Contribution of wide-band oscillations
excited by the fluid excitation functions
to the prediction errors of the pole
coordinates data
W. Kosek1, A. Rzeszótko1 , W. Popiński2
1Space
Research Centre, Polish Academy of Sciences, Warsaw, Poland
2Central Statistical Office, Warsaw, Poland
Journées "Systèmes de référence spatio-temporels"
and X. Lohrmann-Kolloquium 22, 23, 24 September 2008 - Dresden, Germany
DATA

x, y pole coordinates data from the IERS: EOPC04_IAU2000.62-now
(1962.0 - 2008.6), Δt = 1 day,
http://hpiers.obspm.fr/iers/eop/eopc04_05/,

Equatorial components of atmospheric angular momentum from
NCEP/NCAR, aam.ncep.reanalysis.* (1948 - 2008.6) Δt = 0.25 day,
ftp://ftp.aer.com/pub/anon_collaborations/sba/,

Equatorial components of ocean angular momentum (mass + motion):
1) c20010701.oam (gross03.oam) (Jan. 1980 - Mar. 2002) Δt = 1 day,
2) ECCO_kf049f.oam (Mar. 2002 - Mar. 2006), Δt = 1 day,
http://euler.jpl.nasa.gov/sbo/sbo_data.html,

Equatorial components of effective angular momentum function of the
hydrology obtained by numerical integration of water storage data from
NCEP: water_ncep_1979.dat, water_ncep_1980.dat, …,
water_ncep_2004.dat, Δt = 1 day,
ftp://ftp.csr.utexas.edu/pub/ggfc/water/NCEP.
x, y pole coordinates model data computed
from fluid excitation functions
Differential equation of polar motion:
i m (t )  m(t )   (t )

ch
m(t)  x(t)  iy(t)
- pole coordinates,
 (t )   (t )  i (t ) - equatorial excitation functions corresponding to AAM, OAM
1
2

 ch  2 1 i
Tch 
2Q
and HAM,




- complex-valued Chandler frequency,
where Tch  433 days and Q  170
Approximate solution of this equation in discrete time moments can be obtained
using the trapezoidal rule of numerical integration:
m(t  t)  m(t)  expi ch t   i
 ch t 
2

 (t  t)   (t)  expi ch t 
THE MORLET WAVELET TRANSFORM COHERENCE
The WT coefficients of complex-valued signal x(t ) are defined as:
 
1
1
/
2
X (b, a)  | a |  x() (a) exp(ib)d,
2

where
a  0, b are dilation and translation parameters, respectively,

 ()  exp((  2 )2 / 2) is the CFT of complex-valued Morlet wavelet function:
 (t)  exp(t 2 / 2) exp(i2 t) /

x() is the CFT of x(t).
2 and
Spectro-temporal coherence between x(t ) and y(t ) time series is defined as:
 Xˆ (t  b, a)Yˆ (t  b, a)
M
b  M
ˆxy (t, a) 
 Xˆ (t  b, a)
M
b  M
2 M
 Yˆ (t  b, a)
2
,


