Wavelet-based filtering of intermittent events from geomagnetic time-series

Planetary and Space Science 49 (2001) 1219–1231www.elsevier.com/locate/planspasci

Wavelet-based !ltering of intermittent events fromgeomagnetic time-series

P$eter Kov$acsa ;∗, Vincenzo Carboneb, Zolt$an V+or+oscaE�otv�os Lorand Geophysical Institute, H-1440 Budapest, P.O. Box 35, Hungary

bDipartimento di Fisica and Istituto Nazionale di Fisica della Materia, Unita di Cosenza, 87100 Rende (CS), ItalycGeomagnetic Observatory of the Slovak Academy of Sciences, 94701 Hurbanovo, Slovakia

Received 25 October 2000; received in revised form 8 February 2001; accepted 25 February 2001

Abstract

A four-years-long minute-mean geomagnetic time-series recorded in a middle-latitude observatory was used to investigate the macro-scopic dynamics of the magnetosphere–ionosphere complex system. Through probability density and structure function analyses it is shownthat the investigated signal exhibits intermittent 7uctuations in a certain temporal scale range. This property is regarded as being similar tothe turbulent phenomena of magneto-hydrodynamical systems. In this work, discrete orthonormal wavelet transformation and !ltering arecarried out in order to identify and separate the intermittent parts of the signal from the homogeneous noise-like background. The empiricalprobability distributions of the laminar time between and the energy content of intermittent events are computed in the time-scale domainrepresentation of the original signal provided by the wavelet transformation. It is shown that the results obtained are in contradiction witheach other in the context of the phenomenology of the classical self-organized critical state model. We argue that ‘near-SOC’ or chaoticturbulence model can explain the observed features without ambiguity. c© 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction

The time variation of the geomagnetic !eld exhibits7uctuations in a wide range of temporal scales extendingfrom a few seconds to the geological scale. Sharp sep-aration between the frequency ranges belonging to thevariations of di?erent origins cannot be made. It is widelyaccepted, however, that temporal variations of long wave-length (dominantly more than 1–2 years) are driven mainlyby inner-Earth movements whereas the rapid changes aregenerated by coherent or incoherent plasma movementsthat have evolved in the magnetosphere–ionosphere (MI)system. In this paper, our aim is to describe the externalorigin part of the !eld variation in the context of tur-bulence methodology. In order to be able to apply theconcepts of turbulence studies, the following assump-tions have to be made; (1) the magnetosphere behaves asan ideal magneto-hydrodynamical (MHD) system wherecoherent plasma eddies can evolve in a wide range ofspatial scales generating a cascade like energy transfer

∗ Corresponding author. Tel.: +36-1-384-3302; fax: +36-1-384-3306.E-mail addresses: [email protected] (P. Kov$acs), carbone@!s.unical.it

(V. Carbone), [email protected] (Z. V+or+os).

between system-size and dissipative (Kolmogorov) scales,and (2) the time variation of the ground based recordsof the magnetic !eld is a valid observation of describingthe 7uctuating physical processes taking place in the MIsystem. In fact, abrupt changes in speed of the Earthward7ows in the neutral sheet may be observed at a groundbased observatory as it was reported by Shiokawa et al.(1998). Hence, ongoing turbulent plasma processes in theremote regions of the magnetosphere may contribute to thespectrum of geomagnetic 7uctuations. As a matter of fact,however, the previous suggestions can also be invalidatedby several arguments. For instance, one can argue that themagnetosphere is a bounded system in which the coherentstreaming of the plasma cannot take place. It implies thatTaylor’s ‘frozen !eld’ hypothesis cannot hold so that thedirect transformation from temporal to spatial observationsof the geomagnetic !eld may not be accomplished. Also,the speci!c structure of the geomagnetic !eld lines makesthe MI system very inhomogeneous thereby invalidatingthe ideality of the system. Moreover, in contrast to thebehaviour of ideal magneto7uids, the magnetosphere ishighly dispersive as a consequence of the intensive ener-getic interaction of the plasma particles with the geomag-netic lines of force, and the evolving wave-!elds (Dudok

0032-0633/01/$ - see front matter c© 2001 Elsevier Science Ltd. All rights reserved.PII: S 0032 -0633(01)00063 -0

