5 - McGill Physics



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5. Cascades, dimensions and codimensions:

5.1 Multifractals and the codimension function

5.1.1 Probabilities and codimensions

We have given evidence that the atmosphere is scaling over wide ranges of scale (ch. 1), we have argued that the dynamics are also scaling (ch. 2) and lead to multiplicative cascades (ch. 3), and finally we have given empirical support for this (ch. 4). Throughout our approach was to provide the minimal theoretical framework necessary for understanding the most straightforward data analyses (e.g. the trace moments method, [pic]). We mentioned that specification of all of the statistical moments is generally a complete statistical characterization of the process and hence this was equivalent to their specification in terms of probabilities. However, we did not go further to specify the exact relation; the moment characterization was convenient and adequarte for our purposes. We now turn to the complete formalism needed to understand the probability structure of the cascades. In this chapter, we thus continue to study the properties of cascade this time emphasizing their probability distributions and their exponents, the codimensions.

5.1.2 Revisiting the b Model

We have already described the monofractal stochastic b model in ch. 3 that is said to be “monofractal” (sometimes called aor “unifractal”, because it can be defined with the help of a unique codimension); let us examine its probability structure as the cascade develops. Recall from ch. 3 that it has one parameter c>0 and that two states specifally the statistics of the multipliers me:

[pic] (1)

where l0 is the single step (integer) scale ratio. Recall that the magnitude of the boost [pic] >1 is chosen so that at each cascade step the ensemble averaged [pic] is conserved:

[pic] (2)

Indeed at each step in the cascade the fraction of the alive eddies decreases by the factor [pic] (hence the name “b model”) and conversely their energy flux density is increased by the factor [pic] to assure (average) conservation. Rather than follow ch. 3 and consider how the moments change with scale, let us now consider how the probabilities evolve as we increase the number of cascade steps. After n steps, the effect of the single-step dichotomy of “dead” or “alive” is amplified by the total (n step) scale ratio l =l0n:

[pic] (3)

hence either the density diverges en with an (algebraic) order of singularity c, but with an (algebraically) decreasing probability, or is “calmed” down to zero.

Following the discussion (and definitions) given in section 3.2, c is the codimension of the alive eddies, hence their corresponding dimension D is:

[pic] (4)

(d is the dimension of the embedding space, equal to 2 in Figures 3.4, 3.9). This is the dimension of the “support” of turbulence, corresponding to the fact that after n steps the average number of alive eddies in the [pic] is

[pic] (5)

5.1.2 Revisiting the a Model

In ch. 3, we introduced the a model which more realistically allows eddies to be either “more active” or “less active” according to the following binomial process:

[pic] (6)

where g+, g- correspond to boosts and decreases respectively, the b model being the special case where [pic] and [pic] (due to conservation = 1, there are only two free parameters eq. 3.6):

[pic] (7)

Taking [pic], the pure orders of singularity [pic]and [pic] lead to the appearance of mixed orders of singularity, of different orders [pic]. These are built up step by step through a complex succession of [pic] and [pic], values as illustrated in Figure 3.7b.

What is the behaviour as the number of cascade steps, [pic]? Consider two steps of the process, the various probabilities and random factors are:

[pic] (8)

This process has the same probability and amplification factors as a new three-state a model with a new scale ratio of l0 2 defined as:

[pic] (9)

Iterating this procedure, after n = n+ + n- steps we find:

[pic] (10)

where [pic] is the number of combinations of n objects taken n+ at a time. This implies that we may write:

[pic] (11)

The pij’s are the “submultiplicities” (the prefactors in the above), cij are the corresponding exponents (“subcodimensions”) and [pic] is the total ratio of scales from the outer scale to the smallest scale. Notice that the requirement that [pic] implies that some of the [pic] are >1 (boosts) and some are 0 and some gilg) = 1- CDF where CDF= the standard “cumulative probability distribution function” CDF =[pic]. However both are obviously related by Pr(el≥lg) =1-CDF. Here and throughout his book will will always use ≥ and use the term “probability distribution” in the sense of exceedence probability distribution and therefore for events above a given (scaling) threshold.

