Geology 5640/6640 23 Mar 2015
Introduction to Seismology
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 = nv/L:
ux,t


n1
An un x,n ,t

 x 
An sin n cosn t
 v 
n1

• The Amplitudes An in this equation relate to the source
that excited
the string:
 n x s 

An  sin
F  n 
 v 
• 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)
© A.R. Lowry 2015
Geology 5640/6640
Seismology
23 Mar 2015
Last time: Normal Modes (Continued)
• On a sphere, free oscillations are described in terms of
spherical harmonics
as:



u(r, , ,t) 

n
Alm
 , )e
yln (r)xlm (
n
i  lm
t
n 0 l 0 m 0
Here n is radial order (0 for fundamental; > 0 for overtones);
l (colatitude) and m (longitude) are surface orders; Almn
n are eigenfrequencies;
describe
source
displacement;

lm

& yln(r) (at depth) and xlm (surface) are eigenfunctions.
• Spherical harmonics are basis functions on a sphere:
orthonormal and can completely describe any function.
Why study normal modes?
u(r, , ,t) 




n
Alm
 , )e
yln (r)xlm (
n
i  lm
t
n 0 l 0 m 0
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 lmn are known!)
Conversely, given a source function Almn and known lmn,
one can predict u… The modes form the basis vectors
to describe displacements if one wants to model
synthetic seismograms.
The frequencies lmn depend on density, shear modulus,
and compressibility modulus of the Earth… so are used
to get Earth structure.
Recall PREM is derived from normal modes!
Toroidal and spheroidal
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:
u (r, , ) 
T


l
 
 , )e
m
m
A
W
(r)
T
n l n l
l (
i n lm t
n 0 l 0 ml
Radial
eigenfunction
S:
u (r, , ) 
S


l
 
n 0 l 0 ml
m
A
n l
 U (r)
n
l
Surface
eigenfunction
 ,
Rlm (
)nVl (r) Slm

( , ) e
i n lm t
Characteristics of the modes
Toroidal modes nTml :
• 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?)
Spheroidal modes nSml :
• 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 !)
n, l, m …
S:
n : no direct relationship with nodes with depth
l : # nodal planes in latitude
m : # nodal planes in longitude
! Max nodal planes = l
T:
n : nodal planes with depth
l : # nodal planes in latitude
m : # nodal planes in longitude
! Max nodal planes = l - 1
0
0S 2
0
0T 3
Spheroidal normal modes: examples:
...
0S0 :
« balloon » or
« breathing » :
radial only
0S2 :
« football » mode
(Fundamental, 53.9
minutes)
0S3 :
(25.7 minutes)
(20.5 minutes)
...
...
0S29 :
(4.5 minutes)
Rem: 0S1= translation
Animation 0S0/3 from Lucien Saviot
Animation 0S2 from Hein Haak
http://www.u-bourgogne.fr/REACTIVITE/manapi/saviot/deform/
http://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html
0S29
from:
http://wwwsoc.nii.ac.jp/geod-soc/web-text/part3/nawa/nawa-1_files/Fig1.jpg
Toroidal normal modes: examples:
0T2 :
«twisting» mode
(44.2 minutes, observed in
1989 with an extensometer)
1T2
0T3
(12.6 minutes)
(28.4 minutes)
Rem: 0T1= rotation
0T0= not existing
Animation from Hein Haak
Animation from Lucien Saviot
http://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html
http://www.u-bourgogne.fr/REACTIVITE/manapi/saviot/deform/
Geophysics and normal modes
Solid mantle
Fluid outer core
(1906)
Solid inner core
(1936)
•Solidity demonstrated
by normal modes (1971)
•Differential rotation
of the inner core ?
Anisotropy (e.g. crystal
of iron aligned with
rotation)?
Shadow zone
Eigenfunctions
Ruedi Widmer’s home page:
http://www-gpi.physik.uni-karlsruhe.de/pub/widmer/Modes/modes.html
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
shear energy density
compressional energy density
Today, also confirmed by more modes and
by measuring the elusive PKJKP phases
Eigenfunctions : 0Sl
Equivalent to surface Rayleigh waves
l > 20: outer mantle
l < 20: whole mantle
shear energy density
compressional energy density
Ruedi Widmer’s home page:
http://www-gpi.physik.uni-karlsruhe.de/pub/widmer/Modes/modes.html
Eigenfunctions : S vs. T
S can affect the whole Earth (esp. overtones)
T in the mantle only !
shear energy density
compressional energy density
n = 10 nodal lines
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 : 0Sl and 0Tl
0S
equivalent to interfering surface Rayleigh waves
0T equivalent to interfering surface Love waves
www.advalytix.de/ pics/SAWRAiGH.gif
http://www.eas.purdue.edu/~braile/edumod/waves/Lwave.htm
The great Sumatra-Andaman Earthquake
http://www.iris.iris.edu/sumatra/
The great Sumatra-Andaman Earthquake
1300 km
300 km
Sumatra Earthquake: spectrum
Membach, SG C021, 20041226 08h00-20041231 00h00
1
0 S4
0 S0
0 S3
0.8
0.6
0.4
0 S2
1 S2
0.2
0T2
2 S1
0T3
0T4
0
0.0004
0.0008
0.0012
0.0016
0.002
Sumatra Earthquake: time domain
Membach, SG C021, 20041226 - 20050430
Q factor 5327
Q factor 500
M. Van Camp
http://www.iris.iris.edu/sumatra/
Splitting
n
Sl
m
m
T
n l
If 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
No more degeneracy if no more spherical symmetry :
 Coriolis
 Ellipticity
 3D
Different frequencies and eigenfunctions for each l, m
Splitting
 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 2004
Membach SG-C021
0S2 Multiplets m=-2, -1, 0, 1, 2
“Zeeman effect”
M. Van Camp
http://www.iris.iris.edu/sumatra/
Modes and Magnitude
Time after beginning of the rupture:
00:11
8.0 (MW)
00:45
8.5 (MW)
01:15
8.5 (MW)
04:20
8.9 (MW)
19:03
9.0 (MW)
Jan. 2005 9.3 (MW)
April 2005 9.2 (MW)
P-waves 7 stations
P-waves 25 stations
Surface waves 157 stations
Surface waves (automatic)
Surface waves (revised)
Free oscillations
GPS displacements
300-500 s surface waves
http://www.gps.caltech.edu/%7Ejichen/Earthquake/2004/aceh/aceh.html