Introduction

Imaging is the combined process of migration and velocity analysis. The final image provides two important pieces of information about the subsurface: its structure and some of its rock properties. To obtain a reliable image, we need a reliable velocity model. Therefore, the image process becomes a combined procedure between migration and migration velocity analysis.

The final image by itself provides information about the accuracy of the velocity model. This information is present in the redundancy of the seismic data, that is in non-zero-offset images. The information is distributed along a 3-dimensional image space, for 2D seismic data; the coordinates of this space are $I(z_\xi,m_\xi,h)$. The subsets of this image for a fixed image point ($m_\xi$) with coordinates $(z_\xi,h)$ are known as common-image gathers (CIG), or common-reflection-point gathers (CRP). If the CIGs are a function of $(z_\xi,h)$, the gathers are also referred to offset-domain common-image gathers (ODCIG). These gathers can also be expressed in terms of an opening angle $\gamma$, by transforming the offset axis (h) into the opening angle ($\gamma$) to obtain a common image gather with coordinates ($z_\xi$,$\gamma$); these gathers are known as Angle-Domain Common-Image Gathers (ADCIG) Biondi and Symes (2004); Brandsberg-Dahl et al. (1999); Prucha et al. (1999); Rickett and Sava (2002); Sava and Fomel (2003); de Bruin et al. (1990).

There are two kinds of ODCIGs: those produced by Kirchhoff migration, and those produced by wavefield-extrapolation migration, referred to from now on as 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=h_D), and involves the concept of flat gathers. For wave-equation ODCIGs, the offset dimension is a model parameter ($h=h_\xi$), and involves the concept of focused events. In this paper, we will refer to these gathers as subsurface offset-domain common-image gathers (SODCIG).

There are problems observed with ODCIGs, which can be alleviated by parameterizing the offset axis into an angle axis to form angle-domain common image-gathers. Unlike ODCIGs, ADCIGs produced with either method have similar characteristics, since they describe the reflectivity as a function of the angle at the reflector.

Depending on the seismic experiment we are analyzing, the coordinates of the image space possess different information relevant to the experiment. We refer to a conventional seismic reflection experiment, where the source and the receiver have the same type of wave, as single-mode case. The transformation from ODCIGs to ADCIGs is a well-known process in the literature, and in this case the angle axis represents the true reflection angle.

A seismic experiment where the source and the receiver process different types of waves is known as multi-component seismic, throughout this paper, we refer to this experiment as converted-mode case, as for example the conversion from P wave into an S wave at the reflection point. This paper discusses the common-image gathers for this kind of experiment, focusing mainly on SODCIGs and their accurate transformation into ADCIGs. We validate our results by generalizing the concepts of Kirchhoff migration.

A final side product of our analysis is the ability to separate the final image into two parts, each one corresponding to a distinctive wave. Throughout this process, the velocity ratio between the different velocities plays an important role in the transformation. We present and analyze the kinematics of our equations and present some synthetic results.