Length Probability Density Function Research Articles

On the Existence, Uniqueness and Correctness of the Fracture Diameter Distribution Given the Fracture Trace Length Distribution

Based on the fracture trace length distribution, conditions for the existence, uniqueness, and correctness of the fracture diameter distribution are given using Warburton’s fracture model. In particular, a solution for the fracture diameter distribution exists and is unique for all trace length probability density functions, hA(y), such that \(h_{A}(y)/\sqrt{y^{2}-x^{2}}\) is Lebesgue integrable on [x,∞). This condition is met by the uniform, exponential, gamma, lognormal, and power-law trace length distributions as well as by the trace length distributions that arise from a deterministic fracture diameter or from a discontinuous fracture diameter length distribution. Exponential, gamma, lognormal, and power-law trace length distributions satisfy necessary conditions for the diameter distribution to be non-negative, and necessary and sufficient conditions for the distribution to have unit integral over the real line. Negative values of the fracture diameter distribution arise when the trace has a uniform distribution and the lower bound of the fracture trace is greater than zero.

Improving fractured carbonate-reservoir characterization with remote sensing of beds, fractures, and vugs

Many key aquifers and oil reservoirs are in carbonate rocks. Understanding the flow behavior within this commonly complex pore space requires new perspectives and technology in order to improve carbonate aquifer and reservoir characterization. Dissolution of carbonates is related to flow; hence, quantifying the size of dissolution vugs on carbonate outcrops can help characterize controls on flow, namely matrix permeability and fracture connectivity. LIDAR (light detection and ranging) scans, combined with high-resolution photography, enable us to both measure vugs' areas and assess spatial relationships between vugs, beds, and fractures. We developed a method of obtaining and interpreting necessary vug, bed, and fracture data on the basis of these technologies. Application of this method on a Cretaceous Edwards Group outcrop in Texas (United States) revealed a significant correlation between the relative vug area of beds obtained remotely and air permeability measured in plugs extracted from these beds (R2 = 0.94, P = 0.001). The total area of vugs intersected by fractures was used to establish a probability density function of fracture lengths that can improve flow modeling of the reservoir. These findings show the potential of using LIDAR and photo images in reservoir characterization by data analysis of geological features, in addition to their use for accurate mapping.

Geometric figures with the same chord length probability density function: Six examples

The chord length probability density functions for isotropic uniform random chords f( l) have been studied for 12 different geometric figures K i . The detailed analysis shows: six pairs of different K i possess the same f i ( l). In greater detail, for the following six figure-pairs (specific length parameter b), 1. 60°-angle with side b ↔ equilateral triangle with side length b, 2. 90°-angle with side b ↔ square with side length b, 3. 60°-wedge of side length b ↔ triangular rod of side length b, 4. 90°-wedge of side length b ↔ square rod of side length b, 5. 90°-angle with one side b, one infinitely long side e → ∞, ↔ plane stripe of breadth b, 6. 90°-wedge with one breadth b, two sides of length e → ∞, ↔ infinite Layer of constant thickness b, the respective functions f i ( l, b) are identical. Thus, without additional shape information, an identification of such a figure via its chord length probability density function (PDF) is not possible. However, in all the cases considered, the length parameter b, involved in the function f( l), can be recognized from the intrinsic behavior of f( l, b). Furthermore, the agreement of the first moments of the respective functions f i can be verified by use of the extended Cauchy theorem for non-convex figures.

Fiber Length Distribution in Cotton Processing: A Finite Mixture Distribution Model

This paper introduces a new approach to modeling and parameterizing cotton fiber length distribution. The approach uses finite mixture models to derive a parametric expression of the fiber length probability density function. The model was applied to a multitude of empirical length distributions and proved to adequately parameterize the complex distribution patterns, as well as express the intrinsic and process-related factors determining their shape.

A Method to Estimate Operational Fatigue Life of Aeronautical Structural Components and Experimental Verification Thereof - An Outline

A Method to Estimate Operational Fatigue Life of Aeronautical Structural Components and Experimental Verification Thereof - An OutlineThe study presents an attempt to estimate fatigue life of structural components with further verification of the applied method by means of testing the specimens taken. Therefore, it has been assumed that a specimen subjected to tests is a structural component of a real system, and operational loads have been simulated in the form of a predefined loading scheme. All parameters necessary for the method have been derived from the analyses of both the specimens subjected to tests and properties of the preset load spectrum.The applied method of fatigue life estimation represents a probabilistic approach based on the Paris formula for the crack growth rate, and on difference equations that after some transformation result in an equation of the Fokker-Planck type. A probability density function of a crack length is a solution to this equation and depends on either the total time of operating the component in question or the number of load cycles applied.The probability density function of a crack length has been used to find out a formula for the probability of not exceeding the permissible crack length against the number of load cycles. The derived relationship has been applied after normalization to estimate fatigue life as based on results of experimental examination of specimens made from titanium alloy.The nomenclature of the aeronautical engineering is used throughout the paper due to the assumption that the component exposed to tests represents a part of an aircraft. This, in turn, has been intended to show some specific application(s) of the formulated model.

