Elastic Waves

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Introduction to Seismology

Ali [email protected]

Department of Earth SciencesKFUPM

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Chapter 3Body Elastic Waves

http://faculty.kfupm.edu.sa/ES/oncel/geop204chap3.htmChapter 4, Bullen and Bolt

http://faculty.kfupm.edu.sa/ES/oncel/geop204presenta.htm

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http://faculty.kfupm.edu.sa/ES/oncel/oncellinks.htm

http://faculty.kfupm.edu.sa/ES/oncel/geop204link.htmSome Links

Recall: Wavecrests (high points)

troughs (low points)

equilibrium

(middle)

wave speed = wavelength/period = wavelength x frequency.

We often express this as v = f λ

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Wave Equation

α and β are termed for the P-wave and S-wavevelocities. Often, the symbols Vp and Vs are usedinstead of α and β.Θ is the scalar displacement potential.Where µ,λ are the Lamé coefficients

where λ is bulk modulus (incompressibility), µ shear

modulus (rigidity) and r density.Intr

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λ = k - =2µ ν E3 ( 1 + ν ) ( 1 - 2ν )

V = α = =k + ( )µ λ + 2µ

ρ ρp4/3

V = β = µρs

Where µ,λ are the Lamé coefficients and λ is

Seismic velocities

Question: How α and β depend

on density ρ?

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P wave velocity α and S wave velocity β depend on physical properties of medium through which they travel:

Rock Type Density Young's Modulus Poisson's Ratio Vp Vs Vp/Vs Vs as %Vpr E m (m/s) (m/s)

Shale (AZ) 2.00 0.120 0.040 2454 1698 1.44 69.22%Siltstone (CO) 2.00 0.120 0.040 2454 1698 1.44 69.22%

Limestone (PA) 2.00 1.100 0.156 7640 4877 1.57 63.84%Limestone (AZ) 2.00 1.100 0.180 7728 4828 1.60 62.47%Quartzite (MT) 3.00 0.636 0.115 4675 3083 1.52 65.96%

Sandstone (WY) 3.00 0.140 0.060 2169 1484 1.46 68.42%Slate (MA) 3.00 0.487 0.115 4091 2698 1.52 65.96%Schist (MA) 3.00 0.544 0.181 4440 2771 1.60 62.41%Schist (CO) 2.70 0.680 0.200 5290 3239 1.63 61.24%Gneiss (MA) 2.64 0.255 0.146 3189 2053 1.55 64.38%Marble (MD) 2.87 0.717 0.270 5587 3136 1.78 56.13%Marble (VT) 2.71 0.343 0.141 3643 2355 1.55 64.65%Granite (MA) 2.66 0.416 0.055 3967 2722 1.46 68.62%Granite (MA) 2.65 0.354 0.096 3693 2469 1.50 66.85%Gabbro (PA) 3.05 0.727 0.162 5043 3203 1.57 63.51%Diabase (ME) 2.96 1.020 0.271 6569 3682 1.78 56.05%Basalt (OR) 2.74 0.630 0.220 5124 3070 1.67 59.91%

Andesite (ID) 2.57 0.540 0.180 4776 2984 1.60 62.47%Tuff (OR) 1.45 0.014 0.110 996 659 1.51 66.20%

Elastic Coefficients and Seismic Velocities

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A linear relationship between density and seismic velocity where a and b are constants (Birch, 1961).

Velocity and Density “Birch’s law”Crust and mantle rock observations

6km 18km 30km

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V = a ρ + bThree pressures

Nafe-Drake Curve

Cross-plotting velocity and density values of crustalrocks gives the Nafe-Drake curve after itsdiscoverers.

Only a few rocks such as salt (unusually low density)and sulphide ores (unusually high densities) lie offthe curve.

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An important empirical relation, used in joint interpretation of wide angle reflection and refraction data and gravity data, exists between P wave velocity and density.

Nafe-Drake Curve

Figure 3.10 of Lillie, 1999, modified from Birch, 1960

L=limestone; Q=quartz; Sh=shale; Ss=sandstone.

