COMPARISON OF TWO WAVELET TRANSFORM COHERENCE
AND CROSS-COVARIANCE FUNCTIONS APPLIED ON POLAR
MOTION AND ATMOSPHERIC EXCITATION
Waldemar Popiński1, Wiesław Kosek2, Harald Schuh3, Michael Schmidt4
1.
Department of Standards, Central Statistical Office, Warsaw, Poland, w.popinski@stat.gov.pl
Space Research Centre, Polish Academy of Sciences, Warsaw, Poland, kosek@cbk.waw.pl
3.
Technische Universität Wien, Austria, hschuh@luna.tuwien.ac.at
4.
Deutsches Geodätisches Forschungsinstitut, München, Germany, schmidt@dgfi.badw.de
2.
ABSTRACT. The wavelet transform techniques were applied to compute time-frequency
spectra, coherence and cross-covariance functions between complex-valued polar motion and
atmospheric excitation functions. These wavelet transform approaches are based on the
classical wavelet transform with Morlet wavelet and the harmonic wavelet transform. The
computed coherence and cross-covariance functions enable comparison of polar motion and
atmospheric excitation functions data in the chosen frequency band. In the study we
concentrate on short period oscillations with periods ranging from several to about 250 days.
The time lag functions show frequency dependent time lags corresponding to maxima of the
modules of cross-covariance functions between the polar motion and atmospheric excitation
functions.
1. INTRODUCTION
In the last years the wavelet transform has become a very appropriate tool for analyzing the
Earth orientation parameters and atmospheric angular momentum time series. The wavelet
transform enables detection of time-varying amplitudes and frequencies of oscillations present
in such time series. The goal of the investigations presented here is to find time-frequency
relationships between polar motion and its atmospheric excitation using the wavelet
techniques. Recent investigations on that topic were published by e.g. Schmitz-Hübsch and
Schuh (1999).
The wavelet coherence and cross-covariance functions will be applied to examine the
relationship between polar motion and atmospheric excitation functions. Coherence is defined
as correlation coefficient between the wavelet transform coefficients representing two time
series in time-frequency domain (Popiński and Kosek 1994). The cross-covariance function
allows to determine time lags between similar variations occurring in two time series as a
function of frequency (Schmidt and Schuh 2000b). In the present paper the wavelet transform
with Morlet analysing function (MWT) (Chui 1992) and harmonic wavelet transform (HWT)
(Newland 1998) techniques will be applied. Both wavelet techniques enable changing the
frequency resolution in the coherence and cross-covariance functions.
2. THE WAVELET TRANSFORM CROSS-COVARIANCE AND COHERENCE
FUNCTIONS
The continuous wavelet transform of a square integrable complex-valued signal x (t ) is
defined by (Chui 1992):
X (b, a ) | a |1/ 2

 x(t ) ((t  b) / a)dt ,
(1)

where in general  (t ) is a complex-valued wavelet function, b is the translation (shift)
parameter and a  0 is the dilation (scale) parameter.
The wavelet transform is well-suited to detect transient periodic fluctuations, as well as
changes in their parameters, because it can focus on a limited time span of the signal (Foster
1996). If the wavelet function is localised in time near t  0 , then computing coefficients
X (b, a ) we explore the behaviour of x (t ) near t  b .
Using the Plancherel identity (Chui 1992) we easily obtain a formula for coefficients X (b, a )

which involves the continuous Fourier transforms (CFT) of the signal x ( ) and the wavelet

function  ( ) , namely
X (b, a) 
1
| a |1 / 2
2



 x ( ) (a ) exp( ib )d
.
(2)

If the wavelet function is localised in frequency domain, then computing coefficients

X (b, a ) we explore the behaviour of x ( ) in different frequency bands depending on the
scale parameter a . The above formula is important for practical application of the wavelet
transform in time series analysis since we can compute the discrete Fourier transform (DFT)

of the data x (t ) , t  0,1,..., N  1 , multiply it by  (a k ) at discrete Fourier frequencies  k ,

k   N / 2  1, N / 2  2,..., N / 2 , and perform the inverse DFT to obtain estimates X (b, a) of
the coefficients X (b, a ) for a fixed scale a and b  0,1,..., N  1. For simplicity of notation
we assume that the sampling interval of data is t  1 .
Wavelet spectra of the signals x (t ) and y (t ) are defined by (Liu 1994):
Wx (b, a) | X (b, a) |2  X (b, a) X (b, a) ,
(3a)
W y (b, a) | Y (b, a) | 2  Y (b, a)Y (b, a)
(3b)
and the scale dependent signal variances by (Schmidt and Schuh 2000a):








 x2 (a)   | X (b, a) |2 db   Wx (b, a)db ,
 y2 (a)   | Y (b, a) |2 db   W y (b, a)db .
(4a)
(4b)
Moreover, we can define wavelet cross-spectrum and cross-covariance of the signals x (t ) and
y (t ) by the formulae (Schmidt and Schuh 2000b):
Wxy (b, a )  X (b, a )Y (b, a ) ,
(5a)

