Ray Theory: Characteristics and Asymptotics


Andrej Bóna
Department of Exploration Geophysics, Curtain University
Perth, Australia
A.Bona@curtain.edu.au

Michael Slawinski
Department of Earth Science, Memorial University
Newfoundland, Canada
mslawins@mun.ca

In these lecture notes we strive to explain the understanding of the underpinnings of ray theory. These notes are intended for senior undergraduate and graduate students interested in the modern treatment of ray theory expressed in mathematical language. We assume that the reader is familiar with linear algebra, differential and integral calculus, vector calculus as well as tensor analysis. To investigate seismic wave propagation, we often use the concepts of rays and wavefronts. These concepts result from studying the elastodynamic equations using the method of characteristics or using the high-frequency approximation. Characteristics of the elastodynamic equations are given by the eikonal function whose level sets are wavefronts. Characteristic equations of the eikonal equation are the Hamilton ray equations whose solutions are rays. Hence, rays are bicharacteristics of the elastodynamic equations. Characteristics are entities that are associated with differential equations in a way that is invariant under a change of coordinates. This property illustrates the fact that characteristics possess information about the physical essence of a given phenomenon.

This book is intended for graduate students as well as scientists interested in theoretical seismology. We assume that the reader is familiar with undergraduate mathematics and physics. The chapters of this book are intended to be studied in sequence. In that manner, the entire book can be used as a manual for a one-semester graduate course. Each chapter begins with a section called Preliminary remarks, where we provide the motivation for the specific concepts discussed therein, outline the structure of the chapter and provide links to other chapters. Each chapter ends with a section called Closing remarks, which emphasizes the importance of the discussed concepts and show their relevance to other chapters. Each chapter is followed by Exercises and their solutions.

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