Two-dimensional circularly symmetric digital filters are used for smoothing, sharpening, and restoration of images in image processing Lu and Antoniou (1992). Only recently they have been applied to seismic-imaging problems Hale (1991), and Soubaras (1992). These authors have applied it to 3-D depth migration. In 3-D depth migration, wavefields are extrapolated in depth one frequency at a time via a two-dimensional convolution of the data with a circularly symmetric, temporal frequency and velocity dependent filter. The depth extrapolator function in 3-D is a circularly symmetric operator in the wavenumber domain for a given temporal frequency. This property can be utilized in developing an explicit wavefield extrapolation operator using the techniques described in the paper. Hale 1991 has shown that the 3-D depth extrapolator designed using McClellan transformations are efficient and require less storage. The application of the filter to seismic imaging has been the main motivation for studying the design of these filters here.

In this paper, I discuss the elements of the filter design and
illustrate its properties using two examples : A lowpass filter and
a wavefield extrapolation filter.
In migration, we are interested in implementing the 2-D filters in
the space domain rather than in the wavenumber domain since the velocity
variations could be handled more easily than the *k*_{x}-*k*_{y} domain.
Therefore, I show the implementation of the 2-D lowpass filter
in both the
*k*_{x}-*k*_{y} and *x*-*y* domains.
In a companion paper Biondi and Palacharla (1993),
the application of the
filters to 3-D depth migration is dealt with in more detail.

11/17/1997