Conclusions

Implicit extrapolations have the advantages over explicit methods, that they can be unconditionally stable, and shorter filters are required to achieve higher accuracy. Through the helical coordinate system, we have recast the 2-D deconvolution at the heart of implicit 3-D wavefield extrapolation, into a one-dimensional problem that can be solved efficiently by recursion. Lateral variations in velocity are handled by non-stationary deconvolution. While we have demonstrated our method by migrating a simple synthetic example with the 45$^\circ$ approximation, helical boundary conditions may be applied to the full range of implicit methods, making them viable for 3-D applications.