elastic moduli for homogeneous and isotropic...

Wave theory

Because rocks can be regarded as elastic bodies in the range ofsmall deformation, the elastic waves are able to travel through theirmaterial.

The seismic waves are low-frequency elastic waves which travelthrough the rock formations.

The seismic methods are based on the fact that seismic wavescoming from a source are able to propagate through rocks.

Actually, the seismic wave propagation takes place in the form of aspatial and temporal variation of the stress and strain fields inside therocks.

There is a very close interaction between the stress and deformationfields. A change in one of them causes the change of the other, andvice versa.

From the point of view of wave theory, the wave propagation isconsidered as the spatial and temporal propagation of a perturbationcaused by a source.

Elastic waves

Elastic wave is the temporal and spatial propagation of adisturbance in the stress-deformation state of an elasticmedium.

The source of the disturbance can be an impulsive event or aperiodic process which occurs at a point or a relatively smallvolume of an elastic body.

Of course, any change in the stress field is always induced by achange in the magnitude and/or direction of external forcesacting on the body.

The change in the stress field causes the displacement ofparticles inside the body and the deformation patterncorresponding to this change will propagate outward from thelocation of source as an elastic wave.

Elastic waves

It is important to note that not the particles travel through themedium but the change in the stress and deformation fieldsduring 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 impulseor a longer time like a periodical variation.

In the case of an impulsive source, the elastic wave calms downgradually as the time is passing and the distance is increasingfrom the source.

A source operating periodically is able to retain the elastic wavemotion inside the medium for a longer time.

Elastic waves

There are two principal types of elastic waves:

• body wave

• and surface wave.

Body wave is a wave which travels three-dimensionally through anelastic medium (inside the body).

Surface wave is a wave which propagates along and near by thesurface of an elastic medium.

http://www.parkseismic.com/Whatisseismicwave.html

Body waves

Two types of deformation pattern can propagate in the form of bodywaves.

When contractions and expansions are periodically taking placeduring the wave propagation, the particles are oscillating along axesparallel 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 inphysics (similarly to the sound waves).

During this wave motion a dynamical change occurs in both thevolume and shape of the local environments of the medium.

Here, the stress field does not have a shear component during thewave propagation.

This is the reason why a compressional wave can propagate not onlyin solids but fluids.

Compressional waves

This figure illustrates the propagationof compressional wave.

The square prisms symbolize thesame piece of an elastic medium atdifferent time moments.

The small cubes represent theparticles of the medium.

The time passes from up to down.

It can be seen how the compressionand dilatation of particles travel inthe 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) periodicalternations in the shear stress-deformation field produce the wavemotion.

Only the shear component of the stress field plays role in the shearwave motion (there is no normal stress component).

The particles are oscillating along axes perpendicular to the directionof wave propagation.

So, this type of wave belongs to the group of transverse waves inphysics (similarly to the light waves).

During this wave motion the dynamical change affects only the shapeof the local environments of the medium (but not its volume).

A shear wave cannot propagate in fluids because fluids are not ableto resist shear forces.

Shear waves

This figure illustrates the propagationof shear wave in a similar way than itwas shown previously for thecompressional wave.

It can be seen that the motions of theparticles are perpendicular to thedirection of wave propagation.

Do not forget that

not the particles propagate in themedium 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.

While the plane of particles’ oscillation can vary point by pointfor the propagation of a general shear wave, in the case of apolarized shear wave the particles oscillate only in a singleplane.

Depending on the orientation of this plane, we can speak abouthorizontally polarized shear waves (SH) and vertically polarizedshear waves (SV).

Polarized share waves

For horizontally polarized shear waves (SH), the particles oscillateonly in the horizontal plane.

For vertically polarized shear waves (SV) the particles oscillate onlyin 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 thedensity of the material through which the waves travel.

By means of the following relationships, we can calculate the velocityof 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 wavesIn the case of fluids (for example water), the value of is equal tozero.

Consequently, the compressional wave propagates slower in fluidsthan in solids.

It also means that the velocity of compressional waves is significantlylower in a highly porous and/or fractured rock filled with water than ina compacted and consolidated rock.

