Efficient Computation of Layered Medium Green's Function and Its Application in Geophysics
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This dissertation is mainly focused on the efficient computation of Green's function and electromagnetic scattering problems in layered media. The layered medium Green's functions (LMGFs), together with various algorithms based on them, are particular powerful methods to solve electromagnetic problems in layered media. The significance of this work is to understand the physics of the electromagnetic wave propagation and interaction with complex scatterers in multi-layered media, and also to arrive at a few fast and efficient computational algorithms. The LMGFs can be categorized into traditional ones and mixed-potential ones. The first type of Green's function directly calculates the field due to a dipole source and the resulting semi-analytic Sommerfeld integrals (SIs) appearing in the traditional Green's function components can be evaluated extremely fast and accurately, finding various applications in forward and inverse modeling in geophysical prospecting. The mixed-potential LMGF is mostly useful for the formulation of system matrices appearing in method of moments for solving scattering problems in layered media because of its less singular nature. For both types of LMGFs, a few techniques, such as asymptotic analysis, singularity extraction, and the weighted average method are developed for the fast convergence of the resulting generalized Sommerfeld integrals. The calculation of correction term added to the scalar potential for vertical currents, give rise to a few efficient methods for the evaluation of the so-called half-line source potential (HLSP). In order to further reduce matrix fill time when solving 3-D problems using the method of moments, a 3-D simplex interpolation approach is developed for interpolating the spectral-domain integral terms. A simple but efficient approach is to allocate an interpolation table covering all appropriate source and test point combinations, but to populate data points on the fly only as needed. To remove all unbounded singularities a second-level singularity extraction is also needed for curl-type operators to further regularize the spectral integrals, permitting a uniform tabulation density. Three types of problem: geophysical prospecting, antenna, and radiation problems, are employed to demonstrate the accuracy and efficiency of our algorithm. In each case, good agreements are achieved between our results and results completed by independent approaches.