Photonic superdiffusive motion in resonance radiation trapping

In this work we consider the relation between the jump length probability density function and the line shape function in resonance radiation trapping in atomic vapors. The two-sided jump length probability density function suitable for a unidimensional formulation of radiative transfer is also derived. As a side result, a procedure to obtain the Maxwell distribution of velocities from the Maxwell-Boltzmann distribution of speeds was obtained. General relations that give the asymptotic jump length behavior and the Levy flight parameter mu for any line shape are obtained. The results are applied to generalized Doppler, generalized Lorentz, and Voigt line shape functions. It is concluded that the lighter the tail of the line shape function, the less heavy the tail of the jump length probability density function, although this tail is always heavy, with mu < or =1.

Mixing of confined coaxial flows

The paper presents experimental results on the mixing process in a coaxial jet mixer in two mixing regimes. In the first mixing regime, a recirculation zone develops just behind a nozzle near mixer walls, while in the second regime a jet is mixed with the co-flow without developing a recirculation zone. In the both regimes, the mixing process is studied at Re d = 10 000. Behind the nozzle over the range 0.1 < x/ D < 9.1, a velocity field in mixer cross-sections is measured by a one-component laser Doppler velocity meter and a scalar field is detected by the laser image fluorescence (LIF) method. A transverse autocorrelation function, integral length scales and probability density functions (PDF) are calculated using instantaneous distributions of a scalar and its fluctuations. It is shown that the scalar field acquires a homogeneous state faster than the velocity one. A quasi-uniform scalar distribution over the mixer cross-section is completed at the distance x/ D = 5.1 in the first mixing regime, while this distribution has not been yet attained in the second. Analysis of the turbulent statistical moments and the autocorrelation function reveals how unsteady vortex structures exert a dramatic influence on the mixing. When the recirculation zone has developed, long-period antiphase oscillations exist near the mixer walls.

Evaluation of drop size distribution from chord length measurements

AbstractThe relationship between the chord length distribution (CLD) obtained by a point sensor and the drop size distribution (DSD) in a liquid‐liquid dispersion is investigated. Based on analysis of the frequency of drop‐cuttings by the sensor, a physical model is built to derive the probability density function of chord lengths for a given DSD, and vice versa. The effect of biased sampling towards larger drops relevant to point sensors, which is often ignored by investigators, is included in this relationship. A new algorithm is introduced to solve the problem of noisy or even negative DSD values by adding smoothing equations while performing the backward conversion. The effects of parameters, such as the number of drop size groups used, the noise level, and the smoothing factor value, on the backward transform are further studied. Both forward and backward transforms are shown to be in good agreement with ideal data when using continuous (e.g., log‐normal, uniform) distributions and with data obtained from Monte‐Carlo simulations. Good agreement is also found between the DSD obtained from chord length measurements and drop size data determined by direct observation. © 2005 American Institute of Chemical Engineers AIChE J, 2006

The probability density function of completed length of service (CLS) distribution and wastage function for some secondary schools in Enugu State

By investigating the existing relationships between the probability density function of CLS distribution and some other wastage functions this paper estimates the functions for some secondary schools in Enugu State. Wastage probabilities are calculated, survivor functions estimated. The accompanying standard errors are also obtained. Key words: Manpower Planning, Length of Service, Modelling, Survivor Functions. [Global Jnl Mathematical Sci Vol.2(1) 2003: 81-88]

An allometric model for trees

This paper presents a general mathematical model for the morphometric description of trees. This model is based on the introduction of fractal theory, and more particularly of the concept of self-similarity, into a statistical physics rationale. Fractal theory provides the necessary tools to describe the complexity of tree structure. Statistics, when applied to physics, makes it possible to explain the properties of complex objects starting from their components. The combination of both tools allowed us to develop a theoretical model that is the probability density function of the morphometric lengths of trees. An example of validation of this law is given here: the theoretical model of morphometric lengths is compared with experimental data of Cupressocyparis.

