In Chapters 17 and 18 we have considered in some detail the characteristics and elastic displacement field corresponding to point sources. A more complete representation of the seismic source must include its dimensions and fracture propagation, and consider their effects on wave radiation. First considerations of the dimensions of the seismic focus proposed models consisting of spherical cavities of finite radii with uniform distribution of stresses on their surface (Jeffreys, 1931; Nishimura, 1937; Scholte, 1962).

The first models for extended sources of shear fracture were kinematic models, consisting of a slip that propagates with a constant velocity over a surface of finite area. Ari Ben Menahem (1961, 1962) described extended sources in terms of distributions of single and double couples propagating with a certain velocity over a rectangular surface, and determined the corresponding displacements of body and surface waves. Hans Berckhemer (1962) studied the effect on the width of temporal pulses of a circular fracture of finite radius that propagates from its center. Robert Burridge and Knopoff (1964) treated shear dislocations that propagate over a certain area and showed their equivalence to propagating double couples. Haskell (1964, 1966) proposed a rectangular model of fracture, and Brian Savage (1966) proposed an elliptical fault and studied the effects on the spectra of body and surface waves. James Brune (1970) presented a model with shear stresses suddenly applied to a circular fault, and studied elastic displacements in near and far fields. More recent kinematic models include propagating shear fractures on finite faults with variable slip, rupture velocity, and rise time (Hartzell, 1989).

The physical problem of fracture is a dynamic problem in which the slip has to be considered a consequence of stress conditions and the strength of material in the focal region. Dynamic models of the seismic source take these conditions into consideration and are based on the theory of the generation and propagation of fractures in stressed media. A complete dynamic model must, then, include the whole of the fracturing process, its initiation or nucleation, propagation and arrest derived from stress conditions, and properties of the material in the focal region. Two determinant factors are tectonic stresses which are a consequence of lithospheric plate motion and mechanical properties of rocks in the fault region.

Up to this chapter, we have considered the motion produced by earthquakes from the point of view of wave propagation and free oscillations, without considering its origin. Chapters 16 to 20 are dedicated to this subject, that is, the earthquake source, its location, size, and mechanism. As we saw in Chapter 2, most earthquakes are tectonics and caused by sudden motion along faults which releases accumulated tectonic stresses and produces seismic waves. The source of an earthquake can then be considered to be the product of rupturing of a particular fault with a relative displacement of its two sides and release of the accumulated elastic strain that had been produced by tectonic processes. However, as a first approximation, we can consider the source of an earthquake as a point, called its focus, from which seismic waves are generated and propagated. The reduction of the focus to a point is called the point-source approximation, which is valid if observation points are at a sufficiently large distance compared with source dimensions, and wave lengths are also large. The simplest point source model is the isotropic point source, in which the source is considered as a point from which waves propagate with equal amplitude in all directions. In Chapter 2 we have already seen in a descriptive form the problem of the location, time, and size of earthquakes. In this chapter we will quantify this knowledge, defining the focal parameters of earthquakes and their determination, such as location and origin time, the different ways of measuring their size, defining the concepts of intensity and magnitude, and also those of seismic moment and stress drop.

Location of an earthquake focus

The location of the point focus of an earthquake in space and time is given by its geographic coordinates (x0, y0), its depth (z0), and the time of occurrence or origin time (t0). Owing to the dimensions of the actual source, the focal coordinates refer to a certain point, for example, where rupture starts, and the origin time refers to the initiation of the faulting process. The definitions of the parameters of the focal location use this point approximation.

In Chapters 8 to 10, using ray theory, we have seen how travel time curves depend on the characteristics of the media through which the seismic waves propagate and in consequence that the velocity distribution with depth can be deduced from them. In this chapter we will apply these results to observations of wave propagation in the Earth and discuss the results concerning its internal structure. Thus we will see how we obtained the knowledge about the structure of the Earth that we introduced in Section 2.6. For short distances (less than 1000 km) we can use the flat-Earth approximation and plane geometry. Seismic waves for that range of distances give information on depths of about 100 km, that is, of the crust and part of the upper mantle (lithosphere). For this range of distances we can apply the theory derived in Chapters 8 and 9. For greater distances the spherical shape of the Earth must be considered, so the results of Chapter 10 must be applied. The effects due to the deviations of the form of the Earth from a sphere, that is, mainly its flatness, can be taken into account by using corrections to the spherical model. In seismology these effects are not very important.

