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:
![]() |
(1) |
this lateral shift is responsible for the CMP to CRP correction,
the transformation vectors, and
) are:
![]() |
(2) | |
(3) |
The final step of the first operation is to apply a log-stretch along the time axis with the following relation:
![]() |
(4) |
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:
![]() |
(5) |
The filter is given by
![]() |
(6) |
with the phase function defined by
![]() |
(7) |
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.