Photon path length distributions inferred from rotating shadowband spectrometer measurements at the Atmospheric Radiation Measurements Program Southern Great Plains site

An algorithm for retrieving the first two moments of the photon path length probability density function for both the oxygen A‐band and the 0.820 μm water vapor band from measurements of the second generation Rotating Shadowband Spectrometer (RSS) is developed and applied to data from the Atmospheric Radiation Measurements (ARM) Program Southern Great Plains (SGP) site. In the algorithm, solar direct‐beam measurements are used to characterize the instrument response function in pixel (i.e., wavelength) space. By using nonlinear least squares regression with a two‐parameter gamma function constraint, the mean and variance of the photon path length probability density function for cloudy skies are subsequently retrieved from spectral measurements in both bands. Two case studies illustrate that the variance of the photon path length probability density function is more sensitive than the mean of the probability density function to vertical cloud structure. Interestingly, the first two moments of the photon path length probability density function appear to exhibit sufficient sensitivity to detect cirrus that the ARM SGP millimeter‐wave cloud radar failed to detect. Photon path length probability density functions from both the oxygen A‐band and 0.820 μm water vapor band provide additional insights into radiative transfer through a variety of cloudy conditions, improving our understanding of water vapor absorption of solar radiation in these conditions.

Air method measurements of apple vessel length distributions with improved apparatus and theory.

Studies showing that rootstock dwarfing potential is related to plant hydraulic conductance led to the hypothesis that xylem properties are also related. Vessel length distribution and other properties of apple wood from a series of varieties were measured using the 'air method' in order to test this hypothesis. Apparatus was built to measure and monitor conductivity to air of fresh wood segments of different lengths. Theory for determining vessel length distribution was improved to give a single parameter uni-modal vessel length probability density function. The function, derived by combining the exponential extinction (with extinction coefficient k) of wood conductivity to air (C) as wood length (x) increases (i.e. C=Co exp (-kx)) with the differential double difference formula, is Px=xxk2 exp (-kx), where Px is the fraction of vessels of length x. The main parameter of the distribution, k, was found to be the inverse of the mode of the distribution, i.e. the most common vessel length, Lo. Lo for ten apple rootstock and scion varieties varied from 5.6+/-0.1 cm (+/-SE) for MM.111 to 9.0+/-1.0 for Prunifolia (P <0.05). Average maximum vessel length was approximately 50 cm, and differences were not significant. Effective vessel radii ranged from 14 for Prunifolia to 24.3+/-0.7 micro m for M.26, with standard errors less than 12% of the mean. Specific conductivity of a 15 cm wood segment ranged from 2x109-4) to 1.6+/-0.2x10(-2) dm3 s(-1) kPa(-1) m(-1) for maruba and M.26, respectively, with standard errors up to 63% of the mean. Vessel density at the air entry point ranged from 18+/-3 to 42+/-6 vessels mm-2 for M.26 and MM.106, respectively, with standard errors as high as 89% of the mean. It was concluded that there is no general relationship between the wood properties investigated and rootstock size class, and that plasticity increases from vessel lengths to radii to specific conductivity and vessel densities.

A simple triangular approximation of the area function for the calculation of network hydrological response

The hydrological response of a catchment is the result of combined hillslope and channel network delays in the translation of rainfall to discharge at the catchment outlet. In this paper we examine three simple descriptions of the channel network response following an instantaneous hillslope runoff input within a general framework in which the role of network morphology is described using the probability density function (pdf) of network path lengths, and the role of hydrodynamic dispersion is described using a solution of the linear advection dispersion equation. We show that the pdf of path lengths, represented by the area function, can be adequately approximated by a triangular distribution with just two parameters (the mean and maximum network path lengths). The triangular pdf can be combined with a description of residence times in individual paths to derive an analytical expression for the hydrological response of the channel network. This expression provides a better approximation of the network response than both the original geomorphological instantaneous unit hydrograph (GIUH), based on Horton–Strahler ratios, and a single-channel representation, based on the concept of geomorphological dispersion. Using the same expression, we show that geomorphological factors generally prevail in controlling the mean and variance of network residence times. In contrast, hydrodynamic dispersion is shown to have a significant influence on the skewness of the network response over a wide range of catchment characteristics, strongly influencing the hydrograph peak and time to peak values in these cases. Copyright © 1999 John Wiley & Sons, Ltd.

Von Bertalanffy growth model in a random environment

The well-known von Bertalanffy growth model for describing age-length relationship is formulated in a randomly fluctuating environment. The fluctuations in the system are assumed to be described by a Gaussian white noise stochastic process. The resulting model, in terms of a stochastic differential equation, is solved analytically. It is shown that the probability density function of length of a fish is a Gaussian stochastic process. Finally, as an illustration, the methodology is applied to a set of pearl oyster (Pinctada fucata (Gould)) data.

