Implicit extrapolation theory

As described in Chapter , wavefield extrapolation algorithms depend on an operator, R, that marches the wavefield q, at depth z, down to depth $z+\Delta z$. 

$$q_{z+\Delta z}=R \; q_{z}.$$ (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$^\circ$ one-way wave equation Fomel and Claerbout (1997), is given by  

$$R_{\rm implicit}({\bf k_x})=e^{i\omega s} \; \frac{1-4 \ome... ...1-4 \omega^2 \, s^2 - i \omega \, s \, \vert{\bf k_x}\vert^2},$$ (35)

where s=1/v and $\Delta z=1$.

This operator can be implemented numerically (without Fourier transforms) by replacing $\vert{\bf k_x}\vert^2$ with a finite-difference equivalent whose amplitude spectrum, D, in the constant velocity case will also be a simple (non-negative) function of ${\bf k_x}$.Irrespective of the choice of $D({\bf k_x})$, this operator can be written as a pure phase-shift operator,

$$R_{\rm implicit}({\bf k_x}) = e^{i \phi({\bf k_x})},$$ (36)

where $\phi({\bf k_x}) = \omega \, s + 2 \arctan \frac{\omega \, s \, D({\bf k_x})}{1-4\, \omega^2 \, s^2}$. Consequently, in the constant velocity case, this formulation is unconditionally stable for all values of $\omega \, s$.

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

$$R_{\rm explicit}(k)=e^{i \omega s} \; \left( 1+ \gamma_1 k^2 + \gamma_2 k^4 \right)$$ (37)

where complex coefficients $\gamma_1$ and $\gamma_2$ can be calculated using a Taylor series, for example, and $k^2=\vert{\bf k_x}\vert^2$.Although in practice stable explicit operators can be constructed for constant velocity media Hale (1990b), they can never represent a pure phase-shift. Their stability is conditional, and cannot be guaranteed for media with lateral velocity variations.

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.