seismic methods iii - geofizika.uni-miskolc.hu · ray theory is generally used in geometrical...

Geophysics of Exploration for Water

Seismic methods III

edited by Peter Vass

Ray theory

If we want to study in what directions the waves travel and how theboundaries of different media modify the direction of wavepropagation, it is worth applying the approach of ray theory.While in wave theory, the physical description of wave motion isimportant (mechanical quantities as functions of time and space areused), the geometrical relationships of the propagation of waveenergy is studied in ray theory (e.g. tracing the routes of waves in alayered subsurface structure by using geometrical quantities).Ray theory is generally used in geometrical optics where the ray is anidealized model of paths along which light waves travel.The principles used for light waves are also applicable to seismicwaves in ray theory.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 local energy transfer connectedto the wave propagation.

Ray theoryThis figure demonstrates the geometrical relation between wave-fronts and ray paths of a seismic wave.It can be seen that a seismic wave propagates not only in a singledirection and the ray paths may bend because of continuous velocitychange in a medium.Bed boundaries with sudden (step-wise) velocity change will result inbreak points in the ray paths, where the direction of local wavepropagation sharply change.

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Ray theoryThis figure shows a so-called horizontally layered half-space which isa frequently used geophysical model of sedimentary basins.Some of the possible ray paths of seismic waves are also presented.The ray paths penetrate some of the layers at their points ofrefraction and reflect off one of the bed boundaries.The ray paths start from a source point located on the surface andarrive at a receiver which is also planted on the surface.

http://www.crewes.org/ResearchLinks/Converted_Waves/Page2.html

Refractions of waves are necessarilyassociated with scanning deeper partsof the subsurface, but wave arrivals,which carry information on the sitegeology, cannot be detected on thesurface without reflections.

Reflection and refraction of seismic waves

When a seismic body wave (either a compressional or a shear wave)arrives at an interface separating two media with different elasticproperties, some part of the energy will reflect off the interface andthe other part will penetrate into the second medium.

Reflection is a phenomenon which occurs when a wave arrives at aninterface between two different media, the direction of propagationsuddenly changes, and the wave front returns into the medium fromwhich it came (there is no penetration).

Refraction is a phenomenon which occurs when a wave arrives at aninterface between two different media, the direction of propagationsuddenly changes, the wave front passes the interface and penetratethe other medium.


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 two pairs is a compressional wave (Pwave) and the other is always a shear wave (S wave).

Thus, the energy of an incident wave splits among the followingwaves:• a reflected compressional wave,• a reflected shear wave,• a refracted compressional wave,• and a refracted shear wave.

After the occurrence of these boundary events, all the newlyproduced waves travel their own ways with different velocities indifferent directions.


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

p.

S in subscript = shear waveP in subscript = compressional

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

The angles are referred to the axis of incidence (which is perpendicular to the boundary).

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


There is a relationship between the different angles and wavevelocities which is formulated by the so-called generalized Snell'slaw.

This law states that the ratio between the sine of the angle and theappropriate wave velocity yields a constant value which is identicalfor each type of waves for a given bed boundary.This constant value depends on the angle of incident wave and thevelocity of compressional wave in the upper medium.If this value with the velocities of different waves are known in aparticular case, the angles of reflections and refractions can becalculated by the help of the relationship.


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 come into being.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

Apart from some special applications,only the compressional (P-) wavearrivals are required to detected inseismic surveys.So, the normal incident is the mostadvantageous case from an energeticpoint of view (no energy expended onshear (S-) waves).


Unfortunately, the case of normal incidence cannot be implemented in thefield (because the application of a seismic source and a receiver at thesame point and time is impossible).However, a near-normal (or near-vertical) incidence may be carried out. Inthis case, the distance between the seismic source and the receiver ismuch shorter than the depth of boundary (thus the angle of incidence issmall). Due to the small deviation from normal incidence, the energyexpended on shear waves is reduced.

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

In the case of near-surface seismicsurveys, even the near normalincidence is impossible to provide, sosome (not neglectable) energy losscoming from this effect must becalculated upon.


There is another special and important case of boundary phenomena,which is connected to the refracted waves.When the seismic wave travels faster in the lower layer than in theupper one (V

2> V

1), which is often fulfilled in practice, the angle of

refraction is grater than the angle of incidence (r > i).This phenomenon can also be derived from Snell's law.

We can also see that the increase in angle of incidence results in theincrease in angle of refraction.When the angle of incidence reaches a certain value, the angle ofrefraction will be 90°. It means that the direction of wave propagationwill be perpendicular 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:

A wave critically refracted propagates collaterally with the bed boundary. Itis a body wave whose velocity corresponds to the lower layer (V

P2).

Due to the oscillation of material particles, the points of the boundary willbehave as sources of a secondary seismic wave.This secondary wave called head wave leaves the boundary travelsupwards through the upper layer, and its ray paths subtend critical angle tothe vertical.


The wave-front of the head wave is a parallel plane and perpendicular to the ray paths.

Reflection and refraction of seismic wavesThe wave-front of the head wave arrives the surface. The arrival time linearlyincreases with the distance from the source.By detecting the arrivals of the head wave on the surface, we can obtaininformation about the depth of the bed boundary and the wave velocity of lowerlayer. Of course, the application of suitable data processing techniques isrequired to extract this information from the raw data set.Actually, the so-called seismic refraction method exploits the phenomenon ofcritical refraction and is used to derive geological information about the positionof bed boundary (refractor). Its application is typically limited to shallowerintervals of the subsurface.