1220 P. Kovacs et al. / Planetary and Space Science 49 (2001) 1219–1231

de Wit and Krasnoselskikh, 1996). On the other hand,it has long been known that several features of the MIsystem, as they can be deduced from ground observations,are similar to those of hydrodynamical (HD) or MHD sys-tems. Among others, analogously with turbulent activities,the history of the geomagnetic !eld is characterized by theinterchange of quiescent and disturbed periods. The energydissipation is strongly intermittent as was recently showne.g. by multifractal analysis of a singularity spectrum of acertain energy related measure derived from high-latitudegeomagnetic time-series (V+or+os, 2000). The singularityspectra reveal that a multiplicative cascade model coulddescribe the underlying energy dissipation processes onthe time scale of substorms (from 30 min to a few hours).Another implication concerning the existence of multiscaleprocesses in the magnetosphere, or at least in the tail region,was pointed out by Chang (1999), who stated that the en-ergy transport processes between the energy injection anddissipation scales involve ‘characteristic parameters thatdi?er by orders of magnitude’. The power spectral density(PSD) function of AE electrojet activity index (Tsurutaniet al., 1990), as well as high and middle-latitude geomag-netic time-series (Uritsky and Pudovkin, 1998; V+or+oset al., 1998) exhibit a broken power law character, whichimplies (consistently with the intermittency observations)the need of higher-order statistics for describing the systemunder consideration. The spectral break was identi!ed inthe range of 0.05–0:07 mHz depending on the investigatedtime-series (AE or geomagnetic from di?erent latitudes).Tsurutani et al. (1990) have further shown that prior tothe break the spectral exponent of the AE index is similarto that of the southward component of the interplanetarymagnetic !eld (IMF), but this correspondence is not satis-!ed for the high-frequency intervals. This !nding indicatesthat the high-frequency part of the magnetospheric 7uctua-tions are the intrinsic characteristic of the system while thelow-frequency magnetospheric processes are governed bythe solar wind being a real MHD system (see e.g. Usmanov,1995; Carbone et al., 1996a, b). V+or+os et al. (1998) drew thesame conclusion when, using the extended self-similarityconcept introduced by Benzi et al. (1993), they found iden-tical scaling behaviour for structure functions computedfrom low-frequency (wavelength, �¿ 300 min) geomag-netic, and in-situ measured solar wind data.The above observations, i.e. bursty periods in the physical

!eld, power law PSD, close dynamical connection with solarwind turbulence, suggest that it is not unrealistic to investi-gate the magnetosphere dynamics through the well-knownconcepts of HD or MHD studies, such as intermittency, cas-cade processes, or coherent eddies. Of course, a descriptionof the magnetosphere phenomena with MHD equations isnot expected on the basis of the present state of knowledge,also because of the violation of the above-mentioned as-sumptions. Instead, it seems to be much more realistic tosearch for some correspondences with global macroscopicmodels that may be also valid for HD or MHD systems.

In early studies of magnetospheric time-series (AE orgeomagnetic) it was suggested that the magnetosphereis a low-dimensional nonlinear system (e.g. Baker et al.,1990; V+or+os, 1991; Vassiliadis et al., 1993) whose dy-namics can be so studied in the frame of chaotic models.Among others, this hypothesis was fastened by fractal anal-ysis of storm-time geomagnetic signals that demonstratedself-similar underlying dynamics (V+or+os, 1990). On theother hand, Osborne and Provenzale (1989) argued thatlow-dimensionality is not decisive proof for the existenceof chaos, because stochastic coloured noise can share thisproperty. Using di?erent methods, Vassiliadis et al. (1995)have shown that optimized input–output nonlinear predic-tors of magnetospheric activity are always low-dimensionaland the properties of the predictor may bear on the prop-erties of the magnetospheric dynamics. Furthermore, it hasbeen conjectured by Chang (1992) that consistently withthe aforementioned power law PSD functions of AE andgeomagnetic time-series, the low-dimensional behaviourand the fractality suggest the theory by which the magne-tosphere is in forced or self-organised critical (SOC) state(Bak et al., 1987). The hypothesis of SOC was later sup-ported by the observations of scale-free size and lifetimedistributions of bursty periods identi!ed in di?erent lengthAE time-series by the threshold method (Consolini, 1997;Takalo and Timonen, 1999). Moreover, Uritsky and Pu-dovkin (1998) have shown that a simple two-dimensionalsandpile model driven by real solar wind data can exhibitsimilar spectral and fractal properties as the AE time-series.Another computer simulation with the sandpile model(Chapman et al., 1998) could reconstruct the previously re-vealed di?erent distributions of energies dissipated by smallinternal and system-size bursty events in the magnetosphere.Also, Klimas et al. (2000) have presented an SOC model oflocalized reconnection in the magnetospheric plasma sheet,by which they were able to understand how a coherent,predictable (low-dimensional) substorm cycle can developfrom the strong turbulence typical in the plasma sheet. Fur-ther experimental evidence supporting SOC like behaviourand the localized reconnection model was provided byConsolini and Lui (2000), who studied 7uctuations relatedto the disruption of the cross-tail currents phase-transitionand nonlinear re-organization.We will see that the statistical distribution of internal

time between bursty events of a four-years-long geomag-netic signal may reveal a contradiction with the funda-mental phenomenology of SOC systems. In the presentanalysis, the basic tools for identifying the bursty periodsare the wavelet transformation and wavelet !ltering thatwere already used for straightforward studies of HD turbu-lent signals e.g. by Farge et al. (1999) and Schneider et al.(1999). The following chapter deals with the presentation ofthe intermittent nature of the analysed signal. The wavelettransformation and !ltering will be de!ned next, whilethe results and the conclusions are shown in the last twochapters.