C xy (  , a) 
 X (b, a)Y (b   , a)db
(5b)

and further their wavelet scale dependent coherence:
C xy (0, a)
 xy (a ) 
 x2 (a) y2 (a)
.
(6)
We can also define the scale dependent time lag function  (a ) by the condition (Schmidt and
Schuh 2000b)  (a )  arg max C xy (  , a ) .
 R
The wavelet scale dependent coherence detects similarities between two time series in
different frequency bands. Analysing the cross-covariance functions one can learn whether
specific variations in the first time series precede or occur after similar variations in the
second one and estimate their time delay (Schmidt and Schuh 2000b).
If only discrete equidistant data of the time series x (t ) and y (t ) , t  0,1,..., N  1, are
available we have to replace the integration by a summation and in consequence we compute
the cross-covariance and scale dependent coherence estimates according to the formulae:

C xy (0, a)
,
 xy (a)  

 x2 (a) y2 (a)

(7)
where
N 1 


C xy (  , a)   X (k , a)Y (k   , a) ,

N 1

k 0
2
 x2 (a)   X (k , a)
,
k 0
N 1


(8)
2
 y2 (a)   Y (k , a) .
(9)
k 0

Analogously, we define the time lag function estimate  (a ) as the value satisfying the


condition  (a)  arg max C xy (  , a) , where the maximum is determined over some finite set
 D
D   K ,K  1,, K  1, K  of time shifts  .
Since the wavelet transform coefficients characterise local variations or oscillations of the
analysed time series it is natural to compute the running correlation coefficient of such data
for two analysed time series. In this way we obtain spectro-temporal coherence given by the
formula:

R xy (t , a)
,


R x (t , a) R y (t , a)

 xy (t , a) 
(10)
where

R xy (t , a ) 
M


 X (t  k , a)Y (t  k , a) ,
k  M
(11)

R x (t , a) 
M

2

X (t  k , a)
,
k  M

R y (t , a) 
2

 Y (t  k , a) ,
M
(12)
k  M
and M is a positive integer. For a fixed M such spectro-temporal wavelet coherence depends
on the scale parameter a  0 and the central shift parameter t .
For complex-valued time series x(t ) and y (t ) similarity in the prograde ( T  2t ) as well as
retrograde ( T  2t ) oscillations can be detected in spectro-temporal coherence and crosscovariance.
2.1 THE MORLET WAVELET TRANSFORM CONCEPT
The most widely known wavelet function used in time series applications is the Morlet
wavelet which is approximately a gaussian function with harmonic modulation (SchmitzHübsch and Schuh 1999):
 (t ) 
1
2
exp( ipt ) exp( t 2 / 2 2 )  2 exp( t 2 /  2 ) exp(  p 2 2 / 4)
,
(13)
where p is the frequency parameter (usually p  5 ),  is a parameter which controls the
decay of the Morlet wavelet. The variation of  allows the variations of the adaptive
window.
It is easy to see that for p =2  the dilated Morlet wavelet oscillates with a period a , so the
nominal value of the dilation parameter can then be interpreted as an oscillation period
(Popiński and Kosek 1994). The CFT of the Morlet wavelet is given by the formula (Chui
1992):
 ( )   exp( (  p) 2  2 / 2)  exp( (  p) 2  2 / 4) exp(  p 2 2 / 4) ,

(14)
and one can see that this wavelet has quasi-compact support both in time and frequency
domain. Consequently, in order to extract information characterising the analysed signal
behaviour in some chosen limited frequency band other wavelets must be used e.g. harmonic
wavelets described in the sequel.
2.2 THE HARMONIC WAVELET TRANSFORM CONCEPT
In time-frequency analysis by means of the harmonic wavelet transform (HWT) the wavelet
transform coefficients of a signal x(t ) are defined by the following convolution formula
(Newland 1998):
H x b,   

 w (t  b,  ) x(t )dt ,
(15)

where w(t ,  ) is the complex-valued harmonic wavelet function localised in frequency
domain near some central frequency  and b is the translation (shift) parameter.
Analogously as for the classical wavelet transform the frequency domain formula reads
H x b,   



 w( ,  ) x ( ) exp( ib )d
.
(16)

