Basic Seismics

7/29/2019 Basic Seismics

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Basic Seismic



Introduction

The fundamental concept of seismic work issimple.

Energy is generated at a source and transmitted

into the earth as seismic waves, these wavesbounce back from reflecting interfaces within theearth and are detected by geophones spread outon the surface of earth.



Seismic Wave Theory



Elastic Properties of Solids

Stress

Stress is force per unit area. Thus when force is appliedto a body, the stress is the ratio of the force to the areaon which the force is applied.

Normal Stress (S)

If the applied force is perpendicular to the area this iscalled a normal stress.

Consider a rectangular block of some solid object suchas rock.

The shaded area show the block in initial state.

Now if a force F is applied on the bottom surface to pulldown or push up the block, this force per unit area (F/A)is called stress.





Strain

When an elastic body is subjected to stresses, changes inshape and dimensions occur.

These changes are called strains.

Longitudinal Strain (EL)

When the force F is applied upward, the length L of theobject increases by dL and when the force F is applied

downward the length L decreases by dL. The ratio of dL to L is called longitudinal strain.

Transverse Strain (EW)

When the force F is applied downward, with the decreasein length, the width W increases by dW.

Similarly, when the force is applied upward, with theincrease of length the width W decreases by dW.

The ratio of dW is called the transverse strain.





Poisson’s Ration (s)

It is the ratio of the transverse strain (EW) to longitudinal strain(EL).

It is the measure of the amount of transverse distortioncompared to the longitudinal distortion.

If the volume of the block after the stress is applied werecompared to original volume, a limit on the value of “s” wouldbe found.

Consider a tensional force. Relating transverse strain (dW/ W)to longitudinal strain (dL/ L):

LdL sW dW //

LdL

W dW

s /

/

The dimension less quantity “s” (elastic constant) may beinterpreted as a measure of incompressibility.

“s” has a prominent exploration significance.





Elastic Material

By elastic material we mean material, which regains its

shape when the force causing deformation is removed. The phenomenon of elastic compression or expansion is a

displacement of the material such that the originalcondition is restored when the stress is removed.

As larger and larger stresses are applied, the elastic limit iseventually reached, beyond which the material cannotreturn to its original configuration when the stress isremoved.

For stresses from zero up to the proportional limit (smaller

than the elastic limit) most materials obey the relationshipcalled Hooke’s Law.



Hooke’s Law

In an elastic material, stress is proportional to the strain.

When several stresses exist, each produces strainsindependently of the others, hence the total strain is thesum of the strains produced by the individual stresses.

Hooke’s Law states that:

Stress is proportional to Strain





Young’s Modulus

By definition of Hooke’s Law:

Stress = constant X strain

S = E X EL

Where the proportionality constant E is called Young’sModulus relates stress to strain and varies according tolithology.

FA = E X dL/L It is interesting to note here that there is a time

dependence associated with Hooke’s Law and theproportionality limit.

A large stress applied for a short time may not exceed the

proportional limit, while a smaller stress applied for a longtime could permanently deform the material.

This may be one of the reasons that rocks behaveelastically with the passage of seismic waves.





Shear Stress

When the force is tangential to the area (a surface), this is

called shearing stress.Shear Strain (Esh)

By the application of shearing stress, an angular distortionis produced with no associated change of volume.

The angle of distortion is called “phi” and is defined as theshear strain Esh.

Shear Modulus (u)

In case of elastic distortion, the stress is proportional to thestrain.

Ssh = u X phi

Where the constant of proportionality “u” is called the shear modulus or the modulus of rigidity.





Volume Stress

A sphere or cube may be compressed by a uniformpressure (force/unit area); the excess pressure will causethe reduction of the volume, say dv.

Volume stress is the extra pressure per unit volume.

Volume Strain

It is the decrease in volume per unit volume i.e., dv/v.

Bulk Modulus (K)

It was found experimentally that volume stress is linearlyproportional to volume strain.

S : dv/v

K = S/(dv/v)

A large bulk modulus means the material is more pliable.So the bulk modulus is a measure of rigidity againstchanges in volume.





Relations Between Elastic Constants

The constants E, phi, M and K are known as elastic

constants and are different for different materials.

For solids there is a direct relationship between “u” and theconstants E and “s” relating to compressional distortion.

These are : -

u = E / 2(1 + s)K = E [3(1 – 2s)]

Seismic Waves

The seismic wave is the cornerstone in all seismic

activities. The seismic geophysicist generates records andprocesses them.



The Nature of Seismic Waves

The seismic wave is a sound wave, which is a mechanical

wave motion in an elastic medium.

When a pebble is dropped into a pond, water waves travelradially outwards, when a piano is played, the wires vibrateand the sound waves spread through the room.

These are examples of wave motion, and they have twoimportant points in common.

