     Next: 3-D missing data interpolation Up: Plane-wave destruction in 3-D Previous: Plane-wave destruction in 3-D

Factorizing plane waves

Let us denote the coordinates of a three-dimensional space by t, x, and y. A theoretical plane wave is described by the equation (116)
where f is an arbitrary function, and and are the plane slopes in the corresponding direction. It is easy to verify that a plane wave of the form ( ) satisfies the following system of partial differential equations: (117)

The first equation in ( ) describes plane waves on the slices and is completely equivalent to equation ( ). In its discrete form, it is represented as a convolution with the two-dimensional finite-difference filter from equation ( ). Similarly, the second equation transforms into a convolution with filter , which acts on the slices. The discrete (finite-difference) form of equations ( ) involves a blocked convolution operator: (118)
where is the model vector corresponding to P(t,x,y).

As follows from the theoretical analysis of the data regularization problem in Chapter , regularization implicitly deals with the spectrum of the regularization filter, which approximates the inverse model covariance. In other words, it involves the square operator (119)
If we were able to transform this operator to the form , where is a three-dimensional minimum-phase convolution, we could use the three-dimensional filter in place of the inconvenient pair and .

The problem of finding from its spectrum is the familiar spectral factorization problem. In fact, we already encountered a problem analogous to ( ) in the previous section in the factorization of the discrete two-dimensional Laplacian operator: (120)
where and represent the partial derivative operators along the x and y directions, respectively, and the two-dimensional filter is known as helix derivative Claerbout (1999); Zhao (1999).

If we represent the filter with the help of a simple first-order upwind finite-difference scheme (121)
then, after the helical mapping to 1-D, it becomes a one-dimensional filter with the Z-transform (122)
where Nt is the number of samples on the t-axis. Similarly, the filter takes the form (123)
The problem is reduced to a 1-D spectral factorization of (124)
The spectral factorization of ( ) produces a minimum-phase filter applicable for 3-D forward and inverse convolution. Equation ( ) is shown here just to illustrate the concept. In practice, I use the longer and much more accurate plane-wave filters of equation ( ) in place of the simplified filters ( ) and ( ). cube
Figure 25
3-D plane wave construction with the factorized 3-D filter. Left: , . Right: , .      Figure 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 the Wilson-Burg factorization.

Clapp (2000a) 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 ( ). While being a much more efficient approach, splitting suffers from induced anisotropy in the inverse impulse response. Figure 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 26
Two-dimensional inverse impulse responses for filters constructed with spectral factorization (left) and splitting (right). The splitting response is evidently much less isotropic.          Next: 3-D missing data interpolation Up: Plane-wave destruction in 3-D Previous: Plane-wave destruction in 3-D
Stanford Exploration Project
12/28/2000