小波束域波场的分解、传播及在地震偏移成像中的应用
摘要
自上个世纪二十年代至今，波场分解、传播与偏移成像技术经历了将近一个世纪的发展，形成了多种不同的方法，如Kirchhoff波动方程高频近似解方法，频率-波数域的相移(phase-shift)方法和相移-内插(PSPI)方法，以及在混合域中基于单向波动方程的相位屏、广义屏方法等。这些方法均采用一组具有全局特性的基本函数对波场进行分解，如占据整个空间的富氏调谐函数或充满各个方向的Dirac函数(点源)，并通过求解基本函数满足波动方程的精确解或近似解来达到波场外推的目的。传统方法的波场分解和传播不论在空间域或波数域都只能在一个域内具有局域化(localization)的特性。例如Dirac函数(点源)在空间域可以有精确的定位，但在传播方向上(波数域)却毫无确定性；而平面波(波数域的基本解)则具有精确的传播方向，但其波前却是无限延伸的，不具任何空间局域性。这些基本函数的全局特性使其在非均匀空间的演化变得十分复杂，从而影响了传统方法的波场外推精度及效率。
认识到全局性传播算子在波场外推中的困难，近二十年来，许多研究都致力于发展基于波场局部空间分解的局域化传播算子以替代传统的全局算子。先进的数学分析理论和技术，尤其是九十年代以来引入的小波变换以及包含离散窗口富里叶变换(WFT)、离散小波变换等信号表示方法的框架理论，为局域化算子的发展提供了坚实的数学基础。基于射线理论的高频渐近射束(复射束、高斯射束)叠加方法，以窗口富里叶变换(WFT)以及小波变换为基础的局部相位-空间域(小波束域)波场外推方法等相继产生。虽然这些局域性方法在实际应用中还存在着一定的困难，但却为波场演化问题的研究提供了新的思路。
本论文通过对WFT、小波基、小波包，以及相关的框架理论等的分析比较，选择了两组形式简单，且具有适宜于波场外推特性的基本函数集合来进行波场分解、传播及偏移成像问题的研究。其一为将高斯窗函数经平移和调制而构成的一组窗口富里叶框架(Gabor-Daubechies框架，或G-D框架)基本函数，另一种为在富里叶分析和小波包理论基础上发展起来的局部余弦基函数。理论分析和数值计算的重点将放在基于前者的波场外推方法上。
论文共分六章，各章节具体内容如下：
第一章为引论，分析了传统的全局化传播算子以及已经发展的局域化波场外推方法的特点及各自存在的问题，说明了选取合适的波场分解基本函数的必要性。在此基础上，阐述了选取G-D框架和局部余弦基作为基本函数进行小波束域波场分解和外推的原因，并对论文的主要内容做了简要概括。
第二章详细介绍了G-D框架及其相关理论，包括框架的定义、G-D框架分解与重构、构造对偶框架的方法以及选取最佳对偶窗口函数的规则等，分析了G-D框架分解与WFT、Gabor变换、正交小波变换的关系。通过对具体信号的分析，对不同变换方法的信号表示效率进行了对比，并总结了G-D框架及对其进行尺度扩展组成的Gabor函数族在应用于波场相关的研究中时，优于其它正交分解方法的特性。
在单向波算子分解和相位屏近似方法的基础上，第三章发展了以空间－波数参
数化的2维（2D）’、波束域波场外推方法，得到了采用＊－D框架和局部余弦基分解
的小波束域局域化传播算于的解析表达式，并基于局部扰动理论，给出了补偿局部
速度扰动的相位修正算子的近似表示。对于采用G－D框架的小波束方法，讨论了
其波场外推的具体实现算法，并通过不同介质点源脉冲响应的计算以及叠前、叠后
偏移成像的数值实验，说明了该方法在波场传播和地震成像中的可行性和有效性。
地震资料处理的一个重要应用是地震偏移成像。地震偏移，尤其是叠前偏移
处理，在地震数据分析处理中己日趋重要。实际应用对复杂地质结构中地震偏移技
术的精度和效率提出越来越高的要求，传统的单一成像结果己不能满足日益增长的
探测模型物理特性的需要。波场在小波束域的分解和传播具有优良的空间和方向局
域特性。基于这些特性，第四章重点探讨了叠前小波束偏移方法的推广和应用，其
中包括射束源和平面源激发、资料合成及偏移的原理和应用，探测孔径相关的方向
性照射和倾角响应分析，目标定向化地震成像，以及通过叠前偏移计算反射系数矩
阵并提取结构局部方向性信息汐局部角度相关的反射系数，局部反射体倾角等）等
方面。分析结果表明叠前小波束偏移在改善成像质量、目标定向和结构定向的地震
成像，以及提取结构局部方向性信息如局部角度相关的反射系数，局部反射体倾
角等）的研究中都具有明显的潜在应用价值。
二至四章的理论分析和数值计算均是基于ZD模型进行的。从理论上来看，小
波束波场外推方法可以直接推广应用于3D模型，但在具体算法实现中，尤其对于
具有冗余度的G－D框架分解，还需要在技术上做特殊考虑以获得具有实用价值的
3D计算方法。第五章针对3D模型推导了基于G－D框架分解的小波束域波场自由
