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International Journal of Bifurcation and Chaos, Vol. 21, No. 12 (2011) 3417–3456c© World Scientific Publishing CompanyDOI: 10.1142/S0218127411030647

MULTIFRACTALS, GENERALIZED SCALEINVARIANCE AND COMPLEXITY

IN GEOPHYSICS

DANIEL SCHERTZERLEESU, Ecole des Ponts ParisTech, Universite Paris-Est,6-8 Av. B. Pascal, 77455 Marne-la-Vallee Cedex 2, France

[email protected]

SHAUN LOVEJOYPhysics Department, McGill University, 3600,

University St., Montreal, Que. H3A 2T8, [email protected]

Received March 10, 2011; Revised May 6, 2011

The complexity of geophysics has been extremely stimulating for developing concepts andtechniques to analyze, understand and simulate it. This is particularly true for multifractalsand Generalized Scale Invariance. We review the fundamentals, introduced with the help ofpedagogical examples, then their abstract generalization is considered. This includes the char-acterization of multifractals, cascade models, their universality classes, extremes, as well asthe necessity to broadly generalize the notion of scale to deal with anisotropy, which is ratherubiquitous in geophysics.

Keywords : Multifractals; generalized scale invariance; scaling; geophysics.

1. Introduction

Geophysics, especially atmospheric dynamics overvarious scale ranges (i.e. from micro-turbulenceto climate dynamics), has been an inspiring com-plex system for elaborating innovative concepts andtechniques. For instance, everyone has in mind, theLorenz’s paper of 1963 [Lorenz, 1963] and the cor-responding “butterfly effect” paradigm. However,there have been numerous contributions, whichshowed that the complexity of geophysical phenom-ena could be much more considerable and of asomewhat different nature. This includes a ratherdifferent approach to predictability by Lorenz him-self [Lorenz, 1969], fractals [Mandelbrot, 1983],strange attractors [Nicolis & Nicolis, 1984], self-organized criticality [Bak et al., 1987; Bak & Tang,1989], and stochastic resonance [Nicolis, 1981; Benziet al., 1982; Nicolis, 1982; Benzi, 2010]. This paper

is focused on the fundamentals of multifractals andgeneralized scale invariance, and shows that theyare rather at the crossroads of the aforementionedconcepts. This is illustrated with the help of exam-ples to understand, analyze and simulate nonlin-ear phenomena in geophysics over wide ranges ofscales.

In a very general manner, multifractals arespace or space-time fields — the latter being oftencalled processes — that have structures at allscales. They are, therefore, a broad generalizationof the (geometrical) fractals. Indeed, the set indi-cator function of a fractal set is a field that hasonly two values (= 0 or 1), whereas multifractalfields generally have multiple values, which are veryoften continuous. The property to have structuresat all scales is trivially scale invariant since it doesnot depend on the scale of observation. This shows

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3418 D. Schertzer & S. Lovejoy

that a multifractal field can also be defined as beinginvariant for a given scale transform, which is there-fore a symmetry for this field. One says that it is ascale invariant field or a scaling field for short. Wewill give a very general, precise definition of thisinvariance and the corresponding scale transformthat could be either deterministic or stochastic(e.g. involves only equality in probability distri-bution or other statistical equivalences), isotropicor not.

Due to their definition, multifractals aretherefore not only quite general, but also quitefundamental. Indeed, not only are symmetry princi-ples the building blocks of physics and many otherdisciplines [e.g. Weyl, 1952; Zee, 1986], but scalesymmetry is an element of the extended Galileaninvariance. Unfortunately, attention in mechanics,especially in point mechanics, has been initiallyfocused on the space shifts between two (Galilean)frameworks that differ only by a constant rela-tive velocity or a given rotation that define thepure Galilean group. But with extended bodies,therefore continuous mechanics, it broadened toother transforms such as scale dilations. In par-ticular, [Sedov, 1972] pointed out in the wake ofthe Π-theorem of Buckimgam [Buckingham, 1914,1915; Sonin, 2004] the key role of the latter in fluidmechanics, including for many applications (e.g. toestimate the blast wave of a nuclear explosion).More recently, Speziale [1985] demonstrated theirrelevance in selecting and defining relevant sub-grid models in turbulence. This has been widelyused under the denomination of self-similarity, butwith unnecessary limitations. Indeed, the main goalwas to find a unique scale transform under whichthe nonlinear dynamical equations (in particular,the Navier–Stokes equations associated with others,such as the advection-dissipation equation). Thiswas mainly achieved by adimensionalising theseequations with the help of various so-called char-acteristic quantities, including characteristic spaceand time scales. Multifractals are, in fact, invari-ant for a multiple scale symmetry and thereforecorrespond once again to a broad generalization ofproperties that were previously perceived in a toorestrictive framework. We indeed have to go fromscale analysis, a seminal example being the quasi-geostrophic approximation derivation by Charney[1948], to scaling analysis [Schertzer et al., 2011]that we will briefly discuss in Sec. 8.4.

A not-so-trivial consequence of this multiplic-ity of scale symmetries is that these fields could

be understood as an infinite hierarchy of embed-ded fractals, e.g. those supporting singularitieshigher than a given order. This is the source ofthe terminology, but more importantly of a ratherstraightforward manner to understand intermit-tency: higher and higher levels of “activity” ofthe field are concentrated on smaller and smallerfractions of the space. This provides a ratherstraightforward way to quantify and analyze inter-mittency. This easiness is in sharp contrast withthe mathematical nature of a multifractal field: itis a (mathematical) measure, which is furthermoremulti-singular. The fact that it is a measure impliesthat it is not a pointwise field with a given valueat any given point, but it only yields an averagevalue over any given (small) neighborhood of thispoint or an integration of a suitable set of “testfunctions”. The fact it is multisingular with respectto the Lebesgue measure, i.e. does not admit adensity with respect to the latter, means that itsestimated density at larger and larger resolutiondiverges with various power-laws exponents. Thesesingularity orders are called singularities for short.This can be therefore seen as a very broad gen-eralization of the singular Dirac’s measure (oftenimproperly called function), whose mass is concen-trated on a pointwise “atom” and therefore canbe understood as a divergence of its density thatbecomes infinite at this point, whose dimension iszero. For multifractals, theirs mass is distributed onthe embedded fractal sets mentioned above.

As discussed below, these generalizations, aswell as their necessity, step-by-step became patentwith the help of cascade models in turbulence andstrange attractors. These models were first rathercrude (e.g. discrete in scales) and apparently sim-ple, but yielded nontrivial behaviors which werestep-by-step recognized as escaping from the origi-nal framework of analysis, i.e. fractal analysis. Theirpresent refined offsprings (e.g. continuous in scales)are generic models providing multifractal fieldsand related data analysis techniques. These modelsand multifractals have been used in a wide rangeof scientific disciplines, from ecology to financialphysics, including high energy physics, geophysics,astrophysics, etc. A priori, all domains of non-linear science and complex science are concerned,but geophysics remains a preeminent domain ofapplications.

To cover various aspects of multifractals andgeneralized scale invariance, the paper is organizedas follows:

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Multifractals, Generalized Scale Invariance and Complexity in Geophysics 3419

• Section 2 introduces the key notions of singularmeasures with the help of the pedagogical exam-ple of the rainrate at various resolutions,

• Section 3 introduces the cascade models in aphenomenological manner and two pedagogicalmodels,

• Section 4 introduces the general multifractalframework with the codimension and the scalingmoment functions and their duality via theLegendre transform, as well as a comparisonbetween multifractal formalisms,

• Section 5 goes a bit further in formalism byintroducing cascade generators, characteristicfunctions and multifractal scale symmetry in ageneral, yet precise manner,

• Section 6 is devoted to the question of universal-ity, in particular to drastically reduce the numberof parameters characterizing a multifractal,

• Section 7 pursues, in a given way, this discussionto address the question of models that are con-tinuous in scales,

• Section 8 suggests how to broadly generalizethe previous results with the help of general-ized scales and Lie cascades in the frameworkof Generalized Scale Invariance, as well as theirapplication to differential systems.

• Section 9 is focused on the question of extremes,particularly on heavy tails that are easily gen-erated by multifractals, the relationship withself-organized criticality and related multifractaltransitions.

Although this is already a long paper, it doesnot cover all the aspects of multifractals in geo-physics. For instance, the question of multifractalpredictability [Schertzer & Lovejoy, 2004b] is notdiscussed, but only evoked, in spite of the factthat it corresponds to a broad generalization of theresults of [Lorenz, 1969]. More generally, and for thesame reason, although the concepts are illustratedwith the help of geophysical examples, this paperdoes not pretend to review in an exhaustive man-ner their applications to geophysics.

2. Singularities Everywhere inGeophysics?

To better appreciate the underlying physico-mathematical nature of a multifractal, let us con-sider the precipitation intermittency with the helpof common sense intuition: most of the time it doesnot rain, furthermore, when it does rain its intensitycan be extremely variable. In spite of this intuition,

an adequate mathematical framework had beenelusive for a while and has been seriously elabo-rated only during the last twenty five years. Indeed,the variability of precipitation — which occursover a wide range of (space and time) scales andintensities — is beyond the scope of classical geo-physics. A well-known symptom of this basic prob-lem is that the basic hydro-meteorological quantity,the rain rate r, has a strong scale dependency, sinceit depends on the duration on which it is measured.This is illustrated by Fig. 1(a) that displays (fromtop to bottom) the Nımes time series of rainratesrλ(t) from an hour resolution (λ = 365 × 24) downto a year resolution (λ = 1):

rλ(t) =λ

T

∑i

1[τi,τi+1[(t)R([τi, τi+1[);

τi+1 − τi =T

λ(1)

where R([τi, τi+1[) is the rain accumulation (i.e.the rain received) during the time interval [τi, τi+1[and 1A denotes the indicator function of any setA. The intensity scale strikingly decreases with theresolution, e.g. the maximum decreases from 35 to0.1 mm/h, which gives some credence that the vari-ability decreases with decreasing resolution. Nev-ertheless, as emphasized in [Ladoy et al., 1993b]the variability still exists in the yearly resolution.The classical assumption is that the rain accumu-lation is a regular (mathematical) measure dR(t)with respect to the (Lebesgue) volume measure dt,i.e. dR(t) admits a pointwise density r(t), definedalmost everywhere:

dR(t) = r(t)dt. (2)

This density r(t) is called the rainrate andis considered as the basic quantity of interestin hydrometeorology. However, this assumptionimplies that r(t) would correspond to a scale inde-pendent small scale limit of rλ(t)(λ → ∞), whichis in contradiction with the observed strong scaledependency of rλ(t). On the contrary, as discussedat length in [Schertzer et al., 2010], as well as inSec. 4.1, the assumption that dR(t) is a (multiple)singular measure with respect to the Lebesgue mea-sure implies that rλ(t) exhibits power-law singular-ities at larger and larger resolutions:

rλ(t) ≈ λγ . (3)

Figure 1(b) displays these singularities for therainrate rλ(t) displayed in Fig. 1(a), resolution byresolution. The intensity of these singularities γ’s is

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Fig. 1. Rainfall time series at Nımes (1972–1975) from 1h to 1 year duration: (a) the rainrate rλ(t) [Eq. (1)] exhibits a strongscale dependency, since its maximal value decreases from 35 to 0.1 mm/h from 1h to 1 year duration (the unit of the intensityscale corresponds to 0.1 mm/h); (b) the corresponding singularities γ ≈ logλ(rλ) show on the contrary a remarkable scaleindependency over the same range of durations. Reproduced from [Schertzer et al., 2010].

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Fig. 1. (Continued)

strikingly independent of the scale, contrary to therainrate intensity. One may also not note that thereis not a unique value of γ, i.e. the measure dR(t)is a multiple singular measure, with respect to theLebesgue measure dt (which would yield γ = 0).Therefore — contrary to the usual hypothesis —there is no self-consistent definition of an instanta-neous pointwise rain rate defined as a function r(t).

We can proceed to a similar analysis withrespect to space. This can be achieved with thehelp of statistical moments that should, as discussedin Sec. 4.2, display multiple power-laws generated

by singularities. Figure 2 displays the statisticalmoments of the rain surrogate (radar reflectivity)over nearly four orders of magnitude in scale butcould continue over another two orders of magni-tude to smaller scales as observed with the help ofground radar data [Schertzer & Lovejoy, 1987].

