Uncertainty,entropy, scaling and hydrological stochastics. 1. Marginal distributional properties of hydrological processes and state scaling / Incertitude,entropie, effet d'échelle et propriétés stochastiques hydrologiques. 1. Propriétés distributionnelles marginales des processus hydrologiques et échelle d'état

Abstract

The well-established physical and mathematical principle of maximum entropy (ME), is used to explain the distributional and autocorrelation properties of hydrological processes, including the scaling behaviour both in state and in time. In this context, maximum entropy is interpreted as maximum uncertainty. The conditions used for the maximization of entropy are as simple as possible, i.e. that hydrological processes are non-negative with specified coefficients of variation (CV) and lag one autocorrelation. In this first part of the study, the marginal distributional properties of hydrological variables and the state scaling behaviour are investigated. Application of the ME principle under these very simple conditions results in the truncated normal distribution for small values of CV and in a nonexponential type (Pareto) distribution for high values of CV. In addition, the normal and the exponential distributions appear as limiting cases of these two distributions. Testing of these theoretical results with numerous hydrological data sets on several scales validates the applicability of the ME principle, thus emphasizing the dominance of uncertainty in hydrological processes. Both theoretical and empirical results show that the state scaling is only an approximation for the high return periods, which is merely valid when processes have high variation on small time scales. In other cases the normal distributional behaviour, which does not have state scaling properties, is a more appropriate approximation. Interestingly however, as discussed in the second part of the study, the normal distribution combined with positive autocorrelation of a process, results in time scaling behaviour due to the ME principle.     

The influence of dam and lakes on the Yangtze River streamflow: long‐range correlation and complexity analyses

Employing long‐range correlation, complexity features and clustering, this study investigated the influence of dam and lake‐river systems on the Yangtze River flow. The impact of the Gezhouba Dam and the lake systems on streamflow was evaluated by analysing daily streamflow records at the Cuntan, the Yichang and the Datong station. Results indicated no evident influence of the Gezhouba Dam on streamflow changes. Distinct differences in scaling behaviour, long‐range correlation and clustering of streamflow at the Datong station when compared with those at the Cuntan and Yichang stations undoubtedly showed the influence of water storage and the buffering effect of the lake systems between the Datong station and other two hydrological stations on streamflow in the lower Yangtze River basin. Decreased regularity, enhanced long‐range correlation and increased clustering of streamflow in the lower Yangtze River basin due to the effect of water storage of the lake systems were corroborated. Copyright © 2011 John Wiley & Sons, Ltd.     

A probability-weighted moment test to assess simple scaling

We present a statistically robust approach based on probability weighted moments to assess the presence of simple scaling in geophysical processes. The proposed approach is different from current approaches which rely on estimation of high order moments. High order moments of simple scaling processes (distributions) may not have theoretically defined values and consequently, their empirical estimates are highly variable and do not converge with increasing sample size. They are, therefore, not an appropriate tool for inference. On the other hand we show that the probability weighted moments of such processes (distributions) do exist and, hence, their empirical estimates are more robust. These moments, therefore, provide an appropriate tool for inferring the presence of scaling. We illustrate this using simulated Levystable processes and then draw inference on the nature of scaling in fluctuations of a spatial rainfall process.     

西北太平洋热带气旋路径时间标度律的研究

时间标度计算表明，西北太平洋热带气旋路径是一个无标度性的系统，其关联方差谱遵从频率的－２￣－３次方幂律，不同背景下的路径系统均如此。由此得到的不同季节、不同地域的热带气旋路径可预报时间尺度基本上为３￣４ｄ，但异常热带气旋路径的可预报时间尺度则为１￣２ｄ。     

Further Studies Of Atmospheric Turbulence In Layers Near The Surface: Scaling The Tke Budget Above The Roughness Sublayer

The second of two experimental studies of the TKE budget conducted on sites of different roughness is described, and results are compared. The first took place within a shallow layer above a small field of mostly bare, cultivated soil; the second was carried out above a roughness sublayer of significant depth on an extensive plain of tall dry grass. Budget terms observed in the second study were scaled with a modified u which compensated for effects of an unusually large stress gradient and ensured that the _m functions would be collinear. By showing that the modification becomes negligible in smaller gradients, it is demonstrated that in normal conditions, budgets observed above significant roughness sublayers should be normalized by scaling in terms of the unreduced Reynolds stress at the sublayer's upper surface. This procedure is shown to be consistent with the expectation that TKE budgets in layers near the surface all scale in fundamentally the same way.Other findings include: (1) the fact that most _m functions previously reported are not quite collinear is attributed to a type of overspeeding known to affect three-cup anemometers; (2) revised _m functions, collinear and largely free of the effects of overspeeding, are determined from a well-established characteristic of the linear _m relation for the stable case; (3) data that define collinear _m functions can also be represented with single hyperbolic curves; (4) dissipation is found to be 10 to 15% too small to balance total TKE production in unstable and neutral conditions and to decrease with increasing z/L in thestable regime; and (5) new relations for based on the observed behaviour of the dissipation deficit provide an improved closure for the set of equations that express the budget terms as functions of _m and z/L.     

