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:

$$t^2={t_0}^2+\frac{4x^2}{v^2},$$ (2)

while for common-offset sections, the moveout is given by:

$$t={\left[\frac{{(x+s/2)}^2}{v^2} +\frac{{t_0}^2}{4} \right]}... ...ft[ \frac{{(x-s/2)}^2}{v^2} +\frac{{t_0}^2}{4} \right]}^{1/2},$$ (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.