- First, energy is propagated to distant points.

- Second, the disturbance travels through the mediumwithout giving the medium as a whole any permanent

displacement.





Thus the ripples spread outward carrying energy with

them, but as one can see by watching the motion of asmall floating body, the water of the pond does not movewith the waves, only the disturbance.

We can summarize:

“a wave is a progressive disturbance propagated

from point to point in a medium or space without progress or advance by the points themselves”.

The Representation of Seismic Waves

We can represent a wave by simply recording thedisturbance, particle displacement, velocity or particlecompression.



Types of Seismic Waves, their Generation and Propagation

Among the various types of waves, the most familiar are thesound or acoustic waves which provide the closest analogy

to the type of waves most often generated and measured inreflection seismology.

There are two major types of seismic waves, body wavesand surface waves.

Body Waves

These are waves propagating inside the elastic material or we can say that these are the waves traveling within themedium. There are two types of body waves:

a. Longitudinal Waves (Compressional/ Primary)

A type of seismic wave in which particle motion is in thesame direction as the wave is moving.

These are the waves we seek to generate and record inexploration seismology.





They are also referred to as compressional wavessince they are associated with an oscillatory changein volume of a given amount of material.

Yet another name frequently applied to them is P-wave, or primary wave.

They are the waves, which arrive “first”.

The velocity (VL) at which these longitudinal waves

travel through the material is determined by theelastic properties of the material.

It can be shown by the relationship:

d u K VL /3/4

d = density of the material

K = bulk modulus

u = shear modulus



As the density of the material increases (with theother properties constant), the velocity of the elasticwave propagation decreases.

As the rigidity (measured by the bulk modulus)increases, the velocity increases.

Velocity for longitudinal wave propagation also

depends on the shear modulus.

This might not have been expected. Denser rocksusually have higher velocities, so that in general therigidity must tend to increase faster than the density.



b. Transverse waves (Shear/ Secondary)

This is a type of seismic wave in which particlemotion is at right angle to the direction in which thewave is moving.

One way of differentiating transverse waves fromlongitudinal waves is, of course, by the direction of oscillation of the particles in the solid.

Another way of looking at it is that a small materialchanges its volume with the passage of longitudinalwaves, but changes only its shape with the passageof transverse waves.

These transverse waves are also some times called

shear or rotational waves because they cause anangular deformation at constant volume.

They are also called secondary or S-waves becausethey arrive “second” following the longitudinal waves.





If the transverse oscillation is entirely in vertical plane, thewaves are called Sv, waves.

If the oscillation is entirely in horizontal plane, they are

called SH waves. The velocity (VS) of a shear, wave is related to elastic

constants and bulk density of the material in which it ispropagated as:

Longitudinal wave velocity is always greater thantransverse wave velocity.

Transverse waves are rarely used in seismic explorationfor petroleum.

However, a great deal of research has been carried outwith the object of putting it use and combining it with the P-waves to infer the elastic constants and bulk density of thetransmitting media.



Measuring the longitudinal, along with the transverse wavevelocity, we may infer the elastic constants utilizing the

following equations.



Surface waves

If a pebble is thrown into a pond or a strong wind

blows across an open stretch of water, the surface of the water is disturbed and waves that travel awayfrom the source are setup.

The amplitude of the wave is greatest at the surface;at the depth of a wavelength the displacement at anypoint is only a fraction of what is at the surface.

Such waves called surface waves, can occur in asolid in that case they are controlled by elasticity.

Whereas the surface waves in water in the above

example are controlled by gravity. There are two types of surface waves in solids:

Rayliegh waves and Love waves.



Rayliegh waves

The surface waves of most interest in seismic workare called Rayliegh waves and occur as ground roll

along the air-rock interface.

These waves travel along only the free surface of anelastic solid.

The practical motion is in a vertical plane and is

elliptical and retrograde with respect to the directionof propagation.

The amplitude of the motion velocity of Rayleighwaves is slower than for body waves in the samemedium.

It is about nine tenths of that of the S-wave.

The Rayleigh waves can be identified by low-velocity,low frequency. They often obscure reflections onseismic records.





Love waves

These waves are observed only when there is a lowvelocity layer overlying a medium in which elastic

waves have a higher speed. The wave motion is horizontal and is perpendicular to

the direction of propagation.

It has been demonstrated that these waves

propagate by multiple reflection between the top andbottom surfaces of the low velocity layer.

Love waves propagate with a velocity equal to thevelocity of shear waves in the upper layer in the lower medium for a very long wavelengths.

Love waves, being transverse waves, are seldomrecorded in reflection seismic prospecting for whichthe detectors are designed to respond to only thevertical ground motion.





Huygen’s Principle

The basic idea is that at any instant in time, a wavefront can be considered to present a series of pointsources of which each emits a spherical wave.