We will show that the stochastic multifractalfields offer a very convenient and operationalframework to handle such stochastic (multi-)singular measures. They indeed allow us tosystematically characterize, model and under-stand extremely variable fields while avoiding the

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Fig. 2. Statistical moments 〈rqλ〉 of order q of statistical moments (q = 0, 0.1, . . . , 2.9, bottom to top) with respect to their

resolution λ in a log–log plot of the rainrate rλ estimated by the radar of the TRMM satellite (≈ 1100 orbits) during a duration∆t and across a horizontal section ∆x∆y (x along the satellite path, y transverse to it), with ∆y = 4.3 km, ∆x = Learth/λ(Learth = 20 000 km). Straight lines correspond to power law fits. Reproduced from [Lovejoy et al., 2008a].

restrictive homogeneity assumptions implicit inthe conventional approaches. While traditionalnumerical approaches are forced to use drasticscale truncations, to transform partial differen-tial equations (PDE’s) into ordinary differentialequations (ODE’s), to make arbitrary regularityassumptions, and to perform ad-hoc and unjusti-fied parameterizations (if only for the nonexplicit“subgrid” scales), these classical manipulations(and mutilations) violate a fundamental symmetry

of nonlinear PDE’s: their scale invariance (seeFig. 4). Even in spite of these (over) simplify-ing assumptions, the consequences of such choicesare ultimately complex and unwieldy numeri-cal codes. Often the relevance of such codes, isquestionable and furthermore as their scales arequite different from those of observations, theyare often only intercompared with other (simi-lar) models. Multifractal analyses of these modelsare therefore rather indispensable to evaluate their

v1,1

v2,1

v3,1 v3,2

v2,2

v3,4v3,3

vi,j

vi+1,2jvi+1,2j-1

Fig. 3. A rather general scheme of a (deterministic) cascade process, which actually corresponds to a simplification of theNavier–Stokes equations, where only direct interactions between eddies and their offsprings are preserved. Reproduced from[Chigirinskaya & Schertzer, 1996].

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Fig. 4. A polemical illustration of the drastic reductionof Partial Differential Equations to Ordinary DifferentialEquations in General Circulation Models. Reproduced from[Schertzer & Lovejoy, 1993].

performance [Stolle et al., 2009; Lovejoy &Schertzer, 2011; Gires et al., 2011]. Further-more, they could be as well indispensable todevelop stochastic parametrizations, whose neces-sity is more and more recognized [Palmer &Williams, 2010], in order to reach the intrinsic pre-dictability limits, which are themselve multifractal[Schertzer & Lovejoy, 2004a].

Concerning multifractal modeling, cascademodels (see Fig. 3 for an illustration) are genericmultifractal processes. Indeed, they are built upwith the help of an elementary scale invariant pro-cess that is hierarchically repeated scale after scalealong a hierarchical scaling tree of smaller andsmaller structures. They therefore generate scal-ing fields and yield a rather straightforward wayto understand that extreme variability over a verylarge range of scales may merely result from therepetition scale after scale of a given elementaryprocess or interactions between neighboring scales.

This phenomenological idea can be traced backto Richardson’s celebrated poem on self-similar cas-cades [Richardson, 1922] describing atmosphericdynamics as a cascade process. However, it requiredsome time and various developments before provid-ing well-defined models [Yaglom, 1966; Mandelbrot,1974; Frisch et al., 1978] and to become rigorouslyconceptualized with the help of fractals [Mandel-brot, 1977, 1983].

Then it evolved (after 1983), into a multifractalapproach. The earliest scale invariant multifractal

models, which we will review, are superficially quitesimple phenomenological “toy models”. Neverthe-less, they yield exotic phenomena (exotic comparedto conventional smooth mathematical descriptionsof the real world. . .) and have highly nontrivial con-sequences. For example, as we will see later, simplecascade models already give rise to a fundamentaldifference between observables and truncated pro-cesses, and such a difference is a general propertyof the wide class of “hard” multifractal processes(which distinguish between “dressed” and “bare”properties respectively) as discussed in Sec. 9.These models produce hierarchies of self-organizedrandom structures.

3. Phenomenology of Cascades

Starting with [Richardson, 1922], the phenomenol-ogy of (scalar) turbulent cascades was first dis-cussed in the context of hydrodynamic turbulencewhere the structures are “eddies”. Since we sim-ply follow how the “activity”, measured by the tur-bulent energy flux to smaller scales in turbulence,becomes more and more inhomogeneous as largestructures break up into smaller and smaller scales,cascades are a very general paradigm.

The key assumption in phenomenological mod-els of turbulence (which became explicit with thepioneering work of [Novikov & Stewart, 1964;Yaglom, 1966; Mandelbrot, 1974] is that succes-sive steps (independently) define the fraction ofthe flux of energy distributed over smaller scales.It should be clear that the small scales cannot beregarded as adding energy; they only modulate theenergy passed down from larger scales. The explicithypothesis is that the fraction of the energy flux(more generally the “activity”) from a parent struc-ture to an offspring will be determined in a scaleinvariant manner.

In the (pedagogical) case of “discrete (in scale)cascade models” “eddies” are defined by the hier-archical and iterative division of a D-dimensionalcube into smaller subcubes, with a constant ratio ofscales λ1 (greater than 1, very often equal to 2, seeFig. 3). More precisely, for each n ∈ N , the initialD-dimensional cube ∆0

0 of size L is divided step bystep into λn

1 smaller disjoint subcubes ∆in (ij = 0,

1, . . . (λn1−1); j = 1, 2, . . . ,D) of size n = L/λn

1 thatfully cover ∆0

0. In other words, the λ1-base coordi-nates of the corner ci = iλ−n

1 of the subcube ∆in

has only n digits. The density of the flux energy εn

at the step n is taken to be strictly homogeneous

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on each “subeddies” of scale n, i.e. εn is a stepfunction:

εn(x) =λn−1∑i=0

εin1

∆in(x) (4)

where 1∆

in

is the indicator function of the subcube

∆in. The energy density εn−1 at step n − 1 will be

multiplicatively distributed to subeddies:

εn(x) = µεn(x)εn−1(x) (5)

with the help of the following multiplicativeincrement decomposed on the same basis:

µεn(x) =λn−1∑i=0

µεin1

∆in(x) (6)

where the multiplicative increment components µεin

are usually assumed to be identically and inde-pendently distributed (i.i.d.), as well as indepen-dent of the variables ε

in, i.e. µε

in

d= µε. Ensembleconservation, called “canonical conservation, ofthe flux corresponds to (〈·〉 denotes the ensembleaverage):

〈εn〉 = 〈ε0〉 (7)

obviously it requires that the random variableµε defining the multiplicative increments shouldsatisfy:

〈µε〉 = 1. (8)

This is also a sufficient condition, except if thecascade is degenerate, i.e. dies away after a finitenumber of steps (εn → 0 almost surely). This ariseswhen a too strong, unrealistic mean intermittencyis chosen and we will see that this can be easilyquantified.

In spite of their apparently simple yet some-what awkward discretizations, these models arealready able to give key insights into the funda-mentals of cascade processes. This is confirmedusing more realistic (continuous in scale) cascades,which are necessary to take into account other(statistical) symmetries (e.g. translation invariance,see Sec. 7).

3.1. Unifractal insights and thesimplest cascade model(β-model)

The simplest cascade model, often called the β-model, takes the intermittency of turbulence intoaccount by assuming [Novikov & Stewart, 1964;Mandelbrot, 1974; Frisch et al., 1978] that eddiesare either dead (inactive) or alive (active). Thiscorresponds1 to the fact that the random variableµε, which defines the multiplicative increments, hasonly two states (see Fig. 5):

Pr(µε = λc1) = λ−c

1 (alive)

Pr(µε = 0) = 1 − λ−c1 (dead).

(9)

The boost µε = λc1 > 1 is chosen so that the

ensemble averaged ε is conserved [Eq. (8)]. At eachstep in the cascade, the fraction of the alive eddies

L

LL/2

L/2

L/2

L/2

N(L) -D N(L) -D s

Ln 4

Ln 2Ln 3

D = Ln 2

= 2

D = s =1.58

ISOTROPIC

= self similarity

L∝∝ L

Fig. 5. A schematic of an isotropic discrete in scale cascade.The left-hand side shows a nonintermittent (“homogeneous”)cascade, the right-hand side shows how intermittency can bemodeled by assuming that not all sub-eddies are “alive”. Thisis an implementation of the “β-model”. From [Schertzer &Lovejoy, 1993].

1The β-model is often defined more vaguely than this. We follow the more precise stochastic presentation in [Schertzer &Lovejoy, 1984].

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decreases by the factor β = λ−c1 (hence the name

“β-model”) and conversely their energy flux den-sity is increased by the factor 1/β to assure (aver-age) conservation. After n steps, this drastic andsimple dichotomy is merely amplified by the totalscale ratio λn

1 :

Pr(εn = λnc1 ) = λ−nc

1 (alive)

Pr(εn = 0) = 1 − λ−nc1 (dead).

(10)

Hence either the density goes on to divergewith an (algebraic) order of singularity c, or is atonce “killed” (set to zero)! Following the usual def-inition of a geometric codimension C(A) of a setA of dimension D(A) embedded in a space E ofdimension D(E):

C(A) = D(E) − D(A) (11)

c is the codimension of the alive eddies andtheir corresponding (geometrical) dimension Ds is(if c < D):

Ds = D − c (12)

which is often called the dimension of the supportof turbulence. The degenerate case merely corre-sponds to c ≥ D, i.e. when Eq. (10) would yielda negative dimension, which in fact correspondsto the fact that intermittency is so large that any

activity almost surely vanishes after a finite numberof steps.

3.2. The simplest multifractalvariant (the α-model)

We already pointed out the mere occurrence/nonoccurence of rain is not too informative, e.g.in the Nimes time series [Fig. 1(a)] the daily aver-age ≈ 2.1 mm mm/day is negligible compared to a228 mm in a few hours — the October 1988 catas-trophe in Nımes! Furthermore, we already pointedout that this time series does not display a uniquesingularity [see Fig. 1(b) and the corresponding dis-cussion in Sec. 2]. The variability of time series inNimes is so significant that [Ladoy et al., 1993a]and [Bendjoudi et al., 1997] found evidence of diver-gence of high order statistical moments (a subjectwe will discuss more in Sec. 9). Similarly, Fig. 6shows the qualitatively similar spatial distributionof the radar reflectivity of rain. This variabilityseems strikingly analogous to that of the energy fluxcascade in turbulence (see schematic in Fig. 7), ananalogy that turns out to be quite profound.

On the theoretical level the β-model turns outto be a poor approximation to turbulence because itis unstable under perturbation: as soon as one con-siders a more realistic alternative [Schertzer & Love-joy, 1983] to the caricatural dead/alive dichotomy,

Fig. 6. A 1-D subsatellite section of satellite radar reflectivity (Z; units mm6/m3) of rain at a 4.3 km horizontal resolution,one full orbit showing flux-like spikes; the mean Z is 53mm6/m3. From [Lovejoy et al., 2008a].

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CASCADE LEVELS

0 --

1 --

2 -- . . .

n --

xy

ε

0l

l0 / λ1

2

n

l0 / λ

l0 / λ

multiplication by 4independent random(multiplicative) increments

multiplication by 16independent random(multiplicative)increments

Fig. 7. A schematic of few steps of a discrete multiplicative cascade process, here the “α-model” with two “pure” singularitiesγ+ > 0, γ− < 0 (corresponding to the two values taken by the independent random increments λγ+ > 1, λγ− < 1) leading tothe appearance of mixed singularities γ (γ− ≤ γ ≤ γ+). From [Schertzer & Lovejoy, 1989b].

most of the peculiar properties of the β-modelare lost. Indeed, let us consider the more realisticα-instability allowing subeddies to be either “moreactive” or “less active”:

Pr(µε = λγ+

1 ) = λc+1

Pr(µε = λγ−1 ) = 1 − λ

c+1 = λ

c−1

(13)

with γ+ > 0, γ− < 0. The β-model is recoveredwith γ+ = c = c+; γ− = −∞ and the “canonical”conservation [Eq. (7)] implies that there are reallyonly two free parameters out of c+, c−, γ+, γ−, sinceit corresponds to:

λγ+−c+ + λγ−−c− = 1. (14)

At step n, we have “mixed” singularitiesγr, 0 ≤ r ≤ 1 resulting from linear combinationsof the “pure” singularities γ0 = γ−, γ1 = γ+, wherer = n+/n is the fraction of outcomes of γ+ alongthe cascade branch leading to γr:

γr = rγ+ + (1 − r)γ−

Pr(ε = (λn1 )γr) =

(nnr

)(λn

1 )rc++(1−r)c− (λn1 )c(γr)

c(γr) = r(Logλ1(r)+ c+)+(1−r)(Logλ1

(1−r)+ c−)(15)

where(

nnr

)is the number of nr-combinations of n

objects and the Stirling formula is used to deriveasymptotics for large n and for any given (andfixed) ratio r = n+/n. The “p-model” [Meneveau &Sreenivasan, 1987] and the “binomial multifractalmeasure” correspond to microcanonical versions ofthe α-model, which means that the flux of energyis strictly conserved, not only on the ensembleaverage. This constraint fundamentally changes theproperties of the processes, as we shall see below.

