ORIGINAL PAPERA multifractal formalism for vector-valued random fieldsbased on wavelet analysis: application to turbulent velocityand vorticity 3D numerical dataPierre Kestener Alain ArneodoPublished online: 24 April 2007?Springer-Verlag 2007AbstractExtreme atmospheric events are intimately re-lated to the statistics of atmospheric turbulent velocities.These, in turn, exhibit multifractal scaling, which isdetermining the nature of the asymptotic behavior ofvelocities, and whose parameter evaluation is therefore ofgreat interest currently. We combine singular valuedecomposition techniques and wavelet transform analysisto generalize the multifractal formalism to vector-valuedrandom fields. The so-called Tensorial Wavelet TransformModulus Maxima (TWTMM) method is calibrated onsynthetic self-similar 2D vector-valued multifractal mea-suresandmonofractal3Dvector-valuedfractionalBrownian fields. We report the results of some applicationof the TWTMM method to turbulent velocity and vorticityfields generated by direct numerical simulations of theincompressible NavierStokes equations. This study re-vealstheexistenceofanintimaterelationshipDvh 1 Dxh; between the singularity spectra ofthese two vector fields which are found significantly moreintermittent than previously estimated from longitudinaland transverse velocity increment statistics.1 IntroductionSeveral studies over the last decade have shown thatatmospheric extremes are subject to multifractal scaling(Hubert et al. 1993; Burlando and Rosso 1996; Bendjoudiet al. 1997; Veneziano and Furcolo 2002; De Michele et al.2002; Castro et al. 2004), a finding that is heuristicallysupported by the multifractality found in chaotic dynamicalsystems in general (Halsey et al. 1986; Collet et al. 1987;Rand1989),andinturbulentflowsinparticular(Mandelbrot 1974, 1989; Paladin and Vulpiani 1987;Meneveau and Sreenivasan 1991). Estimating the asymp-totic behavior resulting from the observed scaling lawsrelies in a fundamental manner on the correctness of theparameterization of the multifractal singularity spectra inthe domain where statistics are robust. Box-counting andcorrelation algorithms were successfully adapted to resolvemultifractal scaling for isotropic self-similar fractals bycomputation of the generalized fractal dimensions Dq(Grassberger and Procaccia 1983; Badii and Politi 1984;Grassberger et al. 1988; Argoul et al. 1990; Meisel et al.1992). As to self-affine fractals (Mandelbrot 1977; Peitgenet al. 1987), Parisi and Frisch (1985) proposed, in thecontext of the analysis of fully developed turbulencevelocity data, an alternative multifractal description basedon the investigation of the scaling behavior of the so-calledstructure functions (Monin and Yaglom 1975; Frisch1995): Spl dvlp ? lfp(p integer 0), wheredvl(x) = v(x + l) v(x) is an increment of the recordedlongitudinal velocity component over a distance l. Then,P. KestenerCEA-Saclay, DSM/DAPNIA/SEDI,91191 Gif-sur-Yvette, FranceP. Kestener ? A. Arneodo (&)Laboratoire de Physique (UMR 5672),Ecole Normale Supe rieure de Lyon,46 alle e dItalie, 69364 Lyon ce dex 07, Francee-mail: alain.arneodoens-lyon.frA. ArneodoLaboratoire Transdisciplinaire Joliot Curie,Ecole Normale Supe rieure de Lyon,46 alle e dItalie, 69364 Lyon ce dex 07, France123Stoch Environ Res Risk Assess (2008) 22:421435DOI 10.1007/s00477-007-0121-6from the local scaling behavior of the velocity increments,dvl(x) lh(x), the D(h) singularity spectrum is defined as theHausdorff dimension of the set of points x where the localroughness (or Ho lder) exponent h(x) of v is h (Parisi andFrisch 1985; Baraba si and Vicsek 1991; Baraba si et al.1991; Muzy et al. 1991; Arneodo et al. 1995a). In princi-ple, D(h) can be attained by Legendre transforming thestructure function scaling exponents fp(Parisi and Frisch1985; Muzy et al. 1991; Arneodo et al. 1995a). Unfortu-nately, as noticed by Muzy et al (1993), there are somefundamental drawbacks to the structure function method.Indeed, it generally fails to fully characterize the D(h)singularity spectrum since only the strongest singularitiesof the function v itself (and not the singularities present inthe derivatives of v) are a priori amenable to this analysis.Even though one can extend this study from integer to realpositive p values by considering the increment absolutevalue |dvl|, the structure functions generally do not exist forp 1, as well as reg-ular behavior, bias the estimate of fp(Muzy et al. 1991,1993; Arneodo et al. 1995a).In a previous work, Arneodo and collaborators (Muzyet al. 1991, 1993; Arneodo et al. 1995a) have shown thatthere exists a natural way of performing a unified multi-fractal analysis of both singular measures and multi-affinefunctions, which consists in using the continuous wavelettransform (WT) (Goupillaud et al. 1984; Grossmann andMorlet 1984; Meyer 1990; Daubechies 1992; Mallat 1998).By using wavelets instead of boxes, one can take advantageof freedom in the choice of these generalized osci