Angle domain common image gathers for steep reflectors

Introduction

Velocity estimation plays a key role in seismic imaging. A typical migration velocity analysis method includes three steps: (1) migrations using the background velocity are run to obtain angle domain CIGs; (2) curvatures are estimated from angle domain CIGs by residual moveout analysis; (3) curvature information is inverted to velocity update by back projection.

Assumptions about the subsurface are made during most velocity estimation methods, such as a horizontal stratified earth for NMO or flat reflectors by Toldi (1985). Even recently, for a sophisticated tomography, only reflectors that are not very steep are chosen for velocity estimation and are assumed to be flat in residual moveout analysis (Clapp, 2000).

However, CIGs of steeply dipping reflectors are important in velocity analysis. The angular coverage of the rays illuminating near-flat reflectors is very limited. Most rays travel in a direction that is less than $30^\circ$ from the vertical direction. Therefore, seismic reflection tomography is a limited angle tomography, an ill-posed problem (Tam and Perez-Mendez, 1981). Since most rays are almost vertical, the vertical resolution in seismic reflection tomography is very limited (Clapp, 2000). In contrast, most waves illuminating steeply dipping reflectors have a part of wave-path that is almost horizontal. Therefore using angle domain CIGs of steep reflectors improves the angle coverage of rays in tomography. As a consequence, this reduces the poor condition and improves the stability of the problem. It also leads to less artifacts caused by low angular coverage and better vertical resolution of the resulting velocity.

Angle domain CIGs of steeply dipping reflectors are also useful for anisotropy parameter estimation. VSP and check shots are usually used to improve the angular coverage in anisotropy parameter inversion (Bear et al., 2005) in addition to the reflectors picked in conventional reflection tomography. Since angle domain CIGs of steep reflectors also broaden the angular coverage, they help to constrain anisotropy parameter estimation. It is well known that the anisotropy parameter $\delta$ is mainly constrained by waves traveling close to the vertical direction but the anisotropy parameter $\eta$ is mainly constrained by waves traveling close to the horizontal direction. Therefore, CIGs of reflectors that are almost flat is useful for estimating the parameter $\delta$ but the estimation for the parameter $\eta$ estimation needs CIGs of steeply dipping reflectors.

Downward-continuation migration is routinely applied in the industry. However, it is difficult to obtain reliable angle domain CIGs of steeply dipping reflectors by conventional downward continuation migration (Biondi and Shan, 2002; Biondi and Symes, 2004). Downward continuation migration is based on the one-way wave equation, so it can propagate waves traveling almost vertically well but it cannot propagate waves traveling almost horizontally accurately. But waves illuminating steeply dipping reflectors travel almost horizontally or even overturn before or after they bounce. Reverse-time migration can image steeply dipping reflectors and provide robust angle domain CIGs by using the vertical subsurface offset in addition to the horizontal subsurface offset (Biondi and Shan, 2002).

Shan and Biondi (2004) have demonstrated that plane-wave migration in tilted coordinates is an effective tool to image steeply dipping reflectors. In this paper, we discuss how to produce reliable angle domain CIGs using plane-wave migration in tilted coordinates. We use the BP velocity benchmark dataset to compare angle domain CIGs from reverse-time migration and plane-wave migration in tilted coordinates. Before we discuss plane-wave migration in tilted coordinates, we briefly review how to generate angle domain CIGs by downward continuation migration and reverse-time migration.

Angle domain common image gathers for steep reflectors