Theory of 3-D angle gathers in wave-equation seismic imaging

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# Algorithm I: Angle gathers during downward continuation

This algorithm follows from equation (13). It consists of the following steps, applied at each propagation depth :

1. Generate local offset gathers and transform them to the wavenumber domain. In the double-square-root migration, the local offset wavenumbers are immediately available. In the shot gather migration, local offsets are generated by cross-correlation of the source and receiver wavefields (Rickett and Sava, 2002).
2. For each frequency , transform the local offset wavenumbers into the angle coordinates according to equation (13). The angle coordinates depend on velocity but do not depend on the local structural dip. In the 2-D case, each frequency slice is simply the plane, and each angle coordinate corresponds to a circle in that plane centered at the origin and described by equation (14). Figure 3 shows an example of a 2-D frequency slice transformed to angles.
3. Accumulate contributions from all frequencies to apply the imaging condition in time.

angle1
Figure 3.
Constant-depth constant-frequency slice mapped to reflection angles according to the 2-D version of Algorithm I. Zero offset wavenumber maps to zero (normal incidence) angle. The top right corner is the evanescent region.

This algorithm is applicable for targets localized in depth. The local offset gathers need to be computed for all lateral locations, but there is no need to store them in memory, because conversion to angles happens on the fly. The algorithm outputs not angles directly, but velocity-dependent parameters . Alkhalifah and Fomel (2009) extend this algorithm to transversally-isotropic media.

 Theory of 3-D angle gathers in wave-equation seismic imaging

Next: Algorithm II: Post-migration angle Up: Fomel: 3-D angle gathers Previous: Common-azimuth approximation

2013-03-02