Seismology

 The study of the shaking of the Earth's interior caused by natural or artificial sources. The revolution in understanding the Earth brought about by the paradigm of plate tectonics in the early 1960s provided a rationale that has since guided an intense period of investigation into the structure of the Earth's crust, mantle, and core. Plate tectonics provided Earth scientists with a kinematic theory for global tectonics that gave context to many of their studies. Ancient regions of crustal deformation could, for instance, be seen as the result of past plate boundary interactions, and the global distribution of earthquakes could be understood as the result of present plate interactions. The extraordinary insights into the nature of the solid Earth provided by kinematic plate tectonics also indicated new avenues in which to pursue basic questions concerning the dynamic processes shaping the Earth. Throughout the period in which plate tectonics was advanced and its basic tenets tested and confirmed, and into the latest phase of inquiry into basic processes, seismology (and particularly seismic imaging) has provided critical observational evidence upon which discoveries have been made and theory has been advanced. The use of the term image to describe the result of seismic investigations is closely linked mathematically to inversion and wavefield extrapolation and applies both to structures within the Earth and to the nature of seismic sources. See also: Plate tectonics

Theoretical seismology

A seismic source is an energy conversion process that over a short time (generally less than a minute and usually less than 1–10 s) transforms stored potential energy into elastic kinetic energy. This energy then propagates in the form of seismic waves through the Earth until it is converted into heat by internal (molecular) friction. Most earthquakes, for example, release stored elastic energy, nuclear and chemical explosions release the energy stored in nuclear and molecular bonds, and airguns release the pressure differential between a container of compressed air and surrounding water. Large sources, that is, sources that release large amounts of potential energy, can be detected worldwide. Earthquakes above Richter magnitude 5 and explosions above 50 kilotons or so are large enough to be observed globally before the seismic waves dissipate below modern levels of detection. The largest earthquakes, such as the 1960 event in Chile with an estimated magnitude of 9.6, cause the Earth to reverberate for months afterward until the energy falls below the observation threshold. Small charges of dynamite or small earthquakes are detectable at a distance of a few tens to a few hundreds of kilometers, depending on the type of rock between the explosion and the detector. The smallest of these may be detectable for only a few seconds. See also: Earthquake

Seismic vibrations are recorded by instruments known as seismometers that sense the change in the position of the ground (or water pressure) as seismic waves pass underneath. The record of ground motion as a function of time is a seismogram, which may be in either analog or digital form. Advances in computer technology have made analog recording virtually obsolete: most seismograms are recorded digitally, which makes quantitative analysis much more feasible.

Equation of motion

 The dissipation of seismic energy in the Earth generally is small enough that the response of the Earth to a seismic disturbance can be approximated by the equation of motion for a disturbance in a perfectly elastic body. This equation holds regardless of the type of source, and is closely related to the acoustic-wave equation governing the propagation of sound in a fluid. Deviations of the elastic body away from equilibrium or rest are resisted by the internal strength of the body. An elastic material can be thought of as a collection of masses connected by springs, with the spring constants governing the response of the material to an externally applied force or stress. More formally, the strain or distortion of an elastic body when subjected to an applied stress is described with a constitutive relation, which is just a generalization of Hooke's law. Instead of just one spring constant, however, a generalized elastic solid needs 21 constants to completely specify its constitutive relation. Fortunately, the Earth is very nearly isotropic; that is, its response to an applied stress is independent of the direction of that stress. This reduces the complete specification of the constitutive relation from 21 to just 2 constants, the Lamé parameters λ and μ. A well-known result is that the equation of motion for an isotropic perfectly elastic solid separates into two equations describing the propagation of purely dilatational (volume changing, curl-free) and purely rotational (no volume changing, divergence-free) disturbances. These propagate with wave speeds α and β [Eqs. (1) and (2)], respectively,

 

 

 (1)                     

 

 

 

 (2)            

where ρ is the density and the Lamé parameters have units of pressure (force/area).

