JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. Bll, PAGES 27,069-27,089, NOVEMBER 10, 1998
Wavelet analysis of the Chandler wobble
Dominique Gibert
G•osciencesRennes- CNRS/INSU, Universit• de Rennes1, France
Matthias
Holschneider
Centre de Physique Th•orique, CNRS Luminy, Marseille, France
Jean-Louis
Le Mou;51
Institut de Physique du Globe de Paris, France
Abstract.
Wavelet analysisis applied to analyze polar motion spanningthe years
1890-1997.First, the wavelettransformis usedto identifythe components
(prograde
and retrograde)presentin the data. This wavelettransformis subsequently
used
to filter and reconstructeach component. Then we define the ridge of the wavelet
transform and showhow it can be usedto detect rapid phasejumps in a signal. This
techniqueis applied to the reconstructedprogradeChandler wobblecomponent,and
severalfeaturescharacteristicof phasejumps are identified. Syntheticsignalswith
adjustablephasejumps (in terms of their dates,durations,and amplitudes)are
constructedto produceridge functionssimilar to the one obtainedfor the Chandler
wobble. We find that less than 10 phasejumps are necessaryto reproduce the
observedfeatures. All but one phasejump have durations between I and 2 years,
and their dates are found to remarkably follow those of geomagneticjerks with a
delay not exceeding3 years. Elementarystatisticaltests assigna high probability to
the correlation between the dates of the phasejumps and those of the jerks. Simple
physicalmodelsof core-mantlecouplingshowthat the observedphasejumps can be
recovered
with torquesof about1020Nm.
1.
Introduction
the Chandlerwobblechangedby morethan 1500[Danjon and Guinot,1954;Guinot,1972;Pejovi5,1990].No
The motion of the Earth's pole with respect to the
clear interpretation has beengivenof this phasechange
mantle has been observed for more than 100 years. In
up to now. This is not surprisingsincethe mechanism
1905, Chandler showedthat this motion is basicallythe
maintainingthe wobble(freeEuleriannutation)is not
sum of two periodic oscillations,the forced annual osknown itself, althoughdifferenthypotheseshave been
cillation and the Chandler wobble with a period close
proposed(e.g., the atmosphere,earthquakes,or core
to 435 mean solar days [e.g., Guinot, 1972; Lambeck,
motions[Wahr, 1988;Dickman,1993]).As the excita1980; Wahr, 1988; Cazenaveand Feigl, 1994]. The
tion mechanismis not known, it is also hard to estimate
mean pole itself, taken as an averageover a time span
the correspondingattenuationfactor Q of the wobble,
of a few years, presents a more or less linear drift on
a parameterlong-sought
after [e.g.,Dickmanand Nam,
which are superimposedoscillations with a 3-decade
1998].
In this paper, we return to the analysisof the Earth's
polar
motion, despitethe numberof papersalreadydelatter wobble is poorly determined, and only for the
voted to it, for two reasons.First, becausethe wavelet
last 10 years can it be ascertainedwith a much smaller
amplitudethan before(M. Feissel,personalcommuni- analysisthat we haveusedto analyzegeomagnetictime
series[Alexandrescu
et al., 1995, 1996]appearsto us
cation, 1998).
well-adapted
to
decipher
the parametersof the polar
The amplitude of the Chandler wobble is known to
motion time series,which are comprisedof the sum of
vary with time [Lambeck,
1980].Sodoesits phase,and
timescalecalled the Markowitz wobble [e.g., Vicente
and Curie, 1976;Dickman, 1981; Vondrdk,1985].This
from 1926 in a time span of some 10 years, the phaseof
Copyright 1998 by the American G•uphysical Union.
Paper number 98JB02527.
0148-0227/98/ 98JB-02527509.00
two periodicvariations(of about 12 and 14 months)
with variable amplitudes and phasesand an irregular
trend. Second,1926 appearsto be the time of a major
geomagneticjerk. These resultsled us to examinecorrelations betweenpolar motion and geomagneticevents
by applyingto the Chandlerseriesthe sametechniques
27,069
27,070
GIBERT ET AL' WAVELET ANALYSIS OF THE CHANDLER WOBBLE
we had appliedto the geomagnetic
series[Alexandrescuanalysisof oscillatingsignals,and we shall usecomplex
et al., 1995, 1996].Sucha correlationwouldof course analyzing waveletsof the followingform:
supportthe hypothesisthat the coreplaysa part in the
excitation of the Chandler wobble, despite the fact that
up to now no conclusivemechanismto explain it has
been achieved.
•a(5)-- ;A½ ;
whereA½(t) is a real-valuedpositiveenvelopewith a
quasi-compact
support,andq•½(t) is a phaseterm. The
Morlet'swavelets[Goupillaudet al., 1984]belongto the
2. Method of Analysis
This sectionaims at giving the detailsof the method
used to analyze the polar motion series. Three main
above wavelet family, and we shall define them as
+ (t)-- a1exp 8a:•(•2
exp+irra
stepscan be identifiedin the processing
sequence:(1)
•a,M
computationof continuouswavelettransform;(2) filtering by meansof the wavelettransform;and (3) de- The Fourier transform of thesewaveletsis given by
tecting rapid changesin the phaseof the signal. The
wavelet transform is a mathematical tool particularly
well-suitedto study nonstationarysignalsby meansof
timescale or time-frequency analysis. Applications of
the wavelet transform in geophysicsand astrophysics
can be found, for instance, in the papers by Gambis
[1992],Baudin et al. [1994,1996],Alexandrescu
et al.
[1995, 1996], Chao and Naito [1995],and Moteau et
al. [1997]and in the book by Foufoula-Georgiou
and
Kumar [1994]. In this section,we presentthe theoretical backgroundrelevant to the presentpaper and,
•a,M(u) -
•M (t)exp(-2ir•ut)dt
= x/r•rrexp -8•r2(r2 au,
, (4)
This last expressionshowsthat a waveletis typically a
band-passgaussian-likefilter with a central frequency
u• - 1/(2a). Observealsothat for a • 1, the following
inequalities are satisfied'
< 1o-8o
in particular, the use of the wavelet transform as a
time-frequency analysis tool suitable to detect rapid
phase variations. The reader interested in the wavelet
transform from a general point of view is referred to
o) < 2 lo
o.
t'he books by Meyer [1990], Daubechies[1992], and We may then considerthat the •+a,M waveletshave
a band passlimited to positivefrequencies
( and will
Holschneider[1995].
be calledprograde
wavelets),
whilethe •,M havea
2.1.
Continuous
Wavelet
bandpassrestrictedto negativefrequencies
(andwill be
calledretrogradewavelets).The parameter(r > 0 con-
Transform
trols both the duration of the gaussianenvelope,that
is, the time resolutionr, and the dual-frequencyresolution
v which are linked by the uncertainty principle
by GrossmannandMotlet [1984].Forthe presentstudy,
r
x
v
•_
(2•r)-• FortheMorlet's
wavelets
usedin the
we shall write the wavelet transform of a signal s with
present
study,
we
have
respectto the analyzing wavelet 0 in the form of a convolution product,
1
1
2.1.1.
