Geology 5640/6640Introduction to Seismology

20 Mar 2015

© A.R. Lowry 2015

Last time: Love Waves; Group & Phase Velocity• At shorter periods, Love waves can have a fundamental plus higher modes; longer periods have fewer modes

• The wave velocity cx increases with period and with mode number

• Amplitudes uy(z) are ~sinusoidal above the turning depth; decay exponentially below (where the wave is evanescent)

• Dispersive waves have both group velocity U (velocity of the envelope or “beat”) and phase velocity c (velocity of individual peaks) which relate as:

(hence, c > U)Read for Fri 20 Mar: S&W 119-157 (§3.1–3.3)

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U = c + kdc

dk= c − λ

dc

dλ

To measure Group Velocity:• Measure period as the time between successive peaks or troughs• Travel-time is the time at the time of arrival of the wave group minus the origin time• Divide the source-receiver distance by travel-time to get the group velocity

Or can get a bit more sophisticated by filteringthe waveform (multiplyingby a “windowing function”in the frequency domain)to isolate elements of thewaveform that have a particular period, usinga Fourier transform.

To get phase velocity, cantransform to phase ()and e.g. solving for c()from the difference in phaseof the arrivals at two sites.

Gravimeters for seismological Gravimeters for seismological broadband monitoring:broadband monitoring:Earth’s free oscillationsEarth’s free oscillations

Michel Van CampRoyal Observatory of Belgium

Note: These slides borrow heavily from a presentation by Michel van Camp, ROB…

Fundamental(n = 1)

etc.

L

u

Free Oscillations: are stationary waves consisting of interference of propagating waves.

1st

harmonic(n = 2)

2nd

harmonic(n = 3)

3rd

harmonic(n = 4)

Consider a vibrating stringattached at both ends:It can only vibrate atthe eigenfrequenciesfor which displacementu = sin(x/v) cos(t)is always zero at the endpoints: I.e., only forn = nv/L.Thus we can write

And the totaldisplacement is given by:€

un x,t( ) = sinnπ

Lx

⎛

⎝ ⎜

⎞

⎠ ⎟cos

nπv

Lt

⎛

⎝ ⎜

⎞

⎠ ⎟

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u x,t( ) = Anun x,ωn ,t( )n=1

∞

∑ = An sinωn x

v

⎛

⎝ ⎜

⎞

⎠ ⎟cos ωn t( )

n=1

∞

∑

Here An is a weight that dependson the source displacement as:

where F(n) describes the shapeof the source and xs is thelocation. This example from thetext is for xs = 8 and

with = 0.2.

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An = sinnπxs

L

⎛

⎝ ⎜

⎞

⎠ ⎟F ωn( )

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F ωn( ) = exp −ωn

2τ 2

4

⎛

⎝ ⎜

⎞

⎠ ⎟

€

u x,t( ) = An sinωn x

v

⎛

⎝ ⎜

⎞

⎠ ⎟cos ωn t( )

n=1

∞

∑

Seismic normal modesSeismic normal modes

• Periods < 54 min, amplitudes < 1 mm

• Observable months after great earthquakes (e.g. Sumatra, Dec 2004 took about 5 months to decay)

Few minutes after the earthquakeConstructive interferences free oscillations (or stationary waves)

Few hours after the earthquake (0S20)

(Duck from Théocrite, © J.-L. & P. Coudray)

Travelling surface wavesTravelling surface waves

Richard Aster, New Mexico Institute of Mining and Technology http://www.iris.iris.edu/sumatra/

HistoricHistoric

• First theories:

First mathematical formulations for a steel sphere: Lamb, 1882: 78 min

Love, 1911 : Earth steel sphere + gravitation: eigen period = 60

minutes

• First Observations:

Potsdam, 1889: first teleseism (Japan): waves can travel the whole Earth. Isabella (California) 1952 : Kamchatka earthquake (Mw=9.0). Attempt to identify a « mode » of 57 minutes. Wrong but reawakened interest. Isabella (California) 22 may 1960: Chile earthquake (Mw = 9.5): numerous modes are identified Alaska 1964 earthquake (Mw = 9.2) Columbia 1970: deep earthquake (650 km): overtones IDA Network

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vu (r,θ ,φ,t) =

n= 0

∞

∑l= 0

∞

∑ Almn

m= 0

∞

∑ yln (r)

r x lm (θ ,φ)e iω lm

n t

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u(x,t) = An

n= 0

∞

∑ sinωn x

v

⎛

⎝ ⎜

⎞

⎠ ⎟cos ωn t( )

On the sphere…For a vibrating string:

On the sphere:

Here, n is the radial order (n = 0 for the fundamental; n > 0 for overtones)

l and m are surface ordersl is the angular order;–l < m < l is the azimuthal order

Radialeigenfunction

Surfaceeigenfunction

A Quick Digression on Basis FunctionsConsider an arbitrary function f(t). It can be represented as a sum of sine and cosine waves with various frequencies via:

This is the Fourier transform that we keep talking about… basically it “translates” the temporal (or spatial) description of a function into the “language” of frequency and phase.

Given enough frequencies, the Fourier transform can exactly construct any arbitrary function from a sum of sines and cosines.

We call e–it a basis function.

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F ω( ) = f t( )e−iωtdt

−∞

∞

∫

On a sphere, the analogous description of sines and cosines is called spherical harmonics.

Spherical harmonics are described by Legendre polynomials and Legendre functions. Legendre polynomials are:

where l denotes angular order.For a sphere, x = cos sothese describe variations withcolatitude (e.g. from the source in this diagram). Legendre functions are defined by:

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Pl =1

2l l!

d

dxx 2 −1( )

l

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Plm x( ) =1− x 2( )

m

2

2l l!

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

d l +m

dx l +mx 2 −1( )

l ⎡

⎣ ⎢

⎤

⎦ ⎥

The Legendre polynomials and Legendre functions can be combined to create a set of basis functions on a sphere:

Note that l describes harmonics that depend on the colatitude and m describes harmonics that have a longitudinal () dependence.

As is true of all basis functions, spherical harmonics are orthonormal:

€

Ylm θ ,φ( ) = −1( )m 2 l +1( )

4π

⎛

⎝ ⎜

⎞

⎠ ⎟

l − m( )!

l + m( )!

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥Plm cosθ( )e

imφ

€

0

2π

∫ sinθYl 'm'* θ ,φ( )Ylm θ ,φ( )

0

π

∫ dθdφ =δl ' lδm'm€

Ylm

( )