4. Seismic Waves
Introduction
Seismology is the study of the passage of elastic waves (see below) through the earth. Earthquake seismology is the best tool to study the interior of the earth.
When an earthquake or explosion occurs, part of the energy released is as elastic waves that are transmitted through the earth.
The waves are then detected and recorded by seismograms, which measure, amplify and record the motion of the ground.
The information is then used to determine earthquake locations, the subsurface structures and etc.
This pendulum-mounted seismograph records horizontal motion. The mass is coupled to the Earth by means of a pendulum and a pivot is attached to a rod to constrain the mass to move in the horizontal direction only.
The spring-mounted seismograph records the vertical ground motion. A spring is attached to the mass which is connected to a rod. The rod is attached to a pivot to constrain the mass to move in an up and down direction only.
Basic Physics
There is some basic terminology and physics that describe the various aspects of wave form and motion.
The wavelength (λ) is the distance between two adjacent points on the wave that have similar displacements, one wavelength is the distance between successive crest.
Amplitude (A) of the wave is the maximum displacement of the particle motions, or the height of the ripple crest.
Period (T) is the time it takes for two successive waves to pass a reference point or the motion to complete one cycle.
The cycle of seismic waves or repetitions in a given unit of time is called frequency (f). Frequency and period are related by this relationship:
f = 1 / T [unit: hertz (Hz) or 1/s]
The speed in which the wavefront (or ripple crest) travel can be detected if the time the wavefront takes to reach a known distance is recorded:
V = distance / time [unit: m/s]
Or if wavelength and frequency are known:
V= f λ
Elastic Moduli
Elasticity is the behavior of a material that when subjected to a stress (force/area), deforms and changes shape (strain), but returns to original shape when the stress is removed.
The manner and speed of seismic waves travel through material is controlled by their elastic properties.
The linear relationship between applied stress, σ, and resulting strain ε is:
σ = Eε
E is the constant of proportionality called an elastic modulus.
We are concerned with two types of deformation – uniform compression or expansion, and shear deformation:
The original volume (V0) change to final volume (VF) when compared to the pressure change is called bulk modulus (K). The bulk modulus is a measure of the incompressibility of the material:
K = V0(P-P0)/(V0-VF)
When deforming a solid state by simple shear, a shear strain (γ) is induced by applying a shear stress, σ. The ratio of these quantities is the rigidity modulus (μ):
μ = σ/ γ
Units of elastic modulus are the same as pressure – i.e. MPa or GPa.
Seismic Waves
Two different types wave produced by an earthquake: body waves and surface waves.
Body Waves
· Body waves are seismic waves that travel through the body of the earth.
· Body waves are reflected and transmitted at interfaces where seismic velocity and/or density change, and they obey Snell's law.
The two different types of body waves are:
· P-Waves (P = primary or comPressional). They are longitudinal waves involve compression and rarefaction of the material as the wave passes through, but not rotation. P-waves are transmitted by particle movement back and forth along the direction of propagation of the wave.
· P-waves have the greatest speed and appear first on seismograms.
· S-Waves (S = secondary or shear). Also known as transverse waves, because particle motions are transverse to the direction of movement of the wavefront, or perpendicular to the ray. These waves involve shearing and rotation of the material as the wave passes through it, but not volume change.
· VS < VP and appear on seismograms after P-waves.
Surface Waves
· Surface waves are seismic waves that are guided along the surface of the Earth and the layers near the surface.
· These waves do not penetrate the deep interior of the earth, and are normally generated by shallow earthquakes (not nuclear explosions).
· Surface waves have larger amplitude and longer duration than body waves.
· These waves arrive at seismograph after the arrival of P- and S-waves because of their slower velocities. The two different surface waves are:
· Rayleigh waves or descriptively called "ground roll" in exploration seismology. The particle motion of this wave is confined to a vertical plane containing the direction of propagation and retrogrades elliptically. The particle displacements are greatest at the surface and decrease exponentially downward. Rayleigh waves show dispersion, and its velocity is not constant but varies with wavelength. cf. how ocean waves propagate.