P. Kovacs et al. / Planetary and Space Science 49 (2001) 1219–1231 1221

2. Description of the analysed data

During the analyses we used the !rst 2097152(= 221)elements of a four-years-long geomagnetic time-series thatwas recorded in the middle-latitude Tihany observatory(�=46:3◦;�=99:1◦, Hungary) between 1993 and 1996.The data sampling interval is 1 min corresponding to about16:7 mHz sampling frequency. It is widely accepted andcan also be visualized by eye that any disturbances (stormsor substorms) erupting in the MI system are principallyre7ected by the north–south directed Bx component of thegeomagnetic !eld at middle latitudes. For this reason, thispaper deals with the analyses only of the Bx records (the xindex will hereafter be neglected). It is pointed out, how-ever, that the investigations were extended to the east–west(Y ) and downward (Z) directed components, as well, andit was concluded that the results are more or less identicalwith that obtained by the use of the X records. The veryde!nite regular daily variation of the !eld was eliminatedfrom the signal, simply by subtracting the mean daily !eldfrom each one-day long segment of the series. The wrongor missing values whose number do not exceed 1% of thetotal length of the time-series were corrected by linear in-terpolation in order to ensure the continuity that is essential,e.g. in the case of the wavelet transformations.The graph in Fig. 1 shows a two-week segment of the

investigated signal. The curve represents a characteristicscenario in the evolution of the geomagnetic !eld, notablythe rapid change of the !eld from a quiescent period to ahighly 7uctuating one and then the gradual relaxation of theerupted disturbances. It was illustrated in the !rst sectionthat if one disregards the real reason for the evolution of thegeomagnetic disturbances (storms, or substorms), the 7uc-tuating nature of the !eld can be interpreted in the presentcontext as the manifestation of turbulent phenomena thattake place within the plasma of the magnetosphere. It haslong been accepted that turbulence evolves through cascadeprocesses that involve a hierarchy of coherent vortex struc-tures belonging to a wide range of spatial scales. In the caseof inhomogeneous 7ow down (cascade) of the energy fromsystem-size scales to dissipative (Kolmogorov) ones, qui-escent and active regions of the investigated medium areformed that can thus be studied in terms of fractal or multi-fractal terminology (Frisch, 1995). The inhomogeneity in-volves the singular behaviour of the energy distribution inphysical space resulting in strong gradients, or intermittencyin the time-series of the energy related physical quantitiesof the system, e.g. velocity, temperature or magnetic !elds.The intermittent nature of the geomagnetic !eld (B), or of

any other physical quantity, can be investigated through theprobability density functions (PDF) of a set of two-pointsdi?erence time-series, �B�(t)=B(t+ �)−B(t), of the orig-inal !eld. The parameter � is a varying scale that (accept-ing the Taylor hypothesis) represents the spatial size of aneddy contributing to the energy cascade process. Notice thatthe di?erentiation of the original signal means a kind of

high-pass !ltering with its border frequency proportional to�. For ordinary HD or MHD 7uids, intermittency appearsin the heavy tails of the distribution functions at moderatescales implying non-Gaussian statistical behaviour of thesystems (Frisch, 1995). This means that the distribution ofthe vortex strengths, that are quanti!ed by the elements ofthe di?erenced time-series, is non-random, but ruled by de-terministic dynamics. At scales larger than the size of thebiggest contributing eddy, the coherence between the ele-ments of the original signal ceases resulting in Gaussiandistributions of the di?erenced time-series at these scales.Similar analysis with AE index time-series has been per-formed by Consolini and De Michelis (1998) who obtain-ing scale-dependent probability distribution functions for thesignal concluded to its intermittent nature under both quietand disturbed magnetospheric conditions.Before computing the PDF-s for the geomagnetic !eld,

the di?erenced time-series were normalized by means of

�B�(t)=�B�(t)− 〈�B�(t)〉

�; (1)

where 〈�B�(t)〉 and � represent, as usual, the average andthe standard deviation of the data set, respectively. The dis-tributions of the �B�(t) series were constructed for discretescales, �=2n min (n=1 : : : 18). During the numerical com-putation of the PDF-s, the number of samples falling in unitsize bins of the range determined by the extreme values ofthe set were computed and later normalized by the lengthof the series. The results obtained are shown in Fig. 2 forfour selected scales (�=2; 32; 512; 8192 min). It is apparentthat at the smallest scale, the probability of strong events aremuch higher than would be in the case of a pure Gaussianprocess. The tails of the distributions diminish with increas-ing scale parameter, as a consequence of the decrease ofthe probability of coherent 7uctuations between points sepa-rated by increasing distance. On the other hand, at the high-est represented scales the PDF-s are still not really Gaussianbut they seem to comprise a mixture of two or more di?er-ent distributions. It can be assumed then that the geomag-netic time-series are governed by more than one signi!cant,and independent process whose own statistics add togetherforming the PDF-s. This assumption is taken into accountin the course of our further investigations and in the !nalconclusions (see the fourth and !nal Chapters).In order to quantify the degree of deviation from Gaussian

distribution, i.e. the level of intermittency at di?erent scales,we computed the 7atness values of the two-points di?erencetime-series de!ned as

F =〈�B�(t)4〉〈�B�(t)2〉2 : (2)

The 7atness of a normally distributed signal is equal to 3.Adding intermittent 7uctuations to an originally Gaussiansignal implies the spreading of its PDF and, consequently,the increase of its 7atness value. For the di?erenced


Fig. 1. The variation of the X component geomagnetic !eld from 01=11=93 until 14=11=93. A geomagnetic storm broke on 3rd November and lasteduntil about 8th November. The average diurnal variation was eliminated.

