When a disturbance occurs in the interior of a solid body, elastic waves are generated and spread out in all directions. These are of the following distinct types:
Primary or P waves, (dilatational) in which the particles of the earth are displaced radially in relation to the source, i.e. to and fro along the direction of wave propagation.
Secondary or S waves, (rotational or shear) in which the particle displacement is perpendicular to the direction of propagation.
Each wave travels with its own characteristic velocity which, at any given point in the medium,is determined by the density p and elastic moduli lambda and µ at the point in question. It is shown in textbooks of elasticity that the elastic moduli are related to each other by means of the "Poisson's ratio", sigma. Hence the velocities VP and VS of the P and S waves are connected by the equations.
In many minerals lambda is approximately equal to µ, the value of sigma is near to 1/4, and VP/VS is about SQRT(3)
If the medium is homogeneous and perfectly elastic, both P and S waves are propagated without loss of energy. For a spherical wave train, the volume of medium affected is therefore proportional to the area of the diverging wave front, and hence to the square of the distance from the source. The energy density in the material is proportional to the square of the wave amplitude, so the amplitude is inversely proportional to the first power of the distance.
If the advancing wave front encounters a discontinuity, part of the energy is refracted and the rest reflected. The geometry of the process is that if the incoming wave arrives with velocity vi and angle of incidence i, it generates a wavelet which moves along the interface with velocity vi cosec i. In the most general case, an incident P wave can generate a family of conical (or 'head' waves) and nearly-spherical P and S waves, which propagate away from the interface in both media. The possible transformations are discussed in detail in Cagniard (1939 or 1962). Fig. 1.1.2 is a composite made up from several figures in that work. For each of the transformed waves, the velocity vi is related to the angle r of reflection or refraction by the equation:vr cosec r = vi cosec i. 
Equation  has a real solution only if vr < cosec i, so that whereas the high-velocity P waves can always generate some lower-velocity S energy on reflection (except when incidence is exactly normal) S waves can only generate reflected P when cosec i > VP/VS. The value of i for which cosec i = VP/VS is analogous to the 'critical angle' of refraction in optics.
Transformations permitted by Equation  are also governed by symmetry, so that an S wave polarized parallel to the interface can never exchange energy with a P wave, or with a normally polarized component of S. This fact sometimes imparts important diagnostic characteristics to phases which are generated by transformations in the mode of propagation.
The actual velocity pattern in the Earth is governed by the existence of a series of concentric spheroidal zones, in each of which the wave velocity is internally graded. Much steeper gradients, or sharp discontinuities, separate the major zones from each other. Horizontally and vertically polarized S waves are called SH and SV respectively.
As a consequence of this velocity structure, the ray paths are curved within each zone, and develop several branches when reflections and refractions occur at zone boundaries. The times of propagation, expressed as a function of distance, develop into the complicated pattern of travel-time curves which is set out in Fig. 1.1.2a. Propagation paths are shown in Fig. 1.1.2b. The amplitude of oscillation of the free surface is also a complicated function of distance, as expressed in the table of magnitude factors in Section PAR 3.2. The ray paths are further discussed in Section 2 of this chapter.
Next, we must consider the effect of absorption in the real medium and of innumerable minor irregularities in propagation velocity which exist within the Earth, especially in its outer layers. These irregularities extract energy from an advancing wave front and scatter it in the form of 'signal-generated noise'. The scattered energy reaches the surface of the Earth behind the P wave which has travelled by the most direct path, and therefore produces a 'tail' which may obscure later phases. It has been shown that the proportion of energy scattered in a given ray path increases rapidly with decreasing wavelength, and is greater at any given wavelength for S waves than for P. In consequence of their lower velocity, S waves have a shorter wavelength than P for any given period of vibration, and are therefore more seriously attenuated on both counts. The numerical factors are such that P waves with a period of about 1 second can be transmitted through the whole body of the Earth with little loss, whereas S waves need a period of at least 5 seconds for comparable transmission. As the continuous 'microseismic' disturbance of the earth is much more intense for periods of about 5 seconds than for periods of about 1 second, the S waves of small, distant sources are much harder to detect than are the P waves. Figs. 1.1.3a and 1.1.3b illustrate the effect by contrasting the short-period record of a local event with the short-period and long-period records of a distant one.
