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Another way to overcome the problem of phase-wrap-around is to
compute the phase spectrum by integrating its derivative. The phase
spectrum of a signal is related to its Fourier transform as follows:

| |
(8) |

Taking the derivative with respect to on both sides gives
| |
(9) |

If we assume , then
| |
(10) |

The phase spectrum computed in this way is continuous. Furthermore,
it can be shown (Oppenheim and Schafer, 1975) that
| |
(11) |

If we define the group delay as
| |
(12) |

then equation (11) tells us that *n*_{0} is an average group delay
over frequency range .Considering the limited band width of the seismic signals and the presence of
noises, we should include uneven weights in the averaging step of
equation (11) as follows:
| |
(13) |

where is a weighting function. One choice of the weighting
function is
If , the result is the evenly weighted average of the group delay.
If , the result is equal to the group delay of the frequency
corresponding to the peak amplitude spectrum.

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** Up:** THEORY
** Previous:** Factorizing for minimum phase
Stanford Exploration Project

12/18/1997