There is another consequence of the zero value of

Namely the velocity of shear wave becomes zero in fluids.

A velocity value of zero implies that the shear wave cannot propagatein fluids.


Since the value of bulk modulus (K) is always positive, the Vp

isalways greater than V

S. 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 bedetected at first by a receiver located on the surface far enough fromthe source of the elastic waves.

This is the reason why the other name of this wave primary or P-wave.

The second arrival is connected to the shear wave whose othername is secondary or S-wave.

From the point of view of seismic methods mostly the compressionalwave is of importance.


This figure illustrates a seismogram obtained by detecting the arrivalsof different waves with a seismic receiver (geophone) located on thesurface. 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 somerandom noise which is an undesired component of the signal.

http://depthome.brooklyn.cuny.edu/geology/onlinecore/plates/platequiz.htm

Surface wavesThere are two types of surface waves travel near the Earth'ssurface:• Rayleigh wave also known as ground roll,• and Love wave.

Surface waves are slower than body waves, so they will arrivelater at a receiver located on the surface far enough from theseismic source.

RemarkIf the receiver is too close to the source, the arrivals of surfacewaves can overtake the body waves because they have to runa much shorter distance to get the receiver than the bodywaves.Therefore the selection of suitable source-receiver distance isvery 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.


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.


Surface waves

The effect of surface waves can be very destructive in the caseof earthquakes.From the point of view of seismic methods, they are not useful.Actually, the surface waves are considered to be noise and theycan cause difficulties in the recognition of compressional wavearrivals on the recorded seismograms.It is a very important task to decrease the effect of surfacewaves on the recorded data sets by using signal processingtechniques.

https://www.researchgate.net/figure/224353018_fig1_Fig-1-a-An-example-of-a-real-very-noisy-seismogram-case-8-Table-III-with-some

Direct and air waves

There are two other types of waves which are connected to seismicmeasurements but they do not provide any useful information about thesubsurface geology:• direct wave• and air wave.Both of them are generated by the seismic source and detected by thereceivers. So the arrivals of these waves usually appear on theseismograms.

Direct wave is a seismic body wave which travels through the mediumdirectly from the source to the receivers without meeting any subsurfacecontact. Actually, direct waves travel collaterally with the surface. Becauseof its shorter route (between the source and a receiver), it arrives sooner inthe receivers than the body waves reflected off formation boundaries.

Air wave is a wave which travels through the air directly from the source tothe receivers. It is easy to recognize the arrival of air wave on aseismogram because it travels at a speed of 330 m/s,(which is the speed ofsound 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:

𝑢 𝑡 = 𝐴 ∙ sin2𝜋

𝑇𝑡 + 𝜑 = 𝐴 ∙ 𝑠𝑖𝑛 2𝜋𝑓𝑡 + 𝜑

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.


The previous formula may be used to express the oscillation of aparticle during the wave propagation at a given point of a medium.

http://www.doctronics.co.uk/signals.htm

Some basic properties of wavesAmplitude (A) gives the maximum extent of the peaks in a wave. Itsdimension and unit are the same as that of the dependent variable. Inour example the dimension of the particle displacement is length (thedisplacements are as low as nanometres in seismic surveys).Period (T) gives the time interval between successive peaks (ortroughs) in a wave. Its dimension is time.Frequency (more exactly the temporal frequency) denoted by f is thereciprocal of the period (T) and gives the number of periods includedin a unit time. Its dimension is the reciprocal of time, and the unit isHz (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 ofphase within a period varies from 0 to 2(sometimes the interval ofphase ranges from -to).A so-called zero-phase signal means that the periodic change startsat zero-crossing going positive. (The initial time moment chosen foran investigated process is generally zero.)


The so-called phase difference gives the difference between thephase angles of two periodic processes having the same frequency.We can speak that two waves are in-phase if their phase angles arethe same.On the contrary, when the phase angles are different, the waves areout-of-phase.

http://www.doctronics.co.uk/signals.htm


When a quantity varies periodically not only with time but also with theposition along a line, another parameter, the so-called wavelength, isrequired for the description of the variation.Wavelength () is the distance between successive peaks (or troughs) in awave. Actually, it is a spatial period and its dimension is length.

https://simple.wikipedia.org/wiki/Sine_wave

The reciprocal of wavelength is thespatial frequency () which giveshow many times the spatial periodsrepeat in a unit of distance.


