Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology

The need of understanding and modelling the space-time variability of natural processes in hydrological sciences produced a large body of literature over the last thirty years. In this context, multifractal framework provides parsimonious models which can be applied to a wide scale range of hydrological processes, and are based on the empirical detection of some patterns in observational data, i.e. a scale invariant mechanism repeating scale after scale. Hence, multifractal analyses heavily rely on available data series and their statistical processing. In such analyses, high order moments are often estimated and used in model identification and fitting as if they were reliable. This paper warns practitioners for blind use in geophysical time series analyses of classical statistics, which is based upon independent samples typically following distributions of exponential type. Indeed, the study of natural processes reveals scaling behaviours in state (departure from exponential distribution tails) and in time (departure from independence), thus implying dramatic increase of bias and uncertainty in statistical estimation. Surprisingly, all these differences are commonly unaccounted for in most multifractal analyses of hydrological processes, which may result in inappropriate modelling, wrong inferences and false claims about the properties of the processes studied. Using theoretical reasoning and Monte Carlo simulations we find that the reliability of multifractal methods that use high order moments (> 3) is questionable. In particular, we suggest to use the first two moments in all problems as they suffice to define the most important characteristics of the distribution.