Stolt migration is regarded as the fastest post-stack migration method of all the known algorithms. A known price for that speed is the constant velocity assumption. The time-stretching trick proposed in Stolt's classic paper 1978 provides an approximate extension of the method to a variable velocity case. Stolt stretch implicitly transforms reflection traveltime curves to fit an approximate constant velocity pattern Claerbout (1985); Levin (1983, 1985). In other words, the wave equation with variable velocity is transformed by a particular stretch of the time axis to an approximate differential equation with constant coefficients. The two constant coefficients are an arbitrarily chosen frame velocity and a specific nondimensional parameter (W in Stolt's original notation). In the constant velocity case W is equal to 1, and the transformed equation coincides with the exact constant velocity wave equation. In variable velocity media, W is generally assumed to lie between and 1. As shown by Beasley et al. 1988, the cascaded f-k migration approach can move the value of W for each migration in a cascade closer to 1, thus increasing the accuracy of the Stolt stretch approximation.
The W factor was defined by Stolt 1978 as an approximate average of a complicated function. Stolt's definition cannot be used directly for computation because it includes a combined dependence on both time and space coordinates. Therefore, in practice, the estimation of this factor is always replaced by a heuristic guess. That's why Levin 1983 called the W parameter ``infamous'' (joking, of course), and Beasley et al. 1988 called it it ``esoteric.''
This paper develops a method to evaluate the Stolt stretch parameter explicitly. The main idea is to constrain the parameter by fitting the exact and approximated traveltime functions. In the case of isotropic interpretation, the W parameter is connected to the ``parameter of heterogeneity'' Castle (1988); Malovichko (1978); de Bazelaire (1988). In the case of anisotropic (transversally isotropic) interpretation, it can be related to the ``parameter of anellipticity'' Dellinger et al. (1993); Muir and Dellinger (1985).