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# THE PSPM OPERATOR IN CONSTANT VELOCITY MEDIA

Prestack partial migration is a mapping of prestack data to zero-offset data. This mapping can be subdivided in two steps:
1.
Full prestack migration to a depth model. An input data point in a constant offset section, is migrated into an ellipse. In other words a single impulse on a trace can be generated by a reflection in any point on the reflecting elliptic surface.
2.
Zero-offset modeling. Each point on the ellipse can be modeled into a hyperbola, which is obtained by considering the chosen point as a point diffractor.

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:

 (1)
Which can be rewritten as

z2ma2 = a2b2 - x2mb2

The equation of the tangent to the ellipse is where and .

 (2)

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:
 (3)

 (4)

From the previous equation (4) we obtain the value for xm, and introducing this value in equation 1, we obtain zm.

 (5)
 (6)

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:
 (7)
 (8)

Inserting equations (5,6) in (7,8) we obtain:

 (9)

 (10)
which are the parametric equations (function of the dip angle )of the well known DMO ellipse.

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.

Next: PSPM IMPULSE RESPONSE IN Up: Popovici and Biondi: PSPM Previous: DMO
Stanford Exploration Project
1/13/1998