Geology 5640/6640 Introduction to Seismology 23 Mar 2015 A.R. Lowry 2015 Last time: Normal Modes...
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Transcript of Geology 5640/6640 Introduction to Seismology 23 Mar 2015 A.R. Lowry 2015 Last time: Normal Modes...
Geology 5640/6640Introduction to Seismology
23 Mar 2015
© A.R. Lowry 2015
Last time: Normal Modes• Free oscillations are stationary waves that result from interference of propagating waves• For a string (length L, velocity v) fixed at the endpoints, all propagating waves have eigenfrequencies n = nv/L:
• The Amplitudes An in this equation relate to the source that excited the string:
• Propagating waves in the string can be represented by these normal modes.• In the Earth, the equation is a leetle more complicated…
Read for Wed 25 Mar: S&W 119-157 (§3.1–3.3)
€
u x,t( ) = Anun x,ωn ,t( )n=1
∞
∑ = An sin ωn xv
⎛ ⎝ ⎜
⎞ ⎠ ⎟cos ωn t( )
n=1
∞
∑
€
An = sin ωn xs
v ⎛ ⎝ ⎜
⎞ ⎠ ⎟F ωn( )
Geology 5640/6640Seismology
Last time: Normal Modes (Continued)• On a sphere, free oscillations are described in terms of spherical harmonics as:
Here n is radial order (0 for fundamental; > 0 for overtones); l (colatitude) and m (longitude) are surface orders; Alm
n
describe source displacement; lmn are eigenfrequencies;
& yln(r) (at depth) and xlm
(surface) are eigenfunctions.
• Spherical harmonics are basis functions on a sphere: orthonormal and can completely describe any function.
23 Mar 2015
€
v u (r,θ ,φ,t) =n= 0
∞
∑l= 0
∞
∑ Almn
m= 0
∞
∑ yln (r)
r x lm (θ ,φ)e iω lm
n t
Why study normal modes?
Almn are the excitation amplitudes, analogous to An in the 1D
(string) example… So from measurements of u one can get information about the source (provided the eigenfrequencies lm
n are known!)
Conversely, given a source function Almn and known lm
n, one can predict u… The modes form the basis vectors to describe displacements if one wants to model synthetic seismograms.The frequencies lm
n depend on density, shear modulus, and compressibility modulus of the Earth… so are used to get Earth structure.
€
v u (r,θ ,φ,t) =n= 0
∞
∑l= 0
∞
∑ Almn
m= 0
∞
∑ yln (r)
r x lm (θ ,φ)e iω lm
n t
Recall PREM is derived from normal modes!
Toroidal and spheroidalToroidal and spheroidal
€
uT (r,θ ,φ) =n= 0
∞
∑l= 0
∞
∑ n Alm
m=−l
l
∑ nWl (r)Tlm (θ ,φ)e i nω l
m t
Using spherical harmonics (base on a spherical surface), we can separate the displacements into Toroidal (torsional) and spheroidal modes (as done with SH and P/SV waves):
T :
€
uS (r,θ ,φ) =n= 0
∞
∑l= 0
∞
∑ n Alm
m=−l
l
∑ nU l (r) Rlm (θ ,φ)+nVl (r) Sl
m (θ ,φ)[ ]e i nω lm t
S : Radialeigenfunction
Surfaceeigenfunction
Characteristics of the modesCharacteristics of the modes
• No radial component: tangential only, normal to the radius: motion confined to the surface of n concentric spheres inside the Earth.• Changes in the shape, not of volume
Not observable using a gravimeter (but…)
• Do not exist in a fluid: so only in the mantle (and the inner core?)
• Horizontal components (tangential) et vertical (radial)• No simple relationship between n and nodal spheres
• 0S2 is the longest (“fundamental”)
• Affect the whole Earth (even into
the fluid outer core !)
Toroidal modes nTml : Spheroidal modes nSm
l :
n, l, m …n, l, m …
S : n : no direct relationship with nodes with depthl : # nodal planes in latitudem : # nodal planes in longitude
! Max nodal planes = l0S02
T : n : nodal planes with depthl : # nodal planes in latitudem : # nodal planes in longitude
! Max nodal planes = l - 10T03
0S0 : « balloon » or
« breathing » :
radial only
(20.5 minutes)
0S2 : « football » mode
(Fundamental, 53.9 minutes)
0S3 :
(25.7 minutes)
Spheroidal normal modes: examples:Spheroidal normal modes: examples:
Animation 0S2 from Hein Haakhttp://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html
Animation 0S0/3 from Lucien Saviothttp://www.u-bourgogne.fr/REACTIVITE/manapi/saviot/deform/
0S29 from:http://wwwsoc.nii.ac.jp/geod-soc/web-text/part3/nawa/nawa-1_files/Fig1.jpg
0S29 :
(4.5 minutes)
... ...
