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

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

Hydrology and Earth System Sciences Discussions ◽

10.5194/hessd-8-7805-2011 ◽

2011 ◽

Vol 8 (4) ◽

pp. 7805-7843 ◽

Cited By ~ 4

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.

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Solar wind low-frequency magnetohydrodynamic turbulence: extended self-similarity and scaling laws

Nonlinear Processes in Geophysics ◽

10.5194/npg-3-247-1996 ◽

1996 ◽

Vol 3 (4) ◽

pp. 247-261 ◽

Cited By ~ 14

Author(s):

V. Carbone ◽

P. Veltri ◽

R. Bruno

Keyword(s):

Solar Wind ◽

Scaling Laws ◽

Velocity Structure ◽

Fluid Flows ◽

Structure Functions ◽

Point Of View ◽

Self Similarity ◽

Extended Self ◽

Scaling Exponents ◽

Magnetohydrodynamic Turbulence

Abstract. In this paper we review some of the work done in investigating the scaling properties of Magnetohydrodynamic turbulence, by using velocity fluctuations measurements performed in the interplanetary space plasma by the Helios spacecraft. The set of scaling exponents ξq for the q-th order velocity structure functions, have been determined by using the Extended Self-Similarity hypothesis. We have found that the q-th order velocity structure function, when plotted vs. the 4-th order structure function, displays a range of self-similarity which extends over all the lengths covered by measurements, thus allowing for a very good determination of ξq. Moreover the results seem to show that the scaling exponents are the same regardless the various observation periods considered. The obtained scaling exponents have been compared with the results of some intermittency models for Kraichnan's turbulence, derived in the framework of infinitely divisible fragmentation processes, showing the good agreement between these models and our observations. Finally, on the basis of the actually available data sets, we show that scaling laws in Solar Wind turbulence seem to be different from turbulent scaling laws in the ordinary fluid flows. This is true for high-order velocity structure functions, while low-order velocity structure functions show the same scaling laws. Since our measurements involve length scales which extend over many order of magnitude where dissipation is practically absent, our results show that Solar Wind turbulence can be regarded as a testing bench for the investigation of general scaling behaviour in turbulent flows. In particular our results strongly support the point of view which attributes a key role to the inertial range dynamics in determining the intermittency characteristics in fluid flows, in contrast with the point of view which attributes intermittency to a finite Reynolds number effect.

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

Hydrology and Earth System Sciences ◽

10.5194/hess-19-729-2015 ◽

2015 ◽

Vol 19 (2) ◽

pp. 729-745 ◽

Cited By ~ 9

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.

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

Journal of Climate ◽

10.1175/jcli-d-11-00374.1 ◽

2012 ◽

Vol 25 (16) ◽

pp. 5538-5555 ◽

Cited By ~ 16

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.

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Intermittency, non-Gaussian statistics and fractal scaling of MHD fluctuations in the solar wind

Nonlinear Processes in Geophysics ◽

10.5194/npg-4-101-1997 ◽

1997 ◽

Vol 4 (2) ◽

pp. 101-124 ◽

Cited By ~ 119

Author(s):

E. Marsch ◽

C.-Y. Tu

Keyword(s):

Solar Wind ◽

Multifractal Spectrum ◽

Structure Functions ◽

Future Research ◽

Mhd Turbulence ◽

Scaling Exponents ◽

Model Calculations ◽

Fractal Scaling ◽

Gaussian Statistics ◽

Non Gaussian

Abstract. This paper gives a review of some recent work on intermittency, non-Gaussian statistics, and fractal scaling of solar wind magnetohydrodynamic turbulence. Model calculations and theories are discussed and put in their context with the in-situ observations of the solar wind fluctuations, essentially of the flow velocity and magnetic field. Emphasis is placed more on a comparison of the data with the theory than on a complete derivation of the model results, which are treated in a more tutorial fashion. The introduction reminds of some important observations and key aspects of the solar wind turbulence. Then structure functions are defined and observational results discussed. The probability density functions provide a direct means to analyse the statistical properties of the fluctuations. Evidence for non-Gaussian statistics is provided. Intermittency and simple scaling models are discussed, which yield algebraic expressions for the scaling exponents of the structure functions. The concept of the extended self-similarity is presented and corresponding observational evidence for its existence in the solar wind is provided. Subsequently, and extended structure function model, including the p-model scaling and a scale-dependent cascade, is discussed and compared with selected measurements. The basics of the multifractals are presented and applied to solar wind data. The multifractal scaling of the kinetic energy flux as proxy for the unknown cascading rate is established observationally, and the so-called multifractal spectrum is obtained. Finally, the scaling exponents of the associated correlation functions are derived and analysed. The paper concludes with a discussion of the empirical results and prospects for the future research in this field and in solar wind MHD turbulence in general.

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MIXED CAUSAL-NONCAUSAL AR PROCESSES AND THE MODELLING OF EXPLOSIVE BUBBLES

Econometric Theory ◽

10.1017/s0266466618000452 ◽

2019 ◽

Vol 35 (6) ◽

pp. 1234-1270 ◽

Cited By ~ 7

Author(s):

Sébastien Fries ◽

Jean-Michel Zakoian

Keyword(s):

Conditional Distribution ◽

Stable Distribution ◽

Financial Time Series ◽

Domain Of Attraction ◽

Real Data ◽

Autoregressive Processes ◽

Financial Time ◽

Causal Representation ◽

Heavy Tailed ◽

Non Gaussian

Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.

