Geometry

Extension from 2-D to 3-D requires the introduction of new angles in the geometric definition of the problem. However, we will see that a 3-D problem can be formulated as a 2-D problem whose analysis is done by Biondi and Symes (2003).

In 2-D, two angles define the geometry of the problem: the dip of the interface, $\alpha$, and the aperture of the reflection, $\gamma$.In 3-D, the reflector is not only defined by its dip $\alpha$, but also by its azimuth $\phi$.Further, in the full 3-D geometry, the rays are not necessarily coplanar during downward-continuation. As a consequence, the reflection may have a different azimuth than the azimuth of the survey. We call $\beta$ the reflection azimuth. When the correct migration velocity is used, the two rays focus correctly in one point. Near this image point the two rays define defines one plane. The most general configuration considers an incorrect migration velocity. Because the velocity is incorrect, the rays do not focus in one point. Instead, they stop as two different points with the same depth when using downward continuation. The distance between the two points is the offset at constant z. The middle of the segment is the image point in the offset domain domain at constant z (zODCIG). In such configuration, the rays are not necessarily coplanar. We introduce a new angle, $\xi$,accounting for the non-coplanarity of the rays. To make the link with the 2-D case, we seek an apparent propagation plane containing all the information about the actual geometry. In all cases, the image point in the angle domain moves along the normal to the apparent interface by an amount dependent on the migration velocity used, and on the aperture angle. The image point in the angle domain is obtained by transforming of the image point in the offset-domain. We use the post-migration transformation in the Fourier domain introduced in Sava and Fomel (2000) for the 2-D geometry, and its 3-D extensions by Tisserant and Biondi (2003) for the 3-D full prestack migration. Our approach in the next two sections is based on a ray construction. Later, we will present in the third section another approach based on plane-waves.