Theory of 3-D angle gathers in wave-equation seismic imaging

Introduction

Wave extrapolation provides an accurate method for seismic imaging in structurally complex areas (Biondi, 2006; Etgen et al., 2009). Wave extrapolation methods have several known advantages in comparison with direct methods such as Kirchhoff migration thanks to their ability to handle multi-pathing, strong velocity heterogeneities, and finite-bandwidth wave-propagation effects (Gray et al., 2001). However, velocity and amplitude analysis in the prestack domain are not immediately available for wave extrapolation methods. To overcome this limitation, several authors (Xie and Wu, 2002; Rickett and Sava, 2002; Soubaras, 2003; Sava and Fomel, 2005; Prucha et al., 1999; Mosher and Foster, 2000; Sava and Fomel, 2006; de Bruin et al., 1990; Sava and Fomel, 2003) suggested methods for constructing angle gathers from downward-continued wavefields. Angles in angle gathers are generally understood as the reflection (scattering) angles at reflecting interfaces (Brandsberg-Dahl et al., 2003; Xu et al., 2001). Angle gathers facilitate velocity analysis (Stork et al., 2002; Liu et al., 2001) and can be used in principle for extracting angle-dependent reflectivity information directly at the target reflectors (Sava et al., 2001). Stolk and de Hoop (2002) assert that angle gathers generated with wavefield extrapolation are genuinely free of artifacts documented for Kirchhoff-generated angle gathers (Stolk and Symes, 2002,2004).

There are two possible approaches to angle-gather construction with wavefield continuation. In the first approach, one generates gathers at each depth level converting offset-space-frequency planes into angle-space planes simultaneously with applying the imaging condition. The offset in this case refers to the local offset between source and receiver parts of the downward continued prestack data. Such a construction was suggested, for example, by Prucha et al. (1999). This approach is attractive because of its localization in depth. However, the method of Prucha et al. (1999) produces gathers in the offset ray parameter as opposed to angle. As a result, the angle-domain information becomes structure-dependent: the output depends not only on the scattering angle but also on the structural dip.

In the second approach, one converts migrated images in offset-depth domain to angle-depth gathers after imaging of all depth levels is completed. Sava and Fomel (2003) suggested a simple Radon-transform procedure for extracting angle gathers from migrated images. The transformation is independent of velocity and structure. Rickett and Sava (2002) adopted it for constructing angle gathers in the shot-gather migration. Biondi and Symes (2004) demonstrate that the method of Sava and Fomel (2003) is strictly valid in the 3-D case only in the absence of cross-line structural dips. They present an extension of this method for the common-azimuth approximation (Biondi and Palacharla, 1996).

In this paper, I present a more complete analysis of the angle-gather construction in 3-D imaging by wavefield continuation. First, I show how to remove the structural dependence in the depth-slice approach. The improved mapping retains the velocity dependence but removes the effect of the structure. Additionally, I extend the second, post-migration approach to a complete 3-D wide-azimuth situation. Under the common-azimuth approximation, this formulation reduces to the result of Biondi et al. (2003) and, in the absence of cross-line structure, it is equivalent to the Radon construction of Sava and Fomel (2003).

Theory of 3-D angle gathers in wave-equation seismic imaging