Comprehensive multifractal analysis of turbulent velocity using wavelet leaders

International audienceThe multifractal (MF) framework relates the scaling properties of turbulence to its local regu- larity properties through a statistical description as a collection of local singularities. The MF properties are moreover linked to the multiplicative cascade process that creates the peculiar properties of turbulence such as intermittency. A comprehensive estimation of the MF properties of turbulence from data analysis, using a tool valid for all kind of singularities (including oscillating singularities) and mathematically well- founded, is thus of first importance in order to extract a reliable information on the underlying physical processes. The MF formalism based on the wavelet leaders (WL) is a new MF formalism which is the first to meet all these requests. This paper aims at its description and at its application to experimental turbulent velocity data. After a detailed discussion of the practical use of the MF formalism based on the WL the following questions are carefully investigated: (1) What is the dependence of MF properties on the Reynolds number? (2) Are oscillating singularities present in turbulent velocity data? (3) Which MF model does correctly account for the observed MF properties? Results from several data set analyses are used to discuss the dependence of the computed MF properties on the Reynolds number but also to assess their common or universal component. An exact though partial answer (no oscillating singularities are detected) to the issue of the presence of oscillating singularities is provided for the first time. Eventually an accurate parameterization with cumulants exponents up to order 4 confirms that the log-normal model (with c2 = −0.025 ± 0.002) correctly accounts for the universal MF properties of turbulent velocity

Comprehensive Multifractal Analysis of Turbulent Velocity Using the Wavelet Leaders

The multifractal framework relates the scaling properties of turbulence to its local regularity properties through a statistical description as a collection of local singularities. The multifractal properties are moreover linked to the multiplicative cascade process that creates the peculiar properties of turbulence such as intermittency. A comprehensive estimation of the multifractal properties of turbulence from data analysis, using a tool valid for all kind of singularities (including oscillating singularities) and mathematically well-founded, is thus of first importance in order to extract a reliable information on the underlying physical processes. The wavelet leaders yield a new multifractal formalism which meets all these requests. This paper aims at describing it and at applying it to experimental turbulent velocity data. After a detailed discussion of the practical use of the wavelet leader based multifractal formalism, the following questions are carefully investigated: (1) What is the dependence of multifractal properties on the Reynolds number? (2) Are oscillating singularities present in turbulent velocity data? (3) Which multifractal model does correctly account for the observed multifractal properties? Results from several data set analysis are used to discuss the dependence of the computed multifractal properties on the Reynolds number but also to assess their common or universal component. An exact though partial answer (no oscillating singularities are detected) to the issue of the presence of oscillating singularities is provided for the first time. Eventually an accurate parameterization with cumulants exponents up to order 4 confirms that the log-normal model (with $c_2 = -0.025\pm 0.002$) correctly accounts for the universal multifractal properties of turbulent velocity

De l'estimation des exposants des lois d'échelle

- Nous nous intéressons dans cet article à l'estimation multirésolution (boîte, accroissement, ondelette) des exposants ζ(q) des lois d'échelle de processus multiplicatifs. Nous observons, à partir de quatre types de processus présentant des lois d'échelle prescrites a priori, que ces estimateurs ne rendent compte des ζ(q) que dans un intervalle de valeurs de q et se comportent nécessairement comme une fonction linéaire de q en dehors. Nous étudions cet effet et le relions au fait que les cascades multiplicatives sont des martingales

A Unified Formalism for Physical Attacks

Technical reportThe security of cryptographic algorithms can be considered in two contexts. On the one hand, these algorithms can be proven secure mathematically. On the other hand, physical attacks can weaken the implementation of an algorithm yet proven secure. Under the common name of physical attacks, different attacks are regrouped: side channel attacks and fault injection attacks. This paper presents a common formalism for these attacks and highlights their underlying principles. All physical attacks on symmetric algorithms can be described with a 3-step process. Moreover it is possible to compare different physical attacks, by separating the theoretical attack path and the experimental parts of the attacks

