Applied Seismology: Introduction and Principles
4.2 Seismic waves
4.2.1 Stress and strain
When an external force F is applied across an area A of a surface of a body, forces inside the body are established in proportion to the
Table 4.1 Derived information and applications of exploration seismology.
Gross geological features:
Depth to bedrock
Measurement of glacier thickness Location of faults and fracture zones Fault displacement
Location and character of buried valleys Lithological determinations
Stratigraphy
Location of basic igneous dykes Petrophysical information:
Elastic moduli Density Attenuation Porosity
Elastic wave velocities Anisotropy
Rippability Applications:
Engineering site investigations Rock competence
Sand and gravel resources Detection of cavities
Seabed integrity (for siting drilling rigs)
Degassing or dewatering of submarine sediments Preconstruction site suitability for:
new landfill sites major buildings marinas and piers sewage outfall pipes tunnel construction, etc.
Hydrogeology and groundwater exploration Ground particle velocities
Forensic applications:
location of crashed aircraft on land design of aircraft superstructures monitoring Nuclear Test Ban Treaty location of large bore military weapons Location of trapped miners
Seismic hazard zonation
external force. The ratio of the force to area (F/A) is known as stress.
Stress can be resolved into two components, one at right-angles to the surface (normal or dilatational stress) and one in the plane of the surface (shear stress). The stressed body undergoes strain, which is the amount of deformation expressed as the ratio of the change in length (or volume) to the original length (or volume). According to Hooke’s Law, stress and strain are linearly dependent and the body behaves elastically until the yield point is reached. Below the yield point, on relaxation of stress, the body reverts to its pre-stressed shape and size. At stresses beyond the yield point, the body behaves in a plastic or ductile manner and permanent damage results. If further stress is applied, the body is strained until it fractures.
Earthquakes occur when rocks are strained until fracture, when stress is then released. However, in exploration seismology, the amounts of stress and strain away from the immediate vicinity of a seismic source are minuscule and lie well within the elastic be- haviour of natural materials. The stress/strain relationship for any
Before deformation V1 (A)
(B)
(C) (D)
Before deformation V1
Equal pressure in all directions
No lateral strain After deformation V2
After deformation V2
Change in volume –ΔV = V1– V2
ΔP
ΔP
F –F
–F
ΔP
θ
τ
L +ΔL
L + ΔL L
L
Compression
Extension
Triaxial strain F
Figure 4.1 Elastic moduli. (A) Young’s modulus; (B) bulk (rigidity) modulus; (C) shear modulus; (D) axial modulus.
material is defined by various elastic moduli, as outlined in Figure 4.1 and Box 4.1.
4.2.2 Types of seismic waves
Seismic waves, which consist of tiny packets of elastic strain energy, travel away from any seismic source at speeds determined by the elastic moduli and the densities of the media through which they pass (Section 4.2.3). There are two main types of seismic waves:
those that pass through the bulk of a medium are known as body waves; those confined to the interfaces between media with con- trasting elastic properties, particularly the ground surface, are called
surface waves. Other types of waves encountered in some applica- tions are guided waves, which are confined to particular thin bands sandwiched between layers with higher seismic velocities by total internal reflection. Examples of these are channel or seam waves, which propagate along coal seams (Regueiro, 1990a,b), and tube waves, which travel up and down fluid-filled boreholes.
4.2.2.1 Body waves
Two types of body wave can travel through an elastic medium. P- waves, which are the most important in exploration seismology, are also known as longitudinal, primary, push, or compressional waves.
Box 4.1 Elastic moduli
Young’s modulus
E= Longitudinal stressF/A Longitudinal strainL/L = σ
ε (in the case of triaxial strain)
Bulk modulus
k= Volume stressP Volume strainv/v (in the case of excess hydrostatic pressure) Shear (rigidity) modulus (a Lam´e constant)
µ= shear stressτ shear strainε
(µ≈7×104MPa;µ=0 for fluids) Axial modulus
U= Longitudinal stressF/A Longitudinal strainL/L =σ
ε (in the case with no lateral strain)
Relationships between Young’s modulus (E), Poisson’s ratio (σ), and the two Lam´e constants (µandλ)
E = µ(3λ+2µ)
(λ+µ) σ= λ
2(λ+µ) k= 3λ+2µ 3
and λ= Eσ
(1+σ)(1−2σ)
Poisson’s ratio ranges from 0.05 (very hard rocks) to 0.45 (for loose sediments). Elastic constants for rocks can be found in handbooks of physical constants.
Material particles oscillate about fixed points in the direction of wave propagation (Figure 4.2A) by compressional and dilatational strain, exactly like a sound wave. The second type of wave is the S- wave, also known as the transverse, secondary or shear wave. Particle motion is at right-angles to the direction of wave propagation and occurs by pure shear strain (Figure 4.2B). When particle motion is confined to one plane only, the S-wave is said to be plane-polarised.
The identification and use of polarised shear waves in their vertical and horizontally polarised modes (SV and SH respectively) are becoming increasingly important in exploration seismology, as will be discussed later.
