seismic waves: principles

seismic waves: principles When stress builds up in the Earth to such a level that rapid slip occurs on a fracture (i.e. an earthquake takes place) or when an explosion or mechanical device is used to initiate a seismic disturbance artificially, a complex field of seismic waves is generated. This wave field propagates much like the waves that travel away from the point at which a stone is thrown into a pond (see elastic wave propagation). In the Earth, however, even ignoring the non-elastic effects near the source, the situation is much more complex because several types of waves are generated. Also, the Earth is composed of many differing bodies and layers of rock which are heterogeneous at many different scales, and the propagation is in three dimensions. Furthermore, discontinuities in the Earth reflect and refract some of the propagating energy, thus changing its direction of travel, and the Earth simply absorbs some of the seismic energy, effectively converting it to heat. Treating the full range of these complexities mathematically is a major challenge which is the topic of much continuing research.

It is usually sufficient to treat the seismic energy of the source as being divided into three types of waves. These are: (1) P-waves, which are the fastest and are simply sound waves resulting in vibrations in the direction in which the wave is travelling (see seismic body waves); (2) S-waves, which typically travel at about 60 per cent of the speed of P-waves and result in vibrations perpendicular to the direction in which the wave is travelling (see seismic body waves); and (3) surface waves, which typically travel at about 90 per cent of the speed of S-waves and are analogous to waves travelling on the surface of the ocean (see seismic surface waves). These waves travel by different paths, as shown in Fig. 1a. As observed at a point on the Earth's surface they produce a seismogram (a record of ground motion produced by a seismograph) as shown in Fig. 1b.

A number of simplifying assumptions are made in order to make seismic theory mathematically tractable. The first is usually that the observer is far enough from the seismic source for the actual ground displacements to be small enough to be within the elastic limit of rocks: in other words that the seismic waves travel without causing rock breakage and that the Earth returns to its undisturbed state after they have passed. This assumption arises from Hooke's law, which states that there is a linear relationship between stress and strain until the elastic limit is exceeded (i.e. until fracture, when permanent deformation occurs). Hooke's law is discussed in more detail below. A second assumption is that the Earth is composed of materials which are either homogeneous at some scale or whose properties change slowly with respect to the wavelength of a particular seismic wave. If one thinks of a snapshot of a seismic wave at a particular moment in time, as shown in Fig. 1c, the wavelength l is the distance between two adjacent peaks in the wavelet (seismic pulse). This quantity is a very useful measure of scale because inhomogeneities in the Earth which are small relative to a wavelength (one-quarter of a wavelength is commonly used as a relative standard of what is ‘small’) can, for most purposes, be thought of as being averaged out by the seismic wave. Since the Earth naturally absorbs higher frequencies (i.e. the more rapid oscillations generated by the seismic source), the wavelength can be thought of as slowly increasing as the wave travels through the Earth. Thus, the minimum size of a feature that is a directly detectable discontinuity also increases with time. One of the difficulties in applying seismology to study Earth structure (see controlled-source seismology and deep seismic reflection profiling) is appreciating that seismic body waves travelling through the Earth usually have longer wavelengths than we would prefer, thus limiting our ability to ‘see’ features at depth clearly. For example, a P-wave travelling at a velocity of 6km s⁻¹ at a depth of about 10 km in the Earth's crust might have a maximum frequency of about 10Hz, and its wavelength would be about 600m. Clearly, the inhomogeneities represented by the individual mineral grains, which constitute a rock such as granite and have dimensions of up to a centimetre or so, are not an important consideration. Instead, we can think of this granite as a rock having seismic properties which are the average of its constituents. Discontinuities which are large with respect to a wavelength reflect and refract seismic waves according to the classic laws of optics which are discussed below. The third assumption usually made is that the materials through which a seismic wave travels are mechanically isotopic, which means the seismic velocity is independent of the direction of travel through the material. Mechanical anisotropy adds considerable complexity to the treatment of seismic waves but has been the object of much research interest (see seismic anisotropy). Mapping regions of significant anisotropy can provide valuable information on such factors as flow or stress.