The second large family of seismic waves is the surface waves which carry the greatest amount of energy from shallow focus earthquakes.
The main categories of surface waves are as follows:
Rayleigh waves, which run along the surface in a manner somewhat similar to that of gravity waves on the surface of a liquid. As the wave advances, each particle in the medium executes a retrograde elliptical orbit, its displacements being resisted by both shear and compression stresses. In a uniform medium for which Poisson's ratio equals 1/4 the velocity of Rayleigh waves is 0.92 times the velocity of S waves. The particle motion has its greatest amplitude at the surface and decreases with depth.
We see therefore that the velocity of propagation will depend on the average properties of the medium within about a wavelength of the surface. As the P and S velocities tend to increase with depth, wave velocity increases with increasing wave length, and under continental conditions the group velocity shows a minimum for periods of 15-20 seconds. There is a broad maximum for periods in the vicinity of 75 seconds, giving a velocity of about 4.2 km/s.
Channel waves, which occur when the structure of a propagating region includes layers bounded by layers of different velocity or by free surfaces. Waves propagating at a slight angle to the plane of the layer may then be partially or totally deflected inwards whenever they approach the channel boundaries. Reflection at sharp boundaries in solid media can only be total for S waves polarized parallel to the sides of the channel, but P waves can be trapped in liquid channels. When trapping is complete, propagation can extend over long distances with little spreading of energy. 'Love waves' and the 'T phase' are names given to transverse waves trapped in the Earth's crust and upper mantle, and to compressional waves in a low-velocity layer of the ocean, respectively.
Higher-mode waves, which are overtones of fundamental-mode surface waves, and characterized by the existence of one or more nodal planes parallel to the channel boundaries. The difference between energy distribution in depth between various modes enables higher-mode waves to yield important information about horizontally stratified structures, but crustal inhomogeneity often leads to strong attenuation and mode conversion at continental oceanic borders and by mountain structures.
Coupled waves, which occur along undisturbed continental paths in the form of long-period, normally dispersed wave trains, immediately after the P phase, or coupled to S, SS and SSS. These are additional branches of higher-mode surface waves. They are known as 'PL' waves or as 'leaking modes' because energy is progressively lost into the mantle.
In the case of a finite body like the Earth, which has a free surface and numerous internal zones of contrasting physical properties, modes of propagation are observed in which the wave velocity is a function of the frequency. Such modes of propagation are said to exhibit 'dispersion'.
In order to understand the propagation of a dispersed wave train, we consider a packet made up of infinite sinusoidal wave trains, each component being characterized by frequency omega and velocity c. We can also define a wave number k such that omega = kc.
At any time t, and at distance x from the origin, the particle displacement a for any of the harmonic components may be written
and we may consider an initial impulse for which all components are in phase where x = 0, t = 0. Thus we have strong disturbance near the origin, and destructive interference leading to negligible disturbance elsewhere.
At some later time and at a distance from the origin, a disturbance will be observable if any of the wave trains are in phase over a significant band of frequency. The condition for reinforcement is that
within the band. Differentiating with respect to k, we find
We have thus a situation in which waves have appeared at a time and place as though they had been propagated with the 'group velocity' u, although detailed observations over a narrow range of distance would show the crests and troughs moving with the 'wave' or 'phase' velocity c. Fig. 1.2.1 gives a schematic illustration of the phenomenon.
A situation which commonly occurs in the Earth is that the wave velocity is confined between a pair of asymptotes corresponding to short-period and long-period propagation respectively and varies monotonically from one to the other at intermediate periods. If the short-period limit of wave velocity is lower than the long-period limit, dc/dk is negative in the intervening range, and the group velocity may show a minimum (Fig. 1.2.1a). Fig. 1.2.1b shows actual dispersion curves for the principal phases.
Ideally, a wave train corresponding to such a dispersion curve would show the long waves arriving first, with periods getting shorter later in the train, as the group velocity falls towards the minimum. As no energy can travel with less than the minimum group velocity, the motion ends abruptly, and waves having periods less than that of the minimum are superposed on the waves from the longer-period branch. Such an abrupt termination, with confused motion ahead of it, is called an 'Airy phase'.
