Michel Van Camp
Royal Observatory of Belgium

x
L
Fundamental
1 st
Harmonic
2 nd
Harmonic
3 rd
Harmonic d ~ sin
 x cos(
 t ) c

L
 i
 c d j th mode : T

2 L jc
~
 j j
 sin
L
 j
 c j

L x sin x
L cos(
: eigenvalue or eigenfrequ ency

U j j
 ct
L
( x ,

) j
) : eigenfunct ion
 d ( x , t )
 j



0
A j
U j
( x ,
 j
) cos(
 j t ) etc.
Free oscillation = stationary wave
 Interference of two counter propagating waves
(see e.g. http://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html)

(Duck from Théocrite,
© J.-L. & P. Coudray)
Few minutes after the earthquake
Constructive interferences
 free oscillations
(or stationary waves)
 Periods < 54 min, amplitudes < 1 mm
Few hours after the earthquake (
0
S
20
)
 Observable months after great earthquakes (e.g. Sumatra, Dec 2004)

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

 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 reawake 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

Vibrating string: d ( x , t )
 j



0
A j sin(
 c j x
) cos(
 j t )
( r ,

,

)
 n



0 l



0 m



0 n
A l m n y l
( r )
 x l m
(

,

) e i n
 l m t
Radial eigenfunction
Surface eigenfunction n = radial order n = 0 : fundamental n > 0 : overtones l , m = surface orders l = angular order
-l < m < l = azimuthal order

d ( r ,

,

)
 n



0 l



0 m



0 n
A l m n y l
( r )
 x l m
(

,

) e i n
 l m t n
A m l
: excitation amplitude e
 from d one can have info on the source if all n y l known and x m l
Conversely:
 from A one can predict d : modes form the basis vectors, their combination describe the displacement (synthetic sismograms)

 Frequencies of the eigen modes depend on :
 The shape of the Earth
 and its
 density,
(resistance to acceleration)
 shear modulus,
(resistance to a change of shape)
 compressibility modulus
(resistance to a change of volume).

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 : d
T
( r ,

,

)
 n



0 l



0 m l 
  l n
A l m n
W l
( r e
) T l m
(

,

) e i n
 l m t
S : d
S
( r ,

,

)
 n



0 l



0
Radial eigenfunction
Surface eigenfunction
 m
 l
 l n
A l m
 n
U l
( r ) R l m
(

,

)
 n
V l
( r ) S l m
(

,

)
 e i n
 l m t

Toroidal modes n
T m l
: Spheroidal modes n
S m l
:
 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

0
S
2 is the longest (“fundamental”)
 Affect the whole Earth (even into the fluid outer core !)

S :
n : no direct relationship with nodes with depth
l : # nodal planes in latitude
m : # nodal planes in longitude
! Max nodal planes = l
0
S 0
2
T :
n : nodal planes with depth
l : # nodal planes in latitude
m : # nodal planes in longitude
! Max nodal planes = l - 1
0
T 0
3

...
...
0
S
0
:
« balloon » or
« breathing » : radial only
(20.5 minutes)
0
S
2
: « football » mode
(Fundamental, 53.9 minutes)
0
S
3
:
(25.7 minutes)
...
0
S
29
:
(4.5 minutes)
Rem:
0
S
1
= translation
Animation
0
S
0/3 from Lucien Saviot http://www.u-bourgogne.fr/REACTIVITE/manapi/saviot/deform/
Animation
0
S
2 from Hein Haak http://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html
0
S
29 from: http://wwwsoc.nii.ac.jp/geod-soc/web-text/part3/nawa/nawa-1_files/Fig1.jpg

0
T
2
: «twisting» mode
(44.2 minutes, observed in
1989 with an extensometer)
Animation from Hein Haak http://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html
1
T
2
(12.6 minutes)
0
T
3
(28.4 minutes)
Rem:
0
T
1
= rotation
0
T
0
= not existing
Animation from Lucien Saviot http://www.u-bourgogne.fr/REACTIVITE/manapi/saviot/deform/

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

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

0
l
Equivalent to surface Rayleigh waves l < 20: whole mantle l > 20: outer mantle shear energy density compressional energy density
Ruedi Widmer’s home page: http://www-gpi.physik.uni-karlsruhe.de/pub/widmer/Modes/modes.html

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

0
l
0
l
0
S equivalent to interfering surface Rayleigh waves
0
T equivalent to interfering surface Love waves www.advalytix.de/ pics/SAWRAiGH.gif http://www.eas.purdue.edu/~braile/edumod/waves/Lwave.htm

