Plane wave prediction in 3-D |
Let us denote the coordinates of a three-dimensional space by ,
, and . A theoretical plane wave is described by the equation
The first equation in (2) describes plane waves on the
slices. In is discrete form, it is represented as a
convolution with the two-dimensional finite-difference filter
. Similarly, the second equation transforms into a
convolution with filter , which acts on the
slices. The discrete (finite-difference) form of equations
(2) involves a blocked convolution operator:
In many applications, we are actually interested in the spectrum of
the prediction filter, which approximates the inverse spectrum of the
predicted data. In other words, we deal with the square operator
The problem of finding from its spectrum is known as
spectral factorization. It is well understood for 1-D signals
(Claerbout, 1976), but until recently it was an open problem
in the multidimensional case. Helix transform maps multidimensional
filters to 1-D by applying special boundary conditions and allows us
to use the full arsenal of 1-D methods, including spectral
factorization, on multidimensional problems
(Claerbout, 1998b). A problem, analogous to (4),
has already occurred in the factorization of the discrete
two-dimensional Laplacian operator:
If we represent the filter with the help of a simple first-order
upwind finite-difference scheme
All examples in this paper actually use a slightly more sophisticated
formula for 2-D plane-wave predictors:
eplane
Figure 1. 3-D plane wave prediction with a 402-point filter. Left: , . Right: , . |
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shape
Figure 2. Schematic filter shape for a 26-point 3-D plane prediction filter. The dark block represents the leading coefficient. There are 9 blocks in the first row and 17 blocks in the second row. | |
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tplane
Figure 3. 3-D plane wave prediction with a 26-point filter. Left: , . Right: , . |
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Figure 1 shows examples of plane-wave construction. The two plots in the figure are outputs of a spike, divided recursively (on a helix) by , where is a 3-D minimum-phase filter, obtained by Wilson-Burg factorization. The factorization was carried out in the assumption of and ; therefore, the filter had coefficients. Using such a long filter may be too expensive for practical purposes. Fortunately, the Wilson-Burg method allows us to specify the filter length and shape beforehand. By experimenting with different filter shapes, I found that a reasonable accuracy can be achieved with a 26-point filter, depicted in Figure 2. Plane-wave construction for a shortened filter is shown in Figure 3. The predicted plane wave is shorter and looks more like a slanted disk. It is advantageous to deal with short plane waves if the filter is applied for local prediction of non-stationary signals.
Clapp (2000) has proposed constructing 3-D plane-wave destruction (steering) filters by splitting. In Clapp's method, the two orthogonal 2-D filters and are simply convolved with each other instead of forming the autocorrelation (4). While being a much more efficient approach, splitting suffers from induced anisotropy in the inverse impulse response. Figure 4 illustrates this effect in the 2-D plane by comparing the inverse impulse responses of plane-wave filters obtained by spectral factorization and splitting. The splitting response is evidently much less isotropic.
bob
Figure 4. Two-dimensional inverse impulse responses for filters constructed with spectral factorization (left) and splitting (right). The splitting response is evidently much less isotropic. |
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In the next sections, I address the problem of estimating plane-wave slopes and show some examples of applying local plane-wave prediction in 3-D problems.
Plane wave prediction in 3-D |