In practice, the propagation of short-period surface waves is readily perturbed by minor variations such as low-velocity surface sediments in the propagating structure, so the short-period end of the group-velocity curve may contain a number of subsidiary maxima and minima. These waves are therefore relatively hard to observe and interpret. For this reason, and by analogy with the optical case, a wave train in which long waves arrive before the shorter ones is said to exhibit 'normal dispersion'.
The character of a seismogram is strongly influenced by the depth of focus, source mechanism, and whether the shock is local, near, distant or very distant. In general, the deep focus earthquakes can be recognized by their relatively small surface waves and by the occurrence of characteristic reflections from the Earth's surface (pP, sP and sS). Near earthquakes are characterized by short total duration, large high-frequency content and a characteristic shape of wave envelope. Beginners will have to assign other events to the correct range of distance by careful comparison of phase readings with the travel-time curves, but observers experienced in the interpretation of records from a given station develop a remarkable capacity for recognizing characteristic patterns in earthquakes from different active regions of the Earth.
Examples of the appearance of seismograms from different distances are given in Gutenberg (1934, 1935, 1936 and 1939), Vesanen (1942), Lehmann (1954), USGS Special Publication No. 254 (1959), Richter (1958), Simon (1968) and Bolt (1976).
Increased sensitivity of modern seismographs has led to the more frequent identification of later, compound phases at global earthquake observatories. Also the advance of theory has provided at least tentative explanations for previously unnamed waves observed on seismograms. In the interest of comprehension, it is therefore important in observatory practice to follow a common but simple nomenclature as far as feasible.
The nomenclature given at the head of this chapter is believed to be already widely used and understood. Changing practice has led to some omissions of older symbols as well as some changes in the association of symbols with phases. Some phases listed are still a matter of investigation and thus the appropriate nomenclature was selected so as not to preclude alternative explanations of the phases.
Overall the traditional rules followed by seismologists to build up compound phases have been followed. Thus internal lower-case letters denote reflections from a boundary (PcP, etc.). Terminating lower-case letters (not subscripts) are used for the near-earthquake body phases and numerical superscripts for the multiple refractions (PKKKP = PK3P).
Note that the international agencies (Sections OUT 2 and 3) have introduced modifications to restrict nomenclature within the character ranges available for telegraphic and computer-readable input.
Near earthquakes are defined as those which are observed within a distance of 1000 km of the epicenter. At such distances, we can neglect the effects of curvature of the Earth's surface. The characteristics of the seismograms are as follows:
P and S waves are well defined on short-period records, with arrivals separated by not more than 1.5 minutes.
The entire record is much shorter than that of a well-recorded distant earthquake. Do not, however, be misled by the appearance of a small distant event recorded only by a short-period seismograph, for in this case all the later phases may be missed, and the visible part of the record may be quite short.
Periods of P and S waves are all short (commonly between a few tenths of a second and 3 seconds), those of surface waves up to about 10 seconds.
The commencement of surface waves is usually hard to observe, because of the masking effect of Sg waves, which have large amplitudes at short distances. For very shallow sources such as rockbursts, explosions or very shallow earthquakes, surface waves are separated from S waves.
The propagation paths in this range of distance are within the crust or immediately below it, so that the full set of phases which is described below will appear only if the focus is in the upper part of the crust. Arrival times may be disturbed by crustal irregularities, and should be interpreted in the light of the travel-time curves appropriate to the region.
The types of materials which are often reported to exist in sufficient volume to form propagating layers of continental dimensions are as follows:
Sediments, usually less than 10 km thick, velocities of P waves up to 5 km/s.
Upper Crust or 'Granitic Layer', often- including metamorphosed sediments, P velocities in the range 5.5-6.4 km/s, 10-30 km thick on continental plains, disturbed and thickened in mountainous regions, absent from oceans. Waves are described in the literature as Pg and Sg or P and S, observed as first arrivals at ranges extending to 120-170km.
'Intermediate layers', basaltic rock, separated from granitic materials by the 'Conrad discontinuity', with P-wave velocities described in the range of 6.3-7.3 km/s. The waves are usually described as P* and S*, sometimes as Pb and Sb. Some authors suggest several subdivisions within the intermediate zone.
