Conclusions

Implicit extrapolations have several advantages over explicit methods: they have the potential for unconditional stability, and shorter filters are required to achieve higher accuracy. Through the helical coordinate system, I 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. I have demonstrated the algorithm by migrating simple constant velocity synthetic examples with the conventional 45$^\circ$paraxial wave equation. The extension to lateral velocity variations is discussed in the next chapter.