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Statistical Scaling of Randomly Fluctuating Hierarchical Variables  

Shlomo P. Neuman, Monica Riva, Alberto Guadagnini, Martina Siena, and Chiara Recalcati

Environmental variables tend to fluctuate randomly and exhibit multiscale structures in space and time. Whereas random fluctuations arise from variations in environmental properties and phenomena, multiscale behavior implies that these properties and phenomena possess hierarchical structures. Understanding and quantifying such random, multiscale behavior is critical for the analysis of fluid flow as well as mass and energy transport in the environment. The multiscale nature of randomly fluctuating variables that characterize a hierarchical environment (or process) tends to be reflected in the way their increments vary in space (or time). Quite often such increments (a) fluctuate randomly in a highly irregular fashion; (b) possess symmetric, non-Gaussian frequency distributions characterized by heavy tails, which sometimes decay with separation distance or lag; (c) exhibit nonlinear power-law scaling of sample structure functions (statistical moments of absolute increments) in a midrange of lags, with breakdown in such scaling at small and large lags; (d) show extended power-law scaling (linear relations between log structure functions of successive orders) at all lags; (e) display nonlinear scaling of power-law exponent with order of sample structure function; and (f) reveal various degrees of anisotropy in these behaviors. Similar statistical scaling is known to characterize many earth, ecological, biological, physical, astrophysical, and financial variables. The literature has traditionally associated statistical scaling behaviors of the aforementioned kind with multifractals. This is so even though multifractal theory (a) focuses solely on statistical scaling of variable increments, unrelated to statistics of the variable itself, and (b) explains neither observed breakdown in power-law scaling at small and large lags nor extended power-law scaling of such increments. A novel Generalized sub-Gaussian scaling model is introduced that does not suffer from such deficiencies, and some of its key aspects are illustrated on microscale surface measurements of a calcite crystal fragment undergoing dissolution reaction due to contact with a fluid solution.