Definition
The continuous
wavelet
of
the
transform
wavelet
transform.
was first introduced
ß
¾V
[0,s](t,a)- Is,•a](t),
(1)
rxv
2•rwithr-2a'v4rra,
(6)
which states that the time-frequencyresolutionachiev-
where•a (t) - a-• * (-t/a) anda > 0 is the dilation able with the Morlet's wavelet is optimal. Figure 1
parameter(* is for the complexconjugate).The an- showstwoexamplesof progradewaveletsdefinedby (3)
alyzingwavelet0 (t) is a localizedoscillatingfunction correspondingto (r - 1.0, 6.0 and a = 1, togetherwith
with a compactor a quasi-compact
support[Holschnei- their Fourier transforms. In this study, the time unit
der, 1995].The oscillatingcharacteristicis neededto will be i tropical year, and the correspondingfrequency
ensurethat the analyzingwavelethasa vanishinginte- will be expressed
in cycleper tropicalyear (cpy.).
2.1.2. Synthetic examples. Let us now apply
gral in orderto makethe transformation
(1) an isometry, that is, to preserveenergy[Holschneider,
1995]. the wavelet transform to a syntheticsignal constructed
A great property of the wavelet transform is its abil-
ity to capture variationsin a signalat both any time
and any timescale. Although the wavelet transform
possesses
this quality whateverthe analyzingwavelet,
specificapplicationsmay requirethe use of particular
wavelets[see,e.g., Alexandrescu
et al., 1995;Moteauet
al., 1997].In the presentstudy,we are interestedin the
by addingtwo progrademonochromatic
complexsignals
with a variableenvelope.The periodsof the components
are 365 and 435 mean solar days, and the superimposition of the two componentsresults in large beating
phenomena. The samplinginterval, the duration, and
the envelopesare chosenaccordingto the characteristics
of the annual
and the Chandler
wobbles of the actual
GIBERTET AL.: WAVELETANALYSISOF THE CHANDLERWOBBLE
1.0
27,071
6.0
(a)
(b)
0.5
4.0
0.0
2.0
-0.5
-1.0
0.0
_
.0.0
1.0
;0.0
[
(c)
(d)
0.5
:0.0
0.0
0.0
-0.5
-1.0
I , i , , • , i
0.0
-50
-25
0
25
50
0.0
0.3
'
0.5
•
0.8
frequency(c.p.y.)
time (years)
Figure 1. (a) ProgradeMorlet'swavelet(bothimaginaryandrealparts)with c -- 1, a -- 1, and
(b) its Fouriertransform.(c) Progradewaveletwith c - 6, a - 1, and (d) its Fouriertransform.
The time unit is i tropical year.
wavelet
transform
0.8
0.•
0.2
0
-0.2
-0.4
1900
I
I
1920
1940
1960
1980
Time
Figure 2.
Modulusof (top) the progradewavelettransformof (bottom) a signalconstructed
by superimposingtwo progradeoscillatingsignalswith respectiveperiods365 and 435 mean solar
days. The amplitudes of the two componentsare those computed for the annual and Chandler
componentsof the true polar motion (seeFigure 12 below). The Morlet's waveletshavebeen
computedwith • - i and havetoo largea frequencybandwidth(seeFigure 1) to resolvethe
two componentspresent in the total signal. As a result, the modulus of the wavelet transform
oscillateswith a beatingperiodof 6 years.Frequencyis givenin cyclesper year (cpy).
27,072
GIBERT ET AL.: WAVELET
wavelet
ANALYSIS OF THE CHANDLER
WOBBLE
transform
.
0.8
0.2
-0.2
-0.4
1900
'
'
1920
'
1940
'
1960
1980
Time
Figure 3. Sameas Figure2, but with Morlet's waveletscomputedwith rr - 6. The frequency
resolutionissufficient(seeFigure1) to resolvethe twocomponents
presentin the analyzedsignal,
and the modulus of the wavelet transform showstwo horizontal energy bands centeredon the
frequencies
1. and0.840cpy (i.e.,periodsof 365and435meansolardays).
polar motion analyzed in the next sectionof this paper.
The modulus of the prograde wavelet transform of the
signal computed with rr - 1 is displayedin Figure 2 and
clearly showsa beating pattern with a 6-year period resulting from the interference between the annual and
Chandler components. In this figure, the separation
between the two components is impossible to achieve
becausethe energy patchescorrespondingto each component strongly overlap. This difficulty can be removed
by increasing the frequency resolution, that is, by in-
j•0
+•[r(-,
a)ßXa
(')]
(t)da (8)
a
HereagainXa(t) = a-ix (t/a) stands
forthedilationof
X- Despiteits appearance,this integral doesnot blow
up at a = 0 because of the oscillations of the wavelets
(seediscussion
below). The operatorjM mapsa function r over the time-dilation half-spaceonto a function
of timeby superimposing
the wavelets
Xa('- b).Therefore jM is calledthe waveletsynthesis.The following
creasingthe resolutionparameterrr in equation(3), as relation holdsfor arbitrary •b and X:
in Figure 3 whererr = 6. In this figure,onemay observe
that the energypatchesof eachspectralcomponentare
now separated and can be easily identified.
2.2. Filtering in the Time-Frequency
Plane
2.2.1.
Half
Ix,w
= :r
[4},
(o)
and with u • 0,
'
The
reconstruction
formula.
Now that
•,,x(u)- j•0
•b*
-{-c•
^(au)f (au)d___a.(10)
a
the wavelet transform has been proved to be useful
for identifying the nonstationary spectral components Here .T stands for the Fourier transform operator, * is
presentin a signal, we addressthe problemof filtering for the complexconjugate,and •,x actsas a multiin the time-frequency half plane. The wavelet trans- plicationoperator.Note that •,x takesonlytwo val-
(the progradepart,
form (1) can be invertedundercertainconditionswe ues,onefor the positivefrequencies
u
>
0)
and
a
possibly
different
one
for the negative
shall now examine. Let us considerthe followingoperator:
frequencies
(the retrogradepart, u < 0):
;V[[X,r] (t) --
r (b,a) Xa(t -- b)
(7)
•*,x(u)--f0
•p*
(asignu)
d___a
(11)
f/frO
4-•
dadb
{-øø
^(asignu)•
a
a '
GIBERT ET AL.' WAVELET ANALYSIS OF THE CHANDLER WOBBLE
Note that becauseof the oscillationsof •b, we have
with
It'(t)-/•[X,•]
(t•)da. (19)
•(0) - 0, andtheintegral
is convergent.
In particu-
J timin
lar, wemaychoose
>4in sucha waythat
•½,x(u)- 1.
(12)
A sufficient condition for such a reconstruction X to
exist is
27,073
a
Note that this may be usedfor interactivedesignof
filters.
2.2.2.
Synthetic examples.
We now use equa-
tion (15) to isolatethe signalcomponents
identifiedin
the wavelet transform shownin Figure 3. This is done
by zeroing the wavelet transform outside the regions
0<C'½-•(4-a) --<oo,
(13) containingthe energyof the componentto be retained.
a
In practice,the filtering has beendonein an interactive
in whichcase,•b is calledadmissible.Note that a pro- way by surroundingthe retained energy patcheswith
grade or retrogradewaveletcan neverbe admissiblein polygonallines drawn with the computermouse. All
this sensesincefor either sign, the integralis 0. We reasonablechoicesof contouringprovideessentiallythe
sameresults. Figure 4 showsthe reconstructedcompocome back to this in a moment.
nents
together with their discrepancieswith respectto
For an admissible•p, a possiblechoicefor X wouldbe
the original components.One can observethat the reconstructionsare almost perfect. If noiseis addedup to
x(t)- c½(t).
