Appendix H: Evaluation of Chained Integrals
In this appendix, we use the stationary-phase method to evaluate certain stacking integrals that appear when we chain the diffraction-stack integral (DSI) and the isochron-stack integral (ISI), i.e., when we insert them into each other to solve the configuration-transform (CT) and remigration (RM) problems.
Figures & Tables
From Volume Editor’s Preface by Robert H. Stolt
Seismic True-Amplitude Imaging is a ray-theoretical exposition of seismic imaging processes, unapologetic in that it bristles at any suggestion that ray theory is not wave theory. This attitude is justified in part by being right – ray theory does come from an asymptotic approximation to the wave equation – and by the fact that similar asymptotic assumptions tend to sneak like viruses into so-called full-wave-theoretical processes as well.
On other issues, the book is more accommodating. After noting that the original Kirchhoff diffraction integral was devised only for forward-wave propagation, the authors concede that the term Kirchhoff migration has passed into common usage, and they learn to live with it. Similarly, although the term true amplitude is a lexicological minefield, it is widely applied to processes that seek to faithfully preserve amplitude information. The authors are comfortable with that, although they apply a very specific definition of the term in their text.
This book uses the term imaging in the widest possible sense, which was an eye-opener for me. I have tended to use imaging as a synonym for migration, but I can’t do that anymore. The authors note that seismic data in any stage of processing is likely to contain discrete events or images, perhaps geometrically distorted but nevertheless pictures or images of the earth’s interior. This observation is facilitated by the ray-theoretical point of view, because it is largely in asymptopia that such images form. In consequence, any operation that affects or changes the images in the data can be considered an imaging process. This brings under the imaging umbrella a variety of processes, including partial and residual migration, conversion of one experimental configuration into another, and so on.
Strictly, I suppose, Kirchhoff imaging employs a ray-theoretical model for propagation and a far-field diffractive model for reflections. It is best suited for a medium composed of regions where earth properties change slowly, divided by reflecting surfaces where properties change abruptly. Where those conditions are met imperfectly, the method may tend to impose this model, because this is how the method views the world. That is not necessarily bad – geophysicists tend to view the world that way too.
Kirchhoff depth migration has enjoyed a long period of preeminence. Advances in computer power in recent years have allowed contemplation of full-wave-equation depth-migration algorithms, but the authors of Seismic True-Amplitude Imaging show no deep concern for the future of ray-theoretical imaging. In that, they are probably justified, for several reasons. First, migration methods, once established, never die. The inverse seismic problem is so tough and complex that no single technique, however powerful, can be universally successful. New techniques may be added to the arsenal but are not likely to completely displace proven methodologies.
Second, the asymptotic approximation is very well met under most circumstances by seismic data, and even when that is not the case, ray-theoretical techniques often are able to extract useful information. Third, ray theory is well suited to true-amplitude processing because amplitudes can be computed explicitly at every point and related back to the underlying earth properties. Fourth, Kirchhoff-based imaging is uniquely suited to composite operations (e.g., demigration using one velocity structure followed by migration using another velocity structure) because, in asymptopia, a composite operation can be condensed into a single operation.
For extremely complex geology, ray theory might have difficulty providing a complete description of wave propagation, even where it is technically possible to do so. A full-wave-equation method, in contrast, may provide a complete description without extra effort. That might give some advantage to a full-wave method, but the blessings are mixed. In ray theory, one knows exactly which waves are where. If some portion of the complete waveform (perhaps a multiple or a converted wave) does not contribute positively to the desired image, a ray-theoretical method might eliminate it. Even where a full-wave method can produce the better image, one would likely want to run a Kirchhoff algorithm concurrently to aid in analysis.
Seismic True-Amplitude Imaging provides a clear, readable, and reasonably complete presentation of Kirchhoff imaging theory. Although subjects such as beam forming and multipath imaging are not presented in detail, the tools to deal with them are present. For those of us not steeped in ray theory, the book provides a good introduction and tutorial, then digs deeply and profoundly into a theory of generalized imaging.