Time-shift imaging condition for converted waves

Imaging condition

If we make the assumption that seismic data consists of singly-scattered reflections, we can describe migration as a succession of two steps:

Wavefield extrapolation, in which step we construct source and receiver wavefields from synthetic or recorded data. The source and receiver wavefields are four-dimensional objects denoted by the symbols ${ \bf u}_s({ \bf m},t)$ and ${ \bf u}_r({ \bf m},t)$, where ${ \bf m}$ indicates position in a three-dimensional space, and $t$ indicates time. In typical migration procedures, the four-dimensional objects ${ \bf u}_s$ and ${ \bf u}_r$ are not stored explicitly, but they are computed on-the-fly as needed for imaging at a given position ${ \bf m}$ in space.

Imaging condition, in which step we extract reflectivity information by comparing the source and receiver wavefields. A useful imaging condition produces a map of reflectivity at all locations in space, function of angles of incidence and reflection. This information can be employed in migration velocity analysis (MVA) and amplitude-versus-angle analysis (AVA).

We can distinguish two parts of an imaging condition: wavefield comparison and angle-decomposition. In the first part, we explore the match of source and receiver wavefields and build objects containing reflectivity information. Cross-correlation at every location in space is an example of wavefield comparison. In the second part, we extract the actual angle-dependent reflectivity information from images produced by space-time wavefield cross-correlation.

We can look at an imaging procedure formulated in this framework as an exercise in matching of two four-dimensional objects. Fundamentally, there is no difference between the four coordinate axes, except for their physical meaning. We can exploit this similitude of coordinate axes in formulating generic wavefield comparison procedures. Deconvolution or cross-correlation are just two particular options. Angle-decomposition, however, requires physical interpretation of the four coordinate axes to extract meaningful information about reflection angles. We exploit those physical relations to derive the formulas presented in this paper.

All types of migration procedures for converted waves, including Kirchhoff migration, migration by wavefield extrapolation, reverse-time migration etc. can be formulated in this framework.

Time-shift imaging condition for converted waves