Hard and soft multifractal processes

Schertzer, D.; Lovejoy, S.

doi:10.1016/0378-4371(92)90455-y

Cited by 102 publications

(54 citation statements)

References 19 publications

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“…its objects are mathematical functions defined at mathematical points. On the contrary, the generic multifractal processes -cascades -are singular measures, they are "delocalized", see Schertzer and Lovejoy (1992), Schertzer et al (2002Schertzer et al ( , 2010, so that strictly speaking, they are outside the scope of wavelet analysis. Wavelets may therefore not always be appropriate.…”

Section: Fluctuations and Waveletsmentioning

confidence: 99%

Haar wavelets, fluctuations and structure functions: convenient choices for geophysics

Lovejoy

Schertzer

2012

Nonlin. Processes Geophys.

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Abstract. Geophysical processes are typically variable over huge ranges of space-time scales. This has lead to the development of many techniques for decomposing series and fields into fluctuations v at well-defined scales. Classically, one defines fluctuations as differences:and this is adequate for many applications ( x is the "lag"). However, if over a range one has scaling v ∝ x H , these difference fluctuations are only adequate when 0 < H < 1. Hence, there is the need for other types of fluctuations. In particular, atmospheric processes in the "macroweather" range ≈ 10 days to 10-30 yr generally have −1 < H < 0, so that a definition valid over the range −1 < H < 1 would be very useful for atmospheric applications.A general framework for defining fluctuations is wavelets. However, the generality of wavelets often leads to fairly arbitrary choices of "mother wavelet" and the resulting wavelet coefficients may be difficult to interpret. In this paper we argue that a good choice is provided by the (historically) first wavelet, the Haar wavelet (Haar, 1910), which is easy to interpret and -if needed -to generalize, yet has rarely been used in geophysics. It is also easy to implement numerically: the Haar fluctuation ( v) Haar at lag x is simply equal to the difference of the mean from x to x + x/2 and from x + x/2 to x + x. Indeed, we shall see that the interest of the Haar wavelet is this relation to the integrated process rather than its wavelet nature per se.Using numerical multifractal simulations, we show that it is quite accurate, and we compare and contrast it with another similar technique, detrended fluctuation analysis. We find that, for estimating scaling exponents, the two methods are very similar, yet Haar-based methods have the advantage of being numerically faster, theoretically simpler and physically easier to interpret.

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Section: Fluctuations and Waveletsmentioning

confidence: 99%

Haar wavelets, fluctuations and structure functions: convenient choices for geophysics

Lovejoy

Schertzer

2012

Nonlin. Processes Geophys.

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“…Since the atmospheric turbulence, rainfall, and other geophysical processes interact with each other and the external environment, the microcanonical constraint appears highly artificial and such models should be considered essentially academic. The most important consequence of the microcanonical constraint is that the model necessarily returns upper bounded singularities, limiting the occurrence of extreme realizations (e.g., Schertzer et al, 1991;Schertzer and Lovejoy, 1992;Lovejoy and Schertzer, 1995).…”

Section: Introductionmentioning

confidence: 99%

Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models

Serinaldi

2010

Nonlin. Processes Geophys.

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Abstract. Discrete multiplicative random cascade (MRC) models were extensively studied and applied to disaggregate rainfall data, thanks to their formal simplicity and the small number of involved parameters. Focusing on temporal disaggregation, the rationale of these models is based on multiplying the value assumed by a physical attribute (e.g., rainfall intensity) at a given time scale L, by a suitable number b of random weights, to obtain b attribute values corresponding to statistically plausible observations at a smaller L/b time resolution. In the original formulation of the MRC models, the random weights were assumed to be independent and identically distributed. However, for several studies this hypothesis did not appear to be realistic for the observed rainfall series as the distribution of the weights was shown to depend on the space-time scale and rainfall intensity. Since these findings contrast with the scale invariance assumption behind the MRC models and impact on the applicability of these models, it is worth studying their nature. This study explores the possible presence of dependence of the parameters of two discrete MRC models on rainfall intensity and time scale, by analyzing point rainfall series with 5-min time resolution. Taking into account a discrete microcanonical (MC) model based on beta distribution and a discrete canonical beta-logstable (BLS), the analysis points out that the relations between the parameters and rainfall intensity across the time scales are detectable and can be modeled by a set of simple functions accounting for the parameterrainfall intensity relationship, and another set describing the link between the parameters and the time scale. Therefore, MC and BLS models were modified to explicitly account for these relationships and compared with the continuous in Correspondence to: F. Serinaldi (f.serinaldi@unitus.it) scale universal multifractal (CUM) model, which is used as a physically based benchmark model. Monte Carlo simulations point out that the dependence of MC and BLS parameters on rainfall intensity and cascade scales can be recognized also in CUM series, meaning that these relations cannot be considered as a definitive sign of departure from multifractality. Even though the modified MC model is not properly a scaling model (parameters depend on rainfall intensity and scale), it reproduces the empirical traces of the moments and moment exponent function as effective as the CUM model. Moreover, the MC model is able to reproduce some rainfall properties of hydrological interest, such as the distribution of event rainfall amount, wet/dry spell length, and the autocorrelation function, better than its competitors owing to its strong, albeit unrealistic, conservative nature. Therefore, even though the CUM model represents the most parsimonious and the only physically/theoretically consistent model, results provided by MC model motivate, to some extent, the interest recognized in the literature for this type of discrete models.

