Only the simplest seismic wave propagation problems are amenable to a direct analytic solution. As we have seen in Part II, for stratified media the methods of attack have been based on a semi-analytic approach in which transform methods are used to simplify the equations so that attention can be concentrated on behaviour in the frequency-slowness domain; the resulting integrals need to be evaluated numerically. Once we face a three-dimensionally varying medium the simplicity of such transform methods are lost, and coupling between different slowness components has to be taken into account to describe the passage of waves through the 3-D structure (see, e.g., Haines, 1988). This approach can be quite successful for some classes of simple problems where the the medium remains quasi-stratified but the shape of interfaces are distorted (Koketsu et al., 1991). Interaction of the seismic wavefield with isolated heterogeneities can also be tackled by using specific developments based on a multipole representation of the resulting scattered wavefield as in the T-matrix methods of Boström & Karlsson (1984) and Bostock & Kennett (1992). However, multiple interactions between ‘scatterers’ or between the scattered field and interfaces rapidly leads to challenging computational issues.

For fully three-dimensional problems a more direct attack is needed. At high frequencies the main tool is asymptotic ray theory, as introduced in Chapter I:9. These techniques have been progressively developed to allow for full 3-dimensional variations including the influence of anisotropic structures. Červený (2001) provides a comprehensive development of the current state of the theory for local and regional scale problems. The results for global models for both body waves and surface waves are presented by Dahlen & Tromp (1998, Chapters 15, 16). The details of the implementation of ray methods are quite dependent on the way in which the three-dimensional structure is described, and in particular on the specification of interfaces and their interaction.

Ray methods provide travel times and, with extra effort, amplitude information in a high frequency approximation. They are therefore a fundamental tool in understanding the nature of the main propagation processes, in the same way as we have used them for stratified structures.

There is considerable variety in the ways in which events at regional distances are treated in different parts of the world. Most seismically active regions have a well distributed set of stations across the zone where earthquakes are most common, and the tendency in recent years has been for higher concentrations of instrumentation as, e.g., in Southern California and Japan. Where there are a significant number of seismic stations an indication of the likely position of an event is provided by the station recording the first indications of ground motion. This helps to start the process of location which is usually accomplished by a linearised inversion scheme or using some grid search algorithm (see Section 29.1). With relatively close stations it is usually possible to get quite good constraints of the depth of the event by exploiting both P and S information, using a velocity model designed to fit local conditions. The major difficulties in tectonically active areas come with the presence of substantial 3-D structure which introduces systematic errors into simple location procedures. It is frequently difficult to be confident of the association of smaller events with mapped faults.

For the San Andreas fault in central California, the marked contrast in seismic properties between the two sides of the transcurrent fault makes precise location difficult. There appears to be an offset between events at depth and the surface trace of the fault (Thurber et al., 2000) and this has been one facet of efforts to set up an observatory system with drilling through the fault zone at depth (the SAFOD project). Seismometers down the drill hole are planned to allow close observations of small magnitude events and so pin down the region where slip is occurring on this major fault.

In less seismically active areas the networks of stations tend to be rather sparse, so that there is both poorer coverage of events and poorer location capability. Also there are frequently more man-made events, e.g., from quarrying or mining which can be picked up using the networks. Even in such areas the majority of analysis is based on using short-period records for single stations.

ground is large and there is a strong interaction with man-made structures. The first instrument specifically designed to record strong ground motion close to an earthquake was installed in Tokyo at the end of the nineteenth century (Sekiya & Omori, 1894). This instrument was built to withstand severe shaking and employed mechanical recording with no amplification. The record displayed by Sekiya & Omori, for an event that caused some damage in Tokyo, was recorded with 3 separate orthogonal components on a circular plate. The largest amplitude of motion is about 45 mm on the NE/SW component with a period close to 2 s for the main pulse but also a long duration of coda. The large amplitudes and extended shaking reflect the influence of the Kanto sedimentary basin on which Tokyo is built.

Strong ground motion recording has been developed in a variety of ways since these pioneering studies, and now many classes of structures have their own instrumentation, notably on dams. In urban areas, recordings are commonly made with accelerometers which are triggered by movement of their surroundings (e.g. in buildings); in the past, this has often meant that only the S waves have been well recorded. The advent of digital recordings means that the new generation of strong ground motion recorders have sufficient memory that the P wave motion can be preserved for stations out to 100–200 km from the event. The emplacement of many such recorders in buildings, which is important for engineering seismology, means that the ground motion is modulated by the interaction with the man-made structure.

