3-D Wavefield extrapolation filter

The wavefield extrapolator in three dimensions is given by Equation 8. The extrapolator for a given temporal frequency $\omega$ is a circularly symmetric operator in (k_x - k_y) domain.  

$$D(k_x,k_y)= \exp{\left \{ i\frac{\Delta{z}}{\Delta{x}}\left[... ...lta x}{v}}^2 \right ) -k_x^2 - k_y^2 \right]^{1/2} \right \}},$$ (8)

where v is the velocity while k_x and k_y are the normalized wavenumbers such that any distance quantity is measured in terms of the horizontal sampling interval $\Delta{x}$. Throughout this paper k refers to the normalized wavenumber.

 

extrap3d
Figure 7 Ideal 3-D extrapolator in the wavenumber (k_x-k_y) domain for a particular temporal frequency $\omega$. 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 k_x and k_y 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 $\omega$. The parameters used in generating the extrapolator are dx=dz=12.5 m, $\omega =40\pi rad/s$, 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 $\exp{(-x^2/{2\sigma^2}}$) where $\sigma$ 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 $\omega$ 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 (k_x-k_y) domain generated using McClellan transformation. Compare the operator to the ideal operator in Figure 7.