Abstract
Abstract. 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, a 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 against the 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 that, because of estimation problems, the use of moments of order higher than two should be avoided, either in justifying or fitting models. Nonetheless, in most problems the first two moments provide enough information for the most important characteristics of the distribution.
Highlights
A simple way to understand the extreme variability of several geophysical processes over a practically important range of scales is offered by the idea that the same type of elementary process acts at each relevant scale
The multifractal framework provides parsimonious models to study the variability of several natural processes in geosciences, such as rainfall
The classical statistical approaches rely on several simplifying assumptions, tacit or explicit, such as independence in time and exponentially decaying distribution tails, which are invalidated in natural processes causing bias and uncertainty in statistical estimations
Summary
A simple way to understand the extreme variability of several geophysical processes over a practically important range of scales is offered by the idea that the same type of elementary process acts at each relevant scale. Rainfall models of multifractal type have, for a long time been used to reproduce several statistical properties of actual rainfall fields, including the power-law behaviour of the moments of different orders and spectral densities, rainfall intermittency and extremes It has been a common practice to neglect this problem, which is introduced when the process exhibits dependence in time and is magnified when the distribution function significantly departs from the Gaussian form, which itself is an example of an exceptionally light-tailed distribution In their pioneering work on statistical hydrology, Wallis et al (1974) already provided some insight into the sampling properties of moment estimators when varying the marginal probability distribution function of the underlying stochastic process. We show (Sect. 3) that, even in quantities whose estimates are in theory unbiased, the dependence and non-normality affect significantly their statistical properties, and sample estimates based on classical statistics are characterized by high bias and uncertainty
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