Elastic Waves

Introduction to Seismology-KFUPM

Introduction to Seismology
Chapter 3
Body Elastic Waves
http://faculty.kfupm.edu.sa/ES/oncel/geop204chap3.htm
Chapter 4, Bullen and Bolt

Ali Oncel
oncel@kfupm.edu.sa
Department of Earth Sciences
KFUPM

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

http://faculty.kfupm.edu.sa/ES/oncel/oncellinks.htm
Some Links
http://faculty.kfupm.edu.sa/ES/oncel/geop204link.htm

Recall: Wave
crests (high points)

equilibrium
(middle)

troughs (low points)

wave speed = wavelength/period = wavelength x frequency.

We often express this as v=fλ

1


Wave Equation

α and β are termed for the P-wave and S-wave
velocities. Often, the symbols Vp and Vs are used
instead 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.

Seismic velocities
P wave velocity α and S wave velocity β depend on
physical properties of medium through which they
travel:
k + ( 4/3)µ λ + 2µ
V = α = = ρ
p
ρ

µ Question: How α and β depend
Vs = β =
ρ on density ρ?

Where µ,λ are the Lamé coefficients and λ is

λ = k - 2µ = νE
3 ( 1 + ν ) ( 1 - 2ν )

Elastic Coefficients and Seismic
Velocities
Rock Type Density Young's Modulus Poisson's Ratio Vp Vs Vp/Vs Vs as %Vp
r 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%

2


Velocity and Density “Birch’s law”
Crust and mantle rock observations
A linear relationship between density and seismic velocity
where a and b are constants (Birch, 1961). V = a ρ + b
Three pressures

6km 18km 30km

Nafe-Drake Curve
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.
Cross-plotting velocity and density values of crustal
rocks gives the Nafe-Drake curve after its
discoverers.

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

Nafe-Drake Curve

Sediments and
sedimentary rock

Igneous and
metamorphic rock

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

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

3


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)

Factors affecting S-wave velocity
Increases with
mafic mineral content (Nafe-Drake curve)
with pressure (modulus change > density
change)

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

No S waves in
fluids, e.g. water of molten rock. Velocity
zero

Velocity-Geology

Grifts and King, 1981

4


Amplitude Changes of Particle Motion

Reference

Maximum amplitude of particle motion occurs along
the 90 degree phase wave front. Other wave
fronts correspond to positions where the wave goes
from positive to negative amplitude (180 degree)
and at the minimum amplitude (270).

Wave Fronts and Raypaths
Initial wavefronts for
compressional (P),shear
(S), and Rayleigh ( R )
waves.
Changes in velocity cause segments of wave fronts to
speed up or slow down, distorting the wave fronts
from perfect spheres.

Reference

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

Seismic Trace

Reference

Seismic waves radiating from a source to one receiver.

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

5


Chapter 3
Body Elastic Waves

Ali Oncel
oncel@kfupm.edu.sa
KFUPM

Previous Lecture

Wave equation
Elastic Coefficients and Seismic Waves
Birch's Law
Nafe-Drake Curve
Factors affecting P-wave and S-wave velocity
Seismic velocities for Geological Materials
Amplitude Changes of Particle Motions
Animation: Particle Motion in Seismic Waves
Wavefronts and RayPaths
Seismic Trace

Recall: Wavefronts and raypaths
nt
fro ular
v e erpendic t3
P
a

le
ang
W

t2
Ray
p ath t1
t0 Com pressional
Source (P) m otion

Shear (S)
m otion
From: http://web.ics.purdue.edu/~braile/edumod/slinky/slinky.htm

6


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

Seismic Waves A program for the visualization of
wave propagation contributor: Alan Jones
Year: 2006

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

Solution for Homework 2

Write up phases of
from 1 to 6?
1

2

3
6

5 4

7


Body Wave Propagation
P- and S- Waves (propagation along raypath)
Earth’s surface

Seismograph X

Y
Source
* SH
SV P-wave particle
motion -- parallel
to direction of
S-wave particle propagation
motion -- perpendicular
to direction of propagation (usually
approximately in SV and SH
Z (down) directions)

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

Identify the waves of Body and Surface?

