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:
The final step of the first operation is to apply a log-stretch along the time axis with the following relation:
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:
The filter is given by
with the phase function defined by either
To implement this PS-AMO operator, we use the following procedure:
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.