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# Stolt Stretch Theory Review

Stolt time migration can be summarized as the following sequence of transformations:
 (1)
where
 (2)
The function is the dispersion relation and has the following expression in the constant velocity case:
 (3)

The approximation suggested by Stolt 1978 for extending the method to v(z) media involves a change of the time variable (Stolt-stretch):
 (4)
where s(t) is the stretched time variable, v0 is an arbitrarily chosen constant velocity, and vrms(t) is the root mean square velocity along the vertical ray, defined by
 (5)
This change of variable yields a transformed wave-equation for the wavefield extrapolation, in which Stolt replaces a slowly varying complicated function of several parameters (denoted by W) by its average value. Making this approximation yields a new dispersion relation in the transformed coordinate system:
 (6)

This factor W contains all the information about the heterogeneities of the medium. However, it has to be determined a priori, that is, before migration. This empirical choice for W was one of the drawbacks of the Stolt-stretch method. Fomel 1995 derived an analytical formulation of the Stolt-stretch parameter, based on Malovichko's formula for approximating traveltimes in vertically inhomogeneous media Malovichko (1978):
 (7)
where the function S(t) defines the so-called parameter of heterogeneity:
 (8)

Fomel proved that, for a given depth (or vertical traveltime), the optimal value of W is
 (9)
where vrms(t) is the root mean square velocity along the vertical ray, and the vertical traveltime. The value of W used during Stolt migration is the average along the vertical profile of these W(t). In the case of an homogeneous constant-velocity model, W is equal to 1.0, whereas it has to be less than 1.0 if the velocity increases monotonically with depth.

We can sum up the application of Stolt-stretch algorithm with the optimal parameter W by the following sequence of steps:

1. Stretch the time axis and determine
the value of W along the vertical profile
2. Interpolate stretched time to a regular grid
3. 2-D FFT
4. Apply Stolt migration with the dispersion relation (6)
5. 2-D inverse FFT
6. Unstretch the time axis

Next: Application Up: Vaillant & Fomel: On Previous: Introduction
Stanford Exploration Project
10/25/1999