F. Lombardo, E. Volpi, D. Koutsoyiannis, and S.M. Papalexiou, Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology, *Hydrology and Earth System Sciences*, 18, 243–255, doi:10.5194/hess-18-243-2014, 2014.

[doc_id=1343]

[English]

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 widescale 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.

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**Additional material:**

- Discussion version (HESSD, 10, 4627-4654, 2013) (999 KB)
- Comments by Fahim Ashkar (579 KB)
- Reply to Fahim Ashkar’s comments (182 KB)
- Comments by David Rupp (180 KB)
- Reply to David Rupp's comments (572 KB)
- Comments by Pierluigi Furcolo (576 KB)
- Reply to Pierluigi Furcolo’s comments (132 KB)
- "Multifractality: at least three moments" by Schertzer et al. (333 KB)
- Reply to "Multifractality: at least three moments" by Schertzer et al. (DOI: 10.13140/RG.2.1.350) (151 KB)
- "Further (monofractal) Limitations of Climactograms" by Lovejoy et al. (617 KB)
- Is consistency a limitation? —Reply to Lovejoy et al. (DOI: 10.13140/RG.2.1.2391.3207) (519 KB)

**See also:**
http://dx.doi.org/10.5194/hess-18-243-2014

**Remarks:**

Replies to discussions can also be found in:

http://dx.doi.org/10.13140/RG.2.1.3505.4325

http://dx.doi.org/10.13140/RG.2.1.2391.3207

**Our works referenced by this work:**

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**Our works that reference this work:**

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11. | D. Koutsoyiannis, Knowable moments for high-order stochastic characterization and modelling of hydrological processes, Hydrological Sciences Journal, 64 (1), 19–33, doi:10.1080/02626667.2018.1556794, 2019. |

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15. | P. Kossieris, I. Tsoukalas, A. Efstratiadis, and C. Makropoulos, Generic framework for downscaling statistical quantities at fine time-scales and its perspectives towards cost-effective enrichment of water demand records, Water, 13 (23), 3429, doi:10.3390/w13233429, 2021. |

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