5.2 The Codimension Multifractal Formalism

5.2.1 Codimension of Singularities c(g) and its relation to K(q)

In this section we continue our discussion of multifractal fields in term of singularities, but also relate this to its dual representation in terms of statistical moments which we already discussed in ch. 3. Contrary to the popular dimension f(a) formalism which was developped for (low dimensional) deterministic chaos (see section 5.4), we develop a codimension formalism necessary for stochastic processes, it is therefore more general than the dimension formalism.

The measure of the fraction (at resolution [pic] with corresponding scale[pic]) of the probability space with singularities higher than [pic] is given by the probability distribution eq. 16. The previous section underlines this new feature; the exponent c(g) is a function not a unique value. Rather than dealing with just a scaling geometric set of points we are dealing with a scaling function (in the limit [pic], the density of a measure); from this function we can define an infinite number of sets, e.g., one for each order of singularity [pic](fig. 5.1).

Figure 5.1 here

We now derive the basic connection between c(g) and the moment scaling exponent K(q). To relate the two; write the expression for the moments in terms of the probability density of the singularities:

[pic] (17)

(where we have absorbed the c’(g)logl factor into the “[pic]” symbol since it is slowly varying, subexponential). This yields:

[pic] (18)

where we have used [pic] (this is just a change of variables [pic] for [pic], [pic] is a fixed parameter).

Hence:

[pic] (19)

We see that our problem is to obtain an asymptotic expansion of an integral with integrand of the form exp(x f(g)) where x = logl is a large parameter and f(g) = qg - c(g). These expansions can be conveniently preformed using the mathematical technique of “steepest descents” e.g. (Bleistein and Handelsman, 1986) which shows that the dominant term in the expansion for the integral is [pic] (i.e. the integral is dominated by the singularity g which yields the maximum value of the exponent) so that as long as x = log[pic]:

[pic] (20)

This relation between K(q) and c(g) is called a “Legendre transform” (Parisi and Frisch, 1985); see fig. 5.2. We can also invert the relation to obtain c(g) from K(q); just as the inverse Laplace transform used to obtain K(q) from c(g) is another Laplace transform so the inverse Legendre transform is just another Legendre transform. To show this, consider the twice iterated Legendre transform F(q) of K(q):

[pic] (21)

Taking ∂F/∂γ = 0 ⇒ q = q' so that we see that F(q) = K(q). This shows that a Legendre transform is equal to its inverse, hence we conclude:

[pic] (22)

The g which for a given q maximizes qg - c(g) is gq and is the solution of c’(gq) = q (fig. 5.3). Similarly, the value of q which for given g maximizes qg- K(q) is qg so that:

[pic] (23)

This is a one-to-one correspondence between moments and orders of singularities (see figures 5.2 and 5.3). Note that if [pic] is bounded by gmax (for example in microcanonical cascades, g≤d; ch. 3 or for the a model; g ≤ g+) there is a qmax = c’(gmax) such that for q>qmax, K(q) = qgmax -c(gmax), i.e. K(q) becomes linear in q (see Figure 5.4).

Figure 5.2, 5.3, 5.4 here

5.2.2 Properties of codimension functions

We have seen that for each singularity order g, that c(g) is the statistical scaling exponent characterizing how its probability changes with scale. The first obvious property is that due to its very definition (eq. 16) c(g) is an increasing function of g: c’(g)>0. Another fundamental property which follows directly from the Legendre relation with K(q), is that c(g) must be convex: c’’(g)>0.

Many properties of the codimension function can be illustrated graphically. For example, consider the mean, q =1. First, applying K’(q) = g (eq. 23) we find K’(1) = g1 where [pic] is the singularity giving the dominant contribution to the mean (the q = 1 moment). In ch. 3 we have already defined C1 = K’(1), so that this implies C1 = g1; the Legendre relation thus justifies the name “codimension of the mean” for C1. Also at q = 1 we have K(1) = 0 (due to the scale by scale conservation of the flux) so that from eq. 22, C1 = c(C1) as indicated in Figure 5.5 (this is a fixed point relation). C1 is thus simultaneously the codimension of the mean of the process and the order of singularity giving the dominant contribution to the mean. Finally, applying c’(g) = q (eq. 23) we obtain c’(C1) = 1 so that the curve c(g) is also tangent to the line x = y (the bisectrix). If the process is observed on a space of dimension d, it must satisfy d≥ C1, otherwise, following the above, the mean will be so sparse that the process will (almost surely) be zero everywhere; it will be “degenerate”. We will see that when C1>d that the ensemble mean of the spatial averages (the dressed mean) cannot converge.

