Introduction

Most earthquakes can be represented by slip on a fault, which for simplicity will be modeled as a planar feature. When an earthquake occurs, the two sides of the fault suffer a sudden relative displacement with respect to each other, which in turn is the source of seismic waves. Assuming that the Earth is initially at rest and that there are no external forces, the occurrence of an earthquake is the result of a localized and temporary breakdown of the stress–strain relationship, which is the only basic equation of elasticity that is not a fundamental law of physics (Backus and Mulcahy, 1976). Let us compare this situation with that encountered in Chapter 9. There we used (9.4.2) to study the displacement field caused by a given body force, while here the displacement at points away from the fault is caused by a discontinuity in displacement across the fault, and because of the breakdown of Hooke's law, we cannot use (9.4.2) directly. Therefore, we are faced with the following problem: what is the equivalent body force that in the absence of the fault will cause exactly the same displacement field as slip on the fault. Getting an answer to this question took a considerable amount of time and effort (for a review, see Stauder, 1962). As a result of work done during the 1920–50s, two possible models were introduced, based on single and double couples similar to those discussed in Chapter 9.

Introduction

The development of the theory of elasticity took about two centuries, beginning with Galileo in the 1600s (e.g., Love, 1927; Timoshenko, 1953). The most difficult problem was to gain an understanding of the forces involved in an elastic body. This problem was addressed by assuming the existence of attractive and repulsive forces between the molecules of a body. The most successful of the theories based on this assumption was that of Navier, who in 1821 presented the equations of motion for an elastic isotropic solid (Hudson, 1980; Timoshenko, 1953). Navier's results were essentially correct, but because of the molecular assumptions made, only one elastic constant was required, as opposed to the two that characterize an isotropic solid (see §4.6). Interestingly, the results based on the simple molecular theory used by the earlier researchers can be obtained by setting the ratio of P-to S-wave velocities equal to √3 in the more general results derived later. Navier's work attracted the attention of the famous mathematician Cauchy, who in 1822 introduced the concept of stress as we know it today. Instead of considering intermolecular forces, Cauchy introduced the idea of pressure on surfaces internal to the body, with the pressure not perpendicular to the surface, as it would be in the case of hydrostatic pressure. This led to the concept of stress, which is much more complicated than that of strain, and which requires additional continuum mechanics concepts for a full study.

Introduction

After the homogeneous infinite space, the next two simplest configurations are a homogeneous half-space with a free surface and two homogeneous half-spaces (or media, for short) with different elastic properties. The first case can be considered as a special case of the second one with one of the media a vacuum. In either case the boundary between the two media constitutes a surface of discontinuity in elastic properties that has a critical effect on wave propagation. To simplify the problem we will assume plane boundaries and wave fronts. Although in the Earth neither the wave fronts nor the boundaries satisfy these assumptions, they are acceptable approximations as long as the seismic source is sufficiently far from the receiver and/or the wavelength is much shorter than the curvature of the boundary. In addition, the case of spherical wave fronts can be solved in terms of plane wave results (e.g., Aki and Richards, 1980). Therefore, the theory and results described here have a much wider application than could be expected by considering the simplifying assumptions. For example, they are used in teleseismic studies, in the generation of synthetic seismograms using ray theory, and in exploration seismology, particularly in amplitude-versus-offset (AVO) studies.

The interaction of elastic waves with a boundary has a number of similarities with the interaction of acoustic and electromagnetic waves, so that it can be expected that a wave incident on a boundary will generate reflected and transmitted waves (the latter only if the other medium is not a vacuum).

Introduction

In the previous chapters we studied the propagation of plane waves without consideration of the source of the waves. Although this approach is very fruitful, it does not allow investigation of the waves generated by seismic sources, either natural or artificial. Earthquakes are the most important natural sources, and the study of the waves they generate has played a major role in our understanding of the inner structure of the Earth and the nature of the earthquake source, which will be the subject of the next chapter. However, before reaching the point where it can be analyzed it is necessary to start with simpler problems, which will be done in this chapter.

The simplest problem corresponds to a spatially concentrated force (or point source) directed along one of the coordinate axes. Even in this case, however, solving the elastic wave equation is a rather complicated task that requires considerable mathematical background, which is provided below. The starting point is the scalar wave equation with a source term, which is first solved for an impulsive source, in which case the solution is known as Green's function for the problem. Then the Helmholtz decomposition theorem, which we have already encountered in §5.6, is used to reduce the solution of the elastic wave equation to the solution of two simpler ones. After this series of steps, and considerable additional work, the problem of the concentrated force can be solved.

