Relations between the scaling exponents, entropies, and energies of fracture networks

2013 ◽  
Vol 184 (4-5) ◽  
pp. 373-382 ◽  
Author(s):  
Agust Guđmundsson ◽  
Nahid Mohajeri
Keyword(s):  
Power Law ◽  
Scaling Laws ◽  
Plate Boundary ◽  
Normal Faults ◽  
Scaling Exponent ◽  
Size Distributions ◽  
Fracture Networks ◽  
Scaling Exponents ◽  
The Holocene ◽  
Arithmetic Average

Abstract Fracture networks commonly show power-law length distributions. Thermodynamic principles form the basis for understanding fracture initiation and growth, but have not been easily related to the power-law size distributions. Here we present the power-law scaling exponents and the calculated entropies of fracture networks from the Holocene part of the plate boundary in Iceland. The total number of tension fractures and normal faults used in these calculations is 565 and they range in length by five orders of magnitude. Each network can be divided into populations based on ‘breaks’ (abrupt changes) in the scaling exponents. The breaks, we suggest, are related to the comparatively long and deep fractures changing from tension fractures into normal faults and penetrating the contacts between the Holocene lava flows and the underlying and mechanically different Quaternary rocks. The results show a strong linear correlation (r = 0.84) between the population scaling exponents and entropies. The correlation is partly explained by the entropy (and the scaling exponent) varying positively with the arithmetic average and the length range (the difference between the longest and the shortest fracture) of the populations in each network. We show that similar scaling laws apply to other lineaments, such as streets. We propose that the power-law size distributions of fractures are a consequence of energy requirements for fracture growth.

Self-Similar Criticality

Fractals ◽  
2003 ◽  
Vol 11 (03) ◽  
pp. 221-231 ◽  
Author(s):  
Sarah F. Tebbens ◽  
Stephen M. Burroughs
Keyword(s):  
Power Law ◽  
Similar Distribution ◽  
Scaling Exponent ◽  
Cumulative Frequency ◽  
Size Distributions ◽  
Scaling Exponents ◽  
Natural Systems ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Self Similar

Cumulative frequency-size distributions associated with many natural phenomena follow a power law. Self-organized criticality (SOC) models have been used to model characteristics associated with these natural systems. As originally proposed, SOC models generate event frequency-size distributions that follow a power law with a single scaling exponent. Natural systems often exhibit power law frequency-size distributions with a range of scaling exponents. We modify the forest fire SOC model to produce a range of scaling exponents. In our model, uniform energy (material) input produces events initiated on a self-similar distribution of critical grid cells. An event occurs when material is added to a critical cell, causing that material and all material in occupied non-diagonal adjacent cells to leave the grid. The scaling exponent of the resulting cumulative frequency-size distribution depends on the fractal dimension of the critical cells. Since events occur on a self-similar distribution of critical cells, we call this model Self-Similar Criticality (SSC). The SSC model may provide a link between fractal geometry in nature and observed power law frequency-size distributions for many natural systems.

UPPER-TRUNCATED POWER LAW DISTRIBUTIONS

Fractals ◽  
2001 ◽  
Vol 09 (02) ◽  
pp. 209-222 ◽  
Author(s):  
STEPHEN M. BURROUGHS ◽  
SARAH F. TEBBENS
Keyword(s):  
Power Law ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Generalized Function ◽  
Data Sets ◽  
Scaling Exponent ◽  
Cumulative Number ◽  
Size Distributions ◽  
Data Set ◽  
Binned Data

Power law cumulative number-size distributions are widely used to describe the scaling properties of data sets and to establish scale invariance. We derive the relationships between the scaling exponents of non-cumulative and cumulative number-size distributions for linearly binned and logarithmically binned data. Cumulative number-size distributions for data sets of many natural phenomena exhibit a "fall-off" from a power law at the largest object sizes. Previous work has often either ignored the fall-off region or described this region with a different function. We demonstrate that when a data set is abruptly truncated at large object size, fall-off from a power law is expected for the cumulative distribution. Functions to describe this fall-off are derived for both linearly and logarithmically binned data. These functions lead to a generalized function, the upper-truncated power law, that is independent of binning method. Fitting the upper-truncated power law to a cumulative number-size distribution determines the parameters of the power law, thus providing the scaling exponent of the data. Unlike previous approaches that employ alternate functions to describe the fall-off region, an upper-truncated power law describes the data set, including the fall-off, with a single function.

scholarly journals On the Density Variability of Poissonian Discrete Fracture Networks, with application to power-law fracture size distributions

