Zhou et al. (1996) discuss that Hale's 1984 DMO operator via a Fourier transform is computationally expensive because the DMO operator is temporally nonstationary. They use the technique of logarithmic time stretching, first introduced by Bolondi et al. (1982) to present another derivation for the frequency-wavenumber log-stretch DMO operator.

Xu et al. (2001) exploit the idea of computational efficiency of the logarithmic time stretching for the PS-DMO operator. I reformulate the work of Xu et al. (2001) using the PS-DMO smile derived in the previous section and following a procedure similar to Hale (1984) and Zhou et al. (1996). This operation is valid for a constant velocity case.

From equation , and following
Hale's 1984 assumption that
the DMO operator maps each sample of the NMO section (*p*_{n})
from time *t*_{n} to time *t _{0}* without changing its
midpoint location,

(16) |

Equation () implies a change of variable from *t _{0}* to

(17) |

(18) |

which I will represent as *A* or its Fourier equivalent:

(19) |

(20) |

Equation () is the foundation of PS-DMO in the *f-k* domain.
I introduce a time log-stretch transform pair,

(21) | ||

(22) |

where *t*_{c} is the minimum cutoff time introduced to avoid taking the logarithm of zero.
The PS-DMO operator in the *f-k* log-stretch domain becomes

(23) |

where

(24) |

with

(25) |

where is the Fourier representation of the log-stretched time axis . The value
of the function for either *kh* =0 or is obtained as zero by
taking the limit of the function for either or and applying the L'Hopital's rule.

The previous expression is equivalent to the one presented by Xu et al. (2001).
Note that
equation () is based on the assumption that *p _{0}*(

Zhou et al. (1996) solve this problem for PP-DMO in the *f-k* log-stretch
domain by reformulating the *f-k* log-stretch PP-DMO operator
presented by Black et al. (1993). This
operator is based on the
assumption that the midpoint changes its location after the
PP-DMO operator is applied [*p _{0}*(

(26) |

The values of the phase-like function at the points *kh* =0 and are obtained using L'Hopital's rule on the limit of the function , since
the function is singular at those points. Figure shows a series of impulse responses for this
operator.

Figure 6

Note that for a value of , equivalent to , the filter reduces to the known expression for P-wave data Zhou et al. (1996). We can trust the PS results since the PP impulse response, obtained with the filter in equation and , is the same as that obtained by Zhou et al. (1996). Moreover, the amplitude distribution follows Jaramillo's 1997 result. The 3-D representation for this PS-DMO operator is the starting point for the partial-prestack migration operator presented in Chapter 4.

12/14/2006