Wave theory
Because rocks can be considered as elastic bodies in the range of
small deformation, the elastic waves are able to travel through their
material.
The seismic waves are low-frequency elastic waves which travel
through the subsurface layers
The seismic methods are based on the fact that seismic waves
coming from a source are able to propagate through rocks.
Actually, the seismic wave propagation takes place in the form of a
spatial and temporal variation of the stress and strain fields inside the
rocks.
There is a very close interaction between the stress and deformation
fields. A change in one of them causes the change of the other, and
vice versa.
From the point of view of wave theory, the wave propagation is
considered as the spatial and temporal propagation of a perturbation
caused by a source.
Elastic waves
Elastic wave is the temporal and spatial propagation of a
disturbance in the stress-deformation state of an elastic
medium.
The source of the disturbance can be an impulsive event or a
periodic process which occurs at a point or a relatively small
volume of an elastic body.
Of course, any change in the stress field is always induced by a
change in the magnitude and/or direction of external forces
acting on the body.
The change in the stress field causes the displacement of
particles inside the body and the deformation pattern
corresponding to this change will propagate outward from the
location of source as an elastic wave.
Elastic waves
It is important to note that not the particles travel through the
medium but the change in the stress and deformation fields
during the propagation of an elastic wave.
The particles are oscillating about their equilibrium positions.
The wave generation can last a very short time like an impulse
or a longer time like a periodical variation.
In the case of an impulsive source, the elastic wave calms down
gradually as the time is passing and the distance is increasing
from the source.
A source operating periodically is able to retain the elastic wave
motion inside the medium for a longer time.
Elastic waves
There are two principal types of elastic waves:
body waves
and surface waves.
Body wave is a wave which travels three-dimensionally through an
elastic medium (inside the body).
Surface wave is a wave which propagates along and near by the
surface of an elastic medium.
http://www.parkseismic.com/Whatisseismicwave.html
Body waves
Two types of deformation pattern can propagate in the form of body
waves.
When contractions and expansions are periodically taking place
during the wave propagation, the particles are oscillating along axes
parallel to the direction of the wave propagation.
This type of body wave is called compressional wave or P-wave
(primary wave).
Compressional wave belongs to the group of longitudinal waves in
physics (similarly to the sound waves).
This type of wave motion entails both volume change and
deformation.
The stress field does not have a shear component during the wave
propagation.
This is the reason why a compressional wave can propagate not only
in solids but fluids.
Compressional waves
This figure illustrates the propagation
of compressional wave.
The square prisms symbolize the
same piece of an elastic medium at
different time moments.
The small cubes represent the
particles of the medium.
The time increases from up to down.
It can be seen how the compression
and dilatation of particles travel in
the direction of wave propagation
(along the Y-axis).
http://www.geo.mtu.edu/UPSeis/waves.html
Share waves
In the case of a shear wave or S-wave (secondary wave) periodic
alternations in the shear stress-deformation field produce the wave
motion.
Only the shear component of the stress field plays role in the shear
wave motion.
The particles are oscillating along axes perpendicular to the direction
of wave propagation.
So, this type of wave belongs to the group of transverse waves in
physics (similarly to the light waves).
The propagation of shear waves entails only deformation without any
volume change.
The shear wave cannot propagate in fluids because they are not able
to resist shear forces.
Shear waves
This figure illustrates the propagation
of shear wave in a similar way than it
was shown previously for the
compressional wave.
It can be seen that the motions of the
particles are perpendicular to the
direction of wave propagation.
Not the particles move along the
prism but the state of particles'
motion.
http://www.geo.mtu.edu/UPSeis/waves.html
Polarized share waves
There are two special types of shear waves:
• horizontally polarized shear waves,
• and vertically polarized shear waves.
In the case of a polarized shear wave, the particles cannot
oscillate in any direction but only in a single plane.
Depending on the orientation of this plane, we can speak about
horizontally polarized shear waves (SH) and vertically polarized
shear waves (SV).
Polarized share waves
For horizontally polarized shear waves (SH), the particles oscillate in
the horizontal plane.
For vertically polarized shear waves (SV) the particles oscillate in the
vertical plane.
