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At the scattering point, we can define a "distance" or norm as
| |
(39) |

where *R* is the reflectivity, is the wavefield downward extrapolated to a reflector, and is the wavefield downward propagated to the reflector. The scattering wavefield should be equal to or close to the convolution result between the wavefield and the reflectivity.
Letting
| |
(40) |

we have
| |
(41) |

If the incident wavefield equals zero, equation () is satisfied. However this case has no physical meaning. If the incident wavefield does not equal zero, then,
| |
(42) |

In the complex domain, we can rewrite equation () as
| |
(43) |

If the incident wave is quite weak, the following regularization should be introduced:
| |
(44) |

where is the regularization coefficient.
In fact, the reflectivity is related to the incident angle to a reflector of the plane-wave component of a seismic wave. Therefore, we should modify equation () into the following form to reach the angle gathers:
| |
(45) |

where and are a scattering plane wavefield and an incident plane wavefield, respectively.
In fact, the extrapolated wavefield can be defined as
| |
(46) |

Therefore, in the frequency domain, equation () can be rewritten as
| |
(47) |

We will further discuss this topic later to clarify the relationship.

** Next:** Related topics
** Up:** Wang and Shan: Imaging
** Previous:** migration imaging and inversion
Stanford Exploration Project

5/3/2005