(14)the levelpresentin the real data (seethe last synthetic
In thiscase,wehavethe reconstruction
formula[GrossmannandMotlet, 1984;e.g.,Holschneider,
1995]:
exemplebelow),this separation
remainsessentially
unaffected.
2.3. Extraction of the Ridge Function
2.3.1. Ridge function of oscillating signals.
s(t)- •1•o [W
[•b,
s](.,a),Xa
(')]
(t)d___a.
(15)
We now addressthe last step of the processingsequence
a
Thisformulaholdsfor all complex-values
signals(mix- and introduce the ridge of the wavelettransformwhose
ture of progradeandretrograde).Suppose
wearegiven aim is to get information about the variationsof the
a progradewavelet•b.Clearly,formula(13) cannothold phase of the componentspresentin the analyzed sig-
for the minussignsincethe integralis 0. However,sup- nal. There are two concepts of frequency associated
with wavelet transforms.
pose that
The first is the instantaneous
phasevelocity of the waveletcoefficients,
--a < oo'
0<C•p
--f0+øø
I• (a)12da
(16)
d•t,•
_ dtdargW[,p
s](ta)
dt
(20)
then (15) still holdsif s itselfis prograde.For an arbi- The secondconceptassociatesthe scalea with the fretrary mixtureof progradeand retrogradecomponents,quencyuo/a, whereu0 is the centralfrequencyof the
equation(15) extractsthe progradepart of s. The analundilatedwavelet. The ridge is definedto be the set of
ogousstatementis true for retrograde•p.
pointswhereboth conceptscoincide;that is, the points
In the presentstudy,we will usethe followingnonas(t) for whichwe have
stationary
filters.Consider
a functionE (t, a) overthe
d•,•
•r
time-dilationhMf-space.
Typically,E will bethecharac=
(21)
teristicfunctionof somesetof interest;that is, it takes
dt
as(t)'
the value 1 for all points in the set and 0 otherwise.
To illustrate this, let us considerthe particular caseof
Considernow the followinglinearoperator:
a monochromaticsignal s,
,
Ix, E. w [•, d].
(17)
This operator takes s into wavelet space,appliesthe
multiplication operator E., and then transformsback
again. For E, the characteristicfunction of someset in
the half-space,this is a nonstationaryfilter. It allowsthe
extraction of time-varying frequency content. We will
usethis techniqueto separatethe Chandlercomponents
from the total polar motion series.In the specialcase
whereE is the characteristicfunctionof a strip amin_<
a _<amax,the filter becomesstationary,and we have,
s (t) = Asexp[2i•rust],
(22)
with constantamplitude As. The Morlet wavelettransform of this signal reads
W [•b•M,
s](t,a)--
x/•erAs
exp
[i•ta(t)]
exp-8•r2er
2 aus
=F
a
'
where(•t,a(t) = 2•rustis the phaseof thewavelettransform. In this case, the ridge is the line
1
k, E. w [•, d] -
,] (t),
(18)
- --.
as(t) 2us
(24)
27,074
GIBERT ET AL.' WAVELET ANALYSIS OF THE CHANDLER
!
i
i
i
I
I
I
i
i
i
i
i
WOBBLE
0.1
-0.1
-0.3
0.1
'
(b)
Error
-0.1
i
0.1
-0.1
-0.3
I
-
(d)
I
BzTo•
0.1
-0.1
1900
1925
1950
1975
time (years)
Figure 4.
(a and c) Components
of the total signalshownin Figure$ (bottom)recovered
by
filtering in the time-frequencyhalf plane of the wavelettransform whosemodulusis displayedin
•ig.•
• (•op). (b •.d d) •h• •½o.•.aio.
•.o•
•
•l•o •ho,v..
tracted with the help of a Morlet wavelet with •r - 6.
The first exampleis shownin Figure 5 and corresponds
to the an,alysis of the Chandler componentof the synthetic signalshownin Figures2 and 3. Here the wavelet
transform has been computed with •r- i in order to
neousfrequencymodulations
[Sataccoet al., 1990;Del- have the best time resolution. The ridge is a horizonprat et al., 1992]. However,in the presentstudy, we tal line correspondingto an instantaneousfrequency
are interestedin signalswith rapid phasevariationsfor u• (t) = 0.840cpy or, equivalently,
to an instantaneous
which the stationary phase approximation used to ob- periodT• (t) = 435 meansolardayswhichis precisely
tain equation(21) may be too crudeto allowa direct the period used to create the synthetic Chandler comderivationof the instantaneous
frequencyfrom a8(t). ponent analyzed in this example. Let us remark that
Despite this difficulty, and as will be shown in the fol- the ridge function remains straight despitethe amplilowingnumericalexamples,the ridgefunction remainsa tude variations of the input signal.
useful tool to detect rapid variations in the phase of the
The next two examples address the specific case of
signal. In what follows, the ridge of the wavelet trans- sharpvariationsof the phase• (t) whichwe modeled
form can be used as a very sensitive detector of phase by
fluctuations, and we shall considerthe ridge of the polar motion seriesto be the data to be fitted by our synthetic models. The algorithm to obtain the ridge of the
n:l
• Y/-•Tn
wavelettransform of a signalcontainsthen the following
steps: (1) computationof the complexwavelettrans- where T• is the characteristic duration of the phase
form; (2) time derivationof the phaseof the wavelet jump •n occurring at the mean time tn, and To is a
Hence, for a monochromaticsignal, the position of the
ridge allows a precisedetermination of the frequency.
In the case of a signal with slowly varying instantaneousphase velocity, it can be shown using arguments
of stationary phase that the ridge follows the instanta-
q•s
(t)- -•o+
erf
]
+1 ,
(25)
transformto apply equation(21); and (3) identifica- baseperiod(e.g.,435 meansolardays). Figure6 distion of the pointssatisfyingequation(21) in the time- playsthe ridge function of the samesyntheticChandler
signalas the one analyzedin the previousexample,but
dilationhalf planeandformingthe ridgea8(t).
2.3.2. Synthetic examples: Phase jumps. We with N = 6 phasejumps with parameterscorrespondnow discussseveral synthetic examples in order to as- ing to test 1 in Table 1. The ridgefunctionis now consessthe reliability of the method for detecting phase siderablymodifiedby the presence
of thesephasejumps
jumps in oscillating signalslike the Chandler wobble. as can be seen when compared with the constant freRecall that this component has previously been ex- quencycase(Figure5). Overall,the ridgefunctionnow
GIBERTET AL.' WAVELETANALYSIS
OF THE CHANDLER
WOBBLE
wavelet
27,075
transform
1.2
0.2
0
-0.2
o
I
1920
1900
I
I
1940
1960
1980
time
Figure 5.
(top) Wavelettransformmodulusof (bottom) a syntheticChandlercomponent.
The prograde Morlet's waveletshave been computed with cr- 1, and the ridge function of the
wavelettransform(seetext for details)is a horizontalline at frequency
0.840cpy (i.e., periodof
435 meansolardays). Observethat the instantaneous
frequencyremainsconstantdespitethe
amplitude modulation of the analyzedsignal.
wavelet
transform
0.8
'* • .................
•:•••i'•"•:•::•:!
'•:f'"'"'"•:•iiF
"'"'••••'.--'"'""•'••"
-...•:'*:•.
'*"?'""""•.'-'.-'
...................
......
1.2
"•........
,•<•••*•:•:::.':i'.