«balloon» mode:
T = 20.5 min.
 Frequency ~ 0.001 Hertz
Do 256 Hertz
 T= 0.004 s
18 X

?
http://www.iris.iris.edu/sumatra/

300 km
1300 km

1
0.8
Sumatra Earthquake: spectrum
Membach, SG C021, 20041226 08h00-20041231 00h00
0
S
4
0
S
3
0.6
0.4
0
S
2
0.2
0
0
T
2
2
S
1
0.0004
0
T
3
1
S
2
0
T
4
0.0008
0
S
0
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
S l m n
T l m
If SNREI (Solid Not 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

Coriolis

Ellipticity

3D
0
S
0
2
Animation
0
S
2 from Hein Haak http://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html
0
S
2
2

Splitting: Sumatra 2004
Membach SG-C021
0
S
2
Multiplets m=-2, -1, 0, 1, 2
“Zeeman effect”
M. Van Camp http://www.iris.iris.edu/sumatra/

Coupling: Balleny 1998
In an elliptic rotating heterogeneous Earth:
Mode splitting and coupling : the modes no more orthogonal

An eigenfunction can contain perturbation from the eigengfunctions of neighbouring modes e.g. T can present a vertical component or Different modes at the same frequency

Displacement in SNREI
Coriolis force

Modes and Magnitude
Time after beginning of the rupture:
00:11
00:45
01:15
04:20
8.0 (M
W
) P-waves 7 stations
8.5 (M
W
) P-waves 25 stations
8.5 (M
W
) Surface waves 157 stations
8.9 (M
W
) Surface waves (automatic)
19:03 9.0 (M
W
) Surface waves (revised)
Jan. 2005 9.3 (M
W
) Free oscillations
April 2005 9.2 (M
W
) GPS displacements
300-500 s surface waves http://www.gps.caltech.edu/%7Ejichen/Earthquake/2004/aceh/aceh.html

Modes and Magnitude
From aftershocks, free oscillations,
GPS, …
Rupture zone as determined using
300-500 surface waves http://www.ipgp.jussieu.fr/~lacassin/Sumatra/After/AfterNEIC-ALL.gif
Seth Stein and Emile Okal
Calculated vs. observed

3
2
1
0
Modes and Magnitude
SG C021 Membach, same duration:
Sumatra 2004: M w
= 9.1-9.3
Peru 2001: Mw = 8.1
0.001
0.002
0.003
0.004
0.005

Undertones
Oscillations  Restoring force
If rigidity   restoring force  period 
Seismic modes
: restoring force (Elasticity, molecular cohesion) proportional with:

Shear modulus

Incompressibility

Density
Sub-seismic modes (or « undertones »): m,  <<  restoring force proportional to:
 Archimedean force : gravity waves
 Coriolis force : inertial waves
 Lorenz force : hydro magnetic or Alvèn waves
 Magnetic Archimedean Coriolis : MAC waves

Slichter mode (triplet) (pointed out in 1961) :
1
S
1
• Translation of the solid inner core in the liquid outer core
(
1
S
1
, period ~ 4-8 h)
Controlled by the density jump between the inner and outer core, and the Archimedean force of the fluid core

Core modes
• Oscillations in the fluid outer core
(periods in the tidal band (?)
Information on the stratification of the outer core

Undertones
Normal modes of a rotating elliptic Earth
 Nearly Diurnal Free Wobble (NDFW)
 Chandler (~ 435 d)

NDFW
• Nearly Diurnal Free Wobble
(432 days in the Celestial frame = Free Core Nutation)
1.30
1.20
1.10
P
1 K
1
Y
1 j
1
1.00
-0.99
-1.00
Fréquence (cycles par jour)
-1.01
Observing the NDFW / FCN is thus very useful to measure the CMB flattening and to obtain information about the dissipation effect at this interface. Fortunately, the eigenfrequency of the NDFW is located within the tidal band and induces a perturbation of diurnal tides (Unfortunately the amplitude of Y
1 is weak!).
In the space frame, the FCN is measured by the Very Long Baseline Interferometry (VLBI).
Non-Seismic proof of the fluidity of the outer core.