4. The General MultifractalFramework

4.1. The codimension function c(γ)

The pedagogical example of the α-model provideshelpful insights into the general formalism necessaryfor more general cascade processes. For instance,in the α-model as the number of cascade steps nbecomes large, one obtains asymptotic expressions[from Eq. (11)] which depend only on the total ratioof scale (denoted λ = L/ ≥ 1):

Pr(ελ ≥ λγε1) ∼ λ−c(γ) (16)

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This is a basic multifractal relation for mul-tifractal processes, which merely states that themeasure of the fraction of the probability spacecorresponding to the events

Aλ(γ) =(x, ω) ∈ E × Ω | ελ(x, ω) ≥ λγε1

(17)

has a (statistical) codimension c(γ), the precisemeaning of the asymptotic equivalence (λ → ∞)denoted by ∼ will be discussed below. As alreadyemphasized, there is generally no upper bound onc(γ). On the other hand, due to the nested hierarchyof these events (∀λ, γ ≤ γ′ : Aλ(γ) ⊂ Aλ(γ′)) c(γ)is necessarily an increasing function of γ.

Other fundamental properties are that c(γ)must be convex and that if the process is con-servative [Eq. (7)], then c(γ) has the fixed point:c(C1) = C1, where C1 is at the same time a singu-larity corresponding to the mean of the process andits codimension: at this point c(γ) is tangent to thebisectrix. Figure 8(a) illustrates these properties of

c(γ)

γ

C1

rare events

extremeevents

C1

(a)

γ

extremeevents

rareevents

D γs

D

D+Ds

Ns ≈ λDs c(γ)

(b)

Fig. 8. A schematic illustration of a conserved multifrac-tal, showing (a) the relations c(C1) = C1 and c′(C1) = 1,where C1 is the singularity of the mean, (b) how the sam-pling dimension imposes a maximum order of singularitiesγ’s. From [Schertzer & Lovejoy, 1993].

the codimension function c(γ). This graphical rep-resentation helps also to estimate the limitationsdue to the finite size of a sample with the help ofthe “sampling dimension” Ds [Schertzer & Lovejoy,1989b; Lavallee et al., 1991] defined as being thescaling exponent of the number Ns of independentsamples of resolution λ:

Ns,λ ≈ λDs (18)

as well as the corresponding “sampling singular-ity” γs that is the almost sure maximal singularitypresent in a sample of sampling dimension Ds. Thissingularity γs has therefore a codimension equalto the overall effective dimension of sampling [seeFig. 8(b)]:

c(γs) = D + Ds = ∆s; (19)

and it will be discussed further in Sec. 9.4.

4.2. The multiscaling of momentsK(q) and the Legendretransformation

Under fairly general conditions, a random variablecan be specified by either its probability distri-bution or by (all) its statistical moments. For anon-negative random variable x, these two repre-sentations are linked by a Mellin transformation M ,which is:

〈xq−1〉 = M(p)

=∫ ∞

0xq−1p(x)dx (20)

p(x) = M−1(〈xq−1〉)

=1

2πi

∫ c+i∞

c−i∞〈xq−1〉x−qdq (21)

(essentially these are simply the Laplace and inverseLaplace transforms for the logs). In fact, if themoments increase slowly with q (when they sat-isfy the “Carleman criterion” — see [Feller, 1971]),only the knowledge of the integer order moments issufficient. The relevance of the Carleman criterionfor turbulence has been discussed in [Orszag, 1970],but in any case the full Mellin duality [Schertzer &Lovejoy, 1993; Schertzer et al., 2002] will hold.

This is somewhat more general than theLegendre duality pointed out in [Parisi & Frisch,1985], but we can check that the Legendre trans-form is the asymptotic (λ→∞) result linking the

January 19, 2012 9:1 WSPC/S0218-1274 03064


corresponding exponents. Since the codimensionc(γ) is the scaling exponent of the probabilities,let us introduce the corresponding scaling momentfunction K(q):

〈εqλ〉 ≈ λK(q). (22)

For large log λ, we can use the saddle pointapproximation (Laplace’s method, see for example[Bender & Orszag, 1978]) which yields asymptoticapproximations to integrals of exponential form.One obtains that K(q) is related to c(γ) by:

〈εqλ〉 =

∫dPr(ελ)εq

λ ∼∫

dPr(ελ)λqγ

=∫ ∞

−∞Log(λ)dc(γ)λqγλ−c(γ) (23)

which yields the asymptotic behavior (λ → ∞):

λ 1:∫ ∞

−∞dc(γ)eLog(λ)(qγ−c(γ))

∝ eLog(λ)Maxγ(qγ−c(γ)) (24)

as well as the prefactor, which we do not con-sider here. A similar expansion can be done forthe inverse Mellin transform Eq. (21), and we havetherefore the (involutive) Legendre duality for theexponents:

K(q) = maxγ

qγ − c(γ)

⇔ c(γ) = maxq

qγ − K(q). (25)

This demonstrates that both curves are convex(iterating twice the Legendre transform, a noncon-vex curve yields its “convex hull”). One may notethat the convexity of K(q) follows from the factthat it is the “second Laplace, base λ characteristicfunction” of Logλ(ελ). This duality also means thatthe curve c(γ) is the envelop of the tangencies ofK(q) and conversely. Hence there is a simple one-to-one correspondence between moments and ordersof singularities.

4.3. Comparison of multifractalformalisms

Until now, we rather used a codimension multifrac-tal framework [Schertzer & Lovejoy, 1987, 1989a,1992] rather than a dimension multifractal frame-work [Parisi & Frisch, 1985; Halsey et al., 1986]. Thelatter was first introduced in order to explain the

nonlinearity of the scaling exponents of the velocitystructure functions (the statistical moments of thevelocity increments) empirically observed in [Ansel-met et al., 1984]. Parisi and Frisch [1985] consid-ered that the singularities of the velocity incrementsdefined as local Holder exponents should be geo-metrically and rather deterministically distributedover embedded fractals. The f(α) formalism [Halseyet al., 1986], which dealt with multifractal strangeattractors, in many respects further emphasized thisimplicit nonrandom and geometric framework. Atthe notation level, a dimension formalism such asf(α) formalism is formally related to codimensionsaccording to:

αD = D − γ; fD(αD) = D − c(γ). (26)

This is without fundamental formal problemsas far as c(γ) < D, although all these dimensionsdepend on the considered embedding dimension D,as emphasized by the corresponding subindex D.Similarly, the scaling exponent τ(q) of the partitionfunction [Hentschel & Procaccia, 1983] is related tothat of the moments as

τD(q) = (q − 1)D − K(q). (27)

With the help of the so-called refined self-similar hypothesis [Kolmogorov, 1962; Obukhov,1962], the velocity increment singularities can belinearly related to the singularities of the energyflux, therefore, we will keep the energy flux as thebasic field to compare the formalisms. The geo-metric approach considers the following supports ofsingularities:

Sλ(γ) =x ∈ E | ελ(x) = λγε1

(28)

which rather correspond to the boundary of theevents Aλ(γ), which are defined by an inequalitysign [Eq. (17)] instead of an approximate equality,i.e. Sλ(γ) = ∂Aλ(γ) for a given stochastic realiza-tion ω ∈ Ω. As a consequence, there is no compellingreason that the supports Sλ(γ) should be hierar-chically embedded, contrary to the events Aλ(γ).There is neither a compelling reason that their(geometrical) codimensions should be convex andincreasing with respect to the singularity γ. There-fore, Parisi and Frisch [1985] explicitly made thecorresponding hypothesis. Another difference is thelimits (λ → ∞) that are considered. In the geomet-ric approach, one rather considers the set of pointsS(γ) which has the given singularity γ for even-tually all resolutions λ’s (i.e. for large enough λ),

January 19, 2012 9:1 WSPC/S0218-1274 03064


Fig. 9. Schematic diagram of the “pushforward” transformTλ,∗ of a (mathematical) measure µ due to a time contractionTλ with a scale ratio λ (reproduced from [Schertzer et al.,2010]).

therefore the limit inferior:

S(γ) = limΛ→∞

SΛ(λ) ≡⋃λ

⋂Λ>λ

SΛ(γ) (29)

whereas, the asymptotic scaling of the probability[Eq. (16)] is rather related to the limit superior[Schertzer et al., 2002]:

A(γ) = limΛ→∞

AΛ(γ) ≡⋂λ

⋃Λ>λ

AΛ(γ) (30)

which corresponds to singularity γ for infinitelyoften resolutions λ’s. Let us illustrate this ques-tion by pointing out that stochastic multifractal sin-gularities are nonlocal [Schertzer & Lovejoy, 1992].Indeed, when we add more and more cascade stepsin a stochastic cascade, the singularity γλ(x) at agiven location x and increasing resolution λ under-goes a random walk (see e.g. Fig. 10), whereas acomplete localization of the singularity would cor-respond to a pointwise limit: γ(x) = limλ→∞ γλ(x).Therefore, the relevant limit notion is the upperlimit [Eq. (30)] rather than the much more stringentlower limit [Eq. (29)]. For applications, this meansthat the multifractal field is nonlocal, and one can-not always track a given singularity value by locallyrefining the analysis of the field, e.g. with the helpof wavelet analysis. This may yield spurious results.Parisi and Frisch [1985], Frisch [1995] acknowledgedthat within their formalism they could get only abounded range of singularities (in fact c(γ) < D) forthe so-called lognormal model. The practical impor-tance of c(γ) ≥ D will be discussed in Sec. 9.

On the overall, we reviewed the fact thatbeyond their strong communality, various multi-fractal formalisms have important differences dueto basic assumptions on the nature of the process(e.g. stochastic or deterministic), the hypothesis tobe done is more or less stringent, but the resultinghierarchy of fractals can be also rather different: it

Fig. 10. Nonlocal development of singularities γλ(x) (at location x and resolution L/λ) in a perspective view (along thenumber of steps n = Log2[λ]) in a discrete cascade of elementary scale ratio λ1 = 2 and of universal parameters α = 1.5, C1 =0.5 (see Sec. 6.3) over an overall ratio of scale Λ (≥ λ). It is rather obvious that for any given x, γλ(x) fluctuates for increasingλ, contrary to the hypothesis of localness of singularity.

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could be either defined for each realization or withlooser probabilistic constraints.

5. Generators, CharacteristicFunctions and Multifactal ScaleSymmetry

Discrete cascades, i.e. with scale ratio λ ∈ λN1 = 1,

λ1, λ21, λ

31 . . ., point out that the ελ form a multi-

plicative group, a property that will be extended tocontinuous cascades. It is defined by the fact that acascade from resolution 1 to Λ, can be obtained bymultiplying a cascade from resolution 1 to λ ≤ Λby a cascade from λ to Λ. Because, the latter corre-spond to a rescaled cascade from resolution 1 to Λ/λ(see Fig. 11), this group property corresponds to:

∀Λ ≥ λ ≥ 1: εΛ = ελT ∗λ (ε′Λ/λ) (31)

where T ∗λ is the “pullback” transform for any

function f :

∀x ∈ E: T ∗λ (f)(x) = f(Tλ(x)) (32)

defined by a geometric point transform Tλ on thespace E, on which ελ is defined (i.e. time and/orspace). For the moment, up to Sec. 8 devotedto Generalized Scale Invariance, Tλ is taken tobe trivial scale transformation, i.e. the isometriccontraction:

Tλ(x) =x

λ(33)

although it is already clear that the generality of thepullback transform will allow broad generalizationsof the results obtained for the isometric contraction.

The name pullback evokes the fact that this trans-form acts in the opposite direction (“contravari-antly”) to that of the original transform Tλ. Thisgeneral notion is particularly useful when dealingwith differential equations [Schertzer et al., 2012].It is dual to the “pushforward” transform Tλ,∗ act-ing (“covariantly”) on measures according to thefollowing duality equation:∫

fTλ,∗(dµ) =∫

T ∗λ (f)dµ. (34)

The pushforward transform Tλ,∗ is particularlyuseful to mathematically deal with singular mea-sures such as rain accumulation [Schertzer et al.,2010]. It is rather easy to check that the (trivial)group property of the original transform Tλ extendsto both the pullback and pushforward transformsand that both are linear respectively on vectorspaces of functions and their dual spaces of mea-sures.