· VR < VS
· Period is typically ~ 20 s, with wavelength or ~ 100km
· Love waves (named for A.E.H. Love, who discovered them) travel by a transverse motion of particles parallel to the ground surface. cf S-waves.
· Love waves cannot exist in a uniform solid, and can only occur when there is a general increase of S- wave velocity with depth.
· Their existence is another proof of the Earth’s vertical inhomogeneity.
· The particle motion is transverse and horizontal.
· Generally, Love wave velocities are greater than Rayleigh waves, so Love waves arrive before Rayleigh waves on seismograph.
Seismic Wave Velocities
· The velocities of P- and S-waves are given below in terms of the density (ρ) and elastic coefficients of a material:
Vp = √((K+4/3μ)/ρ)
Vs =√(μ/ρ)
· If we note that the bulk modulus (K) and the rigidity modulus (μ) are always positive, then evidently the VP always > VS.
· Shear waves (S-waves) cannot propagate through liquid. This is evident when we substitute G = 0 for liquids, then the velocity of S-waves goes to zero.
· This is how it was determined that the outer core consists of liquid.
· Some times you will come across the bulk sound velocity:
VΦ =√(K/ρ) = √(Vp2 -4/3Vs2)
· Also, Vp and Vs are related via Poisson’s ratio (ν).
· When a rod is stretched it becomes longer but narrower, the ratio to the lateral to longitudinal strain is Poisson’s ratio.
· The ratio of Vp to Vs is given by:
Vp/Vs = [2(1-ν)/(1-2ν)]1/2
· For most rocks, ν ~ 0.25, so Vp ~ 1.7 Vs.
There are a few more general rules to the velocity ranges of common materials:
o Unsaturated sediments have lower values than saturated sediments.
o Unconsolidated sediments have lower values than consolidated sediments.
o Velocities are very similar in saturated, unconsolidated sediments.
o Weathered rocks have lower values than similar rocks that are unweathered.
o Fractured rocks have lower values than similar rocks that are unfractured.
Below is a list of velocity estimation of common waves:
For rocks can plot V v. density:
More generally, Birch observed a general relationship between density and seismic wave velocity which helps us establish the composition of the Earth:
Seismic Ray Theory
· When seismic rays travel through the Earth, they encounter changes in K, μ and ρ. This causes the rays to be reflected and refracted.
· Seismic energy travels in an approximately spherical way.
· The energy at the wave front gets weaker as it moves from its source. This geometrical spreading of the energy causes the amplitude to drop. The energy drops off as x-2, and the amplitude as x-1 – this is called attenuation.
Layered Media - Normal Incidence
· When a ray hits an interface with normal incidence (right angles to boundary), some energy is reflected, while the rest is passed into the lower boundary.
· The reflection coefficient (R) is the ratio of the reflected ray amplitude to the incident ray amplitude:
R= A1/A0
· Energy is prop. to square of amplitude; R=0, all energy transmitted, no reflection; R=-1 or +1, all energy reflected: For normal incidence R is given by:
R= (Z2-Z1)/(Z2+Z1)
Where Z is the acoustic impedance, a material dependent parameter given by
Z = ρV
So Z will determine how much of the wave energy is reflected. Maximum reflection requires greatest difference in Z between the layered media. If Z1=Z2, R=0 and all transmitted. The characteristic impedance of air at room temperature is about 420 Pa s/m. By comparison that of water is about 3400 times higher. atmospheric acoustics vs underwater acoustics.
· The transmission coefficient (T) is: T = A2/A0 = 2Z1/Z2+Z1
Layered media - Oblique Incidence
· When a P-wave is obliquely incident, there is a reflected P wave, and the transmitted ray is refracted in accord with Snell’s Law.
· In addition, some of the compressional energy is converted into shear energy, and a reflected and refracted S-wave is generated too.
Refraction & Reflection Seismology
Reflection seismology is used mostly in exploration methods, while refraction seismology is use more in whole Earth studies.