Fig. 2. PDF of the di?erenced geomagnetic time-series (see text) at four selected scales.

geomagnetic time-series the 7atnesses are shown in Fig. 3ain terms of the scale parameter �. In correspondence withthe quantitative shapes of the PDF-s, the curve obtainedrepresents a continuously decreasing function of �, thatasymptotically approaches but does not actually reach theGaussian value of 3. The minimum is reached at a scaleof about �=2048 min. The deviation of the 7atness from3 at high scales can be explained by the joint occurrencesof independent magnetospheric processes (see before) orby some periodicities of the original signal that implycorrelation between distant signal elements.A complementary way of investigating intermittency is

provided by the well-known structure function analysis. The

structure function of order q is de!ned for the B time-seriesas

Sq(�)= 〈|B(t + �)− B(t)|q〉t ; (3)

where � represents the scale parameter, as before, and theaveraging goes over t. The shapes of the functions are shownin Fig. 4 for four di?erent orders. The inertial range, wherethe structure function scales like Sq(�) ∼ ��(q), can be es-tablished between 16 and 724 min in agreement with the!nding of V+or+os et al. (1998). The intermittent nature of the!eld is apparent in the nonlinear relation between the scalingexponent, �(q) and the order parameter, q (Fig. 5a), imply-ing inhomogeneous streaming of the energy in the inertial


Fig. 3. Flatness curves of the di?erenced time-series as a function of the scale parameter for (a) the original and (b) !ltered geomagnetic time-series. Thecondition factors of the !lter criterion (see text) are shown in the label inset of the bottom panel. Line F =3 represents the 7atness values for a Gaussiansignal. The inset graph on the upper panel shows the 7atnesses for the un!ltered signal using an y axis range smaller than that of the original graph.

Fig. 4. Structure functions of the geomagnetic time-series investigated at four selected order parameters. The dashed vertical lines indicate the scalingregion.

spatial range (Frisch, 1995; Dudok de Wit, 1996). A multi-fractal phase transition is apparent at the moment of aboutq=4 beyond which the curve become linear. This meansthat the investigation of higher-order structure functions is

irrelevant from the point of view of multifractal modeling.On the other hand, to visualize the e?ectiveness of the !l-tering process that will be outlined next, a relatively long�(q) curve will be useful.


Fig. 5. Scaling exponent curves obtained for the structure functions of the (a) original and (b) !ltered geomagnetic signals. The condition factors of the!lter criterion are shown in the inset of (b).

On the basis of the PDF and structure function analy-ses, we conclude that the intermittent behaviour, i.e. the co-herent 7uctuation of the geomagnetic !eld can be tracedup to a temporal scale range of about 724–2048 min. Wewould again emphasize that exact analogies between the be-haviours of ideal magneto7uids and the MI system cannot beexpected on the basis of the time-series analyses presentedhere because of the violation of some necessary conditionshighlighted in the previous section. For our current discus-sions, however, our view is that some essential qualitativecorrespondences must exist. Previously, we argued in anal-ogy with turbulence studies that the observed PDF-s andscaling exponent function refer to the intermittent nature ofthe given geomagnetic !eld. Moreover, we now state thatthe converse is also true; the geomagnetic !eld would beintermittent-free if the distribution functions were Gaussianat every scale or the q dependence of the scaling exponentwas linear, even if the exact Kolmogorov-type (�(q)= q=3)scaling would not be expected. This suggestion will be ap-plied in the following section.

3. Wavelet analysis

Intermittency is a usual phenomenon in 7uid or mag-neto7uid turbulence. It is usually assigned to the coherentvortex structures that can be derived from the 7uid equations(Farge et al., 1999). The coherent eddies represent energiesat various scales that transfer through the entire system inthe course of the cascade process. Any time signal repre-senting this process can be properly analysed by the wavelettechnique (Kumar and Foufoula-Georgiou, 1997) that

enables one to decompose it in terms of scale and time units.Using the statistical properties of intermittency depicted inthe preceding section, Schneider et al. (1999) proposed thewavelet approach as a means of separating the coherent partof a turbulent signal that is responsible for the long tailsof the PDF-s, from the incoherent and homogeneousbackground noise showing Gaussian statistics. The proce-dure enables one to investigate the statistical properties ofthe intermittent and quiescent parts of the signal separatelywithout changing their spectral properties.In analogy with windowed Fourier transformation, the

wavelet decomposition of any !nite energy time signalf(t), is equivalent to its projection to the time-scale ortime-frequency domain. The transformation is de!ned asthe inner product of the analysed signal and the time-shiftedand scaled version of an optional wavelet function, (t)having zero mean and compact support, in accordance withthe following relation:

Wf(�; t) = 〈f; �; t〉=∫

f(u) �; t(u) du

=∫

f(u)1√�P (u− t�

)du; (4)

where �, and t represent the scale and time parameters, re-spectively, and the overbar means complex conjugate. Itcan be shown that the scale parameter of the transforma-tion is inversely proportional to the frequency parameterof the classical Fourier analysis. The basic di?erence be-tween the results of the wavelet and windowed Fourier anal-yses lies in the resolution pattern of the two transformationsover the frequency-time domain. In the case of the wavelet


decomposition the uncertainties in time and frequency deter-minations change over the frequency-time plane so that theirproduct remains constant throughout the entire phase-space.Towards the small scales (i.e. high frequencies) the timeresolution become better at the expense of the increase of theuncertainty in the frequency determination, whereas towardsthe high scales (i.e. low frequencies) the situation is the op-posite. On the other hand, the resolutions of the windowedFourier transform in time and frequencies are uniform inde-pendently of the place on the frequency-time plane. It meansthat at high frequencies the small details can remain hiddenin the Fourier transformation case, whereas with the aid ofwavelet analysis they can be visualized. From this compar-ison it can be seen that the resolution pattern de!ned by thewavelet transformation !ts quiet well to cascade processes,where eddies of continuously changing spatial scales canoccur with time separation decreasing towards the smallestscales.For some of the practical applications, the discretization of

the two parameters of Eq. (4) is prescribed. According to theresolution pattern of the transformation, the discretizationin time and scale must be proportional to each other. Mostoften, the scale parameter (�) is varied on the way of