Sediments and sedimentary rock

Igneous and metamorphic rock

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Reference

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Factors affecting P-wave velocity

Increases with mafic mineral content (Nafe-Drake curve) pressure (modulus change > density change)

Decreases with temperature (modulus change > density change)

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Factors affecting S-wave velocityIncreases with

mafic mineral content (Nafe-Drake curve)with pressure (modulus change > density

change)

Decreases due topresence of fluid, e.g. porous sand or partial melt

No S waves influids, e.g. water of molten rock. Velocity zeroIn

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Grifts and King, 1981

Velocity-Geology

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Maximum amplitude of particle motion occurs alongthe 90 degree phase wave front. Other wavefronts correspond to positions where the wave goesfrom positive to negative amplitude (180 degree)and at the minimum amplitude (270).

Amplitude Changes of Particle Motion

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Reference

Ray paths thus bend (refract) as velocity changes. Seismic energy travels along trajectories perpendicular

to wave fronts.

Initial wavefronts forcompressional (P),shear

(S), and Rayleigh ( R )waves.

Changes in velocity cause segments of wave fronts tospeed up or slow down, distorting the wave frontsfrom perfect spheres.

Wave Fronts and Raypaths

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Reference

Seismic waves radiating from a source to one receiver.

Seismic trace recording ground motion by thereceiver, as a function of the travel time from thesource to the receiver. For controlled source studies(seismic refraction and reflection), the travel time iscommonly plotted positive downward.

Seismic Trace

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Wave equationElastic Coefficients and Seismic Waves Birch's LawNafe-Drake CurveFactors affecting P-wave and S-wave velocitySeismic velocities for Geological MaterialsAmplitude Changes of Particle MotionsAnimation: Particle Motion in Seismic WavesWavefronts and RayPathsSeismic Trace

Previous Lecture

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From: http://web.ics.purdue.edu/~braile/edumod/slinky/slinky.htm

W

avefro nt

t0t1

t2t3

Raypath

Compressional (P) motion

Shear (S) motion

Source

Perpendicular

angle

Recall: Wavefronts and raypaths

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http://www.geol.binghamton.edu/faculty/jones/SeismicWavesSetup.exe

Seismic Waves A program for the visualization of wave propagation contributor: Alan JonesYear: 2006 In

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From: http://www.citiesoflight.net/AlaskaQuake.htmlIntr

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Solution for Homework 2

1

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5 4

Write up phases of from 1 to 6?

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Body Wave PropagationP- and S- Waves (propagation along raypath)

X

Y

S-wave particle motion -- perpendicular to direction of propagation (usually approximately in SV and SH directions)

Earth’s surface

* P-wave particle motion -- parallel to direction of propagation

Source

Seismograph

Z (down)

SV

SH

Modified from http://web.ics.purdue.edu/~braile/edumod/slinky/slinky.htmIntr

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Identify the waves of Body and Surface?

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Three-component seismograms for the M6.5 west coast of Chile earthquake recorded at NNA

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“body waves”

P-waves (“P” for primary)

S-waves (“S” for secondary)

Expansion/compression:push/pull motion

Shear:side-to-side motion

“surface waves” travel on Earth’s surface

travel in Earth’s interior

Recall: Seismic Wave Types

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Surface Waves - Body Waves

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Reference

S-waves do not travel through fluids, so do not exist in Earth’s outer core (inferred to be primarily liquid iron) or in air or water or molten rock magma). S waves travel slower than P waves in a solid and, therefore, arrive after the P wave.

Alternating transverse motions (perpendicular to the direction of propagation, and the raypath); commonly approximately polarized such that particle motion is in vertical or horizontal planes.

S,Shear, Secondary, Transverse

P motion travels fastest in materials, so the P-wave is the first-arriving energy on a seismogram. Generally smaller and higher frequency than the S and Surface-waves. P waves in a liquid or gas are pressure waves, including sound waves.

Alternating compressions (“pushes”) and dilations (“pulls”) which are directed in the same direction as the wave is propagating (along the raypath); and therefore, perpendicular to the wavefront.