Reflection coefficient

Snell's law is a geometrical relationship which does not give any informationabout the amplitudes or the amplitude ratios of the different waves cominginto being at bed boundaries.Since the amplitude and the carried energy of a seismic wave is closelyrelated, the amplitude ratios also inform about the division of incidentenergy into different components transmitted to the newly producedrefracted and reflected waves.In order to resolve this problem, we must get acquainted with a quantitycalled acoustic impedance.Acoustic impedance is an acoustic property of the medium. It is derivedquantity calculated as the product of mass (or bulk) density () and wavevelocity (V):

I = ·V

By means of the acoustic impedance, the so-called reflection coefficient(R) can already be defined. It gives the amplitude ratio of the reflectedcompressional wave to the incident compressional wave in the case ofnormal incidence (when the angle of incidence is equal to zero).


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 bed boundaries is usually asmall 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 the reflection coefficient canreach as high value as 0.7 (70 %) or even higher.Such excellent reflectors are the surface of the oceans and thesurface of the Earth itself. In these cases the contrast between theacoustic impedances (air / water and air / ground) is great.

Refraction coefficient

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 fornormal incidence.It can be expressed by means of the acoustic impedances 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 coefficient

The relationships of both coefficients hold true only for the normalincidence (the angle of incidence is zero).In the case of arbitrary incidence, the solution of the problem is morecomplex. However, the reflected energy of compressional wavescannot be higher in general case than that in the case of normalincidence.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 energetic relations among incident,reflected and refracted waves primarily depend on the velocitycontrast between the two neighbouring media.

Velocity of seismic waves

From the point of view of seismic methods, the most importantproperty of rocks is the velocity of P-wave (or compressional wave).The velocity of P-wave depends on the density and the elasticproperties (elastic moduli) of rocks.

Actually, the elastic moduli also depend on the density.Increasing density entails more intensive increase in elastic modulithan a linear one.This is the reason why the P-wave usually travels faster in denserrocks (while the formula suggests an inverse effect of the density).However, high value of P-wave velocity can associate with averageor low value of density in some special cases (e.g. halite or rock salt,=2.1-2.2 g/cm3, vp= 4000-5500 m/s2).


Since P-waves can propagate not only in solids but also influids, all the factors affecting the bulk density of a rockinfluence the velocity of P-wave.

1. Mineral composition of rock

Different minerals have usually different densities.The density of a mineral depends on its chemical compositionand crystal structure.If a rock is made up of several minerals, the volume fractionsand the density values of mineral components jointly determinethe bulk density of rock.However, the density of mineral components does not alwaysplay determinant play, because the elastic moduli can have highvalue along with density of lower value (e.g. halite).The exception stands beside the rule, but we must know theexception.


2. Porosity of rock

Since the fluids filling the pore space of a rock have significantly lessdensity than that of solid components, the porosity decreases thebulk density of rocks.So, increasing porosity results in decreasing P-wave velocity.

3. Density of fluid in the pore space

When the pore space is occupied by only one fluid phase (e.g. water,or air), the density of the single fluid determines how the porositydecreases the bulk density of rock.The lower the density of fluid is, the lower the bulk density of porousrock is.Note that the density of water slightly depends on its salinity.Decreasing fluid density results in decreasing P-wave velocity.The velocity of P-wave is significantly higher in the case of water filledporosity than in the case of dry (air filled) or gas filled porosity.

Velocity of seismic waves4. Saturation of fluids filling the pore space

When more than one fluid phase are present in the pore space, notonly the densities of fluid phases but also their saturations (ratios) areof importance (such an environment can be the pore space of a rockabove the level of ground-water table).The higher the saturation of a fluid phase having lower density is, thelower the bulk density and P-wave velocity of rock is.

5. Compaction and consolidation

It is generally true that compaction and consolidation increase thedensity of sedimentary rocks.The increase of P-wave velocity with the depth of burial is a normalphenomenon, because increasing lithostatic pressure (or overburdenpressure) results in greater degree of compaction, which entails lessporosity and higher density.


Contrary to P-waves, S-waves (shear waves) are not able to travelthrough fluids, therefore the porosity and the fluid in the pore spacedo not influence the velocity of S-wave.

An S-wave avoids the pore space during its propagation, so itsvelocity depends on the solid rock framework.

Velocity of P-wave in different rocks

Rock or material VP

(m/s)

air 330water 1400-1500ice 3000-4000permafrost 3500-4000weathered layer 250-1000alluvium, sand (dry) 300-1000sand (water saturated) 1200-1900clays 1100-2500glacial moraine 1500-2600coal 1400-1600sandstones 2000-4500slates and shales 2400-5000marls 2000-3000limestones and dolomites 3400-6000anhydrite 4500-5800rock salt (halite) 4000-5500granite and gneiss 5000-6200basalt flow top (highly fractured) 2500-3800basalt 5500-6300gabbro 6400-6800dunite 7500-8400

This table shows the P-wave velocity of some materials and rocks.

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

Velocity of P-wave in different rocks

The P-wave velocity of rocks varies in a wider range than the densityof rocks.The highest velocity values are connected to the igneous and highgrade metamorphic rocks (e.g. granite, gneiss )But fracturing of these rocks decreases the value of velocity (seebasalt flow top).In the case of sedimentary rocks, there can be great differencesamong the velocity values even within the same formation.The degree of compaction and consolidation, the porosity, the fluidcontent, the saturation and the mineral composition are collectivelyresponsible for the P-wave velocity.The lowest velocity values, which are less than the velocity in water,occur in unconsolidated sediments with dry porosity near the surface(see dry sand and weathered layer).The zone of these sediments ranging from the surface to the ground-water table is called weathered layer or low-velocity layer (LVL).

seismic methods iii - geofizika.uni-miskolc.hu · ray theory is generally used in geometrical...

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