Taking ensemble averages we see that = DxH ≈ l-H where we have taken = constant and Dx ≈ l-1, H thus determines the deviation from scale by scale conservation of , it is the basic fluctuation exponent. At the level of random variables, writing el = lg we have Dfl = lg-H so that Dfl has the statistics of a bare cascade process with a translation of singularities by -H (see figure 5.6). This generalizes the Kolmogorov relation (f = v, H = 1/3) although we shall see that such simple change of cascade normalization (g->g-H) is a poor model of the observable f fields; which are best modelled by fractional integration of order H, see section 5.5. Finally, since c(g) is convex with fixed point C1, it is possible (see Figure 5.7) to define the degree of multifractality [pic] by the (local) rate of change of slope at C1, (the singularity corresponding to the mean) its radius of curvature Rc(C1) is:

[pic] (24)

Using the general relation c’(C1)=1 we obtain [pic] hence we can locally (near the mean g = C1) define a curvature parameter a from either of the equivalent relations:

[pic] (25)

These local (q = 1, g = C1) definitions of a are equivalent to the definition via moments a = K’’(1)/K’(1) (ch. 3). For the universal multifractals (ch. 3), this becomes global (i.e., is sufficient to describe the entire c(g) function), and we find an upper bound (maximum degree of multifractality) [pic] (a parabola, eq. 3.40). The [pic] case is the monofractal extreme [pic] whose singularities all have the same fractal dimension.

Figures 5.5, 5.6, 5.7 here

5.2.3 The sampling dimension, sampling singularity and decondsecond-order multifractal phase transitions

The statistical properties we have discussed up until now are valid for (infinite) statistical ensembles. However, real world data are always finite so that sufficiently rare events will always be missed. In order to understand the effect of finite sample sizes, consider a collection of Ns samples each d dimensional and spanning a range of scales l = L/l (= largest / smallest) so that for example there are ld pixels from each sample, (fig. 5.8) and introduce the “sampling dimension” Ds:

[pic] (26)

as well as the singularity gs corresponding to the largest (and rarest) value el,s the "sampling singularity":

[pic] (27)

The relation between Ns and eλ,s is thus equivalent to the relation between their base l logarithms i.e. between Ds and gs (eqs. 26, 27).

In order to relate gs and Ds, consider a collection of satellite images (d = 2). Our question is thus: what is the rarest event with the most extreme gs that we may expect to see on a single picture? On a large enough collection of pictures? The answer to, these questions is straightforward: there are a total of ld+Ds pixels in the sample; hence the rarest event has a probability ≈ l-(d+Ds) (see fig. 5.8). However the probability of finding gs is simply l-c(gs) so that we obtain the following implicit equation for gs:

[pic] (28)

Ds = d+Ds is the corresponding (overall) effective dimension of our sample. More extreme singularities would have codimensions greater than this effective dimension c > Ds and are almost surely not present in our sample.

Figure 5.9, 5.10 here

As an example, we can use the aircraft data shown in fig. 3.1 to estimate the largest singularity that we should expect over a transect 213 points long. We saw that the largest normalized flux value was ≈26.5 which corresponds to an order of singularity of gs = loge/logl = log(26.5)/log(213) = 0.364. Using the estimated multifractal parameters a = 1.8, C1= 0.06 (these are mean C1, a values for 24 flight legs 1120 km long), we find that in a d =1 section that the solution of c(gs) = 1.364 is gs = 0.396 which is very close to the observed maximum.

