Dip and offset together

Prestack migration ellipse

Denoting the horizontal coordinate $x$ of the scattering point by $y_0$ Equation (8.1) 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 (8.1) 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 $y_0$. 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 8.1 and another into $g$. Connect the tacks by a string that is exactly long enough to go through $(y_0 ,z)$. An ellipse going through $(y_0 ,z)$ 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 8.3

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

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

Dip and offset together