Like for all one-parameter groups, we are inter-ested to characterize the infinitesimal generator of acascade, which can be stochastic. Loosely speakingthis allows us to return to an additive group. Letus consider the generator Γλ of the cascade over a(noninfinitesimal) scale ratio λ defined by:

ελ = exp(Γλ). (35)

The additive group property corresponding to themultiplicative property displayed by Eq. (31) is:

∀Λ ≥ λ ≥ 1: ΓΛ = ΓλT ∗λ (Γ′

Λ/λ). (36)

The generator is well defined for any finite res-olution λ, but its limit for asymptotically large

1

λ

Λ

λ

λ

Λdressed cascade

bare cascade

hidden cascade

1 1

X=

T*λ

Λ / λ

Fig. 11. A schematic diagram (horizontal segments schematically represent involved scales) showing a (“dressed”) cascadefrom resolution 1 to a (“bare”) cascade from resolution 1 to λ, multiplied by a (“hidden”) cascade from resolution Λ/λ to Λ,which is the pushback transformed with the help of T ∗

λ of a cascade constructed from resolution 1 to Λ/λ. The terminology of“dressed, bare and hidden” cascades will be discussed in Sec. 9.2. Reproduced from [Schertzer et al., 2002].

January 19, 2012 9:1 WSPC/S0218-1274 03064


resolutions is not trivial. This can be seen withthe help of the moment scaling function K(q),which gains a new and convenient meaning. Indeed,Eq. (22) can be rewritten under the form:

〈eqΓλ〉 = Zλ(q) = eKλ(q) (37)

where Zλ(q) and Kλ(q) are none other than respec-tively the (Laplace) first and second characteristicfunctions — or respectively the moment and cumu-lant generating functions — of the generator Γλ.This further shows that the characteristic functionof Γλ, should be logarithmically divergent (with thescale ratio λ):

Kλ(q) = K(q)Log λ (38)

hence a nontrivial small scale limit of the genera-tor Γλ.

We may now define fractal and multifractalsymmetries in a very general and precise man-ner. Following [Lamperti, 1962], scale invariance forstochastic fractals is usually defined by:

ε(Tλx) d= λHε(x) (39)

where H is a given unique scaling exponent and εcan be an increment of random walk or field, aswell as a measure. Considering the latter case, thiscan be rewritten with the help of the pushforwardtransform into a standard form of a symmetry Sλ:

Sλεd= ε; Sλ = Tλ oTλ,∗ (40)

which corresponds to the product of a pushforwardtransform Tλ,∗ by a codomain contraction, i.e. acontraction Tλ on the space where ε is valued (upuntil now the real space ):

Tλ = λ−H . (41)

A multifractal symmetry is obtained forEq. (40) as soon as Tλ is no longer defined by aunique scaling exponent, but by a full set of singu-larities γ’s:

Tλd= λ−γ ≡ e−Γλ . (42)

Generalized scale invariance (Sec. 8) will fur-ther generalize this symmetry due to nonscalaranisotropic Tλ and/or Tλ,∗.

6. Universality

6.1. The concept of universality

The jump from fractals to multifractals is huge,since it corresponds to a jump from a unique

dimension to an infinite hierarchy of dimensions.This is emphasized by the fact that there is onlya convexity constraint on the nonlinear functionsK(q) and c(γ), therefore a priori an infinity ofparameters are required to determine a multifractalprocess. For obvious theoretical and empirical rea-sons, physics abhors infinity. This is the reason whyin many different fields of physics the theme of uni-versality appears: among the infinity of parameters,it may be possible that only a few are relevant. Forinstance, in critical phenomena most of the expo-nents describing phase transitions depend only onthe dimensionality of the system. This is especiallytrue as soon as we go beyond ideal systems to morerealistic ones that should be robust to perturbationsor self-interactions. Such perturbations or interac-tions may wash out many of the peculiarities of thetheoretical model, retaining only some essential fea-tures. Loosely speaking, a theoretician may concocta model for an isolated system depending on a verylarge number of parameters, but most natural sys-tems are open and the resulting interactions canwash out most of the details, just leaving the (few)essentials. This was the basic idea of the Renormal-izing Group approach [Wilson & Kogut, 1974].

The system can therefore be expected to con-verge to some universal attractor, in the sense thata whole class of models/processes with rather dis-tinct parameters will be nevertheless attracted tothe same process defined by only a small num-ber of relevant parameters: the larger the basin ofattraction, the more universal the attractor.

6.2. Universality in multiplicativeprocesses?

The study of multiplicative random processes has along history [Aitchison & Brown, 1957] going backto at least [McAlsister, 1879], who argued that mul-tiplicative combinations of elementary errors wouldlead to lognormal distributions. Kapteyn [1903] gen-eralized this somewhat and stated what came to beknown as the “law of proportional effect”, which hasbeen frequently invoked since, particularly in biol-ogy and economics (see also [Lopez, 1979] for thislaw in the context of rain). This law was almostinvariably used to justify the use of lognormal dis-tributions i.e. it was tacitly assumed that the log-normal was a universal attractor for multiplicativeprocesses. Although Kolmogorov [1962], Obukhov[1962] did not explicitly give the law of propor-tional effect as motivation, it was almost certainly

January 19, 2012 9:1 WSPC/S0218-1274 03064


the reason why they suggested a lognormal distribu-tion for the energy dissipation in turbulence. Thisclaim seemed to be secured by the explicit “lognor-mal” cascade model developed in [Yaglom, 1966].The claim of universality of the lognormal modelwas first criticized in [Orszag, 1970] and then in[Mandelbrot, 1974].

Whereas Orszag’s criticism was on the groundsthat the (infinite) hierarchy of integer ordermoments would not determine a lognormal process,Mandelbrot’s criticism was based on the fact thateven if the cascade process were lognormal at eachfinite step, that in the small scale limit, the spatialaverages would not be lognormal. This limit alreadyposes a nontrivial mathematical problem, since itcorresponds to a weak limit of random measures[Kahane, 1985], as discussed in Sec. 9.1. Further-more, since the particularities of the discrete models(e.g. the α-model) remain at each scale, therefore inits small-scale limit, this lead to the opposite claim:that multiplicative cascades do not admit any uni-versal behavior [Mandelbrot, 1989, 1991; Gupta &Waymire, 1993].

6.3. Universal multifractals and themultiplicative central limittheorem

As long as we keep the total range of scale fixedand finite mixing (by multiplying them) indepen-dent processes of the same type (identical distribu-tion, therefore the same codimension function c(γ)and scaling moment function K(q)), and then takethe limit Λ → ∞ [Schertzer & Lovejoy, 1987, 1988],we avoid the difficulties that we mentioned aboveand a totally different limiting problem is obtained.

For instance, this may correspond to a (renor-malized) densification of the excited scales byintroducing more and more intermediate scales(see Fig. 12), e.g. to obtain a continuous scalecascade model as discussed in Sec. 7. Alterna-tively, we may also consider the (renormalized)nonlinear mixing of identically independently dis-tributed (i.i.d.) cascade models which correspond totheir multiplication. In both cases, we first look forthe cascade processes that are stable under eithernonlinear mixing or densification, then to theirattractivity and their domain of attraction. Stablecascades are the fixed points ελ

d= ε(1)λ of:

λ−a1ελ =∏

i=1,N

(λ−aN ε(i)λ )1/bN (43)

Fig. 12. Scheme of densification of scales: each horizontalline schematizes to a step of the cascade from large to smallscales (up to bottom). Scale densification of a cascade (left,with a rather large elementary scale ratio) corresponds tomore and more intermediate steps (right, with a smaller ele-mentary ratio). From [Schertzer & Lovejoy, 1997].

where aN , bN (with the nonrestrictive choice b1 = 1)are renormalizing constants and ε

(i)λ are N indepen-

dently identically distributed cascade processes onthe same range of scales with an overall resolutionλ for nonlinear mixing or ε

(i)λ are the successive ele-

mentary cascade steps (of elementary resolution λ)for a scale densification over the same scale ratio λ.Secondly, one has to check that these stable pointsare attractive over a given domain of processes.

A multiplicative central limit theorem [Sch-ertzer & Lovejoy, 1997] is obtained with the helpof the cascade generator. Indeed, Eq. (43) cor-responds to the generalized (additive) stabilityproblem [Levy, 1925, 1937; Gnedenko, 1943; Gne-denko & Kolmogorov, 1954], i.e. the following fixedpoint:

Γλ − a1Log λ =N∑

i=1

Γ(i)λ − aNLog λ

bN. (44)

For instance, iteration of Eq. (43) or (44) showsthat the bN ’s form a multiplicative group withaN = a1:

bNM = bNbM (45)

and one then may introduce the exponent α togenerate this group:

bN = N1/α. (46)

On the other hand, taking ensemble averagesof both sides of Eq. (43) one obtains that the scal-ing moment function K(q) of ελ is the fixed point

January 19, 2012 9:1 WSPC/S0218-1274 03064


K(q) = K(1)(q) of:

K(q) − a1q = N

[K(1)

(q

bN

)− aNq

bN

](47)

which yields the solutions:

K(q) = cqα + a1q. (48)

Attractivity is rather immediate. Indeed, let usconsider the following K(i)(q) (with β > α):

K(i)(q) = K(q) + O(qβ)

= cqα + a1q + O(qβ) (49)

we have therefore:

K(q) − a1q = limN→∞

N

[K(1)

(q

bN

)− aNq

bN

](50)

and therefore the corresponding ελ is attractive:

λ−a1ελ = limN→∞

∏i=1,N

(λ−aNε(i)λ )1/bN . (51)

Finally, with the help of the constraint K(1) =0 of a conservative field, Eq. (48) yields the followingscaling moment function of a conservative universalmultifractal [Schertzer & Lovejoy, 1987] (see Fig. 13for an illustration):

K(q) =C1

α − 1(qα − q); 0 ≤ α ≤ 2 (52)

where C1 is the singularity of the mean field (q = 1),which with the help of the Legendre transform isdefined by:

C1 =d

dqK(q)

∣∣∣∣q=1

. (53)

The constraint on α mentioned in Eq. (52)results from the requirement that the (first) char-acteristic function Zλ(q) [Eq. (37)] of the genera-tor should be positive definite so as to be a Mellintransform of a probability (according to the Schoen-berg’s theorem [Schoenberg, 1938], which comple-ments the Bochner’s theorem [Bochner, 1955] forthe Laplace transform). The two extreme casescorrespond respectively to the (uni/mono) fractalβ-model (α = 0) and the lognormal model (α = 2),whereas the case α = 1 is derived from Eq. (52)with α → 1:

K(q) = C1qLog q; α = 1. (54)

Equation (52) shows that for 0 < α < 2 thegenerator Γλ follows an “extremely asymmetric”or “skewed” Levy distribution (i.e. its skewnessparameter β = −1) so that the fat tail (power-law tail) is present only for negative fluctuations,otherwise the field ελ will have divergent statisticalmoments (K(q) = ∞) for all positive order q. Nev-ertheless, all the negative statistical moments of ελ

diverge, which is neither a problem, nor surprisingdue to its frequent extremely low or even zero values(especially for the β-model).

Due to the fact that qα/α and γα′/α′ are

Legendre duals for (1/α) + (1/α′) = 1, one obtains

2.01.51.00.50.0-1.0

-0.5

0.0

0.5

1.0

2.0

0.0000 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000

K(q)/C1 versus q

q

K(q

)/C

1

α

α = 0.0

α = 2.0

1.5

Fig. 13. The universal multifractal moment scaling exponent K(q) normalized by C1. They lie between the parabolic (α = 2)“log-normal” model and the linear (α = 0) β model.

January 19, 2012 9:1 WSPC/S0218-1274 03064


2.01.51.00.50.0-0.5-1.00.0

0.5

1.0

1.5

2.0

0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000

0.0000

c(γ)/C versus γ/C1

γ / C1

c(γ)

/ C

1

α

α = 0

α = 2.0

Fig. 14. The universal multifractal probability exponent c(γ/C1) normalized by C1. They lie between the parabolic (α = 2)“log-normal” model and the bilinear (α = 0) β model.

from Eq. (52):

c(γ) = C1

(γ

C1α′ +1α

)α′

;1α

+1α′ = 1 (55)

see Figs. 13 and 14 for illustration of the parametersensitivity of the scaling moment and codimensionfunctions.