Let us now calculate the moment exponent Ks(q) for a process with Ns realizations. To do this, we calculate the Legendre transform of c(g) but with the restriction g≤gs; this is the same type of restriction as discussed earlier (fig. 5.4, take gmax = gs):

[pic]

[pic] (29)

hence at q = qs there is a jump/discontinuity in the second derivative of K:

[pic] (30)

due to the existence of formal analogies between multifractal processes and classical thermodynamics, this is termed a “second order multifractal phase transition” (Szépfalusy et al., 1987), (Schertzer and Lovejoy, 1992).

5.2.4 Direct empirical estimation of c(g): the probability distribution multiple scaling (PDMS) technique

In chapter 3, we saw how to empirically verify the cascade structure and characterize the statistics using the moments, and how to determine their scaling exponent, K(q). In this chapter, we saw how – via a Legendre transform of K(q) - this information can be used to estimate c(g). However, it is of interest to be able to estimate c(g) directly; to do this, we start from the fundamental defining eq. 16, take logs of both sides and rewrite it as follows:

[pic] (31)

“[pic]”[pic] corresponds to the logarithm of the sub-exponential (slowly varying) factors that are hidden in the “≈” sign in eq. 16 and the subscript “l” on the probability has been added to underline the resolution dependence of the cumulative histograms. For each order of singularity g, this equation expresses the linearity of log probability with the log of the resolution. The singularity itself must be estimated from the fluxes by:

[pic] (32)

We now see that things are a little less straightforward than when estimating K(q). First, the term “[pic][pic]” may not be so negligible, in particular for moderate l’s, so that using the simple approximation [pic] may not be sufficiently accurate. Second, in eq. 32, we assumed that el is normalized such that =1; if it is not, it is can be normalized by dividing by the ensemble mean: el->el/. However from small samples, there may be factors of the order 2 in uncertainty over this so that even the estimate of g may involve some uncertainty. In comparison, if one wants to estimate K(q), one needn’t worry about either of these issues since (even for the un-normalized el) the linear relation [pic] is exact (at least in the framework of the pure multiplicative cascades): K(q) is simply the slope of the [pic] versus logl graph (and if the normalization is accurate, the outer scale itself can be estimated from the points where the lines cross, see the examples in ch. 4). The relative simplicity of the moment method explains why in practice it is the most commonly used. c(g) can then be estimated from K(q) by Legendre transform (either numerically or using a universal multifractal parametrization).

In order to exploit eqs. 31, 32 to directly estimate c(g), we may thus calculate a series of histograms Prl of the lower and lower resolution fluxes (obtained from el by the usual degrading / “coarse-graining”/ averaging) and use the transformation of variables eq. 32 to write the histograms in terms of g rather than el. There are then two variants on this “probability distribution multiple scale” (PDMS) method (Lavallée et al., 1991). The first and simplest is simply to ignore the typically small “slow” term in eq. 31 and simply use [pic]; the second performs (for each of a series of standard g values) regressions of logPrl versus log l and then estimates c(g) as the (negative) slope.

In fig. 5.10 a, b we show some examples of the first method when applied to the aircraft data discussed in ch. 3; 24 legs, each of 4000 points were used with a resolution of 280 m (i.e. a total of 24 x 4000 = 9.6 x 104 data points for each field), the absolute differences at 280 m resolution were analyzed. In ch. 3 we already considered the temperature flux and showed graphically that the transformation from el’s into g’s did indeed result in an apparently stable dynamic range of the g’s (in comparison, the range of the degraded el’s kept diminishing as l decreased; see fig. 3.2a,b). Figure 5.10a, b simply shows the logl-normalized log probability of the histograms at each resolution. The resolution was degraded systematically by factors of 2; only the first 7 iterations (i.e. a degradation of resolution factor of 128 corresponding to ≈ 36 km) are shown (the corresponding moments are shown in fig. 4.6a,b). From the relatively tight bunching of the curves, we see that the method is reasonably successful: we have effectively removed almost all of the scale dependency from the histograms. Although the curves tend to diverge somewhat at large, (and rare) values of g (where the sample size starts to be inadequate), the basic c(g) shape is discernible. To be a bit more quantitative, we can use the estimate of the sampling dimension Ds = logNs/logl = log24/log4000 ≈ 0.38 to determine that the estimates are reliable only up to c = d+Ds =1.38. Also shown in the figures are straight reference lines since we shall see in the next section that one generally expects high order moments to diverge implying asymptotically linear c(g)’s. Along with both the transverse and longitudinal wind components, and the thermodynamic fields we have also included the essentially aircraft measurement/trajectory specific fields z, p: the fluxes derived from aircraft altitude and pressure (recall we used absolute differences). These both show extreme tails (c.f. the reference slopes ≈ 3), and this in spite of the autopilot attempting to enforce an isobaric trajectory! In fact, as discussed in ch. 6 it seems that due to the aircraft response to wind turbulence, the trajectories only begin to be effectively isobaric for scales around 40 km and greater (i.e. just the resolution of the curves is shown in the figures).

