Restatement of ellipse equations

Recall equation (9) for an ellipse centered at the origin.  

$$0 \eq {y^{2}\over{A^{2}}} + {z^{2}\over{B^{2}}} -1 .$$ (19)

where  

$$A \eq v_{\rm half}\, t_h ,$$ (20)

 

$$B^2 \eq A^2 - h^2 .$$ (21)

The ray goes from the shot at one focus of the ellipse to anywhere on the ellipse, and then to the receiver in traveltime t_h. The equation for a circle of radius $R=t_0 v_{\rm half}$with center on the surface at the source-receiver pair coordinate x=b is  

$$R^2 \eq (y - b)^2 + z^{2} ,$$ (22)

where  

$$R \eq t_0 \, v_{\rm half}.$$ (23)

To get the circle and ellipse tangent to each other, their slopes must match. Implicit differentiation of equation (19) and (22) with respect to y yields:  

$$0 \eq {y \over{A^2}} + {z \over{B^2}} \ {dz \over dy}$$ (24)

 

$$0 \eq (y-b) + z \ {dz \over dy}$$ (25)

Eliminating dz/dy from equations (24) and (25) yields:  

$$y \eq {b\over 1 - {B^{2}\over{A^{2}}}} .$$ (26)

At the point of tangency the circle and the ellipse should coincide. Thus we need to combine equations to eliminate x and z. We eliminate z from equation (19) and (22) to get an equation only dependent on the y variable. This y variable can be eliminated by inserting equation (26).  

$$R^2 \eq B^2 \left( {A^2 - B^2 - b^2 \over{A^2 - B^2}} \right).$$ (27)

Substituting the definitions (20), (21), (23) of various parameter gives the relation between zero-offset traveltime t₀ and nonzero traveltime t_h:  

$$t_0^2\eq \left(t_h^2-{h^{2}\over v_{\rm half}^2}\right) \left(1-{b^2\over h^2}\right).$$ (28)

As with the Rocca operator, equation (28) includes both dip moveout DMO and NMO.