Let *u*(*x*,*t*) be a wavefield.
The *slant stack * of the wavefield
is defined mathematically by

(14) |

The integral across *x* in (14) is done at constant , which
is a slanting line in the (*x*,*t*)-plane.

Slant stack is readily expressed in Fourier space.
The definition of the two-dimensional Fourier transformation of the
wavefield *u*(*x*,*t*) is

(15) |

(16) |

(17) |

(18) |

The inverse Fourier transform of (18) is

(19) |

The result (19) states that a slant stack can be created
by Fourier-domain operations.
First you transform *u*(*x*,*t*) to .Then extract from .Finally, inverse transform from to and
repeat the process for all interesting values of *p*.

Getting from seems easy, but this turns out to be the hard part. The line will not pass nicely through all the mesh points (unless ) so some interpolation must be done. As we have seen from the computational artifacts of Stolt migration, Fourier-domain interpolation should not be done casually.

Both (14) and (19) are used in practice. In (14) you have better control of truncation and aliasing. For large datasets, (19) is much faster.

10/31/1997