�= �m0 ; (5)

where �0 is a !xed dilation step, and m is an integer. Con-sequently, the time parameter is changed as

t= nt0�m0 ; (6)

where t0 ¿ 0 and depends on the selected wavelet function( (t)), and n is an integer. The Dm;n wavelet coeQcientbelonging to the mth and nth discrete parameters is thencomputed by

Dm;n = 〈f; m;n〉= �−m=20

∫f(t) (�−m

0 t − nt0) dt; (7)

where m;n is the discretely scaled and shifted version ofthe original wavelet function. It can be shown that the (t)‘mother wavelet’ can be constructed so that its discretedilates and translates should be orthonormal to each otherforming a complete orthonormal basis for all functionshaving !nite energies. Completing the transformation withthis basis, a series of wavelet coeQcients can be obtainedhaving the same energy (sum of squares of the members)and length as the original signal. This property is essen-tial whenever discrete wavelet transformation is utilizedto statistically analyse any physical observations. A wholeclass of orthonormal wavelet bases was constructed byDaubechies (1988). In the current analysis we applied thefourth member of this class. The question may arise howthe rather irregular shape of this wavelet function can in-7uence our !nal conclusions. For this reason, preliminaryanalyses were completed with the more irregular Haar (!rstmember of the Daubechies wavelet class), and the moresmooth eights Daubechies wavelet functions, as well. It wasconcluded that while the Haar wavelet can result in extrairregularities in the analysed signal through the !ltering

process (see later), it does not occur with the more smoothfourth and eights Daubechies wavelet functions. Further-more, our main purpose, i.e. identi!cation of bursty periodsin the gemagnetic signal (see later), and also our main con-clusions turned out to be independent of the smooth levelof the analysing wavelet functions.Obviously, the strong activity parts of an intermittent sig-

nal that result in the heavy tails of the PDF-s, appear withhigh amplitudes among the wavelet coeQcients, while thequiescent periods are represented by small values. This dis-tinction enables one to determine a criterion by which thebursty and quiet periods of a time-series can be separatedfrom each other. (For a comprehensive study about waveletde-noising see e.g. Donoho et al. (1995).) The !lter condi-tion that we applied in the present analysis was originallyproposed by Katul et al. (1994) in the following way:

Dcm;n =

{Dm;n if D2

m;n ¡C · 〈D2m;n〉n;

0 otherwise;(8)

where 〈::〉n means averaging over the time parameter, andC is a prede!ned condition factor. The !ltered signal arisesthrough the inverse wavelet transformation of the Dc

m;n, !l-tered set of wavelet coeQcients. In their paper, Katul et al.(1994) suggested a method for determining the conditionfactor C, on the basis of the structure function analysis of theoriginal and !ltered signals. Since the !ltering musteliminate any inhomogeneity from a turbulent signal, thestructure functions of the !ltered time-series should exhibitKolmogorov type, �(q)= q=3 scaling (see the precedingsection). It was suggested then that the condition factormust be gradually lowered until the above scaling behaviouris reached. Using a criterion di?erent from (8), Schneideret al. (1999) proposed to eliminate the highest waveletcoeQcients successively so that the 7atness and skewness(third moment of the series) of the !ltered signal shouldapproach the Gaussian values, 3 and 0, respectively (seethe preceding section). Nevertheless, we would stress thatany regular variations of the analysed signal can lead tothe extreme increase of the wavelet coeQcients at scales ofperiodicities resulting in that the regular variations may beconsidered as irregular disturbances or that a lot of coeQ-cients belonging to real geomagnetic disturbances may notbe !ltered out at these scales. That was the principal reason,why the main periodicity of the geomagnetic !eld, i.e. thediurnal variation, was eliminated from the signal prior tobeginning the analysis (see the second chapter).The 7atness and scaling exponent curves of signals ob-

tained by !ltering of the original geomagnetic time-serieswith four di?erent condition factors (C =2; 5; 10; 15) areshown in Figs. 3b and 5b. The scaling region of the structurefunctions investigated were the same as in the case of theoriginal time-series (see the preceding section). Consider-ing the unclear relationship between the MI and real turbu-lent systems, only the linearity and not the exact agreementwith Kolmogorov type scaling is expected in the cases of


the scaling exponent curves of the inhomogeneity-free !l-tered signals. This condition could be satis!ed for C6 5(Fig. 5b). On the other hand, reassuring agreement betweenthe F(�) 7atness curves of the !ltered signals and the Gaus-sian F(�)= 3 linear is reached only if C6 2 (Fig. 3b).Based on the latter result, we decided to apply the C =2basic setting for the de!ned !lter criterion in the forthcom-ing analyses. However, we should not wish to disregard thefact that this choice is quite uncertain because of the con-tradictory results obtained from the methods used to deter-mine C. In order to resolve this confusion we checked therobustness of the results obtained with respect to the condi-tion factor that was utilized. The result of this investigationwill be discussed in the following chapter.The orthonormal wavelet transformation of the analysed

geomagnetic signal resulted in 2097152 (equal to the lengthof the analysed series) wavelet coeQcients. During theC =2!ltering process 168539 coeQcients were annulled, repre-senting about 70% of the total energy (sum of squares ofthe elements) of the original series. The !ltered picture ofthe previously presented part of the geomagnetic variation(see Fig. 1) is shown by Fig. 6.

