Starting from the parametric DMO relations of Black et al. (1993),
Zhou et al. (1996) derives an expression for a DMO applicable on
2D NMO-ed data. In order to extend the expression to 3D, we only have
to replace the product between the wavenumber and half offset
with the dot product of the same quantities, which are vectors in the
case of 3D data. In order to perform AMO from the offset to
the offset , we need to cascade one forward DMO from offset
to zero offset with a reverse DMO from zero offset to
offset . Thus, applying log-stretch, frequency-wavenumber
AMO on a 3D cube of data
in order to obtain
involves the following sequence of
operations:
Apply log-stretch along the time axis on the
cube, with the formula:
(1)
where is the minimum cutoff time introduced to avoid taking the logarithm of zero. All samples from times smaller than are simply left untouched, the rest of the procedure will be applied to the cube
.
3D forward FFT of the
cube. The 3D forward Fourier Transform is defined as follows:
(2)
It can be seen that the sign of the transform along the -axis is opposite to that over the midpoint axes.
For each element of the cube, perform the AMO shift:
(3)
(4)
(5)
and j can take the values 1 or 2.
The frequency domain variables must have incorporated in their value a constant (they are defined according to equation (2))
Do reverse 3D FFT in order to obtain the
cube.
Do reverse log stretch along the time axis and affix to the top of the cube the slices from times smaller than . The final result is a
cube.
Figure 1 shows the impulse response of the above
described AMO.