:.....,
',<•.•,•,e'
,,,"q-•"**'
'"'"'"
""'"""''"':':••'••"•::'"'"•-'.'.'.'.'.".•:•.,:-•.'::.::•-:.:.'"'
.........
':'
............
.*.*,2;f'.'•-.'-•;;.-':.-'-.:':>.--:'..*:'.:::,:':-:.•..--';•-..,;,:.•:,•:-:-•
0.2
•
0
•
-0.2
;•
1900
1920
1940
1960
1980
time
Figure 6. Sameas Figure5, but for (bottom) a signalhavingsix phasejumps with varying
amplitude(seetest 1 in Table1). The ridgefunctiondepartsfroma horizontallineandpossesses
troughsand peakscenteredat the dates of the jumps and with amplitude(but not width)
controlled by the size •b• of the jumps.
27,076
GIBERT ET AL.: WAVELET ANALYSIS OF THE CHANDLER WOBBLE
illustrated in Figure 7 where the phasejumps now have
Table 1. Phase Jumps for the Test Models
Test
I
the sameamplitudebut differentdurations(test 2 in
Table 1). One canobservethat the wigglesof the ridge
Test 2
function are gaussian-likecurveswith increasingwidth
and decreasingamplitude when the duration T• of the
1910
1.5
-180
0.5
90
jump increases.
1925
1.5
-90
1.0
90
A last example is now consideredto examine the in1940
1.5
-30
1.5
90
fluenceof the noise.This is done by adding a gaussian
1955
1.5
30
2.0
90
white noiseto the signalusedin the exampleof Figure
1970
1.5
90
2.5
90
5. The noise level has been adjusted in order to be com1985
1.5
180
3.0
90
parable to the one observedin the energy spectrumof
the real polar motion data to be processedlater. BeHere t,•, dates(years)of the phasejumps;Tn, duration
causeof the presenceof the noisewhich locally perturbs
(years)of the phasejumps;4n, amplitude(in degrees)of
the phaseof the wavelettransform, the ridge functionof
the phase jumps.
the noisysyntheticsignalnowwiggles(Figure8) around
the theoreticalfrequency(0.840 cpy). The peaksand
appears as a horizontal line with gaussian-likepeaks
and troughs centeredon the dates tn and whoseamplitude is controlledby •n. Negative phasejumps can be
interpreted as a local deceleration of the pulsation of
the signal, and the ridge is distorted toward the lower
instantaneousfrequencies. Conversely,positive phase
jumps produce a local accelerationof the pulsationproducing a distortion of the ridge toward the higher instantaneousfrequencies.
Of course,for a givenphasejump •n, the magnitude
of the deceleration or acceleration of the pulsation will
troughsso created are randomly distributed alongthe
time axis with an amplitudecontrolledby the strength
of the signal to noise ratio. Indeed, the largestwiggles appears around 1930 where the wobble is small.
However, despitethe quite large amount of noiseadded
to the synthetic signal, the amplitude of the wiggles
remains small with respect to the peaks and troughs
produced by the phasejumps of the previousexamples
(seeFigures6 and 7) and the onespresentin the true
date (seenext section).
The four examplesdiscussedin this sectionshowthat
the ridgefunction
of thewavelet
transform
is particularly sensitiveto phasejumps and can be usedto detect
depend
onthedurationT• of theevent.Consequently,
the distortion of the ridge will also depend on T•, as
wavelet
rapid phasevariations in a signal.
transform
::•i•:i'•:ii•.•::-;.:',..:::•:•:t..'..---i::
?.?'-'•:i•i."•:.-':½,.
!.::.::,,-.'..•:•i•..•.;•-'.i'
...:?•..:i•..`....`...::.•:ii:i:•i...•:ir*...:•i.:iii.;:.:.:::•:.::.:.;.t•...•.•.....:i•`.,•,.•:•.*.•:.:...•.•...•i•ir.•.•
.-:•.•:•i::;½:•,,.,.;t--•,:•
.....
'
''
'
•:.. ' •'4•":":'•'•:'•;::•'"'"::"'"'":***:'"'""
i• .....'":"*'•:'"'":'*":""
.............
:"t•
b
Ei =.½•:.:.'::=:•':+•:::•::
:". 4•Zf:•..````•i•:.:.•i...&q;::•i•..;:iiii$L...•i.L...•.L...•i•!4;..L...•.•..;...•i•-!(8•iq•;i:•:
..
. .:.•:$•]•;•&.:•..``.•;...`..i•;if;:%:...:•.:.•..`.c*%....Z`:$.:.i•!.•i=/•;•?
•....... .....
i-•½:.
::• k•-'.'.-•
..............
•-,:•', , ..................................
-'-'-:.-•-<
...............................
'...............
.,.'•-.•---'::
.........-'-'-.'
s..........
.".•½.-•.:,'
-½.'½......................
:..:.-,'.:.'+-?
........
>.........
:...>.:-'.:-:.-'.-'->•..".............
:..... -.•"-";•
0.2
0
-0.2
! 900
1920
0
1940
! 960
1980
t.ime
Figure ?. S,me ,s Figure5. but for (bottom), sign,1h,ving six phasejumpswith different
dur,½ions
(see½es½
2 in Table1). Both•hewidth,nd •he ,mplitudeof the troughsof the ridge
function are affectedby the duration of the ph**sejumps.
GIBERT ET AL.- WAVELET ANALYSIS OF THE CHANDLER WOBBLE
wavelet
27,077
transform
0.8
0
-0.2
1900
1920
1940
1960
1980
time
Figure 8.
Sameas Figure5, but for (bottom)a signalpollutedwith a Gaussianwhitenoise
with a standard
deviation
of 0.035 arc sec.
3. Wavelet Analysis of Polar Motion
ergy packetsof both the progradeannual and Chandler
Data
components stand out above the noise level at all time.
On the contrary, the retrogradeenergypacket of the
Chandler component is not so continuousand someThe polar motion data series used in the present times disappearsin the noise. The annualretrograde
Study is distributed by the International Earth Rota- component,however,is clearly visible'and physically
tion Service(http://hpiers.obspm.fr/fileEOP97C01) relevant[Dickman,1981].
3.1.
Data
and spans a time interval from 1890 to 1997 with a
samplinginterval of 0.05 years. The X and Y compo- 3.3. Reconstruction of the Components
nents of the polar motion are shownin Figure 9. Let
The filtering of the retrogradeand progradecompous recall that the X axis coincides with the Greenwich
nents was done as for the synthetic signal previously
meridian, while the Y axis is on the 90ø W meridian. discussed.The processingsequencewas alsocarriedout
This coordinate system is left-handed. We now form with differentvaluesof the adjustableparameters(e.g.,
the complexvalued signalX- iY. Then the prograde er - 8 or 10), and no significantchangeswerefound.
componentsof the polar motion will be analyzed with The reconstructedsignals are displayedin Figures 12
progressivewavelets,that is, a wavelethavingonly pos- and 13. The common vertical scale allows for an easy
itive frequencies.The retrogrademotion corresponds
to comparisonof the amplitudes of each signal, and one
negative frequencies.
can observethat the retrograde componentsare much
smaller than the prograde components.The maximum
3.2. Wavelet Analysis of the Total Complex
amplitudeof the annualsignalsare 0.037 (retrograde)
Signal
and 0.106 (prograde)arc seconds;
andfor the Chandler
The wavelet transforms of the prograde and retro- signals,the maximumamplitudesare 0.016(retrograde)
The progradeChangradecomponents
of the data (computedwith (r - 6) and 0.280 (prograde)arc seconds.
are shownin Figures 10 and 11, respectively.As for the dler componenthas then the largest amplitude among
synthetic signal analyzed in a preceding section, one the four reconstructed signals and also appears to be
can observe that the frequency resolution is sufficient the most irregular one, with a minimum amplitude of
to separatethe energy patchesof the annual and Chan- 0.035 arc seconds,8 times smaller than the maximum.
dler components.The amplitude of the progradecom- The annual prograde signal is much more regular with
ponentsappearsmuchlarger (by a factor of 10) than a minimumamplitudeof about two thirds (i.e., 0.070
the amplitude of the retrograde components.The en- arc seconds)of the maximum[Dickman,1981]. This
27,078
GIBERT ET AL- WAVELET ANALYSIS OF THE CHANDLER WOBBLE
X Componel•t
0.6
0.4
0.2
0.0
-0.2
-0.4
I
.......
