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.

10/25/1999