Parameters and observations

In general the determination of the source mechanisms of earthquakes consists in using seismic wave observations to determine by inversion the parameters of the model assumed to represent the source. The number of parameters depends on the complexity of the model that is used. The location of the source in space and time, usually given by the hypocenter and the origin time, is assumed to be previously known. The hypocenter parameters are the geographical latitude ϕ0 and longitude λ0 of its projection onto the surface of the Earth at the epicenter, the depth h of the focus and the origin time t0. These parameters are determined from the first arrivals of seismic waves and they correspond to the place and time of nucleation of the faulting process. For point models the source is usually assumed to be located at the hypocenter, except for the models that make use of the source centroid (section 5.6). In certain methods some location parameters, such as the depth, or, in the case of the centroid methods, all the location parameters are obtained simultaneously with the source mechanism. The size of the earthquake, given by its surface wave magnitude Ms (see (1.26)) and its body wave magnitude mb (see (1.25)), is independently determined. However, the scalar seismic moment M0 can be either independently determined or evaluated at the same time as the mechanism. From the seismic moment one can obtain the moment magnitude MW (1.30).

Kinematic and dynamic models

Kinematic models for the source of earthquakes, the type of model that we have considered up to this point, describe the time-dependent distribution of the slip Δui(t) on a fault and are a simplified representation of the real fracture process. In kinematic models the rupture propagates at a constant or variable speed and is made to stop at the fault limits. A number of arbitrary factors are introduced into these models, some of which lead to physical inconsistencies at, for example, the borders of the fault. In spite of these limitations, kinematic models provide essential information about the seismic source, such as the fault orientation, source dimensions and slip distribution on the fault plane. However, the physical occurrence of an earthquake provides a dynamic problem: the slip on the fault has to be considered as a consequence of the stress conditions and the strength of the material in the focal region. Dynamic models of the generation of earthquakes take into consideration these conditions and are based on the theory of the generation and propagation of fractures or cracks in stressed media. From this point of view, the mechanism of tectonic earthquakes may be represented by a shear fracture produced when the stress acting overcomes the strength of the material and/or the friction between the two sides of a pre-existing fault. A fracture initiates at a point of the fault where the stress acting on the fault plane exceeds a critical value, then it propagates with a certain rupture velocity and finally stops when the mechanical conditions impede its further propagation. A complete dynamic model must, then, include the whole fracture process, that is, its initiation or nucleation, propagation and arrest, and must be derived from the stress conditions and the properties of the material in the focal region. The two determining factors are the stresses acting on the focal region due to tectonic processes derived from the motion of lithospheric plates and the mechanical properties of the rocks in the region. Since seismic fractures generally take place on pre-existing faults, important factors are the conditions between the two sides of the fault, mainly friction.

Preface

A key problem in seismology is the study of the processes that give rise to earthquakes. It is well established now that earthquakes, with rare exceptions, are caused by shear fracture on pre-existing faults in the Earth. Thus, the modern study of earthquakes and their source mechanisms should be based on the application of dislocation theory and fracture mechanics; this is the point of view taken in this book. General textbooks of seismology have dedicated chapters to this subject, for example, Aki and Richards (1980), Ben Menahem and Singh (1981), Dahlen and Tromp (1998), Gubbins (1990), Lay and Wallace (1995) and Udías (1999). A few books have been written specifically on the subject, for example, Kasahara (1981), Kostrov and Das (1988) and Scholz (1990). Each of these three books has a different approach: Kasahara presents the state of the study of earthquake mechanisms as it was 30 years ago, before modern developments, and does not give details of the mathematical developments. Scholz' approach is that of rock mechanics, with an emphasis on qualitative descriptions and applications to Earth faulting. Finally, Kostrov and Das (1988) gives an excellent presentation of earthquake dynamics, but it may be difficult for students to follow. None of these books includes a detailed presentation and discussion of practical methods for the determination of the earthquake mechanisms together with the theory on which they are based.

Griffith's fracture model

In the previous chapter we studied a simple circular earthquake model proposed by Brune (1970). In that model the amount of radiated energy was about 45% of the strain energy released by the elastic body, so where does the rest of the energy go? It goes into the creation of the fracture surface or fault. This energy is released near the edge or tip of the fault, that is, the place where the material passes from being unbroken to being broken (Kostrov and Das, 1988, pp. 53–62). In the case of a homogeneous, continuous, perfectly elastic material, the material ahead of the fracture front is continuous and elastic and behind the front it is discontinuous (broken). In a tensional fracture (Mode I, see section 9.3), when the two sides of the fracture become separated by a crack-opening displacement Δu, the stress behind the front is zero and the stress drop is total. This situation is usually the one considered in engineering applications. Alan Griffith in 1921 introduced the first elements of brittle fracture in his work on the fracture of metals. At that time it was generally accepted that the strength of a material was about one tenth of the value of Young's modulus. However, in practice, from laboratory experiments it was known that the critical strength at which material fracture occurred was about a thousand times lower. Griffith made the hypothesis that the presence of small cracks lowered the strength of the material, causing it to fracture long before it reached its theoretical breaking strength. He assumed that the surface energy dissipated by the forming of a new crack surface is equal to the resistance to crack growth. To create a new unit area of fracture a certain amount of energy per unit surface, γ, was necessary, which he called the effective specific surface energy. He assumed that γ was a material constant. According to Griffith, for the crack to grow by a new surface element δ S, it was necessary to provide it with an amount of energy equal to 2γδS (the factor 2 is for the two sides of the fracture).

Introduction

The use of geophysical methods in an exploration programme or during mining is a multi-stage and iterative process (Fig. 2.1). The main stages in their order of application are: definition of the survey objectives, data acquisition, data processing, data display and then interpretation of different forms of the data. The geologist should help to define the objectives of the survey and should have a significant contribution during interpretation of the survey data, but to ensure an optimum outcome, an understanding of all the other stages highlighted by Fig. 2.1 is required. Survey objectives dictate the geophysical method(s) to be used and the types of surveys that are appropriate, e.g. ground, airborne etc. Data acquisition involves the two distinct tasks of designing the survey and making the required measurements in the field. Data processing involves reduction (i.e. correcting the survey data for a variety of distorting effects), enhancement and display of the data, all designed to highlight what is perceived to be the most geologically relevant information in the data. The processed data can be displayed in a variety of ways to suit the nature of the dataset and the interpreter’s requirements in using the data. Data interpretation is the analysis of the geophysical data and the creation of a plausible geological model of the study area. This is an indeterminate process; an interpretation evolves through the iterative process as different geological concepts are tested with the data. It is often necessary to revise aspects of the data enhancement as different characteristics of the data assume greater significance, or as increased geological understanding allows more accurate reduction.

The interpreter needs to have a good understanding of the exploration strategy which was the basis for defining the survey objectives. Ideally the interpreter should also have a working knowledge of the geophysical acquisition–processing sequence since this impinges on the evolving interpretation of the data. The type of survey and the nature of the data acquisition affect the type and resolution of the geological information obtainable, whilst the interpretation of geophysical data is dependent on the numerical methods applied to enhance and display the data. Analysis of the data involves their processing and interpretation. We emphasise that interpretation is not a task to be undertaken in isolation; it is an inextricable part of the iterative and multi-stage analysis shown in Fig. 2.1.