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“…On peut montrer (SCHERTZER et LOVEJOY, 1993) qu'il peut exister une valeur critique de l'ordre des moments q, q = qD, telle que les moments statistiques divergent dès que q > qD Un ordre de singularité critique y D correspond à l'ordre critique qD relatif aux moments et on démontre que pour les valeurs de l'ordre de singularité y supé-rieures à YD» la fonction de codimension C(y) devient linéaire, sa pente étant égale à qD Des analogies thermodynamiques ont conduit à qualifier de « transition de phase » le passage au-delà de l'ordre de singularité critique y D (SCHERTZER et LOVEJOY, 1992).…”

Section: Approche Multifractaleunclassified

À propos de la distribution statistique des cumuls pluviométriques annuels. Faut-il en finir avec la normalité?

Benjoudi¹,

Hubert²

2005

rseau

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Il est communément admis que la distribution statistique des précipitations cumulées annuelles suit une loi de Laplace-Gauss. Les écarts entre cette loi et les distributions empiriques sont cependant un fait d'expérience : au-delà d'une probabilité au non dépassement correspondant à une période de retour d'une vingtaine d'années et pour les valeurs les plus fortes de pluie, l'ajustement n'est plus acceptable. Ce décrochage par rapport à la loi normale est mieux mis en évidence par l'étude des longues séries pluviométriques, plus riches en événements extrêmes. Pour étudier le comportement statistique de ces derniers, il est fait appel à un formalisme multifractal qui permet de mettre en évidence que, contrairement à ce qui est généralement admis, la décroissance de la probabilité au dépassement est de nature hyperbolique plutôt qu'exponentielle. Les probabilités des événements catastrophiques sont donc plus importantes que l'on ne le croyait jusqu'ici, ce qui peut avoir des conséquences particulièrement importantes. Cette approche appliquée à un ensemble de séries pluviométriques de longue durée permet de cerner le paramètre caractérisant la décroissance de la probabilité au dépassement. Les résultats obtenus jusqu'ici laissent à penser que ce paramètre pourrait être universel.Up to now, annual rainfall accumulation have generally been modelled according to the Laplace-Gauss probability distribution. After a brief survey of the arguments for using this distribution to describe annual rainfall, which are mainly to consider annual rainfall accumulation as the sum of many independent individual rainfall events of similar magnitude identically distributed, we question its capacity to take into account the various characteristics of the rainfall events, in particular their magnitude, their number and their possible correlation or persistence. We have studied an alternative model based on the multifractal theory, well suited to model phenomena where matter and/or energy concentrate on a more and more sparse domain as the observation scale is decreasing. Some elements of the multifractal theory are briefly described. The main feature of the probabilistic model based on the assumption of a multifractal behavior is that for large enough accumulations, the probability distribution tail would have at any time scale an algebraic rather than an exponential behavior as it is the case for the Laplace-Gauss distribution. It is of value to note that such an algebraic behavior corresponds to a probability of occurrence decreasing much more slowly than in the case of an exponential one. It is also important to note that, unlike exponential distribution laws, all statistical moments of algebraic distribution laws are not defined, those of order greater than the exponent of the algebraic law diverging. This fact is of main importance in relation with sampling, as we cannot a priori be sure of the convergence of all statistical estimators and that the convergence of these estimators is likely to much more slow than that of exponential ...

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Hard and soft multifractal processes

Cited by 102 publications

References 19 publications

Haar wavelets, fluctuations and structure functions: convenient choices for geophysics

Haar wavelets, fluctuations and structure functions: convenient choices for geophysics

Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models

À propos de la distribution statistique des cumuls pluviométriques annuels. Faut-il en finir avec la normalité?

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