In numerical computations we multiply the DFT of a time series x(t ) by the CFT of the
harmonic wavelet function, which is of boxcar type tapered by the gaussian window for better
frequency resolution (Newland 1998). Next, applying the inverse DFT, the wavelet transform
coefficients are estimated according to the following formula (Newland 1998):

H x (t ,  )  FFT 1 W ( k ,  ) FFT x(t )  ,
(17)
where x (t ) , t  0,1,..., N  1 , is a time series,  k  k / N , k   N / 2  1, N / 2  2,..., N / 2 ,
are normalized discrete Fourier frequencies,  1 / 2    1 / 2 is the normalised central
frequency of the tapered boxcar window

exp  (    k ) 2 (2 2 )
W  k ,    
0


if
  k  
otherwise,
with window halfwidth  and smoothing parameter  .
As one can see, changing the central frequency  we can analyse the signal variations in
different frequency bands [    ,    ] (Newland 1998). For an input signal f (t ) which is
sampled N times to give the sequence f 0 , f1 ,..., f N 1 , the above described calculation
produces N wavelet coefficient estimates. They approximate coefficients calculated by the
convolution formula for the harmonic wavelets with centres at each of the N positions that
correspond to the sampled values f 0 , f1 ,..., f N 1 (Newland 1998).
From the definition of the spectro-temporal coherence it follows that the HWT spectrotemporal coherence at frequency  and time t estimates the correlation between the HWT
coefficients of time series x(t ) and y (t ) , corresponding to central frequency  and harmonic
wavelets with centers close to time moment t (Newland 1998).
3. COMPARISON OF THE MWT AND HWT COHERENCE AND CROSSCOVARIANCE FUNCTIONS
The MWT and HWT time-frequency coherence is the running correlation coefficient between
the wavelet transform coefficients which means that changing the signal value at one time
moment influences the coherence functions near the moment of change. Both methods can be
applied to real or complex-valued equidistant time series so there is a possibility to apply the
Fast Fourier Transform algorithm (Singleton 1969) to speed up relevant computations. The
characteristics of the MWT and HWT frequency dependent and time-frequency coherence
and cross-covariance functions are summarised in Table 1. These two techniques can be
applied to investigate the influence of atmospheric angular momentum on polar motion
excitation.
Table 1. Characteristics of the MWT and HWT coherence and cross-covariance functions.
MWT
Morlet wavelet is defined by time domain
analytic formula;
HWT
Harmonic wavelets are defined by
frequency domain formula;
Morlet wavelet function has quasi-compact Harmonic wavelet functions have
support both in time and frequency domain; unbounded support in time domain but their
CFT are localized in given spectral bands;
Using the Morlet wavelet function one can
not analyse the signal behaviour in some
chosen spectral band and the MWT
frequency domain localization depends on
the frequency parameter a ;
Frequency resolution increases/decreases
with the increase/decrease of the 
parameter and time resolution decreases
with the increase of the M parameter.
Using harmonic wavelet functions one can
analyse the signal behaviour in chosen
spectral bands of width 2;
Frequency/time resolution increases/
decreases with the decrease/increase of the
= parameter and time resolution
decreases with the increase of the M
parameter.
4. DATA SETS
In this paper the following time series were used:
1) The x, y pole coordinates IERSC04 data in 1962.0-2000.2 years with 1-day sampling
interval (IERS 1998),
2) the equatorial components  w p ib of the effective atmospheric angular momentum
(EAAM) reanalysis data in 1958.0-2000.2 computed by U.S. National Centers for
Environmental Prediction / National Center for Atmospheric Research (NCEP/NCAR),
being the sum of the wind and pressure modified by inverted barometric correction, the
sampling interval is equal to 0.25 day, the top of the model is 10 hPa (Barnes et al. 1983;
Kalnay et al. 1996; Salstein et al. 1986; Salstein and Rosen 1997).
The atmospheric influence on polar motion is described by the equatorial components of the
EAAM excitation functions (Barnes et al. 1983). In this paper these functions called the
atmospheric excitation functions are the sum of the wind and pressure modified by inverted
