Claerbout 1985 describes implicit 2-D wavefield extrapolation based on the Crank-Nicolson formulation. This rational operator is unitary, and so represents a pure phase-shift. Unfortunately, however, the simple extension to 3-D leads to prohibitive computational complexity.
In practice, wavefield extrapolation in 3-D is usually accomplished with explicit operators and McClellan transforms Hale (1990a,b). Unfortunately again, however, accuracy at high dips (large spatial wavenumbers) can only be achieved with long explicit filters.
In this paper, we construct a finite-difference approximation to the Helmholtz operator in the domain. The helical coordinate system allows us to remap the multi-dimensional operator into one-dimensional space, where we can find two minimum-phase factors using a conventional spectral factorization algorithm.
Each minimum-phase factor provides a recursive filter that we can use to extrapolate the wavefield in depth. Recursive filters move energy longer distances than explicit filters of the same length. By developing a purely recursive wavefield extrapolator, the hope is to achieve accuracy at high dips with shorter filters than is possible with explicit methods.