All the frequencies contained within body waves travel through a given material at the same velocity, subject to the consistency of the elastic moduli and density of the medium through which the waves are propagating.
Compressions
Dilatations (A) P-wave
(B) S-wave
Undisturbed medium
Figure 4.2 Elastic deformations and ground particle motions associated with the passage of body waves. (A) A P-wave, and (B) an S-wave. From Bolt (1982), by permission.
4.2.2.2 Surface waves
Waves that do not penetrate deep into subsurface media are known as surface waves, of which there are two types, Rayleigh and Love waves. Rayleigh waves travel along the free surface of the Earth with amplitudes that decrease exponentially with depth. Particle motion is in a retrograde elliptical sense in a vertical plane with respect to the surface (Figure 4.3A) and, as shear is involved, Rayleigh waves can travel only through a solid medium. Love waves occur only where a medium with a low S-wave velocity overlies a layer with a higher S-wave velocity. Particle motion is at right-angles to the direction of wave propagation but parallel to the surface. These are thus polarised shear waves (Figure 4.3B).
Surface waves have the characteristic that their waveform changes as they travel because different frequency components propagate (A)
(B)
Figure 4.3 Elastic deformations and ground particle motions with the passage of surface waves. (A) A Rayleigh wave, and (B) a Love wave. From Bolt (1982), by permission.
at different rates, a phenomenon known as wave dispersion. The dispersion patterns are indicative of the velocity structure through which the waves travel, and thus surface waves generated by earthquakes can be used in the study of the lithosphere and as- thenosphere. Surface wave dispersion has been described in de- tail by Grant and West (1965, pp. 95–107), by Gubbins (1990, pp. 69–80) and by Sheriff and Geldart (1982, p. 51). Body waves are non-dispersive. In exploration seismology, Rayleigh waves manifest themselves normally as large-amplitude, low-frequency waves called ground roll which can mask reflections on a seis- mic record and are thus considered to be noise. Seismic sur- veys can be conducted in such a way as to minimise ground roll, which can be further reduced by filtering during later data processing.
4.2.3 Seismic wave velocities
The rates at which seismic waves propagate through elastic media are dictated by the elastic moduli and densities of the rocks through which they pass (Box 4.2). As a broad generalisation, velocities increase with increasing density. Examples of P-wave velocities for a range of geological materials are listed in Table 4.2. The seismic wave velocities in sedimentary rocks in particular increase both with depth of burial and age (cf. Chapter 2, Section 2.2.4.1; see Box 4.3).
From the last equation (asterisked) in Box 4.2, it is clear that Poisson’s ratio has a maximum value of 0.5 (at which value the denominator becomes zero). When Poisson’s ratio equals 0.33,
Box 4.2 Seismic wave propagation velocity
Velocity of propagation V through an elastic material is:
V=(Appropriate elastic modulus/densityρ)1/2 Velocity of P-waves is:
VP=
k+4µ/3 ρ
1/2
Velocity of S-waves is:
VS=(µ/ρ)1/2.
The ratio VP/VSis defined in terms of Poisson’s ratio (σ) and is given by:
VP
VS
=(1−σ)1/2 (1/2−σ)(∗)
Note thatµ=0 for a fluid, as fluids cannot support shear, and the maximum value of Poisson’s ratio is 0.5;σ≈0.05 for very hard rocks,≈0.45 for loose, unconsolidated sediments, average
≈0.25.
Table 4.2 Examples of P-wave velocities.
Material VP(m/s)
Air 330
Water 1450–1530
Petroleum 1300–1400
Loess 300–600
Soil 100–500
Snow 350–3000
Solid glacier ice∗ 3000–4000
Sand (loose) 200–2000
Sand (dry, loose) 200–1000
Sand (water saturated, loose) 1500–2000
Glacial moraine 1500–2700
Sand and gravel (near surface) 400–2300
Sand and gravel (at 2 km depth) 3000–3500
Clay 1000–2500
Estuarine muds/clay 300–1800
Floodplain alluvium 1800–2200
Pemafrost (Quaternary sediments) 1500–4900
Sandstone 1400–4500
Limestone (soft) 1700–4200
Limestone (hard) 2800–7000
Dolomites 2500–6500
Anhydrite 3500–5500
Rock salt 4000–5500
Gypsum 2000–3500
Shales 2000–4100
Granites 4600–6200
Basalts 5500–6500
Gabbro 6400–7000
Peridotite 7800–8400
Serpentinite 5500–6500
Gneiss 3500–7600
Marbles 3780–7000
Sulphide ores 3950–6700
Pulverised fuel ash 600–1000
Made ground (rubble, etc) 160–600
Landfill refuse 400–750
Concrete 3000–3500
Disturbed soil 180–335
Clay landfill cap (compacted) 355–380
∗Strongly temperature-dependent for polar ice (Kohnen, 1974).
S-wave velocities are half P-wave velocities. Of the surface waves, Love waves travel at approximately the same speed as S-waves, as they are polarised S-waves, but Rayleigh waves travel slightly slower at about 0.92VS(for Poisson’s ratio=0.25).
