The PS-AMO operator is conceived of as a cascade of forward and reverse PS-DMO; therefore, the accuracy and speed of the PS-DMO operator is important. The PS-DMO operator in the frequency-wavenumber domain Alfaraj (1992) is accurate and conceptually simple, but is computationally expensive because the operator is nonstationary in time.
The technique of logarithmic time-stretching, introduced by Bolondi et al. (1982), increases the computational efficiency, because the PP-DMO operator is stationary in the log-stretch domain. Fast Fourier Transforms (FFT) also can be used instead of the slower Discrete Fourier Transforms (DFT). Zhou et al. (1996) create a PP-DMO operator that considers variations of the traveltime as well as variations in the midpoint position before and after PP-DMO; therefore, the operator has the main properties of handling steeply dipping reflectors properly, and producing slightly stronger amplitudes for steep reflectors. Xu et al. (2001) introduce a log-stretch frequency-wavenumber PS-DMO operator that is computationally efficient and kinematically correct; moreover, their implementation performs a correction for the transformation from CMP to CRP. However, this implementation does not consider the variation along CMP as does the PP-DMO Zhou's et al. 1996 PP-DMO operator does. Rosales (2002) follows a procedure similar to the one Zhou et al. (1996) used for the derivation of PP-DMO to create a 3-D PS-DMO operator that that considers both time shift and spatial shift. This 3-D PS-DMO operator is computationally efficient and kinematically correct.
Rosales and Biondi (2002) introduces the PS-AMO operator that we use in this paper. This PS-AMO operator is computationally efficient because it performs in the frequency-wavenumber log-stretch domain. 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:
this lateral shift is responsible for the CMP to CRP correction, the transformation vectors, and ) are:
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. The data set 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
The set of equations 1-7 compose the f-k log-stretch PS-AMO operator. The next session describes the practical implementation of this operator in order to reduce the dimensionality of the data set.