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) |

(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) |

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) |

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) |

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

(121) |

(122) |

(123) |

(124) |

Figure 25

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.

Figure 26

12/28/2000