(Edited by Yoram Rubin and Susan S. Hubbard: Hydrogeophysics, Springer 2005)
Velocity of body waves
The velocity of body waves depends on the elastic properties and the
density of the material through which the waves travel.
By means of the following relationships, we can calculate the velocity
of body waves:
where VP is the velocity of compressional wave (P-wave),
VS is the velocity of shear wave (S-wave),
 is the density of the material,
K is the bulk modulus,
and  is the shear modulus.
Velocity of body waves
In the case of fluids (for example water), the value of  is equal to
zero.
It results from this that the compressional wave propagates slower in
fluids than in solids.
It also means that the velocity of compressional waves in a highly
porous and/or fractured rock filled with water is significantly slower
than that in a compacted and consolidated rock.
There is another consequence of the zero value of  Namely the
velocity of the shear wave becomes zero in fluids.
A velocity value of zero implies that the shear wave cannot propagate
in fluids.
Velocity of body waves
Since the value of the bulk modulus (K) is positive, the Vp is always
greater than VS. So, the compressional wave travels faster than the
shear wave through the same solid material.
Due to this important property, the compressional wave arrival can be
detected at first by a receiver located on the surface far enough from
the source of the elastic waves.
This is the reason why the other name of this wave primary or Pwave.
The second arrival is connected to the shear wave whose other
name is secondary or S-wave.
From the point of view of seismic methods mostly the compressional
wave is of importance.
Velocity of body waves
This figure illustrates a seismogram obtained by detecting the arrivals
of different waves with a seismic receiver (geophone) located on the
surface. We can see the usual order of the arrivals:
• arrival of compressional or P-wave,
• arrival of shear or S-wave,
• arrival of surface wave components.
A detected and recorded seismic signal always contains some
random noise which is an undesired component of the signal.
http://depthome.brooklyn.cuny.edu/geology/onlinecore/plates/platequiz.htm
Surface waves
There are two types of surface waves travel near the Earth's
surface:
Rayleigh wave also known as ground roll
and Love wave.
Surface waves are slower than body waves, so they will arrive
later at a receiver located on the surface far enough from the
seismic source.
If the receiver is too close to the source, the arrivals of surface
waves can overtake the body waves because they have to
travel a shorter distance to get the receiver than the body
waves.
Therefore the selection of suitable source-receiver distance is
very important for a seismic measurement.
Surface waves
The particle motion is more
complex in the case of surface
waves than body waves.
The Rayleigh waves are
characterized by elliptical
retrograde particle motions in a
vertical plane.
This figure tries to illustrate the
propagation of Rayleigh wave in
the function of time and distance.
http://depthome.brooklyn.cuny.edu/geology/onlinecore/plates/platequiz.htm
Surface waves
Love waves are horizontally
polarized shear waves (SH
waves) which exist only in the
case when the subsurface
structure includes at least two
layers and the velocity of wave
propagation is higher in the lower
layer.
The particle motion in a Love
wave is horizontal and transverse
to the direction of wave
propagation.
http://depthome.brooklyn.cuny.edu/geology/onlinecore/plates/platequiz.htm
Surface waves
The effect of surface waves can be very destructive in the case
of earthquakes.
From the point of view of seismic methods, they are not useful.
Actually, the surface waves are considered as noise and they
can cause difficulties in the recognition of compressional wave
arrivals on the recorded seismograms.
It is a very important task to decrease the effect of surface
waves on the recorded data sets by using signal processing
techniques.
https://www.researchgate.net/figure/224353018_fig1_Fig-1-a-An-example-of-a-realvery-noisy-seismogram-case-8-Table-III-with-some
Direct and air waves
There are two other types of waves which are connected to seismic
measurements but they do not provide any useful information about the
subsurface geology:
• direct wave
• and air wave.
Both of them are generated by the seismic source and arrive in the
receivers. So the arrivals of these waves usually appear on the
seismograms.
Direct wave is a seismic body wave which travels through the medium
directly from the source to the receivers without meeting any subsurface
contact. Actually, direct waves travel collaterally with the surface. Because
of its shorter route (between the source and a receiver), it arrives sooner in
the receivers than the body waves reflected off formation boundaries.
