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Shot gathers can also be synthesized to a new dataset to represent a physical experiment that does not occur
in reality. One of the most important examples is to synthesize shot gathers to plane-wave source gathers.
The plane-wave source gathers represent experiments that planar sources originate from all angles at the surface.
They can also be regarded as the accurate phase-encoding of the shot gathers Liu et al. (2006).
The plane-wave source dataset can be generated by delaying the shot in shot gathers
or slant-stacking in receiver gathers as follows:
| |
(4) |

where *p*_{x} and *p*_{y} are ray parameters in the in-line and cross-line directions respectively.
Its corresponding plane-wave source wavefield at the surface is
| |
(5) |

Similar to the Fourier transformation, we can transform the plane-wave source data back to shot gathers by the inverse
slant-stacking Claerbout (1985) as follows
| |
(6) |

In contrast to the inverse Fourier transformation, the kernel of the integral is weighted by the square of the frequency .
The source wavefield *S*_{p} and receiver wavefield *R*_{p}
are extrapolated into the subsurface independently using the one-way wave equations 1 and 2.
The image of a plane-wave source with a ray parameter pair (*p*_{x},*p*_{y}) is formed by cross-correlating
the source and receiver wavefields weighted with the square of the frequency :

| |
(7) |

The final image is generated by stacking the images of all possible plane-wave sources:
| |
(8) |

Because both slant-stacking and migration are linear operators, the image of the plane-wave migration *I*_{p} is equivalent to the image obtained by shot-profile migration Liu et al. (2002); Zhang et al. (2005).

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Stanford Exploration Project

5/6/2007