Nature of the local wavefield

For larger events where the area of faulting is comparable to the distance of stations fromthe fault, the details of the source process have a strong influence on the nature of the ground motion. The rupture of a major fault occurs progressively and is modified by the presence of stronger portions of the fault. Such ‘asperities’ often serve as nuclei for the distribution of aftershocks.

The nature of the local conditions can also have a profound influence on the recordings of ground motion, both through amplification and reverberation in soft surface sediment but also through interaction with 3-D structure in the vicinity of the fault.

Studies of the uppermost part of the mantle have largely been undertaken with man-made sources. Long-range profiles using conventional explosive sources rarely achieve data coverage beyond 1000 km. In consequence, much of our knowledge of the structures below 100 km depth in the deeper mantle lithosphere and mantle transition zone come from studies using earthquake sources, with composite results from many events. Nuclear explosions can provide sufficient seismic energy to yield records spanning the complex returns from the upper mantle discontinuities. Nuclear tests in the United States of America formed the basis of the work by Helmberger & Wiggins (1971), Wiggins & Helmberger (1973) which used synthetic seismograms to control the nature of the wavespeed profile in the upper mantle. In the former Soviet Union an extensive campaign of ultra-long-range profiles was undertaken using peaceful nuclear explosions, often specifically detonated for the experiments. Seismometers were deployed out to 3000 km from the source with multiple coverage along a number of profiles (see Section 22.2.1).

The uppermost mantle

The times of arrivals for stations from 200 to 1000 km away from the source show little variation in apparent slowness for P waves. A typical phase slowness for Pn is around 0.124 s/km, corresponding to a phase velocity of 8.15 km/s. However, once the opportunity arose to examine the nature of the seismograms in detail, it became apparent that this relatively uniform slowness for the first arrivals did not correspond to a single phase branch but rather to a sequence of en echelon P contributions. Such behaviour has emerged in all cases where there is sufficient data density to follow the details of the arrivals returned from the uppermost mantle.

For example, in the 1971 long-range profile across France (Hirn et al., 1973), a Pd arrival was correlated from 150 to 250 km and is then superseded by a set of further phases PI from 300 to 500 km and PII from 450 to 600 km; a further PIII phase was correlated by Kind (1974). On the basis of the amplitude behaviour (Hirn et al., 1973; Kind, 1974) and systematic travel time inversions (Kennett, 1976), the retrograde

In the previous chapter we have seen how the global seismic wavefield evolves with time after source initiation, and the way in which the complex pattern of internal wavefronts is reflected in the nature of seismograms at surface receivers. The increase in seismic wavespeed with depth through the mantle plays an important role in determining the character of seismograms. The refraction of body waves back to the surface yields the distinctive features of teleseisms through the P and S arrivals. Multiple surface reflections of the body wave arrivals carry phases such as PP, PPP to great distances. The high wavespeed gradients in the upper part of the mantle create a waveguide that traps shear wave energy between the surface and reflection from the increase in wavespeeds leading to complex sets of multiple S reflections grading into the surface wave trains. Surface reflections with conversion, such as PS, have rather asymmetric propagation paths, but can play an important role at greater distances.

For SH waves, the whole mantle acts as a waveguide because these waves cannot penetrate into the core. The boundary conditions on the SH wavefield is thus a requirement of vanishing traction at both the Earth's surface and the core mantle boundary, with complete reflection of SH waves at each surface. The core reflection ScS is therefore both more prominent for SH waves than SV waves and its multiple reflections (ScSH)n continue for a long time. In Section 25.3 we will show how the long term reverberations can be used to investigate the presence of discontinuities in mantle structure.

Mantle phases

The P and S body waves become simpler in character beyond 30°, once their turning points lie in the lower mantle and so no longer feel the influence of the upper mantle discontinuities that we have discussed in Chapter 22. The surface multiples PP, SS etc. turn at shallower depths and so extend the influence of the transition zone to 60° and beyond for higher order multiples (see Section 21.3). The core reflections cut across other phases and so there are limited intervals in which they appear as distinct arrivals.

When we come to study a specific region of the Earth we can exploit both seismic events within and surrounding the region, and those at teleseismic distances, by looking at different aspects of the wavefield.

Where waveform inversion is used for fundamental and higher mode surface waves, the results of global studies of the mantle can provide a useful starting point for the characteristics of large-scale heterogeneity. Particularly where temporary stations have been deployed to enhance coverage it is possible in a regional survey to obtain much higher horizontal resolution as has been demonstrated, e.g., by the Skippy experiment in Australia (van der Hilst, Kennett & Shibutani, 1998).

