(1) |

(2) |

However, to apply this operator directly requires spatial Fourier
transforms, and an assumption of constant lateral velocity. To
overcome this limitation, short finite-difference approximations to
*W*(*k*) are applied in the domain.

An implicit finite-difference formulation approximates *W*(*k*) with a
convolution followed by an inverse convolution. For example, a simple
implicit approximation to equation (1) that corresponds
to the Crank-Nicolson scheme for the 45 one-way wave equation,
is given by

(3) |

An explicit approach approximates *W*(*k*) directly with a single
convolutional filter. For example, a three-term expansion of
equation (1) yields

(4) |

Although in practice stability is not usually a problem for explicit operators, they can never represent a pure phase-shift. Hence, stability cannot be guaranteed for all velocity models Etgen (1994).

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.

7/5/1998