The velocity of wave propagation (V) is related to the (temporal)frequency and wavelength in the following way:

V = f ·

It means that the frequency of wave is reciprocally proportional to thewavelength.The higher the frequency the shorter the wavelength and vice versa.It was shown earlier that the velocity of body waves (P-wave and S-wave) depends on the elastic moduli and the density of materials.So, the velocity of wave is a material property which has a constantvalue for each material.

The frequency range of seismic waves

A seismic wave is an elastic wave which travels through the Earth's interior.The net wave emitted by a seismic source can be decomposed into infinitenumber of elementary sine and cosine components having differentfrequencies and phases.The shorter the effect of a source in time (impulse type source) the widerthe range of frequency components.

This figure shows the superpositionof two sine waves with differentamplitudes, frequencies and phases.The amplitude variation of the netwaveform depends on the localamplifications and attenuations of thecomponents.


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

A seismic wave coming from an impulse type sorce resembles a so-called wavelet which has finite extent both in time and space.It is composed of infinite number of elementary harmoniccomponents.This is the reason why we cannot exactly describe a seismic wavewith a single frequency.Only a dominant frequency and a dominant wavelength can bedefined for a wavelet-like seismic wave.


A seismogram, which contains the observation of several seismicwaves arrived at a receiver successively, is made up of differentwavelets shifted by different time values.Each wavelet belongs to an individual seismic wave which hastravelled its own way (air wave, direct wave, surface wave, reflectedwaves, refracted waves).The time differences among the wave arrivals are due to the differentwave paths and different wave velocities.

http://www.wavelet.org/tutorial/wbasic.htm


The frequency range of seismic waves depends on theproperties of seismic sources and the material properties ofrocks through which the waves propagate.The detectable frequency components of seismic waves causedby earthquakes are typically in the ranges of 0.01 Hz to 2 Hz.The seismic waves generated by artificial sources can becharacterized by a higher frequency range of 10 to about 100Hz.However, sources can produce seismic waves with higherfrequency but the high-frequency components decay quicklyduring the propagation by the effect of rock formations.Because of this quick attenuation, they are not able to arrive atthe receivers.

The attenuation of seismic waves

The energy of a wave component is proportional to thesquare of its amplitude (E ~ A2).As a seismic wave is travelling farther and farther fromthe source, its amplitude along with its energy attenuatesgradually.

The loss of energy is the consequence of two effects:• the geometrical spreading• and intrinsic attenuation.


Geometrical spreadingis 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 ahomogeneous medium, the points being in identical phase ofoscillation form spherical wave fronts 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 wave front may notincrease. Each wave front spreads with its initial energy.It means that the same energy must be distributed over anincreasing surface, so the energy per unit area graduallydecreases.This is the reason why the energy of wave decreases in alldirections with the distance.


This figure shows how the surface of the wave front belongingto a given spatial angle increases with the distance from thesource.

http://www.performing-musician.com/pm/apr09/articles/technotes.htm?print=yes


The other cause of wave energy loss is the so-called intrinsicattenuation which has a physical reason.The displacements of particles during a wave motion entailfrictional dissipation which converts some part of elastic energyinto heat.

The combination of these two effects results in the attenuationof wave amplitude.The attenuation of amplitude can be expressed for ahomogeneous medium by the following relationship:

A = (A0·e- r) / r

where A is the amplitude at distance r from the source,A

0is the initial amplitude and

is the absorption coefficient of the material.


The higher the value of absorption coefficient the higher theattenuation of amplitude and the absorption of seismic energy.The value of absorption coefficient in rocks ranges from 0.2 to0.75 dB/wavelength.The absorption coefficient depends on the frequency and thetype of rock. For higher frequencies its value is greater.For compacted and consolidated rocks the value of absorptioncoefficient is lower (the rate of intrinsic attenuation is less).An ideally elastic medium would have an absorption coefficientof zero value. (It means that there is no intrinsic attenuationinside an ideally elastic medium.)The more compacted and consolidated a rock is, the better itapproximates the behaviour of an ideally elastic medium.


