均匀介质中的F-K反偏移：基本概念、基本公式及其在非均匀介质中的应用

F-K Demigration in Media with Constant Velocity:Basic Concepts,Formulas,and Applications in Inhomogeneous Media

从经典的常速度F-K偏移成像理论公式出发,通过由垂直波数到角频率映射的途径首先建立了常速度F-K反偏移的基本理论公式和基本实现算法,然后再将其用于解决非均匀介质中的反偏移问题。与同样条件下的Kirchhoff型反偏移理论相比,在建立常速度F-K反偏移理论时没有引入任何近似。因此,所提出的是一个在均匀介质中严格精确的反偏移理论。对于一般条件下的非均匀介质,利用了在现代反射地震偏移理论研究中常用的局部化处理方法和相移加插值(PSPI)偏移的基本思想。具体地讲,在处理非均匀介质中的反偏移问题时假设偏移场的形成完全由局部薄板(层)中的速度结构决定。此外,还假设偏移场是速度的连续函数。因此,可以在局部薄板(层)中利用关于垂直坐标的Fourier变换和均匀介质中的频散关系,以及一组均匀参考速度来近似地构造出在不均匀速度模型中任意一个给定深度上的反偏移场。在算法上,在均匀介质中的F-K反偏移是一个一步过程,而在非均匀介质中基于PSPI的F-K反偏移是一个多步递归过程。为了实现反偏移必须要从速度模型的最大深度开始逐层地进行插值和反向相移计算,直至到达地表为止。

Starting from the formulas in classical F - K migration, we present an F - K demigration theory in constant velocity media and the corresponding implementation steps as well as its applications in inhomogeneous media, by way of mapping wavenumbers into angular frequencies. In comparison to the Kirchhoff type demigration theory under the same conditions, the theory given here contains no approximations. Therefore, it is an exact demigration theory for models with constant velocity. For general inhomogeneous media, we use the method of the localization, which is widely used in establishing migration theory in inhomogeneous media, and the basic idea of the phase shift plus interpolation （PSPI） migration. Specifically, for treating the demigration problem in inhomogeneous media we assume that the migrated image is thoroughly determined by the velocity distribution in a thin slab corresponding to the current downward extrapolation interval. Furthermore, it is assume that the demigrated wavefield is a continuous function of the velocity. As a result, we can use the local （windowed） Fourier transform with respect to the depth interval under consideration and a set of constant reference velocities to construct the approximate solution to the demigration problem in inhomogeneous media. In algorithmic aspect, the constant velocity F-K demigration is a one-step procedure without recursion, and the varying velocity F - K demigration is a multi-step recursive procedure. Specifically, in a varying velocity F - K demigration the computation should be started at the bottom of the model and realized in a layer-by-layer way from the bottom of the model towards the surface, until the surface is reached.