Kirchhoff modeling, inversion for reflectivity, and subsurface illumination

Because of its computational efficiency, prestack Kirchhoff depth migration is currently one of the most popular algorithms used in 2-D and 3-D subsur face depth imaging. Nevertheless, Kirchhoff algorithms in their typical imp lementation produce less than ideal results in complex terranes where multi pathing from the surface to a given image point may occur, and beneath fast carbonates, salt, or volcanics through which ray-theoretical energy cannot penetrate to illuminate underlying slower-velocity sediments. To evaluate the likely effectiveness of a proposed seismic-acquisition program, we coul d perform a forward-modeling study, but this can be expensive. We show how Kirchhoff modeling can be defined as the mathematical transpose of Kirchhof f migration. The resulting Kirchhoff modeling algorithm has the same low co mputational cost as Kirchhoff migration and, unlike expensive full acoustic or elastic wave-equation methods, only models the events that Kirchhoff mi gration can image. Kirchhoff modeling is also a necessary element of constrained least-squares Kirchhoff migration. We show how including a simple a priori constraint du ring the inversion (that adjacent common-offset images should be similar) c an greatly improve the resulting image by partially compensating for irregu larities in surface sampling (including missing data), as well as for irreg ularities in ray coverage due to strong lateral variations in velocity and our failure to account for multipathing. By allowing unstacked common-offse t gathers to become interpretable, the additional cost of constrained least -squares migration may be justifiable for velocity analysis and amplitude-v ariation-with-offset studies. One useful by-product of least-squares migration is an image of the subsurf ace illumination for each offset. If the data are sufficiently well sampled (so that including the constraint term is not necessary), the illumination can instead be calculated directly and used to balance the result of conve ntional migration, obtaining most of the advantages of least-squares migrat ion for only about twice the cost of conventional migration.