Calculating Common Image Gather (CIG) offsets involves the evaluation of an imaging
condition at all acceptable values of and . This gives rise
to two new image space variables: the horizontal image coordinate for
the earth model,
, and the subsurface horizontal offset coordinate, . These
variables have much similarity to the data space variables midpoint
and offset. In a strictly *v*(*z*) medium, these axes overlay.
However, in more complicated media, the midpoint variable
is somewhat misleading. This is because the wavefield continuation
extrapolates energy from a midpoint on the surface to a different
midpoint as the wavefields are successively downward-continued. Thus,
mixing these two ideas is inappropriate.
Source and receiver coordinates and horizontal and sub-surface offset
coordinates are related through transforms and .The new coordinate, , a derived parameter with a magnitude
equal to an integer multiple of , is naturally represented as the
product of an integer multiplication factor *h* and horizontal image
space discretization interval (i.e. ) that will most
commonly be unity.

Using these definitions, the image cube may be constructed by applying
the general correlation imaging condition to wavefield *W*,

(17) |

(18) |

(19) |

Continuing our derivation, we now reintroduce the lattice in equation (16) to the imaging condition which yields,

(20) |

(21) |

and applying a Fourier transform over coordinates , , and yields,

The Rect functions of coordinates and are collapsed back to a single Rect function in , where the frequency limit is given by

The summations of the delta functions over

(22) |

(23) |

(24) |

Notice that for the simplified case of zero-offset migration, the
pre-supposed notion that there are no operator aliasing artifacts
introduced can be shown conclusively within the presentation above.
Without two different sampling intervals, be they source/receiver or
orthogonal surface coordinates, there are no choices for the *min*
operator in equation () nor the *max* operator of
equation (25). Instead, the sole variable available,
surface location *x*, dictates the sampling of the model space.

Notice that for the simplified case of zero-offset migration, the pre-supposed notion that there are no operator aliasing artifacts introduced can be shown conclusively within the presentation of the above results. Without two possibly different sampling intervals, for source and receiver grids, there are no choices for the operator in equation (24) nor the operator of equation (25). Instead, the sole variable available, surface location, dictates the sampling of the model space. This does not however release zero-offset migrations from the ramifications of image condition aliasing, as the implied correlation of the source wavefield associated with source-receiver migrations is still present.

5/23/2004