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II.1 Theoretical Seismology 2 Wave Propagation
Rays Snells Law Structure of the
Earth Seismic Waves Near-Field Terms
(Static Displacements) Far-Field Terms (P, S,
Surface waves) Normal modes Free
oscillations of the Earth
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Faulting
Seismic waves
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Homogeneous Earth
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Structure in the Earth
Crust-Mantle Core-Mantle 440 km 660 km
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Snells Law Fermats Principle
Rays
q1
Air
Water
q2
sin q1 / sin q2 n21
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Ray Paths in a Layered Medium
a1
Faster
q1
Slower
Slower
q2
Faster
a2
a1 lt a2
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Ray Paths in a Layered Medium
a1
a2
a3
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Moho
Andrija Mohorovicic (1857-1936) Found seismic
discontinuity at 30 km depth in the Kupa Valley
(Croatia). Mohorovicic discontinuity or
Moho Boundary between crust and mantle
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Backward Branch
Forward Branch
PKP
Forward Branch
PcP
Shadow Zone
P
Forward Branch
Backward Branch
Forward Branch
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PcP
Core Reflections
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Seismic Waves
Aspects of Waves not Explained by Ray Theory
Different types of waves (P, S) Surface
Waves Static Displacements Frequency
content

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Wave Equation
1-D wave equation c propagation speed
Slinky constant velocity wave
propagation, no mass transfer, different from
circulation eq.
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1-D Wave Equation
Solution
T wave period w angular frequency
LW 3.2.1
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Wave Period and Wavelength
Velocity 6 km/s
Space
x
wavelength 300 km
wavelength
Time
t
period 50 s frequency 1/period 0.02 hz
period
Velocity Wavelength / Period
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Period
Wavelength
Body waves (PS) 0.01 to 50 sec 50 m to 500 km
Surface waves 10 to 350 sec 30 to 1000 km
Free Oscillations 350 to 3600 sec 1000 to 10000 km
Static Displacements -
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3-D Wave Equation with Source
source
spatial 2nd derivative
Near-field Terms (Static Displacements)
Solution
Far-field Terms (P, S Waves)
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Near-field terms

• Static displacements
• Only significant close to the fault
• Source of tsunamis

t ?
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Static Displacements
Bei-Fung Bridge near Fung-Yan city, 1999 Chi-Chi,
Taiwan earthquake
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Static displacements
Co-seismic deformation of 2003 Tokachi-oki Earthqu
ake (M8.0)
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Generation of Tsunami from Near-field Term
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Far-field Terms

• Propagating Waves
• No net displacement
• P waves
• S waves

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Surface Waves
Group Velocity (km/sec)
Love
Rayleigh
Period (sec)
S
Shearer, Fig. 8.1
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January 26, 2001 Gujarat, India Earthquake (Mw7.7)
vertical
Rayleigh Waves
radial
transverse
Love Waves
Recorded in Japan at a distance of 57o (6300 km)
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Normal Modes
(Stein and Gellar 1978)
Free Oscillations of the Earth 1960 Chile
Earthquake
(Daishinji, Fukui Prefecture)
Useful for studies of Interior of the Earth
Largest earthquakes
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Toroidal and Spheroidal Modes
Toroidal
Spheroidal
Dahlen and Tromp Fig. 8.5, 8.17
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Natural Vibrations of the Earth
Shearer Ch.8.6 Lay and Wallace, Ch. 4.6
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Free Oscillations l1 m1
Houseman http//earth.leeds.ac.uk/greg/?Sphar/ind
ex.html
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Free Oscillations l1 m2
Houseman http//earth.leeds.ac.uk/greg/?Sphar/ind
ex.html
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Free Oscillations l1 m3
Houseman http//earth.leeds.ac.uk/greg/?Sphar/ind
ex.html
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Summary
Rays Earth structure causes complicated ray
paths through the Earth (P, PKP,
PcP) Wave theory explains P and S waves
Static displacements Surface waves
Normal Modes The Earth rings like a bell
at long periods
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Why are observed seismograms so messy ?
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Structure Free Surface
Earth is a not homogenous whole-space Free
surface causes many complications -
surface waves - reflections (pP, sP, sS)