7. Continuous Scale or “InfinitelyDivisible” Cascades

7.1. The limitations of discretescale cascades

An important consequence of universality is thepossibility of starting with a discrete cascade model(e.g. one with an elementary scale ratio λ1 = 2) andwith the help of a scale densification to obtain a con-tinuous in-scale process λ1 → 1. Such cascades arenecessary since physically we expect real systems toinvolve a continuum of scales and there is usuallyno physically based quantization rule that wouldrestrict scale ratios to integer values. Furthermorethe discrete models involve a hierarchical splittingrule of structures into substructures which is basedon a notion of distance which is not a metric, butrather an “ultra-metric”. More precisely, it cor-responds to the λ-adic ultrametric: the distancebetween two structures at a given level of a discretecascade process is defined by the level of the cascadewhere they first share a common ancestor, i.e. notby the usual distance. This implies that the distancebetween the centers of two contiguous eddies is

not uniformly distributed. Since all the statisticalinterrelations between different structures dependon this ultra-metric, not on the usual metric, thishas drastic consequences. In particular, there is nohope of obtaining a (statistically) translation invari-ant cascade, since such invariance depends on themetric, not the ultra-metric.

In summary, discrete cascades are useful forgrasping many of the fundamentals, but one hasto avoid being blocked by their artifacts. As afinal note on discrete scale cascades, let us empha-size that almost all rigorous mathematical resultson cascade processes have been derived in thisrestricted framework; this is not only becauseit is convenient, but also for complex historicalreasons — including the debate on universalitydiscussed above. As a consequence, the issue of con-tinuous scale or infinitely divisible cascades has notbeen discussed enough.

7.2. Continuous scale cascades andtheir generators

The general idea of cascades as one-parametermultiplicative groups discussed in Sec. 5, especiallyEq. (31) becomes essential for continuous scalecascades. For instance, the corresponding addi-tive property for generators [Eq. (36)] show thatthey should be infinitely divisible, i.e. they can bedivided into smaller and smaller additive compo-nents. On the other hand, we also saw that scalingrequires a logarithmic divergence of the generator.

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For a concrete and generic example, let us con-sider the case of universal multifractals (Sec. 6.3),whose generators are not only infinitely divisible,but also stable and attractive. They are thereforerobust under nonlinear interactions. To satisfy thelogarithmic divergence their generators [Eq. (33)]should be (colored) stable Levy noises [Schertzer &Lovejoy, 1987] obtained by a fractional integrationover a given “subgenerator”, which is white Levynoise γ

(α)0 with a Levy stability index α and fur-

thermore “unitary” in the sense that its (Laplace)second characteristic function is:

K0(q) = sign(α − 1)qα · 1q≥0 + ∞ · 1q<0 (56)

which means that γ(α)0 is extremely asymmetric

[Schertzer et al., 1988] for α < 2, i.e. with askewness β = −1, to have only strong negativeextremes. This is no longer relevant for the Gaus-sian case (α = 2), which is necessarily symmetric

(β = 0,K(q) = q2) and has therefore no divergencefor q < 0. Figure 15 compares the subgenerators inthe α < 2, α = 2 cases.

This can be seen under a (fractional) differen-tial form of order D/α′ using a fractional power ofthe Laplacian ∆x (with a scale resolution L/λ):

−(−∆x,λ)D/2α′Γλ(x) ≈ γ

(α)0 (x) (57)

or under the corresponding and more explicit (frac-tional) integration form:

Γλ(x) =∣∣∣∣ var(α)mD−1(∂BL)

∣∣∣∣1/α∫

BL\BL/λ(x)

dDγ(α)0 (x′)

|x − x′|D/α

− var(α)Log λ;

var(α) =C1

α − 1(58)

Fig. 15. A schematic showing a gaussian and extremal Levy subgenerator (with α = 1.6); note that there are extremes onlyfor the negative fluctuations. From [Wilson, 1991].

January 19, 2012 9:1 WSPC/S0218-1274 03064


where BL(x) is the ball of center x and size L (whichcould be arbitrarily chosen), mD−1(∂BL) is the(D − 1)-dimensional measure of its (hyper-)surface∂BL. For isotropic cases, BL(x) = x′ | |x − x′| ≤L/2, BL\BL/λ(x) = x′ |L/2λ < |x − x′| ≤L/2 and mD−1(∂BL) = 2(L/2)D−1πD/2/ΓE(D/2)(where ΓE is the Euler Gamma function). Inorder to get some convergent moments (finiteK(q)), the subgenerator γ

(α)0 must be extremely

assymmetrical. It is also interesting to note thatελ due to Eq. (58) is the solution of the followingdifferential equation:

dελ = ελdγλ (59)

where dγλ is the infinitesimal generator of themultiplicative group of ελ:

dγλ =∣∣∣∣ var(α)mD−1(∂BL)

∣∣∣∣1/α ∫

∂BL/λ(x)

dDγ(α)0 (x′)

|x − x′|D/α

(60)

where var(α) corresponds to a generalization of the(quadratic) variation of a (semi-) martingale (e.g.[Metivier, 1982]) and this explains why:

dγλ − dΓλ =var(α)dλ

λ. (61)

7.3. Nonconservative multifractalsand the Fractionally IntegratedFlux model (FIF )

Integration of Eq. (59) [or equivalent use of the gen-erator defined by Eq. (58)] yields a conservative field

Fig. 16. Isotropic (i.e. self-similar) multifractal simulations showing the effect of varying the parameters α and H (C1 = 0.1in all cases). (From left to right) H = 0.2, 0.5 and 0.8. (From top to bottom) α = 1.1, 1.5 and 1.8. As H increases, the fieldsbecome smoother and as α decreases, one notices more and more prominent “holes” (i.e. low smooth regions). The realisticvalues for topography (α = 1.79, C1 = 0.12, H = 0.7) correspond to the two lower right-hand simulations. All the simulationshave the same random seed. Reproduced from [Gagnon et al., 2006].

January 19, 2012 9:1 WSPC/S0218-1274 03064


ελ, which is not always a desirable property. Theclassic example of such nonconsevative processes isthe turbulent velocity field; in Kolmogorov three-dimensional isotropic turbulence, whose structurefunction is not scale invariant: 〈∆v2(L/λ)〉 ≈λ−2H〈∆v2(L)〉, with: H = 1/3. This constraint iseasily satisfied with the help of a fractional integra-tion of order H:

νλ(x) ∝∫

BL\BL/λ(x)

dDε(x′)|x − x′|D−H

. (62)

In the framework of this Fractionally IntegratedFlux model [Schertzer et al., 1997], the singularitiesare redistributed with an average shift of −H asseen on the structure functions, i.e. the statistical

moments of the increments ∆ν(x, x′) = |ν(x) −ν(x′)|:

〈|ν(x) − ν(x′)|q〉 ∝ |x − x′|qH−K(q). (63)

These increments have therefore a scale depen-dent mean, with H = 0 as scaling exponent (seeFigs. 16 and 17). Figures 15 and 18 display theeffect of changing the order of this scale invariantsmoothing of order H. The convolution involved inEq. (63) corresponds to a fractional integration of(fractional) order H over ε. However, there is nota unique definition of such a noninteger integra-tion. For instance, to respect causality for processesin time, the corresponding kernel or Green func-tion should be totally asymmetric in order not to

Fig. 17. All simulations have H = 0.35 (this is a scale invariant smoothing, see Sec. 7.3). The parameter C1 increases from0.05 to 0.8 in steps of 0.15 from left to right and α increases from 0.4 from top to bottom in units of 0.4 to 2.0 (bottom).

January 19, 2012 9:1 WSPC/S0218-1274 03064


Fig. 18. All simulations have C1 = 0.05. The parameter H increases from 0.05 to 0.8 in steps of 0.15 from left to right andα increases from 0.4 from top to bottom in units of 0.4 to 2.0 (bottom).

integrate over the future [Marsan et al., 1996]. Thisis particularly important for predictability studies[Schertzer & Lovejoy, 2004a] and multifractal fore-casts [Macor et al., 2007]. In fact, it suffices to intro-duce a Heaviside function (1t≥0) as a prefactor tothe noncausal kernel to obtain a causal Green func-tion g(t, t′), e.g.:

g(t, t′) =1t−t′≥0

|t − t′|D−H. (64)

Furthermore, “delocalized” kernels can bedefined for space-time processes to have a givenscaling exponent D−H and at the same time singu-larities (e.g. poles) in their Fourier transforms thatgenerate waves. This will be discussed further in thenext section that introduces the necessary conceptand tools of Generalized Scale Invariance.

8. Generalized Scale Invariance (GSI)

8.1. Generalized scales

Up until now we have considered “self-similar” frac-tals and multifractals i.e. geometric sets and fieldsthat are not only scaling, but also statistically rota-tionally invariant. However real world systems arenot isotropic so that it is important to generalizethis to strongly anisotropic systems. The resulting“Generalized Scale Invariance” (GSI) was actuallymotivated by the need to account for the strongscaling stratification of the atmosphere [Schertzer &Lovejoy, 1985b, 1985a]: Fig. 19 gives a (scaling)anisotropic version of the isotropic cascade scheme(Fig. 5) originally developed to model atmosphericstratification, which has gained increasing empiri-cal confirmation (Fig. 20) with the help of various

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ANISOTROPIE

= COMPRESSION et REDUCTION

L y

L x/4L x

L y /2 L y /2

/4L x

sD ___Ln 4

Ln 6= 1.29=

DelLn 8Ln 4___ = 1.5=

N(L) - L- DsN(L) - L - Del

Fig. 19. Anisotropic cascade scheme: compare with Fig. 5.From [Schertzer et al., 2002].

atmospheric measurements [Chigirinskaya et al.,1994; Lazarev et al., 1994; Lilley et al., 2004; Love-joy & Schertzer, 2007; Lovejoy et al., 2007].

The usual approach to scaling is to first posit(statistical) isotropy and only then to consider scal-ing. This approach is so prevalent that the terms“scaling” and “self-similarity” are frequently usedinterchangeably! The most famous example of thisis Kolmogorov’s hypothesis of “local isotropy” fromwhich he derived the k−5/3 spectrum for the windfluctuations (k is a wavenumber). The GSI approachis the converse: it first posits scale invariance (scal-ing), as the main symmetry and then considersthe remaining nontrivial symmetries. One may eas-ily check that the type of anisotropic constructionshown in Fig. 19 reproduces itself from scale to scalewithout introducing any characteristic scale. Sinceit involves scaling anisotropy in fixed directions, itis called “self-affinity”. This anisotropic scheme isapparently the first explicit model of a physicalsystem involving a fundamental self-affine fractalmechanism [Schertzer & Lovejoy, 1984, 1985a].

GSI corresponds to the fact that the contrac-tion operators Tλ [Eq. (33)] or Tλ [Eq. (34)], whichdefine the scaling symmetry [Eq. (40)], are no longerisotropic [Eq. (33)] nor merely scalar [Eq. (41)]. Forsimplicity sake we first focus on the anisotropy ofTλ, keeping Tλ scalar, but mutadis muntandis thesame apply to an anisotropic Tλ and a scalar Tλ,and finally both being anisotropic. In an abstractmanner, Tλ is a generalized contraction on a vectorspace E, if it is a one-parameter (semi-) group for

Fig. 20. The symbols show the mean absolute fluctuations(the first order structure functions) of the lidar backscatterratio ρ (a surrogate for the passive scalar aerosol density).The points near the shallow slope (slope 1/3, Kolmogorovvalue) are the horizontal fluctuations, the points near thesteeper line (slope 3/5, Bolgiano–Obukhov value) are for thevertical fluctuations. The averages are over an ensemble ofnine vertical airborne lidar cross-sections spanning the rangeof ≈ 100 m–100 km in the horizontal and 3 m–4 km in the ver-tical. ∆r is either the vertical or horizontal distance measuredin meters. The lines have the theoretical slopes 3/5, 1/3, theyintersect at the sphero-scale here graphically estimated fromthe intersection point as ≈ 10 cm. From [Lilley et al., 2004].

the positive real scale ratio λ (λ ≥ 1 for a semi-group), i.e.

∀λ, λ′ ∈ R+: Tλ′ Tλ = Tλ′λ (65)

and admits a generalized scale denoted ‖x‖ (todistinguish it from the usual Euclidean metric|x|), which satisfies the following three properties[Schertzer et al., 1999] further to that of being non-negative:

(i) nondegeneracy, i.e.

