Surface Waves and Free Oscillations in Seismology
Seismology and the Earth’s Deep Interior
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Potentials
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Free surface boundary conditions
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Solutions propagating along the surface, decaying with depth
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Lamb’s problem
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Constructive interference in a low-velocity layer
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Dispersion curves
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Phase and Group velocity
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- Spherical Harmonics
- Modes of the Earth
- Rotational Splitting
Surface Waves and Free Oscillations
Seismology and the Earth’s Deep Interior
The 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
:
scalar potential
vector potential
displacement
These potentials are solutions to the wave equation
P-wave speed
Shear wave speed
What particular geometry do we want to consider?
Seismology and the Earth’s Deep Interior
Rayleigh 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
Seismology and the Earth’s Deep Interior
Surface waves: Geometry
We are looking for plane waves traveling along one horizontal coordinate
axis, so we can - for example - set
As we only require
y
we set
y
=
from now on. Our trial
solution is thus
And consider only wave motion in the x,z plane. Then
z
y
x
Wavefront
Seismology and the Earth’s Deep Interior
Surface waves: Dispersion relation
With this trial solution we obtain for example coefficients a for
which travelling solutions exist
Together we obtain
In order for a plane wave of that form to decay with depth a has to be
imaginary, in other words
So that
Seismology and the Earth’s Deep Interior
Surface 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)
In isotropic media we have
and
where
Seismology and the Earth’s Deep Interior
Rayleigh waves: solutions
This leads to the following relationship for c,
the phase velocity:
For simplicity we take a fixed relationship between P and shear-
wave velocity
… to obtain
… and the only root which fulfills the condition
is
Seismology and the Earth’s Deep Interior
Displacement
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
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Some remarkable facts can be drawn from this particular form:
Seismology and the Earth’s Deep Interior
Lamb’s Problem
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the two components are out of phase by p
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for small values of z a particle describes an
ellipse and the motion is retrograde
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at some depth z the motion is linear in z
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below that depth the motion is again elliptical but
prograde
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Right Figure: radial and vertical motion for a
source at the surface
theoretical
experimental
Seismology and the Earth’s Deep Interior
Particle Motion (1)
How does the particle motion look like?
theoretical
experimental
Seismology and the Earth’s Deep Interior
Particle motion
Seismology and the Earth’s Deep Interior
Data Example
theoretical
experimental
Seismology and the Earth’s Deep Interior
Data Example
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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?
Seismology and the Earth’s Deep Interior
Love 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) ?
Seismology and the Earth’s Deep Interior
Love 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.
Seismology and the Earth’s Deep Interior
Love Waves: Trapping
The formal derivation is very similar to the derivation of the Rayleigh waves.
The conditions to be fulfilled are:
1.
Free surface condition
2.
Continuity of stress on the boundary
3.
Continuity of displacement on the boundary
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... indicating that there are only solutions if ...
Seismology and the Earth’s Deep Interior
Exercise Result
Seismology and the Earth’s Deep Interior
Love 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?
Seismology and the Earth’s Deep Interior
Love Waves: modes
Some modes for Love waves
Seismology and the Earth’s Deep Interior
Waves around the globe
Seismology and the Earth’s Deep Interior
Stacks
Seismology and the Earth’s Deep Interior
Phase and group velocity
Seismology and the Earth’s Deep Interior
Dispersion
The typical dispersive behavior of surface waves
solid – group velocities; dashed – phase velocities
Seismology and the Earth’s Deep Interior
Love wave dispersion
Seismology and the Earth’s Deep Interior
Love wave dispersion
Seismology and the Earth’s Deep Interior
Love wave dispersion
Seismology and the Earth’s Deep Interior
Love wave dispersion
Seismology and the Earth’s Deep Interior
Love waves, rotations and translations
Seismology and the Earth’s Deep Interior
Love waves, rotations and translations
Seismology and the Earth’s Deep Interior
Higher mode Love waves
Seismology and the Earth’s Deep Interior
Observed Phase and Group Velocities
Group velocity maps
Seismology and the Earth’s Deep Interior
35-second Love wave group velocity map of perturbations with respect to PREM
Larson and Ekstrom 2001
Seismology and the Earth’s Deep Interior
Surface wave paths
Seismology and the Earth’s Deep Interior
Tohoku-oki M9.0
Seismology and the Earth’s Deep Interior
Other data representation
Seismology and the Earth’s Deep Interior
Free 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
Seismology and the Earth’s Deep Interior
Spectra
Seismology and the Earth’s Deep Interior
Principle of modes on a string
Fundamental mode and overtones
Seismology and the Earth’s Deep Interior
Examples
Seismology and the Earth’s Deep Interior
Naming convention of modes
Mode names
Seismology and the Earth’s Deep Interior
spheroidal
toroidal
n - zero-crossings in depth
l - angular order – overall zero crossings on surface
m - azimuthal order – zero crossings through pole
Seismology and the Earth’s Deep Interior
Spherical harmonics
Seismology and the Earth’s Deep Interior
O
S
2
T=54 mins
Seismology and the Earth’s Deep Interior
Eigenmodes 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.
