scholarly journals On the sedimentological character of Alpine basal ice facies

1996 ◽  
Vol 22 ◽  
pp. 187-193 ◽  
Author(s):  
Bryn Hubbard ◽  
Martin Sharp ◽  
Wendy J. Lawson
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Size Fraction ◽  
Fractal Dimensions ◽  
Western Alps ◽  
Similar Distribution ◽  
Self Similarity ◽  
Apparent Contradiction ◽  
Basal Ice ◽  
Self Similar

Seven basal ice facies have been defined on the basis of research at eleven glaciers in the western Alps. The concentration and texture of the debris incorporated in these facies are described. Grain-size distributions are characterised in terms of their: (i) mean size and dispersion, (ii) component Gaussian modes, and (iii) self-similarity.Firnified glacier ice contains low concentrations (≈0.2 g 1−1) of well-sorted and predominantly fine-grained debris that is not self-similar over the range of particle diameters assessed. In contrast, basal ice contains relatively high concentrations (≈4–4000 g 1−1by facies) of polymodal (by size fraction against weight) debris, the texture of which is consistent with incorporation at the glacier bed. Analysis by grain-size against number of particles suggests that these basal facies debris textures are also self-similar. This apparent contradiction may be explained by the insensitivity of the assessment of self-similarity to variations in mass distribution. Comparison of typical size–weight with size–number distributions indicates that neither visual nor statistical assessment of the latter may be sufficiently rigorous to identify self-similarity.Apparent fractal dimensions may indicate the relative importance of fines in a debris distribution. Subglacially derived basal facies debris has a mean fractal dimension of 2.74. This value suggests an excess of fines relative to a self-similar distribution of cubes, which has a fractal dimension of 2.58. Subglacial sediments sampled from the forefield of Skalafellsjökull, Iceland, have fractal dimensions of 2.91 (A-horizon) and 2.81 (B-horizon). Debris from the A-horizon, which is interpreted as having been pervasively deformed, both most closely approaches self-similarity and has the highest fractal dimension of any of the sample groups analyzed.

scholarly journals On the sedimentological character of Alpine basal ice facies

1996 ◽  
Vol 22 ◽  
pp. 187-193 ◽  
Author(s):  
Bryn Hubbard ◽  
Martin Sharp ◽  
Wendy J. Lawson
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Size Fraction ◽  
Fractal Dimensions ◽  
Western Alps ◽  
Similar Distribution ◽  
Self Similarity ◽  
Apparent Contradiction ◽  
Basal Ice ◽  
Self Similar

Seven basal ice facies have been defined on the basis of research at eleven glaciers in the western Alps. The concentration and texture of the debris incorporated in these facies are described. Grain-size distributions are characterised in terms of their: (i) mean size and dispersion, (ii) component Gaussian modes, and (iii) self-similarity.Firnified glacier ice contains low concentrations (≈0.2 g 1−1) of well-sorted and predominantly fine-grained debris that is not self-similar over the range of particle diameters assessed. In contrast, basal ice contains relatively high concentrations (≈4–4000 g 1−1 by facies) of polymodal (by size fraction against weight) debris, the texture of which is consistent with incorporation at the glacier bed. Analysis by grain-size against number of particles suggests that these basal facies debris textures are also self-similar. This apparent contradiction may be explained by the insensitivity of the assessment of self-similarity to variations in mass distribution. Comparison of typical size–weight with size–number distributions indicates that neither visual nor statistical assessment of the latter may be sufficiently rigorous to identify self-similarity.Apparent fractal dimensions may indicate the relative importance of fines in a debris distribution. Subglacially derived basal facies debris has a mean fractal dimension of 2.74. This value suggests an excess of fines relative to a self-similar distribution of cubes, which has a fractal dimension of 2.58. Subglacial sediments sampled from the forefield of Skalafellsjökull, Iceland, have fractal dimensions of 2.91 (A-horizon) and 2.81 (B-horizon). Debris from the A-horizon, which is interpreted as having been pervasively deformed, both most closely approaches self-similarity and has the highest fractal dimension of any of the sample groups analyzed.