4. Statistical analysis of coherent events identi�ed by thewavelet �ltering

In our case, the fundamental purpose of the wavelet !lter-ing outlined above is to identify the bursty events or coher-ent 7uctuations along the geomagnetic time-series that wereanalysed. It should be stressed that—in agreement with thespirit of the wavelet technique—the bursty periods will beconsidered in the time-scale domain and not in the physicalspace in the forthcoming analysis. This means that a geo-magnetic bursty event (e.g. storm or substorm) that appearsin increased amplitude of wavelet coeQcients on a certaintemporal and scale (or frequency) range can be investigatedseparately in each given scale. In the present context, coher-ent events are de!ned as those segments of the scale-timedomain image of the wavelet transformed time-series thatshould be eliminated in order to obtain a pure Gaussian sig-nal after the inverse transformation of the !ltered waveletcoeQcients. Each segment includes one or (generally) moreconsecutive wavelet coeQcients described by Eq. (7).Once the coherent events are identi!ed, one can perform

any statistical analyses relating to the dynamical behaviourof the system to be investigated, e.g. the numerical compu-tation of the distribution of laminar times between events, orthe distribution of energies released along the disturbed pe-riods. Any statistical properties will be considered in termsof the temporal scales. The laminar or waiting time is thetime that elapses from the end of an individual event untilthe beginning of the next one. This will be measured in theunits of the actually investigated temporal scale. As the en-ergy of a particular burst interval at a given scale, the sumof squares of the wavelet coeQcients belonging to the con-

sidered period has been taken. Due to the energy conservingbehaviour of the given orthonormal wavelet transformation,it can be assumed that the energy content of a set of waveletcoeQcients belonging to a certain temporal range is directlyproportional to the energy included by the transformed sig-nal in the same period. Moreover, if we accept that the vari-ation of the geomagnetic !eld is the direct manifestation ofmagnetosphere processes, we can also suppose that the en-ergy represented by the wavelet coeQcients is more or lessrepresentative of the magnetic part of the energy density ofthe magnetosphere electromagnetic !eld, too.Starting with the statistical distributions of energies that

are released in the course of individual events we cansee that they exhibit a very de!nite power-law character,P(x)=Ax−b, over maximum 1.5 decades of magnitude(Fig. 7) in the temporal scale range of 1–256 min. Note thatthe scaling exponents of the above power-law relation arevery similar implying that the statistics obtained is indepen-dent of the temporal scale. As a general phenomenon, it ismentioned that reliable statistics cannot be drawn above acertain temporal scale in the cases of any investigated quan-tities. The reason is that in the phase-space of a discreteorthonormal wavelet transformation the wavelet coeQcientsare distributed unevenly among the given scales so thattheir numbers decrease towards the higher scales. Thus,the number of !ltered coeQcients also decreases, resultingin a statistically non-representative sample of intermittentevents at high temporal scales.The distributions of released energies obtained support

the theory by which the magnetosphere is in a state ofSOC (Chang, 1992; Consolini, 1997; Uritsky and Pudovkin,1998). SOC is a mechanism through which certain systemsdriven by external energy-injection evolves towards an at-tractive !xed point in the phase space without any tuning oftheir system parameters (self-organization) (Bak et al. 1987;Jensen, 1998). The criticality of the system lies in that a verysmall perturbation exerted on the system can generate energyrelease processes (avalanche) of any spatial or temporal sizedepending on the time and place of the perturbation. Froma practical point of view, in that state the system exhibits nocharacteristic time or spatial scales that appear in the powerlaw distributions of any quantities related to the system’savalanches (size, energy, duration). Consequently, if the dis-turbed geomagnetic events are regarded as the manifesta-tions of avalanche processes that take place in theMI system,the scale-free statistical distribution of their energies provideclear evidence for the self-organized criticality of the system.On the other hand, it was implicitly suggested by Bo?etta

et al. (1999) that the observation of the mentioned classicalSOC phenomenology (scale-free energy, size, duration dis-tributions of avalanches) in the case of any physical systemdoes not necessarily imply the real existence of the stateof self-organized criticality. For the case of solar dynamicsthe latter authors used the statistical distributions of laminartimes between bursts of solar 7are activity to be able to dis-tinguish between the SOC and any other possible models of


Fig. 6. A segment of the wavelet !ltered version of the original signal. Same period and vertical-axis range were used as on Fig. 1.