Seismology and the Earth’s Deep Interior
Energy of modes at depth
Seismology and the Earth’s Deep Interior
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
Seismology and the Earth’s Deep Interior
Modes and their meaning
Seismology and the Earth’s Deep Interior
Modes and their meaning
Seismology and the Earth’s Deep Interior
Equations for free oscillations
Seismology and the Earth’s Deep Interior
The Earth’s Eigenfrequencies
Seismology and the Earth’s Deep Interior
Effects of Earth’s Rotation
Seismology and the Earth’s Deep Interior
Splitting modes
Seismology and the Earth’s Deep Interior
Effects of Earth’s Rotation: seismograms
observed
synthetic no splitting
synthetic
Seismology and the Earth’s Deep Interior
Work flow for mode analysis
Seismology and the Earth’s Deep Interior
Surface 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.
Exploring surface waves, potentials, and free oscillations in seismology with a focus on Rayleigh waves, Love waves, and wave equations. Understand the dispersion relation, geometry, and solutions for waves propagating in elastic half spaces and media with depth-dependent properties.
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Presentation Transcript
Surface Waves and Free Oscillations Surface waves in an elastic half spaces: Rayleigh waves 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 Surface waves in media with depth- -dependent properties: Love waves dependent properties: Love waves - Constructive interference in a low-velocity layer - Dispersion curves - Phase and Group velocity Free Oscillations Free Oscillations - Spherical Harmonics - Modes of the Earth - Rotational Splitting Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
The 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: These potentials are solutions to the wave equation = ( + , ) u = 2 2 2 i t = , = 2 2 2 x y z t i i u displacement P-wave speed i scalar potential Shear wave speed vector potential i What particular geometry do we want to consider? Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Rayleigh 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 Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Surface waves: Geometry We are looking for plane waves traveling along one horizontal coordinate axis, so we can - for example - set (.) = y 0 And consider only wave motion in the x,z plane. Then = u Wavefront x x z y y = + u z z x y x As we only require y we set y= from now on. Our trial solution is thus = exp[ ( )] A ik ct az x z Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Surface waves: Dispersion relation With this trial solution we obtain for example coefficients a for which travelling solutions exist 2 c = 1 a 2 In order for a plane wave of that form to decay with depth a has to be imaginary, in other words c Together we obtain = 2 2 exp[ ( / 1 )] A ik ct c z x = 2 2 exp[ ( / 1 )] B ik ct c z x So that c Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Surface 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 xz zz In isotropic media we have = + + 2 = ( ) u u u u zz x x z z z z x x z y where = 2 u = + u xz x z z z x y and = 2 2 exp[ ( / 1 )] A ik ct c z x = 2 2 exp[ ( / 1 )] B ik ct c z x Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Rayleigh waves: solutions This leads to the following relationship for c, the phase velocity: = 2 2 2 2 2 / 1 ) 2 2 2 / 1 ) 2 2 ( / ) 1 ( 4 / 1 ( / c c c For simplicity we take a fixed relationship between P and shear- wave velocity 3 = to obtain + = 6 6 4 4 2 2 / 8 / 56 / 3 / 32 / 2 0 c c c c and the only root which fulfills the condition is 9194 . 0 = c Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Displacement 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 8475 . 0 3933 . 0 = kz kz ( . 0 5773 ) sin ( ) u C e e k ct x x 8475 . 0 3933 . 0 = . 0 . 1 + kz kz ( 8475 4679 ) cos ( ) u C e e k ct x z 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: Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Lambs Problem theoretical -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 experimental - 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 Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Particle Motion (1) How does the particle motion look like? theoretical experimental Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Particle motion Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Data Example theoretical experimental Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
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? Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love 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) ? Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love 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. Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love Waves: Trapping The formal derivation is very similar to the derivation of the Rayleigh waves. The conditions to be fulfilled are: 1. 2. 3. Free surface condition Continuity of stress on the boundary 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 2 / 1 2 / 1 c 2 2 / 1 = 2 2 tan( / 1 ) H c 1 / 1 2 2 / 1 c 1 1 ... indicating that there are only solutions if ... c 1 2 Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Exercise Result 4500 4400 4300 4200 Phase velocity (m/s) 4100 4000 3900 3800 3700 3600 3500 0 10 20 30 40 Period (s) 50 60 70 80 90 Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love 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? Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love Waves: modes Some modes for Love waves Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Waves around the globe Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Stacks Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Phase and group velocity Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Dispersion The typical dispersive behavior of surface waves solid group velocities; dashed phase velocities Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love wave dispersion Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love wave dispersion Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love wave dispersion Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love wave dispersion Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love waves, rotations and translations Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Love waves, rotations and translations Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Higher mode Love waves Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Observed Phase and Group Velocities Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Group velocity maps 35-second Love wave group velocity map of perturbations with respect to PREM Larson and Ekstrom 2001 Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Surface wave paths Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Tohoku-oki M9.0 Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Other data representation Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Free 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 Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Spectra Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Principle of modes on a string Fundamental mode and overtones Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Examples Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Naming convention of modes Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Mode names m l S n spheroidal m T toroidal n l n - zero-crossings in depth l - angular order overall zero crossings on surface m - azimuthal order zero crossings through pole Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Spherical harmonics Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
OS2 T=54 mins Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Eigenmodes 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. Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Energy of modes at depth Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Free oscillations Torsional mode, n=0, =5, |m|=4. period 18 minutes Source: http://icb.u-bourgogne.fr/nano/MANAPI/saviot/terre/index.en.html Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Modes and their meaning Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Modes and their meaning Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
Equations for free oscillations Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
The Earths Eigenfrequencies Surface Waves and Free Oscillations Seismology and the Earth s Deep Interior
