Applying a low-pass filter to each input trace before performing migration is a common antialiasing method used in Kirchhoff 3D migration. The spatial directional derivative that gives the tangent plane direction of the reflector is key in determining the antialiasing frequency limits. In this paper, we present the formula of the traveltime gradient for the diffraction hyperboloid. We also deduce the formula of the vertical traveltime gradient for the reflection ellipsoid, which can be used to construct the antialiasing low-pass filter after migration. The migration stretch is easily computed from the ratio of gradient expressions obtained for the reflection ellipsoid and the diffraction hyperboloid.
The gradient formula includes magnitude and direction. The correct direction of the gradient function is from an image point to the dip-moveout image point ρ∗ direction) on the surface. The gradient is not in the direction from the image point to the midpoint (ρ direction) as has been previously concluded. The correct spatial directional derivative along ρ is given, and it is slightly less than the gradient norm. We also prove that the true direction of the tangent plane on the reflection ellipsoid can be determined by the gradient.
We introduce the directional derivative value along ρ that can be approximately applied to calculate the high-frequency limit. Compared with the maximum directional derivative value (gradient norm), the directional derivative is relatively accurate and is much easier to calculate for use in antialiasing. The different results from the new method and the former method are compared. Numerical examples compare the performance of the different formulae.