Materials which obey Hooke's law are, by definition, elastic. Since seismic waves represent small strains (deformations), we can for most purposes assume that Earth materials are perfectly elastic. This means that once the stress (force per unit area) causing the strain is removed, the material returns to its original state. In its simplest form, the linear relationship which expresses Hooke's law is s = kx, where s is the stress, x is the strain, and k is an elastic constant. Several elastic constants are commonly employed and can be measured in a laboratory. One is Young's modulus, which describes the elongation (or shortening) a sample of material experiences when it is pulled (or pushed) by a force applied in a single direction. Another is the bulk modulus, which describes the volume change a sample undergoes when it experiences hydrostatic compression. Yet another is the shear modulus, which is a measure of the material's resistance to deformation by a shearing stress (i.e., one applied across a face of the sample in a laboratory apparatus). Fluids have no ability to resist shearing, so they have a shear modulus of zero. Poisson's ratio is the ratio of the unit lateral contraction to the unit longitudinal extension when the material is stretched. It can be determined by measuring the ratio of the elongation to ‘necking’ when the material is stretched in a single direction. This ratio relates V_p and V_s; when it has its typical value of 0.25, V_p = √3V_s.

If we can assume that a material is homogeneous and isotropic, only two elastic constants are needed to describe its elastic behaviour because there are algebraic expressions from which one can calculate other constants given any two. In this case, it is also relatively easy to show that two, and only two, body waves (P and S) travel through the material and that their velocities are given by two simple equations.

Anisotropy introduces considerable complexity to the problem. Three body waves exist in this case, and the fastest one (the quasi P-wave) is, for example, no longer analogous to a simple sound wave. In the least isotropic natural material, 21 elastic constants are required to describe its behaviour. In the most general case, the expression for Hooke's law becomes a tensor equation involving stress and strain tensors of rank 2 and a tensor of rank 4 involving these 21 elastic constants.

When seismic waves encounter an interface between two materials with distinctly different seismic properties, some of the energy is reflected from the interface and some is transmitted through it (Fig. 2). Snell's law is a simple relationship which predicts the angles involved. An obvious but important relationship in seismology (and in billiards) is that the angle of reflection equals the angle of incidence.

Some P-wave energy incident on an interface can be converted to S-waves and vice versa. The partition of energy can be calculated by setting up a boundary-value problem. The result is an algebraically complex set of equations derived by Knott and Zoepprittz at the turn of the century. These equations not only reveal the relative strengths of the waves reflected from and transmitted through the interface but also reveal the extent to which energy conversion has occurred. These converted waves are particularly interesting, as are the amplitude variations that occur as the angle of incidence changes. From an applied perspective, the amount of energy reflected from an interface is of particular interest (see seismic exploration methods). This is normally only a few per cent, and the larger the contrast in seismic properties the larger the amount of reflected energy.

The amplitude of a seismic wave also naturally diminishes as the wave travels through the Earth. As the wave front travels, it expands, and thus the energy is spread over a larger and larger surface area. This effect is called geometrical spreading and it results in amplitude being inversely related to the distance travelled. Another effect on amplitude is anelastic attenuation, which refers to a number of phenomena which cause a material to absorb energy. This effect is strongly dependent on frequency: high frequencies are most strongly attenuated. This fact is a fundamental limitation in seismology which limits resolution because high frequencies are needed to ‘see’ features that are small relative to a wavelength. For typical seismic frequencies, anelastic attenuation is an approximately linear function of frequency which results in Q (the Quality Factor) being a constant. This factor is the quantity most often used to describe the anelastic attenuation in rocks. High-Q materials transmit seismic waves efficiently (i.e. they ring like a bell) while low Q materials absorb considerable energy.

G. R. Keller

Bibliography

Aki, K. and and Richards, P. G. (1980) Quantitative seismology: W. H. Freeman, San Francisco.
Bullen, K. E. and and Bolt, Bruce A. (1985) An introduction to the theory of seismology. Cambridge University Press.
Lay, T. and and Wallace, T. C. (1995) Modern global seismology. Academic Press, San Diego.