P,CompressionalPrimary, Longitudinal

Other CharacteristicsParticle MotionWave Type(and names)

Seismic Body Waves

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Rayleigh waves are also dispersive and the amplitudes generally decrease with depth in the Earth. Appearance and particle motion are similar to water waves. Depth of penetration of the Rayleigh waves is also dependent on frequency, with lower frequencies penetrating to greater depth. Generally, Rayleigh waves travel slightly slower than Love waves.

Motion is both in the direction of propagation and perpendicular (in a vertical plane), and “phased” so that the motion is generally elliptical – either prograde or retrograde.

R,Rayleigh, Surface waves, Long waves, Ground roll

Love waves exist because of the Earth’s surface. They are largest at the surface and decrease in amplitude with depth. Love waves are dispersive, that is, the wave velocity is dependent on frequency, generally with low frequencies propagating at higher velocity. Depth of penetration of the Love waves is also dependent on frequency, with lower frequencies penetrating to greater depth.

Transverse horizontal motion, perpendicular to the direction of propagation and generally parallel to the Earth’s surface.

L,

Love, Surface waves, Long waves

Other CharacteristicsParticle MotionWave Type

(and names)

Seismic Surface Waves

From: www.eas.purdue.edu/~braile/edumod/waves/WaveDemo.htmIntr

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http://www.geol.binghamton.edu/faculty/jones/AmaSeis.htmlIntr

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The Using AmaSeis Tutorial:http://web.ics.purdue.edu/~braile/edumod/as1lessons/UsingAmaSeis/UsingAmaSeis.htm

Homework due to March, 19: Plot Seismic Trace for one of available recent earthquakes given by program and try to explain your observations for Seismic Waves such as Picking Body Waves, time for S-P and values of maximum amplitude?

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Seismic Waves of Argentina EQ

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Previous Lecture

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Seismic Wave TypesRevisit: Wavefronts and raypaths

Seismic Waves A program for the visualization of wave propagation contributor: Alan Jones

Body Wave PropagationExample: M6.5, 1998 West Coast of Chile EarthquakeExample: Ms7.8, 1999 Izmit Earthquake, Turkey

Revisit: Seismic Wave Types Downloading the AmaSeis Software

Homework: Seismic Trace Exercise by AmaSeis, Due to March, 19

Term Paper: Refraction

SeismologyDue to March 21

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For more detail, visit to Project Page of Geop204

http://faculty.kfupm.edu.sa/ES/oncel/geop204termproject.htm

•Travel time graph. The seismic traces are plotted according to the distance (X) from the source to each receiver. The elapsed time after the source is fired is the travel time (T).

Travel-Time Graph

X distance from source to the receiver,T total time from the source to the receiverV seismic velocity of the P, S, or R arrival.

Initial wave fronts for P, S and R waves, propagating across several receivers at increasing distance from the source.

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Reference

T=X/V

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Estimates of Seismic Velocity

B) The slope of the travel time for each of the P,S, and R arrivals (see earlier figure) is the inverse of velocity.

A) The slope of line for each arrival is the first derivative (dT/dX).

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Reference

© John F. HermanceSeptember 05, 2002

Model CalculationSimple, Horizontal Two Layers

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Ray paths

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Ray paths for direct, reflected, and critically refracted waves, arriving at receiver a distance (X) from the source. The interface separating velocity (V1) from velocity (V2) material is a distance (h) below the surface.

From: www.aug.geophys.ethz.ch/teach/seismik1/03_geometry.pdfIntr

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Huygen’s Principle

See pp.75 and 152 of Bullen&Bolt

pp. 20 of Burger’s book.

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Fermat’s Principle

pp. 20 of Burger’s book.

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Fermat's principle leads to Snell's law;

Snell’s Law

For a wave traveling from material of velocity V1 into velocity V2 material, ray paths are refracted according to Snell’s law.

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Reflection/Refraction The angle of incidence equals the angle of reflection θ i =θ r , where both angles are measured from the normal:Note also, that all rays lie in the “plane of incidence”.