Air wave is a wave which travels through the air directly from the source to
the receivers. It is easy to recognize the arrival of air wave on a
seismogram because it travels at a speed of 330 m/s,(which is the speed of
sound in air).
Some basic properties of waves
Let us start from a simple sine function to review the most important
properties of waves shortly.
The sine function is a periodic function which repeats over intervals of
2.
This function is suitable to describe a physical quantity varying
periodically as a function of time by the following form:
2𝜋
𝑢 𝑡 = 𝐴 ∙ sin
𝑡 + 𝜑 = 𝐴 ∙ 𝑠𝑖𝑛 2𝜋𝑓𝑡 + 𝜑
𝑇
where t denotes the time (which is the independent variable of the
function),
u means the observed physical quantity (which is the dependent
variable of the function),
A is the amplitude,
T is the period,
f is the frequency,
and  is the phase.
Some basic properties of waves
The previous formula may be used to express the oscillation of a
particle during the wave propagation at a given point of a medium.
http://www.doctronics.co.uk/signals.htm
Some basic properties of waves
Amplitude (A) gives the maximum extent of the peaks in a wave. Its
dimension and unit are the same as that of the dependent variable. In
our example the dimension of the particle displacement is length (the
displacements are as low as nanometres in seismic surveys).
Period (T) gives the time interval between successive peaks (or
troughs) in a wave. Its dimension is time.
Frequency (more exactly the temporal frequency) denoted by f is the
reciprocal of the period (T) and gives the number of periods included
in a unit time. Its dimension is the reciprocal of time, and the unit is
Hz (1/s) in SI.
Phase or phase angle () gives the initial state of a periodic change.
A whole cycle of the periodic change takes a period. The value of
phase within a period varies from 0 to 2(sometimes the interval of
phase ranges from -to).
A so-called zero-phase signal means that the periodic change starts
at zero-crossing going positive. (The initial time moment chosen for
an investigated process is generally zero.)
Some basic properties of waves
The so-called phase difference gives the difference between the
phase angles of two periodic processes having the same frequency.
We can speak that two waves are in-phase if their phase angles are
the same.
On the contrary, when the phase angles are different, the waves are
out-of-phase.
http://www.doctronics.co.uk/signals.htm
Some basic properties of waves
When a quantity varies periodically not only with time but also with the
position along a line, another parameter, the so-called wavelength, is
required for the description of the variation.
Wavelength () is the distance between successive peaks (or troughs) in a
wave. Actually, it is a spatial period and its dimension is length.
The reciprocal of wavelength is the
spatial frequency () which gives
how many times the spatial periods
repeat in a unit of distance.
https://simple.wikipedia.org/wiki/Sine_wave
Some basic properties of waves
The velocity of wave propagation (V) is related to the (temporal)
frequency and the wavelength of the wave in the following way:
V=f·
It was shown earlier that the velocity of body waves (P-wave and Swave) depends on the elastic moduli and the density of materials.
So, the velocity of wave is a material property which has a constant
value for each material.
It means that the frequency of wave is reciprocally proportional to the
wavelength.
The higher the frequency the shorter the wavelength and vice versa.
The frequency range of seismic waves
A seismic wave is an elastic wave which travels through the Earth's interior.
A seismic source emits periodic waves with different frequencies,
amplitudes and phases all at once. These wave components are
superposed and form a net wave.
The shorter the effect of a source in time the wider the range of frequency
components.
Due to the superposition and the frequency-selective attenuation of different
components, a seismic wave can be represented by not a periodic signal
but a non-periodic one.
This figure shows the superposition
of two sine waves with different
amplitudes, frequencies and phases.
The amplitude variation of the net
waveform contains the effects of both
waves.
The frequency range of seismic waves
A seismic wave resembles a so-called wavelet which has finite
extent both in time and space.
This is the reason why we cannot give an exact frequency and
wavelength for a seismic wave.
Only a dominant frequency and a dominant wavelength can be
defined for a wavelet-like seismic wave.
A wavelet is a wave-like
oscillation with an amplitude that
begins at zero and decreases
back to zero at the end.
http://www.lohninger.com/wavelet.html
The frequency range of seismic waves
A seismogram, which contains the observation of several seismic
waves arrived at a receiver successively, is made up of different
wavelets shifted by different time values.
