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

$$H(y) \delta ({{t-t_0} \over p} - y)$$

where

$$\left \{ \begin{array} {ll} H(y)=1; & y \in [a,b] \\ \\ H(y)=0; & y \in (-\infty,a) \cup (b,\infty)\end{array} \right.$$

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

$$p={\tan \alpha}={{\Delta t} \over {\Delta y}}.$$

In the corresponding depth model p becomes

$$p={{\Delta t} \over {\Delta y}}={{ 2 \sin \theta} \over v}$$

as seen in Figure . The angle $\theta$ represents the slope of the reflector in the depth model and v is the velocity of the medium. The function $\delta ({{t-t_0} \over p}-y)$ has unitary amplitude when the argument is zero or $y={{t-t_0} \over p}$.

 

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 ${\Delta t / \Delta y} = 2 \sin \theta / v$,where $\theta$ 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

$$\begin{array} {lcl} S(\omega,k_y) & = & \displaystyle{ {\int... ...{{k_y t_0} \over p}}} \delta(\omega- {k_y \over p})}\end{array}$$

which in the $\omega,k_y$ space represents a line with slope  

$${k_y \over \omega} = p = {dt \over dy} = {{ 2 \sin \theta} \over v}.$$ (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 $\omega,k_y$ space.