The circular symmetry of the operator enables it to be realized using McClellan filter. Figure shows the ideal-extrapolator in 3-D for a certain temporal frequency . The parameters used in generating the extrapolator are dx=dz=12.5 m, , and v = 1000. m/s. A tapered 1-D extrapolator was designed by applying a Gaussian taper to the ideal explicit wavefield extrapolation filter. This leads to a stable extrapolation filter as described in Nautiyal et al, 1993. The Gaussian taper is given by ) where is a parameter to be chosen. Figure shows the amplitude spectrum for the ideal and the tapered extrapolator . The extrapolation filter in two dimensions is a symmetric filter with complex coefficients. Figure shows the absolute magnitude of the filter coefficients. Using the filter coefficients for the tapered extrapolator, I generated the corresponding 3-D extrapolator (Figure ) at the frequency using the McClellan transformation. Compare this 3-D extrapolator with the ideal one shown in Figure .
Figure 8 The ideal 2-D wavefield extrapolator (continuous curve) and the tapered extrapolator (dashed curve) in wavenumber domain.
Figure 9 The magnitude of the tapered extrapolator in space domain. The filter coefficients are complex. It is a nineteen coefficient symmetric operator. The spatial wavelet magnitude is shown for one side.