The imaging operator transforms the data, which is in data-midpoint position, data-offset, and time coordinates [(mD,hD,t)], into an image that is in image-midpoint location, offset, and depth coordinates [()]. This image provides information about the accuracy of the velocity model. This information is present in the redundancy of the prestack seismic image, (i.e. non-zero-offset images). The subsets of this image for a fixed image point () with coordinates are known as common-image gathers (CIGs), or common-reflection-point gathers (CRPs). If the CIGs are a function of , the gathers are also referred as offset-domain common-image gathers (ODCIGs). The common-image gathers can also be expressed in terms of the opening angle , by transforming the offset axis (h) into the opening angle () to obtain a common-image gather with coordinates (,); these gathers are known as Angle-Domain Common-Image Gathers (ADCIGs) Biondi and Symes (2004); Brandsberg-Dahl et al. (1999); de Bruin et al. (1990); Prucha et al. (1999); Rickett and Sava (2002); Sava and Fomel (2003).
There are two kinds of ODCIGs: those produced by Kirchhoff migration, and those produced by wave-equation migration. There is a conceptual difference in the offset dimension between these two kinds of gathers. For Kirchhoff ODCIGs, the offset is a data parameter (h=hD), and involves the concept of flat gathers. For wave-equation ODCIGs, the offset dimension is a model parameter (), and involves the concept of focused events. In this chapter, I will refer to these gathers as subsurface offset-domain common-image gathers (SODCIGs). Imaging artifacts due to multipathing are present in ODCIGs. However, an event in an angle section uniquely determines a ray couple, which in turn uniquely locates the reflector. Hence, the image representation in the angle domain does not have artifacts due to multipathing Stolk and Symes (2002); Clapp (2005). Unlike ODCIGs, ADCIGs produced with either Kirchhoff methods or wave-equation methods have similar characteristics, since the ADCIGs describe the reflectivity as a function of the reflection angle.
This chapter also presents the option to transform the PS-ADCIGs into two angle-domain common-image gathers. The first angle-gather is function of the P-incidence angle, the second one is function of S-reflection angle. I refer to these two angle gathers as P-ADCIGs and S-ADCIGs, respectively. Throughout this process, the ratio between the different velocities plays an important role in the transformation. I present the equations for this mapping and show results on a synthetic data set. I also present results on a 2-D real data set from the Mahogany field in the Gulf of Mexico. For this exercise a comparison between the PZ-ADCIGs and the PS-ADCIGs yield information to improved the PS image.