Plane-wave migration in tilted coordinates

Next: Wavefield extrapolation in tilted Up: Shan and Biondi: Plane-wave Previous: One-way wave equation migration

# Plane-wave source migration

Shot gathers can also be synthesized into a new dataset to represent a physical experiment that is not performed in reality. One of the most important examples is to synthesize shot gathers into plane-wave source gathers. A plane-wave source gather represents what would be recorded if a planar source were excited at the surface with geophones covering the whole area. It can also be regarded as the accurate phase-encoding of the shot gathers (Liu et al., 2002). Plane-wave source gathers can be generated by slant-stacking receiver gathers. The process can be described as follows:
 (6)

where is the ray parameter for the -axis, is the source location, and is the receiver location at the surface. Its corresponding plane-wave source wavefield at the surface is
 (7)

As with the Fourier transformation, we can transform the plane-wave source gathers back to shot gathers by inverse slant-stacking (Claerbout, 1985) as follows:
 (8)

In contrast to the inverse Fourier transformation, the kernel of the integral is weighted by the angular frequency . This inverse transformation weighting function is also called filter in Radon-transform literature.

As with shot-profile migration, there are two steps to migrate a plane-wave source gather by a typical plane-wave migration method. First, the source wavefield and receiver wavefield are extrapolated into all depths in the subsurface independently, using the one-way wave equations 2 and 3, respectively. Second, the image of a plane-wave source with a ray parameter is constructed by cross-correlating the source and receiver wavefields weighted with the angular frequency :

 (9)

where is the conjugate complex of the source wavefield . The whole image is formed by stacking the images of all possible plane-wave sources:
 (10)

Because both slant-stacking and migration are linear operators, the image of the plane-wave migration is equivalent to the image obtained by shot-profile migration (Liu et al., 2002; Zhang et al., 2005). In the discrete form, in practice we need a sufficient number of to make the two images equivalent.

 Plane-wave migration in tilted coordinates

Next: Wavefield extrapolation in tilted Up: Shan and Biondi: Plane-wave Previous: One-way wave equation migration

2007-09-18