传播算于的解析表达式，给出了局部空间小波束混合域补偿局部速度扰动的相位
修正公式，并通过均匀模型和分层模型点源脉冲响应的计算对传播算子进行了检
验。实际模型的叠后偏移成像结果
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 the
Heisenberg 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
认识到全局性传播算子在波场外推中的困难，近二十年来，许多研究都致力于发展基于波场局部空间分解的局域化传播算子以替代传统的全局算子。先进的数学分析理论和技术，尤其是九十年代以来引入的小波变换以及包含离散窗口富里叶变换(WFT)、离散小波变换等信号表示方法的框架理论，为局域化算子的发展提供了坚实的数学基础。基于射线理论的高频渐近射束(复射束、高斯射束)叠加方法，以窗口富里叶变换(WFT)以及小波变换为基础的局部相位-空间域(小波束域)波场外推方法等相继产生。虽然这些局域性方法在实际应用中还存在着一定的困难，但却为波场演化问题的研究提供了新的思路。
本论文通过对WFT、小波基、小波包，以及相关的框架理论等的分析比较，选择了两组形式简单，且具有适宜于波场外推特性的基本函数集合来进行波场分解、传播及偏移成像问题的研究。其一为将高斯窗函数经平移和调制而构成的一组窗口富里叶框架(Gabor-Daubechies框架，或G-D框架)基本函数，另一种为在富里叶分析和小波包理论基础上发展起来的局部余弦基函数。理论分析和数值计算的重点将放在基于前者的波场外推方法上。
论文共分六章，各章节具体内容如下：
第一章为引论，分析了传统的全局化传播算子以及已经发展的局域化波场外推方法的特点及各自存在的问题，说明了选取合适的波场分解基本函数的必要性。在此基础上，阐述了选取G-D框架和局部余弦基作为基本函数进行小波束域波场分解和外推的原因，并对论文的主要内容做了简要概括。
第二章详细介绍了G-D框架及其相关理论，包括框架的定义、G-D框架分解与重构、构造对偶框架的方法以及选取最佳对偶窗口函数的规则等，分析了G-D框架分解与WFT、Gabor变换、正交小波变换的关系。通过对具体信号的分析，对不同变换方法的信号表示效率进行了对比，并总结了G-D框架及对其进行尺度扩展组成的Gabor函数族在应用于波场相关的研究中时，优于其它正交分解方法的特性。
在单向波算子分解和相位屏近似方法的基础上，第三章发展了以空间－波数参
数化的2维（2D）’、波束域波场外推方法，得到了采用＊－D框架和局部余弦基分解
的小波束域局域化传播算于的解析表达式，并基于局部扰动理论，给出了补偿局部
速度扰动的相位修正算子的近似表示。对于采用G－D框架的小波束方法，讨论了
其波场外推的具体实现算法，并通过不同介质点源脉冲响应的计算以及叠前、叠后
偏移成像的数值实验，说明了该方法在波场传播和地震成像中的可行性和有效性。
地震资料处理的一个重要应用是地震偏移成像。地震偏移，尤其是叠前偏移
处理，在地震数据分析处理中己日趋重要。实际应用对复杂地质结构中地震偏移技
术的精度和效率提出越来越高的要求，传统的单一成像结果己不能满足日益增长的
探测模型物理特性的需要。波场在小波束域的分解和传播具有优良的空间和方向局
域特性。基于这些特性，第四章重点探讨了叠前小波束偏移方法的推广和应用，其
中包括射束源和平面源激发、资料合成及偏移的原理和应用，探测孔径相关的方向
性照射和倾角响应分析，目标定向化地震成像，以及通过叠前偏移计算反射系数矩
阵并提取结构局部方向性信息汐局部角度相关的反射系数，局部反射体倾角等）等
方面。分析结果表明叠前小波束偏移在改善成像质量、目标定向和结构定向的地震
成像，以及提取结构局部方向性信息如局部角度相关的反射系数，局部反射体倾
角等）的研究中都具有明显的潜在应用价值。
二至四章的理论分析和数值计算均是基于ZD模型进行的。从理论上来看，小
波束波场外推方法可以直接推广应用于3D模型，但在具体算法实现中，尤其对于
具有冗余度的G－D框架分解，还需要在技术上做特殊考虑以获得具有实用价值的
3D计算方法。第五章针对3D模型推导了基于G－D框架分解的小波束域波场自由
传播算于的解析表达式，给出了局部空间小波束混合域补偿局部速度扰动的相位
修正公式，并通过均匀模型和分层模型点源脉冲响应的计算对传播算子进行了检
验。实际模型的叠后偏移成像结果
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 the
Heisenberg 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
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