Multifractal

The fractal dimension of an object (field) measure the way this object fills the space. Hentscel and Procaccia (1983), Grassberger (1983), Schertzer and Lovejoy (1983) showed in the case of precipitation, that when we arise the threshold value of the singularity for some processes like the intensity, the fractal dimension decreases. In this case, one fractal dimension is not sufficient to describe the phenomenon. From this ascertainment comes the necessity of the generalization of the fractal concept.

The multifractal generalization for the function corresponds to the introduction of the multi-scale invariance which substitute the scale invariance. In order to describe a time series or a multifractal fields, we can say that the multifractal formalism describes an irregular field using two functions and which contain the information of the probability distribution at all scales.