In the case of near-surface sedimentary structures, the degree ofcompaction and consolidation is generally low.It means that high intrinsic attenuation of seismic energy can beexpected for shallow seismic surveys.The high attenuation of amplitude may cause problems in thedetection of wave arrivals.If the amplitude of a detected seismic wave is too small, itsidentification on a seismogram can be problematic (it cannot bedistinguish from the noise component of the signal).In addition, the absorption coefficient depends on the frequency ofwaves which means that higher-frequency waves attenuate fasterthan lower-frequency ones.Therefore, the higher frequency components gradually disappear ina seismic wave as it is propagating through rocks.Consequently, the seismic waves penetrated the deeper parts of thesubsurface structure contain rather lower-frequency components(which has an unfavourable consequence for the vertical resolution ofthe seismic reflection method).

Ray theory

If we want to study in what directions the waves travel and how theboundaries modify the direction of wave propagation, it is worthapplying the approach of ray theory.Contrary to the quantitative description used in wave theory, the raytheory provides a qualitative outlook rather by which we can trace theroutes of waves in a layered subsurface structure (or half-space).Ray theory is generally used in geometrical optics where a ray is theidealized model of a light wave.The principles used for light waves are also applicable to seismicwaves.So, the ray (or ray path) of a seismic wave is a line which follows thedirection of wave propagation and is perpendicular to the wave frontsof the wave in its each point.This line also represents the path of kinetic energy flow connected tothe wave propagation.

Ray theoryThis figure demonstrates the meaning of the ray path for seismicwaves.It can be seen that a seismic wave propagates not only in a singledirection.From the perspective of seismic surveys the ray paths which arrive atformation boundaries, reflect off the contacts and return to the surfaceare of importance.

https://1f308d6acfa3dcbec8dfb7385adde1ece594768d.googledrive.com/host/0B6TvZfgdBGQ8Mzg2MC1nSFpYazQ/dissertation/2%20Methodology.html

Ray theoryThis figure shows a so-called horizontally layered half-space which isa 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 theformation boundaries.The ray paths start from a source located on the surface and arrive ata 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 elasticproperties, some part of the energy will reflect off the contact and theother part will penetrate into the second medium.

Reflection is a process which occurs when a wave front arrives at aninterface between two different media, the direction of propagationchanges, and the wave front returns into the medium from which itcame.

Refraction is a process which occurs when a wave front arrives at aninterface between two different media, the direction of propagationchanges, the wave front passes the interface and penetrate the othermedium.


In general, when an incident seismic wave encounters a boundary, itdivides 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 ownways in different directions and with different velocities.


This figure illustrates what happens when an incident compressionalwave arrives at a boundary with an angle of incidence i

p.

S in a subscript: shear waveP in a subscript: compressional

waveR: angle of reflectionr: angle of refractionV: seismic wave velocity: density1 in a subscript: layer 12 in a subscript: layer 2

The angles are referred to the axis of incidence.

Prem V. Sharma: Environmental and engineering geophysics, Cambridge University Press


There is a relationship between the different angles and wavevelocities 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 wavevelocity yields a constant value which is identical for each type ofwaves.This constant value depends on the angle of incident wave and thevelocity of compressional wave in the upper medium.It can also be seen that the angles of reflections and refractions aredetermined by this constant value as well as the velocities of differentwaves.


