V(x,y,z) and non-stationary inverse convolution

 Implicit extrapolation with the helical factorization discussed in the previous chapter can be easily extended to smoothly-variable velocities. Stationary filtering and inverse filtering can be replaced by their non-stationary counterparts, and the spectral factorization becomes a problem of LU decomposition. I develop a solution the cost of which remains proportional to the number of grid nodes [O(N)]. This solution is also exactly equivalent to the constant velocity factorization in the smooth-velocity limit. Unfortunately, however, I will show that the Godfrey/Muir 1979 bulletproofing cannot be simply applied to the helical factorization, and so the method is susceptible to stability problems. Numerical examples confirm the accuracy of the method in models with smoothly-varying velocity; however, I observe instability in models with strong lateral velocity variations.