 

Elastic-wave propagation

 These velocities are also known as the compressional or primary (P) and shear or secondary (S) velocities, and the corresponding waves are called P and S waves. The compressional velocity is always faster than the shear velocity; in fact, it can be shown that mechanical stability requires that α2 ≥ 4/3β2. In a fluid, μ = 0 and there are no shear waves. The wave equation then reduces to the common acoustic case, and the compressional wave is just the ordinary sound or pressure wave. In the Earth, α can range from a few hundred meters per second in unconsolidated sediments to more than 13.7 km/s (8.2 mi/s) just above the core–mantle boundary. (The fastest spot within the Earth is not its center, but just above the core–mantle boundary in the silicate mantle. The core, being predominantly iron, is in fact relatively slow.) Wave speed β ranges from zero in fluids (ocean, fluid outer core) to about 7.3 km/s (4.4 mi/s) at the core–mantle boundary. See also: Hooke's law; Special functions

A P wave has no curl and thus only causes the material to undergo a volume change with no other distortion. An S wave has no divergence, thus causing no volume change, but right angles embedded in the material are distorted. Explosions are relatively efficient generators of compressional disturbances, but earthquakes generate both compressional and shear waves. Compressional waves, by virtue of the mechanical stability condition, always arrive before shear waves.

 

 

Fig. 1  Relationship between wavefronts and geometrical rays for a constant velocity gradient. 1 km = 0.6 mi.

 

 

 

 Fig. 2  Diagram showing seismic rays associated with a source at the Earth's surface bouncing off the base of a layer and being returned to the surface. The take-off angle  for the ray that returns to the surface at a range of about 3.5 km (2.1 mi) is shown. 1 km = 0.6 mi.

 

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Compressional and shear waves can exist in an elastic body irrespective of its boundaries. For this reason, seismic waves traveling with speed α or β are known as body waves. A third type of wave motion is produced if the elastic material is bounded by a free surface. The free-surface boundary conditions help trap energy near the surface, resulting in a boundary or surface wave. This in turn can be of two types. A Rayleigh wave combines both compressional and shear motion and requires only the presence of a boundary to exist. A Love wave is a pure-shear disturbance that can propagate only in the presence of a change in the elastic properties with depth from the free surface. Both are slower than body waves.

Solutions of the elastic-wave equation in which a wave function of a particular shape propagates with a particular speed are known as traveling waves. An important property of traveling waves is their causality; that is, the wave function has no amplitude before the first predicted arrival of energy. The complete seismic wavefield can be constructed by summing up every possible traveling wave, usually a countably infinite set. A traveling-wave formalism is useful especially when it is desirable to concentrate on particular arrivals of energy. An alternative set of solutions to the wave equation can be constructed by considering standing-wave functions. Individual standing waves by definition do not propagate, but rather oscillate throughout the body with a fixed spatial pattern. However, the sum of all possible standing waves must be equivalent to the sum of all possible traveling waves, because they must represent the same total wavefield. Standing waves, also known as free oscillations or normal modes, are useful representations of the wavefield in the frequency domain.

 

Ray theory

 Traveling-wave or full-wave theory provides the basis for a very useful theoretical abstraction of elastic-wave propagation in terms of the more common notions of wavefronts and their outwardly directed normals, called rays (Fig. 1). The validity of the ray treatment can be deduced from full-wave theory by taking infinite frequency approximations that treat the seismic disturbances as pure impulses having essentially no time duration and unit amplitude. Ray theory makes the prediction of certain kinematic quantities such as ray path, travel time, and distance by a simple geometric exercise involving what is essentially a generalization of Snell's law. Ray theory can be developed in the context of an Earth comprising flat-lying layers of uniform velocities; this is a very useful approximation for most problems in crustal seismology and can be extended to spherical geometry for global studies. Expressions will be developed for travel time and distance in both geometries.

 

 

Fig. 3  Seismic ray paths. (a) A single ray passing through a multilayered Earth comprising a stack of uniform velocity layers will be reflected from each layer and also be refracted as it passes from one layer into the layer below in a manner that obeys Snell's law. Each ray therefore is considered to give rise to a new system of rays. (b) Ray diagram for a cross section of the spherical Earth. At the point labeled v(r), r = radial distance and v = velocity. r0 = radial distance to the turning point.  is the angle of incidence.