A special case of the reflection is the normal (or vertical) incidence.It occurs when the ray path of the incident compressional wave isperpendicular to the boundary (that is the angle of incidence is zero).In such a case, neither reflected nor refracted shear waves aregenerated.The whole energy of the incident wave will be shared between thereflected and refracted (or transmitted) compressional waves.

http://www.ukm.my/rahim/Seismic%20Refraction%20Surveying.htm


In practice, only the near-normal (or near-vertical) incidence may beproduced.In this case, the distance between the seismic source and the receiver ismuch shorter than the depth of boundary (that is the angle of incidence is alow value).However not only compressional waves come into being at the boundarythe energy of reflected and shear waves is very low. So they may beneglected.This is an advantageous situation because generally the compressionalwaves play a dominant role in seismic surveys.

http://www.ipims.com/data/gp13/P0535.asp?UserID=&Code=3776


There is another important case which is connected to the refractedwaves.If the wave travels faster in the lower layer than in the upper layer (V

2

> 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 angle of incidence results in theincrease in angle of refraction.If the angle of incidence reaches a certain value, the angle ofrefraction will be 90 °. So, the direction of wave propagation will beperpendicular to the axis of incidence.This phenomenon is called critical refraction and the angle ofincidence belonging to it is referred to as critical angle.


The critical angle can be expressed from Snell's law as follows:

The critically refracted wave propagates along the bed boundary with thevelocity of the lower layer (V

P2).

Due to the oscillation of particles the points of the interface will behave assources of secondary seismic waves.These secondary waves will travel upwards through the upper layer, andtheir ray paths subtend critical angle to the normal of the interface.


The superposition of secondary waves produces a net wave called head wave.Its wave fronts are parallel planes.


By detecting the arrivals of these secondary waves on the surface,we can obtain arrival time data which contain information about thedepth of the bed boundary and wave velocities.Of course, we have to apply suitable data processing techniques toget the desired information from the raw data.Actually, the seismic refraction method exploits the phenomenon ofcritical refraction.


Reflection coefficient

Snell's law is a geometrical relationship which does not give anyinformation about the relations of the amplitudes belonging to thedifferent types of waves.In order to investigate the question of amplitudes, we must introducethe so-called acoustic impedance.Acoustic impedance is an acoustic property of the medium whichcan be calculated by the product of the density () and wave velocity(V):

I = ·V

The acoustic impedance of a given type of waves can be obtained bysubstituting its velocity in the formula above.In the case of normal incidence (when the angle of incidence is equalto zero) the so-called reflection coefficient (R) gives the amplituderatio of the reflected compressional wave to the incidentcompressional wave.


The reflection coefficient (R) can be expressed by the followingequation:

where AR

is the amplitude of reflected compressional wave,

Aiis the amplitude of incident compressional wave,

I1

is the acoustic impedance of the upper layer,and I

2is the acoustic impedance of the lower layer.

Since the density and the wave velocity (which determine theacoustic impedance) generally increase with depth (due to thecompaction and consolidation) the value of reflection coefficient ispositive in most cases.A positive reflection coefficient means that there is no phase reversalbetween the reflected and incident waves. (Phase reversal meansthat the phase of the wave is shifted by radian (or 180°).


The value of reflection coefficient for the bed boundaries is usually alow value.It is rarely greater than 0.2 (20%).The ratio of reflected energy to the incident energy is proportional tothe square of the reflection coefficient.Its vale is generally less than 0.01 (1%).It means that only a small part of the incident energy reflects off aformation boundary.There are some special cases when this ratio can reach 70 % oreven higher values.Such excellent reflectors are the surface of the oceans and thesurface of the Earth itself.

Refraction coefficients

Similarly to the reflection coefficient, another quantity is defined forthe case of refraction.Refraction coefficient (T) of an interface separating two media givesthe amplitude ratio of the refracted wave to the incident wave.It can be expressed by means of the acoustic impedance in thefollowing way:

where AT

is the amplitude of refracted compressional wave,A

iis the amplitude of incident compressional wave,

R is the reflection coefficient,I1

is the acoustic impedance of the upper layer,and I

2is the acoustic impedance of the lower layer.

Refraction coefficients

The relationship of the refraction coefficient holds true only for thenormal incidence (the angle of incidence is zero).Both of these coefficients (reflected and refracted) depend on thecontrast between the acoustic impedances of upper layer and lowerlayer.Since the density of rocks varies over a relatively narrow range, thewave velocity influences the value of acoustic impedance rather thanthe density.So, it can be stated that the energy relation between the reflected andrefracted waves primarily depends on the velocity contrast betweenthe two media.

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