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The velocity is assumed to be constant in a small volume around the image points. That is why the rays are straight lines in the next few pictures.

Biondi and Symes (2003) noted that, in 2-D, image points in the offset domains (at constant x, xODCIG, and constant z, zODCIG) lie on an apparent interface. In 2-D the apparent interface is a line, whereas it is a plane in 3-D. Not only the image points belong to the interface, but they are also all collinear (Figure ). From this property we can define a new plane that includes the normal and the common line of the image points in the offset domains. This plane has special properties since it contains all the image points: it is the link to the 2-D case. Let us further study this plane.

 OCIGs_alignment Figure 4 The endpoints and image points of all the offset-domain gathers are displayed (dotted lines). All the image points are defined in the interface and are collinear.

If the velocity is incorrect, the rays do not focus at the actual image point but at an apparent image point located in the middle of the end points of the two rays. The apparent image point is the image point in the offset domain. Further, in the case of full prestack migration, the rays are not coplanar, so there is no propagation plane. Physics requires one though. In the absence of a propagation plane, we define an apparent propagation plane. To find it, we start from the actual rays and find a rotation that makes their image coplanar. The rotation is done around the normal at the interface (Figure ). Note that the new rays are parallel to the original ones but have different end points. The plane in which the rays are coplanar after rotation is the apparent propagation plane. The geometric location of the image points in the offset-domain is at the intersection between the propagation plane and the interface. The rotated rays thus define the same plane-wave as the original rays. The angle of the rotation, , is equal to the azimuth defined in the interface of both source and receiver rays. The reflection azimuth, , is equal to the azimuth of the apparent propagation plane.

 rot_copl Figure 5 Transformation of the original rays into coplanar rays. The source an receiver endpoints (the 2 big dots) are rotated around the normal until the rays are coplanar. Note that the new rays and azimuth of the rays measured in the apparent interface. the normal cross at the same point. The two new rays and the normal define the apparent propagation plane.

We now analyze how the image in the angle domain is obtained from the image in the offset domain.

Next: Offset-to-angle transformation. Up: Incorrect migration velocity Previous: Discussion
Stanford Exploration Project
10/14/2003