 

 

 

 

Figure 2 is a diagram showing the rays of a seismic wavefront in a single layer expanding away from a source, a single layer. It is apparent from this simple ray diagram that the rays will intersect the base of the layer and be reflected to the surface. Each ray leaves the source at an angle , known as the take-off angle or angle of incidence. This angle can be used to characterize each of the rays by Eq. (3), where p

 

 

 (3)                    

is the ray parameter. This term characterizes uniquely each ray in a system of rays. Its physical meaning is very important; if the term slowness is used to decribe the inverse of velocity, then p is the horizontal component of a ray's slowness. A vertical component q can also be defined, because unlike components of velocity the quantities p and q behave as vectors. This result makes it possible to cast the calculation of travel time and distance as geometric problems. Equation (4) gives the time T(x) required to reach a

 

 

 (4)                         

given distance x from the source, and Eq. (5) gives

 

 

 (5)                         

the distance reached by a ray of given ray parameter, X(p).

The first term on the right-hand side of Eq. (4) is the travel time of the vertical ray having a take-off angle of 0°, or p = 0. Equation (5) employs notation u to describe the slowness of the medium, rather than its velocity. Both Eqs. (4) and (5) are parametrized by layer thickness h and layer velocity α (or slowness u); and these are the unknowns needed to describe the Earth (α depends on the Lamé parameters and the density). Equation (4) is hyperbolic, which is sometimes observed in practice. This indicates that a simple representation of the Earth as a stack of uniform velocity layers might at times be reasonably close to reality.

Kinematic equations have been developed to describe what happens to rays as they impinge on the boundaries between layers. Figure 3a is a diagram of a single ray propagating in the stack of horizontal layers that define the model Earth. At each interface, part of the ray's energy is reflected, but a portion also passes through into the layer below. The transmitted portion of the ray is refracted; that is, it changes the angle at which it is propagating. The relationship between the incident angle and the refracted angle is exactly the same as that describing the refraction of light between two media of differing refractive index, as shown in Eq. (6). This is the seismic analog of

 

 

 (6)                 

Snell's law. The left-hand side of Eq. (6) is the ray parameter for the ray in the upper layer, and the right-hand side is the ray parameter in the lower layer. Thus Snell's law for seismic energy is equivalent to stating that the ray parameter or the horizontal slowness is conserved on refraction. See also: Refraction of waves

Equation (7)

 

 

 (7)                     

for multilayers describes the distance reached by a ray of given p after passing through n layers, each of which is associated with a slowness uk and thickness hk before being reflected to the surface. It is the sum of the individual contributions to X(p) from each of the layers. The equation for T(x) cannot be written as a simple sum of hyperbolic contributions, although doing so in some special situations can be a reasonable approximation to the real travel-time behavior. Because of this, complex seismograms are generally analyzed by using a “parametric” form such as T(x) = T[X(p)].

The generalization of these kinematic equations to the case where the medium slowness varies continuously with depth is straightforward. More care must be taken, however, in defining what happens to the ray at its deepest point of penetration into the medium. In Eq. (5), the ray is reflected from the base of the deepest layer. More generally, information is sought for the ray traveling downward, then reversing direction and traveling upward. In a medium with continuously varying velocities, this happens when the ray is instantaneously horizontal, that is,  = 90° and the ray parameter or horizontal slowness p is equal to the medium slowness 1/α(zmax); the term zmax is defined to be the turning point of the ray and is the deepest point in the medium sampled by the ray. The integral form of the distance equation is given by Eq. (8),

 

 

 (8)                                

where u(z) = 1/α(z) represents any relationship of slowness versus depth.

The case of a spherical Earth is treated by replacing the flat-Earth ray parameter by

 

 

 (9)                                

Eq. (9), where r is the distance from the center of the Earth, v (r) is the velocity at that radius, and  is the angle of intersection between the ray and the radius vector (Fig. 3b). The velocity function υ(r) denotes either α(r) or β(r), depending on the circumstance. The radius r0 at which  = 90° is known as the turning radius of the ray and denotes the point at which the ray begins its journey back toward the surface. If υ(r) is a constant independent of radius, the ray path is straight but the turning point is still well defined. In a spherical Earth, the ray travels in the plane containing the source, the receiver, and the center of the Earth. Like its flat-Earth counterpart, the spherical-ray parameter must be conserved along the ray.

 

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