Systematic corrections to quadratic approximations for power-law structure functions: the δ expansion

1981 ◽  
Vol 71 (3) ◽  
pp. 321 ◽  
Author(s):  
Stephen M. Wandzura
Keyword(s):  
Power Law ◽  
Structure Functions ◽  
Quadratic Approximations

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.

scholarly journals Scalable statistics of correlated random variables and extremes applied to deep borehole porosities

2015 ◽  
Vol 19 (2) ◽  
pp. 729-745 ◽  
Author(s):  
A. Guadagnini ◽  
S. P. Neuman ◽  
T. Nan ◽  
M. Riva ◽  
C. L. Winter
Keyword(s):  
Power Law ◽  
Heavy Tails ◽  
Structure Functions ◽  
Separation Distance ◽  
Depositional Environments ◽  
Frequency Distributions ◽  
Scale Mixtures ◽  
Log Structure ◽  
Linear Relationships ◽  
Non Gaussian

Abstract. We analyze scale-dependent statistics of correlated random hydrogeological variables and their extremes using neutron porosity data from six deep boreholes, in three diverse depositional environments, as example. We show that key statistics of porosity increments behave and scale in manners typical of many earth and environmental (as well as other) variables. These scaling behaviors include a tendency of increments to have symmetric, non-Gaussian frequency distributions characterized by heavy tails that decay with separation distance or lag; power-law scaling of sample structure functions (statistical moments of absolute increments) in midranges of lags; linear relationships between log structure functions of successive orders at all lags, known as extended self-similarity or ESS; and nonlinear scaling of structure function power-law exponents with function order, a phenomenon commonly attributed in the literature to multifractals. Elsewhere we proposed, explored and demonstrated a new method of geostatistical inference that captures all of these phenomena within a unified theoretical framework. The framework views data as samples from random fields constituting scale mixtures of truncated (monofractal) fractional Brownian motion (tfBm) or fractional Gaussian noise (tfGn). Important questions not addressed in previous studies concern the distribution and statistical scaling of extreme incremental values. Of special interest in hydrology (and many other areas) are statistics of absolute increments exceeding given thresholds, known as peaks over threshold or POTs. In this paper we explore the statistical scaling of data and, for the first time, corresponding POTs associated with samples from scale mixtures of tfBm or tfGn. We demonstrate that porosity data we analyze possess properties of such samples and thus follow the theory we proposed. The porosity data are of additional value in revealing a remarkable cross-over from one scaling regime to another at certain lags. The phenomena we uncover are of key importance for the analysis of fluid flow and solute as well as particulate transport in complex hydrogeologic environments.

scholarly journals Characterizing the i-band variability of YSOs over six orders of magnitude in time-scale

2019 ◽  
Vol 491 (4) ◽  
pp. 5035-5055 ◽  
Author(s):  
Darryl J Sergison ◽  
Tim Naylor ◽  
S P Littlefair ◽  
Cameron P M Bell ◽  
C D H Williams
Keyword(s):  
Time Scales ◽  
Power Law ◽  
Time Scale ◽  
Fourier Transforms ◽  
Simulated Data ◽  
Class Ii ◽  
Structure Functions ◽  
Young Stellar Objects ◽  
Class Iii ◽  
Stellar Objects

ABSTRACT We present an i-band photometric study of over 800 young stellar objects in the OB association Cep OB3b, which samples time-scales from one minute to 10 yr. Using structure functions we show that on all time-scales (τ) there is a monotonic decrease in variability from Class I to Class II through the transition disc (TD) systems to Class III, i.e. the more evolved systems are less variable. The Class Is show an approximately power-law increase (τ0.8) in variability from time-scales of a few minutes to 10 yr. The Class II, TDs, and Class III systems show a qualitatively different behaviour with most showing a power-law increase in variability up to a time-scale corresponding to the rotational period of the star, with little additional variability beyond that time-scale. However, about a third of the Class IIs shows lower overall variability, but their variability is still increasing at 10 yr. This behaviour can be explained if all Class IIs have two primary components to their variability. The first is an underlying roughly power-law variability spectrum, which evidence from the infrared suggests is driven by accretion rate changes. The second component is approximately sinusoidal and results from the rotation of the star. We suggest that the systems with dominant longer time-scale variability have a smaller rotational modulation either because they are seen at low inclinations or have more complex magnetic field geometries. We derive a new way of calculating structure functions for large simulated data sets (the ‘fast structure function’), based on fast Fourier transforms.

scholarly journals Extreme value statistics of scalable data exemplified by neutron porosities in deep boreholes