Refraction Seismology
o Studying subsurface velocity and layer interface structure by analysing the first arrival times of P-waves at the surface of the earth.
o Applications of subsurface imaging include:
1. locating buried archeological sites,
2. assessing subsurface geological hazards,
3. defining aquifer geometry
4. exploring for fossil fuel and other natural resources.
Seismic P-Wave Behavior
· When a ray encounters an inhomogeneity in its travels, eg rock-type 1 to rock-type 2, the incident ray transforms into several new rays. A reflected wave enters and exits at the same angle measured to the normal of the boundary - angle of incidence equals angle of reflection.
NB schematic – figure not correct
o ray path is dependent on the wave velocities through different layers.
o For refraction seismology, the critical angle is the most important angle value to understand. If r = 90o, the refracted wave propagates along the boundary interface.
o If r = 90, then sin(r) = 1, and the critical angle (ic) is given by: ic = sin-1(V1/V2)
o As the critically refracted wave propagates along the boundary, according to Huygen's Theory of Wavelets, the primary critically refracted wave acts as a source for new secondary wave fronts and ray paths.
o These secondary ray paths exit back to the surface at the critical angle.
Simple Refraction Model
o Two Horizontal Layers - In the ideal world (of engineering), refraction seismology is most easily understood through a horizontal two layer model.
Seismic waves are generated from a source (sledge hammer, explosion, air gun….).
o Geophone receivers record seismic signals received along the survey profile.
o Since P-waves travel at the fastest speeds, the first seismic signal received by a geophone represents the P-wave arrival.
o Five P-waves are of interest in refraction seismology:
o direct
o diving
o reflected
o head
o refracted
o The direct wave propagates along the atmosphere-upper layer (called layer1) boundary.
o A transmitted wave through lower layer (layer 2) is termed a diving wave.
o A reflected wave enters with the same angle of incidence as exit angle.
o If the incident wave hits at the critical angle, the critically refracted head wave travels along the layer 1-layer 2 interface.
o Refracted waves propagate from the interface as the head wave progresses, with exit angles equal to the critical angle.
o With arrival time data collected, arrival times for P-waves are noted or computed from the seismographs.
o Arrival times can be represented on a travel-time graph or T-X plot, that is P-wave arrival times (usually in milliseconds) verses distance (geophone location).
o This plot shows that at small distances (x) from the source, the direct wave arrives first.
o At distances up to the critical distance only the direct ray, and weakly (sub-critically) reflected rays arrive at the geophone. The reflected rays are always later than the direct ray.
o At the critical distance, direct waves and the first refracted ray arrives. Its amplitude is stronger than the reflected ray, but is still later than the direct ray.
o At some distance (the cross over distance), the refracted ray arrives first, since it has traveled at V2 for long enough in the interface so as to catch up the direct ray.
o From the travel-time curve we can calculate:
o velocities of P-wave propagation through layers 1 and 2 (V1 and V2)
o thickness of layer 1 (H1).
o To obtain these values, combination of equations and interpretation from the T-X plot is required.
o The travel time of the direct wave is given by:
t DIRECT = x/V1
o So V1 can be obtained from the slope of the direct arrivals, which passes through the origin.
o The travel time for a reflected ray is given by:
tREFLECTION = (x2 + 4H12)1/2/V1
o This is the equation for a hyperbola, where H1 is the layer thickness.
o The travel time for the refracted wave is given by:
tREFACTED = x/V2 + 2H1(V22 –V12)1/2/(V1V2)
o See detailed notes and Fowler for full derivations.
o The equation for t REFRACTED is that of a straight line ( y = mx + c). The slope gives 1/V2 and the intercept on the t axis (i.e. when x=0) enables H1 to be determined from:
H1 = t(x=0)(V1V2)/2(V22 –V12)1/2
Two Layer Dipping Model
o When discussing dipping layers, one wants to quantify the amount of dip. For a simple case of two dipping layers, seismic refraction can be used to calculated dip of the layers.
o For a given survey profile, sources must be located at the beginning of the profile (forward shot) and at the end of the profile (reverse shot).
o P-wave arrival times for both forward and reverse shots can be plotted on a T-X plot.