‖x‖ = 0 ⇔ x = 0 (66)

(ii) linearity with the contraction parameter 1/λ,i.e.

∀x ∈ E, ∀λ ∈ R+: Tλ‖x‖ ≡ ‖Tλ · x‖ = λ−1‖x‖(67)

(iii) balls defined by this scale, i.e.

B = x | ‖x‖ ≤ (68)

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must be strictly decreasing with the contrac-tion Tλ:

∀L ∈ R+, ∀λ > 1: BL/λ ≡ Tλ(BL) ⊂ BL

(69)

and therefore:

∀L ∈ R+, ∀λ′ ≥ λ ≥ 1: BL/λ′ ⊂ BL/λ. (70)

The usual Euclidean norm |x| of a metric spaceis the scale associated to the isotropic contractionTλx = x/λ. The two first properties are ratheridentical to those of a norm, whereas the last oneis weaker than the triangular inequality, which isrequired for a norm.

8.2. Linear GSI

This corresponds to the simplest, but alreadystartling generalization of scale invariance: Tλ isno longer a scalar transform, but remains a lineartransform. In this case, the group property Eq. (65)corresponds to the fact that this linear trans-form is power law of a given linear (infinitesimal)generator G:

Tλ = λ−G ≡ exp(−G · Log(λ)). (71)

This can be easily understood with the help ofthe corresponding matrices. This “integral” form isequivalent to the following differential one:

λd

dλTλ = −G Tλ. (72)

This differential equation also governs the evo-lution (λ ≥ 1) of the point xλ = Tλx1, whose tra-jectory starts from an initial ball BL and crossesthe balls BL/λ. The classical isotropic contractioncorresponds to the scalar case: G = 1, where 1denotes the identity application. The next simplestcase corresponds to the “self-affine” case [Mandel-brot, 1985], where G is diagonalizable, with realeigenvalues µi and eigenvectors ei. Figure 21 showsthat “zooming” into such a multifractal leads to sys-tematic changes in shapes of the basic structures.This graphically demonstrates the “phenomenolog-ical fallacy” [Lovejoy & Schertzer, 2007] in whichdifferences in morphology are confounded with dif-ferences in mechanism. Figure 22 shows the effectof varying the “sphero-scale”; the scale where struc-tures are roughly “roundish”.

It is straightforward to check that forSpec(G) > 0, where Spec(·) denotes the spectrum,the scale can be defined for any given positive0 < α ≤ ∞ as:∥∥∥∥∥

∑i

xiei

∥∥∥∥∥ =

(∑i

‖xiei‖α

)1/α

(73)

with:

∀ i, xi : ‖xiei‖ = ||xi|1/µiei| (74)

where | · | is a given norm on the vector space. Theseequations are very convenient to define anisotropicprocesses, including space-time processes. Indeed,the Green functions defining either a generator ora FIF model could be defined either in the physicalspace or the Fourier space with the help of thesescales, e.g. for a (causal) space-time process:

g(k, x) ≡∫

dDxdtei(k x+ωt)g(x, t)

= (iω + ‖k‖Ht)−H/Ht (75)

where Ht is the scaling anisotropy exponentbetween time and space. Furthermore, to obtain apole, therefore waves, still keeping the same scal-ing behavior, it suffices to consider [Lovejoy et al.,2008b]:

gw(k, x) = (iω − ‖k‖Ht)−H/Ht (76)

as well any weighted products of g(k, x) andgw(k, x), i.e. weighted convolutions of g(k, x) andgw(k, x). This already provides a wide, althoughpresumably not exhaustive framework to study tur-bulence and waves. Equation (74) also confirms thatthe transformation of the norm | · | to the general-ized scale | · | corresponds to a nonlinear transform(except for the scalar case: G = 1), which may bedefined as:

xi → sign(xi)|xi|1/µi . (77)

Nevertheless, the most interesting cases cor-respond to complex eigenvalues, which obviouslyintroduce rotations, and nondiagonalizable genera-tors (Jordan matrices). This is illustrated in Figs. 23and 24 for a two-dimensional generator G. The real-ity of the eigenvalues corresponds to:

a2 =(

Tr(G)2

)2

− Det(G) > 0 (78)

and a dominant stratification effect, whereas a2 <0 corresponds to a dominant rotation effect. For

January 19, 2012 9:1 WSPC/S0218-1274 03064


Fig. 21. A sequence from a zoom of a stratified universal multifactal cloud model with α = 1.8, C1 = 0.1, H = 1/3, Hz = 5/9.(From top left to bottom right) Each represents a blow-up by a factor 1.31 (total blow-up is a factor ≈ 24 000 from beginningto end). If the top left simulation is an atmospheric cross-section 8 km left to right, 4 km thick, then the final (lower left)image is about 32 cm wide by 16 cm high; the sphero-scale is 1m as can be roughly visually confirmed since the left-rightextent of the simulation second from bottom on the right is 1.02 m where structures can be seen to be roughly roundish. From[Lovejoy & Schertzer, 2010].

reasons discussed below, G is parametrized in thefollowing way:

G =(

d + c f − e

f + e d − c

); a2 = c2 + f2 − e2. (79)

All the simulations have α = 1.8, C1 = 0.1,H =0.33 (the empirical parameters for clouds), and aresimulated on 256 × 256 grids with the same start-ing seed so that the differences are only due to theanisotropy (the colors go from blue to white indi-cating values low to high). However, there are muchmore general properties. For instance, the Jacobianof any contraction is Tλ:

det(Tλ) = λ−Del (80)

with:Del = Tr(G) (81)

which corresponds to an effective dimension called“elliptical dimension” in reference to the elliptic-like shape of the balls under a GSI contraction.It is merely the topological dimension d = Tr(1)of the vector space for the isotropic contraction.On the other hand, differentiation of the “integral”property of linearity with the contraction parameter[Eq. (67)] yields:

λd

dλ‖xλ‖ = ∇x‖xλ‖ · G · xλ (82)

which rather corresponds to a generalizationof Euler’s theorem for homogeneous function(obtained for a self-affine G). This differential form

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Fig. 22. The effect of the sphero-scale, Cloud parameters: α = 1.8, C1 = 0.1, H = 1/3, r = vertical/horizontal aspect ratiotop to bottom: r = 1/4, 1, 4, left to right, sphero-scale = 1, 8. From [Lovejoy & Schertzer, 2010].

can be used to assure that balls are strictly decreas-ing with the contraction Tλ. This can be achievedfor:

BL = x | (x,Ax)1/2 ≤ L (83)

where A is a given bilinear application, Eq. (82)indeed yields [Schertzer & Lovejoy, 1985a; Schertzeret al., 1999] the following necessary and sufficientcondition that the balls Tλ(BL) ≡ BL/λ are strictly

decreasing with the contraction group Tλ:

Spec(sym(AG)) > 0 (84)

where sym(·) denotes the symmetric part of a lin-ear application. When A is furthermore positive andsymmetric, i.e. the ball BL is an ellipsoid, this con-dition [Eq. (84)] reduces to:

Spec(sym(G)) > 0 ⇔ Re(Spec(G)) > 0. (85)

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Fig. 23. (Top row) k = 0, we vary c (denoted i) from −0.3,−0.15, . . . , 0.45 left to right and e (denoted j) from−0.5,−0.25, . . . , 0.75 top to bottom. On the right we show the contours of the corresponding scale functions. (Middle row)Same except that k = 10. (Bottom row) e = 0 the c is increased from −0.3,−0.15, . . . , 0.45 left to right, from top to bottom,k is increased from 0, 2, 4, . . . , 10. See text for more details. From [Lovejoy & Schertzer, 2007].

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Fig. 24. (Top row) The same as the bottom row of Fig. 23 except that e = 0.75. (Middle row) c = 0 and e left to rightis: −0.5,−0.25, . . . , 0.75. (Bottom row): Same as the middle row except that c = 0.15. In all rows, from top to bottom, k isincreased (0, 2, 4, . . . , 10), the right hand shows the corresponding scale functions. From [Lovejoy & Schertzer, 2007].

January 19, 2012 9:1 WSPC/S0218-1274 03064


For the two-dimensional case [and parame-trization given by Eq. (79)], this reduces to thefollowing:

Tr(G) > 0 and

det(G) > 0 ⇔ d > 0 and d2 > a2.(86)

8.3. Nonlinear GSI and Liecascades

We will use the example of the two-dimensional gen-erator G with its above parametrization [Eq. (79)]that corresponds to a “pseudo-quaternions” repre-sentation:

G = d1 + eI + fJ + cK;

1 =⌊1 00 1

⌋, I =

⌊0 −11 0

⌋,

J =⌊0 11 0

⌋, K =

⌊1 00 −1

⌋ (87)

to point out that in a rather general manner linearGSI results can be extended, albeit given technicaldifficulties briefly mentioned below, to the nonlinearGSI and lead to the general notion of Lie cascades.Let us first point out that there are at least two rea-sons to look for nonlinear GSI. The first one is thatone has often to work with manifolds rather thanwith vector spaces, the second one [Schertzer &Lovejoy, 1991], not necessarily independent of theprevious one, is that one has often to deal withlocal symmetries, e.g. the original (Weyl’s) localgauge invariance, rather than with global ones. Themain point is that the generators of the (local)symmetries define Lie algebra whose structure isessential to understand the interrelations betweenvarious symmetries.

Indeed, let us consider that the symmetriesTλ and Tλ, therefore the whole scale symmetrySλ [Eq. (40)], together with all other potentialsymmetries (e.g. more classical symmetries suchas rotations) smoothly vary with respect to theirparameters. These symmetries not only form agroup with respect to their composition, but alsoa smooth manifold and therefore a Lie group G(e.g. [Sattinger & Weaver, 1986]). In a rather gen-eral manner, this group is generated from thesymmetries that are infinitesimally close to theidentity (for infinitesimally small parameter vari-ation), which spans the tangent space to the iden-tity transformation. In fact, these generators forma Lie algebra g, i.e. a vector space with a (bilinear)

skewed product called the Lie bracket [·, ·] that fur-thermore satisfies the Jacobi identity. For matrices,the Lie bracket is defined to be the commutator:[

X,Y]

= XY − Y X (88)

In the example of the two-dimensional generator Gwe have:

2I =[J,K

], 2J =

[I,K

], 2K =

[J, I]

(89)

whereas 1 obviously commutes with any elementof this Lie algebra l(2, R) of the two-dimensionalreal matrices. Recall that any Lie algebra g is saidto be abelian if its bracket, whatever is its expres-sion, vanishes (i.e. ∀X,Y ∈ g : [X,Y ] = 0, inshort: [g, g] = 0), a Lie subalgebra s of g is a sub-space of g that is closed under the Lie bracket (i.e.[s, s] ⊂ s), a subspace l of g is an ideal of g if s isnot only closed with itself but with g (i.e. [g, l] ⊂ l),the largest abelian ideal of g is called its radicaland if it is zero g is called semi-simple. The cru-cial importance of abelian (sub-)algebra is due tothe fact that the corresponding Lie (sub-)groups areindeed commutative, i.e. the symmetries commute.

With the help of these definitions, it is ratherstraightforward to check that the one-dimensionalsubalgebra R generated by 1 is the radical of l(2, R),whereas s spanned by I,J,K is semi-simple. Thelatter is classically known as sl(2, R), the specialtwo-dimensional real linear algebra of matrices withzero trace. We have:

l(2, R) = R ⊕ sl(2, R) (90)

which is merely a particular example of the Levidecomposition of any Lie algebra into its radicaland a semi-simple subalgebra. It is also importantto note that the three two-dimensional subalgebrasi (i = 1, 3) spanned respectively by 1, I, 1,J,1,K are all abelian (s1 is merely equivalent to theset of complex numbers), but they are not ideals ofl(2, R). It means that the corresponding subgroupsare commutative, but do not commute with any ele-ment of the full group generated by l(2, R). A simpleconsequence is that if the generator G of Tλ (resp.G of Tλ) belongs to s3, then it will commute withany symmetry generated by K, but not with thosegenerated by I, i.e. rotations.