barometric correction terms since then they get the highest correlation with polar motion data
(Nastula 1995). To find the influence of atmospheric excitation functions on polar motion the
corresponding geodetic excitation function were computed from the IERSC04 pole
coordinates data (IERS 1998) using time domain Wilson and Haubrich (1976) deconvolution
formula, in which the Chandler period is equal to 435 days and the quality factor Q=100.
5. THE MWT AND HWT TIME-FREQUENCY SPECTRA OF THE ATMOSPHERIC
EXCITATION FUNCTIONS
The MWT and HWT time-frequency spectra of the complex-valued 1  i 2 EAAM
excitation functions are shown in Figure 1. The increase of the σ parameter value in the MWT
increases the frequency resolution of the spectrum, while increase of the λ and/or σ parameter
values in the HWT decreases the frequency resolution of the corresponding spectrum. The
frequency resolution of the HWT spectra depends on both λ and σ parameter values so to
reduce the number of parameters it has been assumed that λ=σ in computations. The MWT
and HWT time-frequency spectra of short period oscillations of the equatorial components of
the EAAM excitation functions are very similar, however the frequency resolution becomes
greater for the HWT than for the MWT when the periods become smaller. For periods longer
than 60 days, the amplitudes of the prograde and retrograde oscillations in the equatorial
components of the EAAM excitation functions are of the same order. For the very short
periods in the EAAM excitation functions i.e. shorter than 60 days, the prograde oscillations
are dominant, however. The increase of frequency resolution in the wavelet spectra reveals
the retrograde semi-annual oscillation in the EAAM excitation function, which excites the
prograde semi-annual oscillation in polar motion (Kosek 1995).
240
MWT
=1.0
160
HWT
24
0
21
18
80
15
0
12
= =0.005
20
16
0
18
15
12
0
12
8
0
4
0
9
period (days)
-80
3
6
-1 2
0
3
-1 6
0
-2 0
-2 4
0
1975
1985
1995
1965
=2.0
240
24 0
. 0
30
27
24
21
18
15
12
9
6
3
160
80
0
-80
-160
-240
1965
- 4
0
- 8
0
6
-160
-240
1965
9
0
1975
1985
1995
= = 0.002
24
20 0
. 0
16 0
. 0
21
12 0
. 0
18
80
. 0
15
40
. 0
. 0
0
12
-4 0
. 0
9
-8 0
. 0
-1 2 0
. 0
6
-1 6 0
. 0
3
-2 0 0
. 0
-2 4 0
. 0
1975 1985
YEARS
1995
1965
1975 1985
YEARS
1995
Fig. 1. The MWT (left) and HWT (right) time-frequency spectra of the complex-valued 1  i 2
atmospheric excitation functions for different values of σ (MWT) and λ=σ (HWT), respectively.
6. THE MWT AND HWT COHERENCES, CROSS-COVARIANCE AND TIME LAG
FUNCTIONS BETWEEN THE ATMOSPHERIC AND GEODETIC EXCITATION
FUNCTIONS
The MWT and HWT frequency dependent coherence functions are shown in Figure 2. It can
be noticed that improved frequency resolution can be obtained by increasing the σ parameter
value in the MWT and by decreasing the λ and/or σ parameter values in the HWT. This holds
for all other applications of MWT and HWT shown later on. The peaks of coherence
functions computed by these two wavelet techniques correspond to similar prograde and
retrograde oscillations, however, greater number of peaks can be noticed for the HWT
coherence than for the MWT one in the shorter period band from –70 to 70 days.
MWT
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
2.0
1.0
-200
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-150
-100
-50
0
50
period (days)
100
150
200
HWT
-200 -150 -100
=
0.010
0.015
-50
0
50
period (days)
100
150
200
Fig. 2. The MWT and HWT coherence functions between the complex-valued atmospheric 1  i 2
and geodetic  1  i 2 excitation functions in 1984.0-2000.0 for different values of σ (MWT) and
λ=σ (HWT), respectively.
The MWT and HWT time-frequency coherence functions (eq.10) between the atmospheric
and geodetic excitation functions are shown in Figure 3. The time-frequency coherence
functions computed by the two wavelet methods are very similar except for shorter period
oscillations with periods from –70 to 70 days, where the frequency resolution of the HWT
coherence is greater than for the MWT one. The values of time-frequency coherence
functions between the geodetic and atmospheric excitation functions become greater after
1980 in the shorter period band from –70 to 70 days due to increased accuracy of polar
motion determination. When increasing the frequency resolution of time-frequency coherence
functions both wavelet methods reveal a common retrograde semi-annual oscillation in the
