GARDNER'S SMEAR OPERATOR

A task, even in constant velocity media, is to find analytic expressions for the travel time in the Rocca operator. This we do now.

The Rocca operator $\bold R = \bold C'_0 \bold C_h$says to spray out an ellipse and then sum over a circle. This approach, associated with Gerry Gardner, says that we are interested in all circles that are inside and tangent to an ellipse, since only the ones that are tangent will have a constructive interference.

The Gardner formulation answers this question: Given a single nonzero offset impulse, which events on the zero-offset section will result in the same migrated subsurface picture? Since we know the migration response of a zero and nonzero offset impulses (circle and ellipse) we can rephrase this question: Given an ellipse corresponding to a nonzero offset impulse, what are the circles tangent to it that have their centers at the earth's surface? These circles if superposed will yield the ellipse. Furthermore, each of these circles corresponds to an impulse on the zero-offset section. The set of these impulses in the zero offset section is the DMO+NMO impulse response for a given nonzero offset event.

 

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Figure 23 The nonzero offset migration impulse response is an ellipse. This ellipse can be mapped as a superposition of tangential circles with centers along the survey line. These circles correspond to zero offset migration impulse responses.