err ˆ xy (t , a ) 
b M
where M is a positive integer and Δt is the sampling interval.
a
t (2M  1)
The MWT spectro-temporal coherence between IERS x, y pole coordinates
data and x, y pole coordinates model data computed from AAM, OAM and
HAM excitation functions
x - iy
400
200
IERS, AAM
-200
-400
0.9
0.8
0.7
0.6
0.5
period (days)
1965 1970 1975 1980 1985 1990 1995 2000 2005
400
200
IERS, OAM
-200
-400
1980 1985 1990 1995 2000 2005
0.4
400
0.3
200
IERS, HAM
0.2
0.1
-200
0
-400
1980 1985 1990 1995 2000 2005
YEARS
The MWT spectro-temporal coherence between IERS x, y pole coordinates
data and x, y pole coordinates model data computed from AAM, AAM+OAM
and AAM+OAM+HAM excitation functions
x - iy
400
200
IERS, AAM
-200
-400
1965 1970 1975 1980 1985 1990 1995 2000 2005
0.9
0.8
0.7
0.6
0.5
period (days)
400
200
IERS, AAM+OAM
-200
-400
1980 1985 1990 1995 2000 2005
0.4
400
0.3
200
IERS, AAM+OAM+HAM
0.2
0.1
-200
0
-400
1980
1985
1990
1995
YEARS
2000
2005
Prediction of x, y pole coordinates data
by the LS+AR method
x, y
x, y
LS model
(Chandler circle + annual and
semiannual ellipses + linear trend)
LS extrapolation
Prediction of
x, y
x, y
LS residuals
AR prediction
Prediction of
x, y
LS extrapolation
x, y
LS residuals
LS+AR prediction errors of IERS x, y pole coordinates data and of x, y pole
coordinates model data computed from AAM, OAM and HAM excitation
functions
y (IERS)
x (IERS)
300
300
200
200
100
100
0
1980
1984
1988
1992
1996
2000
2004
2008
0
1980
days in the future
300
300
200
200
100
100
1984
1988
1992
1996
2000
2004
2008
300
200
200
100
100
1988
1992
1996
2000
2004
2004
2008
arcsec
0.1
0.04
1984
1988
1992
1996
2000
2004
300
200
200
100
100
1988
1992
YEARS
1996
2000
2004
2008
0.02
0
1984
1988
1992
1996
2000
2004
1988
1992
1996
2000
2004
y (HAM)
300
1984
2000
0.06
0
1980
x (HAM)
0
1980
1996
y (OAM)
300
1984
1992
0.08
0
1980
x (OAM)
0
1980
1988
y (AAM)
x (AAM)
0
1980
1984
0
1980
1984
YEARS
The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y
pole coordinates model data computed from AAM (orange), OAM (blue) and HAM
(green) excitation functions
arcsec
arcsec
x
IERS
0.03
y
0.03
OAM
0.02
0.02
AAM
HAM
0.01
0.00
0.01
0.00
0
100
200
300
days in the future
0
100
200
300
days in the future
The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y
pole coordinates model data computed from AAM+OAM (red) and AAM+OAM+HAM
(purple) excitation functions
arcsec
arcsec
x
IERS
0.03
y
AAM+OAM 0.03
AAM+OAM+HAM
0.02
0.02
0.01
0.01
0.00
0.00
0
100
200
300
days in the future
0
100
200
300
days in the future
DISCRETE WAVELET TRANSFORM BAND PASS FILTER
The DWT j-th frequency component of the complex valued signal x(t) is given by:
2 j11
p
x j (t)   S j,k j,k (t) for t  0,1,...,n 1,
j  j0, j0 1,..., p 1, n  2 ,
k 2 j1
Signal reconstruction:
n1
p1
S j,k   x(t ) j,k (t ) - the DWT coefficients,
 x j (t )  x(t )
j j
t 0
0
 j,k (t)  n 2 j / 2 j (t  n / 2  2 j kn) - discrete Shannon wavelets.
For fixed lowest frequency index
0  j  p  2 and time index k  2 0 ,2 0 1,...,2 0 1
0
j
j 1
0
 j (t )  1n exp[i (t  n / 2) / n] sin[2  (t  n / 2) / n],
sin[ (t  n / 2) / n]
0
j
j
j 1
 j (n / 2)  2 0 / n
0
For higher frequency index j  j0 1, j0  2,..., p 1 and time index k  2 j 1,2 j 1 1,...,2 j 1 1
j
j
 j (t )  1n exp[i (t  n / 2) / n]sin[2  (t  n / 2) / n](2cos[2  (t  n / 2) / n]1) ,
sin[ (t  n / 2) / n]
 j (n / 2)  2 j / n
The DWT frequency components of x pole coordinate data
arcsec
0.04
0.00
-0.04
0.04
0.00
-0.04
0.04
0.00
-0.04
0.04
0.00
-0.04
0.30
0.00
-0.30
0.04
0.00
-0.04
0.02
0.00
-0.02
0.01
0.00
-0.01
0.01
0.00
-0.01
0.01
0.00
-0.01
0.01
0.00
-0.01
0.01
0.00
-0.01
j= 0
j= 1
j= 2
longer period
j= 3
j= 4
Chandler + Annual
j= 5
Semiannual
j= 6
j= 7
j= 8
j= 9
j=10
j=11
1962
40000
44000
48000
MJD
52000
2008
shorter period
The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y
pole coordinates model data computed by summing the chosen DWTBPF components
arcsec
0.04
x
0.03
arcsec
IERS
0.04
Ch + An + Sa
Ch + An + shorter period
Ch + An + longer period
Ch + An
0.03
0.02
0.02
0.01
0.01
0.00
0.00
0
50 100 150 200 250 300 350
days in the future
0
y
50 100 150 200 250 300 350
days in the future
The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y
pole coordinates model data computed from AAM+OAM (red) excitation functions as
well as by summing the DWTBPF components corresponding to Chandler, annual and
shorter period oscillations (green)
arcsec
arcsec
x
IERS
0.03
AAM+OAM
y
0.03
Ch + An + shorter period
0.02
0.02
0.01
0.01
0.00
0.00
0
100
200
300
days in the future
0
100
200
300
days in the future
CONCLUSIONS





The contributions of atmospheric or ocean angular momentum excitation
functions to the mean prediction errors of x, y pole coordinates data
from 1 to about 100 days in the future is similar and of the order of 60%
of the total prediction error.
The contribution of ocean angular momentum excitation function to the
mean prediction errors of x, y pole coordinates data for prediction
lengths greater than 100 days becomes greater than the contribution of
the atmospheric excitation function.
The contribution of the joint atmosphere and ocean angular momentum
excitation to the mean prediction errors of x, y pole coordinates data is
almost equal to the contribution of the sum of Chandler + annual and
shorter period frequency components. Both contributions explain about
80÷90% of the total prediction error.
Big prediction errors of IERS x, y pole coordinates data in 1981-1982
and in 2006-2007 are mostly caused by wide-band ocean and atmospheric
excitation, respectively.
The contribution of the hydrologic angular momentum excitation to the
mean prediction errors of x, y pole coordinates data is negligible.
Acknowledgements
This paper was supported by the Polish Ministry of Education
and Science, project No 8T12E 039 29 under the leadership of
Dr. W. Kosek. The authors of this poster are also supported by
the Organizers of Journées "Systemes de référence spatiotemporels" and X. Lohrmann-Kolloquium.
poster available: http://www.cbk.waw.pl/~kosek