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.     

Influence of source region properties on scaling relations forM<0 events

Excavation induced seismic events with moment magnitudesM<0 are examined in an attempt to determine the role geology, excavation geometry, and stress have on scaling relations. Correlations are established based on accurate measurements of excavation geometry and methodology, stress regime, rock mass structure, local tectonics, and seismic locations. Scaling relations incorporated seismic moments and source radii obtained by spectral analysis, accounting for source, propagation, and site effects, and using Madariaga's dynamic circular fault model. Observations suggest that the interaction of stresses with pre-existing fractures, fracture complexity and depth of events are the main factors influencing source characteristics and scaling behaviour. Self-similar relationships were found for events at similar depths or for weakly structured rock masses with reduced clamping stresses, whereas a non-similar behaviour was found for events with increasing depth or for heavily fractured zones under stress confinement. Additionally, the scaling behaviour for combined data sets tended to mask the non-similar trends. Overall, depth and fracture complexity, initially thought as second order effects, appear to significantly influence source characteristics of seismic events withM<0 and consequently favour a non-similar earthquake generation process.     

Is the ocean floor a fractal?

The topographic structure of the ocean bottom is investigated at different scales of resolution to answer the question: Can the seafloor be described as a fractal process? Methods from geostatistics, the theory of regionalized variables, are used to analyze the spatial structure of the ocean floor at different scales of resolution. The key to the analysis is the variogram criterion: Self-similarity of a stochastic process implies self-similarity of its variogram. The criterion is derived and proved here: it also is valid for special cases of self-affinity (in a sense adequate for topography). It has been proposed that seafloor topography can be simulated as a fractal (an object of Hausdorff dimension strictly larger than its topological dimension), having scaling properties (self-similarity or self-affinity). The objective of this study is to compare the implications of these concepts with observations of the seafloor. The analyses are based on SEABEAM bathymetric data from the East Pacific Rise at 13°N/104°W and at 9°N/104°W and use tracks that run both across the ridge crest and along the ridge flank. In the geostatistical evaluation, the data are considered as a stochastic process. The spatial continuity of this process is described by variograms that are calculated for different scales and directions. Applications of the variogram criterion to scale-dependent variogram models yields the following results: Although the seafloor may be a fractal in the sense of the definition involving the Hausdorff dimension, it is not self-similar, nor self-affine (in the given sense). Mathematical models of scale-dependent spatial structures are presented, and their relationship to geologic processes such as ridge evolution, crust formation, and sedimentation is discussed.     

Multifractal modeling and spatial point processes

The multifractal model can be applied to spatial point processes. It provides new, approximately power-law type, expressions for their second-order intensity and K (r) functions. The box-counting and cluster dimensions are different but mutually interrelated according to multifractal theory. This approach is used to describe the underlying spatial structure of gold mineral occurrences in the Iskut River area, northwestern British Columbia. The box-counting and cluster dimensions for the example are estimated to be 1.335±0.077 and 1.219±0.037, respectively. The relatively strong clustering of the gold deposits is reflected by the fact that both values are considerably less than the corresponding Euclidean dimension (=2).