Abstract

Several attempts have been made recently to generalize the multifractal formalism, originally introduced for singular measures, to fractal signals. We report on a systematic comparison between the structure-function approach, pioneered by Parisi and Frisch [in 2 Proceedings of the International School on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, edited by M. Ghil, R. Benzi, and G. Parisi (North-Holland, Amsterdam, 1985), p. 84] to account for the multifractal nature of fully developed turbulent signals, and an alternative method we have developed within the framework of the wavelet-transform analysis. We comment on the intrinsic limitations of the structure-function approach; this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases. We demonstrate that our method, based on the wavelet-transform modulus-maxima representation, works in most situations and is likely to be the ground of a unified multifractal description of self-affine distributions. Our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations.

Highlights

  • The multifractal formalism [1,2,3,4,5,6,7,8,9] has been established to account for the statistical scaling properties of singular measures arising in various physical situations [10,11,12,13,14,15,16,17,18]

  • We have proposed an alternative method, based on wavelettransform modulus-maxima tracking, that gives direct access to the D (h) singularity spectrum of any fractal signal

  • From the direct comparison of the wavelet-transform modulus-maxima (WTMM) method with the SF method for specific examples, we have demonstrated that the former does not introduce any bias in the estimate of the scaling exponents of some partition functions which are at the heart of these statistical analyses of multifractal signals

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Summary

INTRODUCTION

The multifractal formalism [1,2,3,4,5,6,7,8,9] has been established to account for the statistical scaling properties of singular measures arising in various physical situations [10,11,12,13,14,15,16,17,18]. Notable examples include the invariant probability distribution on a strange attractor, the distribution of voltage drop across a random resistor network, the distribution of growth probabilities on the interface of a diffusionlimited aggregate, and the dissipation field in fully developed turbulent flows This formalism lies upon the determination of the f (a) singularity spectrum [1] which associates the Hausdorff dimension f (a) to the subset of the support of the measure JL where the singularity strength is a: j(a)=dimH{xiJL

11. THE WAVELET-TRANSFORM MODULUS-MAXIMA METHOD
LIMITATIONS
SIGNALS NOT SINGULAR EVERYWHERE
CONCLUSION
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