Tomography is a technique whereby three-dimensional images are derived from the processing of the integrated properties of the medium that rays encounter along their paths through it. Tomography is perhaps best known in its medical applications, in which images of specific plane sections of the body are obtained using X-rays. Seismic tomography refers to the derivation of the three-dimensional velocity structure of the Earth from seismic waves. It is considerably more complex than medical tomography in that the natural sources of seismic waves (earthquakes) are of uncertain location, the propagation paths of the waves are unknown, and the receivers (seismographs) are of restricted distribution. These difficulties can be overcome, however, and since the late 1970s seismic tomography has provided important new information on Earth structure. The method was first described by Aki et al. (1977) and has been reviewed by Dziewonski & Anderson (1984), Thurber & Aki (1987), and Romanowicz (2003).

Seismic tomography makes use of the accurately recorded travel times of seismic waves from geographically distributed earthquakes at a distributed suite of seismograph stations. The many different travel paths from earthquakes to receivers cross each other many times. If there are any regions of anomalous seismic velocity in the space traversed by the rays, the travel times of the waves crossing this region are affected. The simultaneous interpretation of travel time anomalies for the many criss-crossing paths then allows the anomalous regions to be delineated, providing a three-dimensional model of the velocity space.

Both body waves and surface waves (Section 2.1.3) can be used in tomography analysis. With body waves, the actual travel times of P or S phases are utilized. The procedure with surface waves is more complex, however, as they are dispersive; that is, their velocity depends upon their wavelength. The depth of penetration of surface waves is also wavelength-dependent, with the longer wavelengths reaching greater depths. Since seismic velocity generally increases with depth, the longer wavelengths travel more rapidly. Thus, when surface waves are utilized, it is necessary to measure the phase or group velocities of their different component wavelengths. Because of their low frequency, surface waves provide less resolution than body waves. However, they sample the Earth in a different fashion and, since either Rayleigh or Love waves (Section 2.1.3) may be used, additional constraints on shear velocity and its anisotropy are provided.

The normal procedure in seismic tomography is to assume an initial "one-dimensional" model of the velocity space in which the velocity is radially symmetrical. The travel time of a body wave from earthquake to seismograph is then equal to the sum of the travel times through the individual elements of the model. Any lateral velocity variations within the model are then reflected in variations in arrival times with respect to the mean arrival time of undisturbed events. Similarly, the dispersion of surface waves across a heterogeneous model differs from the mean dispersion through a radially symmetrical model. The method makes use of a simplifying assumption based on Fermat's Principle, which assumes that the ray paths for a radially symmetrical and laterally variable velocity model are identical if the heterogeneities are small and that the differences in travel times are caused solely by heterogeneity in the velocity structure of the travel path. This obviates the necessity of computing the new travel path implied by refractions at the velocity perturbations.

There are two main approaches to seismic tomography depending upon how the velocity heterogeneity of the model is represented. Local methods make use of body waves and subdivide the model space into a series of discrete elements so that it has the form of a three-dimensional ensemble of blocks. A set of linear equations is then derived which link the anomalies in arrival times to velocity variations over the different travel paths. A solution of the equations can then be obtained, commonly using matrix inversion techniques, to obtain the velocity anomaly in each block. Global methods express the velocity variations of the model in terms of some linear combination of continuous basic functions, such as spherical harmonic functions.

Local methods can make use of either teleseismic or local events. In the teleseismic method (Fig. 2.11) a large set of distant seismic events is recorded at a

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Figure 2.12 Geometry of the local inversion method. Figure 2.12 Geometry of the local inversion method. network of seismographs over the volume of interest. Because of their long travel path, the incident wave fronts can be considered planar. It is assumed that deviations from expected arrival times are caused by velocity variations beneath the network. In practice, deviations from the mean travel times are computed to compensate for any extraneous effects experienced by the waves outside the volume of interest. Inversion of the series of equations of relative travel time through the volume then provides the relative velocity perturbations in each block of the model. The method can be extended by the use of a worldwide distribution of recorded teleseismic events to model the whole mantle. In the local method the seismic sources are located within the volume of interest (Fig. 2.12). In this case the location and time of the earthquakes must be accurately known, and ray-tracing methods used to construct the travel paths of the rays. The inversion procedure is then similar to that for teleseisms. One of the uses of the resulting three-dimensional velocity distributions is to improve focal depth determinations. Global methods commonly make use ofboth surface and body waves with long travel paths. If the Earth were spherically symmetrical, these surface waves would follow great circle routes. However, again making use of Fermat's Principle, it is assumed that ray paths in a heterogeneous Earth are similarly great circles, with anomalous travel times resulting from the heterogeneity. In the single-station configuration, the surface wave dispersion is measured for the rays traveling directly from earthquake to receiver. Information from only moderate-size events can be utilized, but the source parameters have to be well known. The great circle method uses multiple circuit waves, that is, waves that have traveled directly from source to receiver and have then circumnavigated the Earth to be recorded again (Fig. 2.13). Here the differential dispersion between the first and second passes is measured, eliminating any undesirable source effects. This method is appropriate to global modeling, but can only use those large magnitude events that give observable multiple circuits. |

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