atmospheric and geodetic excitation functions. It means that the prograde semi-annual
oscillation in polar motion is mainly excited by the equatorial components of the EAAM
excitation functions.
period (days)
MWT
220
180
140
100
60
20
-20
-60
-100
-140
-180
-220
220
180
140
100
60
20
-20
-60
-100
-140
-180
-220
= 1.0
19
70
19
80
1 90
1970
1980
1990
220
180
140
100
60
20
-20
-60
-100
-140
-180
-220
HWT
19
70
1970
1970
19
80
1 90
1980
1990
YEARS
19
80
1980
1 90
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1990
= = 0.002
= 2.0
19
70
= = 0.005
220
180
140
100
60
20
-20
-60
-100
-140
-180
-220
19
70
1970
19
80
1 90
1980
1990
YEARS
Fig. 3. The MWT and HWT time-frequency coherence functions (eq. 10) between the complex-valued
atmospheric 1  i 2 and geodetic  1  i 2 excitation functions for different values of σ (MWT) ,
=σ (HWT), and M=500.
The MWT and HWT time delay or phase lag functions between the complex-valued
atmospheric 1  i 2 and geodetic  1  i 2 excitation functions for different σ and λ=σ
parameter values, respectively, are shown in Figure 4. The time delay depends on the choice
of σ and λ=σ in the MWT and HWT, respectively. There is a good agreement between these
time delay functions in the frequency band from 50 to 150 days. Usually, the frequency
resolution of time delay functions grows faster for the HWT than for the MWT when the
absolute value of the oscillation period decreases. Negative time delay for oscillations with
periods of ±180 days in the MWT and +120, -80 days in MWT/HWT, respectively, means
that these oscillations in the equatorial components of the EAAM excitation functions seem to
happen before similar oscillations in the geodetic excitation functions (since the EAAM
excitation functions were x(t ) and geodetic excitation functions y (t ) time series). It is
difficult to determine the errors of the time delays so it is also questionable whether a positive
HWT time delay for the oscillation with a 60-day period really means that this oscillation
occurs first in polar motion and later in the atmospheric excitation functions.
lag (days)
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
-120
MWT
lag (days)
-200
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
-120
1.2
-150 -100
HWT
1.0
-50
0
50
100
150
200
period (days)
=
0.010
0.015
-200 -150 -100
-50
0
50
100
150
200
period (days)
Fig. 4. The MWT and HWT time lag functions between the complex-valued atmospheric 1  i 2
and geodetic  1  i 2 excitation functions in 1984.0 – 2000.0 for different values of σ (MWT) and
λ=σ (HWT), respectively.
The absolute values of the MWT and HWT cross-covariance functions between the complexvalued atmospheric 1  i 2 and geodetic  1  i 2 excitation functions are shown in Figure
5. Also in this case the frequency resolution of the MWT and HWT cross-covariance
functions depends on the choice of σ and λ=σ parameter values, respectively. This frequency
resolution grows faster again for the HWT than for the MWT when the period becomes
smaller.
7. CONCLUSIONS
The MWT and HWT techniques enable computation of time-frequency coherences and crosscovariance functions between two complex-valued time series. Time-frequency resolution of
the spectra and coherence as well as the frequency resolution of the cross-covariance and time
delay functions can be varied by changing the σ and λ and/or σ parameter values in the MWT
and HWT, respectively. Usually, the frequency resolution of the spectra, coherence, crosscovariance and time delay functions grows faster with decrease of the period for the HWT
than for the MWT.
Fig. 5. The absolute value of MWT and HWT cross-covariance between the complex-valued
atmospheric 1  i 2 and geodetic  1  i 2 excitation functions in 1984.0-2000.0.
The computed time delay depends on the choice of the σ and λ and/or σ parameter values in
the MWT and HWT, respectively. A negative time delay for oscillations with periods of 180
days in the MWT and 120 days in the MWT/HWT seems to indicate that these oscillations in
the equatorial components of the atmospheric excitation functions precede analogous
oscillations in the geodetic excitation function by about 20 to 60 days.
Results indicate that geophysical phenomena other than atmospheric excitation contribute to
short period polar motion oscillations.
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