Evaluating the Stolt-stretch parameter

Introduction

Although Stolt migration is regarded as the fastest of all the known seismic migration algorithms, it has a limited applicability because of the intrinsic constant velocity assumption. The time-stretching trick proposed in Stolt's classic paper (Stolt, 1978) provides an approximate extension of the method to a variable velocity case. Implicitly, Stolt stretch transforms reflection traveltime curves to fit an approximate constant velocity pattern (Levin, 1985; Claerbout, 1985; Levin, 1983). 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 special non-dimensional 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 0 and $1$ . As shown by Larner and Beasley (1987), 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 is defined by Stolt (1978) as an approximate average of a complicated function, which depends on both time and space coordinates and cannot be computed directly. Therefore, in practice, the estimation of $W$ is always replaced by a heuristic guess. That is why Levin (1983) jokingly called the $W$ parameter ``infamous'', and Larner and Beasley (1987) called it ``esoteric.''

In this paper, we use an analytic technique to evaluate the Stolt stretch parameter explicitly. The main idea is to constrain this parameter by fitting the Stolt-stretch traveltime function to the exact one. It turns out that in the isotropic case, the $W$ parameter is connected to the ``parameter of heterogeneity'' (Sword, 1987; Castle, 1988; de Bazelaire, 1988; Malovichko, 1978). The definition of heterogeneity is modified for the case of an anisotropic (transversally isotropic) media.

We demonstrate an application of the Stolt stretch analytical expression on a real data example from the North Sea. The velocity profile is optimally partitioned for the method of cascaded migration, which allows us to image steeply dipping reflectors at the accuracy comparable to that of the phase-shift method but at a much smaller cost.

Although Stolt migration is not currently at the forefront of geophysical research, it is still widely used in practice (Yilmaz et al., 2001; Yilmaz, 2001) and keeps recurring in different contexts. Popovici et al. (1996) propose a new interpolation scheme for improving the practical accuracy of the method. Sava (2000) uses a variation of Stolt migration - Stolt residual migration (Stolt, 1996) - in the context of wave-equation migration velocity analysis.

The growth in computer speed does not automatically make fast algorithms obsolete, because the amount of processed data tends to grow at the same rate or even faster. The researchers working in the field of seismic imaging are often interested in the following questions: What is the fastest possible migration algorithm? How accurate can it get? Stolt migration answers the first question. The answer to the second question is developed in this paper.

Evaluating the Stolt-stretch parameter