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Theoretical Seismology 2: Wave Propagation. Based on a lecture by James Mori of the Earthquake Hazards Division, Disaster Prevention Research Institute, Kyoto University . Contents. ・ Rays Snell ’ s Law Structure of the Earth ・ Seismic Waves

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Theoretical seismology 2 wave propagation l.jpg

Theoretical Seismology 2: Wave Propagation

Based on a lecture by James Mori of the Earthquake Hazards Division,Disaster Prevention Research Institute,Kyoto University


Contents l.jpg
Contents

・ Rays

Snell’s 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

USGS


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Homogeneous Earth

If the Earth had constant velocity the wave paths would be very simple.


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Structure in the Earth results in complicated paths

Lowrie, 1997, fig 3.69

USGS

Bolt, 2004, fig 6.3


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Snell’s Law

Fermat’s Principle

Rays

q1

Air

Water

q2

sin q1 / sin q2 = n21


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a1

q1

q2

a2

a1 > a2

Ray Paths in a Layered Medium

a = velocity of seismic energy in the layer

Faster

a1

q1

Slower

Slower

q2

Faster

a2

a1 < a2


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Time

1/a3

1/a2

1/a1

Distance

Ray Paths in a Layered Medium

a1

a2

a3


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The 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

The Moho

The Moho

Copywrite Tasa Graphic Arts


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Forward Branch

Backward Branch


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Forward Branch

Forward Branch

Backward Branch

Shadow Zone


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PcP

Shadow

Zone

・ 1912 Gutenberg observed shadow zone 105o to 143o

・ 1939 Jeffreys fixed depth of core at 2898 km

(using PcP)

Backward

Branch

Forward

Branch

PKP

Forward

Branch

PcP

Shadow

Zone

P

Forward Branch

Forward

Branch

Backward 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

This is the equation that explains the waves on a spring: 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


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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


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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

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

Earthquake (M8.0)


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Generation of Tsunami

from Near-field Term

UNESCO-IOC (Great waves)


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Far-field Terms

  • ・ Propagating Waves

  • ・ No net displacement

  • ・ P waves

  • ・ S waves

 Copyright 2004. L. Braile.


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surface waves

 Copyright 2004. L. Braile.


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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)


Amplitude and intensity l.jpg

A(t) = A0e -ω0t/2Q

Amplitude and Intensity

Seismic waves loose amplitude with distance traveled - attenuation

So the amplitude of the waves depends on distance from the earthquake. Therefore unlike magnitude intensity is not a single number.


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Modified Mercalli Intensity

I Barely felt

II Felt by only few people

III Felt noticeably, standing autos rock slightly

IV Felt by many, windows and walls creak

V Felt by nearly everyone, some dished and windows broken

VI Felt by all, damaged plaster and chimneys

VII Damage to poorly constructed buildings

VIII Collapse of poorly constructed buildings,

slight damage to well built structures

IX Considerable damage to well constructed buildings,

buildings shifted off foundations

X Damage to well built wooden structures, some masonry

buildings destroyed, train rails bent, landslides

XI Few masonry structure remain standing, bridges

destroyed, ground fissures

XII Damage total


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Normal Modes

Liberty Bell

(USA)

Useful for studies of

・ Interior of the Earth

・ Largest earthquakes

l=1 m=3

l=1 m=1

l=1 m=2

Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html


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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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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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