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# Fundamentals of Seismic Wave Propagation

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This book makes no pretense at being a comprehensive text. Many impor- tant topics are not even mentioned – surface waves apart from interface waves, normal modes, source functions apart from impulsive point sources, attenuation, etc. – and no real data or interpretation methods are included. Many excellent texts, some recent, already cover these subjects comprehensively, e.g. Aki and Richards (1980, 2002), Dahlen and Tromp (1998) and Kennett (2001). This book also assumes a basic understanding of seismology and wave propagation, although these are briefly reviewed. Again many recent excellent undergraduate texts exist, e.g. Shearer (1999), Pujol (2003), etc. The book concentrates on the theoretical development of methods used to model high-frequency, body waves in realistic, three-dimensional, elastic Earth models, and the description of the types of signals generated. Even so in the interests of brevity, some theoretical results that could easily have been included in the text have been omitted, e.g. body-wave theory in a sphere. Further reading is suggested at the end of each chapter, often in the form of exercises. This book is intended as a text for a graduate or research level course, or ref- erence book for seismologists. It has developed from material I have presented in graduate courses over the years at a number of universities (Alberta, Toronto, ix x Preface California and Cambridge) and to research seismologists at Schlumberger Cambridge Research. All the material has never been presented in one course – there is probably too much – and some of the recent developments, particularly in Chapter 10, have never been in my own courses. The book has been written so that each theoretical technique is introduced using the simplest feasible model, and these are then generalized to more realistic situations often using the same basic notation as the introductory development. When only a limited amount of material can be covered in a course, these generalizations in the later part of each chapter can be omitted, allowing the important and powerful techniques developed later in the book to be included. Thus the mathematical techniques used in ray theory, reflection and transmission coefficients, transform methods and generalizations of ray theory are all first developed for acoustic waves. Although isotropic and aniso- tropic elastic waves introduce extra algebraic complications, the basic techniques remain the same. Only a few types of signals, particularly interface waves, specif- ically require the complications of elasticity. The material in this text belongs to theoretical seismology but the results should be useful to all seismologists. Some knowledge of physics, wave propagation and applied mathematics is assumed. Most of the mathematics used can be found in one of the many undergraduate texts for physical sciences and engineering – an excel- lent example is Riley, Hobson and Bence (2002) – and references are not given to the ‘standard’ results that are in such a book. Where results are less well known or non-standard, some details or references are given in the text. Particular empha- sis is placed on developing a consistent notation and approach throughout, which highlights similarities and allows more complicated methods and extensions to be developed without difficulty. Although this book does not cover seismic interpre- tation, the types of signals caused by different model features are comprehensively described. Where possible these canonical signals are described by simple, stan- dard time-domain functions as well as by the classical spectral results. Many of the diagrams were drawn using Matlab. Programming exercises sug- gest using Matlab and solutions have been written using Matlab. Matlab is a trade- mark of MathWorks, Inc. I would like to thank various people at Schlumberger Cambridge Research for providing the time and facilities for me to write this book, in particular my depart- ment heads – Phil Christie, Tony Booer, Dave Nichols and James Martin – and manager – Mike Sheppard – who introduced personal research time which I have used to complete the manuscript. I am also indebted to Schlumberger for permis- sion to publish. Finally, I would like to thank my family particularly Heather, my daughter, who helped me with some of the diagrams and Lillian, my wife, for her infinite patience and support. Preliminaries Unfortunately, the nomenclature, symbols and terms used in theoretical seis- mology have not been standardized in the literature, as they have in some other subjects. It would be a vain hope to rectify this situation now, but at least we can attempt to use consistent conventions, adequate for the task, throughout this book. While it would be nice to use the most sophisticated notation to allow for complete generality, rigour and developments in the future, one has to be a realist. Most seismologists have to use and understand the results of the- oretical seismology, without being mathematicians. Thus the phrase adequate or fit for the task is adhered to. Unfortunately, some mathematicians will find the methods and notation naive, and some seismologists will still not be able to follow the mathematics, but hopefully the middle ground of an audience of typical seismologists and physical scientists will find this book useful and at an appropriate level. 0.1 Nomenclature 0.1.1 Homogeneous and inhomogeneous The words homogeneous and inhomogeneous are used with various meanings in physics and mathematics. They are overused in wave propagation with at least four meanings: inhomogeneous medium indicating a medium where the physical parameters, e.g. density, vary with position; inhomogeneous wave when the ampli- tude varies on a wavefront; inhomogeneous differential equation for an equation with a source term independent of the field variable; and, homogeneous boundary conditions where either displacement or traction is zero. To avoid confusion, as the four usages could occur in the same problem, we will use it in the first sense and avoid the others except in some limited circumstances. xi xii Preliminaries 0.1.2 Order, dimensions and units The term dimension is used with various meanings in physics and mathematics. In dimensional analysis, it is used to distinguish the dimensions of mass, length and time. In vector-matrix algebra, the dimension counts the number of components, e.g. the vector (x1, x2, x3) has dimension 3. We have found dimensional analysis extremely useful to check complicated algebraic expressions which may include vectors. In order to avoid confusion, we use the term units to describe this usage, e.g. the velocity has units [LT−1] (we appreciate that this usage is less than rig- orous – in dimensional analysis, the dimensions of velocity are [LT−1] while the units are km/s or m/s, etc. – but have been unable to find a simple alternative). We also use the term order to describe the order of a tensor, i.e. the number of indices. The velocity is a first-order tensor, of dimension 3 with units [LT−1]. The elastic parameters are a fourth-order tensor, of dimension 3 × 3 × 3 × 3 with units [ML−1T−2]. 0.1.3 Vectors and matrices Mathematicians would rightly argue that a good notation is an important part of any problem. Generality and abstractness become a virtue. We will take a some- what more pragmatic approach and would also argue that the notation should suite the intended audience. Most seismologists are happy with vector and matrix alge- bra, but become less comfortable with higher-order tensors, regarding a second- order tensor as synonymous with a matrix. The learning curve to understand fully, higher-order tensors and their notation is probably not justified by the elegance or intellectual satisfaction achieved. The ability to ‘read’ or ‘visualize’ the nota- tion of an equation outweighs the compactness and