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![Page 1: Gravimeters for seismological broadband monitoring: Earth’s free oscillations Michel Van Camp Royal Observatory of Belgium.](https://reader034.fdocuments.us/reader034/viewer/2022052510/56649c6d5503460f9491f064/html5/thumbnails/1.jpg)
Gravimeters for seismological Gravimeters for seismological broadband monitoring:broadband monitoring:Earth’s free oscillationsEarth’s free oscillations
Michel Van CampRoyal Observatory of Belgium
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What is a free oscillation?What is a free oscillation?
3rd
Harmonicetc.
2nd Harmonic
1st Harmonic
)cos(),(),(
ioneigenfunct :),(sin
encyeigenfrequor eigenvalue :
)cos(sin~
2 :mode
)cos(sin~
0
txUAtxd
xUL
xjL
cj
L
ctj
L
xjd
jc
LTj
ic
L
tc
xd
jjjj
j
jj
j
th
Fundamental
L
x
Free oscillation = stationary waveInterference of two counter propagating waves
(see e.g. http://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html)
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Seismic normal modesSeismic normal modes
Periods < 54 min, amplitudes < 1 mm
Observable months after great earthquakes (e.g. Sumatra, Dec 2004)
Few minutes after the earthquakeConstructive interferences free oscillations (or stationary waves)
Few hours after the earthquake (0S20)
(Duck from Théocrite, © J.-L. & P. Coudray)
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Travelling surface wavesTravelling surface waves
Richard Aster, New Mexico Institute of Mining and Technology http://www.iris.iris.edu/sumatra/
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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 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
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On the sphere…On the sphere…
timlln
m
mln
ln
mlnexryArd ),()(),,(
000
n = radial ordern = 0 : fundamentaln > 0 : overtones
l, m = surface ordersl = angular order-l < m < l = azimuthal order
Radialeigenfunction
Surfaceeigenfunction
)cos()sin(),(0
tc
xAtxd j
j
jj
Vibrating string:
On the sphere:
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Why studying normal modes?Why studying normal modes?
enAml : excitation amplitude
from d one can have info on the source if all nyl and xml
known
Conversely: from A one can predict d : modes form the basis vectors, their combination describe the displacement (synthetic
sismograms)
timlln
m
mln
ln
mlnexryArd ),()(),,(
000
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Why studying normal modes ?Why studying normal modes ?
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).
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Toroidal and spheroidalToroidal and spheroidal
etim
lln
l
lm
mln
ln
T mlneTrWArd ),()(),,(
00
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 :
timlln
mlln
l
lm
mln
ln
S mlneSrVRrUArd ),()(),()(),,(
00
S :Radial
eigenfunctionSurface
eigenfunction
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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 :
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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 = l
0S02
T : n : nodal planes with depthl : # nodal planes in latitudem : # nodal planes in longitude
! Max nodal planes = l - 1
0T03
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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
...
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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
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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)?
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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
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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
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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
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Eigenfunctions : Eigenfunctions : 00SSll and and 00TTl l
0S equivalent to interfering surface Rayleigh waves
0T equivalent to interfering surface Love waves
http://www.eas.purdue.edu/~braile/edumod/waves/Lwave.htm
www.advalytix.de/ pics/SAWRAiGH.gif
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Music and seismic normal modesMusic and seismic normal modes
«balloon» mode:
T = 20.5 min.
Frequency ~ 0.001 Hertz
Do 256 Hertz
T= 0.004 s
18 X18 X
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The great Sumatra-Andaman EarthquakeThe great Sumatra-Andaman Earthquake
http://www.iris.iris.edu/sumatra/
?
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300 km
The great Sumatra-Andaman EarthquakeThe great Sumatra-Andaman Earthquake
1300 km
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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
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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
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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 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
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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
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SplittingSplitting
020 S 2
20 S
Coriolis
Ellipticity
3D
Animation 0S2 from Hein Haakhttp://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html
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Splitting: Sumatra 2004Splitting: Sumatra 2004
http://www.iris.iris.edu/sumatra/M. Van Camp
Membach SG-C021
0S2 Multiplets m=-2, -1, 0, 1, 2
“Zeeman effect”
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Coupling: Balleny 1998Coupling: 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 componentor Different modes at the same frequency
Coriolis force
Displacement in SNREI
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Modes and MagnitudeModes and Magnitude
Time 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
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Modes and MagnitudeModes and Magnitude
Seth Stein and Emile OkalCalculated vs. observed
http://www.ipgp.jussieu.fr/~lacassin/Sumatra/After/AfterNEIC-ALL.gif
Rupture zone as determined using 300-500 surface waves
From aftershocks,free oscillations,GPS, …
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nm/s
²
0.001 0.002 0.003 0.004 0.005
0
1
2
3
Modes and MagnitudeModes and Magnitude
SG C021 Membach, same duration:Sumatra 2004: Mw = 9.1-9.3Peru 2001: Mw = 8.1
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UndertonesUndertones
Seismic modes : restoring force (Elasticity, molecular cohesion) proportional with:
Shear modulus Incompressibility Density
Sub-seismic modes (or « undertones »): << restoring force proportional to:
Archimedean force : gravity waves Coriolis force : inertial waves Lorenz force : hydro magnetic or Alvèn waves Magnetic Archimedean Coriolis : MAC waves
Oscillations Restoring force
If rigidity restoring force period
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Slichter mode (triplet) (pointed out in 1961) : Slichter mode (triplet) (pointed out in 1961) : 11SS11
• Translation of the solid inner core in the liquid outer core(1S1, period ~ 4-8 h)
Controlled by the density jump between the inner and outer core, and the Archimedean force of the fluid core
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Core modesCore modes
• Oscillations in the fluid outer core(periods in the tidal band (?)