2014 ◽  
Vol 11 (10) ◽  
pp. 11637-11686
Author(s):  
A. Guadagnini ◽  
S. P. Neuman ◽  
T. Nan ◽  
M. Riva ◽  
C. L. Winter
Keyword(s):  
Power Law ◽  
Heavy Tails ◽  
Structure Functions ◽  
Separation Distance ◽  
Depositional Environments ◽  
Extreme Value Statistics ◽  
Frequency Distributions ◽  
Scale Mixtures ◽  
Log Structure ◽  
Deep Boreholes

Abstract. Spatial statistics of earth and environmental (as well as many other) data tend to vary with scale. Common manifestations of scale-dependent statistics include a tendency of increments to have symmetric, non-Gaussian frequency distributions characterized by heavy tails that decay with separation distance or lag; power-law scaling of sample structure functions (statistical moments of absolute increments) in midranges of lags; linear relationships between log structure functions of successive orders at all lags, known as extended self-similarity or ESS; and nonlinear scaling of structure function power-law exponents with function order, a phenomenon commonly attributed in the literature to multifractals. Elsewhere we proposed, explored and demonstrated a new method of geostatistical inference that captures all of these phenomena within a unified theoretical framework. The framework views data as samples from random fields constituting scale-mixtures of truncated (monofractal) fractional Brownian motion (tfBm) or fractional Gaussian noise (tfGn). Important questions not addressed in previous studies concern the distribution and statistical scaling of extreme incremental values. Of special interest in hydrology (and many other areas) are statistics of absolute increments exceeding given thresholds, known as peaks over thresholds or POTs. In this paper we explore for the first time the statistical behavior of POTs associated with samples from scale-mixtures of tfBm or tfGn. We are fortunate to have at our disposal thousands of neutron porosity values from six deep boreholes, in three diverse depositional environments, which we show possess the properties of such samples thus following the theory we proposed. The porosity data are of additional value in revealing a remarkable transition from one scaling regime to another at certain lags. The phenomena we uncover are of fundamental importance for the analysis of fluid flow and solute as well as particulate transport in complex hydrogeologic environments.

Structure functions and intensities if ocean circulation off east and west Australia

1972 ◽  
Vol 23 (2) ◽  
pp. 99 ◽  
Author(s):  
BV Hamon ◽  
GR Cresswell
Keyword(s):  
Ocean Circulation ◽  
Power Law ◽  
Structure Functions ◽  
Horizontal Scale ◽  
Dynamic Height ◽  
East And West ◽  
Latitudinal Trend ◽  
Surface Dynamic ◽  
Scale Length

The intensity of ocean circulation off east Australia was found to be about five times that off west Australia, as measured by the variances of surface dynamic height, after allowing for latitudinal trend. A dominant horizontal scale length of about 500 km was found off both coasts, from structure functions of surface dynamic height. The structure functions are shown for the range 25 < l < 1200 km, where I = horizontal spacing between stations. In this range, the structure functions do not fit a 2/3 power law.

Energy spectra of stably stratified turbulence

2012 ◽  
Vol 698 ◽  
pp. 19-50 ◽  
Author(s):  
Y. Kimura ◽  
J. R. Herring
Keyword(s):  
Power Law ◽  
Large Scale ◽  
Wave Spectrum ◽  
Structure Functions ◽  
Direct Numerical Simulations ◽  
Energy Spectra ◽  
Stratified Turbulence ◽  
Uhlenbeck Process ◽  
Small Scales ◽  
Incompressible Turbulence

AbstractWe investigate homogeneous incompressible turbulence subjected to a range of degrees of stratification. Our basic method is pseudospectral direct numerical simulations at a resolution of $102{4}^{3} $. Such resolution is sufficient to reveal inertial power-law ranges for suitably comprised horizontal and vertical spectra, which are designated as the wave and vortex mode (the Craya–Herring representation). We study mainly turbulence that is produced from randomly large-scale forcing via an Ornstein–Uhlenbeck process applied isotropically to the horizontal velocity field. In general, both the wave and vortex spectra are consistent with a Kolmogorov-like ${k}^{\ensuremath{-} 5/ 3} $ range at sufficiently large $k$. At large scales, and for sufficiently strong stratification, the wave spectrum is a steeper ${ k}_{\perp }^{\ensuremath{-} 2} $, while that for the vortex component is consistent with ${ k}_{\perp }^{\ensuremath{-} 3} $. Here ${k}_{\perp } $ is the horizontally gathered wavenumber. In contrast to the horizontal wavenumber spectra, the vertical wavenumber spectra show very different features. For those spectra, a clear ${ k}_{z}^{\ensuremath{-} 3} $ dependence for small scales is observed while the large scales show rather flat spectra. By modelling the horizontal layering of vorticity, we attempt to explain the flat spectra. These spectra are linked to two-point structure functions of the velocity correlations in the horizontal and vertical directions. We can observe the power-law transition also in certain of the two-point structure functions.

