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.

Prestack depth migration is the most glamorous step of seismic processing because it transforms mere data into an image, and that image is considered to be an accurate structural description of the earth. Thus, our expectations of its accuracy, robustness, and reliability are high. Amazingly, seismic migration usually delivers. The past few decades have seen migration move from its heuristic roots to mathematically sound techniques that, using relatively few assumptions, render accurate pictures of the interior of the earth. Interestingly, the earth and the subjects we want to image inside it are varied enough that, so far, no single migration technique has dominated practical application. All techniques continually improve and borrow from each other, so one technique may never dominate. Despite the progress in structural imaging, we have not reached the point where seismic images provide quantitatively accurate descriptions of rocks and fluids. Nor have we attained the goal of using migration as part of a purely computational process to determine subsurface velocity. In areas where images have the highest quality, we might be nearing those goals, collectively called inversion. Where data are more challenging, the goals seem elusive. We describe the progress made in depth migration to the present and the most significant barriers to attaining its inversion goals in the future. We also conjecture on progress likely to be made in the years ahead and on challenges that migration might not be able to meet.

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.

We generalize the pseudo-spectral method for the acoustic wave equation to create analytical solutions to the constant velocity acoustic wave equation in an arbitrary number of space dimensions. We accomplish this by modifying the Fourier Transform of the Laplacian operator so that it compensates exactly for the error due to the second-order finite-difference time marching scheme used in the conventional pseudo-spectral method. Of more practical interest, we show that this modified or pseudo-Laplacian is a smoothly varying function of the parameters of the acoustic wave equation (velocity most importantly) and thus can be further generalized to create near-analyticallyaccurate solutions for the variable velocity case. We call this new method the pseudo-analytical method. We further show that by applying this approach to the concept of acoustic anisotropic wave propagation, we can create scalar-mode VTI and TTI wave equations that overcome the disadvantages of previously published methods for acoustic anisotropic wave propagation. These methods should be ideal for forward modeling and reverse time migration applications.