At some later instant in time, the new location of theactual wave front is where these overlapping

spherical waves interfere constructively. The inner solid lines represent snapshots of a wave

front.

After a unit of time passes, a new snapshot is theouter solid line.

Note that a fair degree of resolutions or description of the irregularly shaped initial wave front is lost on later wave fronts.



Plane Waves, Ray Path and Wavefront

At large distances from the source, the spherical

wavefronts are so large that they can be treated as aplane and are sometimes called “plane waves”.

It is often convenient to illustrate a wave by drawingthe line perpendicular to the wave front.

This line is called a “ray” or “wave path”.

Huygen’s wavelets travel equal distance in a giventime from all points on a wavefront, so wavefrontsshown at equal time intervals are concentric spheresand the ray are straight lines.

Such displays of wavefronts on a graph of horizontaldistance versus depth are called wavefront charts.



Law of Reflection States that the angle of incidence (between the ray

and the perpendicular to the reflecting surface) isequal to the angle of reflection (between the ray andthe perpendicular to the reflecting surface).



Snell’s Law

States that a wave traversing a boundary between twomedia of velocity v1 and v2 is refracted in such a way

that:

2

sin

1

sin

v

r

v

i

When Sin i = v1/v2, Sin r becomes 1 and r becomes 90

degree.

This means that the refracted wave path does notpenetrate the medium but travels along the interface.

The angle “ic” whose Sin is v1/v2 is called the “criticalangle of incidence”.

For all values of “i” greater than the critical ang le, thereis no refraction into the second layer and the wave istotally reflected.



Head Waves

Waves entering and leaving the high-speed layer at thecritical angle are sometimes called headwaves.

Diffraction of Waves

Diffraction will occur in the vicinity of any irregularity in anelastic interface, especially at a discontinuity.

In geology, such irregularities are common.

Diffractions associated with geologic faults are often of practical importance in seismic prospecting.

At distances from the diffracting surface, which are, largecompared to the seismic wavelength, one can use

Huygen’s method to analyze diffraction phenomena. It is interesting to note that a reflection can be thought of

as the interference result of diffractions from the variouspoints lying all along the reflector.



The incident energy could be a reflected or refractedseismic wave coming from any direction.

However, when the interface terminates, there is anabsence of sources and therefore an absence of interferingwavelets on of side of the termination.

The resulting wavefronts then appear as if the termination

were a point source. The amplitude of a diffraction falls off rapidly with distance

from the source.



Reflection Coefficient (R)

For the specific case of normal incidence (ray pathperpendicular to interface) there is a simple equation

relating the incident amplitude Ai to the reflected amplitude Ar.

The ratio Ar / Ai is called the “reflection coefficient”, R.

The equations which relate these quantities are quite

simple. If the seismic velocity is v, the rock density is rho( ), the

product, *v, defines the “acoustic impedance” of the rock.

The particle velocity is the time rate of change of thedisplacement, and the acoustic pressure is equal to the

product of acoustic impedance and particle velocity.

If R is defined by:

)11()22(

)11()22(

vd vd

vd vd R



Then the reflection coefficient for displacement and particlevelocity is equal to –R while that for acoustic pressure is+R.

The parameter R ranges between –1 to +1.

A positive value means that the wave travels from a

material of lower impedance into a material of higher impedance.



Transmission Coefficient ( T )

The ratio of transmitted energy in a P-wave (Et) to incidentenergy (Ei) is:

2

1122

112

d vd v

d v

Ei

Et

The square root of this ratio is termed as transmission

coefficient (T). The transmission coefficient also expresses the relative

amplitude of the transmitted to the incident waves.

1122

112

d vd v

d v

Ai

At T

For displacement or particle velocity, it is equal to 1-R,while for pressure it is equal to 1+R.



The transmission coefficient for displacement and particlesvelocity is different from that for pressure because of thedifference in sign of their reflected waves.

Both forms of the transmission coefficient can have valuesfrom zero to two.

For any value greater than one the transmitted amplitude islarger than the incident amplitude.



Mode Conversion

When a wave is incident obliquely, angular stresses occur at the interface.

There must be some reaction of the rock to these forces.

The reaction is the generation of shear waves at theinterface where only compressional waves were incident – an effect known as mode conversion.

Both reflected and refracted shear waves are generated,and they carry away part of the energy.

The variation also depends strongly on the values of theelastic constants of the rocks.

It can be seen that the reflector coefficient for compressional waves decreases slowly as the angle of incidence increases until near the critical angle, where allthe energy which was refracted at smaller angles suddenlyis available for reflection.



Notice that the ocean bottom and the base of the

weathering can reflect huge amounts of energy bycomparison, which can cause serious problems in seismicexploration efforts.

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Basic Seismics