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Physica A 295 (2001) 215218

www.elsevier.com/locate/physa

Multiscale analysis from turbulent time serieswith wavelet transform

C. Rodrigues Neto; 1 , A. Zanandrea1 , F.M. Ramos1 , R.R. Rosa1 ,

M.J.A. Bolzan1 , L.D.A. Sa1

Instituto Nacional de Pesquisas Espaciais (INPE) Sao Jose dos Campos, Brazil

Abstract

We present a multiscale signal analysis based on the multifractal spectrum obtained by the

Wavelet Transform Modulus Maxima technique. We analyze time series from turbulent data:

the rst step is to obtain the PDF of the utuations for velocities records and then to t them

by means of the Tsallis generalized thermodynamics (Tsallis, J. Stat. Phys. 52 (1998) 479) the

second step is to obtain the multifractal spectra of the time series by the wavelet transform

(Muzy et al., Phys. Rev. Lett. 67 (1991) 3515). The aim of this approach was to investigate apossible phenomenological connection between the entropic parameter (q) and the multifractal

spectrum for turbulence. c 2001 Elsevier Science B.V. All rights reserved.

PACS: 02.70.Bf; 47.20.Ky; 67.55.Fa; 05.70.Ln; 05.60.Cd

Keywords: Turbulence; Non-extensive statistics; Wavelets; Multifractal formalism

In the last years, the scaling and non-extensivity behaviors have been found in several

kinds of phenomena, like those observed in nancial [1], biological [2] and physicalsystems [3]. These systems cover a wide range of scales, and it has showed that the

scaling properties of the probability distribution function moments of the observed time

series are an important aspect, from what it is possible to determine the fractality or

the multifractality [4] of the attractor support of the signal recurrent plots. Beyond

the dimension of the support, the time series singularities are also very important

for the comprehension of the system, and one of the best way to access both, the

fractal dimensions and the singularities, is the multifractal spectrum [5]. Despite the

diverse nature of the underlying generating system, one of the best characteristics of

Corresponding author: Rua 4, 2789, 13.504-092 Rio Claro, SP, Brazil. Fax: +55-12-3456375.

E-mail address: camilo neto@yahoo.com (C. Rodrigues Neto).1 The authors CRN, AZ, FMR and RRR are supported by FAPESP. FMR, MJAB and LDAS are also

supported by CNPq.

0378-4371/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 0 7 6 - 0

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this approach is a unied view of the dynamical systems as a binomial multiplicative

process [6] or, in a more general way, as a multifractal process [7].

Nowadays, the turbulence problem is one of the most important in classical physics

and has studied in the past years for several authors [8]. In the present work, we con-centrated ourself in the atmospheric surface layer velocity experiments [9]. Following

our previous article [10], we started by performing a non-extensive statistics analysis

[11] at several scales and, from which time series, we got the multifractal spectrum.

The main question that we addressed here was to investigate how the fractality changes

with the observation scale of the problem and, how the multifractal spectrum can be

related with the Tsallis q parameter.

The aim of the multifractal formalism was to determine the f() singularity spec-

trum of the velocity time series. Here, is the Holder coecient and f() is the fractal

dimension of the time series support with singularity . To understand the meaning ofthe multifractal spectrum, let us introduce the Hurst coecient. The Hurst coecient

H is used in general as a measure of the persistence of a time series: 06H 0:5 indi-

cates an antipersistent time series while 0:5 H61:0, an persistent one. Antipersistent

time series reverses itself more frequently and covers less distance than a random walk

while persistent time series has long memory eects. The Hurst coecient is a statisti-

cal measure of the time series, being calculated over all the individual, time localized,

Holder coecient . If the time series is homogeneous, = H all over; on the other

hand, = H in general. A very ecient way to obtain the multifractal spectrum is uti-

lizing the Wavelet Transform Modulus Maxima technique described by Arneodo [12].We utilized this technique with the Morlet wavelet, that has zero moments until the

second order. In this article, the wavelet transform was used only as a tool to obtain

the multifractal spectra.

In the Fig. 1, the rst 4096 time steps of the original signal are shown in the left and,

in the right, we see its multifractal spectrum. The Holder coecient at the maximum

of the multifractal spectrum of the Fig. 1 determines the general behavior of the time

series, like an overall Hurst coecient. It is compatible with the multifractal spectum

showed, e.g., in Feder [5]. In the Fig. 2 we see the time series sampled at several

scales in the left and, in the right, the non-extensive statistics at every scale: basically,it shows that at the small scales the turbulence is non-extensive (see [10] for details).

The multifractal spectra in the Fig. 3 show that center of the spectum displaces

from the right to the left as we change the observation scale: the smaller the scales,

the greater the displacement. But the shape of the spectrum does not changes very

much! To understand this, we remind that the dierentiation of a signal reduces by

one the order of Holder coecient. Here, we congecture that the displacement of the

spectrum center is a mere consequence of the dierentiation like operation that the

dierent scale sampling indeed is. If this is the case, then the fractality character of

the turbulence as can be gured out by the f() spectrumdoes not change fromscale to scale.

Turbulence has challenged generations of scientists. The recent techniques of mul-

tifractal dimension spectrum obtained by the modulus maxima wavelet transform [12]

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C. Rodrigues Neto et al. / Physica A 295 (2001) 215218 217

Fig. 1. Left: First 4096 time steps from turbulent time series of the Amazonian surface boundary layer [9].

The time interval between consecutive time steps is 1=60 s; right: multifractal spectrum of the left time

series.

Fig. 2. Left: First 4096 time steps from time series at dierent scales. The time series are constructed taking

the velocity dierence at each time steps. From bottom to top: = 1, = 10, = 100, = 500 and

= 5000; right: dierent statistics for the dierent observation scales [10] of the left time series.

had shed some light to this dicult problem. This article was an attempt to provide

an interconnection between the Tsallis generalized statistics [11] and the multifractalspectrum [5] obtained nowadays by usual wavelet transform. Our main conclusion is

that, although the non-extensive statistic changes from scale to scaleas can be seen

by the dierent curves in the left of the Fig. 2, the fractality character of the time series

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Fig. 3. Multifractal spectra of the left time series of the Fig. 2.

at each scale does not changeas can be seen by the very similar multifractal spectra

in the Fig. 3. Many others aspects remain to be studied, e.g. how the q parameter isrelated with the persistence statistics of the multifractal spectra f().

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