Prestack migration ellipse

Equation (1) in (y,h)-space is  

$$t\,v\ \eq \ \sqrt { z^2\ +\ {( y \ -\ y_0 \ - \ h) }^2} \ +\ \sqrt { z^2\ +\ {( y \ -\ y_0 \ + \ h) }^2}$$ (5)

A basic insight into equation (1) is to notice that at constant-offset h and constant travel time t the locus of possible reflectors is an ellipse in the (y ,z)-plane centered at y₀. The reason it is an ellipse follows from the geometric definition of an ellipse. To draw an ellipse, place a nail or tack into s on Figure 1 and another into g. Connect the tacks by a string that is exactly long enough to go through (y₀ ,z). An ellipse going through (y₀ ,z) may be constructed by sliding a pencil along the string, keeping the string tight. The string keeps the total distance tv constant as is shown in Figure 3

 

ellipse1 Figure 3 Prestack migration ellipse, the locus of all scatterers with constant traveltime for source S and receiver G.

Replacing depth z in equation (5) by the vertical traveltime depth $\tau = 2z/v=z/v_{\rm half}$ we get  

$$t \eq {1 \over 2}\ \left( \sqrt { \tau^2\ +\ [( y-y_0)-h]^... ...rt { \tau^2\ +\ [( y-y_0)+h]^2 / v_{\rm half}^2 } \ \right)$$ (6)