2005-09-16 - 4:50 p.m.

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 . (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)
where s=1/v and .

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)
where . Consequently, in the constant velocity case, this formulation is unconditionally stable for all values of .

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

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 . (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)
where s=1/v and .

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)
where . Consequently, in the constant velocity case, this formulation is unconditionally stable for all values of .

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

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