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8/13/2019 Teori Seismik32

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Seismolo and the Earths Dee 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- Lambs 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- Lambs 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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The Wave Equation: PotentialsThe Wave Equation: Potentials

Do solutions to the wave equation exist for an elastic half space, whichtravel 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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Surface waves: GeometrySurface waves: Geometry

We are looking for plane waves traveling along one horizontalcoordinate axis, so we can - for example - set

0(.)=y

And consider only wave motion in the x,z plane. Then

As we only require y weset y= from now on. Our

trial solution is thus

yxzz

yzxx

u

u

+=

=

)](exp[ xazctikA =

z

y

x

Wavefront


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Surface waves: Disperion relationSurface waves: Disperion relation

With this trial solution we obtain for example coefficientsa for which travelling solutions exist

12

2

=

ca

Together we obtain

In order for a plane wave of that form to decay with depth a has to beimaginary, in other words


6/46Seismolo and the Earths Dee 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 theboundary 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

uu

+=

=

zxxz

zzzzxxzz

uuuu

=

++=

22)( where


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DisplacementDisplacement

Putting this value back into our solutionswe finally obtain the displacement in thex-z plane for a plane harmonic surfacewave propagating along direction x

)(cos)4679.18475.0(

)(sin)5773.0(

3933.08475.0

3933.08475.0

xctkeeCu

xctkeeCu

kzkz

z

kzkz

x

+=

=

This development was first made by Lord Rayleigh in 1885. Itdemonstrates that YES there are solutions to the wave

equation propagating along a free surface!

Some remarkable facts can be drawn from this particular form:



Lambs ProblemLambs Problem

-the two components are out of phase by

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 ellipticalbut prograde

- the phase velocity is independent of k: thereis no dispersion for a homogeneous half space

- the problem of a vertical point force at thesurface of a half space is called Lambs

problem (after Horace Lamb, 1904).

- Right Figure: radial and vertical motion for asource at the surface

theoretical

experimental



Particle Motion (1)Particle Motion (1)

How does the particle motion look like?

theoretical experimental



Data ExampleData Example

theoretical experimental




Question:

We derived that Rayleigh waves are non-dispersive!But in the observed seismograms we clearly see ahighly dispersed surface wave train?

We also see dispersive wave motion on bothhorizontal components!

Do SH-type surface waves exist?Why are the observed waves dispersive?



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 betweenan evanescent P wave and a phase shifted SV wave as in thecase of Rayleigh waves. What happens if we have layer over a

half space (Love, 1911) ?



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 thereis a critical angle beyon which SH reverberations will be totally trapped.


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Love Waves: TrappingLove Waves: Trapping

The formal derivation is very similar to the derivation of the Rayleighwaves. The conditions to be fulfilled are:

1. Free surface condition

2. Continuity of stress on the boundary3. Continuity of displacement on the boundary

Similary we obtain a condition for which solutions exist. This time weobtain a frequency-dependent solution a dispersion relation

22

11

2

2

2

222

1

/1/1

/1/1)/1/1tan(

c

ccH

=

... indicating that there are only solutions if ...

2


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Love Waves: SolutionsLove Waves: Solutions

Graphical solution ofthe previous equation.Intersection of dashed

and solid lines yielddiscrete modes.

Is it possible, now, to

explain the observeddispersive behaviour?


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Love Waves: modesLove Waves: modes

Some modes for Love waves

d h l bW d h l b


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Waves around the globeWaves around the globe

D E lD t E l


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Surface waves travelling around the globe

Li id l h lfLi id l h lf


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Liquid layer over a half spaceLiquid layer over a half space

Similar derivation for Rayleigh type motion leads to dispersive behavior

A lit d A liA lit d A li


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Amplitude AnomaliesAmplitude Anomalies

What are the effects on the amplitude of surface waves?

Away from source or antipode geometrical spreading is approx. prop. to (sin)1/2


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VelocityVelocity


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VelocityVelocity

Interference of two waves at two positions (2)

DispersionDispersion


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The typical dispersive behavior of surface wavessolid group velocities; dashed phase velocities

Wave PacketsWave Packets


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Seismograms of a Lovewave train filtered withdifferent central periods.

Each narrowband tracehas the appearance of awave packet arriving atdifferent times.



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Group and phasevelocity measurementspeak-and-troughmethod

Phase velocities fromarray measurement



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Stronger gradients cause greater dispersion


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Love wave dispersionLove wave dispersion


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Modal SummationModal Summation


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Mo a Summat onmm

Free oscillations - DataFree oscillations - Data


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20-hour long recording of agravimeter recordind thestrong earthquake near

Mexico City in 1985 (tides

removed). Spikes correspondto Rayleigh waves.

Spectra of the seismogramgiven above. Spikes atdiscrete frequencies

correspond toeigenfrequencies of the Earth

Eigenmodes of a stringEigenmodes of a string


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g gg g

Geometry of a string underntension with fixed end points.Motions of the string excited

by any source comprise aweighted sum of theeigenfunctions (which?).

Eigenmodes of a sphereEigenmodes of a sphere


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g pg p

Eigenmodes of a homogeneoussphere. Note that there are modeswith only volumetric changes (like P

waves, called spheroidal) and modeswith pure shear motion (like shear

waves, called toroidal).

- pure radial modes involve no nodal

patterns on the surface- overtones have nodal surfaces atdepth

- toroidal modes involve purelyhorizontal twisting

- toroidal overtones have nodalsurfaces at constant radii.


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Bessel and LegendreBessel and Legendre


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Solutions to the wave equation on spherical coordinates: Bessel functions (left)and Legendre polynomials (right).

Spherical HarmonicsSpherical Harmonics


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Examples of spherical surface harmonics. There are zonal,sectoral and tesseral harmonics.

The Earths EigenfrequenciesThe Earths Eigenfrequencies


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Spheroidal modeeigenfrequencies

Toroidal mode

eigenfrequencies

Effects of Earths RotationEffects of Earths Rotation


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non-polar latitude

polar latitude

Effects of Earths Rotation: seismogramsEffects of Earths Rotation: seismograms


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observed

synthetic no splitting

synthetic

Lateral heterogeneityLateral heterogeneity


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Illustration of the distortion of standing-waves

due to heterogeneity. The spatial shift of thephase perturbs the observed multiplet amplitude

ExamplesExamples


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Sumatra M9, 26-12-04Sumatra M9, 26-12-04


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Surface Waves: SummarySurface Waves: Summary


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Rayleigh waves are solutions to the elastic wave equation given

a half space and a free surface. Their amplitude decaysexponentially with depth. The particle motion is elliptical andconsists 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-velocitysurface layer, so-called Love waves exist which aredispersive, propagate along the surface. Their amplitude alsodecays exponentially with depth.

Free oscillations are standing waves which form after bigearthquakes inside the Earth. Spheroidal and toroidaleigenmodes correspond are analogous concepts to P and shearwaves.

Rayleigh waves are solutions to the elastic wave equation givena half space and a free surface. Their amplitude decaysexponentially with depth. The particle motion is elliptical andconsists of motion in the plane through source and receiver.

SH-type surface waves do not exist in a half space. Howeverin layered media, particularly if there is a low-velocitysurface layer, so-called Love waves exist which aredispersive, propagate along the surface. Their amplitude alsodecays exponentially with depth.

Free oscillations are standing waves which form after bigearthquakes inside the Earth. Spheroidal and toroidaleigenmodes correspond are analogous concepts to P and shearwaves.

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