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While we have only described the factorization for *v*(*z*) velocity
models, the method can also be extended to
handle lateral variations in velocity.
For every value of , we precompute the factors
of the 1-D helical filters, *a*_{1} and *a*_{2}.
Filter coefficients are stored in a look-up table.
We then extrapolate the wavefield by non-stationary convolution,
followed by non-stationary polynomial division.
The convolution is with the spatially variable filter pair
corresponding to *a*_{2}. The polynomial division is with the filter pair
corresponding to *a*_{1}.
The non-stationary polynomial division is exactly analogous to
time-varying deconvolution, since the helical boundary conditions have
converted the two-dimensional system to one-dimension.

Since we interpolate filters, not downward continued wavefields as in
`split-step' migration Stoffa et al. (1990), the number of reference
velocities used has minimal effect on the overall cost of the
migration.

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Stanford Exploration Project

7/5/1998