Each wavelet belongs to a seismic wave which has travelled its own
way and reflected off some point of a formation boundary.
The time differences among the wave arrivals are owing to the
different routes.
http://www.wavelet.org/tutorial/wbasic.htm
The frequency range of seismic waves
The frequency range of seismic waves depends on the
properties of seismic sources and the material properties of
rocks through which the waves propagate.
The frequency components of seismic waves caused by
earthquakes are typically in the ranges of 0.01 Hz to 2 Hz.
The seismic waves generated by artificial sources can be
characterized by a higher frequency range of 10 to about 100
Hz.
However, sources can produce seismic waves with higher
frequency but the high-frequency components decay quickly
during the propagation by the effect of rock formations.
Because of this quick attenuation, they are not able to arrive at
the receivers.
The attenuation of seismic waves
The energy of a wave is proportional to the square of its
amplitude (E ~ A2).
As a seismic wave is travelling farther and farther from
the source, its amplitude along with its energy attenuates
gradually.
The loss of energy is the consequence of two effects:
• the geometrical spreading
• and the intrinsic attenuation.
The attenuation of seismic waves
Geometrical spreading is simply a geometrical effect without
any physical reason.
The process of geometrical spreading is the following.
When a wave coming from a source point propagates in a
homogeneous medium, the points being in identical phase of
oscillation form spherical wavefronts at any time moment.
As the wave is getting farther and farther the sizes of wavefronts
are becoming larger and larger.
But the overall energy belonging to a wavefront may not
increase. Each wavefront spreads with its initial energy.
It means the same energy scatters over an increasing surface,
so the energy per unit area gradually decreases.
This is the reason why the energy of wave decreases in all
directions with the distance.
The attenuation of seismic waves
This figure shows how the surface of the wave front belonging
to a spatial angle increases with the distance from the source.
http://www.performingmusician.com/pm/apr09/articles/technotes.htm?print=yes
The attenuation of seismic waves
The other cause of wave energy loss is the so-called intrinsic
attenuation which has a physical reason.
The displacements of particles during a wave motion entail
frictional dissipation which converts some part of elastic energy
into heat.
The combination of these two effects results in the attenuation
of wave amplitude.
The attenuation of amplitude can be expressed for a
homogeneous medium by the following relationship:
A = (A0·e- r) / r
where A is the amplitude at distance r from the source,
A0 is the initial amplitude and
 is the absorption coefficient of the material.
The attenuation of seismic waves
The higher the value of absorption coefficient the higher the
attenuation of amplitude and the absorption of seismic energy.
The value of absorption coefficient in rocks ranges from 0.2 to
0.75 dB/wavelength.
The absorption coefficient depends on the frequency and the
properties of rock.
The more compacted and consolidated a rock the lower the
value of its absorption coefficient.
The ideally elastic medium would have an absorption coefficient
of zero value. (It means that there is no intrinsic attenuation
inside an ideally elastic medium.)
The more compacted and consolidated a rock the better it
approximates the behaviour of ideally elastic medium.
The attenuation of seismic waves
In the case of near-surface sedimentary structures, the degree of
compaction and consolidation is generally low.
It means that high intrinsic attenuation of seismic energy can be
expected for shallow seismic surveys.
The high attenuation of amplitude may cause problems in the
detection of wave arrivals.
If the amplitude of a detected seismic wave is too small, its
identification on a seismogram can be problematic (it cannot be
distinguish from the noise component of the signal).
In addition, the absorption coefficient depends on the frequency of
waves which means that higher-frequency waves attenuate faster
than lower-frequency ones
Therefore, the higher frequency components gradually disappear in
a seismic wave as it is propagating through rocks.
It means that the seismic waves penetrated the deeper parts of the
subsurface structure contain rather lower-frequency components
(which has an unfavourable consequence for the vertical resolution of
the seismic reflection method).
Ray theory
If we want to study in what directions the waves travel and how the
boundaries modify the direction of wave propagation, it is worth
applying the approach of ray theory.
Contrary to the quantitative description used in wave theory, the ray
theory provides a qualitative outlook rather by which we can trace the
routes of waves in a layered subsurface structure (or half-space).
