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 subsurface depth imaging. Nevertheless, Kirchhoff algorithms in their typical implementation produce less than ideal results in complex terranes where multipathing 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 could perform a forward-modeling study, but this can be expensive. We show how Kirchhoff modeling can be defined as the mathematical transpose of Kirchhoff migration. The resulting Kirchhoff modeling algorithm has the same low computational cost as Kirchhoff migration and, unlike expensive full acoustic or elastic wave-equation methods, only models the events that Kirchhoff migration can image. Kirchhoff modeling is also a necessary element of constrained least-squares Kirchhoff migration. We show how including a simple a priori constraint during the inversion (that adjacent common-offset images should be similar) can greatly improve the resulting image by partially compensating for irregularities in surface sampling (including missing data), as well as for irregularities in ray coverage due to strong lateral variations in velocity and our failure to account for multipathing. By allowing unstacked common-offset gathers to become interpretable, the additional cost of constrained least-squares migration may be justifiable for velocity analysis and amplitude-variation-with-offset studies. One useful by-product of least-squares migration is an image of the subsurface 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 conventional migration, obtaining most of the advantages of least-squares migration for only about twice the cost of conventional migration.