Introduction

McClellan transformations offer a computationally efficient scheme for the depth extrapolation of 3-D seismic data in the frequency domain Hale (1991); McClellan and Chan (1977); Palacharla (1993). The computational efficiency is achieved by exploiting the circular symmetry of the wavefield extrapolator. Using McClellan transformations the convolution with the 2-D ($N\times N$) filter that is required for depth extrapolating the wavefield Blacquiere et al. (1989) is reduced to N convolutions with a compact 2-D ``transformation'' filter. The accuracy of the McClellan transformation depends on the accuracy of the approximation used for the ``transformation'' filter; this transformation filter is defined in the wavenumber domain $\vec k$ as $\cos\mid\vec k\mid$.Hale 1991 gives two different approximations: the original McClellan filter, which has nine terms in the space domain, and a higher-order approximation with seventeen terms. However, the spectrum of even the higher-order McClellan filter deviates from the circular response at high wavenumbers. The error depends on the azimuthal direction and it is maximum along the $45^{\circ}$ line to the orthogonal coordinate axes. This azimuthal dependence of the error causes the wavefield extrapolator to be anisotropic.

Blacquiere 1991 suggested a way for improving the accuracy of the transformation by designing optimum McClellan filters as a function of the frequencies and dips to be imaged. The filters that are designed by this approach are optimum within the limitations of the stencil-pattern used for the optimization and they do not take advantage of the particular structure of the filter to be approximated. The most accurate ones are defined on a $5\times 5$ stencils, thus they are more expensive to apply than the higher-order McClellan filter proposed by Hale.

In this paper we suggest a method for improving the accuracy of the transformation filter, without increasing its computational cost. The method takes advantage of the particular structure of the $\cos\mid\vec k\mid$ filter; namely, its being close to a separable filter. In order to improve the accuracy, we rotate the 9-point McClellan filter by 45 degrees, and then rescale the wavenumber axis so that the rotated filter matches the desired filter exactly along the orthogonal axes. The average of this rotated filter with the original McClellan one is a filter which is more isotropic, and closer to the ideal transformation filter.

Hedley 1992 has suggested an improvement in the circular response of the transformation filters by migrating the data on a hexagonal grid Woodward and Muir (1983). The hexagonal grid leads to a more accurate filter response for higher wavenumbers, because the hexagonal grid is more isotropic than the Cartesian grid. We think that our idea of rotating the McClellan filter for correcting the anisotropy could be extended to hexagonal grids, as well. We expect it to be more isotropic on the hexagonal grid compared to the corresponding McClellan filter.