Numerical Integration for Kirchhoff Migration

Abstract

In 3D seismic surveys, common offset data often involve an irregular distribution of midpoints. In such a situation, common offset Kirchhoff migration cannot be correctly performed by means of a mere discrete diffraction stack algorithm. Such an algorithm indeed corresponds to an inconsistent numerical integration formula. To overcome this difficulty, genuine numerical integration formulas (yielding a consistent approximation of the continuous diffraction stack) have to be used, e.g. numerical integration formulas based on polynomials leading to the so-called Lagrange or Hermite interpolations. A numerical integration formula based on Lagrange interpolation can cope with irregularly sampled midpoints provided that the density of midpoints involved in the common offset gather is sufficient. Besides, Hermite interpolation, more accurate in theory than the former, also provides relative improvement in the images at the vicinity of the migrated event,. Both techniques can be implemented by means of a simple preprocessing (adequate scaling) of the data. Thus they are quite easy to implement in any existing diffraction stack code. In addition, they can be used in combination with filters to prevent aliasing of the migration operator. The additional computation cost is negligible compared with the cost of running the diffraction stack itself.