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## Water-bottom multiples

Consider a model with a dipping water-bottom in 2D. The raypath of the primary reflection can be easily computed using the concept of the image source as illustrated in Figure .

 rayprim Figure 1 Construction of the primary reflection from a dipping interface.

The moveout of the primary reflection in the CMP domain is given by:
 (1)
where tp is the time of the primary, is the dip angle of the reflector, ZD is the perpendicular distance between the surface and the reflector at the CMP location, hD is half the source-receiver offset, V is the propagation velocity above the dipping reflector, is the normal moveout velocity and t0 is the traveltime of the zero-offset reflection. This is obviously the equation of a hyperbola, as illustrated in Figure .

The raypath of the multiple reflection can be considered as a cascaded of two primary reflections as SRME methods do (Figure ), but the traveltime of the multiple, tm, can be computed more easily as the traveltime of an equivalent primary from a reflector dipping at twice the dip angle of the actual reflector as illustrated in Figure . That is,
 (2)
where is the perpendicular distance between the surface and the equivalent reflector with twice the dip at the CMP location and is now . Figure  corresponds to a CMP showing the primary and the multiple reflection. Obviously, they are both hyperbolas since the multiple has the same kinematics as a primary from a reflector dipping at twice the dip as indicated above.

 raymul1 Figure 2 Decomposition of the water-bottom multiple as a cascaded of two primary reflections.

 raymul2 Figure 3 Multiple reflection as a primary from an equivalent reflector with twice the dip angle.

 moveouts1 Figure 4 Moveout curves of primary and water-bottom multiple from a dipping interface on a CMP gather.

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Stanford Exploration Project
5/3/2005