FRACTAL DIMENSION AND LACUNARITY OF TRACTOGRAPHY IMAGES OF THE HUMAN BRAIN

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 181-189 ◽  
Author(s):  
P. KATSALOULIS ◽  
D. A. VERGANELAKIS ◽  
A. PROVATA
Keyword(s):  
Fractal Dimension ◽  
Human Brain ◽  
Fractal Dimensions ◽  
Resonance Imaging ◽  
Self Similarity ◽  
Healthy Human ◽  
Box Counting ◽  
Self Similar ◽  
Box Method ◽  
The Brain

Tractography images produced by Magnetic Resonance Imaging scans have been used to calculate the topology of the neuron tracts in the human brain. This technique gives neuroanatomical details, limited by the system resolution properties. In the observed scales the images demonstrated the statistical self-similar structure of the neuron axons and its fractal dimensions were estimated using the classic Box Counting technique. To assess the degree of clustering in the neural tracts network, lacunarity was calculated using the Gliding Box method. The two-dimensional tractography images were taken from four subjects using various angles and different parts in the brain. The results demonstrated that the average estimated fractal dimension of tractography images is approximately Df = 1.60 with standard deviation 0.11 for healthy human-brain tissues, and it presents statistical self-similarity features similar to many other biological root-like structures.

A Modified Box-Counting Method to Estimate the Fractal Dimensions

2011 ◽  
Vol 58-60 ◽  
pp. 1756-1761 ◽  
Author(s):  
Jie Xu ◽  
Giusepe Lacidogna
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
The Self ◽  
Whole Body ◽  
Counting Method ◽  
Border Effect ◽  
Self Similarity ◽  
Box Counting ◽  
Self Similar ◽  
Box Counting Method

A fractal is a property of self-similarity, each small part of the fractal object is similar to the whole body. The traditional box-counting method (TBCM) to estimate fractal dimension can not reflect the self-similar property of the fractal and leads to two major problems, the border effect and noninteger values of box size. The modified box-counting method (MBCM), proposed in this study, not only eliminate the shortcomings of the TBCM, but also reflects the physical meaning about the self-similar of the fractal. The applications of MBCM shows a good estimation compared with the theoretical ones, which the biggest difference is smaller than 5%.

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

scholarly journals A laboratory study of sediment deformation: stress heterogeneity and grain-size evolution

1996 ◽  
Vol 22 ◽  
pp. 167-175 ◽  
Author(s):  
Neal R. Iverson ◽  
Thomas S. Hooyer ◽  
Roger Leb. Hooke
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Normal Stress ◽  
Size Distribution ◽  
Grain Size Distribution ◽  
Local Stress ◽  
Large Strains ◽  
Self Similar ◽  
Deformation Stress ◽  
Stress Heterogeneity

In shearing sediment beneath glaciers, networks of grains may transiently support shear and normal stresses that are larger than spatial averages. Consistent with studies of fault-gouge genesis, we hypothesize that crushing of grains in such networks is responsible for surrounding larger grains with smaller grains. At sufficiently large strains, this should minimize stress heterogeneity, favor intergranular sliding and abrasion rather than crushing, and result in a self-similar grain-size distribution.This hypothesis is tested with a ring-shear device that slowly shears a large annular sediment sample to high strains. Shearing and comminution of weak equigranular (2.0–3.3 mm) sediment resulted in a self-similar grain-size distribution with a fractal dimension that increased with shear strain toward a steady value of 2.85. This value is significantly larger than that of gouges produced purely by crushing, 2.6, but it is comparable to values for tilts thought to be deforming beneath modern glaciers, 2.8 to nearly 3.0. At low strains, under a steady mean normal stress of 84 kPa, variations in normal stress measured locally ranged in amplitude from 50 to 300 kPa with wavelengths that were 100 times larger than the initial grain diameter. Crushing of grains, observed through the transparent walls of the device, apparently caused the failure of grain networks. At shearing displacements ranging from 0.7 to 1.0 m, the amplitude of local stress fluctuations decreased abruptly. This change is attributed to fine sediment that distributed stresses more uniformly and caused grain networks to fail primarily by intergranular sliding rather than by crushing of grains. Sliding between grains apparently produced silt by abrasion and resulted in a fractal dimension that was higher than if there had been only crushing.A size distribution with a fractal dimension greater than 2.6 is probably a necessary but not sufficient condition for determining whether a basal till has been highly deformed. Stress heterogeneity in subglacial sediment that is shearing through its full thickness should contribute to the erosion of underlying rock.

