** Next:** Stretching and aliasing
** Up:** Vlad and Biondi: Log-stretch
** Previous:** The Azimuth Moveout

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 *kh* 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:
- 1.
- Apply log-stretch along the time axis on the cube, with the formula:
| |
(1) |

where *t*_{c} is the minimum cutoff time introduced to avoid taking the logarithm of zero. All samples from times smaller than *t*_{c} are simply left untouched, the rest of the procedure will be applied to the cube . - 2.
- 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.
- 3.
- 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))
- 4.
- Do reverse 3D FFT in order to obtain the cube.
- 5.
- Do reverse log stretch along the time axis and affix to the top of the cube the slices from times smaller than
*t*_{c}. The final result is a cube.

Figure 1 shows the impulse response of the above
described AMO.
**impresp1
**

Figure 1 AMO impulse response

** Next:** Stretching and aliasing
** Up:** Vlad and Biondi: Log-stretch
** Previous:** The Azimuth Moveout
Stanford Exploration Project

9/18/2001