Observations and methods

The first seismic waves used for the study of the Earth's structure were those produced by earthquakes. Even today this is the main source of information, especially for the deep interior. Among the first tables and curves of travel times of seismic waves were those of Oldham, who in 1906 deduced the existence of the Earth's core. These tables were completed by Zöppritz and Herbert H. Turner and later, in 1914, by Gutenberg. In 1940, Jeffreys and Bullen published their tables of travel times which are very widely used, even today (Jeffreys and Bullen, 1940). In 1968, Eugene Herrin published tables of travel times for P waves only (Herrin, 1968). On a recommendation from the IASPEI, travel time tables have been derived from a spherically symmetric velocity model, known as iasp91, based on modern data (the ISC catalog 1964–88) (Kennett and Engdahl, 1991).

Ground motions produced by earthquakes and recorded by seismographs, as explained in Chapter 3, constitute the basic observations for the quantitative study of earthquakes. In order to formalize and quantify the information about the motion produced by earthquakes we use the principles and equations of the mechanics of continuous media. For this purpose, we approximate the Earth by a continuous deformable medium to which the equations of mechanics can be applied.We start by recalling some of the basic ideas on the mechanical behavior of continuous media. These ideas constitute the fundamentals necessary to understand the motion produced by earthquakes. It is, then, very important to understand these in order to quantify and formalize our knowledge. First we explain the basic concepts of displacement, strain, stress, and elasticity coefficients in order to proceed to presentation of the fundamental equations of motion, with consideration of energy and heat release.

Displacement, strain, and stress

A continuous medium is an idealization of a material in which the distance between two contiguous points can be made infinitesimally small. In this idealization, the granular structure of the materials of the Earth and their molecular and atomic nature are not considered. In a continuous medium the term particle is used to mean a geometric point without dimensions. Density and mechanical properties of continuous media are considered to be continuous functions of the spatial coordinates and time. Motion of points in a continuous medium can be described by what are known as Lagrangian and Eulerian descriptions (after Leonhard Euler and Joseph Louis Lagrange). In seismology we use the linearized theory of elasticity adopting the Lagrangian viewpoint, which considers points in the medium as individual moving particles (Dahlen and Tromp, 1998, 26–36).

The study of the mechanical behavior of continuous media can be traced back to the discovery in 1660 by Robert Hooke of a linear law that relates applied forces and deformations in an elastic body and its applications formulated by Edmé Mariotte at around 1680. The first studies on the behavior of elastic materials were those of Daniel Bernoulli, Euler, Lagrange, and Charles Coulomb. In 1827, Claude-Louis Navier established the general equations for equilibrium and vibration in elastic solids. This work was continued by Augustine Louis Cauchy, Siméon-Denis Poisson, Gabriel Lamé, Gustav Robert Kirchhoff, William Thomson (Lord Kelvin), John William Strutt (Lord Rayleigh), Augustus E. H. Love, and Horace Lamb, among others.

A second edition of this textbook, first published 17 years ago, is a wonderful opportunity to review its contents and improve its pedagogical orientation, in view of themany comments and interactions received, teaching experience, and experimental progress in seismology over the intervening years. Seismology is the science of earthquakes, which are both natural disasters profoundly affecting human lives, and a subject of study through application of the principles of the physical sciences. These two aspects are linked, since an important aim of the study of seismology is to mitigate the terrible effects of earthquakes through a more complete knowledge of their nature. Seismology provides us also with a powerful instrument to study the constitution and dynamics of the Earth. To emphasize these different aspects of seismology, Chapter 2 has been added to present, as an introduction, the complex phenomenon of earthquakes from a narrative point of view. As a physical science, the fundamentals of seismology are based on analysis of the seismic waves produced by earthquakes and registered by seismographs. The importance of this aspect is shown by presenting the analysis of seismographic digital data in Chapter 3, so that it can be used in subsequent chapters. This is a unique feature not present in other texts on seismology. Thus, this approach has been used in the new edition. The text is at an introductory level for students in the last years of the European Licentiate and the first year of Masters programs or American upper-division undergraduate courses and first graduate courses, and at similar levels of study in other countries. As a first book, no previous knowledge of seismology, as such, is assumed of the student. The book's emphasis, as indicated by its title, is on the fundamental physical principles which constitute the basis of the analysis of seismic waves and their basic development. In consequence, a number of topics have been selected. It has been noticed that sometimes even graduate students lack a true grasp of the fundamental physical principles underlying some aspects of seismology. In this book, the most fundamental concepts are, therefore, developed in detail, with their mathematical developments fullyworked out. Simple cases, such as one-dimensional problems and those in liquid media, are used as introductory topics. In some instances, more difficult subjects are introduced, although not fully developed.