2019 ◽  
Vol 49 ◽  
pp. 77-83 ◽  
Author(s):  
Etienne Lavoine ◽  
Philippe Davy ◽  
Caroline Darcel ◽  
Romain Le Goc
Keyword(s):  
Size Distribution ◽  
Power Law ◽  
Analytical Solutions ◽  
Three Dimensional ◽  
Size Distributions ◽  
Fracture Density ◽  
Fracture Networks ◽  
Discrete Fracture Networks ◽  
Discrete Fracture ◽  
Fracture Size

Abstract. This paper presents analytical solutions to estimate at any scale the fracture density variability associated to stochastic Discrete Fracture Networks. These analytical solutions are based upon the assumption that each fracture in the network is an independent event. Analytical solutions are developed for any kind of fracture density indicators. Those analytical solutions are verified by numerical computing of the fracture density variability in three-dimensional stochastic Discrete Fracture Network (DFN) models following various orientation and size distributions, including the heavy-tailed power-law fracture size distribution. We show that this variability is dependent on the fracture size distribution and the measurement scale, but not on the orientation distribution. We also show that for networks following power-law size distribution, the scaling of the three-dimensional fracture density variability clearly depends on the power-law exponent.

scholarly journals Population is the main driver of war group size and conflict casualties

2017 ◽  
Vol 114 (52) ◽  
pp. E11101-E11110 ◽  
Author(s):  
Rahul C. Oka ◽  
Marc Kissel ◽  
Mark Golitko ◽  
Susan Guise Sheridan ◽  
Nam C. Kim ◽  
...  
Keyword(s):  
Group Size ◽  
Power Law ◽  
Scaling Laws ◽  
Small Scale ◽  
Scaling Exponent ◽  
Group Conflict ◽  
Sectarian Violence ◽  
Main Driver ◽  
Power Law Function ◽  
Group Population

The proportions of individuals involved in intergroup coalitional conflict, measured by war group size (W), conflict casualties (C), and overall group conflict deaths (G), have declined with respect to growing populations, implying that states are less violent than small-scale societies. We argue that these trends are better explained by scaling laws shared by both past and contemporary societies regardless of social organization, where group population (P) directly determines W and indirectly determines C and G. W is shown to be a power law function of P with scaling exponent X [demographic conflict investment (DCI)]. C is shown to be a power law function of W with scaling exponent Y [conflict lethality (CL)]. G is shown to be a power law function of P with scaling exponent Z [group conflict mortality (GCM)]. Results show that, while W/P and G/P decrease as expected with increasing P, C/W increases with growing W. Small-scale societies show higher but more variance in DCI and CL than contemporary states. We find no significant differences in DCI or CL between small-scale societies and contemporary states undergoing drafts or conflict, after accounting for variance and scale. We calculate relative measures of DCI and CL applicable to all societies that can be tracked over time for one or multiple actors. In light of the recent global emergence of populist, nationalist, and sectarian violence, our comparison-focused approach to DCI and CL will enable better models and analysis of the landscapes of violence in the 21st century.

Urban scaling in rapidly urbanising China

Urban Studies ◽  
2021 ◽  
pp. 004209802110178
Author(s):  
Weiqian Lei ◽  
Limin Jiao ◽  
Gang Xu ◽  
Zhengzi Zhou
Keyword(s):  
Urban Population ◽  
Economies Of Scale ◽  
Scaling Laws ◽  
Temporal Evolution ◽  
Urban System ◽  
Scaling Analysis ◽  
Scaling Exponent ◽  
Scaling Exponents ◽  
Economic Output ◽  
Urban Scaling

Understanding the scaling characteristics in China is critical for perceiving the development process of rapidly urbanising countries. This paper conducts a comprehensive scaling analysis with quantitative assessment of a large number of diverse urban indicators of 275 Chinese cities. Our findings confirm that urban scaling laws can also be applied to rapidly urbanising China but demonstrate some unique features echoing its distinct urbanisation. Chinese urban population agglomeration results in more effective economic production but the economies of scale for infrastructure are less obvious. Some urban indicators associated with infrastructure and living facilities surprisingly scale super-linearly with urban population size, contrary to expected sublinear scaling behaviours. In developing countries, different-sized cities have diverse agglomeration, industrial and resource allocation advantages, which can be reflected by scaling exponents. We characterise these unique features in detail, exploring the spatial disparities and temporal evolution of scaling exponents ( β). Strong regional variations and differences are particularly pronounced in Northeast China and the Beijing-Tianjin-Hebei Urban Agglomeration. Scaling exponent variations over time reflect the temporal evolution of the urban system and measure the coordination and balance of urbanisation. Economic output was most efficient in 2009 and β of GDP was slightly greater than 1.15 in recent years. Urban land expansion has been accelerating since 2000 with β remaining around 0.85–0.90. The study of urban scaling in China is enlightening in elaborating the uniqueness and coordination of urban development in rapidly urbanising countries and provides support in formulating differentiated urban planning for different-sized cities to promote coordinated development.

