Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves;...
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Transcript of Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves;...
![Page 1: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/1.jpg)
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)
€
U = c + kdc
dk= c − λ
dc
dλ
![Page 2: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/2.jpg)
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
![Page 3: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/3.jpg)
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.
![Page 4: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/4.jpg)
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…
![Page 5: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/5.jpg)
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
⎛
⎝ ⎜
⎞
⎠ ⎟
€
u x,t( ) = Anun x,ωn ,t( )n=1
∞
∑ = An sinωn x
v
⎛
⎝ ⎜
⎞
⎠ ⎟cos ωn t( )
n=1
∞
∑
![Page 6: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/6.jpg)
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.
€
An = sinnπxs
L
⎛
⎝ ⎜
⎞
⎠ ⎟F ωn( )
€
F ωn( ) = exp −ωn
2τ 2
4
⎛
⎝ ⎜
⎞
⎠ ⎟
€
u x,t( ) = An sinωn x
v
⎛
⎝ ⎜
⎞
⎠ ⎟cos ωn t( )
n=1
∞
∑
![Page 7: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/7.jpg)
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)
![Page 8: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/8.jpg)
Travelling surface wavesTravelling surface waves
Richard Aster, New Mexico Institute of Mining and Technology http://www.iris.iris.edu/sumatra/
![Page 9: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/9.jpg)
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
![Page 10: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/10.jpg)
€
vu (r,θ ,φ,t) =
n= 0
∞
∑l= 0
∞
∑ Almn
m= 0
∞
∑ yln (r)
r x lm (θ ,φ)e iω lm
n t
€
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
![Page 11: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/11.jpg)
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.
€
F ω( ) = f t( )e−iωtdt
−∞
∞
∫
![Page 12: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/12.jpg)
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:
€
Pl =1
2l l!
d
dxx 2 −1( )
l
€
Plm x( ) =1− x 2( )
m
2
2l l!
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
d l +m
dx l +mx 2 −1( )
l ⎡
⎣ ⎢
⎤
⎦ ⎥
![Page 13: Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves.](https://reader036.fdocuments.net/reader036/viewer/2022062315/56649e995503460f94b9ca97/html5/thumbnails/13.jpg)
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
( )