Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion

Our goal is present a theory for determining the characteristics of the interior of a body bases only on observations made on some boundary surface. In particular, we are interested in finding ways of imaging structures inside a body. In addition to imaging, we hae the more ambitious goal of actually determining values of certain material parameters characteristic to these structures. This problem is encountered in many branches of applied science. This include such diverse disciplines as Earth science, medicine, materials science, archaeology, and the ocean sciences-just to name a few. These disciplines all face the same problem of mapping structures in environments where it is either impossible or impractical to make direct observations.

While presenting solutions to the migration/inversion/imaging problem we also intend to familiarize the reader with the philosophy of research that has proven to be effective in leading to these solutions. We hope that the student will benefit from this text both by gaining insights into the specific mathematical issues associated with the seismic inversion problem and by acquiring a “feeling” for how to decide what issues take precedence in the stages of a research project.

In Chapter 1 we discussed the historical aspects of migration. In that context, migration was viewed as the back propagation of the recorded data to its “correct” position on the image. This view was modified in Chapter 2 and 3, where we encouraged the reader to view this process as being the solution of an inverse problem. In this chapter, we will examine the imaging problem from yet another perspective, that of band- or aperture-limited Fourier transforms. In doing so, we show that there are solid mathematical reasons for choosing to invert for reflectivity. We will also see why extracting smooth information using migration-like techniques will not be easy. (Readers who wish to immediately see the methods of Chapter 3 extended to the problem of inverting seismic data collected with nonzero source-receiver offset in a variable-wavespeed medium, may skip directly to Chapters 5 and 6, and return to this chapter later.)

In Chapter 3, we created our first inversion formulas for the three-dimensional inverse-scattering imaging problem. We formulated the problem for the special case of data collected with a zero-offset recording geometry, and assumed a constant-background wavespeed. The formulas we derived yielded a bandlimited representation of the singular function of a reflector surface scaled by the normal-incidence reflection coefficient of the reflector—the reflectivity function—β(x). These formulas were further shown to correspond to the classic Fourier transform—based migration formula created by Stolt [1978] and the Kirchhoff-based formula created by Schneider [1978], with the latter deduced from the former.

The formulas of Chapter 5 were derived for general 3D data sets, but, as we have mentioned in previous chapters, a large amount of oil industry seismic data is still in the form of seismograms collected along single lines. In Chapter 3, we introduced the concept of the two-and-one-half-dimensional (2.5D) geometry to compensate for the fewer degrees of freedom available in such single-line experiments. The two-and-one-half-dimensional formulas will allow for the inversion of a single line of data while accounting for many aspects of three-dimensional wave propagation. It is the method of choice when data are gathered along a line in a direction of dominant lateral changes in the subsurface (dip direction), providing an amplitude- consistent inversion of the medium in the vertical plane containing the source-receiver line. The method cannot, however, account for scattered energy that arrives from outside the vertical plane, except by imaging these arrivals at equivalent, but incorrect, in-plane locations. It is also possible to perform 2.5D migration/inversion on lines making oblique angles with the dip direction [French, 1975], again, as long as that direction is fixed in the subsurface. The main point to be stressed is that the two-and-one-half- dimensional assumption allows a reduction of the processing of the three- dimensional inversion of Chapter 5, just as it did for the simpler constant- background, zero-offset inversion of Chapter 3. (See Bleistein, Cohen, and Hagin [1987].)

Throughout this text we have treated the seismic imaging problem as an inverse problem. To realize the goal of creating inversion formulas, we began with a simple idea. We created approximate forward modeling formulas, which we wrote as Fourier-like integrals. We then deduced their inverses, relying on the invertibility property of the Fourier transform. The inversion formulas that we obtained were also Fourier-like integrals owing to this procedure, with many of the classical Fourier-based migration techniques “falling out” as special cases of these more general formulas.