Theory of differential offset continuation

Example 3: elliptic reflector

The third example (the right side of Figure 4) is the curious case of a focusing elliptic reflector. Let $y$ be the center of the ellipse and $h$ be half the distance between the foci of the ellipse. If both foci are on the surface, the zero-offset traveltime curve is defined by the so-called ``DMO smile'' (Deregowski and Rocca, 1981):

$$t_0\left(y_0\right)={t_n \over h}\,\sqrt{h^2-\left(y-y_0\right)^2}\;,$$ (40)

where $t_n=2\,z/v$, and $z$ is the small semi-axis of the ellipse. The time-ray equations are

$$y_1\left(t_1\right)=y+{h^2\over {y-y_0}}\,{{t_1^2-t_n^2} \ov... ... \left(y-y_0\right)^2}\,{{t_1^2-t_n^2} \over t_n^2} \right)\;.$$ (41)

When $y_1$ coincides with $y$, and $h_1$ coincides with $h$, the source and the receiver are in the foci of the elliptic reflector, and the traveltime curve degenerates to a point $t_1=t_n$. This remarkable fact is the actual basis of the geometric theory of dip moveout (Deregowski and Rocca, 1981).

Theory of differential offset continuation