Gravimeters for seismological Gravimeters for seismological broadband monitoring:broadband monitoring:Earth’s free oscillationsEarth’s free oscillations

Michel Van CampRoyal Observatory of Belgium

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)

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)

Travelling surface wavesTravelling surface waves

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

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

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:

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

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).

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

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 = l

0S02

T : n : nodal planes with depthl : # nodal planes in latitudem : # nodal planes in longitude

! Max nodal planes = l - 1

0T03

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 waves

0T equivalent to interfering surface Love waves

http://www.eas.purdue.edu/~braile/edumod/waves/Lwave.htm

www.advalytix.de/ pics/SAWRAiGH.gif

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

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 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

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

SplittingSplitting

020 S 2

20 S

Coriolis

Ellipticity

3D

Animation 0S2 from Hein Haakhttp://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html

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”

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

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

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, …

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

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

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

Core modesCore modes

• Oscillations in the fluid outer core(periods in the tidal band (?)

Information on the stratification of the outer core

UndertonesUndertones

Normal modes of a rotating elliptic Earth

Nearly Diurnal Free Wobble (NDFW)

Chandler (~ 435 d)

• 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

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

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

Observing normal modesObserving normal modes

Extensometres (Isabella, 1960)

Long period seismometers

Spring and superconducting gravimeters

!!! Not able to monitor toroidal modes (but…)

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

Different design of seismometersDifferent design of seismometers

Garden gate Inverted pendulum

Leaf springLaCoste

Bifilar (Zöllner)

Spring

g

g

mvc

Spring gravimeter Superconducting gravimeter (magnetic levitation)

Principle of the superconducting gravimeterPrinciple of the superconducting gravimeter

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) !!!

STS-1 VerticalSTS-1 Vertical

STS-1 VerticalSTS-1 Vertical

Hinge

Leaf spring

BoomSeismic

mass(m)

g

STS-1 HorizontalSTS-1 Horizontal

Allows us to measure Toroidal AND Spheroidal modesAllows us to measure Toroidal AND Spheroidal modes

Garden gate suspension

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

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

“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

Evolution of the acquisition systems used by the IDA network

Presently: 1 accelerometer + BB seismometer (STS-1, Güralp)

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

-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

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

The IRIS data baseThe IRIS data base

Membach SG C021 on the IRIS data baseMembach SG C021 on the IRIS data base

Pressure

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

Elsewhere?Elsewhere?

Helioseismology and Astroseismology (or Asteroseismology): Spheroidal modes

On the other planets: Mars, Venus? Modes could be excited by the atmosphere (« humming »)