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Downward continuation is the process in which we recursively extrapolate in depth
the wavefield recorded at the surface. Mathematically, this operation amounts to a
phase shift applied to the wavefield Claerbout (1985)
where U_{z} is the wavefield at depth z, and is the extrapolated wavefield
at depth . The downward continuation operator at depth z is
 
(1) 
with the vertical wavenumber, k_{z}, given by the oneway wave equation,
also known as the single square root (SSR) equation
where is the temporal frequency, s is the laterally variable slowness of
the medium, and is the horizontal wavenumber.
Since downward continuation by phase shift can be applied for slowness models
that only vary with depth, we need to split the operator into two parts:
a constant slowness continuation operator applied in the domain,
which accounts for the propagation in depth,
and a screen operator applied in the domain,
which accounts for the wavefield perturbations due to the lateral slowness variations.
In essence, we approximate the vertical wavenumber k_{z} with its constant
slowness counterpart k_{z}_{o}, corrected by a term describing the spatial
variability of the slowness function (Figure ).
screen
Figure 1 A sketch of mixeddomain migration. The wavefield at depth
z is downward continued to depth trough a variableslowness screen.

 
Next: Fourier finitedifference
Up: Sava: Mixeddomain operators
Previous: Introduction
Stanford Exploration Project
9/5/2000