated numerically. The straightforward solution by the method of moments (MoM) for the volume integral equation method is computationally expensive. Due to the shift-invariance and correlation properties of the layered-medium Green's functions, the Stabilized Bi-Conjugate Gradient Fast Fourier Transform (BCGS-FFT) has been developed to greatly reduce the computational complexity of the MoM, but so far this method is limited to objects located in a homogeneous background or in the same layer of a layered medium background. For those problems with objects located in different layers, FFT cannot be applied directly in the direction normal to the layer interfaces, thus the MoM solution requires huge computer memory and CPU time. To overcome these difficulties, the BCGS-FFT method combined with the domain decomposition method (DDM) is proposed in this work. With the BCGS-FFT-DDM, the objects or different parts of an object are first treated separately in several subdomains, each of which satisfies the 3D shift-invariance and correlation properties; the couplings among the different objects/parts are then taken into account, where the coupling matrices can be built to satisfy the 2D shift-invariance property if the objects/subdomains have the same mesh size on the xy-plane. Hence, 3D FFT and 2D FFT can respectively be applied to accelerate the self- and mutual-coupling matrix-vector multiplications. By doing so, the impedance matrix is explicitly formed as one including both the self- and mutual-coupling parts, and the solver converges well for problems with considerable conductivity contrasts. The computational complexity in memory and CPU time can be significantly reduced. Using the BCGS-FFT-DDM as the forward solver, the inversion algorithm based on the Born approximation method and Born iterative method are developed to reconstructed the size, location and properties of the targets buried underground.

NMM-SCBC: The NMM method is widely employed in well-logging problems, because it can transform the original 2.5D problem to a 1D eigenvalue problem at the radial direction, which is usually treated with finite element method (FEM), and a semi-analytical problem at the z direction, which can be easily dealt with the mode matching strategy, and therefore the computational load is significantly reduced. However, more and more well-logging problems are equipped with carbon steel casing, which is extremely thin but with extremely high conductivity. With the conventional NMM method, the extremely thin casing will make the mesh for the FEM tremendously dense and the extremely high conductivity will make the matrices for the eigenvalue problem near ill-posed, both of which will make the solution of the eigenvalue problem inefficient and inaccurate. To overcome this problem, we proposed to apply the SCBC to substitute the extremely thin and highly conductive casing. To employ the NMM-SCBC, the mixed-order FEM isdeveloped to treat the 1D eigenvalue problem, in which the SCBC is deliberately applied. After the eigenvalues and the eigenvectors are solved for each horizontal layer, the mode matching strategy will be applied across the horizontal layers as the conventional NMM method.

MM-VIE-DDM: In many applications, electromagnetic scattering from inhomogeneous objects embedded in multiple layers with cylindrical geometry needs to be simulated numerically. The Numerical Mode Matching (NMM) method has long been The numerical mode matching (NMM) method has long been demonstrated to be the most efficient algorithm for the application under the axial symmetric background, compared with the full 2D methods (such as finite element, finite difference, integral equation method etc.) However, it only works well for the problems with axial symmetric background. For the problem with both cylindrical geometry and 3D objects (such as borehole with reservoirs, fractures), full 3D solvers (such as BCGS-FFT-DDM) can be applied, but will require tremendously large memory and CPU time. The combination of the NMM method and the BCGS-FFT-DDM, or NMM-BCGS-FFT-DDM, is proposed in this paper to deal with the limitations of each method alone. Following the general work flow of the conventional BCGS-FFT-DDM, we substitute the incident field and the dyadic Green's function from the objects to the receivers in the layer media with those considering the cylindrical structures, which are obtained from the NMM method, and neglect the impacts from the cylindrical structures when calculating the total fields inside the objects. Reciprocity and interpolation will be utilized to speed up the calculation when obtaining the Green's function from the objects to the receivers. With the proposed method, the problems with objects in layer media with cylindrical structure can be treated efficiently and accurately. Some numerical results are presented to show the capability of this method.