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Universality

While a fractal set is primarily characterized by its fractal dimension, a general multifractal would require an infinite number of parameters: without some simplifying principle, it would be unmanageable. Fortunately, Schertzer and Lovejoy 1987 (see also 1997) show that under fairly general circumstances, the underlying dynamical multifractal processes have certain stable, attractive behaviours. That means that independent of many of the details, if the dynamical mechanism is repeated scale after scale (or interacts with enough indpendent processes over a fixed range of scales), the result is a special particulary simple "universal" multifractal behaviour.

The three universal parameters:

Universal multifractals are characterized by only three parameters:

a: This is the index of multifractality; it varies from 0 to its maximum value 2, and it describes how rapidly the fractal dimensions vary as we leave the mean. It is not very intuitive; the accompanying simulations may be the best way to visualize the effect of varying a.

C1: This describes the sparseness of the level of activity which gives the dominant contribution to the mean field. It is a "codimension"; the corresponding fractal dimension is d-C1 where d is the dimension of the observing space. It is bounded between 0 and d. For a multifractal developed over a range of scales l, this mean level is . Most geophysical multifractals have C1<0.2; many, including the turbulent wind field have C1<0.1. Although this value may seem small, if l is large enough, it can already imply huge variability ("intermittency"). For example in the atmosphere, l may be as large as 10^10 (10 000 km-1mm) so that for C1=0.1 we have 1- =90% of the field would be too weak to give a significant contribution to the mean. If we consider higher order moments, the situation is much more dramatic; for example, if a=2 (close to the observed values), then the corresponding fraction of the variance is 99.99%, while for the skewness (third order moment) the fraction is 1-10^9.

H: This is a kind of smoothness parameter; it denotes the order of the integration (H>0) or differentiation (H<0) needed to obtain the observed field from a (direct) multifractal cascade process. It can be, (and generally is) fractional. It corresponds to the exponent of a power law filter. Sometimes, such as turbulence, its value can be obtained by dimensional reasoning; 1/3 corresponds to the famous Kolmogorov value.

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