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