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In the previous chapter, I described the process of applying helical
boundary conditions to facilitate the factorization of implicit
extrapolators.
However, I only covered the case where the velocity was constant
within each depth layer, i.e. constant velocity and *v*(*z*) earth
models.
The advantage of working in the space domain, as opposed to
the spatial-frequency domain, is that method can be adapted to handle
operators changing laterally.
Indeed, the strength of conventional implicit finite-difference
methods comes in areas with strong lateral velocity variations, where
the small filters can accurately model the rapid velocity changes, and
the implicit formulation can guarantee unconditional stability
Godfrey et al. (1979).
In this chapter, I describe how recursive filtering can be extended to
handle non-stationarity. This allows implicit depth migration with
the helix factorization to be applied in areas with lateral velocity
variations.
Unfortunately, however, I am unable to formulate the helical
factorization in such a way that maintains the unconditional stability
of the conventional implicit schemes.
Therefore, the stability of helical extrapolators in laterally
variable media cannot be guaranteed.

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Stanford Exploration Project

5/27/2001