A structural model for the seismicity of the Arudy (1980) epicentral area (Western Pyrenees, France)

The Western Pyrenees presents a diffuse and moderate ( M ≤ 5.7) instrumental seismicity. It nevertheless historically suffered from strong earthquakes (I = IX MSK). The seismic sources of these events are not yet clearly identified. We focus on the Arudy (1980) epicentral area ( M = 5.1) and propose here the reactivation of early Cretaceous normal faults of the Iberian margin as a potential source. The late Cretaceous inversion of this basin, first in a left-lateral strike-slip mode and then in a more frontal convergence, resulted in a pop-up geometry. This flower structure attests of the presence of a deep crustal discontinuity.
The present-day geodynamic arrangement suggests that this accident is reactivated in a right lateral mode. This reactivation leads to a strain partitioning between the deep discontinuity that accommodates the lateral component of the motion and shallow thrusts, rooted on this discontinuity. These thrusts accommodate the shortening component of the strain. The distribution of the instrumental seismicity fits well the structural model of the Arudy basin. Whatever the compressive regional context, the structural behaviour of the system explains too the extensive stress tensor determined for the Arudy crisis if we interpret it in terms of strain ellipsoid. Indeed numerical modelling has shown that this concomitant activity of strike-slip and thrust faulting results in an extensive component that can rise 50 per cent of the finite strain.
We identify too a 25–30 km long potential seismic source for the Arudy area. The size of the structure and its potential reactivation in a strike-slip mode suggest that a maximum earthquake magnitude of ∼6.5 could be expected. The extrapolation of this model at the scale of the Western Pyrenees allows to propose other potential sources for major regional historical earthquakes.     

Scale Invariances in the Morphology and Evolution of Braided Rivers

This paper presents an overview and synthesis of an extensive research effort to characterize and quantify scale invariances in the morphology and evolution of braided rivers. Braided rivers were shown to exhibit anisotropic spatial scaling (self-affinity) in their morphology, implying a statistical scale invariance under appropriate rescaling of the axes along and perpendicular to the main direction of flow. The scaling exponents were found similar in rivers of diverse flow regimes, slopes, types of bed material and braid plain widths, indicating the presence of universal features in the underlying mechanisms responsible for the formation of their spatial structure. In regions where predominant geologic controls or predominant flow paths were present, no spatial scaling was found. Regarding their spatiotemporal evolution, braided rivers were found to exhibit dynamic scaling, implying that a smaller part of a braided river evolves identically to a larger one provided that space and time are appropriately normalized. Based on these findings, and some additional analysis of experimental rivers as they approach equilibrium, it was concluded that the mechanism bringing braided rivers to a state where they show spatial and temporal scaling is self-organized criticality and inferences about the physical mechanisms of self-organization were suggested.     

Characterization of Geochemical Distributions Using Multifractal Models

The use of multifractals in the applied sciences has proven useful in the characterization and modeling of complex phenomena. Multifractal theory has also been recently applied to the study and characterization of geochemical distributions, and its relation to spatial statistics clearly stated. The present paper proposes a two-dimensional multifractal model based on a trinomial multiplicative cascade as a proxy to some geochemical distribution. The equations for the generalized dimensions, mass exponent, coarse Lipschitz–Hölder exponent, and multifractal spectrum are derived. This model was tested with an example data set used for geochemical exploration of gold deposits in Northwest Portugal. The element used was arsenic because a large number of sample assays were below detection limit for gold. Arsenic, however, has a positive correlation with gold, and the two generations of arsenopyrite identified in the gold quartz veins are consistent with different mineralizing events, which gave rise to different gold grades. Performing the multifractal analysis has shown problems arising in the subdivision of the area with boxes of constant side length and in the uncertainty the edge effects produce in the experimental estimation of the mass exponent. However, it was possible to closely fit a multifractal spectrum to the data with enrichment factors in the range 2.4–2.6 and constant K₁ = 1.3. Such parameters may give some information on the magnitude of the concentration efficiency and heterogeneity of the distribution of arsenic in the mineralized structures. In a simple test with estimated points using ordinary lognormal kriging, the fitted multifractal model showed the magnitude of smoothing in estimated data. Therefore, it is concluded that multifractal models may be useful in the stochastic simulation of geochemical distributions.     

Fractures: Finite-size scaling and multifractals

The distributions of contact area and void space in single fractures in granite rock have been determined experimentally by making metal casts of the void spaces between the fracture surfaces under normal loads. The resulting metal casts on 52 cm diameter core samples show a complex geometry for the flow paths through the fracture. This geometry is analyzed using finite-size scaling. The spanning probabilities and percolation probabilities of the metal casts are calculted as functions of observation scale. Under the highest stresses of 33 MPa and 85 MPa there is a significant size-dependence of the geometric flow properties for observation scales smaller than 2 mm. Based on this data, the macroscopic percolation properties of the extended fracture can be well represented by relatively small core samples, even under normal stresses larger than 33 MPa. The metal casts also have rich multifractal structure that changes with changing stress.