the energy release processes. It was shown that besides thepower law distributions of energies and the durations of solar7ares, the distribution of lengths of inter-burst intervals alsoexhibit scale-free behaviour referring to a certain correla-tion among the individual events. The authors argue that this!nding contradicts the classical mechanism of SOC, wherethe successive events must be totally uncorrelated with eachother thereby involving exponential (e.g. Poissonian) wait-ing time statistics (Wheatland et al., 1998). As an alternativeto the SOC mechanism, Bo?etta et al. (1999) conjecturedthat dissipative events in turbulence can describe the dynam-ics of the process underlying solar 7ares. They showed thatthe statistics of total energy, duration and peak intensity ofsolar 7ares, as well as the statistics of inter-7are durations,can be correctly described by the shell model of MHD tur-bulence.The statistical distributions of waiting times between co-

herent intermittent events of geomagnetic processes resolvedinto di?erent scales is analogous with the observations ob-tained for the solar 7are dynamics (Fig. 8a, b). The distri-butions were plotted in log–log scale for the temporal scalerange of 1–512 min. In any distribution, one unit on the hor-izontal axis is equal to the length of the investigated scale.As in the released energy case, the scale-free behaviour ofthe statistics is represented by the similar scaling exponents(see the insets of Fig. 8a, b) of the !tted power law curves.Accepting the argumentation of Bo?etta et al. (1999),

the classical SOC mechanism seems to be unrealistic as ameans of describing the dynamics of the magnetosphere onthe basis of the outlined waiting time statistics. On the otherhand, the previously investigated PDF-s of the di?erencedtime-series demonstrated that the geomagnetic !eld is af-fected basically by at least two major physical activities.Each activity can result in extreme events having di?erentenergy content, spectral behaviour, statistical parameters,etc. Thus, before deciding to reject the SOC model for themagnetosphere as a whole, an attempt has to be made to sep-arate these events on the basis of a speci!c type of behaviour.

This kind of distinction may also be of extreme importanceregarding a recent sandpile model developed by Chapman etal. (1998) that was able to demonstrate that small internal,and big system-wide magnetosphere processes can behavedi?erently in a dynamical sense showing SOC and non-SOCfeatures in the former and latter cases, respectively. It is ev-ident that a given value of C in our !lter condition (Eq. (8))corresponds to an arti!cial energy limit that discriminatesbetween quiet and disturbed events of the geomagnetic his-tory. For this reason, in the framework of the current !l-ter process we have the possibility to eliminate events ofsuccessively lowering energies by the gradual decreasing ofthis factor. The dependence of the statistical results on theapplied energy limit were thus investigated using three sup-plementary condition factors (C =1:1; 5; 10). The results aresummarized in Fig. 9 where the distributions of the ener-gies released and the waiting times are plotted for the previ-ously applied scales as functions of the condition factors. Itturns out that apart from the reliability of the results, noneof the major statistical behaviours deviates to any great ex-tent from that obtained for the previously presented C =2case. It means that the scale-free character of the distribu-tions of the laminar times and released energies seem to berobust in terms of the applied energy limit. We concludethen that the present analysis has not revealed any statisti-cal di?erences between geomagnetic processes of di?erentenergies.

5. Summary—conclusions

During the course of our analysis we supposed that themagnetosphere behaves as an ideal magneto7uid in whichcoherent magnetic structures can arise in a wide range ofspatial scales. Furthermore, we suggested that the tempo-ral variation of the geomagnetic !eld is representative ofthe spatial structure of the investigated system. We showedthat the geomagnetic !eld is highly intermittent in the sense


Fig. 7. Distributions of released signal energies computed from the wavelet coeQcients of bursty signal periods at the temporal scales shown in theinset. The bursty periods were identi!ed by wavelet !ltering (Eq. (8)) with C =2 setting. The scaling exponents, b of the !tted P(x)=Ax−b power lawcurves are also shown in the inset.

Fig. 8. Distributions of laminar times between intermittent events identi!ed by !ltered wavelet coeQcients (see text) at temporal scales presented in theinsets of graphs (a) and (b). The wavelet !ltering (Eq. (8)) was accomplished with C =2 setting. The scaling exponents, b of the !tted P(x)=Ax−b powerlaw curves are shown in the insets. The units on the horizontal axis correspond to the length of the investigated temporal scale for each represented curve.

that the 7atness of its high-pass !ltered time-series increaseswithout bounds as the border frequency of the !lter increases(Frisch, 1995). In analogy with turbulent studies we pro-

posed an inertial range of energy 7ow of the system by utiliz-ing the structure function analysis between 16 and 724 min.Inertial range cascade processes and intermittency of the


Fig. 9. Distributions of laminar times (left) between and released signal energies (right) of bursty events identi!ed by wavelet !lterings with theapplications of three di?erent condition factors (C =1:1 (top panels), C =5 (middle panels), and C =10 (bottom panels)). The temporal scales and thesymbols used are the same as in Figs. 6 and 7a for the released energy and laminar time statistics, respectively.

signal were assigned to turbulent phenomenology that iscommon in ordinary hydro7uid systems. It was emphasizedthat the wavelet approach is an excellent tool for analysing7uid or magneto7uid turbulence as a consequence of its abil-ity to resolve the given signals in terms of spatial or tempo-ral scales. Using the wavelet technique and a special !ltercriterion, the coherent structures—regarded as the ingredi-ents of the supposed cascade processes—can be identi!ed,and eliminated from the analysed signal.We stressed that the scale-free statistical distributions of

released energies and laminar times related to intermittentevents are in contradiction with each other if we want tounderstand the dynamics of the investigated system in thelight of the classical (true) SOC mechanism. Bo?etta et al.(1999), and Freeman et al. (2000b) obtained similar statis-tical !ndings when investigating solar 7are and solar winddynamics, respectively, but their conjectures were di?erent.Bo?etta et al. (1999) argued that the time correlation be-tween 7ares evidenced by the power law waiting time statis-tics excludes the possibility of the existence of the SOCstate so they proposed the chaotic shell model of turbu-lence as a means of describing the solar 7are dynamics.Freeman et al. (2000b), however, conjecture that the scalefree laminar time distribution can indeed co-exist with asystem-state that they called ‘near-SOC’ referring to thelack of real criticality. They suggest then that an apparentpower law waiting time statistics is not informative enoughto distinguish between the turbulence model proposed byBo?etta et al. (1999) and the ‘near-SOC’ description of a

