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## 3-D Wavefield extrapolation filter

The wavefield extrapolator in three dimensions is given by Equation 8. The extrapolator for a given temporal frequency is a circularly symmetric operator in (kx - ky) domain.
 (8)
where v is the velocity while kx and ky are the normalized wavenumbers such that any distance quantity is measured in terms of the horizontal sampling interval . Throughout this paper k refers to the normalized wavenumber.

extrap3d
Figure 7
Ideal 3-D extrapolator in the wavenumber (kx-ky) domain for a particular temporal frequency . The circular region in the center is the propagating region and the region external to it is the evanescent region. The two horizontal axes are the normalized wavenumbers kx and ky and they go from -0.5 to 0.5 cycles/s.

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 .

 explicit Figure 8 The ideal 2-D wavefield extrapolator (continuous curve) and the tapered extrapolator (dashed curve) in wavenumber domain.

 migfil 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.

mclextrap3d
Figure 10
3-D wavefield extrapolator in (kx-ky) domain generated using McClellan transformation. Compare the operator to the ideal operator in Figure 7.

Next: CONCLUSIONS Up: EXAMPLES Previous: Lowpass Filter
Stanford Exploration Project
11/17/1997