Multifractal behaviour of long‐term karstic discharge fluctuations

Karstic watersheds are highly complex hydrogeological systems that are characterized by a multiscale behaviour corresponding to the different pathways of water in these systems. The main issue of karstic spring discharge fluctuations consists in the presence and the identification of characteristic time scales in the discharge time series. To identify and characterize these dynamics, we acquired, for many years at the outlet of two karstic watersheds in South of France, discharge data at 3‐mn, 30‐mn and daily sampling rate. These hydrological records constitute to our knowledge the longest uninterrupted discharge time series available at these sampling rates. The analysis of the hydrological records at different levels of detail leads to a natural scale analysis of these time series in a multifractal framework. From a universal class of multifractal models based on cascade multiplicative processes, the time series first highlights two cut‐off scales around 1 and 16 h that correspond to distinct responses of the aquifer drainage system. Then we provide estimates of the multifractal parameters α and C₁ and the moment of divergence q_D corresponding to the behaviour of karstic systems. These results constitute the first estimates of the multifractal characteristics of karstic spingflows based on 10 years of high‐resolution discharge time series and should lead to several improvements in rainfall‐karstic springflow simulation models. Copyright © 2012 John Wiley & Sons, Ltd.     

Analyses multifractales et spatio-temporelles des précipitations du modèle Méso-NH et des données radar

Résumé

Dans le cadre des multifractals universels, il est possible de caractériser la variabilité spatio-temporelle de la pluie sur une grande gamme d’échelle à l'aide de trois paramètres invariants d’échelles. Dans cette étude, nous avons estimé ces paramètres multifractals sur des simulations numériques effectuées avec le modèle méso-échelle Méso-NH, développé par Météo-France et le Laboratoire d'Aérologie (Univ. P. Sabatier, Toulouse, France), et des images radar composites, couvrant le même événement pluvieux, à savoir un orage particulièrement violent, dit de type Cévenol, ayant eu lieu sur la partie sud de la France du 5 au 9 Septembre 2005. La comparaison des résultats montre que les deux types de données présentent des domaines d'invariance d’échelle relativement similaires, et dont les propriétés sont en accord avec les modèles de précipitation spatio-temporels unifiés et scalants les plus simples. Néanmoins l’évaluation de leurs exposants conduit à des valeurs parfois fortement différentes.

Citation Gires, A., Tchiguirinskaia, I., Schertzer, D. & Lovejoy, S. (2011) Analyses multifractales et spatio-temporelles des précipitations du modèle Méso-NH et des données radar. Hydrol. Sci. J. 56(3), 380–396.     

Multifractal comparison of the outputs of two optical disdrometers

ABSTRACT

In this paper a universal multifractals comparison of the outputs of two types of collocated optical disdrometers installed on the roof of the Ecole des Ponts ParisTech is performed. A Campbell Scientific PWS100 which analyses the light scattered by the hydrometeors and an OTT Parsivel² which analyses the portion of occluded light are deployed. Both devices provide a binned distribution of drops according to their size and velocity. Various fields are studied across a range of scales: rain rate (R), liquid water content (ρ), polarimetric weather radar quantities such the horizontal reflectivity (Z_h) and the specific differential phase (K_dp), and drop size distribution (DSD) parameters such as the total drop concentration (N_t) and the mass-weighted diameter (D_m). For both devices, good scaling is retrieved on the whole range of available scales (2?h–30?s), except for the DSD parameters for which the scaling only holds down to few minutes. For R, the universal multifractal parameters are found to equal 1.5 and 0.2 for α and C₁, respectively. Results are interpreted with the help of the classical Z_h–R and R–K_dp radar relations.

Editor D. Koutsoyiannis; Associate editor E. Volpi     

Multifractal versus monofractal analysis of wetland topography

The land surface elevation distribution will serve as fundamental input data to any wetland flow model. As an alternative to the traditional smooth function approach to represent or interpolate elevation data, we explore the use of Levy monofractals and universal multifractals as a means for defining a statistically equivalent topography. The motivation behind this effort is that fractals, like natural topography, are irregular, they offer a way to relate elevation variations measured at different scales, and the relationships are of a statistical nature. The study site was a riparian wetland near Savannah, GA, that contained beavers, and a total of four elevation transects were examined. The elevation increments showed definite non-Gaussian behavior, with parameter values, such as the Hurst coefficient and Lévy index (α), depending on the question of presence of beaver activity. It was obvious that the data were highly irregular, especially the transects influenced by beavers. Significantly different α values were obtained depending on whether the entire data set or just the tails were examined, which demonstrated inability of the monofractal model to reflect fully the irregularity of wetland data. Further analysis confirmed definite multifractal scaling, and it is concluded that the multifractal model is superior for this data set. Universal multifractal parameters are calculated and compared to those obtained previously for more typical terrain. Although it is difficult to consider a unique universal multifractal parameter α for the entire wetland, multifractal-like scaling was evident in each transect as reflected by the nonlinear behaviors of the scaling functions. We demonstrate a good agreement between theory and measurements up to a critical order of statistical moments, q _D , close to 3.5 and obtain realistic unconditioned simulations of multifractal wetland topography based on our parameter estimates. Future work should be devoted to conditioning multifractal realizations to data and to obtaining larger data sets so that the question of anisotropy may be studied.     

Blunt extension of discrete universal multifractal cascades: development and application to downscaling

ABSTRACT

Scale issues are ubiquitous in geosciences. Because of their simplicity and intuitiveness, and despite strong limitations, notably its non-stationarity features, discrete random multiplicative cascade processes are very often used to address these scale issues. A novel approach based on the parsimonious framework of Universal Multifractals (UM) is introduced to tackle this issue while preserving the simple structure of discrete cascades. It basically consists in smoothing at each cascade step the random multiplicative increments with the help of a geometric interpolation over a moving window. The window size enables to introduce non-conservativeness in the simulated fields. It is established theoretically,] and numerically confirmed, that the simulated fields also exhibit a multifractal behaviour with expected features. It is shown that such an approach remains valid over a limited range of UM parameters. Finally, we test downscaling of rainfall fields with the help of this blunt discrete cascade process, and we discuss challenges for future developments.     

The flood probability distribution tail: how heavy is it?

This paper empirically investigates the asymptotic behaviour of the flood probability distribution and more precisely the possible occurrence of heavy tail distributions, generally predicted by multiplicative cascades. Since heavy tails considerably increase the frequency of extremes, they have many practical and societal consequences. A French database of 173 daily discharge time series is analyzed. These series correspond to various climatic and hydrological conditions, drainage areas ranging from 10 to 10⁵ km², and are from 22 to 95 years long. The peaks-over-threshold method has been used with a set of semi-parametric estimators (Hill and Generalized Hill estimators), and parametric estimators (maximum likelihood and L-moments). We discuss the respective interest of the estimators and compare their respective estimates of the shape parameter of the probability distribution of the peaks. We emphasize the influence of the selected number of the highest observations that are used in the estimation procedure and in this respect the particular interest of the semi-parametric estimators. Nevertheless, the various estimators agree on the prevalence of heavy tails and we point out some links between their presence and hydrological and climatic conditions.