scholarly journals Rock Mechanical Property Influenced by Inhomogeneity

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yunliang Tan ◽  
Dongmei Huang ◽  
Ze Zhang
Keyword(s):  
Mechanical Property ◽  
Fractal Dimension ◽  
Rock Type ◽  
Fractal Dimensions ◽  
Rock Material ◽  
The Self ◽  
Self Similarity ◽  
Red Sandstone ◽  
Variation Patterns ◽  
Rock Microstructure

In order to identify the microstructure inhomogeneity influence on rock mechanical property, SEM scanning test and fractal dimension estimation were adopted. The investigations showed that the self-similarity of rock microstructure markedly changes with the scanned microscale. Different rocks behave in different fractal dimension variation patterns with the scanned magnification, so it is conditional to adopt fractal dimension to describe rock material. Grey diabase and black diabase have high suitability; red sandstone has low suitability. The suitability of fractal-dimension-describing method for rocks depends on both investigating scale and rock type. The homogeneities of grey diabase, black diabase, grey sandstone, and red sandstone are 7.8, 5.7, 4.4, and 3.4, separately; their average fractal dimensions of microstructure are 2.06, 2.03, 1.72, and 1.40 correspondingly, so the homogeneity is well consistent with fractal dimension. For rock material, the stronger brittleness is, the less profile fractal dimension is. In a sense, brittleness is an image of rock inhomogeneity in macroscale, while profile fractal dimension is an image of rock inhomogeneity in microscale. To combine the test of brittleness with the estimation of fractal dimension with condition will be an effective approach for understanding rock failure mechanism, patterns, and behaviours.

Preliminary Evidence for a Theory of the Fractal City

1996 ◽  
Vol 28 (10) ◽  
pp. 1745-1762 ◽  
Author(s):  
M Batty ◽  
Y Xie
Keyword(s):  
Fractal Dimension ◽  
Urban Form ◽  
Scaling Laws ◽  
Fractal Dimensions ◽  
Power Laws ◽  
Preliminary Evidence ◽  
Residential Development ◽  
Self Similarity ◽  
Estimation Techniques ◽  
Data Source

In this paper, we argue that the geometry of urban residential development is fractal. Both the degree to which space is filled and the rate at which it is filled follow scaling laws which imply invariance of function, and self-similarity of urban form across scale. These characteristics are captured in population density functions based on inverse power laws whose parameters are fractal dimensions. First we outline the relevant elements of the theory in terms of scaling relations and then we introduce two methods for estimating fractal dimension based on varying the size of cities and the scale at which their form is detected. Exact and statistical estimation techniques are applied to each method respectively generating dimensions which measure the extent and the rate of space filling. These methods are then applied to residential development patterns in six industrial cities in the northeastern United States, with an innovative data source from the TIGER/Line files. The results support the theory of the fractal city outlined in books by Batty and Longley and Frankhauser, but with the clear conclusion that different scale and estimation techniques generate different types of fractal dimension.

scholarly journals CHARACTERISTICS OF SELF‐SIMILARITY OF SEISMICITY AND THE FAULT NETWORK OF THE SIKHOTE ALIN OROGENIC BELT AND THE ADJACENT AREAS

2019 ◽  
Vol 10 (2) ◽  
pp. 541-559
Author(s):  
V. S. Zakharov ◽  
A. N. Didenko ◽  
G. Z. Gil’manova ◽  
T. V. Merkulova
Keyword(s):  
Fractal Dimension ◽  
Quantitative Characteristic ◽  
Recurrence Plot ◽  
Orogenic Belt ◽  
Sikhote Alin ◽  
Self Similarity ◽  
Crustal Earthquakes ◽  
Self Similar ◽  
Fault Network ◽  
Bureya Massif

