Angle gathers in wave-equation imaging for transversely isotropic media

Introduction

Angle gathers have gained prominence as they provide wave equation imaging methods with an outlet to perform velocity analysis. Angle gathers also alleviate the limitations that offset gathers have 2in handling with multi pathing (Stolk and de Hoop, 2006; Mosher and Foster, 2000; de Bruin et al., 1990). An angle gather decomposition for anisotropic media will allow us to export these features to the anisotropic world, and this is especially important considering the number of parameters we need to deal with in anisotropic media and the prevalent multi-pathing that takes place in such media.

Downward wave extrapolation provides an accurate method of seismic imaging in structurally complex areas. Downward wave extrapolation is also naturally formulated to produce angle gathers (Soubaras, 2003; Xie and Wu, 2002; Rickett and Sava, 2002; Mosher and Foster, 2000; de Bruin et al., 1990; Sava and Fomel, 2005; Biondi and Symes, 2004; Sava and Fomel, 2003). Fomel (2004) showed that structural dependence can be removed in a depth-slice approach to extracting angle gathers. Specifically, one can generate gathers at each depth level, converting offset-space-frequency planes into angle-space planes and applying simultaneously the imaging condition. The improved mapping retains velocity dependence but removes the effect of the structure. Because of its ray-parameter-based (Fourier) formulation, this approach lends itself naturally to an anisotropic phase-velocity extension.

Migration velocity analysis in anisotropic media remains a challenging and open issue. Multiple parameters are needed to represent the anisotropic model. For a transversely isotropic medium with vertical axis of symmetry (VTI media), only NMO velocity ($v$ ) and the non-elliptic parameter ($\eta$ ) predominantly influence imaging (Alkhalifah et al., 2001; Alkhalifah and Tsvankin, 1995). Vertical velocity ($v_z$ ) controls mainly placement of the image in depth. Nevertheless, estimating even two parameters that can vary laterally and vertically from image gathers is difficult. Angle gathers provide an opportunity to use residual moveout to help update anisotropic parameters. Biondi (2007) suggested an approach to extracting angle gathers in anisotropic media from post-migration data. Biondi's formulation is based on numerical calculation of angle gathers and relies on ray information that is hard to examine analytically.

In this paper, we develop an analytical formulation for extracting angle gathers in VTI 2D media. We use the depth-slice approach to angle gathers as a platform for the extension 2to VTI. Angle gather mapping depends strongly on anisotropic parameters. We analyze this dependency using numerical computations. Next, 2we show synthetic data examples that confirm the theoretical analysis. Finally, we explain the possible extension of our approach to TTI (tilted transversely isotropic) media. An extension to 3D can be achieved along the lines of Fomel (2004).

Angle gathers in wave-equation imaging for transversely isotropic media