scholarly journals First-Order Structure Function Analysis of Statistical Scale Invariance in the AIRS-Observed Water Vapor Field

2012 ◽  
Vol 25 (16) ◽  
pp. 5538-5555 ◽  
Author(s):  
Kyle G. Pressel ◽  
William D. Collins
Keyword(s):  
Water Vapor ◽  
Structure Function ◽  
Power Law ◽  
Scale Invariance ◽  
Structure Functions ◽  
Scale Dependence ◽  
Global Maximum ◽  
Order Structure ◽  
Scaling Exponents ◽  
First Order

Abstract The power-law scale dependence, or scaling, of first-order structure functions of the tropospheric water vapor field between 58°S and 58°N is investigated using observations from the Atmospheric Infrared Sounder (AIRS). Power-law scale dependence of the first-order structure function would indicate that the water vapor field exhibits statistical scale invariance. Directional and directionally independent first-order structure functions are computed to assess the directional dependence of derived first-order structure function scaling exponents (H) for a range of scales from 50 to 500 km. In comparison to other methods of assessing statistical scale invariance, the methodology used here requires minimal assumptions regarding the homogeneity of the spatial distribution of data within regions of analysis. Additionally, the methodology facilitates the evaluation of anisotropy and quantifies the extent to which the structure functions exhibit scale invariance. The spatial and seasonal dependence of the computed scaling exponents are explored. Minimum scaling exponents at all levels are shown to occur proximate to the equator, while the global maximum is shown to occur in the middle troposphere near the tropical–subtropical margin of the winter hemisphere. From a detailed analysis of AIRS maritime scaling exponents, it is concluded that the AIRS observations suggest the existence of two scaling regimes in the extratropics. One of these regimes characterizes the statistical scale invariance the free troposphere with H approximately = 0.55 and a second that characterizes the statistical scale invariance of the boundary layer with H approximately = ⅓.

scholarly journals Conditional and unconditional second-order structure functions in bubbly channel flows of power-law fluids

2021 ◽  
Vol 33 (5) ◽  
pp. 055121
Author(s):  
E. Trautner ◽  
M. Klein ◽  
F. Bräuer ◽  
J. Hasslberger
Keyword(s):  
Power Law ◽  
Structure Functions ◽  
Second Order ◽  
Channel Flows ◽  
Order Structure ◽  
Power Law Fluids

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

2011 ◽  
Vol 8 (4) ◽  
pp. 7805-7843 ◽  
Author(s):  
M. Siena ◽  
A. Guadagnini ◽  
M. Riva ◽  
S. P. Neuman
Keyword(s):  
Power Law ◽  
Spatial Data ◽  
Structure Functions ◽  
Air Permeability ◽  
Self Similarity ◽  
Scaling Exponents ◽  
Theoretical Support ◽  
Sample Structure ◽  
Extended Power ◽  
Permeability Data

Abstract. We use three methods to identify power law scaling of (natural) log air permeability data collected by Tidwell and Wilson (1999) on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), extended power-law scaling also known as Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on qth-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. Most analyses of this type published to date concern time series or one-dimensional transects of spatial data associated with a unique measurement (support) scale. We consider log air permeability data having diverse support scales on the faces of a cube. 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 (Guadagnini and Neuman, 2011; Guadagnini et al., 2011) 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. ESS and G-ESS ratios between scaling exponents ξ(q) associated with various orders q show no distinct dependence on support volume or on two out of three Cartesian directions (there being no distinct power law scaling in the third direction). The finding by Tidwell and Wilson (1999) 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 size scales show nonlinear (multifractal) 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, in accord with Neuman (2010a, b, 2011), 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 is 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. Our estimate of lower cutoff is consistent with a theoretical support scale of the data.

Analysis of power law assumptions for short-period comets and Kuiper bodies

1999 ◽  
Vol 173 ◽  
pp. 289-293 ◽  
Author(s):  
J.R. Donnison ◽  
L.I. Pettit
Keyword(s):  
Power Law ◽  
Pareto Distribution ◽  
Limited Data ◽  
Short Period ◽  
Probability Plots ◽  
Exponential Probability

AbstractA Pareto distribution was used to model the magnitude data for short-period comets up to 1988. It was found using exponential probability plots that the brightness did not vary with period and that the cut-off point previously adopted can be supported statistically. Examination of the diameters of Trans-Neptunian bodies showed that a power law does not adequately fit the limited data available.

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