(34) |

An implicit finite-difference formulation approximates *R* with a
convolution followed by an inverse convolution. For example, a
rational approximation to equation () that corresponds
to the Crank-Nicolson scheme for the 45 one-way wave equation
Fomel and Claerbout (1997), is given by

(35) |

This operator can be implemented numerically (without Fourier
transforms) by replacing with a finite-difference
equivalent whose amplitude spectrum, *D*, in the constant velocity
case will also be a simple (non-negative) function of .Irrespective of the choice of , this operator
can be written as a pure phase-shift operator,

(36) |

An explicit approach approximates *R* directly with a single
convolutional filter. For example, a three-term expansion of
equation () yields

(37) |

Also in order to preserve high angular accuracy for steep dips, explicit filters need to be longer than their implicit counterparts. The advantage of finite-difference methods over Fourier methods is that the effect of the finite-difference convolution filters is localized, leading to accurate results for rapidly varying velocity models. This is less of an advantage for long filters.

- Matrix representation of implicit operators
- Helical boundary conditions
- Cross-spectral factorization
- Polynomial division

5/27/2001