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PRACTICAL IMPLEMENTATION

In practice, 2-D Kirchhoff migration is carried out by summing the amplitudes in (x,t) space along the diffraction curve that corresponds to Huygen's secondary source at each point in (x,z) space. For zero-offset sections, the moveout at a source-receiver point a distance x from the apex of a diffraction hyperbola at time t0 is given by:

\begin{displaymath}
t^2={t_0}^2+\frac{4x^2}{v^2},\end{displaymath} (2)
while for common-offset sections, the moveout is given by:
\begin{displaymath}
t={\left[\frac{{(x+s/2)}^2}{v^2}
+\frac{{t_0}^2}{4} \right]}...
 ...ft[ \frac{{(x-s/2)}^2}{v^2} 
+\frac{{t_0}^2}{4} \right]}^{1/2},\end{displaymath} (3)
where s is the value of offset and v is the velocity at the apex of the hyperbola.

Before summation, each sample is scaled by the obliquity factor, $cos \theta$, and the spherical spreading factor (1/vt)1/2. After summation the output section is convolved with a filter whose amplitude-frequency response is proportional to the square root of frequency, with a constant phase delay of $\pi /4$. The latter corresponds to the half-derivative that was mentioned earlier on.



 
previous up next print clean
Next: Some practical considerations Up: Mallia-Zarb: Kirchhoff migration Previous: THEORETICAL APPROACH
Stanford Exploration Project
11/11/1997