Approximate analytical formulas for Kirchhoff migration operator

Two important tools in seismic processing--Kirchhoff migration and demigration operators--are the basis for many imaging problems solution. Due to high numerical and computational requirements, the use of those tools for three dimensions are very computation-costly. This fact has motivated us to investigate Kirchhoff migration operations for simpler types of media in order to provide faster results to be used as an approximation for more realistic media. To obtain results with lower computational effort, a convenient environment is the so called 2.5D situation, i.e., considering 3-D wave propagation in a medium that does not vary in the horizontal direction perpendicular to the seismic line. In this case, 2-D ray-tracing is sufficient to describe the 3-D propagation effects, particularly geometric-spreading. In a medium where the parameters depend only on the depth component (1-D situation), the imaging operations only require the solution of semi-analytical integrals, which can be both precisely and immediately implemented. For some particular cases of vertical velocity distributions, approximate analytical formulas are devised for migration stacking-lines and weight functions. Several imaging algorithms present very efficient computational performance by using those models. Thus, it is possible to establish a set of cases which may be useful for validating the implementation of more complex situations.