The 'Upper Mantle' immediately below the "Mohorovicic discontinuity" is ultrabasic rock with P-wave velocities quoted in the range of 7.5-8.2 km/sec. The depth of the Mohorovicic discontinuity varies from 25-60 km in continental areas, and from 5-15 km below the floor of the oceans. The waves in this material are usually described as Pn and Sn. Ray paths in a schematic section of the Earth's crust are shown in Fig.2.1.
The practical procedure of recognition is to look for the first P arrival, and for the beginning of the group of larger amplitudes which represents the Sg waves. If the S-P interval is less than 12 seconds, the first waves to arrive are probably Pg and Sg. If the interval exceeds 25 seconds, the first waves to arrive are probably Pn. In this case, Sn may be seen before the large Sg and surface waves. Between these limits the various branches of the travel-time curves cross over each other, and careful observation will need to be combined with detailed knowledge of the area to produce identification.
The commencement of Pn is at about two-thirds of the range at which Pn and Pg intersect. Near this point Pn is reinforced by a wave reflected from the Mohorovicic discontinuity (PmP), and may give rise to a very strong phase a few seconds after Pg.
Beyond the intersection, Pg and Sg may be observed after Pn and Sn. The relative amplitude of Pg declines with increasing distance.
The possibility of observing P* depends critically on the range of observation, and on the velocities and thicknesses of the layers in the region concerned. Sometimes it appears as a fairly clear first arrival for some tens of kilometers on either side of the Pn-Pg intersection. In other regions, the P* line runs very close to the intersection and the whole picture is confused. Nowadays, velocity filtering by seismological array stations is improving the situation and P* can often be observed as a distinct peak on a correllogram, when it could not have been observed on the output of any one of the array elements.
The paths of different phases and their range of occurrence have been illustrated on Fig. 1.1.2b. Although there are peculiarities in some regions, along some paths or for some seismographs, a general description can now be given of the appearance of seismograms at various distances.
At some distance, not far from 1000 km from the source, Pn and Sn become hard to observe (except in some shield regions) but the teleseismic phases P and S, with rather longer periods, are often reported instead. For the next 15°, the records are somewhat confused because the waves are being refracted through a region of the Earth's mantle which contains substantial gradients in velocity, and the corresponding travel-time curve has a number of overlapping branches accompanied by varying amplitudes. P is usually more clearly defined on the vertical component and S on the horizontal components. All arrival times of clear onsets should be reported.
With increasing epicentral distance, the curved paths of the seismic waves carry them nearer to the center of the Earth until, at an epicentral distance of about 100°, the ray path for the P wave touches the core of the Earth. At this point, the so-called 'shadow zone' commences. The shadow has quite a sharp edge for the shortest P waves, although the components with longer periods are diffracted readily around the core boundary, and may be observed out to distances of 160° or more.
Waves which leave the source with a steeper descent than that of the grazing Pc wave, strike the Earth's core and divide into reflected and refracted portions. The reflected energy is itself divided into two parts, which we call PcP and PcS, according as to whether or not the mode of propagation is transformed on reflection. Similarly, the S wave which strikes the core gives rise to the phases ScS and ScP.
Waves which enter the core are refracted sharply downwards, and undergo a further sharp refraction when they leave from the other side. As a result, the point of emergence is more than 180° from the epicenter, if the measurement of distance is carried round in the direction of the initial departure of the ray. As the initial ray path plunges more steeply into the earth, the core refractions become less abrupt and the point of emergence moves back towards the epicenter. The minimum distance of emergence is near 144°, at which point the seismic rays form a cusp, where very large amplitudes are observed. Further steepening of the angle of entry first increases the distance of the point of emergence, and then leads to complicated further behavior which reflects the structure of deeper parts of the core.
Fig. 2.2 is a simple schematic diagram which illustrates the pattern of normal and retrograde branches, corresponding to a simplified version of the Earth's structure. More detailed interpretations of observations from modern short-period seismographs have revealed additional features of the travel-time curve (see, for example, Bolt, 1964, figure 2, produced here as Fig. 2.2a).
