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.

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 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 Spatial Heterogeneity of the Incidence of Powdery Mildew on Hop Cones

Plant Disease ◽  
2006 ◽  
Vol 90 (11) ◽  
pp. 1433-1440 ◽  
Author(s):  
David H. Gent ◽  
Walter F. Mahaffee ◽  
William W. Turechek
Keyword(s):  
Powdery Mildew ◽  
Spatial Heterogeneity ◽  
Power Law ◽  
Binomial Distribution ◽  
Disease Incidence ◽  
Disease Assessment ◽  
Cluster Sampling ◽  
Data Sets ◽  
Ratio Test ◽  
Frequency Distributions

The spatial heterogeneity of the incidence of hop cones with powdery mildew (Podosphaera macularis) was characterized from transect surveys of 41 commercial hop yards in Oregon and Washington from 2000 to 2005. The proportion of sampled cones with powdery mildew ( p) was recorded for each of 221 transects, where N = 60 sampling units of n = 25 cones assessed in each transect according to a cluster sampling strategy. Disease incidence ranged from 0 to 0.92 among all yards and dates. The binomial and beta-binomial frequency distributions were fit to the N sampling units in a transect using maximum likelihood. The estimation procedure converged for 74% of the data sets where p > 0, and a loglikelihood ratio test indicated that the beta-binomial distribution provided a better fit to the data than the binomial distribution for 46% of the data sets, indicating an aggregated pattern of disease. Similarly, the C(α) test indicated that 54% could be described by the beta-binomial distribution. The heterogeneity parameter of the beta-binomial distribution, θ, a measure of variation among sampling units, ranged from 0.01 to 0.20, with a mean of 0.037 and a median of 0.015. Estimates of the index of dispersion ranged from 0.79 to 7.78, with a mean of 1.81 and a median of 1.37, and were significantly greater than 1 for 54% of the data sets. The binary power law provided an excellent fit to the data, with slope and intercept parameters significantly greater than 1, which indicated that heterogeneity varied systematically with the incidence of infected cones. A covariance analysis indicated that the geographic location (region) of the yards and the type of hop cultivar had little effect on heterogeneity; however, the year of sampling significantly influenced the intercept and slope parameters of the binary power law. Significant spatial autocorrelation was detected in only 11% of the data sets, with estimates of first-order autocorrelation, r1, ranging from -0.30 to 0.70, with a mean of 0.06 and a median of 0.04; however, correlation was detected in only 20 and 16% of the data sets by median and ordinary runs analysis, respectively. Together, these analyses suggest that the incidence of powdery mildew on cones was slightly aggregated among plants, but patterns of aggregation larger than the sampling unit were rare (20% or less of data sets). Knowledge of the heterogeneity of diseased cones was used to construct fixed sampling curves to precisely estimate the incidence of powdery mildew on cones at varying disease intensities. Use of the sampling curves developed in this research should help to improve sampling methods for disease assessment and management decisions.

The breakdown of the power-law frequency distributions for the hard X-ray peak count rates of solar flares

2013 ◽  
Vol 13 (12) ◽  
pp. 1482-1492 ◽  
Author(s):  
You-Ping Li ◽  
Wei-Qun Gan ◽  
Li Feng ◽  
Si-Ming Liu ◽  
A. Struminsky
Keyword(s):  
Power Law ◽  
Solar Flares ◽  
Frequency Distributions ◽  
X Ray ◽  
Peak Count

scholarly journals The Observational Error of Automated Wind Reports from Aircraft

1986 ◽  
Vol 67 (2) ◽  
pp. 177-185 ◽  
Author(s):  
Lauren L. Morone
Keyword(s):  
Measurement Error ◽  
Structure Function ◽  
Error Variance ◽  
Structure Functions ◽  
Separation Distance ◽  
Observational Error ◽  
Random Measurement Error ◽  
Measurement Error Variance ◽  
Flight Level ◽  
Types Of Error

Data collected from aircraft equipped with AIDS (Aircraft Integrated Data System) instrumentation during the Global Weather Experiment year of 1979 are used to estimate the observational error of winds at flight level from this and other aircraft automated wind-reporting systems. Structure functions are computed from reports that are paired using specific criteria. The value of this function extrapolated to zero separation distance is an estimate of twice the random measurement-error variance of the AIDS-measured winds. Component-wind errors computed in this way range from 2.1 to 3.1 m · s−1 for the two months of data examined, January and August 1979. Observational error, specified in optimum-interpolation analyses to allow the analysis to distinguish among observations of differing quality, is composed of both measurement error and the error of unrepresentativeness. The latter type of error is a function of the resolvable scale of the analysis-prediction system. The structure function, which measures the variability of a field as a function of separation distance, includes both of these types of error. If the resolvable scale of an analysis procedure is known, an estimate of the observational error can be computed from the structure function at that particular distance. An observational error of 5.3 m · s−1 was computed for the u and v wind components for a sample resolvable scale of 300 km. The errors computed from the structure functions are compared to colocation statistics from radiosondes. The errors associated with automated wind reports are found to compare favorably with those estimated for radiosonde winds at that level.

