Antialiasing of Kirchhoff operators by reciprocal parameterization |

Integral (Kirchhoff-type) operators are widely used in seismic imaging and data processing for such tasks as migration, dip moveout (Hale, 1991), azimuth moveout (Biondi et al., 1998), and shot continuation (Bagaini and Spagnolini, 1996). In theory, the operators correspond to continuous integrals. In practice, the integration is replaced by summation and becomes prone to sampling errors. A common problem with practical implementation of integral operators is the operator aliasing, caused by spatial undersampling of the summation path (Lumley et al., 1994). When the integration path is parameterized in the spatial coordinate, as it is commonly done in practice, the steeper part of the summation path becomes undersampled.

The operator aliasing problem, as opposed to the data aliasing and image aliasing problems, is discussed in detail by Lumley et al. (1994) and Biondi (2001). It arises when the slope of the operator traveltime exceeds the limit, defined by the time and space sampling of the data - the Nyquist frequencies (Claerbout, 1992a). Even if the input data are not aliased, operator aliasing can cause severe distortions in the output. Several successful techniques have been proposed in the literature to overcome the operator aliasing problem. Different versions of the temporal filtering method were suggested by Gray (1992) and Lumley et al. (1994) and further enhanced by Abma et al. (1999) and Biondi (2001). This method reduces the aliasing error by limiting the rate of change in the integrand (the input data) with temporal filtering. Unfortunately, this approach is suboptimal in the case of rapid changes in the summation path gradient. A different approach to antialiasing was suggested by Hale (1991) for the integral dip moveout. Hale's approach provides accurate results by parameterizing the operator in the time coordinate rather than the space coordinate. Unfortunately, this approach requires an additional expense of interpolation in both space and time coordinates for computing the flat part of the operator.

In this paper, I propose a new antialiasing method derived from the time-slice technique, developed by Hale (1991). The method switches between interpolation in time and interpolation in space depending on the local operator dips. It is particularly attractive for computing 3-D operators with rapidly varying dips and limited aperture (Fomel and Biondi, 1995). Synthetic examples show the superior performance of the new method in comparison the temporal filtering approach.

Antialiasing of Kirchhoff operators by reciprocal parameterization |

2013-03-03