Weak-contrast edge and vertex diffractions in anisotropic elastic media

Diffraction phenomena caused by edges and vertices are important in a numbe r of studies and applications of wave propagation methods. Diffractions pla y a fundamental role, for example, in the interpretation of seismic data fr om faults and other complex geological structures in which petroleum reserv oirs are located. The geometric theory of diffraction provides a good description of the diff raction of scalar waves from an edge in a homogeneous medium assuming free or rigid boundary conditions. The solution becomes infinite at the shadow b oundary between the illuminated zone and the shadow zone, so that, within t his region, a boundary layer solution must be used. This can be achieved by analytic continuation of the geometrical ray solution or, equivalently, us ing a one-dimensional or two-dimensional diffusion equation for modelling t he amplitude of the diffracted wave. An alternative, older method for obtaining asymptotic expressions for diffe rent parts of the wavefield is by asymptotic expansion of the Kirchhoff int egral. The authors have developed a new reciprocal surface-scattering integ ral by applying the divergence theorem to the Born volume integral. This ne w integral is called the Born-Kirchhoff (BK) integral. The scattering surfa ce is a finite sum of smooth surfaces separated by smooth curves with a fin ite number of corners. The BK integral is now a sum of integrals over each smooth surface. These integrals are evaluated by the method of stationary p hase, resulting in specularly reflected waves and boundary diffracted waves represented by a line integral along the boundary of each surface. Further asymptotic evaluation of these line integrals results in expressions for e dge-and corner-diffracted waves. Smooth approximations of the wavefield are given for the case that a reflection point and an edge-diffraction point a re close to each other, and when an edge-diffraction point and a corner-dif fraction point are close to each other. These formulas can be cascaded to provide asymptotic expressions for multip le converted, reflected, transmitted and diffracted waves in anisotropic, i nhomogeneous, elastic media. In the case that the rays are well separated f rom all shadow zones, the new expressions satisfy reciprocity. (C) 1999 Els evier Science B.V. All rights reserved.