Lateral velocity variations

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 $\omega/v$, we precompute the factors of the 1-D helical filters, a₁ and a₂. 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 a₂. The polynomial division is with the filter pair corresponding to a₁. 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.