Finally, we show a similar analysis of hourly raingauge data from Nime France (1972- 1975, ≈35,000 points). Again we see a good collapse to a unique c(g), and this time there is evidence for an asymptotic linear behaviour with slope qD ≈3 (see section 5.4).

Figure 5.11 a, b, c here

5.2.5 Codimensions of Universal multifractals, cascades

When discussing the moment characterization of the cascades, we have already noted that the two parameters C1, a are of fundamental significance. C1 characterizes the order and codimension of the mean singularities of the corresponding conservative flux, it is the local trend of the normalized K(q) near the mean; K(q) = C1 (q-1) is the best monofractal “b model approximation near the mean” (q ≈ 1). Finally, a = K’’(1)/K’(1) characterizes the curvature near the mean. The curvature parameter [pic] can also be defined directly from the probability exponent c(g) by using the local radius of curvature Rc(C1) of c(g) at the point g = C1, i.e. the corresponding singularity (eq. 25). Finally, for the observed field f, there is a third exponent H which characterizes the deviation from conservation of the mean fluctuation Df ≈ DxH ≈ DxH since =constant, it is a “fluctuation” exponent.

In ch. 3 we discussed how the universal attractors of additive processes can be used to deduce those of the multiplicative processes by studying their “generators”, Gl = log[pic]. Since multiplying fields [pic] is equivalent to adding generators [pic] (for a fixed scale ratio [pic]); the generators which are stable and attractive under addition are the Lévy generators discussed in ch.3 and to which we return in section 5.5. The moment exponent K(q), (see eq. 3.40) is given by:

[pic] (33)

[pic] (34)

The top formula is valid for 0≤a≤2; however as discussed in section 5.5, K diverges for all q 0, g+>0 and the relation between g+ and g- (top line) is from the conservation requirement of the a model (eq. 7; see (Schertzer et al., 1995)). Clearly, g+ is the highest order singularity and c is the corresponding codimension so that the log-Poisson cascade has instrinsically a maximum singularity that it can produce. The log-Poisson cascade shares with the Levy generator universal multifractals the possibility of “densififying” the cascade - i.e. it can be made continuous in scale (“infinitely divisible”; section 5.6, take the limit l0->1), so that it could be said to have “weak” universality properties. However, the Levy generator cascades, could be termed “strongly” universal since the generator is stable and attractive as well.

The limitations of the log-Poisson cascade can be seen as soon as one considers applications. For example, in hydrodynamic turbulence, (She and Levèsque, 1994) assumed non fractal, d =1 filament-like structures for the highest order singularity along with homogeneous eddy turn over times, this leads to the parameter choice c = 2, g+ = 2/3 (implying l1g- = 3/2). In Magneto-Hydrodynamic turbulence one can argue that extreme events occur on current sheets and select c =1, g+ = 1/2 (Grauer et al., 1994). In both cases, the use of a model with maximum order of singularity seems difficult to justify, as do the particular parameter values. In the atmosphere, (Deidda, 2000) and (Onof and Arnbjerg-Nielsen, 2009) have used the log-Poisson model for modelling rainfall, where they found - perhaps unsurprisingly - that the parameters c, g+ vary from realization to realization, exactly as predicted by models with unbounded singularities (such as the strongly Levy generator universal cascades with a>1 where on any realization, the maximum singularity is itself simply a random variable rather than a fixed parameter).