1890
1915
1940
1965
1990
'
'
•
1915
1940
1965
1990
0.6
0.4
0.2
0.0
-0.2
-0.4
1890
,
years
Figure 9. IERSpolarmotionseries
usedin thepresent
study.Thesampling
intervalis 0.05
yr. The X component is measuredalong the Greenwichmeridian and counted positive toward
Greenwich,and the Y componentis measuredalong the 90ø W meridian and positive westward.
wavelet
transform
1.2
0.5
0
1900
1920
1940
1960
1980
Time
Figure 10. Progradewavelettransformof the polarmotiondata shownin Figure9. The
waveletshavebeencomputedwith er- 6 asfor the syntheticexampleshownin Figure3. The
energybandscorresponding
to the annualandChandler
components
areclearlyvisible(compare
with Figure3).
GIBERT ET AL.- WAVELET ANALYSIS OF THE CHANDLER WOBBLE
wavelet
27,079
transform
0.8
1.2
0.5
o
1900
1920
1960
1940
1980
Time
Figure 11. Retrogradewavelet transform of the polar motion data shown in Figure 9. The
waveletshave been computed with cr- 6 as for the synthetic example shownin Figure 3. Note
that the rangeof (right) the gray scaleof this figureis about 10 timessmallerthan the scaleof
Figure 10.
-0.2
Chandler
prograde
•t1•
0.2
-0.2
,
,
,
,
I
,
,
,
•
I
,
,
•
,
I
•
•
•
,
I
,
Annualretrograde
0.2
-
.
0.0
-0.2
,
0.2
•
•
I
I
Chandlerretrograde
0.0
-0.2
1890
I
I
I
I
1915
1940
1965
1990
,
years
Figure 12. Reconstructedretrogradeand progradesignalsobtainedby filtering the wavelet
transformsshownin Figures11 and 10.
27,080
GIBERT ET AL.: WAVELET ANALYSIS OF THE CHANDLER WOBBLE
(a)retrograde
annual
-0.3
(b)prograde
annual
ß 9o
0.0
0.3
0.6
-0.3
0.0
I
I
t
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
(½)
retrograde
Chandler
I (d)
prograde
Chandler
0.3
0.6
I I •I I (I I I I I I •I I I I I•I l
(e)
to•l
po•
moti•
-0.3
0.0
W90
(fi
residual
W90
.......
..%•.:•?•:
ß...•
....
0.3
0.6
-0.6
-0.3
0.0
arc seconds
0.3
-0.6
-0.3
0.0
0.3
arc seconds
Figure 13. (top and middle) Reconstructed
retrogradeand progradecomponents
(seealso
Figure 12) displayedin a right-handedcoordinatesystem.The regularityof the progradeannual
motion is conspicuousin contrast with the more irregular prograde Chandler wobble. Observe
the much smaller amplitudes of the retrogradecomponents.The original polar motion is shown
at the bottomleft togetherwith the residuals(bottomright) obtainedby removingthe retrograde
and prograde componentsfrom the total motion. The gray trajectory is for the whole data set,
and the circlesrepresentthe residual pole position every 10 years starting in 1890 at the top
right. Observethe fast westwardtrend in the residualsaround1960 (the eighthcirclefrom the
top right).
differencebetween the Chandler and the annual prograde componentsreflects in particular the fact that
the annual componentis a forcedoscillation,while the
Chandlerwobbleis a freeoscillationmaintainedagainst
dissipation.
4. Interpretation
of
Chandler
Prograde Ridge Function
moidal phasejump with a duration of 6 months to 3
years(Table1) produces
a trough
ora peakoftheridge
functioncenteredon the date of the phasejump. We
shallnowlookif it is possible
to reproduce
the experimental ridge function by meansof a small number of
phasejumpsdistributedthroughoutthe wholetime interval considered.We shall try to make the numberof
jumps as small as possible.
We now focus on the analysisof the prograde Chan- 4.1. Construction of the Signal Model
dler componentshownin Figure 12. The ridge function
As shownin a preceding
section,the ridgeof the
of this signal is shown in Figure 14 together with the wavelet transform is very sensitiveto nonlinear vari-
wavelettransform(rr- 1) from whichit wasextracted ationsin the phaseof the signal.However,
we already
usingequation(21). The inspectionof the synthetic statedthat the definitionof the ridgefunctionis only
ridge functions in Figures 6 and 7 revealsthat a sig- approximate
if too sharpvariationsoccurin the phase
GIBERT ET AL.: ¾VAVELET ANALYSIS OF THE CHANDLER WOBBLE
wavelet
27,081
transform
0.9
0.2
0
-0.2
1900
1920
1940
1960
o
1980
time
Figure 14. Prograde
wavelet
transform
of thereconstructed
Chandler
component
shown
in
Figures12 and13. The wavelets
havebeencomputed
with • - 1 in orderto reachthefinest
time resolution.The ridgefunctionis alsodisplayed.
sothat equation(24) is not exactlysatisfied.In sucha the jumps t,,, their duration T•, and their number N
case,the phaseof the signalcannot be safelydeduced are adjustable parameters.
from the ridge for further quantitative interpretation.
Accountingfor this situation,we have adopteda strat- 4.2. Fitting of the Chandler Prograde Ridge
egywhichwe believepossesses
a largedegreeof robust- Function
nesswith respect to possiblebiasesin the ridge funcThe first half of the experimental ridge function is
tion as well as to any artefact due to edge effects,time
easily
reproduced until 1940 with a base period To =
sampling,etc. This is achievedby analyzingsynthetic
signalsthrough exactly the sameprocedureas the ac- 434 mean solardays and five phasejumps (the acculater). In order
tual data, as detailed in the previoussection,namely: racy of thesevalueswill be discussed
to introduce the smallest number of such events, the
(1) wavelettransformwith rr = 6, (2) filteringin the
time-dilationhalf plane, (3) reconstruction
of the fil- secondhalf of the ridge needs a slow phase variation
teredsignal,(4) wavelettransformwith rr- 1, and (5) with a duration of about 15 years centeredin 1953 to-
ridge extraction. The synthetic signal is constructed
with the envelope
corresponding
to the real (noisy)data
and sampled exactly as the real data. Its phaseis assumed linear in time with superimposedphasejumps
with sigmoidalshapes. This model signal has the following expression:
getherwith severalrapid phasejumpssimilar(in amplitude and duration)to thoseusedfor the first half of the
ridge. The adjustedparametersof the model are given
in Table 2, and Figure 15 showsthe corresponding
ridge
function. We do not claim that this solution is unique,
and it is very difficult to scanthe whole solutionspace.
