Antialiasing Filter for Kirchhoff Migration

The antialiasing condition for the KM operator (Gray, 1992; Lumley et al., 1994; Abma et al., 1999; Biondi, 2001; Zhang et al., 2003) says that the local slope $dt/dx$ (e.g., computed from the traveltime table) of the associated hyperbola diffraction curve in $x-t$ space should satisfy the Nyquist sampling criterion:

$$\left\vert \frac{dt}{dx} \right\vert \leq \frac{T}{2\Delta x}=\frac{1}{2f\Delta x},$$ (10)

where $T$ is the minimum period in the data at frequency $f$ , and $\Delta x$ is the input trace spacing. If this condition is not satisfied for any $(x,t)$ sample in the CSG, the local portion of this trace at time $t$ is high-cut filtered to $f_{cut}=1/(2\Delta x)/\vert dt/dx\vert$ to eliminate the offending high-frequency components.

Figure: Migration kernels plotted in data space as colored hyperbolas for a) primary and b) primary+multiple events associated with shallow (green) and deep (pink) trial image points. The best match between the data (black hyperbolas) and migration curves (pink and green) is when the trial image point is near the actual scatterer's position; the dot-product between the migration kernel and data fingerprints will give the greatest value when the trial image point is at the actual scatterer's location.