Figures 5.12 a, b, c , 5.13 here

5.3 Divergence of Statistical Moments and extremes

5.3.1 Dressed and bare moments

We now consider the effect of spatially integrating a cascade and then taking the limit [pic]. This leads to the fundamental difference between the “bare” and “dressed” cascade properties; the former have all moments finite (since by definition, for bare quantities, [pic] is finite) whereas the latter will generally have divergence for all moments greater than a critical value qD which depends on the dimension of space over which the process is integrated, see fig. 5.14 for a schematic.

Figure 5.14 here

In order to define the dressed flux, start by defining the L resolution flux [pic] over the set A:

[pic] (37)

We can now define the “partially dressed” flux density [pic] as:

[pic] (38)

where vol(Bl) = l-D is the D-dimensional volume of a ball (interval, square, cube etc.) of size L/l and the “(fully) dressed flux density” as:

[pic] (39)

The terms “bare” and “dressed” are borrowed from renormalization jargon and are justified because the “bare” quantities neglect the small scale interactions (D. This shows that microcanonical models - which must have gmax≤ D - cannot display divergence of moments – at least not on the spaces over which the microcanonical constraint is imposed.

A linear cd(g) implies a power law tail on the probability distributions; this is just another way of obtaining the fundamental equivalence between divergence of moments q≥qD and "hyperbolic" or "fat tailed" behaviour of the probability distribution (for large enough thresholds s):

[pic] (50)

In order to observe the algebraic probability tail however, the sample size must be sufficiently large. Let us consider the effect of varying Ds = logNs/logl. Following the argument for eq. 28 section 5.2.3, the maximum observable dressed singularity gd,s is given by the solution of [pic] and by taking the Legendre transform of cd(g) with the restriction [pic] (see fig. 5.16 and 5.17) we obtain the finite sample dressed Kd,s(q):

[pic]

[pic] (51)

In the limit [pic], [pic], and for q>qD, [pic] as expected. This transition corresponds to a jump in the first derivative of the K(q):

[pic] (52)

hence, this is a “first order multifractal phase transition”.

Figure 5.16, 5.17 here

5.3.3 The Physical Significance of the Divergence of Moments: the multifractal butterfly effect, self-organized criticality

In the previous subsection we noted the equivalence between the divergence of moments [pic] (q>qD) and "hyperbolic" or "fat tailed" behaviour of the probability distribution [pic] (eq. 50) i.e. for large enough thresholds s, an algebraic fall-off of the probability distribution. In real cascades, viscosity cuts off the cascade process at the viscous scale so that real observables are only partially dressed (section 5.3.1). This implies that the graphs of logPr versus logs will only be linear over a finite interval; for sufficiently large s, they will be truncated. However, it must not be concluded that the divergence of moments is academic - on the contrary, it has a profound physical significance. To see this, denote the inner scale l, the observing scale L, and the ratio L = L/l>>1. In this case, our previous analysis for dressed moments will be valid, and if we estimate dressed moments with q>qD (i.e. C(q)>D), then the result will be [pic](appendix 5A), which can be very large and whose value will depend crucially on the small scale details, i.e., the exact value of l, etc. On the contrary, when q0 and Lévy index a therefore satisfies:

[pic]

[pic] (69)

(with corresponding exception for a = 1). Due to the additivity of second characteristic functions for any independent, identically distributed random variables (section 3.3.3), this implies that for the sum of two statistically independent Lévy variables A, B we have:

[pic] (70)

where C is also an extremal Lévy with the same a but with amplitude c. Eq. 70 expresses the “stability under addition” property of the Lévy variables and a, b, c are the corresponding amplitudes. Equation 70 shows that for Lévy variables [pic](the second characteristic function) of the random variables [pic] (amplitude f) are “a additive”, a property that generalizes to Lévy noises; we use this below.

To understand why only extremals must to be used to generate cascades (Schertzer and Lovejoy, 1987), (Schertzer and Lovejoy, 1991), and not the more general Lévy variables la with a>0, A+ =0. The latter are required to avoid the divergence of the Laplace transform for q>0, simply because an algebraic fall-off cannot tame an exponential divergence. It turns out that for a1, although most of the values have the same sign as the tail. The qualitative difference between the a>1 and a>1 when a>1 whereas for a ................
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