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Here, we extend prediction filter theory to N plane waves. Assume that the
data have N plane waves with different dipping angles, the dataset can then be
expressed by

| |
(22) |

Fourier transform along the time axis will give us
| |
(23) |

where .
Trace is represented by

| |
(24) |

where propagator .Assuming trace is known, trace *W*(*f*,*x*) can
be predicted by a N points prediction filter
| |
(25) |

Inserting equation (23) into equation (25)
| |
(26) |

For each
| |
(27) |

Equation (27) can be expressed in matrix form
| |
(28) |

Equation (28) is a Van der Monde system. This system guarantees that
there is one solution. So every is a function of .
This means that the prediction filter relies on frequency in the case of
frequency-independent grid; whereas in the case of frequency-dependent
grids, we can still get a prediction filter which is independent from
frequency.

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Stanford Exploration Project

11/11/1997