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## f-k log-stretch PS-AMO

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 tc 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.

Next: Impulse response Up: PS Azimuth Moveout Previous: PS Azimuth Moveout
Stanford Exploration Project
12/14/2006