Exploring three-dimensional implicit wavefield extrapolation with the helix transform |

**Sergey Fomel and Jon F. Claerbout**

Implicit extrapolation is an efficient and unconditionally stable
method of wavefield continuation. Unfortunately, implicit wave
extrapolation in three dimensions requires an expensive solution of
a large system of linear equations. However, by mapping the
computational domain into one dimension via the helix transform, we
show that the matrix inversion problem can be recast in terms of an
efficient recursive filtering. Apart from the boundary conditions,
the solution is exact in the case of constant coefficients (that is,
a laterally homogeneous velocity.) We illustrate this fact with an
example of three-dimensional velocity continuation and discuss
possible ways of attacking the problem of lateral variations.

- Introduction
- Implicit versus Explicit extrapolation
- Spectral factorization and three-dimensional extrapolation

- Three-dimensional implicit velocity continuation
- Depth extrapolation and the v(x) challenge
- Conclusions
- Acknowledgments
- Bibliography
- About this document ...

2014-02-17