given system. The introduction of the ‘near-SOC’ terminol-ogy was motivated by the observation that several systemsthat were originally believed to display true SOC behaviourwere later con!rmed to reach this state only at a certainpoint of the parameter space; in other words tuning was nec-essary. For instance, it was recently shown (Kinouchi andPrado, 1999) that the extremal Feder and Feder and Olami–Feder–Christensen SOC (Jensen, 1998) models show thephenomenology of self-organized criticality only in the con-servative regime, where the coupling coeQcient between thesupercritical site and its nearest-neighbours is tuned exactlyto 0.25 (all the energy subtracted from the supercritical siteshould be evenly distributed to its nearest neighbours). Onthe other hand, if one introduces random parameters into themodels by inserting an explicit noise term into the drivingalgorithm in the EFF model case or inhomogeneities throughthe open boundaries in the OFC model case, the e?ect is toenlarge the interval of the coupling coeQcient by which thesystem’s behaviour is ‘almost critical’. The ‘almost critical’state means that the dimensions of the scaling regions of theavalanche distributions depend not only on the system sizebut also on the level of dissipation.For clarity, it should be noted that the identi!cation of

the necessary and unique conditions for self-organizationand criticality is still the subject of debate in the current lit-erature. For instance, Bo?etta’s de!nition for criticality ornon-criticality can be questioned by a paper of Paczuski et al.(1996) who gave implications for temporal correlation apartfrom the evidence for criticality in the case of a certain class


of SOCmodels. It is also not evident that the dissipative OFCmodel is only ‘almost’ and not really critical (see e.g. Liseand Paczuski, 2000) and thus the ‘near-SOC’ terminology isvalid. Due to the uncertainties that have been highlighted wedo not consider the experimental results presented in this pa-per satisfactory to make an unambiguous decision whetherthe dynamics of the magnetosphere and ionosphere can beillustrated by the SOC or near-SOC mechanisms, or by anyother (e.g. shell) model of turbulence. There is no doubt,because of the obvious evidence for self-organization to acritical state in the cases of various theoretical models andnatural systems, it is tempting to believe that the apparentscale-free energy distributions of avalanche like processesin the magnetosphere are informative enough to explain thewhole investigated dynamics in the context of SOC termi-nology. On the other hand, the current analysis proves alsothat the application of turbulence methodology to geomag-netic time-series provides results that can relate the magne-tosphere dynamics to a very speci!c model of turbulence,i.e. the shell model, too, even if the real conditions of turbu-lent behaviour (e.g. continuously streaming, homogeneous,undispersive medium) are not fully satis!ed in the formercase.Of course, we are aware that the former !nding does not

mean a clear evidence to the turbulent nature of the magne-tosphere dynamics. It should also be considered that thereare a lot of implications that the magnetosphere is in strongdynamical interaction with a real turbulent system—the so-lar wind (Tsurutani et al., 1990; V+or+os et al., 1998; Freemanet al., 2000a). The questions arise then; to what degree maythe recent results be determined by the dynamics of the solarwind and the magnetosphere itself, or consequently, whichparts of the obtained results can be regarded as the speci!ccharacteristics of the investigated system, i.e. the magne-tosphere? Another question, viz. the origin of the bimodalcharacter of the obtained PDF-s of di?erenced time-seriesat high temporal scales, has remained unresolved in thispaper. Using some prede!ned energy limits in the !ltercondition (Eq. (8)) an e?ort has been made to investigateseparately the di?erent geomagnetic activities on the basisof their characteristic energies. Relevant di?erences in theinvestigated statistical parameters of the energetically sortedgeomagnetic events, however, could not be observed in thisway. It is emphasized that the fundamental purpose of thispaper was not to identify the origin of activities a?ecting thegeomagnetic !eld. In the following, however, it is planned toinvestigate this problem using not only the energetic but alsothe higher-order statistical properties of the various physi-cal processes. (Some preliminary results on this topic havealready been published by V+or+os and Kov$acs (2001).) Fur-thermore, our opinion is that a straightforward analysis isalso needed to determine the e?ects of the dynamics of thesolar wind on the results obtained using in-situ measuredmagnetosphere and solar wind data, and the geomagneticrecords of observatories exposed to solar wind in7uence ofdi?erent strengths along di?erent geomagnetic latitudes.

Acknowledgements

We thank Nicholas Watkins for many valuable sugges-tions. P. Kov$acs wishes to acknowledge the kind hospitalityof the sta? of the Physics Department of Universit$a dellaCalabria of Italy during his stay at their university. The anal-ysed data were obtained from the database of the Intermag-net observatory network. The work was supported by theHungarian Science Research Fund (OTKA) under projectnumber F030331 and by the E+otv+os Scholarship provided bythe Hungarian Scholarship Committee. Z. V+or+os was sup-ported by VEGA grant 2=6040=99.

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Wavelet-based filtering of intermittent events from geomagnetic time-series