Fundamental physics for seismology

If we strike the earth with a hammer, an explosive charge, or an earthquake, the material of the earth near the source becomes compressed (or extended) and twisted. Earth materials are elastic and these deformations propagate away from the source. The speed of propagation and the type of motion propagated depends upon the elastic properties of the material, namely

1. density: 
2. bulk modulus:  (compressibility)
3. shear modulus:  (twistability)

By knowing the time at which the source occurred, and by measuring the travel time that it takes for the various waves to arrive at a receiver, it is possible to obtain information about the velocity and density of the material. This information is then used to provide information about structure and rock type.

Applications of seismic surveying

1. delineation of sedimentary layers
2. determination of depth to bedrock, and topography of the bedrock horizon
3. locating faults
4. large scale structure of earth (using earthquakes)
5. characterization of near surface material properties.

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Stress

Stress can be normal to the surface it is applied on, and hence compressive or tensile (top figures).
Stress can be parallel to the surface. This is shear stress (bottom figure).	shear

Strain

If a body is stressed it will undergo a change in size and shape. The body is strained. If the strains are small then the body will recover its original shape when the stress is removed. This is elastic strain. If the strain is too great so that the yield strength is exceeded, then the body deforms plastically and may ultimately fracture. Plastic strain is not recoverable.

The figure plots how strain (horizontal axis) accumulates as stress (vertical axis) is applied. While behaviour is elastic, increasing or decreasing the stress does nothing to change the material. Once stress is sufficiant to make material behave platically, reducing the stress rsults in reduced strain along a different path on the graph. Once the fracture point is reached, the strain is released by breaking.

Elastic Moduli

The linear relationship between stress and strain is known as Hooke's Law and is specified by five elastic moduli or elastic constants which express the ratio of a particular stress to a resultant strain. Four important constants are explained briefly next. Elastic constants are defined in Wikedia, and in the Schlumberger "Oilfield Glossary".

  Young's Modulus

Consider a rod (figure right) of length l, cross-sectional area A, to which a uniform force F is applied to each end (a tensional force in this case): Young's modulus, E, is given by .

  Poisson's Ratio

  Bulk modulus

Consider an initial volume V which is subjected to a hydrostatic pressure (force per unit area is the same in each direction). Let V be the change in volume.  The bulk modulus is .

  Shear modulus

If we apply a force parallel to a surface the force per unit area is known as the shear stress . Consider a rectangular cube. Application of shear stresses to the top and bottom interfaces produce a strain proportional to tan(Q). The shear modulus is

Elastic model for the Earth rocks

Earth materials are elastic. The easiest way to obtain intuition is to think about rocks as being made up of a set of connected springs. Apply a force (stress) to any part and you will eventually get motion elsewhere. There are two general types of waves that can travel in the elastic material. Body waves which travel through materials Surface waves which travel only along boundaries between materials such as the ground / air boundary. Specific types of waves within these general types are described below. Each wave type travels with a velocity that depends upon the elastic properties.

Body Waves

Compressional waves (P waves) propagate by compression and rarefaction, and the velocity of such waves, Vp is given in terms of elastic moduli introduced above and density , by the equation shown here.
Shear waves (S waves) propagate by a pure shear strain perpendicular to the wave propagation. There are two planes of polarization. SV and SH waves travel at the same speed, Vs, which is given in terms of elastic moduli introduced above and density , by the equation to the right .

Note that if  = 0 then V_s = 0. This tells us that shear waves do not travel in a liquid.

Seismic reflection and refraction surveying is usually carried out by observing how P-waves travel, reflect, and refract within the geologic layers near the surface. We will see that the energy is generated by an artificial source. S-waves can also be used, but they are a little harder to generate artificially (compared to P-waves), and require special ground motion sensors. However, since Vs < Vp, there are situations where it is beneficial to use S-waves instead of P-waves.

Surface waves

Rayleigh waves propagate along a free surface or on the boundary between two materials. Particle motion is a retrograde ellipse, and in the same plane as wave energy propagation. The amplitude of particle motion decays exponentially with depth. Rayleigh wave speed VR < VS. Large earthquakes can generate Rayleigh waves that circumnavigate the globe. This provides information about the velocity structure in the upper few hundred kilometers of the earth.

Love waves exist in a surface layer when the shear wave velocity of the upper layer is less than the shear wave velocity of the lower layer. The waves are trapped in the upper layer and the particle motion is parallel to the free surface and perpendicular to the direction of propagation. VS1 < VLove < VS2

Both Love waves and Rayleigh waves are dispersive. That is, different frequency components travel at different speeds. So the wave changes shape as it travels. Also, the dispersion can be used to provide information about the velocity structure in the upper region of the earth. For shallow work, it is possible to generate surface waves artificially, and then observe the waves at a series of locations at increasing distances from the source. This type of field work is sometimes called multi-channel analysis of surface waves or MASW. This is usually considered an "advanced" topic in applied geophysics.

Waves and Rays

A wavefront indicates the locations at which the phase of the wave has the same value. For example, visualize the peaks (or troughs) of water ripples after a rock has been thrown in. The direction of propagation of the energy is normal to the wavefront. Seismic rays are imaginary lines perpendicular to the wavefront that indicate the path along which the wavefront is travelling. Rays are not physical entities. They exist only to illustrate where the energy travels. If you are connected to the internet, a good introduction, including Java animations, can be found within the notes for seismic surveying at the Colorado School of Mines on-line geophysics course. See the contents for there chapter, or start on the section on Wave Types.

Attenuation The amplitude of seismic waves falls off with distance from the source. There are two primary reasons:  Geometrical spreading - that is, energy falls off as 1/r2 and hence the amplitude falls of as 1/r. Earth materials are not perfectly elastic. Some frictional heating occurs as the waves propagate through the earth. This is often described as "absorption" and the absorption coefficient expresses the proportion of energy lost as the wave travels a distance of one wavelength. The figure here shows the progressive change of shape of an original spike pulse during its propagation through the ground due to the effects of absorption (After Anstey 1977.) The spike's shape changes as well as experiencing reduced amplited. This is because the different frequencies making up the pulse decay at different rates - in fact, higher frequencies decay more rapidely than lower frequencies. This is easily observed on earthquake signals that have been recorded at different locations. As noted above in the context of surface waves, such frequency dependent behaviour is called dispersion.