BI-MULTIFRACTAL ANALYSIS AND MULTI-AFFINE MODELING OF NON-STATIONARY GEOPHYSICAL PROCESSES, APPLICATION TO TURBULENCE AND CLOUDS

In recent years, considerable progress has been achieved in the description of natural variability, largely due to the widespread use of scale-invariant concepts such as fractals and multifractals. In particular, this last concept has been used to clarify the fuzzy notion of "inhomogeneity" by introducing and quantifying the effects of intermittency. In this paper, we present a more comprehensive approach to multifractal data analysis and simulation that includes and combines the currently popular singularity analysis techniques with the more traditional approach based on structure functions. Being related to the new idea of "multi-affinity", these last statistics are regaining favor and constitute the proper framework to address the problem of quantifying and qualifying yet another outstanding fuzzy notion, that of "non-stationarity". This is an important step because non-stationary behavior is ubiquitous in Nature. Using turbulence as an example, we also show how a unified multifractal formalism can help in extracting, from data alone, the "effective constitutive laws" that describe phenomenologically the nonlinearities of the macroscopic transport processes that shape the geophysical field represented by the dataset. Finally, we argue that the essential multifractality of any natural system can be captured on the "q = 1 multifractal plane" and describe ways in which it can be used in practical geophysical problems.