Slant-stack gathers are ellipses

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 = z_j² + 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 $z=vt\cos\theta$ and $x=vt\sin \theta$ or $x=z\tan\theta$.Take the equation for linear moveout  

$$\tau \eq t \ \ -\ \ p \ x$$ (9)

and eliminate t and x with the parametric equations.  

$$\tau \eq {z \over v \ \cos \, \theta } \ \ -\ \ {\sin \, \t... ...\over v }\ \ z \ \tan \, \theta \eq {z \over v }\ \cos\,\theta$$ (10)

 

$$\tau \eq {z \over v }\ \sqrt{ 1 \ -\ p^2 v^2 }$$ (11)

Squaring gives the familiar ellipse equation  

$$\left( {\tau \over z } \right)^2 \ \ +\ \ p^2 \eq {1 \over v^2}$$ (12)

Equation (12) is plotted in Figure 6 for various reflector depths z_j.

 

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Figure 6 Travel-time curves for a data gather on a multilayer earth model of constant velocity before and after slant stacking.