Kirchhoff modeling, inversion for reflectivity, and subsurface illumination

Abstract

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 raytheoretical 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 waveequation 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 leastsquares 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.

Multi‐source least‐squares reverse time migration

“…The least-squares migration method (Lailly, 1984;Cole and Karrenbach, 1992;Schuster, 1993;Nemeth et al, 1999;Duquet et al, 2000) has been shown to sometimes produce migration images with better quality than those computed by conventional migration. When least-squares migration is implemented with the RTM method (Tang and Biondi, 2009;Dai and Schuster, 2010a;Dai et al, 2010;Wong et al, 2011;Dai et al, 2012), it can reduce not only the acquisition footprint but also the artifacts in the RTM image, while enhancing the image resolution.…”

“…Least-squares migration was originally implemented with Kirchhoff migration (KM) method (Nemeth et al, 1999;Duquet et al, 2000), but was later developed for phase shift migration algorithms (Kaplan et al, 2010;Huang and Schuster, 2012a). In this dissertation, I propose to implement LSM with reverse time migration (RTM) (Baysal et al, 1983;Whitmore, 1983;McMechan, 1983), where the Green's functions are calculated by finite-difference solution to the full wave equation instead of asymtotic Green's function in Kirchhoff migration (Bleistein, 1987), or one-way solution in one-way wave equation migration (Claerbout, 1985).…”

“…The least-squares migration method (Lailly, 1984;Cole and Karrenbach, 1992;Schuster, 1993;Nemeth et al, 1999;Duquet et al, 2000) has been shown to sometimes produce migration images with better quality than those computed by conventional migration. Its original implementation was with Kirchhoff migration (Nemeth et al, 1999;Duquet et al, 2000), but was later developed for phase shift migration algorithms (Kaplan et al, 2010;Huang and Schuster, 2012a).…”

“…Duquet et al (2000) tested a Kirchhoff LSM method but they also used a regularization term which penalized the difference in images from those estimated in different offset gathers. Their regularization was effective even with migration velocity errors up to 20%.…”

“…To indirectly account for the inverse Hessian matrix, an iterative conjugate gradient method can be used to solve either the linear (Lailly, 1984;Nemeth et al, 1999;Duquet et al, 2000) or non-linear (Tarantola, 1984;Mora, 1987) optimization problem. In this way, the effects of the source wavelet, limited recording aperture, geometric spreading, etc, are taken into account to produce images with reduced acquisition footprints, balanced amplitudes and improved resolution.…”