For instance,it is possibletc fit the ridge function with
m (t) = Aa (t)exp [i4a(t)] + Ac(t)exp [ic)c(t)], (26) models having a base period very different from 434
mean solar days. Table 2 givesthe parametersobtained
whereAa (t) and Ac(t) are the envelopes
of the profor base periods between432 and 436 mean solar days,
grade annual and Chandlercomponents
reconstructed
an interval which encompasses
the most recent experi-
from the data, • (t) = 2•rt, and
mentaldeterminationof the Chandlerperiod [Vicente
and Wilson,1997].The parametersof the phasejumps
obtained for thesedifferentperiodsfall closetogether.
- 2rrt
+••[erfft-tn
\ )+ (27)
However, the model obtained for To - 434 mean solar
wherethe time unit is the tropical year. The baseperiod days is remarkable sinceit involvesthe smallestnumber
To, the amplitudesof the phasejumps •n, the datesof of phasejumps for the best fit. For this reason,we shall
27,082
GIBERT ET AL'
WAVELET ANALYSIS OF THE CHANDLER WOBBLE
Table 2. Phase Jumps for the Ridge Function Models
To -- 432
To -- 433
To -- 434
To -- 435
To -- 436
1905.3
1912.5
1.0
3.0
23
-46
1905.3
1912.5
1.0
3.0
27
-36
1905.2
1912.0
1.0
1.6
29
-24
1905.2
1912.5
1.0
1.6
32
-20
1905.2
1912.5
1.0
1.6
37
-13
1921.0
1926.8
1.5
1.8
10
149
1924.0
1926.8
1.5
1.8
14
152
.........
1926.8
1.8
164
1920.0
1926.8
1.5
1.8
12
164
1922.0
1926.8
1.5
1.8
21
166
1933.2
1.5
1.8
48
1932.8
1.5
48
1933.0
1.5
1.8
59
1939.8
1952.2
1953.0
1961.3
1970.0
1980.2
1.5
2.6
15.0
1.0
1.0
1.5
70
47
-165
19
25
38
1940.0
1952.3
1953.0
1961.0
1969.9
1980.2
1.5
2.2
15.0
1.0
1.0
1.5
77
47
-140
17
28
44
1940.2
1952.4
1953.5
1961.0
1970.0
1980.3
1.5
2.3
15.0
1.0
1.0
1.6
56
68
45
-105
19
28
49
1933.5
1.5
2.6
15.0
1.0
1.0
1.5
53
62
46
-185
16
23
36
1933.5
1940.2
1952.4
1953.0
1961.3
1970.0
1980.4
1939.8
1952.4
1953.5
1961.4
1970.0
1980.3
1.5
2.7
15.0
1.0
1.0
1.8
72
50
-95
22
32
54
HereTo,baseperiod(meansolardays)of the Chandlerpolarmotion;t,,, dates(years)of thephasejumps;T,,, duration
(years)of the phasejumps;•b•, amplitude(in degrees)
of the phasejumps. The modelspresented
in this tablehavea
goodness
of fit equalto 1.343,1.077,1.017,1.070,and 1.065for baseperiodsfrom432 to 436 meansolardays,respectively.
retain this model as our preferredone in the rest of this of the best solutionin the neighborhood
of a starting
model.
paper.
4.3.1. Method. Insteadof considering
the ridge
function,we look directly at the phaseof the signal.
We now presenta companionmethod designedto as- We extractthe Chandlercomponent
fromthe total sigsessthe reliability of the resultsobtained abovefrom the nal with a stationaryfilter, f (t), constructed
by applyridgefunction.This methodis far lesstime consuming ing the reconstruction
formula(15) restrictedto a fithan the previousone and allowsfor an iterative search nite dilationinterval [amin,
amax],to the wavelettrans4.3.
Analysis of the Instantaneous
1900
Phase
1925
1950
1975
180
0
135
90
45
•
0
o[,
oi
phasejumps
--- phasejumps
-45
-90
o magneticjerks
-135
[] magneticjerk
-180
0.82
0.84
0.86
0.88
0.90
0.92
,
0.02
discrepancy]
0.00
-0.02
1900
1925
1950
1975
time (years)
Figure 15. (top) phasejumps (verticalbars)obtainedfor the wobblemodelwith To = 434
meansolardays.Solidbarsarefor jumpsobtainedwith the ridgefunction(Table2); dottedbars
are for jumpsobtainedwith the instantaneous
phasemethod(Table3). The circlesdisplayedat
the top correspond
to the geomagnetic
jerks foundby Alexar•drescu
et al. [1996],and the open
squareis for a jerk reportedby Golovkov
et al. [1989]andMcLeod[1989].(middle)Experimental
and syntheticridgefunctions.(bottom)Discrepancy
betweenthe ridgesshownabove.
GIBERT ET AL.' WAVELET ANALYSIS OF THE CHANDLER WOBBLE
27,083
0.80
0.85
0.90
IERS data
Filtered model
------
,
1900
I
1920
,
I
,
1940
Unfiltered
I
1960
,
model
I
1980
t
2000
time (years)
Figure 16. Instantaneous
frequencyof the Chandlerprogradecomponent(solidline) and of
the syntheticsignal(dashedline) filteredin the sameway as the data. The curvecorresponding
to the untilfetedsyntheticsignal(dottedline) clearlyshowsthe phasejumpsat isolatedpoints.
0
form of the Dirac • (t) (seeequation(19)). Oncethe
0-•arg[Fx,* sx(t)]-filteredsignals! (t) = Is, f] (t)is computed,
the phase
•b!(t) = arg[s! (t)] is readilyobtained.Next, the experimentalphase•b!(t) is fittedwith the phase•b,•,!(t)
of the signalmodelgivenby equation(26) andfiltered
with the filter f (t). This fitting is doneby minimizing wheref0 and f• are well-localizedfunctions.This shows
1['0(•--r)+
,1(•)+O(•)]•31)
the functional
dt
0
(t)-
arg([f, m] (t))
that the observed,filtered, phasejump has a duration
that is at least given by the envelopeof the analyz(28) ing filter. Thereforeapparentlyslowvariationsof the
instantaneousphasespeedmay be due to rapid phase
variationsin the untilfetedsignal(compareFigure16)
with a conjugate gradient method.
4.3.2. Results. The starting model used in the
To understandthe effectof the filtering on the phase, iterative fitting procedureis the one obtained from the
note that we have the followingexpansion.We consider
ridge function analysisfor a baseperiodTo = 434 mean
the following family of signals
solardays(seeTable 2). The bestresultsobtainedat
the end of the iteration sequenceare given in Table
3 and the corresponding
phaseis shownin Figure 16.
sx
(t)-exp
iwot+ic)(t•)]
(29)
It can be observed that the final model remains close
where 4 is a function growingfrom -a/2
to +a/2.
to the starting one. The dates of the phasejumps are
Thus, in some sense, ,k is the duration of the phase modifiedby lessthan 5 monthson average.The values
jump from -a/2
to +a/2 located at the central fre- of the phasejumps are globally 20% smallerthan the
valuesobtainedfrom the ridge functionanalysis.The
durationsof the jumps differ by lessthan 3 monthson
average,with three largerchanges(.-• 1 year) for the
1913, 1926, and 1980jumps.
quency coo.We now consider a family of filters of the
form
M is a complex function, not necessarilythe envelope. 5. Discussion
Here ,k' determinesthe sizeof the filter or its inversethe
frequencywidth. Then the followingasymptoticdevelIn the model adopted above, the Chandler compoopment
holds
for,k/,k'<<1,
nent time serieshas a constantperiod(434 meansolar
27,084
GIBERT ET AL'
WAVELET ANALYSIS OF THE CHANDLER
WOBBLE
107
10o
10-1
10-2
I
0.5
model
3:
data
envelopes
+phase
jumps
-
---- model 2: data envelopes
........ model 1' constantenvelopes
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
-
1.4
frequency(c.p.y.)
