What do we expect from a theory? Efficiency, prediction, and explanation. Certainly the first aim seems to be the most important. Many specialists apply a theory mainly with the purpose to construct effective algorithms. But I believe the prediction and explanation by no means are less important. Though we do use computers, we ourselves are not computers at all. And an investigator who possesses algorithms but does not understand why and how it works cannot estimate proper results of processing and realize all possibilities of algorithms.
There is no comprehensive theory. Any theory describes only a part of a whole picture from some particular point of view. It especially concerns theories pretending to give explanations. Any analysis orientated towards understanding a specific class of transformations is connected with the decomposition of a wave field under transformation into two parts: (1) the main part and, (2) all other oscillations. In many cases a set of individual waves is considered the main part of the field. Such an analysis can't be but an approximation. As a rule, formulas describing propagation of waves are asymptotic. But it is not bad! A brilliant example is the ray theory which represents a basement of our understanding of the wave propagation in a wide class of inhomogeneous media. Of course, the approximate character of the ray method also has a pragmatic significance since it generates a lot of simple algorithms for dynamic forward modeling. For many years this pragmatic aspect was estimated as the most essential. Many geophysicists believed that after some time (as a result of the appearance of big computers) finite-difference techniques would monopolize forward modeling. But this has not happened! If something did happen, it was that results of numerical modeling in many cases were analyzed with the help of the ray method because sometimes this is the only way to connect particularities of a wave field with particularities of a model.
A very important advantage of the ray theory is its geometrical character. All main notions of the theory -- fronts, rays, caustics, critical angles, diffraction, etc. -- are geometrical by nature. It is the geometry that stimulates our imagination and, therefore, improves our understanding of wave phenomena. Geometrical character of the ray theory is essentially connected with the fact that the theory factually describes propagation of wave field discontinuities and their interaction with discontinuities of functions describing media.
It is natural to use the same approach for the analysis of wave field transformations that we use in the seismic processing. The usual approach is to analyze these transformations in (f, k)-domain. But this technique is adequate in a case where if the transformation under investigation possesses stationarity with respect to variables that are not valid in some other cases. But that's not all. It is not so simple to analyze the location (travel-time curve) and amplitude of a wave after transformation in spectra terms. It is necessary to use a complex domain in the framework of the stationary phase technique. Meanwhile, discontinuities allow one to obtain a desirable result much more simple.
For the first time I used discontinuities about a decade ago while investigating the wave field continuation. I could see that the notion of wave field discontinuity allows one to study a wide class of problems without specifying a particular algorithm and therefore to pose the task of choosing the optimum one. Subsequently, it had become evident that this approach was effective in many other cases, especially when the transformation was expressed by the integral geometry operator.
Publication of these course lectures became possible thanks to Jon Claerbout who invited me to visit Stanford University and to the enthusiasm of the SEP participants who were not only my attentive listeners but helped very much in the preparation of this course. I am especially much obliged to Debbie Levoy and Lin Zhang.