Transformation to dip-dependent Common Image Gathers

Biondo Biondi and William Symes

biondo@sep.stanford.edu

ABSTRACT

We introduce a new transform of offset-domain Common Image Gathers (CIGs) obtained by wavefield-continuation migration methods. This transformation can be applied to either horizontal-offset CIGs or vertical-offset CIGs. It overcomes the limitations that both kinds of CIGs suffer in the presence of a wide range of reflectors' dips. The result of our transformation is an image cube that is equivalent to the image cube that would have been computed if the offset direction were aligned along the apparent geological dip of each event. The proposed transformation applies a non-uniform dip-dependent stretching of the offset axis and can be efficiently performed in the Fourier domain. Because it is dependent on the image's apparent dip, the offset stretching automatically corrects for the image-point dispersal. Tests on a synthetic data set confirm the potential advantages of the transformation for migration velocity analysis of data containing steeply dipping reflectors.

The analysis of Common Image Gathers (CIG) is an essential tool for updating the velocity model after depth migration. When using wavefield-continuation migration methods, angle-domain CIGs (ADCIGs) are usually used for velocity analysis Clapp and Biondi (2000). The computation of ADCIG is based on slant-stack transformation of the wavefield either before imaging Prucha et al. (1999) or after imaging Biondi and Shan (2002); Rickett and Sava (2001); Sava et al. (2001). In either case, the slant stack transformation is usually applied along the horizontal offset axis.

However, when the geological dips are steep, this ``conventional'' way of computing CIGs does not produce useful gathers, even if it is kinematically valid for all geological dips that are milder than 90 degrees. As the geological dips increase, the horizontal-offset CIGs (HOCIGs) degenerate, and their focusing around zero offset blurs. This limitation of HOCIGs led both of the authors to independently propose a partial solution to the problem; that is, the computation of CIG along a different offset direction than the horizontal one, and in particular along the vertical direction Biondi and Shan (2002). Unfortunately, neither set of angle-domain gathers (HOCIG ad VOCIG) provides useful information for the whole range of geological dips, making their use for velocity updating awkward. While VOCIG are a step in the right direction, they are not readily usable for migration velocity analysis (MVA).

In this paper we present a new method to transform a set of CIGs (HOCIGs and/or VOCIGs) into another set of CIGs. The resulting image cube is equivalent to the image cube that would have been computed by aligning the offset direction along the local geological dips. This transformation applies a non-uniform dip-dependent stretching of the offset axis and can be cheaply performed in the Fourier domain. Because the offset stretching is dependent on the reflector's dip, it also automatically corrects for the image-point dispersal. It thus has the potential to improve substantially the accuracy and resolution of residual moveout analysis of events from dipping reflectors. It has been recognized for long time Etgen (1990) that image-point dispersal is a substantial hurdle in using dipping reflections for velocity updating.

The proposed transformation is dependent on the apparent dips in the image cube, and creates an image cube in which the effective offset depends on those apparent dips in an ``optimal'' way. We will thus refer to the resulting CIGs as dip-dependent offset CIGs (DDOCIGs), and to the transformation as the ``transformation to DDOCIGs.''

The proposed method is independent from the particular migration method used to obtain the CIGs. The input offset-domain image cube can be computed by either downward-continuation migration (shot profile or survey sinking) or reverse-time migration Biondi (2002). The proposed transformation should also improve the accuracy and resolution of velocity analysis applied only to ``conventional'' HOCIGs. In its most immediate application, it should also improve the image obtained by stacking after a residual moveout correction.

The next section illustrates the problem of HOCIGs and VOCIGs with a real data set from the North Sea that was recorded above a steeply dipping salt edge. The following section introduces the new transformation, that is then tested on a synthetic data set.