scholarly journals A regional-scale assessment of using metabolic scaling theory to predict ecosystem properties

2019 ◽  
Vol 286 (1915) ◽  
pp. 20192221 ◽  
Author(s):  
James K. McCarthy ◽  
John M. Dwyer ◽  
Karel Mokany
Keyword(s):  
Power Law ◽  
Regional Scale ◽  
Fire Frequency ◽  
Scaling Theory ◽  
Scaling Exponent ◽  
Size Distributions ◽  
Metabolic Scaling ◽  
Metabolic Scaling Theory ◽  
Relative Productivity ◽  
Heath Communities

Metabolic scaling theory (MST) is one of ecology's most high-profile general models and can be used to link size distributions and productivity in forest systems. Much of MST's foundation is based on size distributions following a power law function with a scaling exponent of −2, a property assumed to be consistent in steady-state ecosystems. We tested the theory's generality by comparing actual size distributions with those predicted using MST parameters assumed to be general. We then used environmental variables and functional traits to explain deviation from theoretical expectations. Finally, we compared values of relative productivity predicted using MST with a remote-sensed measure of productivity. We found that fire-prone heath communities deviated from MST-predicted size distributions, whereas fire-sensitive rainforests largely agreed with the theory. Scaling exponents ranged from −1.4 to −5.3. Deviation from the power law assumption was best explained by specific leaf area, which varies along fire frequency and moisture gradients. While MST may hold in low-disturbance systems, we show that it cannot be applied under many environmental contexts. The theory should remain general, but understanding the factors driving deviation from MST and subsequent refinements is required if it is to be applied robustly across larger scales.

scholarly journals Nonlinear stratospheric variability: multifractal de-trended fluctuation analysis and singularity spectra

2016 ◽  
Vol 472 (2191) ◽  
pp. 20150864 ◽  
Author(s):  
Gualtiero Badin ◽  
Daniela I. V. Domeisen
Keyword(s):  
Time Scales ◽  
Power Law ◽  
Scaling Laws ◽  
Low Frequency ◽  
Small Scale ◽  
Fluctuation Analysis ◽  
Scaling Exponent ◽  
Low Frequency Variability ◽  
Generalized Hurst Exponent ◽  
Stratospheric Variability

Characterizing the stratosphere as a turbulent system, temporal fluctuations often show different correlations for different time scales as well as intermittent behaviour that cannot be captured by a single scaling exponent. In this study, the different scaling laws in the long-term stratospheric variability are studied using multifractal de-trended fluctuation analysis (MF-DFA). The analysis is performed comparing four re-analysis products and different realizations of an idealized numerical model, isolating the role of topographic forcing and seasonal variability, as well as the absence of climate teleconnections and small-scale forcing. The Northern Hemisphere (NH) shows a transition of scaling exponents for time scales shorter than about 1 year, for which the variability is multifractal and scales in time with a power law corresponding to a red spectrum, to longer time scales, for which the variability is monofractal and scales in time with a power law corresponding to white noise. Southern Hemisphere (SH) variability also shows a transition at annual scales. The SH also shows a narrower dynamical range in multifractality than the NH, as seen in the generalized Hurst exponent and in the singularity spectra. The numerical integrations show that the models are able to reproduce the low-frequency variability but are not able to fully capture the shorter term variability of the stratosphere.

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

scholarly journals Extended power-law scaling of air permeabilities measured on a block of tuff

2012 ◽  
Vol 16 (1) ◽  
pp. 29-42 ◽  
Author(s):  
M. Siena ◽  
A. Guadagnini ◽  
M. Riva ◽  
S. P. Neuman
Keyword(s):  
Power Law ◽  
Structure Functions ◽  
Self Similarity ◽  
Scaling Exponents ◽  
Multi Scale ◽  
Sample Structure ◽  
Heavy Tailed ◽  
Non Gaussian ◽  
Nonlinear Fashion ◽  
Extended Power

Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.

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