The form of the reflection operator and extended image gathers for a dipping plane reflector in two space dimensions

The reflection operator for a simple flat-lying interface can be thought of as the set of all its plane-wave reflection coefficients or as the set of virtual surveys with sources and receivers along the interface. When there is dip, however, it is necessary to include the varying effects of propagation between the virtual-survey level and the interface. Hence, step one in this paper is to derive the reflection operator for a dipping plane interface as observed at a datum level some distance away. The key assumption is that the aperture at the datum level is sufficient to characterize the reflector properties around a particular point. This translates into an assumption that the dip is moderate, though no explicit small-angle approximation is required. The second step is to find the apparent reflection operator that would relate data that have been extrapolated from the datum towards and possibly beyond the reflector using an assumed migration velocity. This apparent reflection operator is closely related to extended common-image gathers. The apparent reflection operator may be analysed asymptotically in terms of rays and other signals, shedding light on the structure of extended image gathers. In keeping with the virtual-survey idea, the results are considered in a subsurface space-time or slowness-time domain at various extrapolation levels around the interface. An important distinction is drawn between using subsurface midpoint-offset coordinates and the wavefield coordinates of the incident and reflected waves. The latter reveal more clearly the effects of dip, because they lead to a more asymmetric apparent reflection operator. Properties such as an up-dip shift of a traveltime minimum and its associated curvature theoretically provide information about the reflector location and dip and the migration-velocity error. The space-time form of the reflection operator can be highly intricate around the offset-time origin and it was described for a simple flat interface in a background paper. To avoid a layer of mathematics, the reflection-operator formulas presented here are in the intermediate space-frequency domain. They are analysed by considering their stationary-phase and branch-point high-frequency contributions. There is no Born-like assumption of weak reflector contrast and so wide-angle, total reflection and head-wave effects are included. Snell’s law is an explicit part of the theory. It is hoped that the work will therefore be a step towards the goal of unifying amplitude-versus-offset, imaging and waveform inversion.