Geophys. J. Int. zyxwvutsrqponm (1995) 123,277-290 Asymptotic edge-and-vertex diffraction theory zyx Andrzej Hanyga Institute for Solid Earth Physics, University of Bergen, Alligaten 41, N5007 Bergen, Norway Accepted 1995 May 9. Received 1995 May 9; in original form 1994 July 21 SUMMARY Uniformly asymptotic formulae for edge-and-vertex diffraction in the time-domain, involving elementary functions of time, traveltimes and GTD amplitudes, are derived. Explicit expressions for diffraction at a pyramid and a triangle are constructed. They can be applied to the numerical calculation of a field reflected and diffracted at 3-D objects with sharp edges and to reflection from triangulated surfaces. The computational cost is very low. Key words: diffraction, edge waves, seismic modelling. zyx 1 INTRODUCTION Diffracted waves account for a substantial part of the visible energy in a seismic wavefield reflected from a complex subsur- face. They allow correct identification of faults and disconti- nuities in layering. On the other hand, for reasons related to computer capacity, 3-D seismic modelling still relies on asymp- totic ray theory (ART). It is therefore important to be able to include diffraction in ray-theoretical computational schemes. A general approach to the asymptotic solution of diffraction problems is laid out in Hanyga (1993, 1994, 1995). It is based on a coupling of point-to-point ray tracing with uniformly valid asymptotic formulae in the time or frequency domain. Ray tracing must include diffracted rays, accounting for evan- escent fields in deep shadow (and complex rays in caustic shadows; Hanyga zyxwvutsrqp & Seredynska 1991; Hanyga & Helle 1990, 1994). This approach leads to computationally effective algor- ithms which combine powerful ray-tracing algorithms (Hanyga 1988, 1991) with simple analytic formulae for the evaluation of the traces. In Hanyga (1995), it is shown that this approach is very general, and that in all the cases of practical interest- with the exception of umbilic caustics-the time-domain asymptotic formulae involve elementary functions and elliptic integrals. This paper is devoted to the diffraction effects associated with shadows caused by sharp objects (edge-and-vertex diffrac- tion). The method was, however, originally applied to caustics. Frequency-domain asymptotic expressions for caustic diffrac- tion were introduced in Ludwig (1966). Time-domain asymp- totic expressions for cuspoid caustics were introduced in Burridge (1962) and Stickler, Ahluwalia & Ting (1981), and in a more complete form in Hanyga & Seredynska (1991). The numerical test described in Hanyga & Helle (1990, 1994) provides a good example of their unexpected accuracy. The first step towards the development of ray theory for edge diffraction was made in Keller (1958,1966). More detailed formulae were obtained by Lewis & Boersma (1969), and a generalization to higher-order edges and vertices was presented in Kaminetzky & Keller (1972). The theory developed by Keller is called the geometrical theory of diffraction (GTD). In order to describe the evanescent field in the penumbra, Keller introduced edge- and vertex-diffracted rays. Diffracted rays are always associated with a specific shadow boundary of a specific ray congruence. As a concrete example, we shall speak about shadows of reflected rays, although the same remarks apply to shadows of other kinds of rays. The amplitudes of diffracted rays are proportional to diffrac- tion coefficients-in the same way that the amplitudes of reflected rays are proportional to reflection coefficients. Diffraction coefficients can be calculated by solving boundary value problems that allow explicit solutions in integral form- for example by separation of variables or by the Wiener-Hopf method. It is then assumed that the formulae for diffraction coefficients remain valid in more general boundary value problems with inhomogeneities and curved interfaces and edges (Keller 1958; James 1976; Achenbach, Gautesen & McMaken 1982). This is a version of the ‘locality principle’ for edge and vertex diffraction. Edge and vertex diffraction coefficients depend on the angle between the launching direction of the diffracted ray and the shadow boundary. At the shadow boundary, the diffracted rays coincide with reflected rays and the diffraction coefficients assume infinite values. It is shown in Hanyga (1989a, 1994) that the coalescence of two different kinds of rays and the blow-up of diffraction coefficients at a point in the wavefield are closely interrelated. Furthermore, it is possible to determine the singularity of diffraction coefficients at the shadow boundary without even calculating them for any specific problem. In order to overcome this difficulty, Keller and his co-workers applied boundary-layer methods to edge diffraction (Buchal & Keller 1960 Bakker 1990; Klem-Musatov & Aizenberg 1984). A different approach was developed in Orlov (1974), and independently in Hanyga (1989a), in the frequency z 0 1995 RAS 277 Downloaded from https://academic.oup.com/gji/article-abstract/123/1/277/570680 by guest on 14 June 2020