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Seismic ray theory is analogous to optical ray theory and has been applied for over 100 years to interpret seismic data. It continues to be used extensively today, owing to its simplicity and applicability to a wide range of problems. These applications include most earthquake location algorithms, body-wave focal mechanism determinations, and inversions for velocity structure in the crust and mantle. Ray theory is intuitively easy to understand, simple to program, and very efficient. Compared to more complete solutions, it is relatively straightforward to generalize to three-dimensional velocity models. However, ray theory also has several important limitations. It is a high-frequency approximation, which may fail at long periods or within steep velocity gradients, and it does not easily predict any “non-geometrical” effects, such as head waves or diffracted waves. The ray geometries must be completely specified, making it difficult to study the effects of reverberation and resonance due to multiple reflections within a layer.
In this chapter, we will be concerned only with the timing of seismic arrivals, deferring the consideration of amplitudes and other details to later. This narrow focus is nonetheless very useful for many problems; a significant fraction of current research in seismology uses only travel time information. The theoretical basis for much of ray theory is derived from the eikonal equation (see Appendix C); however, because these results are not required for most applications we do not describe them here.
In the preceding chapter we examined the problem of tracing rays and calculating travel time curves from a known velocity structure. We derived expressions for ray tracing in a one-dimensional (1-D) velocity model in which velocity varies only with depth; ray tracing in general three-dimensional (3-D) structures is more complex but follows similar principles. We now examine the case where we are given travel times obtained from observations and wish to invert for a velocity structure that can explain the data. As one might imagine, the inversion is much more complicated than the forward problem. The main strategy used by seismologists, both in global and crustal studies, has generally been to divide the problem into two parts:
A 1-D “average” velocity model is determined from all the available data. This is generally a non-linear problem but is tractable since we are seeking a single function of depth. Analysis often does not proceed beyond this point.
If sufficient 3-D ray coverage is present, the 1-D model is used as a reference model and a travel time residual is computed for each datum by subtracting the predicted time from the observed time. A 3-D model is obtained by inverting the travel time residuals for velocity perturbations relative to the reference model. If the velocity perturbations are fairly small, this problem can be linearized and is computationally feasible even for large data sets. This is the basis for tomographic inversion techniques.
For many years the most widely used 1-D model of Earth's seismic velocities has been the Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981). This model was designed to fit a variety of different data sets, including free oscillation center frequency measurements, surface wave dispersion observations, travel time data for a number of body-wave phases, and basic astronomical data (Earth's radius, mass, and moment of inertia). In addition to profiling the P and S velocities, PREM specifies density and attenuation as functions of depth. Although these parameters are known less precisely than the seismic velocities, including them is important because it makes the model complete and suitable for use as a reference to compute synthetic seismograms without requiring additional assumptions. In order to simultaneously fit Love and Rayleigh wave observations, PREM is transversely isotropic between 80 and 220 km depth in the upper mantle. This is a spherically symmetric form of anisotropy in which SH and SV waves travel at different speeds. For simplicity, the table here lists only values from an isotropic version of PREM. The true PREM model is also specified in terms of polynomials between node points; linear interpolation between the 100 km spacing of values in this table will produce only approximate results. All current Earth models have values that are reasonably close to PREM; the largest differences are in the upper mantle, where, for example, a discontinuity at 220 km is not found in most models.
This book provides an approachable and concise introduction to seismic theory, designed as a first course for undergraduate students. It clearly explains the fundamental concepts, emphasizing intuitive understanding over lengthy derivations. Incorporating over 30% new material, this second edition includes all the topics needed for a one-semester course in seismology. Additional material has been added throughout including numerical methods, 3-D ray tracing, earthquake location, attenuation, normal modes, and receiver functions. The chapter on earthquakes and source theory has been extensively revised and enlarged, and now includes details on non-double-couple sources, earthquake scaling, radiated energy, and finite slip inversions. Each chapter includes worked problems and detailed exercises that give students the opportunity to apply the techniques they have learned to compute results of interest and to illustrate the Earth's seismic properties. Computer subroutines and datasets for use in the exercises are available at www.cambridge.org/shearer.
Despite their usefulness as a research tool for illuminating Earth structure, earthquakes are generally considered harmful because of their potential for causing death and destruction. It is therefore unfortunate that the most useful thing that seismologists could do – predict earthquakes – is what they are least able to do. Although many ideas for earthquake prediction have been explored, the sad truth is that reliable prediction of damaging earthquakes is not currently possible on any time scale. In this section we will present some of the terminology and concepts in earthquake occurrence modeling and discuss possible reasons why major earthquakes are so difficult to predict.
The earthquake cycle
The idea that earthquakes represent a sudden release of accumulated stress in the crust was first documented by H. F. Reid, who examined survey lines taken before and after the 1906 earthquake in San Francisco. His results led to the elastic rebound theory of earthquake occurrence, in which stress and strain increase gradually and are then released during an earthquake by sudden movement along a fault (Fig. 10.1). This mechanism is now recognized to be the primary cause of earthquakes in the crust. Earthquakes occur mostly along the boundaries between Earth's surface plates (see Fig. 1.2), releasing the stress that results from the relative tectonic motion between the different plates. Observations of surface deformations, using ground- and satellite-based surveying techniques, can be used to monitor the slow strain changes that are seen in seismically active regions (often termed the secular strain rate) and the sudden change that occurs in the deformation field during earthquakes (these are termed co-seismic changes).