Ray theory is generally used in geometrical optics where a ray is the
idealized model of a light wave.
The principles used for light waves are also applicable to seismic
waves.
So, the ray (or ray path) of a seismic wave is a line which shows the
direction of wave propagation and is perpendicular to the wavefronts
of the wave.
This line also represents the path of energy flow connecting to the
wave propagation.
Ray theory
This figure demonstrates the meaning of the ray path for seismic
waves.
It can be seen that a seismic wave propagates not only in a single
direction.
From a seismic survey point of view, those ray paths are important
which arrive at formation boundaries, reflect off the contacts and
return to the surface.
https://1f308d6acfa3dcbec8dfb7385adde1ece594768d.googledrive.com/host/0
B6TvZfgdBGQ8Mzg2MC1nSFpYazQ/dissertation/2%20Methodology.html
Ray theory
This figure shows a so-called horizontally layered half-space which is
a frequently used geophysical model of sedimentary basins.
Some possible ray paths of seismic waves are also presented.
The ray paths penetrate some of the layers and reflect off one of the
formation boundaries.
The ray paths start from a source located on the surface and arrive at
a receiver which is also located on the surface.
http://www.crewes.org/ResearchLinks/Converted_Waves/Page2.html
Reflection and refraction of seismic
waves
When a seismic body wave (either a compressional or a shear wave)
arrives at a contact separating two media with different elastic
properties, some part of the energy will reflect off the contact and the
other part will penetrate into the second medium.
Reflection is a process which occurs when a wavefront arrives at an
interface between two different media, the direction of propagation
changes, and the wave front returns into the medium from which it
came.
Refraction is a process which occurs when a wavefront arrives at an
interface between two different media, the direction of propagation
changes, the wave front passes the interface and penetrate the other
medium.
Reflection and refraction of seismic
waves
In general, when an incident seismic wave encounters a boundary, it
divides into two reflected waves and two refracted waves.
One of the waves in these pairs is a compressional wave (P wave)
and the other is a shear wave (S wave).
So, the energy of an incident wave splits among the following waves:
• a reflected compressional wave,
• a reflected shear wave,
• a refracted compressional wave
• and a refracted shear wave
After the reflection and refraction, all of these waves travel their own
ways in different directions.
Reflection and refraction of seismic
waves
This figure illustrates what happens when an incident compressional
wave arrives at a boundary with an angle of incidence ip.
S in a subscript: shear wave
P in a subscript: compressional
wave
R: angle of reflection
r: angle of refraction
V: seismic wave velocity
: density
1 in a subscript: layer 1
2 in a subscript: layer 2
The angles are referred to the axis
of incidence.
Prem V. Sharma: Environmental and engineering
geophysics, Cambridge University Press
Reflection and refraction of seismic
waves
There is a relationship between the different angles and wave
velocities which is given by the so-called generalized Snell's law.
By this law, the ratio of the sine of the angle to the appropriate wave
velocity yields a constant value which is identical for each type of
waves.
This constant value depends on the angle of incident wave and the
velocity of compressional wave in the upper medium.
It can also be seen that the angles of reflections and refractions are
determined by this constant value as well as the velocities of different
waves.
Reflection and refraction of seismic
waves
A special case of the reflection is the normal (or vertical) incidence.
It occurs when the ray path of the incident compressional wave is
perpendicular to the boundary (that is the angle of incidence is zero).
In such a case, neither reflected nor refracted shear waves are
generated.
The whole energy of the incident wave will be shared between the
reflected and refracted (or transmitted) compressional waves.
http://www.ukm.my/rahim/Seismic%20Refraction%20Surveying.htm
Reflection and refraction of seismic
waves
In practice, only the near-normal (or near-vertical) incidence may be
produced.
In this case, the distance between the seismic source and the receiver is
much shorter than the depth of boundary (that is the angle of incidence is a
low value).
However not only compressional waves come into being at the boundary
the energy of reflected and shear waves is very low. So they may be
neglected.
This is an advantageous situation because generally the compressional
waves play a dominant role in seismic surveys.
http://www.ipims.com/data/gp13/P0535.asp?UserID=&Code=3776
Reflection and refraction of seismic
waves
There is another important case which is connected to the refracted
waves.
