Angle-domain parameters computed via weighted slant-stack

Introduction

Angle-domain common image gathers (ADCIGs), created from downward-continuation or reverse time migration, can provide useful lithological and velocity information (Prucha et al., 1999). In geologically complex areas, poor illumination causes undesirable kinematic effects and amplitude variations along the angle axis (Prucha et al., 2000; Valenciano, 2006).

The subsurface-offset-to-angle transformation consists of a radial trace transform in the Fourier space with some regularization in the angle direction (Sava and Fomel, 2000) or slant-stack in the physical space plus an additional transformation from offset ray-parameter to reflection angle (Prucha et al., 1999). The regularization, to some extent, can diminish the amplitude variation caused by poor illumination. The more accurate solution to the illumination problem, however, is achieved by computing a regularized least-squares inverse image (Clapp, 2005) rather than the simply the adjoint (migration). The inverse image problem can be solved either by computing the Hessian implicitly (Clapp, 2005) or explicitly Valenciano (2006), preferentially, in the reflection-angle domain or, without any physically meaningful regularization direction, in the subsurface-offset domain.

In the reflection-angle domain, the inverse image problem which explicitly computes the Hessian can be performed according to two different strategies. First, by computing the angle-domain Hessian. Valenciano and Biondi (2006) proposed to obtain the angle-domain Hessian by applying the slant-stack technique to compute ADCIGs on the subsurface-offset Hessian. They noticed that the resulting angle-domain Hessian for a model with a Gaussian velocity anomaly lacked the resolution to determine which angles were more illuminated. Recently, Fomel (2003) introduced the theoretical framework of the oriented wave equation, under which computing the angle-domain Hessian could be promising. In the other approach, the angle-domain Hessian can be evaluated by chaining the offset-to-angle operator and the subsurface-offset Hessian (Valenciano and Biondi, 2005). Valenciano (2007), in this report, shows good results obtained by applying this strategy in Sigsbee dataset.

Here, I propose a general framework to map any information computed in the subsurface-offset domain to the angle-domain. The proposed approach relies on the asymptotic nature of the slant-stack transformation from subsurface-offset to angle domain. I first show the validity of the stationary-phase assumption for the offset-to-angle transformation, then describe a weighted transformation from subsurface-offset to reflection-angle domain, and finally illustrate the technique with the transformation of the diagonal of the Hessian in the subsurface-offset domain to the angle domain, yielding amplitude factors to compensate for illumination problems in ADCIGs. Additionally, I show the transformation of some off-diagonal terms which, at present, does not have a direct application in the amplitude correction problem.

Angle-domain parameters computed via weighted slant-stack