Figure 17.
Amplitudespectraof the progradeChandlerand annualcomponents
(the data)
and of severalmodels. Observethe important wideningof the spectralpeaks, especiallythe one
corresponding
to the Chandlerfrequency(0.814cpy),for themodelsincludingthe dataenvelopes.
days),a variableamplitude,and severalphasejumps. netic jerks found in previousstudies[Alexandrescu
et
The power spectrumof this model is very similar to that al., 1995,1996].Sevenjerks, andonlyseven,havebeen
of the actual data and, as Figure 17 shows,we observe
that includingthe phasejumps increasesthis similarity.
The combined effectsof both the amplitude variations
and the phase jumps considerablywiden the spectral
line of the Chandler term, suggestingthat the determination of the Q factor from the width of the Chandler
identified accordingto the criteria definedin Alexan-
drescuet al. [1996]sincethe beginningof the century
Table 3. Phase Jumps Obtained by Instantaneous
PhaseFitting
spectralpeakis stronglybiased(Figure17).Deconvolving from the excitation term is absolutelynecessaryto
obtainreliableestimatesof Q [O'ConnellandDziewonski, 1976; Dickman, 1981; Dickmanand Nam, 1998].
Another result is that the absenceof a largephasejump
around 1986eliminatesthe possibilityof the largephase
jump of 1926beingthe result of beatingphenomenabetween closefrequencies,as expectedfor a two-frequency
modelof the Chandlerwobble[McCarthy,1974].
The most novel result of this study is the remarkable coincidenceof the dates of the phasejumps, introducedto explain the experimentalridgefunctionsof the
Chandler wobble series, and the dates of the geomag-
1905.5
1913.5
1926.4
1932.1
1939.5
1952.6
1953.7
1961.3
1972.0
1980.5
1.0
0.8
3.3
1.5
1.5
2.4
15.0
1.3
1.0
2.5
24.8
-23.5
152.7
50.2
64.3
34.6
-72.0
10.8
16.1
33.3
Here t,, dates(years)of the phasejumps;Tn, duration
(years)of the phasejumps; qbn,amplitude(in degrees)of
the phase jumps.
GIBERT ET AL.' WAVELET ANALYSIS OF THE CHANDLER WOBBLE
27,085
(1901,1913,1925,1932,1949,1970,and 1980),all of to exert strongenoughequatorialtorques,wouldrequire a conductanceof the mantle muchhigherthan
believed
nowadays[Shanklandet al., 1993;Greff-Lefftz
large area at the surfaceof the Earth. Golovkovet al.
and
Legros,
1994]. The topographiccouplingcan be
[1989]andMcLeod[1989]pointedout anotherjerk event
much
more
efficient
[Hindereret al., 1990;Hulot et al.,
in 1939 and 1940 respectively.We did notice a jerk behavior in some observatories of the Pacific Ocean, and 1996;Hide et al., 1996].
them observedin a number of observatoriescovering a
the occurrenceof a jerk at this time, with a large geo-
For decadetime constants,the secularvariationof the
fieldis generated
by the advectionof the
graphicalextension,cannotbe discardedor established geomagnetic
andScott,
accordingto our criteria. We indicatetheir 1940event fieldby theflowat thetopof thecore[Roberts
1965;
Backus,
1968].
The
pressure
field
linked
to this
by a differentsymbolon Figure15,in orderto makeit
clear that it is not one of the eventsreported by Alexan- flow is typicallyof the orderof 200 Pa, [e.g.,Hinderer
drescu
et al. [1996].Golovkov
et al. [1989]listedstill et al., 1987, 1990;Hulot et al., 1996].The geomagnetic
of the
anotherjerk in 1958,the existence
of whichseems
to be jerks can be attributedto a suddenacceleration
flow.
The
pressure
field
presents
the
same
acceleration
supportedby a studyof Jackson[1997].But Alexandrescuet al. [1996]didnotfindanyconclusive
evidence andcandifferby 100percents(aftera fewyears)[Hulot
for it.
et al., 1993]from the valueobtainedby extrapolating
The coincidencebetweenthe phasejumps dates and
the geomagneticjerks dates illustrated by Figure 15 is
rather striking, and some elementary statistical tests
do not reject the hypothesis of a causal relation between the jerks and phase jumps. In fact, we claim
that the analysissupportsthe idea that a magneticjerk
is followed within at most 3 years by a phase jump in
the Chandler component. This statement is true for
all jerks except the ones in 1901 and 1913, and holds
true for both methodsusedfor the dating of the phase
its trend before the jerk date. Let us then considerthe
effectof a changeof pressureAP of 200 Pa actingon the
ellipsoidalcore-mantleboundary(CMB); the resulting
changeAL in the equatorial torque is
AL-
3Acc•c
izx*,
whereAP* is the efficientpart of AP (only the har-
moniccomponent
P•(cosO)e
i• of AP is efficient
when
jumps (ridge and instantaneous
phasevelocity). The there are no bumps, but other componentsof the exlarger time delay between the 1901 jerk and the cor- pansionintervenewhenthe CMB is bumpy),Ac is the
respondingphasejump may be attributed to an edge moment of inertia of the core, ac is its dynamicalflateffect. For the 1913 jerk, the instantaneousphase ve- tening,b is the radius,and pcis the coredensity[Hinlocity method for estimating the date of the phasejump deter et al., 1990]. AL is • 1020Nmfor AP = 200
placesit after the correspondingjerk, whereasthe ridge Pa.
Let us then examinethe effectof a torqueof the form
method givesa date slightly beforethe magneticevent.
In addition, the precedinganalysispoints to a phase
jump in 1960with no corresponding
magneticevent(we
L (t) - A0H(t) [tH (T - t) + TH (t - T)]
(33)
have not retained the 1958 event, however beneficial to
the proposedcorrelationit wouldbe). Its amplitudeis on the ridgefunctiondefinedas previously.The equahardly larger than the amplitude of the fluctuationsdue tion of polar motionis [Guinot,1973;Lainbeck,
1980;
to the noise(Figure8). The long-duration
phasechange Wahr, 1988;Dickman,1993],
occurring from 1945 to 1960 is of a different nature. It
is to be noted that it coincideswith a major changein
i d
----m
+ m- g,,
f2dt
the trend of the meanpolarmotion(seeFigure9). The
(34)
interpretation of this phasejump remains ambiguous
with • the Chandlerangularfrequencyand wherethe
and, in fact, if oneusesthe more recentseriesof Vondrdk
torquecomponentof the excitationfunctionq• is given
(personalcommunication),
this phasejump disappears, by
but not the others.
i
One should assesswhether any mechanismrelated to
the geomagneticjerks could explain the suggestedcorrelation. We will not make any complete treatment of
this question in the present paper, but limit ourselves
= •I'0H(t) [tH (T - t) + TH (t - T)], (36)
to a preliminary discussionand estimating someorders
of magnitude. For the short-time constants involved where w is the Earth angular velocity and C- A is the
herefor the phasejumps (Table2), the corecan act on difference between the principal moments of inertia of
the mantle rotation (lengthof the day and polar mo- the Earth's mantle. We use the usual complex notation
=
(35)
i•u wherethe subscripts
tion) throughelectromagnetic
[Stix and Roberts,1984; m: m• -im u and ß - •Holme, 1998]or topographic[Hide, 1986; Jault and Le denote the referenceaxes defined in a previous section.