The records obtained at various distances can now be described in more detail as follows:
The following basic errors of interpretation are common, and should be avoided:
The misinterpretation of PP and PS waves as P and S at distances between about 1 15° and 120°, leading to a false estimate of about 80° for the distance.
Misinterpretation of SKS as S at distances in excess of 84°.
Earthquake foci may occur at depths of as much as 700 km below the surface, but most of the waves from such sources travel along ray paths which coincide with those of surface events at an earlier time and somewhat greater epicentral distance (Fig. 2.3). Deep-focus traveltime curves can therefore be derived theoretically from the values appropriate to surface foci. The process leads to families of P and S curves in which the short-range arrivals are late and the core-phase times early in comparison with the surface-focus curve which matches the travel time at the middle of the range of distance.
In the case of surface reflections, depth of focus introduces the possibility of a distinct reflection taking place at a point on the surface rather near the source. The short branch of these reflected paths is described by a lowercase letter, so the entire paths have designations pP, sP and sS respectively. These are sharp in character and often of considerable amplitude, and the time intervals between their arrival and the preceding P and S phases are sensitive indicators of depth of focus.
Thus, the most obvious mark of a large deep teleseism is the reduplication of the principal phases and the small amplitude or even absence of surface waves. In comparison with records of shallow shocks, individual phases have sharper onsets and are well developed.
At short distances P and S are sharp and short and strong and ScS is recorded on horizontal seismographs following a few minutes after S. The most confusing seismograms are those from intermediate-depth foci in distances around 35° to 40° and 115°. Near the former distance range the waves reflected from the core are very strong and are duplicated by phases pPcP, sPcP, etc., whereas S is relatively weak and other phases may be misinterpreted for it; near 115° the seismogram begins with PKIKP followed by PP, SKP, PPP, SKS, etc. all again duplicated and under those circumstances the identification becomes very difficult. The complication with close arrivals of S, SKS, PS at distances ranging from 80° to 85° is the same as for shallow shocks.
Estimates of depth based on pP or sP should be used with caution, as these phases are often mistaken for each other. Sometimes pP and sP are misinterpreted as S phases of local events. Sometimes also P'P' is reported as a separate arrival of another earthquake.
The main characteristics of Rayleigh waves, Love waves and the T phase were outlined in Section 1.2 and the purpose at this stage is to draw attention to certain additional characteristics of particular seismic paths.
The most important of these characteristics arises from the difference between oceanic and continental paths for the propagation of Rayleigh waves. When a layer of water lies on top of rock the effect of its inertia and lack of rigidity is to reduce the wave velocity at the shorter periods. When the water is of oceanic depth, the slope of the dispersion curve becomes extremely steep at periods between 10 and 20 seconds, and the wave train develops a long tail, or 'coda' of almost sinusoidal oscillations. With continental paths, the group velocity has a minimum value between 2.8 and 3.2 km/s, and the wave train is a comparatively compact packet terminated, in theory, by an Airy Phase. In practice, scattering within the continental layers often generates a coda behind the Airy Phase. The coda therefore appears as a diminution of amplitude for wave periods of about 20 seconds, instead of total cessation of motion.
In continental areas, short-period waves can be trapped in the upper layers, to produce a channel wave called Lg. The commencement of this wave has the velocity of the short-range body wave Sg, but the method of propagation permits it to propagate for much longer distances if the essential layering is not interrupted.
The surface wave group of large distant shocks often begins with waves of large amplitude and very long period ranging from about 1 to 4 minutes; these waves are usually horizontal and transverse as predicted for Love waves and are denoted by G. With decreasing magnitude the amplitude of G waves decreases much more rapidly than that of the 20-sec LR waves in the maximum group of surface waves. This long-period movement is best detected by instruments of class B (see section on instruments). Instruments belonging to classes C and E record somewhat later a transverse motion of periods 30 to 50 seconds, denoted by LQ. Several minutes after G, vertical component seismographs and the horizontal component instruments operating in the direction of propagation record Rayleigh waves (LR) of smaller period than LQ; both LQ and LR usually show a clear dispersion. The identification of individual surface wave phases may become difficult if the wave front has been refracted at a coastline or tectonically complex region, as it may then approach from a direction appreciably different from that of the epicenter.
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