Conclusions

The helical factorization scheme outlined in Chapter  can be extended to deal with non-stationarity by replacing stationary convolution and inverse convolution with their non-stationary counterparts. I factorize non-stationary filters by making an assumption of local stationarity. Although this is only strictly valid for smoothly-varying media, is has the advantage that for a fixed number of filter coefficients the cost remains linear in the number of grid nodes.

Since I factorize the entire implicit system, it is difficult to ensure that the extrapolator embedded inside has the non-negative definite property necessary for unconditional stability. Unfortunately, this means the helical factorization is not robust to the presence of strong lateral velocity variations in the model, and the solution may diverge.