Whereas the mapping from Lie groups to Liealgebra is rather one-to-one, the inverse is oftenmore complex: different Lie groups may have thesame Lie algebra. Nevertheless, the exponentialmap allows to fully capture the local structure of thegroup from its algebra. Here, the exponential map

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−5 5 10

−20

−10

10

20

−10 −5 5 10

−10

−5

5

10

(a) (b)

Fig. 25. GSI balls for λ = 1 to 10 by steps δλ = 0.5: (a) linear case for g(x) = Gx [Eq. (91)] where the matrix is definedaccording to Eq. (79) with the set of parameters (d, c, f, e) = (1, 0.5, 0.5, 1), i.e. stratification being dominant, (b) nonlinear

case: the parameter e now depends on the location x : e(x) = |x|3/4. This introduces an extreme differential rotation.

from a tangent space to its manifold correspond tothe generalization of the exponential function, par-ticularly seen under its differential form as:

dxλ

dλ= −g(xλ)

λ. (91)

Loosely speaking, one has only to carefullydistinguish the tangent space from its manifold,whereas both are confused in the (linear) case ofvector spaces. The above mentioned limitation tothe local structure is that the integration can intro-duce an external outer scale, i.e. the inverse of Tλ =T1/λ, i.e. a dilation, for large λ is not always defined.

As simple examples, we can consider the inte-gration of the nonlinear differential equation (91),with the initial condition x1 belonging to a givenball BL, still on a vector space E, but where gis nonlinear. Its gradient ∇xg is equivalent to alocal (infinitesimal) generator G. Therefore, thecondition of (local) growth of the balls for linearGSI applies now to ∇xg. For the two-dimensionalcase, the same parametrization as before for G[Eq. (87)] corresponds to spatially variable coef-ficients (d, c, e, f) and if BL is a symmetric ellip-soid, ∇xg needs only to satisfy the same condition[Eq. (86)] as G.

Figure 25(b) is one among many figures thatcan be obtained in this nonlinear manner. Note dueto nonlinearity the balls Bλ, are no longer convexsets, contrary to the linear case.

8.4. GSI, scaling analysis anddifferential systems

GSI has a huge potential of applications whendealing with anisotropy and classical differential

systems. Indeed, the classical manner to deal withsuch problems was to first proceed to a scale anal-ysis, i.e. to define so-called characteristic quanti-ties that help to select a few corresponding rele-vant terms that will be retained to build up approx-imations of the original equations. However, as dis-cussed by analysis [Schertzer et al., 2012], on theparticular example of the quasi-geostrophic approx-imation [Charney, 1948], there are many reasons toexpect that the scaling of the approximation will bedifferent from the original equations. GSI allows toproceed in a different manner: an anisotropic analy-sis enables us to select the relevant interactions thatoccur on a wide range of scales (contrary to the scaleanalysis, which is local in scale) and obtain a newset of equations, which corresponds to a statisticalbreaking of an isotropic statistical symmetry, whichcan be seen as a broad generalization of the spon-taneous symmetry breaking. Here, the anisotropicscaling splits a vector equation that admits isotropicscaling into separated component equations. In thecase of the vorticity equation this can be obtainedby first decomposing the fields and operators intohorizontal and vertical components (with respectiveindices h and v), e.g. for the velocity field u and gra-dient operator ∇:

u = uh + uv; ∇ = ∇h + ∇v (92)

and with the help of the following pushback trans-forms [Schertzer et al., 2012]:

T ∗λ (u) = λγ(uh + λhuv);

T ∗λ (∇) = λγ(∇h + λ−h∇v)

(93)

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where h is the scaling anisotropy exponent of thevertical vs. the horizontal, as well as the correspond-ing decomposition of the vorticity ω into:

ω = ωv + σ + τ ; ωv = ∇h × uh,

σ = ∇h × uv, τ = ∇v × uh

(94)

where ωh = σ+τ is the horizontal vorticity, which isassumed to be negligible at large scales where the(almost vertical) Earth rotation Ω is assumed tobe dominant. A careful examination of the scalingexponents resulting from Eq. (93) involved in thevorticity equation (D./Dt denotes the Lagrangiantime derivative):

Dω

Dt= ω · ∇u (95)

points out that the latter splits into the three fol-lowing equations:

Dσ

Dt= σ · ∇huh

Dτ

Dt= (τ · ∇h + ωv · ∇v)uh

Dωv

Dt= (τ · ∇h + ωv · ∇v)uv

(96)

Conversely, these three equations, togetherwith the large scale condition ωv ≈ Ω, allow ananisotropic scaling like that displayed by Eq. (93),as well as a nonlinear growth of the horizontalvorticity. The latter mechanism is absent fromthe quasi-geostrophic approximation, which corre-sponds to σ ≡ τ ≡ 0 and the approximation:

Dωv

Dt≈ Ωv · ∇vuv. (97)

Contrary to the quasi-geostrophic equations,the fractional vorticity equations (96) are notan approximation of the vorticity equations (95),because solutions of the former are also solutions ofthe latter. They rather correspond to select relevantinteractions to provide solutions having a scaling asprescribed by Eq. (93). This explains the extremecontrast between Eq. (96) with a 3D nonlinear vor-ticity stretching and Eq. (96) with only a weak, lin-ear stretching. We believe that this scaling analysisis rather illustrative of the potential of GSI to bet-ter investigate the nonlinear generating equationsof anisotropic scaling fields, well beyond the scaleanalysis and the resulting classical approximations.

9. The Extremes: Self-OrganizedCriticality and MultifractalPhase Transitions

9.1. The singular small-scale limitof a cascade process

The small-scale limit λ → ∞ of a cascade process isvery singular since for any positive singularity γ,the density ελ ≈ λγ diverges. These divergencesare statistically significant for γ > C1, since itcorresponds to q > 1 and K(q) > 0, thereforeto 〈ελ〉 = λK(q) → ∞. This singular small-scaledivergence means that if a limit exists, it is not inthe sense of functions. This is similar to the Diracδ-“function”, in fact the Dirac δ-measure, which isnot a function at all but a “generalized function”defined as a limit of functions. The Dirac δ-measureis indeed only meaningful if we integrate over it. Itis rather obvious that the β-model corresponds toa (random) generalization of the Dirac δ-measurefor points belonging to a fractal set of codimen-sion c = D − Ds. Conversely, the Dirac δ-measurecan be understood as the particular (deterministic)case corresponding to a codimension c = D anddimension Ds = 0, i.e. to isolated points.

As a consequence, one has to consider the limitof the corresponding measures Πλ(A) → Π∞(A)over compact sets A of dimension D, i.e. the D-dimensional integration of the density ελ over A:

Π∞(A) = limλ→∞

Πλ(A) = limλ→∞

∫A

ελdDx. (98)

In agreement with turbulent nomenclature, theintegral Πλ can be termed a “flux” (of energy)through the scale l = L/λ, whereas ελ can be called(energy) flux density at this scale. Therefore, weanticipate that the fluxes (but not the densities)converge as λ → ∞. Still due to the singular natureof the limit, we may expect that convergence canonly occur over a limited range of order q of thestatistical moments:

∃ qD > 1, ∀ q ≥ qD : 〈Π∞(A)q〉 = ∞ (99)

whereas:

∀λ < ∞ : 〈Πλ(A)q〉 < ∞. (100)

The subscript D of the critical order q under-scores its dependence on the dimension of the spaceover which it is integrated. This dependence canbe used [Schertzer & Lovejoy, 1984] to demonstratethat cascade processes are generically multifractals:the dependence of qD on D defines a hierarchy of

January 19, 2012 9:1 WSPC/S0218-1274 03064


fractal sets. An important consequence of the diver-gence of moments [Eq. (99)] is also the exponent ofthe power-law fall-off of the probability distribution:

∃ qD > 1, π 1 : Pr(Π∞(A) > π) ≈ π−qD .

(101)

Equations (99)–(101) are equivalent and thedivergence of moments has many practical impli-cations that we discuss below. The divergence ofmoments coupled with fractal structures is oftenconsidered the hallmark of self-organized critical-ity (SOC; [Bak et al., 1987, 1988]) so that cascadeprocesses can be considered a nonclassical route toSOC. In this respect, we could note an importantdifference: classical SOC is a cellular-automatonmodel (the prototype being the “sandpile” and itsavalanches) that requires a “zero-flux” limit: in thesandpile each grain must be added one at a timeonly after the activity induced by the previous grainis over. In contrast, the cascades require a quasi-constant flux, and this is generally more physical.

To graphically appreciate the propensity formultifractal processes to generate extremes at allscales, see Fig. 26. In order to make the point thatoccasionally very rare events occur, a single value(the maximum) of the subgenerator was boosted bya factor N (placed at the centre of the simulation).

9.2. Bare and dressed cascades

The singular limit of the cascade process requiresus to distinguish between the properties of a cas-cade stopped at a finite resolution λ, (all of whosepositive moments are finite) from those correspond-ing to integrals (at the same scale) of the processtaken to the small scale limit. Schertzer and Lovejoy[1987] argued that physically this difference is dueto the effect of the small scale interactions, which“dress” the former “bare” process. This is remi-niscent of what happens in renormalization whenhigher and higher order interactions are taken intoaccount. In the same way, we distinguish betweenthe “bare” cascade quantities obtained after the cas-cade has proceeded down to a finite resolution λ,and the corresponding “dressed” quantity obtainedafter integrating a completed cascade over the samescale (l = L/λ; see Fig. 11 for an illustration ofa finite resolution Λ). Due to the group propertyof a multiplicative cascade discussed in Sec. 5, adressed cascade factors into its bare part and thehidden part, which corresponds to a flux of a cas-cade from L to Lλ/Λ rescaled with the help of

the pushback operator T ∗λλ. For small enough order

moments, the flux prefactor remains finite in thelimit Λ → ∞, the bare and dressed scaling proper-ties are the same. However for q ≥ qD this prefactorbegins to scale with Λ/λ, so that a drastic changeoccurs; it diverges with Λ.

In a real system, the scaling is cutoff by dissi-pation so that while the moments q < qD dependon the large scales in contrast, the value of themoments q > qD will depend on the small scaledetails. Since it is this small scale activity thatcauses the divergence, in analogy with the classical(chaotic) metaphor for sensitive dependence, thishas been called the “multifractal butterfly effect”[Lovejoy & Schertzer, 1998].

9.3. Scale dependence anddivergence of the flux : Theheuristic argument

Let us first consider some simple heuristics whosemain interest is that they are model independent.They are based on the fact that a D-dimensionalintegration of a singularity γ just corresponds toshifting the latter by −D, which corresponds tothe scaling exponent of the elementary volume ofintegration. As a consequence, all singularities oforder γ < D will be smoothed out. This alreadyexplains why the question of statistical divergencesis beyond the scope of deterministic multifractalformalisms (see Sec. 4.3). On the contrary, diver-gences arising from γ ≥ D will not be smoothedout and therefore the scale of observation is irrel-evant: the flux will scale with the inner scale ofactivity of the cascade and therefore will divergeas Λ → ∞. However due to its low statisticalweight, this divergence may remain statisticallyinsignificant. Nevertheless, one may reach a criticalγD ≥ D where it becomes significant. Above γD,the observed dressed codimension function cd(γ) nolonger corresponds to c(γ): the dressed quantitieswill have much larger fluctuations than the barequantities. cd(γ) can therefore be estimated by con-sidering that it should maximize the occurrences ofhigh singularities, while nevertheless respecting theconvexity constraint. This means that cd(γ) shouldfollow the tangency of c(γ) at γD:

γ ≤ γD : cd(γ) = c(γ)

γ ≥ γD : cd(γ) = c(γD) + qD(γ − γD)(102)

Since qD = c′(γD) is the critical order corre-sponding to γD in the framework of the Legendre

January 19, 2012 9:1 WSPC/S0218-1274 03064


(a) (b)

Fig. 26. This shows two realizations (a) of a random multifractal process with a single value of the maximum of the subgen-

erator (at the centre of a 512 × 512 grid) boosted by factors of N from 16 to 64 (increasing by 2(1/2) from top to bottom) inorder to simulate very rare events. The scaling is anisotropic with complex eigenvalues of G, the scale function is shown in (b).

duality (Sec. 4.2), we see that the divergence of thestatistical moments for q ≥ qD is a consequenceof the fact that a straight line is singular for theLegendre transform. We therefore obtain:

q < qD: Kd(q) = K(q); q ≥ qD: Kd(q) = ∞(103)

Kd(q) = K(q); q < qD

Kd(q) = ∞; q ≥ qD.(104)

This heuristic argument can be made rigor-ous by introducing [Schertzer & Lovejoy, 1987]the Trace Moments of the flux which are sim-pler to handle than the statistical moments of theflux.