θi θr

How is the angle of refraction related to the angle of incidence?

Unlike reflection, θ 1 cannot equal θ 2!!• Why?? Remember v = fλv1 ≠ v2

Therefore, θ 2 must be different from θ 1 !!

θ1

θ2

n1

n2

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A) A compressional wave, incident upon an interface at an oblique angle, is split into four phases: P and S waves reflected back into the original medium; P and S waves refracted into other medium.

Reflected/Refracted Waves

See pp.140-152 of Bullen&BoltIntr

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θ2

θ1

•Wave fronts are distortedfrom perfect spheres as energy transmitted into material of different velocity. Ray paths thusbend (“refract”) across an interface where velocity changes.

The angles for incident and refracted are measured from a line drawn perpendicular to the interface between the two layers.

Seismic Refraction

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Behavior of Refracted Ray on Velocity Changes

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Behavior of Seismic Waves Penetrating the Earth

At the mantle-outer core (fluid) boundary the decrease in velocity causes those rays refracted into the core to bend towards the normal

In the mantle and inner core, the velocities increase with depth, so the ray bend away from the normal

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Modules of Bulk (k) and Shear (µ)Bulk Moduluswhere Θ = dilatation = ∆V/V

and P = pressurek= (∆P/Θ)

Ratio of increase in pressure to associated volume change

shear stress = (∆F /A)

µ = shear stressshear strain

shear modulusshear strain = (∆l /L)

Force per unit area to change the shape of the material

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Recall

Ratio Vp and Vs depends on Poisson ratio:

where

Poisson’s ratio varies from 0 to ½. Poisson’s ratio has the value ½ for

fluidsSee pp.32 of Bullen&Bolt

∆LLεxx =

∆WWεyy =

σ= (εyy / εxx)

Ε = (∆F /A)(∆L/L)

Young Module

Poisson’s Ratio

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Recall

Poisson’s Ratio/Young Module

The elastic constants E, σ, µ are mostly used in works of engineering seismology because they are easily measured by simple experiments.

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Seismic Velocities (P-wave)

See pp.318 and 471 of Bullen&Bolt

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Recall

Rock Velocities (m/sec)

pp. 18-19 of Berger

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Influences on Rock Velocities

• In situ versus lab measurements• Frequency differences• Confining pressure• Microcracks• Porosity• Lithology• Fluids – dry, wet• Degree of compaction•……………

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Recall

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Refraction and Reflection

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Travel-time GraphEstimates of Seismic Velocity Huygens's Principle Fermat's Principle Calculation of Travel TimesSnell's law-Critically Refracted Arrival Reflection/RefractionReflected/Refracted wavesSeismic Refraction Behavior of refracted ray on velocity changesBehavior of seismic waves refracted ray penetrating

the EarthRepresentative P-wave Velocities for various RocksInfluences on Rock Velocities

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Refracted Ray and Angle

The angle of refraction increases as the angle of incidence increases.

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Energy Return and Critical Angle

A critically refracted wave, traveling at the top of the lower layer with velocity V2, leaks energy back into the upper layer at the critical angle (θ2)

θc θc θc θc θc θc

Lillie, Whole Earth Geophysics, Fig 3.25

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Angle of For Incident P wave For Incident S waveIncidence Reflected Refracted Reflected Refracted