8

Introduction to Seismology-KFUPM Introduction to Seismology-KFUPM Introduction to Seismology-KFUPM

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

9


Recall: Seismic Wave Types
“body waves” travel in Earth’s interior

P-waves (“P” for primary)
Expansion/compression:
push/pull motion

S-waves (“S” for secondary)
Shear:
side-to-side motion

“surface waves” travel on Earth’s surface

Surface Waves - Body Waves

Reference

Seismic Body Waves
Wave Type Particle Motion Other Characteristics
(and names)
P, Alternating compressions P motion travels fastest in
Compressional (“pushes”) and dilations materials, so the P-wave is the
Primary, (“pulls”) which are directed in first-arriving energy on a
Longitudinal the same direction as the seismogram. Generally smaller and
wave is propagating (along the higher frequency than the S and
raypath); and therefore, Surface-waves. P waves in a liquid
perpendicular to the or gas are pressure waves,
wavefront. including sound waves.

S, Alternating transverse S-waves do not travel through
Shear, motions (perpendicular to the fluids, so do not exist in Earth’s
Secondary, direction of propagation, and outer core (inferred to be
Transverse the raypath); commonly primarily liquid iron) or in air or
approximately polarized such water or molten rock magma). S
that particle motion is in waves travel slower than P waves in
vertical or horizontal planes. a solid and, therefore, arrive after
the P wave.

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

10

Seismic Surface Waves

Wave Particle Motion Other Characteristics
Type
(and names)
L, Transverse horizontal Love waves exist because of the Earth’s
motion, perpendicular surface. They are largest at the surface
Love, to the direction of and decrease in amplitude with
Surface propagation and depth. Love waves are dispersive, that is,
waves, Long generally parallel to the the wave velocity is dependent on
waves Earth’s surface. 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.

R, Motion is both in the Rayleigh waves are also dispersive and the
Rayleigh, direction of propagation amplitudes generally decrease with depth in
Surface and perpendicular (in a the Earth. Appearance and particle motion
waves, Long vertical plane), are similar to water waves. Depth of
waves, and “phased” so that penetration of the Rayleigh waves is also
Ground roll the motion is generally dependent on frequency, with lower
elliptical – either frequencies penetrating to greater depth.
prograde or Generally, Rayleigh waves travel slightly
retrograde. slower than Love waves.
From: www.eas.purdue.edu/~braile/edumod/waves/WaveDemo.htm

Downloading the AmaSeis software

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

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

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?

11


Seismic Waves of Argentina EQ

Next Class: Class Presentation

Chapter 3
Body Elastic Waves

Ali Oncel
oncel@kfupm.edu.sa
KFUPM

12


Previous Lecture

Seismic Wave Types
Revisit: Wavefronts and raypaths
Seismic Waves A program for the visualization of
wave propagation contributor: Alan Jones
Body Wave Propagation
Example: M6.5, 1998 West Coast of Chile Earthquake
Example: 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
Seismology
Due to March 21

For more detail,
visit to Project
Page of Geop204

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

Travel-Time
Graph
Initial wave fronts for P, S and R waves, propagating
across several receivers at increasing distance from the
source. •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). T=X/V
X distance from source to the receiver,
T total time from the source to the Reference
receiver
V seismic velocity of the P, S, or R
arrival.

13


Estimates of Seismic Velocity

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

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

Model Calculation

Simple, Horizontal Two Layers

© John F. Hermance
September 05, 2002

Ray paths

© John F. Hermance
September 05, 2002

14


© John F. Hermance
September 05, 2002

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.pdf

Huygen’s Principle

pp. 20 of Burger’s book.

See pp.75 and 152 of Bullen&Bolt

15


Fermat’s Principle

pp. 20 of Burger’s book.

Fermat's principle leads to Snell's law;

Snell’s Law

© John F. Hermance
September 05, 2002

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

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

How is the angle of refraction related to
the angle of incidence?
Unlike reflection, θ 1 cannot equal θ 2
θ1
!! n1
• Why?? Remember v = fλ n2
v1 ≠ v2 θ2
Therefore, θ 2 must be different from
θ 1 !!

16


Reflected/Refracted Waves
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.

See pp.140-152 of Bullen&Bolt

Seismic Refraction
•Wave fronts are distorted
from perfect spheres as
energy transmitted into
material of different
velocity. Ray paths thus
bend (“refract”) across an
θ1 interface where velocity
θ2
changes.