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Centered error entropy Kalman filter with application to satellite attitude determination

Transactions of the Institute of Measurement and Control ◽

10.1177/01423312211019867 ◽

2021 ◽

pp. 014233122110198

Author(s):

Baojian Yang ◽

Lu Cao ◽

Dechao Ran ◽

Bing Xiao

Keyword(s):

Kalman Filter ◽

Gaussian Noise ◽

Attitude Determination ◽

Complexity Analysis ◽

Iteration Algorithm ◽

Robust Algorithms ◽

Convergence Conditions ◽

Satellite Attitude ◽

Heavy Tailed ◽

Non Gaussian

Due to unavoidable factors, heavy-tailed noise appears in satellite attitude estimation. Traditional Kalman filter is prone to performance degradation and even filtering divergence when facing non-Gaussian noise. The existing robust algorithms have limited accuracy. To improve the attitude determination accuracy under non-Gaussian noise, we use the centered error entropy (CEE) criterion to derive a new filter named centered error entropy Kalman filter (CEEKF). CEEKF is formed by maximizing the CEE cost function. In the CEEKF algorithm, the prior state values are transmitted the same as the classical Kalman filter, and the posterior states are calculated by the fixed-point iteration method. The CEE EKF (CEE-EKF) algorithm is also derived to improve filtering accuracy in the case of the nonlinear system. We also give the convergence conditions of the iteration algorithm and the computational complexity analysis of CEEKF. The results of the two simulation examples validate the robustness of the algorithm we presented.

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Temperature Resistance of Mo3Si: Phase Stability, Microhardness, and Creep Properties

Metals ◽

10.3390/met11040564 ◽

2021 ◽

Vol 11 (4) ◽

pp. 564 ◽

Cited By ~ 1

Author(s):

Olha Kauss ◽

Susanne Obert ◽

Iurii Bogomol ◽

Thomas Wablat ◽

Nils Siemensmeyer ◽

...

Keyword(s):

Power Law ◽

Phase Stability ◽

Gas Turbines ◽

Constitutive Models ◽

Creep Properties ◽

Multi Scale ◽

Scale Modeling ◽

Significant Difference ◽

Before And After ◽

Multi Phase

Mo-Si-B alloys are one of the most promising candidates to substitute Ni based superalloys in gas turbines. The optimization of their composition can be achieved more effectively using multi-scale modeling of materials behavior and structural analysis of components for understanding, predicting, and screening properties of new alloys. Nevertheless, this approach is dependent on data on the properties of the single phases in these alloys. The focus of this investigation is Mo3Si, one of the phases in typical Mo-Si-B alloys. The effect of 100 h annealing at 1600 °C on phase stability and microhardness of its three near-stoichiometric compositions—Mo-23Si, Mo-24Si and Mo-25Si (at %)—is reported. While Mo-23Si specimen consist only of Mo3Si before and after annealing, Mo-24Si and Mo-25Si comprise Mo5Si3 and Mo3Si before annealing. The latter is then increased by the annealing. No significant difference in microhardness was detected between the different compositions as well as after annealing. The creep properties of Mo3Si are described at 1093 °C and 1300 °C at varying stress levels as well as at 300 MPa and temperatures between 1050 °C and 1350 °C. Three constitutive models were used for regression of experimental results—(i) power law with a constant creep exponent, (ii) stress range dependent law, and (iii) power law with a temperature-dependent creep exponent. It is confirmed that Mo3Si has a higher creep resistance than Moss and multi-phase Mo-Si-B alloys, but a lower creep strength as compared to Mo5SiB2.

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Improved Process Monitoring Scheme Using Multi-Scale Independent Component Analysis

Arabian Journal for Science and Engineering ◽

10.1007/s13369-021-05822-1 ◽

2021 ◽

Author(s):

K Ramakrishna Kini ◽

Muddu Madakyaru

Keyword(s):

Fault Detection ◽

Independent Component Analysis ◽

Component Analysis ◽

Independent Component ◽

Noisy Environment ◽

Process Data ◽

Multi Scale ◽

Efficient Information ◽

Non Gaussian

AbstractThe task of fault detection is crucial in modern chemical industries for improved product quality and process safety. In this regard, data-driven fault detection (FD) strategy based on independent component analysis (ICA) has gained attention since it improves monitoring by capturing non-gaussian features in the process data. However, presence of measurement noise in the process data degrades performance of the FD strategy since the noise masks important information. To enhance the monitoring under noisy environment, wavelet-based multi-scale filtering is integrated with the ICA model to yield a novel multi-scale Independent component analysis (MSICA) FD strategy. One of the challenges in multi-scale ICA modeling is to choose the optimum decomposition depth. A novel scheme based on ICA model parameter estimation at each depth is proposed in this paper to achieve this. The effectiveness of the proposed MSICA-based FD strategy is illustrated through three case studies, namely: dynamic multi-variate process, quadruple tank process and distillation column process. In each case study, the performance of the MSICA FD strategy is assessed for different noise levels by comparing it with the conventional FD strategies. The results indicate that the proposed MSICA FD strategy can enhance performance for higher levels of noise in the data since multi-scale wavelet-based filtering is able to de-noise and capture efficient information from noisy process data.

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