scholarly journals Wind Power Persistence Characterized by Superstatistics

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Juliane Weber ◽  
Mark Reyers ◽  
Christian Beck ◽  
Marc Timme ◽  
Joaquim G. Pinto ◽  
...  
Keyword(s):  
Wind Power ◽  
Heavy Tails ◽  
Energy System ◽  
Extreme Value Statistics ◽  
Exponential Distributions ◽  
Weather Types ◽  
Atmospheric Circulation Patterns ◽  
Circulation Patterns ◽  
Wind Velocities ◽  
Renewable Electricity Generation

AbstractMitigating climate change demands a transition towards renewable electricity generation, with wind power being a particularly promising technology. Long periods either of high or of low wind therefore essentially define the necessary amount of storage to balance the power system. While the general statistics of wind velocities have been studied extensively, persistence (waiting) time statistics of wind is far from well understood. Here, we investigate the statistics of both high- and low-wind persistence. We find heavy tails and explain them as a superposition of different wind conditions, requiring q-exponential distributions instead of exponential distributions. Persistent wind conditions are not necessarily caused by stationary atmospheric circulation patterns nor by recurring individual weather types but may emerge as a combination of multiple weather types and circulation patterns. This also leads to Fréchet instead of Gumbel extreme value statistics. Understanding wind persistence statistically and synoptically may help to ensure a reliable and economically feasible future energy system, which uses a high share of wind generation.

Effects of initial conditions on the coalescence of micro-bubbles

2017 ◽  
Vol 232 (3) ◽  
pp. 457-465 ◽  
Author(s):  
Rou Chen ◽  
Huidan(Whitney) Yu ◽  
Likun Zhu
Keyword(s):  
Lattice Boltzmann Method ◽  
Power Law ◽  
Lattice Boltzmann ◽  
Development Time ◽  
Initial Conditions ◽  
Separation Distance ◽  
Bubble Coalescence ◽  
Half Power ◽  
Boltzmann Method ◽  
Method Model

The effects of initial conditions on the coalescence of two equal-sized air micro-bubbles ( R0) in water are studied using the lattice Boltzmann method. The focus is on effects of two initial set-ups of parent bubbles, separated by a small distance d and connected with a neck bridge radius r0, on the neck bridge growth at the early stage of the bubble coalescence. A sophisticated free energy lattice Boltzmann method model based on the Cahn-Hilliard diffuse interface approach is employed. This lattice Boltzmann method model has been demonstrated suitable for handling a large density ratio of two fluids up to 1000 and capable of minimizing the nonphysical spurious current. In both initial scenarios, the neck bridge evolution exhibits a half power-law scaling, [Formula: see text] after a development time. The half power-law agrees with the recent analytical prediction and experimental results. It has been found that smaller initial separation distance or smaller initial neck bridge radius results in faster growth of neck bridge and bubble coalescence, which is similar to the effects of these two initial scenarios on droplet coalescence. The physical mechanism behind each behavior has been explored. For the initial connected case, faster neck growth and longer development time corresponding to smaller initial neck radius is due to the significant bias between the capillary forces contributed by the meniscus curvature and the neck bridge curvature, whereas in the case of initial separated scenario, faster growth and shorter development time corresponding to shorter separation distance is due to the formation of elongated neck bridge. The prefactor A0 that represents the growth of neck bridge radius at the characteristic time ti captured in each case is in good agreement with the experimental results.

MODELING CROSS-CORRELATIONS OF TRAFFIC FLOW

2010 ◽  
Vol 20 (10) ◽  
pp. 3323-3328 ◽  
Author(s):  
PENGJIAN SHANG ◽  
KEQIANG DONG ◽  
SANTI KAMAE
Keyword(s):  
Time Series ◽  
Traffic Flow ◽  
Power Law ◽  
Cross Correlation ◽  
Heavy Tails ◽  
Fluctuation Analysis ◽  
Cross Correlation Analysis ◽  
Cross Correlations ◽  
Large Increment ◽  
Active Research

The study of diverse natural and nonstationary signals has recently become an area of active research for physicists. This is because these signals exhibit interesting dynamical properties such as scale invariance, volatility correlation, heavy tails and fractality. The focus of the present paper is on the intriguing power-law autocorrelations and cross-correlations in traffic series. Detrended Cross-Correlation Analysis (DCCA) is used to study the traffic flow fluctuations. It is demonstrated that the time series, observed on the Anhua-Bridge highway in the Beijing Third Ring Road (BTRR), may exhibit power-law cross-correlations when they come from two adjacent sections or lanes. This indicates that a large increment in one traffic variable is more likely to be followed by large increment in the other traffic variable. However, for traffic time series derived from nonadjacent sections or lanes, we find that even though they are power-law autocorrelated, there is no cross-correlation between them with a unique exponent. Our results show that DCCA techniques based on Detrended Fluctuation Analysis (DFA) can be used to analyze and interpret the traffic flow.

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