Multi-scale digital soil mapping with deep learning

We compared different methods of multi-scale terrain feature construction and their relative effectiveness for digital soil mapping with a Deep Learning algorithm. The most common approach for multi-scale feature construction in DSM is to filter terrain attributes based on different neighborhood sizes, however results can be difficult to interpret because the approach is affected by outliers. Alternatively, one can derive the terrain attributes on decomposed elevation data, but the resulting maps can have artefacts rendering the approach undesirable. Here, we introduce ‘mixed scaling’ a new method that overcomes these issues and preserves the landscape features that are identifiable at different scales. The new method also extends the Gaussian pyramid by introducing additional intermediate scales. This minimizes the risk that the scales that are important for soil formation are not available in the model. In our extended implementation of the Gaussian pyramid, we tested four intermediate scales between any two consecutive octaves of the Gaussian pyramid and modelled the data with Deep Learning and Random Forests. We performed the experiments using three different datasets and show that mixed scaling with the extended Gaussian pyramid produced the best performing set of covariates and that modelling with Deep Learning produced the most accurate predictions, which on average were 4–7% more accurate compared to modelling with Random Forests

Analyse multifractale pratique : coefficients dominants et ordres critiques. Applications à la turbulence pleinement développée. Effets de nombre de Reynolds fini

Multifractal description of signals has been developed during the last 20 years, mainly in the fully developed turbulence. Pointwise regularity properties are characterized by the singularity spectrum. Multifractal analysis consists in the singularity spectrum measurement using multifractal formalisms. Simultaneous introduction of wavelet transforms allowed finer practical multifractal analysis, sometimes without receiving well-founded mathematical basis. S. Jaffard recently introduced the wavelet leaders yielding a mathematical well-founded multifractal formalism, that allows the measurement of the whole singularity spectrum and remains valid when performing the analysis of signals with oscillating singularities. This new tool is for the first time implemented, numerically characterized and applied to turbulent velocity data. On the other hand this work rises the important issue of the practical use of multifractal formalisms, that reduce in fine in structure function scaling exponent measurement. Extensive numerical study using reference synthetical multifractal processes illustrates and characterizes the existence of a critical order. An estimator of the critical order is built and numerically characterized. Previous results in turbulence then receive rereading. Finally, the universality of structure function scaling exponents in fully developed turbulence is tackled. Third order structure function scaling exponent modelling is proposed and compared to experimental results, emphasizing its non-universal value.La description multifractale des signaux a été initiée au cours des vingt dernières années, notamment dans le domaine de la turbulence pleinement développée. Les propriétés de régularité ponctuelle des signaux étudiés sont caractérisées à l'aide d'un spectre de singularités. L'analyse multifractale de ces signaux consiste à mesurer ce spectre de singularités, à l'aide de formalismes multifractals. L'apparition des transformées en ondelette, à la même époque, a permis d'affiner la pratique de l'analyse multifractale, sans pour autant toujours reposer sur des bases mathématiques solides. S. Jaffard a récemment introduit les coefficients dominants, qui permettent de construire un formalisme multifractal mathématiquement bien fondé, et au cadre d'application large : il rend possible la mesure de l'ensemble du spectre de singularités, et reste valide lorsque les signaux analysés contiennent des singularités oscillantes. Ce nouvel outil est pour la première fois mis en oeuvre, numériquement caractérisé et appliqué à des signaux de vitesse turbulente. La question du bon usage pratique des formalismes multifractals, qui reposent sur la mesure d'exposants de fonctions de structure, est essentielle. Le travail présenté se propose d'y apporter des éléments de réponse. Une étude numérique, sur un large panel de processus multifractals synthétiques, a permis d'illustrer et de caractériser un aspect essentiel de l'analyse multifractale pratique, l'existence d'un ordre critique. Un estimateur de cet ordre critique est construit et numériquement caractérisé. Une relecture des résultats obtenus en turbulence est alors effectuée. Enfin, la question de l'universalité des exposants des fonctions de structure en turbulence pleinement développée est abordée. Une modélisation de l'exposant de la fonction de structure d'ordre trois est proposée et comparée à des résultats expérimentaux, mettant en évidence le caractère non universel de sa valeur

Analyse multifractale pratique : coefficients dominants et ordres critiques (Applications à la turbulence pleinement développée)

