The PS-DMO operator in the *f-k* log-stretch domain, discussed in
Chapter 2, is easily extended to 3-D, and it is the basis to build
the *f-k* log-stretch PS-AMO operator.
By performing PS-DMO in the frequency-wavenumber
log-stretch domain in cascade with its inverse,
the PS-AMO operator is computationally efficient.
This PS-AMO operator consists of two main operations.
In the first operation,
the input data, , is transformed to the
wavenumber domain () using FFT. Then,
a lateral-shift correction is applied using the transformation vectors ( and
) as follows:

(40) |

The final step of the first operation is to apply a log-stretch along the time axis with the following relation:

(41) |

where *t*_{c} is the minimum cutoff time, introduced to avoid taking the logarithm of zero.
Therefore, the dataset after the first operation is .
In the second operation, the log-stretched time domain () section is transformed
into the frequency domain () using FFT. Then,
the filters and
are applied as follows:

(42) |

The filter is given by

(43) |

with the phase function defined by either

(44) |

or

(45) |

To implement this PS-AMO operator, we use the following procedure:

- 1.
- Calculate the FFT along the midpoint axis for each input offset cube.
- 2.
- Compute the transformation vectors and with equations and .
- 3.
- Apply the lateral shift of the transformation vectors as a phase shift with equation .
- 4.
- Perform the log-stretch transformation over the time axis using equation .
- 5.
- Calculate the FFT along the transformed time axis.
- 6.
- Compute the filters and .
- 7.
- Apply these filters to the data in the log-stretch frequency-wavenumber domain with equation .
- 8.
- Perform inverse FFT and inverse log-stretch.

The lateral shift correction, third step on the above procedure, involves a forward and inverse 2-D Fourier transform on both the inline and crossline CMP axes. Therefore, this step increases condirebly the cost of the PS-AMO operator compared with the conventional log-stetch implementation of the single mode AMO operator.

12/14/2006