Let us denote the coordinates of a three-dimensional space by *t*,
*x*, and *y*. A theoretical plane wave is described by the equation

(1) |

(2) |

The first equation in (2) describes plane waves on the slices. In the discrete form, it can be represented as a convolution with a two-dimensional finite-difference filter . Similarly, the second equation transforms into a convolution with filter , which acts on the slices. The discrete form of equations (2) involves a blocked convolution operator:

(3) |

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

(4) |

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:

(5) |

If we represent the filter with the help of a simple first-order upwind finite-difference scheme

(6) |

(7) |

(8) |

(9) |

All examples in this paper actually use a slightly more sophisticated formula for 2-D plane-wave predictors:

(10) |

Figure 1

shape
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.
Figure 2 |

Figure 3

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 *N*_{t}=20 and
*N*_{x}=20; therefore, the filter had *N*_{t} *N*_{x} +2 = 402 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.

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.

10/25/1999