| Abstract: | The standard Kirchhoff algorithm can be generalized for migration of pre-stack finite-offset data from variable-velocity media. The concentric ellipses over which the data are spread in constant velocity media become significantly distorted (even multi-valued) in the variable velocity case. The specific shapes can be explicitly defined by kinematic extrapolation of the source and recorded wave fields with the ray equation. The use of Kirchhoff migration with a surface source and a subsurface recorder requires that two sets of Kirchhoff loci be superimposed. For each trace, the first set of loci is computed with the source and the actual recorder position as foci; the second set is computed with the source and the virtual recorder position as foci. This dual procedure explicitly incorporates the primary diffracted energy and the free-surface reflections, respectively. Implementation involves the construction of a virtual medium, lying above the free surface, with a velocity distribution that is the mirror image of the actual distribution below the free surface. Ray-equation extrapolation is performed through the real/virtual boundary. The resulting image is produced in a split form, with all the contributions of the primary reflected and diffracted energy lying in the lower ‘real’ half and all the contributions of the energy that was reflected at the free surface lying in the upper ‘virtual’ half. The final image is produced by folding the split image about the free surface and adding the two halves. A practical advantage is that the origin of various contributions (and artifacts) can be more readily identified (for interpretation or removal) in the split images. The ray-equation pre-stack migration algorithm is very general. It is applicable to all source-recorder geometries and variable velocity media and reduces exactly to the standard Kirchhoff algorithm when applied to zero or finite-offset surface survey data. The algorithm is illustrated by application to VSP data. For the VSP geometry, the algorithm does not require any specific trace spacing (in depth) and can be used for data from deviated as well as vertical holes. |
