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
o
o
o
o
Unsaturated sediments have lower values than saturated sediments.
Unconsolidated sediments have lower values than consolidated
sediments.
Velocities are very similar in saturated, unconsolidated sediments.
Weathered rocks have lower values than similar rocks that are
unweathered.
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 rocktype 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
o
o
o
o
The direct wave propagates along the atmosphere-upper layer (called
layer1) boundary.
A transmitted wave through lower layer (layer 2) is termed a diving wave.
A reflected wave enters with the same angle of incidence as exit angle.
If the incident wave hits at the critical angle, the critically refracted head wave
travels along the layer 1-layer 2 interface.
Refracted waves propagate from the interface as the head wave progresses,
with exit angles equal to the critical angle.
o
o
With arrival time data collected, arrival times for P-waves are noted or
computed from the seismographs.
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
o
o
o
o
This plot shows that at small distances (x) from the source, the direct wave
arrives first.
At distances up to the critical distance only the direct ray, and weakly (subcritically) reflected rays arrive at the geophone. The reflected rays are always
later than the direct ray.
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.
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.
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
o
To obtain these values, combination of equations and interpretation from the
T-X plot is required.
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
o
See detailed notes and Fowler for full derivations.
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.
o
From the Principle of Reciprocity, time required for a ray to travel along the
forward and reverse shot should be the same, since the ray pathways are the
same.
o
From the T-X plot, V1 and V2 velocities for forward and reverse shots can be
calculated, as well as the time-intercepts for forward and reverse refracted
waves.
Kearey & Brooks (1984) show how this geometry can be analyzed to get h, θ,
etc.
Horizontal Multi-Layer Model
o
Why only stop with interpretation of two horizontal layers?
NB figure below not exactly correct
o
Calculation of layer velocities and thicknesses for multi-layers requires patience
with many equations chock full of algebra and trigonometry.
Each layer yields an interpolated refracted wave slowness, and time intercept
used to calculate layer thickness.
o
o
This approach leads to understanding why seismic rays are reflected back to the
surface on Earth as V increases generally with depth:
Problems and Limitations
o
The preceding models assume planar boundary interfaces. Conformable
sequences of sedimentary rock may form planar boundaries. However, erosion
and uplift easily produce irregular boundary contacts. More sophisticated
algorithms can process refraction surveys where irregular interfaces might be
expected.
o
Profile length and source energy limit the depth penetration of the refraction
method. Typically, a profile can only detect features at a depth of one-fifth
survey length.
Thus, refraction imaging of the Moho would require profile lengths of over one
hundred kilometers; an difficult experiment.
o
Larger sources could be utilized for greater depth detection, but certain sources
(e.g. explosives) may cause problems in urban areas.
o
Refraction depends on layers to increase in velocity with depth. In the hidden
slow layer senario, a buried layer is overlain by a faster layer. No critical
refraction will occur along the boundary interface.
Thus, refraction will not easily detect the slow layer. All is not lost
since reflection seismology could detect the slower layer.
o
Seismograms require careful analysis to pick first arrival times for layers. If a
thin layer produces first arrivals which cannot easily be identified on a
seismogram, the layer may never be identified. Thus, another layer may be
misinterpreted as incorporating the hidden layer. As a result, layer thicknesses
may increase.
Reflection Seismology
Reflection
seismology
began
to
take
prominence in the 1920s to begin to locate salt
domes, an indication where oil would be found.
The reflection method soon replaced the
refraction after it was proved with numerous
successes, the most visible in the petroleum
industry.
Lets Start With a Single Subsurface Interface
The key is to develop an equation which represents the time it takes for a particular ray
to travel through this single layer. First, the seismic velocity through the layer of material
that the wave is propagating needs to be lower than the layer directly below, which we
will assume is infinitely thick.
Therefore, just by simple time-velocity relation and geometry as before:
This can be re-written (dropping the subscripts) as:
V2t2 = x2 +4h2
so
V2t2/4h2 – x2/4h2 = 1
which has a hyperbolic form:
Now, What Does That Arrival Time Mean Anyway?
Well, the first thing to note is what you can do with the hyperbola.
A hyperbola has an asymptote along which the hyperbola approaches. The
equation of this line is
Therefore, the asymptote for the travel time curve has a slope of the reciprocal of
the velocity (as before).
Another approach to analysing the data is to get velocity and thickness from a
plot of x2 v t2. Now recall:
By squaring both sides, the equation resembles closely the equation of a straight
line.
The slope of the line is the reciprocial of the square of the velocity. The intercepts
gives h via:
Exploration Seismology
Introduction
Reflection seismology is a technique for imaging the geological structure
beneath the earth's surface using sound energy. The technique is used
primarily for oil exploration. An acoustic energy source at the surface
transmits an acoustic signal into the earth, which reflects some of the
energy back toward the surface at each geological interface. An array of
geophones detects the faint signals reflected back to the surface, which
are recorded for later processing. The raw data is very noisy and
uninterpretable, requiring extensive processing to produce an image of
the earth's interior.
Survey ship trails an acoustic source (usually
air 'guns') and a string of hydrophones,
called a streamer.
The streamer is usually
hydrophones every 15m.
~4000m
with
When the air guns are fired, pressure pulse
radiates with ~spherical wavefront through
the water and into the earth. The raypaths
are perpendicular to the wavefronts.
When the wavefront reaches a reflecting
geological boundary, some reflected.
This echoed acoustic energy is sensed
by the hydrophones and recorded on
the ship for later processing.
Figure 1. Marine Seismic Data Acquisition.



Reveals layering, structural features, folding, faulting
Can also be done on land but more difficult
Results in two-way-travel times vs. horizontal distance
Imaging Problems and Solutions.
a) Noise -- the reflection energy is usually small after travelling a large
distance and bouncing off a weak reflector. Spurious noise in the earth,
air and recording electronics can swamp the reflection signal.
b) Multiples -- the raypaths not only travelled from source to receiver with
one bounce off a reflector, but also followed paths making several
intermediate bounces between reflectors and producing a travel time out
of proportion to the reflector depth. Events on the image associated with
raypaths making multiple bounces are called 'multiples' and should be
removed from the image.
c) Source Pulse Shape -- the source pulse may not be sharp enough to
produce a high resolution image and may vary in shape from shot to shot.
(The activation of the source to produce a pulse is termed a 'shot'.)
d) Positioning of Dipping Reflectors -- the acoustic image is produced
by displaying the trace at each record location vertically on the image. If a
reflector is dipping, the raypath reflection point does not lie vertically
below the record location, but is offset to one side. Further processing is
required to correctly position the acoustic image.
See detailed notes and books for more....