For body wave studies, we can use the waves refracted back from the upper mantle within the region if we can disentangle the triplications induced by the presence of the mantle discontinuities. Such regional arrivals travel at shallow angles through the structure. We can also use the arrivals from distant events for which the ray paths travel relatively steeply through the upper part of the mantle and so provide a different class of information. Where we have local seismicity, we can also exploit the observations of these events at distant stations with a knowledge of the global 3-D structure outside the region of interest.

Regional surface-wave tomography

In order to obtain high resolution images of the shallower part of the Earth we need to use seismological probes whose sensitivity is greatest in this region. Fortunately we are able to exploit the surface wave portion of the shear wavefield which commonly represents the largest amplitude portion of broad-band seismograms.

The fundamental mode Love waves and Rayleigh waves are preceded by higher mode surface waves which represent the superposition of multiply reflected S waves interacting with the free surface (cf. § I:16.2). The fundamental mode energy is concentrated near the surface, but the penetration increases as the frequency is reduced because the S wavelength is longer. The higher modes provide both deeper penetration at comparable frequency and complementary information on shallow structure because the pattern of sampling is rather different.

The influence of Earth structure on the surface waves appears through the frequency dispersion of the different surface wave modes so that the character and frequency content changes along the wavetrain as it is recorded at an individual station.

The propagation of seismic phases at regional distances has been a topic of continuous interest since the work of Mohorovičić (1909) on the Kulpatal earthquake in Croatia, from which he proposed the presence of a discontinuity in seismic wavespeeds at a depth near 50 km. The study of crustal structure was pushed forward, mostly using information from natural sources, by many authors including Jeffreys and Conrad. But, it was only when a significant number of records were available from a single source that the full nature of the crustal wavefield could be appreciated.

The first major experiment designed to exploit the refracted waves through the continental crust and mantle was the Heligoland explosion in 1947 (Willmore, 1949) recorded out to 350 km from the source on portable stations, and beyond 700 km at permanent stations. The analysis of crustal and mantle properties using explosive sources to tackle structural problems became a major tool during the 1960s. Initial analysis was based on the interpretation of travel times of the various phases. To try to resolve ambiguities in the models, amplitude information began to be exploited and this lead to the development of both the computation of theoretical seismograms for realistic models (Fuchs & Müller, 1971) and more sophisticated ray-tracing tools (see, e.g., Červený, Pšenčí k & Molotkov, 1977).

Such studies of seismic structure are based on an analysis of the details of seismic arrivals, with emphasis on the correlations between successive seismograms in a record section display. The use of multiple seismograms, with a close spacing in distance, is very helpful in enabling specific types of arrival to be tracked across the record section, so that the influence of noise on individual seismograms can be reduced. The notation used to mark arrivals in such structural studies is designed to provide a detailed correlation between the classes of arrivals and the structural features which give rise to them.

A further impetus to understand the detailed characteristics of the regional wavefield has come from efforts to monitor underground explosions of low yield and to discriminate them from earthquakes (see, e.g., Pomeroy, Best & McEvilly, 1982). In work directed towards source characterisation, the available data commonly consists of a set of seismograms from widely separated arrays or three-component stations.

As we have seen in the previous chapter, it is nowpossible to undertake investigations of the nature of seismic wave propagation in 3-D models. We now turn to the way in which information about such 3-D structures is extracted from seismic observations that provide direct coverage of the region of interest with multiple crossing propagation paths. This process has come to be known as “seismic tomography”. The original meaning of the term tomography implies reconstruction from multiple sections through the feature of interest. Such a configuration occurs in medical applications, where equipment is also designed to provide rather uniform sampling. In studies of the Earth the sampling is much more irregular, being dictated by the location of available sources and receivers. Further, our sensors sit at, or close to, the surface of the Earth so that it is rare to achieve all-round coverage of individual features. As a result we face a situation with somewhat limited sampling and hence significant differences in the potential resolution of different parts of the 3-D structure. The majority of studies have employed fixed grids or basis functions, but in the future we are likely to see increased emphasis on adaptive inversion schemes designed to exploit the characteristics of individual data sets.

Elements of seismic tomography

Seismic tomography depends on the presence of contrast in seismic properties. Such differences in three-dimensional structure are reflected directly in the times of arrival of seismic phases or through the shape and amplitude of seismic waveforms.

The normal procedure is to examine the departures of the observed properties of the seismic wavefield from the predictions for a reference model. In most cases the significant quantity is a time shift in the arrival of a phase, which can be measured directly for a short-period body wave, by correlation techniques for long-period body waves and surface waves, or by waveform inversion where the record contains a number of arrivals as, e.g., the shear wavefield containing both body wave and surface wave components. The travel-time residuals for body waves or modified dispersion characteristics for surface waves can then be used as the basis for recovering a three-dimensional model.