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A slant stack of a data gather yields a single trace
characterized by the slant parameter p.
Slant stacking at many p-values yields a
slant-stack gather. (Those with a strong mathematical-physics background
will note that slant stacking transforms travel-time curves
by the Legendre transformation.
Especially clear background reading is found in
Thermodynamics,
by H.B. Callen, Wiley, 1960, pp. 90-95).
Let us see what happens to the familiar family of hyperbolas
t2 v2 = zj2 + x2 when we slant stack.
It will be convenient to consider the circle and hyperbola
equations in parametric form,
that is,
instead of t2 v2 = x2 + z2,
we use
and
or
.Take the equation for linear moveout
| ![\begin{displaymath}
\tau \eq t \ \ -\ \ p \ x\end{displaymath}](img26.gif) |
(9) |
and eliminate t and x with the parametric equations.
| ![\begin{displaymath}
\tau \eq {z \over v \ \cos \, \theta } \ \ -\ \
{\sin \, \t...
...\over v }\ \ z \ \tan \, \theta
\eq {z \over v }\ \cos\,\theta\end{displaymath}](img27.gif) |
(10) |
| ![\begin{displaymath}
\tau \eq {z \over v }\ \sqrt{ 1 \ -\ p^2 v^2 }\end{displaymath}](img28.gif) |
(11) |
Squaring gives the familiar ellipse equation
| ![\begin{displaymath}
\left( {\tau \over z } \right)^2 \ \ +\ \ p^2 \eq {1 \over v^2}\end{displaymath}](img29.gif) |
(12) |
Equation (12) is plotted in Figure 6
for various reflector depths zj.
sstt
Figure 6
Travel-time curves for a data gather on a multilayer earth model of
constant velocity before and after slant stacking.
Next: Two-layer model
Up: SLANT STACK
Previous: Slant stacking and linear
Stanford Exploration Project
10/31/1997