Geometrics.com

Seismic Theory

Q. What is a seismic wave?

A. A seismic wave is the transfer of energy through elastic earth materials by way of particle oscillation/vibration.

Q. What are the different types of seismic waves?

A. There are four main types of seismic waves, each characterized by its specific particle motion:

  • Compressional or “p” waves are identical to sound waves – the particle motion is parallel to the propagation direction:
p-wave animation by L.R. Braille, Purdue University

p-wave animation by L.W. Braille, Purdue University, http://web.ics.purdue.edu/~braile/edumod/waves/Pwave.htm

  • Shear or “s” waves are characterized by particle motion that is perpendicular to the propagation direction:
s-wave animation by L.R. Braille, Purdue University

s-wave animation by L.W. Braille, Purdue University, http://web.ics.purdue.edu/~braile/edumod/waves/Swave.htm

Taken collectively, p- and s-waves are known as “body” waves. The velocities of both can be measured via seismic refraction.

  • Surface waves, as the name implies, travel primarily along the ground surface; amplitudes decrease rapidly with depth. There are two types of surface waves. Like body waves, they are characterized by particle motion.

"Rayleigh" waves are characterized by elliptical motion perpendicular to the surface:

r-wave animation by L.R. Braille, Purdue University

Raleigh-wave animation by L.W. Braille, Purdue University, http://web.ics.purdue.edu/~braile/edumod/waves/Rwave.htm

In the near surface, this motion is “retrograde”, meaning that it is counter-clockwise when the propagation is left-to-right. At depth, the motion can reverse to prograde.

Particles affected by "Love" waves vibrate perpendicular to the propagation direction:

l-wave animation by L.R. Braille, Purdue University

Love-wave animation by L.W. Braille, Purdue University, http://web.ics.purdue.edu/~braile/edumod/waves/Lwave.htm

While the particle motion is similar to that of shear waves, Love wave amplitude is much higher and decreases rapidly with depth. Love waves are the most destructive waves in earthquakes because of their high amplitude and transverse particle motion.

While the various wave types shown above have been isolated for illustration purposes, all are present to some degree whenever seismic energy is travelling through a solid medium. Hence actual particle motion is extremely complex.

Q. How fast do seismic waves travel and what controls this?

A. Different types of seismic waves travel at different velocities through any given material. In addition, different materials have different seismic properties, meaning that any one wave type can have a wide range of velocities, depending on the material properties. For instance, the p-wave velocity of shale can range from 800-3,700 m/s. Granite can range from 4,800-6,700 m/s. Because of this, by themselves, seismic velocities alone are not particularly diagnostic with regard to rock type.

Ultimately, seismic velocity depends on the density and elastic properties of the material, whatever its composition. Specifically,

  • Compressional-wave velocity depends on the “incompressibility” of the material, as embodied in the bulk modulus. The higher the bulk modulus, the less compressible the material, and the higher the p-wave velocity. Sound travels through water about four times faster than it does through air. Similarly, shear-wave velocity depends on the rigidity of the material, or the resistance to shear. The higher the shear modulus, the higher the s-wave velocity. Mathematically,

where

K = bulk modulus

µ = shear modulus

ρ = density

Note that Vp depends on both the bulk and shear modulus, while Vs depends only on the shear modulus. This observation implies two things:

  1. Shear waves always travel slower than compressional waves through a given material, and
  2. Materials with zero rigidity – i.e., fluids – do not carry shear waves at all. Therefore, the absence or presence of groundwater has no effect on the shear wave velocity. It is interesting to note that, in general, seismic velocity increases with density – denser rocks tend to be much harder and faster. Yet in the above equations, density is in the denominator. This is known as the “velocity-density paradox”, the answer to which can be found in the fact that the elastic moduli tend to increase with density as well, and at a faster rate.

Q. What is a [seismic] wave front?

A. A seismic wave front emanates from an energy source like ripples on a pond, but in three dimensions. It is the surface connecting points of equal travel time from the source. In a homogeneous medium, a surface drawn through points of equal travel time is spherical, as depicted in the animation below.



The seismic trace, which is what is recorded by the seismograph, represents particle motion vs. time. In a homogeneous medium, the wave front can also be described as a surface of constant phase.



Q. What is a [seismic] ray?

A. A seismic ray, or “wave front normal”, is an arrow drawn perpendicular to the seismic wave front to indicate the propagation direction at that point on the wave front. It is a convenient tool to help understand wave propagation through layered media; it is not something that exists in a physical sense.


Q. What is Huygen's Principle?

A. Huygen's Principle can be stated in many ways. The simplest definition states that every point on a wave front can be thought of as a new point source for seismic waves.




With a little thought, this may seem obvious, but this realization is extremely useful for understanding the concept of refraction. This, in turn, is essential to becoming a good seismic practitioner.


Q. What is the Reflection Coefficient?

A. The Reflection Coefficient R between two velocity layers is expressed as

R = (ρ2V2 - ρ1V1) / (ρ2V2 + ρ1V1),


Where ρ = density and V = velocity. The quantity ρV is the seismic impedance of the material. The Reflection Coefficient is therefore the difference in seismic impedance over the sum of seismic impedance of two materials. From the above equation, it is apparent that R will be a positive number when V2 > V1, and a negative number when V2 < V1. A positive R means that the polarity of the reflected wave will be the same as that of the incident wave. A negative R means that the polarity of the reflected wave will be the opposite of the incident wave.

It should also be apparent that the larger the contrast in seismic impedance, the larger the amount of incident energy that is reflected (and the smaller the amount that is transmitted).

The above assumes normal incidence. For incident angles other than 90o, the equation is more complex.

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