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In order to understand how DMO by Fourier transform works we
have to examine some properties of the 2-D Fourier transform.
In particular the fact that all events with a particular
slope in the time-space domain are mapped to a single
radial line (a line passing through the origin)
in the frequency-wavenumber domain.
A two-dimensional function representing a segment of constant amplitude
in a zero-offset section can be described by

where
where *t* and *y* are the zero-offset coordinates,
*t*_{0} is the intersection point on the time axis and
the value *p* is the tangent of the slope
In the corresponding depth model *p* becomes
as seen in Figure .
The angle represents
the slope of the reflector in the depth model and
*v* is the velocity of the medium.
The function has unitary amplitude
when the argument is zero or .
**dipkxomega
**

Figure 10 Dipping bed geometry in a constant-offset section and the
corresponding constant velocity depth model. In a zero-offset
section the slope of the reflection is
,where is the dipping angle in the depth model
and *v* is the velocity.

**FFT2Ddips
**

Figure 11 The effect of 2-D Fourier transform on events with same dip.

**a**. Several reflectors with the same dip in space-time coordinates.

**b**. The 2-D Fourier transform of the same section. Note that all the
events are mapped on a radial line.

I include in the Appendix a formal derivation for
the analytic solution of a 2-D Fourier transform of a constant
length segment. The 2-D Fourier transform of
an infinite segment is

which in the space represents a line
with slope
| |
(9) |

In Figure .a several dipping segments with the
same slope are all mapped after the Fourier transformation
to a radial line
in Figure .b in space.

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Stanford Exploration Project

11/17/1997