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Stolt time migration can be summarized as the following sequence of
transformations:
| ![\begin{displaymath}
p_0(x,t_0) \rightarrow P_0(k_x,\omega_0) \rightarrow P(k_x,\omega)
\rightarrow p(x,t) \; ,\end{displaymath}](img1.gif) |
(1) |
where
| ![\begin{displaymath}
P_0(k_x,\omega_0) = P(k_x,\omega(k_x,\omega_0)) \left\vert
\frac{d\omega(k_x,\omega_0)}{d\omega_0} \right\vert\end{displaymath}](img2.gif) |
(2) |
The function
is the dispersion relation and has
the following expression in the constant velocity case:
| ![\begin{displaymath}
\fbox {$
\omega(k,\omega_0) = \mbox{sign}(\omega_0) \sqrt{\omega_0^2 - v^2
k_x^2}
$}
\end{displaymath}](img4.gif) |
(3) |
The approximation suggested by Stolt 1978 for
extending the method to v(z) media involves a change of the time
variable (Stolt-stretch):
| ![\begin{displaymath}
s(t) = \sqrt{\frac{2}{v_0^2} \int_0^t \tau v_{rms}^2(\tau) d\tau} \; ,\end{displaymath}](img5.gif) |
(4) |
where s(t) is the stretched time variable, v0 is an arbitrarily chosen constant velocity, and vrms(t)
is the root mean square velocity along the vertical ray,
defined by
| ![\begin{displaymath}
v_{rms}(t) = \frac{1}{t} \int_0^t v^2(\tau) d\tau \;.\end{displaymath}](img6.gif) |
(5) |
This change of variable yields a transformed wave-equation for the
wavefield extrapolation, in which Stolt replaces a slowly
varying complicated function of several parameters (denoted by W)
by its average value. Making this
approximation yields a new dispersion relation in the transformed
coordinate system:
| ![\begin{displaymath}
\fbox {$
\hat{\omega}(k_x,\hat{\omega}_0) = \left( 1 - \fra...
...at{\omega}_0)}{W}
\sqrt{\hat{\omega}_0^2 - W v_0^2 k_x^2 }
$}
\end{displaymath}](img7.gif) |
(6) |
This factor W contains all the information about the heterogeneities
of the medium. However, it has to be determined a priori, that is,
before migration.
This empirical choice for W was one of the drawbacks of the
Stolt-stretch method. Fomel 1995 derived an
analytical formulation of the Stolt-stretch parameter, based on
Malovichko's formula for approximating traveltimes in vertically
inhomogeneous media Malovichko (1978):
| ![\begin{displaymath}
t_0 = \left( 1 - \frac{1}{S(t)}\right) t + \frac{1}{S(t)}
\sqrt{t^2 + S(t) \frac{(x-x_0)^2}{v_{rms}^2(t)}} \; ,\end{displaymath}](img8.gif) |
(7) |
where the function S(t) defines the so-called parameter of heterogeneity:
| ![\begin{displaymath}
S(t) = \frac{1}{v_{rms}^4 t} \int_0^{t} v^4(t) dt\end{displaymath}](img9.gif) |
(8) |
Fomel proved that, for a given depth (or vertical traveltime), the
optimal value of W is
| ![\begin{displaymath}
\fbox {$
W(t) = 1 - \frac{v_0^2 s^2(t)}{v_{rms}^2(t) t^2} \left(
\frac{v^2(t)}{v_{rms}^2(t)} - S(t)\right) $}
\; ,\end{displaymath}](img10.gif) |
(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
5/1/2000