


 Planewave migration in tilted coordinates  

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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 planewave source gathers.
A planewave 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 phaseencoding of the shot gathers (Liu et al., 2002).
Planewave source gathers can be generated by slantstacking 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 planewave source wavefield at the surface is

(7) 
As with the Fourier transformation, we can transform the planewave source gathers back to shot gathers by inverse
slantstacking (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 Radontransform literature.
As with shotprofile migration, there are two steps to migrate a planewave source gather by a typical planewave migration method.
First, the source wavefield and receiver wavefield
are extrapolated into all depths in the subsurface independently, using the oneway wave equations 2 and 3, respectively.
Second, the image of a planewave source with a ray parameter is constructed by crosscorrelating
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 planewave sources:

(10) 
Because both slantstacking and migration are linear operators, the image of the planewave migration is equivalent
to the image obtained by shotprofile 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.



 Planewave migration in tilted coordinates  

Next: Wavefield extrapolation in tilted
Up: Shan and Biondi: Planewave
Previous: Oneway wave equation migration
20070918