surface waves and free oscillationsseismology and the earths deep interior surface waves in an...
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Surface waves in an elastic half spaces: Rayleigh waves
- Potentials- Free surface boundary conditions- Solutions propagating along the surface, decaying with depth- Lamb’s problem
Surface waves in media with depth-dependent properties: Love waves
- Constructive interference in a low-velocity layer- Dispersion curves- Phase and Group velocity
Free Oscillations
- Spherical Harmonics - Modes of the Earth - Rotational Splitting
Surface waves in an elastic half spaces: Rayleigh waves
- Potentials- Free surface boundary conditions- Solutions propagating along the surface, decaying with depth- Lamb’s problem
Surface waves in media with depth-dependent properties: Love waves
- Constructive interference in a low-velocity layer- Dispersion curves- Phase and Group velocity
Free Oscillations
- Spherical Harmonics - Modes of the Earth - Rotational Splitting
Surface Waves and Free OscillationsSurface Waves and Free Oscillations
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
The Wave Equation: PotentialsThe Wave Equation: Potentials
Do solutions to the wave equation exist for an elastic half space, which travel along the interface? Let us start by looking at potentials:
i
i
zyx
i
u
u
),,(
scalar potential
vector potential
displacement
These potentials are solutions to the wave equation
iit
t
222
222
P-wave speed
Shear wave speed
What particular geometry do we want to consider?
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Rayleigh WavesRayleigh Waves
SV waves incident on a free surface: conversion and reflection
An evanescent P-wave propagates along the free surface decaying exponentially with depth. The reflected post-crticially reflected SV wave is totally reflected and phase-shifted. These two wave types can only exist together, they both satisfy the free surface boundary condition:
-> Surface waves
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Surface waves: GeometrySurface waves: Geometry
We are looking for plane waves traveling along one horizontal coordinate axis, so we can - for example - set
0(.) y
As we only require Yy we set Yy=Y from now on. Our trial solution is thus
And consider only wave motion in the x,z plane. Then
yxzz
yzxx
u
u
)](exp[ xazctikA
z
y
x
Wavefront
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Surface waves: Dispersion relationSurface waves: Dispersion relation
With this trial solution we obtain for example coefficients a for which travelling solutions exist
12
2
c
a
Together we obtain
In order for a plane wave of that form to decay with depth a has to be imaginary, in other words
c
)]1/(exp[
)]1/(exp[
22
22
xzcctikB
xzcctikA
So that
c
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Surface waves: Boundary ConditionsSurface waves: Boundary Conditions
Analogous to the problem of finding the reflection-transmission coefficients we now have to satisfy the boundary conditions at the free surface (stress free)
0 zzxz
In isotropic media we have
)]1/(exp[
)]1/(exp[
22
22
xzcctikB
xzcctikA
and
yxzz
yzxx
u
u
zxxz
zzzzxxzz
u
uuu
2
2)(where
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Rayleigh waves: solutionsRayleigh waves: solutions
This leads to the following relationship for c, the phase velocity:
2/1222/122222 )/1()/1(4)/2( ccc
For simplicity we take a fixed relationship between P and shear-wave velocity
3
… to obtain
02/32/3/56/8/ 224466 ccc
… and the only root which fulfills the condition
is
c
9194.0c
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
DisplacementDisplacement
Putting this value back into our solutions we finally obtain the displacement in the x-z plane for a plane harmonic surface wave propagating along direction x
)(cos)4679.18475.0(
)(sin)5773.0(3933.08475.0
3933.08475.0
xctkeeCu
xctkeeCukzkz
z
kzkzx
This development was first made by Lord Rayleigh in 1885. It demonstrates that YES there are solutions to the wave equation propagating along a free surface!
Some remarkable facts can be drawn from this particular form:
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Lamb’s ProblemLamb’s Problem
-the two components are out of phase by p
- for small values of z a particle describes an ellipse and the motion is retrograde
- at some depth z the motion is linear in z
- below that depth the motion is again elliptical but prograde
- the phase velocity is independent of k: there is no dispersion for a homogeneous half space
- the problem of a vertical point force at the surface of a half space is called Lamb‘s problem (after Horace Lamb, 1904).
- Right Figure: radial and vertical motion for a source at the surface
theoretical
experimental
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Particle Motion (1)Particle Motion (1)
How does the particle motion look like?
theoretical experimental
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Particle motionParticle motion
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Data Example Data Example
theoretical experimental
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Data Example Data Example
Question:
We derived that Rayleigh waves are non-dispersive!
But in the observed seismograms we clearly see a highly dispersed surface wave train?
We also see dispersive wave motion on both horizontal components!
Do SH-type surface waves exist?Why are the observed waves dispersive?
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love Waves: GeometryLove Waves: Geometry
In an elastic half-space no SH type surface waves exist. Why? Because there is total reflection and no interaction between an evanescent P wave and a phase shifted SV wave as in the case of Rayleigh waves. What happens if we have layer over a half space (Love, 1911) ?
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love Waves: TrappingLove Waves: Trapping
Repeated reflection in a layer over a half space.
Interference between incident, reflected and transmitted SH waves.
When the layer velocity is smaller than the halfspace velocity, then there is a critical angle beyon which SH reverberations will be totally trapped.
