As a result, an impulse in a constant offset section will be moved in a zero offset section accounting also for the effect of the dipping reflector. The element which fixes the position of the migrated point on the ellipse is the dipping angle . The coordinates of the points on the ellipse are functions of the dip angle .
The mapping of the input point (th,xh) from a constant offset section into the point (t0,x0) in a zero-offset section can be expressed as
Given the geometry in Figure 1, the coordinates of the point P can be expressed as a function of the semiaxes of the ellipse (a and b ) and the dip angle .The semiaxis are a = thV and b , where V is half the earth velocity. The equation of the ellipse is:
z2ma2 = a2b2 - x2mb2
The equation of the tangent to the ellipse is where and .
Squaring equation 2 and multiplying both sides with we obtain:
As the intersection of the tangent to the ellipse should satisfy both equation (1) and (2), by equalizing both sides of zm2a2 we obtain:
From the previous equation (4) we obtain the value for xm, and introducing this value in equation 1, we obtain zm.
Each point on the ellipse corresponding to a certain parameter (dipping angle), can be considered a point diffractor and therefore generate a hyperbola. For a zero-offset section the point will be situated on the modeling hyperbola with the following coordinates:
Inserting equations (5,6) in (7,8) we obtain:
In the end the DMO operator contoured for only half space will look like in Figure 3. We demonstrate in Appendix A that the operator follows the envelope of the hyperbolas.