Mou•l, 1989]coupling. The electromagnetic
coupling, The solution for the polar motion is
27,086
GIBERT ET AL.' WAVELET ANALYSIS OF THE CHANDLER WOBBLE
0.80
o.os
-
2'1/
-
0.90
•
•
•
••
• ----- m0= 0.035arcseconds
• ----- m0= 0.053arcseconds
• .... m0= 0.070arcseconds
•
m0= 0.088arcseconds
•
-10.0
mo=O.
140
arc
seconds
0.0
10.0
20.0
time
FiEure •8.
Ridge fU•c[io•s ob[•i•ed fo• sF•[•e[ic po]•r too[ionscorrespo•di• [o
for sever•]v•]uesof [he i•i[i•] •mpli[udem0 (see[ex[ for [he chosen
v•]uesof the •em•i•i•
p•r•meters). Observethai [he •mpli[udeof [he deformation
of [he •id•esis s[ro•]F controlled
m0.
re(t)
-- rnoe
ir•t t<0
re(t)
-
re(t)
-
(37) phasejump (ibrtunately,asalreadysaid,the amplitude
of the wobblevariesby a factorof 8 duringthe century).
Figure 20 shows,despitesomescattering,that the data
0_<t_<Tcan
be considered
to fit law (38).
+•0 (1)
t+•
(m0-•-•
•O)emt
+•oT t>T.
[mo-• (1_
Phasejumps inferred from the actual data look then
compatiblewith a torque origin. Consequently,
should
the magneticjerks generatean equatorial torqueon the
We have computedthe ridge function of this solution mantlereaching1020Nm in a few months,onewould
for different values of T and m0 and for an adjusted hold a nice interpretation of the data. However, there
value•0T - 1.7 x 10-7 corresponding
to a maximum we meetseriousdifficulties.First, 1020Nm is probably
torqueA0T- 1020Nm(theestimate
givenabove).The an upper limit of the possibletorque [Hindereret al.,
efficiencyof the torque decreasesrapidly with T, as is
well known, and we took T - 4 months. Figure 18
shows the ridge functions obtained for several initial
amplitudesof the polar motion 0.023 _<m0 _<0.140 arc
seconds. These ridge functions exhibit peaks with an
amplitude comparableto the amplitudeof thosepresent
1990; Hulot et al., 1996]. The elasticmantlereactsto
thetorquein sucha wayasto lessen
itseffect[Hulotet
al., 1996]. Second,a step changeof the pressureat the
core-mantle boundary reached in a few months is not
directly supportedby the data. The changeof pressure
is rather
of the form
in the real ridgeline (Figure 15).
(39)
AP(t)-- a(t - t0)1/2H (t)
It also appears that the amplitude 5• of the phase
jumps determined as previously for a given •0T deaccordingto the time structure of the jerks derivedby
pends on m0, that is, decreaseswhen m0 increasesas
Alexandrescuet al. [1995, 1996] for a jerk occurring
expectedfrom the form of the solution(37). Figure19 at t = to, and it would take a few years to reach a
illustrates this dependenceand showsthat
(38)
with 1.5 _</• _<2. Let us draw the same graph in the
case of real data, that is, the phase jump amplitude
versus the amplitude of the wobble at the time of the
changeof 200 Pa. With such a time constant,the correspondingtorque is much lessefficient,muchtoo weak
to accountfor the observedphasejumps. In fact, we
meet of course the same difficulty as when trying to
maintain the Chandlerwobbleby the coreflow [Jault
and Le Mou•l, 1989]. However,the magneticfield is
GIBERT ET AL.: WAVELET ANALYSIS OF THE CHANDLER WOBBLE
180
0 synthetic
data
-1.0
y = YoX_.,.
s
___ y = YoX_2.o
y = yox
135
90
45
I
0.05
I
,
0.10
I
0.15
Amplitudeof the Chandlermotion(arc seconds)
Figure 19. Correlationbetweenthe initial amplitudeof the Chandlerpolar motion m0 and
the value of the phasejump obtained from the ridge functionsshownon Figure 18. Also shown
are power laws with exponents-1.0,-1.5, and-2.0.
180
135
_•data
-,.o
y = YoX_,.s
-- y = YoX_2.o
-----=
,,=-•
y = yox
90
45
i
i
i
0.10
0.20
0.30
Amplitudeof theChandlermotion(arcseconds)
Figure 20. Correlationbetweenthe amplitudeof the reconstructed
progradeChandlerpolar
motion (seeFigure 12) and the valueof the phasejumpsobtainedfor a baseperiodTo - 434
meansolardays(seeTable2). Alsoshownarepowerlawswith exponents
-1.0, -1.5, and -2.0.
27,087
27,088
GIBERT ET AL.: WAVELET ANALYSIS OF THE CHANDLER WOBBLE
observed
at the Earth'ssurface,and (dueto screening)
we missboth spatial and temporal short scales,evenfor
a weakly conductingmantle. Furthermore, the energy
of the external origi, ionosphericand magnetospheric,
variations is large for time constantssmaller than, perhaps, i year, making the separationof a possibleinternal signal difficult. The correlation between the dates
of the magneticjerks and those of the Chandler wobble phasejumps suggestedin the presentpaper at least
urgesus to revisit the physicsof the jerks and more generally the fluid flow at the top of the core, as well as its
action on the mantle's
rotation.
It must be recalled that
periods from Earth's rotation, Geophys. Res. Left., 25,
211-214, 1998.
Eubanks, T.M., Variation in the orientation of the Earth, in
Contributions of Space Geodesy to Geodynamics: Earth
Dynamics, Geodyn. Set., vol. 24, edited by D.E. Smith
and D.L. Turcotte, pp. 1-54, AGU, Washington, D.C.,
1993.
Foufoula-Georgiou,
E., and P. Kumar (Eds.), Waveletsin
Geophysics,362 pp., Academic, San Diego, Calif., 1994.
Furuya, M., Y. Hamano, and I. Naito, Quasi-periodic wind
signal as a possibleexcitation of Chandler wobble, J. Geophys. Res., 101, 25537-25546, 1996.
Gambis, D., Wavelet transform analysisof the length of the
day and the E1-NinS/Southernoscillationvariationsat in-
traseasonal and interannuel time scales, Ann. Geophys.,
at the presenttime no convincingother mechanismhas
10, 429-437, 1992.
been proposedfor exciting the Chandler free oscillation Golovkov, V.P., T.I. Svereva, and A.O. Simourian, Common
[Wart, 1988; Dickman, 1993; Eubanks,1993; Wilson,
1993;Furuya et al., 1996]
Acknowledgments.
M. Feisselgave us detailed informations about the preprocessingsof the data series,and J.
Vondr•k provided us with his series of data. S. Dickman
and Associate Editor A. Jacksonmade very constructive reviews which considerablyimproved the original manuscript.
This study was partly granted by the CNRS-INSU Scientific
Program "Int•rieur de la Terre". This is IPGP contribution
1554.
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