Every day there are about fifty earthquakes worldwide that are strong enough to be felt locally, and every few days an earthquake occurs that is capable of damaging structures. Each event radiates seismic waves that travel throughout Earth, and several earthquakes per day produce distant ground motions that, although too weak to be felt, are readily detected with modern instruments anywhere on the globe. Seismology is the science that studies these waves and what they tell us about the structure of Earth and the physics of earthquakes. It is the primary means by which scientists learn about Earth's deep interior, where direct observations are impossible, and has provided many of the most important discoveries regarding the nature of our planet. It is also directly concerned with understanding the physical processes that cause earthquakes and seeking ways to reduce their destructive impacts on humanity.
Seismology occupies an interesting position within the more general fields of geophysics and Earth sciences. It presents fascinating theoretical problems involving analysis of elastic wave propagation in complex media, but it can also be applied simply as a tool to examine different areas of interest. Applications range from studies of Earth's core, thousands of kilometers below the surface, to detailed mapping of shallow crustal structure to help locate petroleum deposits. Much of the underlying physics is no more advanced than Newton's second law (F = ma), but the complications introduced by realistic sources and structures have motivated sophisticated mathematical treatments and extensive use of powerful computers.
Why another book on seismology? Several excellent texts already exist that cover most parts of the field. None, however, is ideal for the purposes of an introductory class. Most simply present far more material than can be adequately covered in a single quarter or semester. My goal for this book is to produce a readable, concise introduction to the quantitative aspects of seismology that is designed specifically for classroom instruction. The result is not as rigorous or comprehensive as Aki and Richards (1980) or Lay and Wallace (1995), but I hope that it is more suited for teaching an overview of seismology within a limited time period.
To quicken the pace, many results are described without detailed proofs or derivations of equations. In these cases, the reader is usually referred to Aki and Richards or other sources for more complete explanations. Generally I have attempted to provide practical descriptions of the main concepts and how they are used to study Earth structure. Some knowledge of physics and vector calculus is assumed, but in an effort to make the book self-contained most of the key concepts are reviewed in the Appendices.
Any book to some extent reflects the prejudices of its author. In this regard, I have perhaps included more material on ray theory and body wave travel times, and less on surface waves and normal modes, than a truly balanced book would require.
Presented here are some important topics that do not easily fit into the structure of the first ten chapters.
Throughout this book, we have often discussed Earth motion in terms of the displacement field, u(x, t), but have not mentioned how these movements are actually measured. A device that detects seismic wave motion is termed a seismometer; the entire instrument package, including the recording apparatus, is called a seismograph. The most common type of seismometer is based on the inertia of a suspended mass, which will tend to remain stationary in response to external vibrations.
As an example, Figure 11.1 shows a simple seismometer design that will detect vertical ground motion. A mass is suspended from a spring and connected to a lever such that it can move only in the vertical direction. Motions of the lever are damped using a “dashpot” to prevent excessive oscillations near the resonant frequency of the system. The differential motion between the mass and the seismometer case (which is rigidly connected to Earth) is measured using the voltage induced in a coil by the motion of a magnet. The induced voltage is proportional to the velocity of the mass for the instrument shown in Figure 11.1. In alternative seismometer designs, the displacement or acceleration of the mass may be recorded. As we will see later, the frequency response of the seismometer is a strong function of whether the displacement, velocity, or acceleration of the mass is measured.
During the last ten years, I have continued teaching the beginning seismology class at University of California, San Diego, and have received feedback from my students, as well as other instructors who have been using the book. The second edition is my attempt to expand on some subjects, clarify parts of the book that have proven confusing, and update the discussion of current research results. The biggest changes are to the Source Theory chapter, which now provides a more complete discussion of non-double-couple sources, stress drop, earthquake scaling, radiated energy, energy partitioning, and magnitude saturation. However, I have also tried to remain concise enough that the book can still be used for a one-quarter or one-semester class, although depending upon the pace of the class it may be necessary to skip some of the material. Sections flagged with a † are suggestions for possible areas to skip without much compromise in understanding of the remaining subjects.
The computer subroutines and data for some of the exercises can now be obtained from www.cambridge.org/shearer, which also contains links to any errors found in the text and other supplemental information that I plan to add in the future.
One of the most important applications of seismology involves the probing of Earth's internal structure by examining energy reflected at steep incidence angles from subsurface layers. This technique may loosely be termed reflection seismology and has been used extensively by the mining and petroleum industries to study the shallow crust, generally using portable instruments and artificial sources. However, similar methods can be applied to the deeper Earth using recordings of earthquakes or large explosions. Because reflected seismic waves are sensitive to sharp changes in velocity or density, reflection seismology can often provide much greater lateral and vertical resolution than can be obtained from study of direct seismic phases such as P and S (analyses of these arrivals may be termed refraction seismology). However, mapping of reflected phases into reflector depths requires knowledge of the average background seismic velocity structure, to which typical reflection seismic data are only weakly sensitive. Thus refraction experiments are a useful complement to reflection experiments when independent constraints on the velocity structure (e.g., from borehole logs) are unavailable.
Reflection seismic experiments are typically characterized by large numbers of sources and receivers at closely spaced and regular intervals. Because the data volume generally makes formal inversions too costly for routine processing, more practical approximate methods have been widely developed to analyze the results. Simple time versus distance plots of the data can produce crude images of the subsurface reflectors; these images become increasingly accurate as additional processing steps are applied to the data.