We performed a comprehensive analysis of the characteristics of self‐similarity of seismicity and the fault network within the Sikhote Alin orogenic belt and the adjacent areas. It has been established that the main features of seismicity are controlled by the crustal earthquakes. Differentiation of the study area according to the density of earthquake epicenters and the fractal dimension of the epicentral field of earthquakes (De) shows that the most active crustal areas are linked to the Kharpi‐Kur‐Priamurye zone, the northern Bureya massif and the Mongol‐Okhotsk folded system. The analysis of the earthquake recurrence plot slope values reveals that the highest b‐values correlate with the areas of the highest seismic activity of the northern part of the Bureya massif and, to a less extent, of the Mongol‐Okhotsk folded system. The increased fractal dimension values for the fault network (Df) correlate with the folded systems (Sikhote Alin and Mongol‐Okhotsk), while the decreased values conform to the depressions and troughs (Middle Amur, Uda and Torom). A comparison of the fractal analysis results for the fault network with the recent stress‐strain data gives evidence of their general confineness to the contemporary areas of intense compression. The correspondence between the field of the parameter b‐value for the upper crustal earthquakes and the fractal dimension value for the fault network (Df) suggests a general consistency between the self‐similar earthquake magnitude (energy) distribution and the fractal distribution of the fault sizes. The analysis results demonstrate that the selfsimilarity parameters provide an important quantitative characteristic in seismotectonics and can be used for the neotectonic and geodynamic analyses.

scholarly journals Exploring the Fractal Parameters of Urban Growth and Form with Wave-Spectrum Analysis

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Yanguang Chen
Keyword(s):  
Fractal Dimension ◽  
Urban Growth ◽  
Urban Form ◽  
Wave Spectrum ◽  
Fractal Dimensions ◽  
Wave Number ◽  
Similar Distribution ◽  
Image Dimension ◽  
Radial Dimension ◽  
The Fourier Transform

The Fourier transform and spectral analysis are employed to estimate the fractal dimension and explore the fractal parameter relations of urban growth and form using mathematical experiments and empirical analyses. Based on the models of urban density, two kinds of fractal dimensions of urban form can be evaluated with the scaling relations between the wave number and the spectral density. One is theradial dimensionof self-similar distribution indicating the macro-urban patterns, and the other, the profile dimension of self-affine tracks indicating the micro-urban evolution. If a city's growth follows the power law, the summation of the two dimension values may be a constant under certain condition. The estimated results of the radial dimension suggest a new fractal dimension, which can be termed “image dimension”. A dual-structure model namedparticle-ripple model(PRM) is proposed to explain the connections and differences between the macro and micro levels of urban form.

scholarly journals Subglacial sediment textures: character and evolution at Haut Glacier d’Arolla, Switzerland

1999 ◽  
Vol 28 ◽  
pp. 241-246 ◽  
Author(s):  
Urs H. Fischer ◽  
Bryn Hubbard
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Fine Fraction ◽  
Particle Diameter ◽  
Fractal Dimensions ◽  
Textural Evolution ◽  
Grain Size Distributions ◽  
Percentage Weight ◽  
Sediment Deformation

AbstractFourteen subglacial debris samples have been recovered from the margins of, or beneath, Haut Glacier d’Arolla, Switzerland. The grain-size distributions of these sediments are presented and compared with each other as bivariate plots of percentage weight against (sieve-defined) particle size and log number of particles against log particle diameter. All of the samples recovered are composed of a broad range of clast sizes and approach self-similarity over the four orders of magnitude of grain-sizes analysed. Fractal dimensions range from 2.47 to 2.77. Sample intercomparison reveals the operation of at least two processes of textural evolution: the production of fines by in-situ weathering, interpreted in terms of abrasion associated with subglacial sediment deformation, and the loss of fines, interpreted in terms of eluviation by percolating subglacial meltwaters. These interpretations are supported and refined through comparison of the grain-size fractions gained (in the case of deformation) and lost (in the case of eluviation) with those fractions respectively generated in a laboratory-based simulation of sediment deformation and exiting the glacier suspended in the proglacial meltwater stream. While sediment deformation has the effect of increasing the fine fraction between 0 and 10ϕ and of raising the fractal dimension of undeformed sediments from 2.47 to 2.77, eluviation removes particles between 2 and 100, driving the fractal dimension of deformed sediments down from 2.77 to 2.54. These fractal dimensions are generally lower than those recorded at other comparable glaciers, consistent with the relatively low rates of sediment deformation inferred from other studies at Haut Glacier d’Arolla.

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