P-wave S-wave P-wave S-wave S-wave P-wave S-wave P-wave10 10.0 6.0 27.6 16.1 10.0 16.8 27.6 50.511 11.0 6.6 30.6 17.8 11.0 18.5 30.6 58.012 12.0 7.2 33.7 19.4 12.0 20.3 33.7 67.513 13.0 7.8 36.9 21.1 13.0 22.0 36.9 88.814 14.0 8.3 40.2 22.8 14.0 23.8 40.2 #NUM!15 15.0 8.9 43.6 24.5 15.0 25.6 43.6 #NUM!16 16.0 9.5 47.3 26.2 16.0 27.3 47.3 #NUM!17 17.0 10.1 51.2 27.9 17.0 29.2 51.2 #NUM!18 18.0 10.7 55.5 29.6 18.0 31.0 55.5 #NUM!19 19.0 11.3 60.2 31.4 19.0 32.9 60.2 #NUM!20 20.0 11.8 65.8 33.2 20.0 34.8 65.8 #NUM!21 21.0 12.4 72.9 35.0 21.0 36.7 72.9 #NUM!22 22.0 13.0 87.4 36.8 22.0 38.6 87.4 #NUM!23 23.0 13.6 #NUM! 38.7 23.0 40.6 #NUM! #NUM!24 24.0 14.1 #NUM! 40.6 24.0 42.7 #NUM! #NUM!25 25.0 14.7 #NUM! 42.5 25.0 44.8 #NUM! #NUM!26 26.0 15.2 #NUM! 44.5 26.0 46.9 #NUM! #NUM!27 27.0 15.8 #NUM! 46.6 27.0 49.2 #NUM! #NUM!28 28.0 16.4 #NUM! 48.7 28.0 51.5 #NUM! #NUM!29 29.0 16.9 #NUM! 50.9 29.0 53.9 #NUM! #NUM!30 30.0 17.5 #NUM! 53.1 30.0 56.4 #NUM! #NUM!

Modified Table

V1-P (m/s) 1500V1-S (m/s) 900V2-P (m/s) 4000V2-S (m/s) 2400In

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Table 2.3 after Berger, pp.29.

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22

21

1

2

21 VVVVhX

Vtrefraction −+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

Total Time of Refraction

Ttotal= T1+T2+T3

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Travel time for Direct/Refracted Waves

12

2112

VVVVhxcr −

+=

Xc=critical distanceXcr=crossover distanceT1= Intercept time

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Seismic Reflection

Reflection occurs when Z1 differs from Z2, where ZAcoustic impedance which is product of density and velocity

V-shaped ray paths for a compressional wave from asource to 6 receivers, reflected from a horizontal interface.

=Z1

=Z2

Lillie, Whole Earth Geophysics, Fig 3.28

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Reflection equation for a reflection hyperbolae:

1

2/122 )4(V

hXt r

+=

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Tim

e

Distance

?

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Dire

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Reflec

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?Refracted or Head Wave

?Crossover distanceti

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IRIS Deployment in Venezuela, 2001

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The source of energy:Betsy M3 Seisgun "l

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The source of energy:Betsy M3 Seisgun

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That is what named as “Model 130-01” which was ordered for ESD in 2006. From: http://www.reftek.com/productshome.html#Seismic%20RecordersIn

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21 “Texans” from Refraction Technology, Inc.

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From: http://www.seismo.unr.edu/geothermal/

200 “Texans”

From: http://www.seismo.unr.edu/geothermal/Intr

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The N. Walker Lane Experiment, 2002

From: http://www.seismo.unr.edu/geothermal/Intr

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How Thick is the Crust?

Journal Publication: Louie, J. N., W. Thelen, S. B. Smith, J. B. Scott, M. Clark, and S. Pullammanappallil, 2004, The northern Walker Lane refraction experiment: Pn arrivals and the northern Sierra Nevada root: Tectonophysics, 388, 253-269.

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?

?Horizontal Rays

Refraction“Tunneling”

7.2 km/s Moho

?

?Horizontal Rays


?

?Horizontal Rays


7.2 km/s Moho

Elev

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180 meter

KFUPM BEACH-2005

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The length of profile, which is 180 meter in this case, provided a depth of resolution to 60 meter but note that velocity in shallow is not detailed due to increased spacing of receivers (=15 meter)?

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10 meter

KFUPM-2006

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In this case, the receiver distance is about 0.4 meter but provided detail information in depth of very shallow even we could not have info about the detail.

Ele

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Class Feedback

What is the most important thing you learned this week?

What is one thing you still do not understand?

Respond to me anonymously over a piece of small paper within 1 or 2 minutes.

Intr

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KFU

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Elastic Waves

Technology

Transcript of Elastic Waves