Introduction

In the previous chapter, I described the process of applying helical boundary conditions to facilitate the factorization of implicit extrapolators. However, I only covered the case where the velocity was constant within each depth layer, i.e. constant velocity and v(z) earth models. The advantage of working in the space domain, as opposed to the spatial-frequency domain, is that method can be adapted to handle operators changing laterally. Indeed, the strength of conventional implicit finite-difference methods comes in areas with strong lateral velocity variations, where the small filters can accurately model the rapid velocity changes, and the implicit formulation can guarantee unconditional stability Godfrey et al. (1979).

In this chapter, I describe how recursive filtering can be extended to handle non-stationarity. This allows implicit depth migration with the helix factorization to be applied in areas with lateral velocity variations. Unfortunately, however, I am unable to formulate the helical factorization in such a way that maintains the unconditional stability of the conventional implicit schemes. Therefore, the stability of helical extrapolators in laterally variable media cannot be guaranteed.