Beamspace Wave Field Decomposition, Propagation, And Seismic Migration Imaging Applications

From 1920s up to Now, various techniques and methods for wave field decomposition, propagation and migration/imaging have been well developed, such as the Kirchhoff asymptotic method, frequency-wavenumber domain phase-shift and phase-shift-plus-interpolation methods, and the one-way wave equation based phase-screen and generalized screen methods, etc. Wave field extrapolation in these methods is implemented based on the expansion of the wave field by sets of basic functions like spatial Fourier harmonies, modes, and Green's functions. Essential to their utility is the requirement that the evolution of the basic functions through the propagation environment constitutes a simplified problem with an exact or approximate closed form solution to the original wave equation. The evolution of a spatial Fourier harmonic through a homogeneous medium is governed by a reduced wave equation, obtained by applying the Fourier transform to the Helmholtz equation, with a simple and well-known solution - the plane wave. The Green's function for point-source excitation is also very simple in homogeneous media. However, since global basic functions like plane waves occupy the entire domain and point source excitation radiates to all directions, their evolution through a non-homogeneous medium constitutes a problem that may become at least as difficult to solve as that of the propagation of the total field. From the mathematical point of view, the difficulty stems from the inability to get, with the traditional global transform like the Fourier transform, a simplified differential equation that governs the evolution of the basic function through the non-homogeneous medium.The recognition of the global nature of propagators as the difficulties' physical genesis led researchers to develop and investigate propagation methods based on the expansion of the total field by sets of spatially confined wave functions, or in other words, to construct localized wave field propagators instead of the traditional global propagators. Advanced mathematical technologies, especially the newly developed wavelet transform and the frame theory, provide a solid foundation for such an effort. The ray-theory based beam-summation method, such as the complex source-generated beam and the Gaussian beam methods, and the local phase-space domain (beamlet domain) wave field extrapolation methods employing windowed Fourier transform (WFT) or wavelet transform are proposed consequently. Although there still remain some considerable difficulties in the applications of these localized methods to the practical use, a new framework tailored for the development of localized propagators has been constructed thereafter.In this thesis, we follow the idea of the beamlet-domain wave field extrapolation methods to construct localized propagators. Through comparative study of signal decomposition efficiency using different representation schemes, we select two groups of basic functions with simple expressions and good localization properties for wave field decomposition, propagation and imaging. One is the non-orthogonal Gabor-Daubechies frame, or G-D frame, a complete set of discrete window Fourier functions which are constructed by space-shifting and harmonically modulating a Gaussian window. Although a G-D frame is not an orthogonal basis, it bears considerable advantages for the study of physical problems, especially those related to the wave field extrapolation, due to the optimal localization properties of the Gaussian window function under theHeisenberg uncertainty principle. The other is the local cosine bases developed as a kind of orthogonal basis based on the Fourier analysis and wavelet-packet theory. In this thesis, theoretical analysis and numerical applications are mainly focused on the beamlet-domain wave field extrapolation using G-D frame propagators.The whole thesis consists of six chapters. The essential content of each chapter is outlined as follows:Chapter One: Introduction. The properties and difficulties of the traditional global propa...