Next: Hale's DMO Up: DMO BY FOURIER TRANSFORM Previous: DMO BY FOURIER TRANSFORM

## 2-D Fourier transforms of dipping events

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, t0 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.

Next: Hale's DMO Up: DMO BY FOURIER TRANSFORM Previous: DMO BY FOURIER TRANSFORM
Stanford Exploration Project
11/17/1997