Analysis of Chandler wobble excitation, reconstructed from observations of the polar
motion of the Earth
Leonid Zotov
Sternberg Astronomical InstituteLomonosov Moscow State University
Journees 2010, September 20 – 22, 2010, Paris Observatory
Earth’s pole motion from the bulletin IERS EOP C01
Since 1846 yr, step 0.05 yr
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
m(t)
1860 1880 1900 1920 1940 1960 1980 2000years
-0.4
-0.2
0
0.2
0.4
arc
sec
X-coordinate of the pole
X-coordinate
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
SSA method – “caterpillar”
[ ]KXXX ...1=Trajectorymatrix
[ ]KL×
Principal components grouping
Hankelization
http://www.gistatgroup.com/gus/books.html
Tddd
TT VUVUUSVX λλ ++== ...111
Singular value decomposition SVD�����
0 200 400 600 800 1000
-5
0
5
10
15
SSA exampleinitial signaltrendoscillationoscillation with modulation
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1860 1880 1900 1920 1940 1960 1980 2000years
-0.2
-0.1
0
0.1
0.2
arc
sec
SSA-decomposition of X-coordinate of the poleChandler componentannual componenttrend
Result of SSA of the 2-D pole coordinate time serieson the example of X-coordinate
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Amplitude spectrum of initial signaland SSA-components
0
0.02
0.04
0.06
0.08
0.1
arcs
ec
0 0.2 0.4 0.6 0.8 1 1.2 1.4cycles per year
PM spectrumSSA ChandlerSSA annualEOP CO1
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Dynamical system of the rotating Earth
433
1=cf 175=QParameters used days-1
)(ˆ)()(ˆ ffLfm χ⋅=
To find - inverse problem)(tχ
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Frequency response
0.6 0.7 0.8 0.9 1 1.1 1.2cycles per year
0
100
200
300 amplitude responseQ=175, fc=1/433Q=100, fc=1/425
0.6 0.7 0.8 0.9 1 1.1 1.2cycles per year
-200
-160
-120
-80
-40
0
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frequency responseQ=175 fc=1/433Q=100, fc=1/425
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Amplitude frequency response of inverse operators
0 0.2 0.4 0.6 0.8 1 1.2 1.4cycles per year
0
0.01
0.02
0.03
0.04
0.05Inversion amplitude response
corrective smoothingregularizationinverse operator
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Wilson-Jeffreys filter
Jeffreys H. (1940) The variation of latitude, Mon Not Roy Astr. Soc., Vol. 100, 139-155Wilson C. (1985), Discrete polar motion equations, Geophys J. Roy. Astr. Soc., Vol. 80, 551-554
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Tikhonov regularization
500=αparameter chosen
LSA-model of the annual oscillation
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Panteleev corrective smoothing
Parameter used
04.00 =fJournées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Comparison with different processes
1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010years
-0.004
-0.002
0
0.002
0.004
arc
sec
X-component of reconstructed Chandler excitationSSA + Wilson filterregularizationcorrective smoothing
1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010years
-0.08
-0.04
0
0.04
0.08
arcs
ec
OAMAAM
-40
-20
0
20
40
Southern Oscillation Index
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Chandler excitation and Tidal model for the Length of day LOD
1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010years
-0.004
-0.002
0
0.002
0.004
arc
sec
X-component of reconstructed Chandler excitationSSA + Wilson filterregularizationcorrective smoothing
-0.8
-0.4
0
0.4
0.8
msIERS tidal variations in the length of the day (LOD) model
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Spectrum of reconstructed excitation,amplitude modulation
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
0
0.5
1
1.5
2
2.5
SSA + Wilson filterRegularizationCorrective smoothing
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
0
0.5
1
1.5
2
2.5
Full moon in perigee repetition cycle and ocean
days
Avsyuk Yu. N. Tidal forces and natural processes, 1996, Moscow, Schmidt Institute of Physics of the Earth, Russian Academy of Sciences.Anisimova E P., Pokazeev K.V. Introduction to physics of hydrosphere. Moscow, MSU, 2002.
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Gabor window-transform
ωt
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Panteleev filters impulse response
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Phase changes
1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010year
-270
-180
-90
0
90
180
270
degr
eese
phaseinitial Chandler PMSSA + Wilson filterregularizationpanteleev filtering
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
Conclusions1) Chandler excitation was reconstructed by three methods for inverse problems solving:
Panteleev corrective smoothing, Wilson-Jeffreys filter after SSA, Tikhonov regularization with annual component subtraction
The results are similar. It gives hope they are reliable.
2) 18,6-year modulation of Chandler excitation, synchronous with the saros tidal cycle and related LOD changes has been found. It could prove that tidal energy is transferred to chandler excitation, probably, through the ocean and atmosphere.The mechanism is still to be found.
This work is performed with support of the President of Russia grantМК-4234.2009.5 and RFBI grant 10-05-00215-а.
Journées 2010 "Systèmes de référence spatio-temporels", September 20-22, 2010, Paris, France
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