La description multifractale des signaux a été initiée au cours des vingt dernières années, notamment dans le domaine de la turbulence pleinement développée. Les propriétés de régularité ponctuelle des signaux étudiés sont caractérisées à l'aide d'un spectre de singularités. L'analyse multifractale de ces signaux consiste à mesurer ce spectre de singularités, à l'aide de formalismes multifractals. L'apparition des transformées en ondelette, à la même époque, a permis d'affiner la pratique de l'analyse multifractale, sans pour autant toujours reposer sur des bases mathématiques solides. S. Jaffard a récemment introduit les coefficients dominants, qui permettent de construire un formalisme multifractal mathématiquement bien fondé, et au cadre d'application large : il rend possible la mesure de l'ensemble du spectre de singularités, et reste valide lorsque les signaux analysés contiennent des singularités oscillantes. Ce nouvel outil est pour la première fois mis en oeuvre, numériquement caractérisé et appliqué à des signaux de vitesse turbulente. La question du bon usage pratique des formalismes multifractals, qui reposent sur la mesure d'exposants de fonctions de structure, est essentielle. Le travail présenté se propose d'y apporter des éléments de réponse. Une étude numérique, sur un large panel de processus multifractals synthétiques, a permis d'illustrer et de caractériser un aspect essentiel de l'analyse multifractale pratique, l'existence d'un ordre critique. Un estimateur de cet ordre critique est construit et numériquement caractérisé. Une relecture des résultats obtenus en turbulence est alors effectuée. Enfin, la question de l'universalité des exposants des fonctions de structure en turbulence pleinement développée est abordée. Une modélisation de l'exposant de la fonction de structure d'ordre trois est proposée et comparée à des résultats expérimentaux, mettant en évidence le caractère non universel de sa valeur.Multifractal description of signals has been developed during the last 20 years, mainly in the fully developed turbulence. Pointwise regularity properties are characterized by the singularity spectrum. Multifractal analysis consists in the singularity spectrum measurement using multifractal formalisms. Simultaneous introduction of wavelet transforms allowed finer practical multifractal analysis, sometimes without receiving well-founded mathematical basis. S. Jaffard recently introduced the wavelet leaders yielding a mathematical well-founded multifractal formalism, that allows the measurement of the whole singularity spectrum and remains valid when performing the analysis of signals with oscillating singularities. This new tool is for the first time implemented, numerically characterized and applied to turbulent velocity data. On the other hand this work rises the important issue of the practical use of multifractal formalisms, that reduce in fine in structure function scaling exponent measurement. Extensive numerical study using reference synthetical multifractal processes illustrates and characterizes the existence of a critical order. An estimator of the critical order is built and numerically characterized. Previous results in turbulence then receive rereading. Finally, the universality of structure function scaling exponents in fully developed turbulence is tackled. Third order structure function scaling exponent modelling is proposed and compared to experimental results, emphasizing its non-universal value.LYON-ENS Sciences (693872304) / SudocSudocFranceF

Wavelet leaders in multifractal analysis

The properties of several multifractal formalisms based on wavelet coefficients are compared from both mathematical and numerical points of view. When it is based directly on wavelet coefficients, the multifractal formalism is shown to yield, at best, the increasing part of the weak scaling exponent spectrum. The formalism has to be based on new multiresolution quantities, the wavelet leaders, in order to yield the entire and correct spectrum of Hölder singularities. The properties of this new multifractal formalism and of the alternative weak scaling exponent multifractal formalism are investigated. Examples based on known synthetic multifractal processes are illustrating its numerical implementation and abilities

Limitation of scaling exponents estimation in turbulence

A significant characteristic of fully developed turbulence is scale invariance, i.e., in a wide range of scale ratios a, usually known as the inertial range, the moments of order $q > 0$ of the increments of the velocity field $v(x)$ or of the aggregated dissipation field $r (x)$ behave as power laws with respects to scale ratios. A key issue in the analysis of turbulence data lies in accurately and precisely measuring the scaling exponents. This yields two questions: how can efficient estimators for the $ζ(q)$ be defined and what are their statistical performance? While the former question owns classical answers, the latter has been mostly overlooked. In the present work, we address carefully this question