Time migration remains a very fast imaging process compared to prestack depth migration and therefore is still commonly used by seismic imaging contractors. Such an economical technique reveals itself useful as a first approach to a problem or for producing accurate images when the interval velocity varies only with depth. Among the many algorithms avaible for post-stack time migration, Stolt's is known as the fastest of all. It is derived from a wavefield downward-continuation in constant velocity. This constant velocity assumption yields the well-known shortcoming of Stolt's algorithm. In his classic paper, Stolt 1978 proposed as an approximation for v(z) media a stretching of the time axis that is commonly called ``Stolt-stretch'' migration. In that context, the vertical heterogeneities of the velocity model are represented by a single nondimensional parameter W, substituted for a complicated function of several parameters. In the constant velocity case, W is equal to 1.0. In a medium where the velocity is increasing with depth, its value is constrained to lie between 0.0 and 1.0.
In practice, a frustrating drawback of the technique is that there was no constructive way to choose the parameter W. To overcome this heuristic guess, Fomel 1995 derived an explicit formulation for W based on Malovichko's formula for approximating traveltimes in vertically inhomogeneous media Castle (1988); Malovichko (1978); Sword (1987); de Bazelaire (1988). In this paper, we implement Stolt-stretch time migration with this optimal choice for W and discuss its accuracy.