Asymptotic pseudounitary stacking operators |

(50) |

where and are the lengths of the incident and reflected rays:

(51) | |||

(52) |

For simplicity, the vertical component of the midpoint is set here to zero. Evaluating the second derivative term in formula (46) for the common-offset geometry leads, after some heavy algebra, to the expression

Substituting (53) into the general formula (46) yields the weighting function for the common-offset true-amplitude constant-velocity migration:

Equation (54) is similar to the result obtained by Sullivan and Cohen (1987). In the case of zero offset , it reduces to equation (49). Note that the value of in (54) corresponds to the two-dimensional (cylindric) waves recorded on the seismic line. A special case is the 2.5-D inversion, when the waves are assumed to be spherical, while the recording is on a line, and the medium has cylindric symmetry. In this case, the modeling weighting function (42) transforms to (Deregowski and Brown, 1983; Bleistein, 1986)

(55) |

and the time filter is . Combining this result with formula (53) for , we obtain the weighting function for the 2.5-D common-offset migration in a constant velocity medium (Sullivan and Cohen, 1987):

(56) |

The corresponding time filter for 2.5-D migration is .

In the common-offset case, the pseudo-unitary weighting is defined from (47) and (53) as follows:

(57) |

where

(58) |

Asymptotic pseudounitary stacking operators |

2013-03-03