Introduction

Tensors play a fundamental role in theoretical physics. The reason for this is that physical laws written in tensor form are independent of the coordinate system used (Morse and Feshbach, 1953). Before elaborating on this point, consider a simple example, based on Segel (1977). Newton's second law is f = ma, where f and a are vectors representing the force and acceleration of an object of mass m. This basic law does not have a coordinate system attached to it. To apply the law in a particular situation it will be convenient to select a coordinate system that simplifies the mathematics, but there is no question that any other system will be equally acceptable. Now consider an example from elasticity, discussed in Chapter 3. The stress vector T (force/area) across a surface element in an elastic solid is related to the vector n normal to the same surface via the stress tensor. The derivation of this relation is carried out using a tetrahedron with faces along the three coordinate planes in a Cartesian coordinate system. Therefore, it is reasonable to ask whether the same result would have been obtained if a different Cartesian coordinate system had been used, or if a spherical, or cylindrical, or any other curvilinear system, had been used. Take another example. The elastic wave equation will be derived in a Cartesian coordinate system.

Introduction

Elasticity theory, which lies at the core of seismology, is most generally studied as a branch of continuum mechanics. Although this general approach is not required in introductory courses, it is adopted here for two reasons. First, it affords a number of insights either not provided, or not as easily derivable, when a more restricted approach is used. Secondly, it is essential to solve advanced seismology problems, as can be seen from the discussion in Box 8.5 of Aki and Richards (1980), and references therein, or from the book by Dahlen and Tromp (1998).

Continuum mechanics studies the deformation and motion of bodies ignoring the discrete nature of matter (molecules, atoms, subatomic particles), and “confines itself to relations among gross phenomena, neglecting the structure of the material on a smaller scale” (Truesdell and Noll, 1965, p. 5). In this phenomenological approach a number of basic concepts (among them, stress, strain, and motion) and principles (see Chapter 3) are used to derive general relations applicable, in principle, to all types of media (e.g., different types of solids and fluids). These relations, however, are not enough to solve specific problems. To do that, additional relations, known as constitutive equations, are needed. These equations, in turn, are used to define ideal materials, such as perfect fluids, viscous fluids, and perfect elastic bodies (Truesdell and Toupin, 1960).

Introduction

The elastic wave equation admits exact solutions in relatively few cases involving simple variations in elastic properties. When these properties vary in a two- or three-dimensional way, the equation cannot be solved exactly and either numerical (e.g., finite-difference, finite-element) or approximate solutions must be sought. Ray theory is one of the possible approaches introduced to solve the equation approximately. This theory is traditionally associated with optics, where it originated (for reviews see, e.g., Kline and Kay, 1965; Cornbleet, 1983; Stavroudis, 1972). The extension of the theory to the propagation of electromagnetic waves is due to Luneburg (1964) (work done in the 1940s) while the application to elastic waves is due to Karal and Keller (1959), although simpler ray-theoretic concepts had been used earlier (Cerveny et al., 1977). Russian authors also contributed to the solution of the elastic problem (see, e.g., Cerveny and Ravindra, 1971).

Over the last three decades elastic wave ray theory has grown enormously in scope and complexity and for this reason in this chapter only the most fundamental aspects will be discussed. Two important topics not addressed here are numerical solutions to the ray equations and the considerably more difficult problem of computing amplitudes. The former is well treated by Lee and Stewart (1981), and the latter by Cerveny (2001), who present a thorough discussion of ray theory and includes an extensive reference list.

Introduction

Surface waves are waves that propagate along a boundary and whose amplitudes go to zero as the distance from the boundary goes to infinity. There are two basic types of surface waves, Love and Rayleigh waves, named after the scientists who studied them first. Love's work was directed to the explanation of waves observed in horizontal seismographs, while Rayleigh predicted the existence of the waves with his name. The main difference between the two types of waves is that the motion is of SH type for Love waves, and of P–SV types for Rayleigh waves. A related type of wave, known as Stoneley waves, consists of P–SV inhomogeneous waves that propagate along the boundary between two half-spaces. In this chapter we will consider these three types of waves. As we shall see, the presence of a layer introduces the phenomenon of dispersion, which is characterized by the existence of two velocities, known as the phase and the group velocity, with the property that they are functions of frequency. In §7.6 a detailed analysis of dispersion is presented. The problem of multilayered media will not be considered here, but the groundwork for its analysis has been introduced in §6.9.2.

To solve problems involving surface waves it is necessary to go through the steps described in §6.1, namely, write the equation for the displacement at any point in the medium and then apply appropriate boundary conditions.