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love Waves: TrappingLove Waves: Trapping
The formal derivation is very similar to the derivation of the Rayleigh waves. The conditions to be fulfilled are:
1. Free surface condition2. Continuity of stress on the boundary3. Continuity of displacement on the boundary
Similary we obtain a condition for which solutions exist. This time we obtain a frequency-dependent solution a dispersion relation
2211
22
2222
1/1/1
/1/1)/1/1tan(
c
ccH
... indicating that there are only solutions if ...
21 c
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Exercise ResultExercise Result
0 10 20 30 40 50 60 70 80 903500
3600
3700
3800
3900
4000
4100
4200
4300
4400
4500
Period (s)
Pha
se v
eloc
ity (
m/s
)
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love Waves: SolutionsLove Waves: Solutions
Graphical solution of the previous equation. Intersection of dashed and solid lines yield discrete modes.
Is it possible, now, to explain the observed dispersive behaviour?
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love Waves: modesLove Waves: modes
Some modes for Love waves
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Waves around the globeWaves around the globe
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
StacksStacks
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Data ExampleData Example
Surface waves travelling around the globe
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Liquid layer over a half spaceLiquid layer over a half space
Similar derivation for Rayleigh type motion leads to dispersive behavior
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Amplitude AnomaliesAmplitude Anomalies
What are the effects on the amplitude of surface waves?
D
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Phase and group velocityPhase and group velocity
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Group-velocitiesGroup-velocities
Interference of two waves at two positions (1)
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
VelocityVelocity
Interference of two waves at two positions (2)
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
DispersionDispersion
The typical dispersive behavior of surface wavessolid – group velocities; dashed – phase velocities
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love wave dispersionLove wave dispersion
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love wave dispersionLove wave dispersion
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love wave dispersionLove wave dispersion
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love wave dispersionLove wave dispersion
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love waves, rotations and translationsLove waves, rotations and translations
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Love waves, rotations and translationsLove waves, rotations and translations
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Higher mode Love wavesHigher mode Love waves
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Observed Phase and Group VelocitiesObserved Phase and Group Velocities
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Wave PacketsWave Packets
Group and phase velocity measurementspeak-and-trough method
Phase velocities from array measurement
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
DispersionDispersion
Stronger gradients cause greater dispersion
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Free oscillations - DataFree oscillations - Data
20-hour long recording of a gravimeter recordind the strong earthquake near Mexico City in 1985 (tides removed). Spikes
correspond to Rayleigh waves.
Spectra of the seismogram given above. Spikes at discrete
frequencies correspond to eigenfrequencies of the Earth
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Surface wave pathsSurface wave paths
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Principle of modes on a stringPrinciple of modes on a string
Fundamental mode and overtones
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
ExamplesExamples
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Naming convention of modesNaming convention of modes
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
OS2 T=54 minsOS2 T=54 mins
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Free oscillations Free oscillations
Source: http://icb.u-bourgogne.fr/nano/MANAPI/saviot/terre/index.en.html
Torsional mode, n=0, ℓ=5, |m|=4. period ≈ 18 minutes
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Eigenmodes of a sphereEigenmodes of a sphere
Eigenmodes of a homogeneous sphere. Note that there are modes with only volumetric changes (like P waves, called spheroidal) and modes with pure shear motion (like shear waves, called
toroidal).
- pure radial modes involve no nodal patterns on the surface- overtones have nodal surfaces at depth - toroidal modes involve purely horizontal twisting- toroidal overtones have nodal surfaces at constant radii.
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Energy of modes at depthEnergy of modes at depth
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Tohoku-oki M9.0Tohoku-oki M9.0
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Other data representationOther data representation
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
SpectraSpectra
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Modes and their meaningModes and their meaning
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Modes and their meaningModes and their meaning
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Equations for free oscillationsEquations for free oscillations
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Spherical harmonicsSpherical harmonics
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
The Earth’s EigenfrequenciesThe Earth’s Eigenfrequencies
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Effects of Earth’s RotationEffects of Earth’s Rotation
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Splitting modesSplitting modes
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Effects of Earth’s Rotation: seismogramsEffects of Earth’s Rotation: seismograms
observed
synthetic no splitting
synthetic
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Work flow for mode analysisWork flow for mode analysis
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Seismology and the Earth’s Deep Interior Surface Waves and Free Oscillations
Surface Waves: SummarySurface Waves: Summary
Rayleigh waves are solutions to the elastic wave equation given a half space and a free surface. Their amplitude decays exponentially with depth. The particle motion is elliptical and consists of motion in the plane through source and receiver.
SH-type surface waves do not exist in a half space. However in layered media, particularly if there is a low-velocity surface layer, so-called Love waves exist which are dispersive, propagate along the surface. Their amplitude also decays exponentially with depth.
Free oscillations are standing waves which form after big earthquakes inside the Earth. Spheroidal and toroidal eigenmodes correspond are analogous concepts to P and shear waves.
Rayleigh waves are solutions to the elastic wave equation given a half space and a free surface. Their amplitude decays exponentially with depth. The particle motion is elliptical and consists of motion in the plane through source and receiver.
SH-type surface waves do not exist in a half space. However in layered media, particularly if there is a low-velocity surface layer, so-called Love waves exist which are dispersive, propagate along the surface. Their amplitude also decays exponentially with depth.
Free oscillations are standing waves which form after big earthquakes inside the Earth. Spheroidal and toroidal eigenmodes correspond are analogous concepts to P and shear waves.