If the wave travels faster in the lower layer than in the upper layer (V2
> V1), which is often fulfilled in practice, the angle of refraction will be
grater than the angle of incidence (r > i).
This relation is the consequence of Snell's law.
We can also see that the increase in the angle of incidence results in
the increase in the angle of refraction.
If the angle of incidence reaches a certain value, the angle of
refraction will be perpendicular to the axis of incidence.
This phenomenon is called critical refraction and the angle of
incidence belonging to it is referred to as critical angle.
Reflection and refraction of seismic
waves
The critical angle can be expressed from Snell's law as follows:
The critically refracted wave also called head wave propagates along the
boundary with the velocity of the lower layer (VP2).
During its propagation, the points of the lower layer located near by the
interface behave as sources of secondary waves.
These secondary waves will travel upwards through the upper layer, and
their ray paths subtend critical angle to the normal of the interface.
http://www.ukm.my/rahim/Seismic%20Refraction%20Surveying.htm
Reflection and refraction of seismic
waves
By detecting the arrivals of these secondary waves on the surface,
we can obtain arrival time data which contain information about the
depth of the formation boundary. Of course, we have to apply suitable
data processing techniques to get the desired information from the
raw data.
Actually, the seismic refraction method is based on the critical
refraction.
http://www.ukm.my/rahim/Seismic%20Refraction%20Surveying.htm
Reflection coefficient
Snell's law is a geometrical relationship which does not give any
information about the relations of the amplitudes belonging to the
different types of waves.
In order to investigate the question of amplitudes, we have to
introduce the so-called acoustic impedance.
Acoustic impedance is an acoustic property of the medium which
can be calculated by the product of the density () and the wave
velocity (V):
I = ·V
The acoustic impedance of a given type of waves can be obtained by
substituting its velocity in the formula above.
In the case of normal incidence (when the angle of incidence is equal
to zero), the relation between the amplitudes of reflected and
refracted compressional waves is characterized by the so-called
reflection coefficient (R ):
Reflection coefficient
The reflection coefficient (R ) is expressed by the following equation:
where AR is the amplitude of reflected compressional wave,
Ai is the amplitude of incident compressional wave,
I1 is the acoustic impedance of the upper layer,
and I2 is the acoustic impedance of the lower layer.
Since the density and the wave velocity (which determine the
acoustic impedance) generally increase with depth (due to the
compaction and consolidation) the value of reflection coefficient is
positive in most cases.
A positive reflection coefficient means that there is no phase reversal
between the reflected and incident waves. (Phase reversal means
that the phase of the wave is shifted by  radian (or 180°).
Reflection coefficient
The value of reflection coefficient for interfaces between rock layers is
usually a low value.
It is rarely greater than 0.2.
Since the ratio of reflected energy to the incident energy is
proportional to the square of the reflection coefficient, its vale is
frequently less than 1%.
It means that only a smaller part of the incident energy reflects off a
formation boundary.
There are some special cases when this ratio can reach 70 % or
even 100%.
Such excellent reflectors are the surface of the oceans and the
surface of the Earth itself.
Refraction coefficients
Similarly to the reflection coefficient, another quantity is defined for
the case of refraction.
Refraction coefficient (T) of an interface separating two media gives
the ratio of the amplitude of refracted wave to the amplitude of
incident wave.
It can be expressed by means of the acoustic impedance in the
following way:
where AT is the amplitude of refracted compressional wave,
Ai is the amplitude of incident compressional wave,
R is the reflection coefficient,
I1 is the acoustic impedance of the upper layer,
and I2 is the acoustic impedance of the lower layer.
Refraction coefficients
The relationship of the refraction coefficient is valid for only the case
of normal incidence.
Both of these coefficients (reflected and refracted) depend on the
contrast between the acoustic impedance of upper layer and the
acoustic impedance of lower layer.
Since the density of rocks varies over a relatively narrow range, the
wave velocity influences the value of acoustic impedance rather than
the density.
So, it can be stated that the energy relation between the reflected and
refracted waves primarily depends on the velocity contrast between
the two media.