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In one of its most general forms Ristow and Ruhl (1994), we can write *k*_{z} as
| |
(2) |

where are binomial coefficients for integer
numbers *n*, *s* represents the spatially variable slowness function at depth *z*,
*s*_{o} is a constant approximation to *s*,
and is a sum of terms derived from *s* and *s*_{o}.
The FFD equation is obtained using two Taylor series expansions of the SSR equations
written for *k*_{z} and *k*_{z}_{o}.
I give a full derivation of Equation (2) in Appendix A.
Higher accuracy can be achieved by replacing the Taylor expansion in
Equation (2) with Muir's continuous fraction expansion
Claerbout (1985). The equivalent form of the general Fourier
finite-difference propagator is:

| |
(3) |

where *a*_{n} and *b*_{n} are coefficients that, in general, depend on the medium
and the constant reference slownesses, *s* and *s*_{o}. The coefficients
*a*_{n} and *b*_{n} are derived either by identification of terms between Equation (2) and
Equation (3), after the approximation , or
by an optimization problem as described by Ristow and Ruhl (1997).

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** Up:** Mixed-domain migration theory
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Stanford Exploration Project

9/5/2000