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

Behavior of
Refracted Ray on
Velocity Changes

17

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

Recall

Modules of Bulk (k) and Shear (µ)
Bulk Modulus
where Θ = dilatation = ∆V/V
and P = pressure
k= (∆P/Θ)
Ratio of increase in pressure to associated volume change
shear stress = (∆F /A)
shear strain = (∆l /L)
shear modulus
shear stress
µ = shear strain
Force per unit area to change the shape of the material

Recall

Poisson’s Ratio/Young Module
Poisson’s Ratio
∆L
εxx = σ= ( εyy / εxx )
L
∆W Young Module
εyy = W (∆F /A)
Ε = (∆L/L)
Ratio Vp and Vs depends on Poisson ratio:

where

Poisson’s ratio varies from 0 to ½. The elastic constants E, σ, µ are
Poisson’s ratio has the value ½ for mostly used in works of engineering
fluids seismology because they are easily
measured by simple experiments.
See pp.32 of Bullen&Bolt

18


Recall
Seismic Velocities (P-wave)

See pp.318 and 471 of Bullen&Bolt

Rock Velocities (m/sec)

pp. 18-19 of Berger

See pp.319 of Bullen&Bolt

Recall

Influences on Rock Velocities

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

19



Refraction and Reflection

Ali Oncel
oncel@kfupm.edu.sa
KFUPM

Previous Lecture
Travel-time Graph
Estimates of Seismic Velocity
Huygens's Principle
Fermat's Principle
Calculation of Travel Times
Snell's law-Critically Refracted Arrival
Reflection/Refraction
Reflected/Refracted waves
Seismic Refraction
Behavior of refracted ray on velocity changes
Behavior of seismic waves refracted ray penetrating
the Earth
Representative P-wave Velocities for various Rocks
Influences on Rock Velocities

Refracted Ray and Angle

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

20


Energy Return and Critical Angle

θc θc θc θc θc θc

Lillie, Whole Earth Geophysics, Fig 3.25

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)

Modified Table
Table 2.3 after Berger, pp.29.
Angle of For Incident P wave For Incident S wave
Incidence Reflected Refracted Reflected Refracted
P-wave S-wave P-wave S-wave S-wave P-wave S-wave P-wave
10 10.0 6.0 27.6 16.1 10.0 16.8 27.6 50.5
11 11.0 6.6 30.6 17.8 11.0 18.5 30.6 58.0
12 12.0 7.2 33.7 19.4 12.0 20.3 33.7 67.5
13 13.0 7.8 36.9 21.1 13.0 22.0 36.9 88.8
14 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
V1-P (m/s) 29.0
1500 16.9 #NUM! 50.9 29.0 53.9 #NUM! #NUM!
V1-S (m/s) 900
30
V2-P (m/s)
30.0
4000
17.5 #NUM! 53.1 30.0 56.4 #NUM! #NUM!
V2-S (m/s) 2400

Total Time of Refraction

⎛ 1 ⎞ 2 h1
⎜ ⎟
t refraction = ⎜ ⎟ X + V 22 − V12
⎝ V2 ⎠ V1V2

Ttotal= T1+T2+T3

21


Travel time for
Direct/Refracte
d Waves
Xc=critical distance
Xcr=crossover distance V1 +V 2
T1= Intercept time xcr = 2h1
V2 −V 1

Seismic
=Z1
Reflection
=Z2

Lillie, Whole Earth Geophysics, Fig 3.28

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

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

Reflection equation for a reflection hyperbolae:

(X 2
+ 4h 2
)1 / 2
tr =
V 1

22


?

ed
?
ct
fle
ave
Re ad W
Time

do r He
acte
Refr

ti ?
Crossover distance

?
ct
re
Di

Distance

IRIS Deployment in Venezuela, 2001
"line" of fifteen Reftek 125 "Texan" recorders

The source of energy:
Betsy M3 Seisgun
From: http://www.passcal.nmt.edu/%7ebob/passcal/venezuela

23


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

21 “Texans” from Refraction
Technology, Inc.

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

200 “Texans”


24


The N. Walker Lane Experiment, 2002


How Thick is the Crust?

?
Horizontal Rays
Refraction ?
“Tunneling”

7.2 km/s Moho

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.

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)?

KFUPM BEACH-2005
Elevation (m)

180 meter

25


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.

KFUPM-2006

Elevation, m
10 meter

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.

26

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