Information on the stratification of the outer core
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UndertonesUndertones
Normal modes of a rotating elliptic Earth
Nearly Diurnal Free Wobble (NDFW)
Chandler (~ 435 d)
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• NNearly DDiurnal FFree WWobble (432 days in the Celestial frame = Free Core Nutation)
P1 K1
-1.01-1.00-0.99Fréquence (cycles par jour)
1.00
1.10
1.20
1.30
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 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.
NDFWNDFW
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Chandler wobble (« polar motion ») (1891)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
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Period
qu
art
-diu
rnal
ter-
diu
rnal
diu
rnal
Fort
nig
htl
y
mon
thly
sem
i-diu
rnal
Tidal band
10 s 100 s 1 h
0.1 nm/s²
1 nm/s²
0.01 nm/s²
10 nm/s²
100 nm/s²
1000 nm/s²
1 s 12 h 1d 1 month14 d 1 yr 435 d
Seismic
normal modes
Induced by the
atmosphere (« humming
»)
Mic
roseis
mS
urf
ace w
aves
6 h (?)
Slich
ter
trip
let
Pola
r m
oti
on
, te
cto
nic
s
NDFW
Liquid outer core modes
Hydrology
Spectrum of the ground acceleration (T > 1 s)Spectrum of the ground acceleration (T > 1 s)
Undertones
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Observing normal modesObserving normal modes
Extensometres (Isabella, 1960)
Long period seismometers
Spring and superconducting gravimeters
!!! Not able to monitor toroidal modes (but…)
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How measuring an earthquake ?How measuring an earthquake ?
Seismogram
Seismometer
Seismograph
inertial pendulum (same idea since 130 years !)
@ 10 km: M = 3 2 µm M = 5 0.2 mm
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Different design of seismometersDifferent design of seismometers
Garden gate Inverted pendulum
Leaf springLaCoste
Bifilar (Zöllner)
Spring
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g
g
mvc
Spring gravimeter Superconducting gravimeter (magnetic levitation)
Principle of the superconducting gravimeterPrinciple of the superconducting gravimeter
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Superconducting gravimeterSuperconducting 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 % MW = 8.4 (largest event between 1965 and 2001) !!!
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STS-1 VerticalSTS-1 Vertical
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STS-1 VerticalSTS-1 Vertical
Hinge
Leaf spring
BoomSeismic
mass(m)
g
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STS-1 HorizontalSTS-1 Horizontal
Allows us to measure Toroidal AND Spheroidal modesAllows us to measure Toroidal AND Spheroidal modes
Garden gate suspension
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Atmospheric effects Atmospheric effects (also affecting Earth tide analysis)(also affecting Earth tide analysis)
Newtonian effects : -4 nm/s²/hPa
(+ buoyancy)
Loading : +1 nm/s²/hPa
+ local deformations
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0 .0 3
0 .0 8
0 .1 3
0 .1 8
C 0 26 S trasbourg
0 .0 3
0 .0 8
0 .1 3
0 .1 8 C 0 21 M embach
0 .4 0 .8 1 .2 1 .6Fréquence (mH z)
Spectra after correction of the barometric effectSpectra after correction of the barometric effect
Balleny Islands 1998, Mw=8.1
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“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
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Evolution of the acquisition systems used by the IDA network
Presently: 1 accelerometer + BB seismometer (STS-1, Güralp)
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The Global Geodynamics Project GGPThe 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
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-No data on-line (“live”); delay of 6 months: seismologists do not use it-Format not used by seismologists-Transfer function not always known
The Global Geodynamics Project GGPThe Global Geodynamics Project GGP
-Standardized format-Stability of data acquisition systems and calibration factors-Exchange of gravity data-Detailed logbooks
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A world première: the SG at the IRIS data baseA world première: the SG at the IRIS data base
NASA/Goddard Space Flight Center Scientific Visualization Studio
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The IRIS data baseThe IRIS data base
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Membach SG C021 on the IRIS data baseMembach SG C021 on the IRIS data base
Pressure
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The future of a geophysical The future of a geophysical stationstation
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
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Elsewhere?Elsewhere?
Helioseismology and Astroseismology (or Asteroseismology): Spheroidal modes
On the other planets: Mars, Venus? Modes could be excited by the atmosphere (« humming »)