No abstract

Historically, seismic migration has been the practice (science, technology, and craft) of collapsing diffraction events on unmigrated records to points, thereby moving (“migrating”) reflection events to their proper locations, creating a true image of structures within the earth. Over the years, the scope of migration has broadened. What began as a structural imaging tool is evolving into a tool for velocity estimation and attribute analysis, making detailed use of the amplitude and phase information in the migrated image. With its expanded scope, migration has moved from the final step of the seismic acquisition and processing flow to a more central one, with links to both the processes preceding and following it. In this paper, we describe the mechanics of migration (the algorithms) as well as some of the problems related to it, such as algorithmic accuracy and efficiency, and velocity estimation. We also describe its relationship with other processes, such as seismic modeling. Our approach is tutorial; we avoid presenting the finest details of either the migration algorithms themselves or the problems to which migration is applied. Rather, we focus on presenting the problems themselves, in the hope that most geophysicists will be able to gain an appreciation of where this imaging method fits in the larger problem of searching for hydrocarbons.

Until recently, the term “elastic” usually implied two‐dimensional (2-D) and isotropic. In this limited context, the divergence and curl operators have found wide use as wave separation operators. For example, Mora (1987) used them in his inversion method to allow separate correlation of P and S arrivals, although the separation is buried in the math and not obvious. Clayton (1981) used them explicitly in several modeling and inversion methods. Devaney and Oristaglio (1986) used closely related operators to separate P and S arrivals in elastic VSP data.

The elastic properties of layered rocks are often measured using the pulse through‐transmission technique on sets of cylindrical cores cut at angles of 0, 90, and 45 degrees to the layering normal (e.g., Vernik and Nur, 1992; Lo et al., 1986; Jones and Wang, 1981). In this method transducers are attached to the flat ends of the three cores (see Figure 1), the first‐break traveltimes of P, SV, and SH‐waves down the axes are measured, and a set of transversely isotropic elastic constants are fit to the results. The usual assumption is that frequency dispersion, boundary reflections, and near‐field effects can all be safely ignored, and that the traveltimes measure either vertical anisotropic group velocity (if the transducers are very small compared to their separation) or phase velocity (if the transducers are relatively wide compared to their separation) (Auld, 1973).

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.