Chandler wobble (« polar motion ») (1891)
This motion, due to the dynamic flattening of the Earth, appears when the rotation axis does not coincide anymore with the polar main axe of inertia.
Without any external torque, the total angular momentum remains constant in magnitude and direction, but the Earth twists so that related to its surface, the instantaneous rotation axis moves around the polar main inertia axis.
Period : 435 days (~14 months)(Chandler 1891)
– 305 days if the Earth was rigid (Euler)
Most probably excited by atmospheric forcing

1000 nm/s²
Spectrum of the ground acceleration (T > 1 s)
Hydrology
Undertones
100 nm/s²
10 nm/s²
Tidal band
1 nm/s²
0.1 nm/s²
0.01 nm/s²
1 s 10 s
Seismic normal modes
Induced by the atmosphere
(« humming »)
100 s 1 h 6 h (?)
Period
12 h
Liquid outer core modes
1d 14 d 1 month 1 yr 435 d

Observing normal modes

Extensometres (Isabella, 1960)

Long period seismometers
 Spring and superconducting gravimeters
!!! Not able to monitor toroidal modes (but…)

How measuring an earthquake ?
 inertial pendulum (same idea since 130 years !)
Seismometer
Seismogram
Seismograph
@ 10 km: M = 3  2 µm
M = 5  0.2 mm

Different design of seismometers
LaCoste Spring Leaf spring
Garden gate Inverted pendulum Bifilar (Zöllner)

Principle of the superconducting gravimeter g
Spring gravimeter g
Superconducting gravimeter
(magnetic levitation) mvc

Superconducting gravimeter
Advantage: stable calibration factor (phase [<0.1 s] and amplitude [0.1 %])
Sumatra 2004: some seismometers suffer 5 to 10 % deviation
(Park et al., Science, 2005)

10 %

M
W
= 8.4 (
 largest event between 1965 and 2001) !!!

STS-1 Vertical

STS-1 Vertical
Leaf spring g
Seismic mass
(m)
Boom
Hinge

STS-1 Horizontal
Garden gate suspension
 Allows us to measure Toroidal AND Spheroidal modes

Atmospheric effects
(also affecting Earth tide analysis)
Loading :
+1 nm/s²/hPa
Newtonian effects : -4 nm/s²/hPa
(+ buoyancy)
+ local deformations

0.08
0.03
0.18
0.13
0.08
0.03
Spectra after correction of the barometric effect
0.18
C026 Strasbourg
0.13
C021 Membach
Balleny Islands 1998, M w
=8.1
0.4
0.8
Fréquence (mHz)
1.2
1.6

“International Deployment of Accelerometers” (Cecil and IDA
Green)
Late ’60ies: First idea after a LaCoste gravimeter provided nice data
The original network 1975-1995 was a global network of digitally recorded La Coste gravimeters
They could provide valuable constraints on earth structure and earthquake mechanisms, but a shortage of data limited further progress. During the same period, low-noise feedback seismometers were developed that allowed such data to be obtained from relatively small (and hence frequent) earthquakes.
A complete description of the IDA network can be found in Eos (1986, 67 (16))
C. & I. Green

Presently: 1 accelerometer + BB seismometer (STS-1, Güralp)
Evolution of the acquisition systems used by the IDA network

The Global Geodynamics Project GGP
Network of ~ 20 superconducting gravimeters

Goal: Extract global signal disturbed by local effects
(« Stacking »)
 Study of undertones, tides, hydrology, …(Crossley et al.,
EOS, 1999)

Study of seismic normal modes : recent investigations have showed they are the best < 1 mHz: important to constrain Earth’s density profile

The Global Geodynamics Project GGP
-Standardized format
-Stability of data acquisition systems and calibration factors
-Exchange of gravity data
-Detailed logbooks
-No data on-line (“live”); delay of 6 months: seismologists do not use it
-Format not used by seismologists
-Transfer function not always known

A world première: the SG at the IRIS data base
NASA/Goddard Space Flight Center Scientific Visualization Studio

The IRIS data base

Membach SG C021 on the IRIS data base
Pressure

The future of a geophysical station
 One instrument, 240 dB dynamics (  A/D 40 bits)
 Noise level: 0.1 nm/s² (frequency dependent)
 Frequency band : 10 -8 to 1000 Hz (1 yr to 0.001 s)
This is what we do in Membach…but with 3 instruments
-1 broadband seismometer Güralp (> 1990): 100 s to 0.02 s (50 Hz)
-1 accelerometer Kinemetrics ETNA (>2003): 10 s to 0.01 s (100 Hz)
-1 superconducting gravimeter (>1995): 20 s to years
-1 absolute gravimeter (>1996): 12 h to centuries (?)
+ 1 L4-3D “historic” (>1985): 0.2 to 50 Hz

Elsewhere?
Helioseismology and Astroseismology (or Asteroseismology): Spheroidal modes
On the other planets: Mars, Venus?
 Modes could be excited by the atmosphere (« humming »)