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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
*t*^{2} *v*^{2} = *z*_{j}^{2} + *x*^{2} when we slant stack.
It will be convenient to consider the circle and hyperbola
equations in *parametric * form,
that is,
instead of *t*^{2} *v*^{2} = *x*^{2} + *z*^{2},
we use and or .Take the equation for linear moveout

| |
(9) |

and eliminate *t* and *x* with the parametric equations.
| |
(10) |

| |
(11) |

Squaring gives the familiar ellipse equation
| |
(12) |

Equation (12) is plotted in Figure 6
for various reflector depths *z*_{j}.
**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