1. Initial integral-Chapter Vdealswith the information on the edge-diffractedwaves that can be obtained by an asymptotic analysis of the integral IV.2(7), i.e.,
Weare going to obtain a high-frequency asymptotic representation of this integral with by using a traditional technique for a contour integral of a rapidly oscillating function (Felsen and Marcuvitz, 1973). We will deform the contour of integration to the form that allows us to obtain the asymptotic formulas by integrating over its separate particular parts. Such parts of the contour must pass through the neighborhoods of those points of the complex plane of z, which give the essential contribution in the integral, when
The asymptotic value of the integral (1) with is formed by contributions from the neighborhoods of the singular points, where the integrand loses its analytic character, and the saddle points, which are the minimax points of the modulus of the integrand. Let us consider the behavior of the integrand in the neighborhoods of the mentioned points.
Figures & Tables
This book is devoted to one important aspect of development of physical foundations of the seismic method — the theory of edge diffraction phenomena. Thoese phenomena occur when conditions of the regular wave reflection/transmission change sharply. Though these phenomena drew the attention of many scientist for many decades, their real influence on the resolution ability of the seismic method was truly understood rather recently due to interpretation of seismic data in block structures. Clearly, to develop seismic method for investigation of such structures without developing the theory of edge diffraction phenomena is impossible. The latter is the aim of this book.
The seismic method is based on the fundamental laws of continuum mechanics. These laws describe the behavior of wavefields on the microscopic level, i.e., in the form of differential equations of motion. Integrating these equations under some initial conditions or boundary conditions, makes possible acquisition of all necessary information on the wavefield in the given situation. However, the working base of the seismic method consists of not only the differential equations of motion themselves but of some general and simple enough consequences of their solutions, which are formulated in the form of physical principles and l aws. The latter include the concepts of wave, Fermat’s principle, the law of conservation of the energy flux, and the reflection/transmission laws. Essentially these laws and principles must form a system of concepts sufficient for the solution of some class of typical interpretation problems. In fact, these principles and laws form the physical fo ndation of the seismic method.