9.4. Finite sample size effects

In practice, we are only able to examine finite sam-ples, hence instead of computing the theoretical

January 19, 2012 9:1 WSPC/S0218-1274 03064


moments:

〈Xq〉 =∫

xqdPX (105)

we only deal with estimates, the most usual onesbeing an average over the Ns independent samples:

Xqs =1

Ns

Ns∑i=1

Xqi . (106)

As long as the law of large numbers applies,these estimates converge towards the theoreticalmoments:

〈Xq〉 = limNs→∞

Xqs. (107)

One may also consider space/time averages andergodicity assumptions. In our case, we must con-sider a combination of statistical and space/timeaveraging. A first consequence of finite Ns is thatonly a limited range of q’s can in fact be safelyexplored: as we now show, estimates of momentsof higher order give no real information about theprocess and may lead to erroneous interpretations,e.g. there had been speculations on the significanceof the q → ∞ limit on the basis of finite empiricalsamples of turbulence data.

The limits imposed by finite sample sizes canbest be understood with the help of the samplingdimension Ds [Eq. (19) and Sec. 4.1]. As we increaseDs (i.e. the number Ns of samples), we graduallyexplore the entire probability space encounteringextreme but rare events that would almost surelybe missed on any finite sample (Fig. 27), i.e. larger

A

N s

Ds

Independent Realizations

Physical space

Physical space

Probability Space

~ λ

Fig. 27. Illustration showing how in random processes theeffective dimension of space (D) can be augmented by con-sidering many independent realizations Ns. As Ns → ∞, theentire (infinite dimensional) probability space is more andmore explored. From [Schertzer & Lovejoy, 1993].

and larger sampling singularity γs [Eq. (19) andFig. 8(b) for illustration].

The Legendre transform of cs(γ) = c(γs) withγ ≥ γs leads to a spurious linear estimate Ks

instead of the nonlinear K for q > qs where qs =c′(γs) is the maximum moment that can accuratelybe estimated:

q ≥ qs : Ks(q) = γs(q − qs) + K(qs);

q < qs; Ks(q) = K(q).(108)

In Sec. 9.5.2 we show that this linear behaviorcorresponds to the multifractal analogue of a sec-ond order phase transition and therefore is rathermodel-independent.

9.5. Multifractal phase transitions

9.5.1. Flux dynamics and thermodynamics

As discussed by various authors [Schuster, 1988;Tel, 1988; Schertzer & Lovejoy, 1992, 1994;Schertzer et al., 1993], there are formal analogiesbetween multifractal exponents and standard ther-modynamic variables. However, depending on themultifractal framework there are notable differencesof appreciation. For the codimension multifractalformalism, Table 1 displays the analogues betweenwhat can be called (statistical) “fluxdynamics”[Schertzer & Lovejoy, 1991] and classical thermody-namics. The expression “flux dynamics” comes fromthe fact that the main quantity of interest is theenergy flux for systems out of equilibrium insteadof the energy for systems in thermodynamic equi-librium. Since the analogies are based on the expo-nents of the probability and number densities theseanalogues are more easily and naturally obtainedin the codimension framework, which define respec-tively the codimension c(γ) of a singularity γ andthe D−S(E) entropy of a state energy E. The Leg-endre conjugate variables of the singularity and theenergy are respectively the moment of order q andthe (reciprocal) temperature β = 1/T . Similarly,the scaling moment function K(q) is the analogueof a (Massieu) potential (= the free energy/T ).

Discontinuities in the analogues of the freeenergy (the dual codimension function C(q) =K(q)/(q − 1)) and the thermodynamic potential(K(q)) correspond to multifractal phase transitions(see Fig. 28 for illustrations). However, thermody-namics applies to systems in equilibrium and with-out dissipation, whereas flux dynamics applies tosystems far from thermodynamic equilibrium whichare strongly dissipative. A practical consequence is

January 19, 2012 9:1 WSPC/S0218-1274 03064


Table 1. Correspondence between fluxdynamics and thermodynamics (setting for notation simplicity kB = 1 forthe Boltzman’s constant kB): Σ(β) being the Massieu potential, F (β) the Helmhotz free energy. From [Schertzer &Lovejoy, 1993].

Flux Dynamics Thermodynamics

Probability space Phase space

Moment order: q (Reciprocal) Temperature: β = T−1

Singularity order: γ (Negative) Energy: −E

Generator (Negative) Hamiltonian

Singularity codimension: c(γ) Codimension of entropy: D − S(E)

Scaling moment function: K(q) = maxγ

(qγ − c(γ)) (Negative) Massieu potential: −Σ(β) = −minE

(βE − S(E))

Dual codimension function: C(q) = K(q)/(q − 1) (Negative) Free energy −F (β) = −Σ(β)/β

Dimension of integration: D External field: h

Ratio of scales: λ Correlation length: ξ

that in order to define its statistics, a multifractalprocess fundamentally requires an infinite hierarchyof temperatures, rather than a single temperature.Therefore observing (i.e. with a finite Ns,Ds) a mul-tifractal process at a given temperature (i.e. q) onlygives very limited information about the process.Similarly, a multifractal phase transition is associ-ated with a qualitative change of observation of thesame system when one changes the observation tem-perature (i.e. changes q), whereas a thermodynamicphase transition corresponds to a qualitative changeof the behavior of the system under observation.

9.5.2. Second order phase transition

Finite sample size effects (Sec. 9.4) can be nowunderstood as corresponding to a phase transitionof second order and in fact a “frozen free energy”transition which has been discussed in various con-texts [Derrida & Gardner, 1986; Mesard et al., 1987;Brax & Pechanski, 1991]. Indeed, we saw that thealmost sure highest order singularity (γs) which canbe observed on Ns realizations, yields with the helpof the Legendre transform a linear behavior of theobserved Ks [Eq. (108)] for q > qs, whereas it isnonlinear as K(q) for q < qs. Therefore, Ks hasa discontinuity of second order at γs (see Fig. 28for an illustration). On the other hand, this linearbehavior implies that the observed analogue of thefree energy Cs(q) seems to be “frozen” for low tem-perature (q → ∞), since we have:

Cs(q) ≡ Ks(q)q − 1

≈ γs

(1 + q−1

(1 − qs

γs

)). (109)

Further to the heuristics derivation we havepresented here, some exact mathematical resultshave been obtained, which are however restricted

to discrete cascades and furthermore to Ds = 1.On the other hand, the notion of second orderphase transition is interesting, because it is rathermodel-independent as it is based on the analogies ofthe statistical exponents of the cascade. Indeed, itshould occur as soon as there are no bounds on thesingularities or their range exceeds the critical γs.

9.5.3. First order phase transition

We can now revisit the question of the divergenceof moments (Sec. 9.3) taking care of the sample

6543210-1

1

3

5

7

9

11

13

15

17

q

K(q

)

Fig. 28. A numerical simulation of multifractal phase tran-sitions for a universal multifractal (α = 2, C1 = 0.9, Λ =210, 29 realizations): K(q) (theoretical, parabola), Ks(q)(observed second order transition — along a tangency toK(q) for large q’s — corresponding to the finite size ofthe sample at the highest resolution, bare, open triangles),Kd,s(q) (observed first order transition, partially dressed

(λ = 27) above K(q) for large q’s, closed triangles), Kd,s(q)(theoretical first order transition, fully dressed, straight line).Reproduced from [Schertzer et al., 1993].

January 19, 2012 9:1 WSPC/S0218-1274 03064


size finite effects, in the heuristic and very generalframework we discussed in Sec. 4. We pointed outthat above a critical singularity γD, the dressedcodimension cd(γ) becomes linear [Eq. (102)]. Dueto the definition of the codimension [Eq. (16)],this corresponds to a power-law for the probabil-ity distribution [Eq. (101)], and by consequenceto a divergence of statistical moments [Eq. (99)].However, due to the finite size of the samples, oneobviously cannot observe with the help of the sta-tistical moment estimates this divergence, but infact only a first order transition, instead of the sec-ond order transition discussed above (Sec. 9.5.2).Indeed, following the argument for Eq. (19), themaximum observable dressed singularity γd,s is thesolution of:

cd(γd,s) = ∆s. (110)

By taking the Legendre transform of cd withthe restriction γd ≤ γd,s, we no longer obtain thetheoretical Kd(q) = ∞ for q > qD [Eq. (103)], butthen obtain the finite sample dressed Kd,s(q):

q ≤ qD : Kd,s(q) = K(q);

q ≥ qD : Kd,s(q) = γd,s(q − qD) + K(qD).(111)

As expected, Eq. (103) is recovered for Ns →∞, due to the fact that γd,s → ∞. For Ns largebut finite, there will be a high q (low tempera-ture) first order phase transition, whereas the scalebreaking mechanism proposed for phase transitionsin strange attractors [Szepfalusy et al., 1987; Csor-das & Szepfalusy, 1989; Barkley & Cumming, 1990]is fundamentally limited to high or negative tem-peratures (small or negative q). This transition cor-responds to a jump in the first derivative K ′(q) ofthe potential analogue [Schertzer et al., 1993]:

∆K ′(qD) ≡ K ′d,s(qD) − K ′(qD)

= γd,s − γD =∆s − c(γD)

qD. (112)

On small sample size (∆s ≈ c(γD)), this transi-tion will be missed, the free energy simply becomesfrozen and we obtain: Kd,s(q) ≈ (q−1)D, which wasalready discussed with the help of some empiricaldata [Schertzer & Lovejoy, 1984], whereas Eq. (111)corresponds to an improvement of earlier workson “pseudo scaling” [Schertzer & Lovejoy, 1984,1987]. The above relations, especially Eq. (112),were tested numerically with the help of lognormal

543210-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

q

K(q)

qD

Fig. 29. The empirical scaling exponent moment functionKd,s(q) of the atmospheric turbulence energy flux usingrespectively 4 (crosses) and 704 (dots) 10 Hz time series,compared to the (bare) K(q) (continuous line) with univer-sal multifractal parameters α = 1.45 and C1 = 0.24. Allthese curves collapse together until qD ≈ 2.4, whereas forlarger moment orders q’s the empirical estimates become lin-ear. The curve Kd,s(q) corresponding to an average of 704realizations is clearly above Kd,s(q), which is only possiblefor a first order multifractal transition, whereas a secondorder transition curve would correspond to a tangent to K(q).Reproduced from [Schmitt et al., 1994].

universal multifractals (see Fig. 28), as well onatmospheric data (see Fig. 29).

10. Future Directions

The goal of this paper was to present the maintheoretical developments of multifractals and gen-eralized scale invariance during the two and a halfdecades of their explicit existence, including numer-ical simulations and data analysis. We hope that itreveals the awesome simplification that multifrac-tals can bring to complex systems by reducing theircomplexity with the help of a quite diverse andwidely applicable scale symmetries.

Although multifractals are increasingly under-stood as a basic framework for analyzing andsimulating extremely variable processes, in manyscientific disciplines and in particular in geophysics,there is still a wide gap between their potentialand their actual use. Beyond the familiar slowand inhomogeneous pace of the diffusion of scien-tific knowledge, an additional reason for this couldbe obstacles inherited from older, preexisting and

January 19, 2012 9:1 WSPC/S0218-1274 03064


highly restrictive concepts of scaling: for examplethat they are only geometrical descriptors or thatthey are inherently isotropic. The latter had beena particularly important obstacle for the develop-ment of multifractals in geophysics due to privilegeddirections generated by gravity and the Earth’srotation: a generalization of the scale notion wasnecessary to overcome it.

We therefore expect that future developmentsof multifractals will build upon their most recentadvances so as to better tackle issues arising in geo-physical systems. This should include new advanceson model independent characteristics, such as uni-versality classes and multifractal phase transitions(beyond the scalar case), to substantially improvingexisting statistical estimators of multifractal param-eters, as well as their related uncertainties. Anotheraxis concerns the notion of scale itself: the fur-ther exploration of possible features of generalized(space-time) scales in particular in dynamical multi-fractal processes. This raises the questions of multi-fractal predictability and prediction that we did notaddress here, although we may expect a rapid devel-opment of multifractal forecast techniques, as wellas the discovery of stimulating connections with thetheory of random dynamical systems. Our currentunderstanding of multifractal extremes points outthe necessity — as well as the possibility — of devel-oping a new extreme value theory that could dealwith processes having long-range dependencies. Atthe same time, there remains the fundamental ques-tion of establishing a more direct connection, i.e.well beyond the present arguments based on numer-ics, phenomenology or on a too formal symme-try, between multifractals and the deterministic-likenonlinear equations that are supposed to generatethem, in particular the Navier–Stokes equations. Wepointed out current developments in this direction.This would have many fundamental consequencessuch as opening the road to new renormalizationtechniques able to grasp intermittency as well as toa better knowledge of the mathematical propertiesof the solutions of these equations.

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