Investigations du caractère multifractal des débits maximaux annuels de crue / Investigation of multifractality of maximum annual flood discharges

Abstract

Abstract The multifractal analysis of maximum annual flood discharges at 55 stations of the Tunisian gauging network allows one to associate the various statistical moments of surface discharges of the basins through a scale-invariant law. On this basis, a random cascades model is identified. The scale-invariant law obtained for extreme values represents a theoretical foundation for empirical models proposed since the 1960s which link extreme values, and their associated hazards, with surfaces of drainage areas.     

Scale invariance for a heterogeneous fault model

A very simple model of fault fracturing is presented. The fault is envisaged as a series of faultlets whose area is randomly distributed and whose strength have a Gaussian distribution. Faultlet slip occurs when the loading stress reaches its strength. Large earthquakes are generated by coalescence of smaller cracks. Theb value of the Gutenberg-Richter frequency magnitude relation is then calculated. Ab equal to 1 (scale invariance) is obtained only for events with magnitudeM greater than 3. At lower magnitude the number of events decreases with decreasing magnitude. A similar result can be explained by the experimental observation of an anomalous scaling law of real earthquakes. Here the suggestion is advanced that a similar observation could be interpreted by a multifractal approach.     

Multifractal Analysis of the Arnea,Greece Seismicity with Potential Implications for Earthquake Prediction

Two-dimensional multifractal analysis is performed in a seismic area of Northern Greece responsible for recent strong earthquakes, including the Arnea sequence of May 1995, culminating in a M_w 5.3 event on 4/5/1995. It is found that multifractality gradually increases prior to the major seismic activity and that declusterization replaces clusterization not long before its initialization. The fractal dimensions D(q) (q > 0) abruptly drop for aftershocks, reflecting their very strong spatial clustering. The observed seismicity patterns seem to be compatible with a percolation process. Before the main sequence, the fractal dimension is consistently in the range 1.67–1.96 (standard deviation included). Percolation theory predicts 1.9 for 2D percolation clusters and 1.8 for the backbone of 3D percolation clusters. If the observed gradual increase in multifractality is due to multifractality reaching a maximum prior to the major slip (percolation), this may enable us to roughly estimate its time of occurrence.     

Multifractal Modeling and Lacunarity Analysis

The so-called gliding box method of lacunarity analysis has been investigated for implementing multifractal modeling in comparison with the ordinary box-counting method. Newly derived results show that the lacunarity index is associated with the dimension (codimension) of fractal, multifractal and some types of nonfractals in power-law relations involving box size; the exponent of the lacunarity function corresponds to the fractal codimension (E – D) for fractals and nonfractals, and to the correlation codimension (E – lpar;2)) for multifractals. These results are illustrated with two case studies: De Wijs's zinc concentration values from the Pulacayo sphalerite-quartz vein in Bolivia and Cochran's tree seedlings example. Both yield low lacunarities and slightly depart from translational invariance.     

Apparent/spurious multifractality of data sampled from fractional Brownian/Lévy motions

Many earth and environmental variables appear to be self‐affine (monofractal) or multifractal with spatial (or temporal) increments having exceedance probability tails that decay as powers of − α where 1 < α ≤ 2. The literature considers self‐affine and multifractal modes of scaling to be fundamentally different, the first arising from additive and the second from multiplicative random fields or processes. We demonstrate theoretically that data having finite support, sampled across a finite domain from one or several realizations of an additive Gaussian field constituting fractional Brownian motion (fBm) characterized by α = 2, give rise to positive square (or absolute) increments which behave as if the field was multifractal when in fact it is monofractal. Sampling such data from additive fractional Lévy motions (fLm) with 1 < α < 2 causes them to exhibit spurious multifractality. Deviations from apparent multifractal behaviour at small and large lags are due to nonzero data support and finite domain size, unrelated to noise or undersampling (the causes cited for such deviations in the literature). Our analysis is based on a formal decomposition of anisotropic fLm (fBm when α = 2) into a continuous hierarchy of statistically independent and homogeneous random fields, or modes, which captures the above behaviour in terms of only E + 3 parameters where E is Euclidean dimension. Although the decomposition is consistent with a hydrologic rationale proposed by Neuman (2003), its mathematical validity is independent of such a rationale. Our results suggest that it may be worth checking how closely would variables considered in the literature to be multifractal (e.g. experimental and simulated turbulent velocities, some simulated porous flow velocities, landscape elevations, rain intensities, river network area and width functions, river flow series, soil water storage and physical properties) fit the simpler monofractal model considered in this paper (such an effort would require paying close attention to the support and sampling window scales of the data). Parsimony would suggest associating variables found to fit both models equally well with the latter. Copyright © 2010 John Wiley & Sons, Ltd.