Rem: 0S1= translation
...
Toroidal normal modes: examples:Toroidal normal modes: examples:
1T2
(12.6 minutes)
0T2 : «twisting» mode
(44.2 minutes, observed in 1989 with an extensometer)
0T3
(28.4 minutes)
Animation from Hein Haakhttp://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html
Animation from Lucien Saviothttp://www.u-bourgogne.fr/REACTIVITE/manapi/saviot/deform/
Rem: 0T1= rotation 0T0= not existing
Solid inner core (1936)
Fluid outer core (1906)
Solid mantle
Shadow zone
Geophysics and normal modesGeophysics and normal modes
•Solidity demonstrated by normal modes (1971)•Differential rotation of the inner core ? Anisotropy (e.g. crystal of iron aligned with rotation)?
EigenfunctionsEigenfunctionsRuedi Widmer’s home page:http://www-gpi.physik.uni-karlsruhe.de/pub/widmer/Modes/modes.html
shear energy densitycompressional energy density
One of the modes used in 1971 to infer the solidity of the inner core:Part of the shear and compressional energy in the inner core
Today, also confirmed by more modes and by measuring the elusive PKJKP phases
Eigenfunctions : Eigenfunctions : 00SSll
shear energy densitycompressional energy density
l > 20: outer mantlel < 20: whole mantle
Ruedi Widmer’s home page:http://www-gpi.physik.uni-karlsruhe.de/pub/widmer/Modes/modes.html
Equivalent to surface Rayleigh waves
Eigenfunctions : S vs. TEigenfunctions : S vs. T
n = 10 nodal linesshear energy densitycompressional energy density
T in the mantle only !S can affect the whole Earth (esp. overtones)
Ruedi Widmer’s home page:http://www-gpi.physik.uni-karlsruhe.de/pub/widmer/Modes/modes.html
Deep earthquakes excite modes whose eigen functions are large at that depth
Eigenfunctions : Eigenfunctions : 00SSll and and 00TTl l
0S equivalent to interfering surface Rayleigh waves0T equivalent to interfering surface Love waves
http://www.eas.purdue.edu/~braile/edumod/waves/Lwave.htm
www.advalytix.de/ pics/SAWRAiGH.gif
The great Sumatra-Andaman EarthquakeThe great Sumatra-Andaman Earthquake
http://www.iris.iris.edu/sumatra/
300 km
The great Sumatra-Andaman EarthquakeThe great Sumatra-Andaman Earthquake
1300 km
0.0004 0.0008 0.0012 0.0016 0.002
0
0.2
0.4
0.6
0.8
1
Sumatra Earthquake: spectrumSumatra Earthquake: spectrum
0S3
0S2
2S1 0T40T3
0T2
0S4
1S2
0S0
Membach, SG C021, 20041226 08h00-20041231 00h00
Sumatra Earthquake: time domainSumatra Earthquake: time domain
Membach, SG C021, 20041226 - 20050430
Q factor 5327
Q factor 500
http://www.iris.iris.edu/sumatra/M. Van Camp
SplittingSplitting
No more degeneracy if no more spherical symmetry :
Coriolis Ellipticity 3D
Different frequencies and eigenfunctions for each l, m
mln
mln
T
SIf SNREI (Solid Non-Rotating Earth Isotropic) Earth : Degeneracy: for n and l, same frequency for –l < m < l
For each m = one singlet.The 2m+1 group of singlets = multiplet
SplittingSplitting
Rotation(Coriolis)
Ellipticity
3D
Waves in the direction of rotation travel faster
Waves from pole to pole run a shorter path (67 km) than along the equator
Waves slowed down (or accelerated) by heterogeneities
Splitting: Sumatra 2004Splitting: Sumatra 2004
http://www.iris.iris.edu/sumatra/M. Van Camp
Membach SG-C0210S2 Multiplets m=-2, -1, 0, 1, 2
“Zeeman effect”
Modes and MagnitudeModes and MagnitudeTime after beginning of the rupture:
00:11 8.0 (MW) P-waves 7 stations00:45 8.5 (MW) P-waves 25 stations01:15 8.5 (MW) Surface waves 157 stations04:20 8.9 (MW) Surface waves (automatic)19:03 9.0 (MW) Surface waves (revised)Jan. 2005 9.3 (MW) Free oscillationsApril 2005 9.2 (MW) GPS displacements
http://www.gps.caltech.edu/%7Ejichen/Earthquake/2004/aceh/aceh.html
300-500 s surface waves