Care has to be taken in comparing seismic velocities. Velocities can be determined from seismic data (see Sections 6.3.4 and 6.4.1 as to how this is done) and from laboratory analysis. When veloc- ities have been determined using seismic refraction, the range of velocities obtained for a given type of material should be cited, and preferably with the standard deviation. In situ measurements made using refraction studies may yield velocities that are significantly different from those obtained from laboratory measurements. This occurs when the in situ rock is heavily jointed or fractured. The
Box 4.3 Elastic wave velocity as a function of geological age and depth (after Faust, 1951)
For shales and sands, the elastic wave velocity V is given by:
V=1.47(Z T)1/6km/s
where Z is the depth (km) and T the geological age in millions of years.
refraction velocities sample the rock and the discontinuities, whereas a laboratory measurement on a solid sample cannot, by virtue of the scale of the sample. Detailed measurements which may yield more representative in situ velocities on a fine scale are those obtained using a Schmidt hammer system. Two geophones are attached to the exposed material at a small distance apart. A hammer is used to generate a P-wave directly on to the rock at a known distance from the receivers. The velocity is obtained from the difference in travel time between the two receivers relative to their separation. No one velocity is absolute. In the case of labo- ratory measurements, ultrasonic transducers are used to transmit a pulse through a sample of the material in question. From the measured travel time of this pulse through the material, whose length is known, a velocity can be calculated. Ultrasonic frequen- cies (0.5–1.5 MHz) are three to four orders of magnitude higher than typical seismic frequencies, and so the velocities may not be directly comparable.
In addition to knowing the frequency of the transducers, it is also important to determine whether the samples have been mea- sured dry or fully saturated, and if the latter, the salinity of the fluid used and the temperature at which the sample was measured. Of greater significance is the problem over mechanical relaxation of re- trieved core pieces. If a sample of rock is obtained from significant depth below ground where it is normally at substantial pressure, the rock specimen expands on its return to the surface, resulting in the formation of microcracks due to the relaxation of confin- ing pressure. These microcracks thereby increase the porosity and decrease the density of the rock. In the case of saturated samples re- trieved from below the seabed, gas bubbles form within the sample as the pressure is reduced and these can reduce the acoustic velocity significantly compared with the in situ velocity. This may be no- ticeable even if samples are retrieved from less than 10 m of water depth.
In porous rocks, the nature of the material within the pores strongly influences the elastic wave velocity; water-saturated rocks have different elastic wave velocities compared with gas-saturated rocks; sandstones with interstitial clay have different propagation characteristics compared with clean sandstones, for example. Seis- mic velocities can be used to estimate porosity using the time- average equation in Box 4.4. If the P-wave velocities of both the pore fluid and the rock matrix are known, the porosity can be deduced.
The form of this equation applies both for water-saturated and frozen rocks. In the case of permafrost, the velocity depends upon (a) the type of geological material, (b) the proportion of interstitial
Box 4.4 Time-average equation to estimate rock porosity
The P-wave velocity V for a rock with fractional porosity (ϕ) is given by:
1 V = ϕ
Vf +1−ϕ Vm
where Vfand Vm are the acoustic velocities in the pore fluid and the rock matrix respectively (Wyllie et al., 1958).
Typical values: Vf=1500 m/s, Vm=2800 m/s.
ice, and (c) the temperature. The importance of seismic velocities in interpretation is discussed in Chapter 6. The P-wave velocity in water is dependent upon temperature and salinity (Box 4.5), but is normally considered to be around 1500 m/s for a salinity of 35 parts per thousand at 13◦C. In high-resolution surveys conducted where water masses with different temperatures and salinities occur, such as at the mouth of a river where fresh river water flows into and over salt water, the stratigraphy of the water column can become important in determining the correct P-wave velocities for use in subsequent data processing.
In stratified media, seismic velocities exhibit anisotropy. Veloc- ities may be up to 10–15% higher for waves propagating parallel to strata than at right-angles to them. Furthermore, some ma- terials with strongly developed mineral fabrics can also demon- strate anisotropy, as for example in glacier ice and in highly foliated metamorphic rocks. In high-resolution surveys over sediments with marked anisotropy, significant differences in seismic character and data quality may be observed. In situations where such anisotropy is anticipated, it is essential to run test lines orientated at different azimuths in order to identify directions of shooting which either degrade the data quality or provide good resolution and depth pen- etration. For example, at the mouth of a fast-flowing river, lines parallel and at right-angles to water flow would be good directions to try. Once tests have been undertaken, an appropriate survey grid can then be established to ensure that the best-quality data are acquired. Towing the hydrophone streamer against the water flow produces better control and signal-to-noise ratios than across the water currents. Cross-track currents can also lead to excessive feathering of the source–receiver system (see sections 4.6.2 and 6.3.2.2).
Box 4.5 P-wave velocity as a function of temperature and salinity in water
V=1449.2+4.6T−0.055T2+0.0003T3 +(1.34−0.01T)(S−35)+0.016d
where S and T are the salinity (parts per thousand) and the temperature (◦C); d is depth (m) (Ewing et al., 1948; cf. Fofonoff and Millard, 1983).