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