First geometrical path length probability density function derivation of the skylight from high‐resolution oxygen A‐band spectroscopy: 2. Derivation of the Lévy index for the skylight transmitted by midlatitude clouds

For the first time Lévy indices (γ) of the solar light transmitted by cloudy skies at mid latitude (50°N, 8.2°E) are reported. The Lévy index describes the dependence of the mean geometrical paths (&lt; LT &gt; ) of photons transmitted from cloudy skies as a function of the vertical cloud extension (Hc) expressed by the scaling law &lt;LT &gt; ∼ Hcγ. For a set of 33 individual cloudy sky &lt; LT &gt; measurements, reported cloud types and heights, Lévy indices are deduced. It is found that the inferred Lévy indices cluster into the range of 1 ≤ γ ≤ 2 for “all sky” observations, and into a range 1.5 ≤ γ ≤ 2 for optically very thick clouds. The observations provide evidence that the cloudy sky geometrical path lengths are Lévy distributed with the γ value depending on the cloud morphology (the shape of individual clouds and the spatial arrangement of the clouds) but also on the internal cloud inhomogeneities. Because of a particular sensitivity of our method to detect the radiative transfer (RT) caused by clouds (rather than by the clear sky parts of the atmosphere), the inferred type of the path statistics is expected to reflect mostly the path length distribution caused by cloud inhomogeneities rather than by the cloud morphology. Since the cloud inhomogeneities are caused by dynamic processes (besides other factors), the RT transfer is expected to be closely connected to atmospheric dynamics. From this it is concluded that the absorption of solar radiation in cloudy skies is connected to the dynamic state (whether it is stratified or convective) of the cloud cover. In particular, it is expected to be different (mostly larger) than calculated by naively assuming homogeneous horizontally infinite cloud covers in conventional non statistical RT models.

Geometrical path length probability density functions of the skylight transmitted by midlatitude cloudy skies: Some case studies

Examples of inferred probability density functions of geometrical path lengths (PDF‐GP) for zenith‐scattered skylight transmitted by cloudy skies at midlatitude (50°N, 8.2°E) are reported. The method relies on high resolution spectroscopy of the O2 A‐band (760nm–780nm) absorption in the transmitted skylight, and an·inverse Laplace transform to recover the PDF‐GP from the absorption strengths of a set of A‐band absorption lines. Some examples of a larger set of measurements are presented and discussed in detail. These examples include PDF‐GPs for typical midlatitude summer (fair weather Cumulus) and winter (Stratus) cloudiness. Moreover, two cloud observations with largely enhanced O2 A‐band absorptions (up to a factor of 7 implying a mean geometrical path length of about 116 km) are discussed, namely the PDF‐GPs of thunderstorm clouds (Cumulonimbus, Cb) and of clouds formed in a cold front (Stratocumulus, Sc; Altostratus, As; Nimbostratus, Ns; Cirrus, Ci).

DESCRIPTION OF SHORT FATIGUE CRACK GROWTH IN A SHOT=PEENED MEDIUM CARBON STEEL

A probabilistic and simplified approach to the description of short crack growth in shot‐peened, medium carbon steel specimens is presented. In order to model the dynamics of short crack growth, a difference equation has been formulated with coefficients originating from a two‐parameter Weibull distribution of crack advance. The Fokker–Planck partial differential equation is the source of a solution based on the form of a probability density function of crack length. The resolved function allows one to calculate the expected crack length, crack growth rate and standard deviation of crack length. The viability of the probabilistic method has been verified using experimental data gained for shot‐peened specimens fatigued under reversed torsion.

Finite temporal measurements of the statistical characteristics of the atmospheric coherence length

Based on the theory of stationary random processes, the probability density function of the atmospheric coherence length measured in a finite time period is derived analytically and verified experimentally. An iteration method is proposed to obtain Fried's coherence length within a finite time measurement.

Identifying Overturns in CTD Profiles

Abstract The authors propose a scheme to test whether inversions in CTD density profiles are caused by overturning motions (from which mixing rates may be inferred) or by measurement noise. Following a common practice, possible overturning regions are found by comparing the observed profile ρ(z) and an imaginary profile ˆρ(z) constructed by reordering ρ(z) to make. it gravitationally stable. The resulting “reordering regions” are subjected to two tests. • The “Thorpe fluctuation” profile ρ′(z) = ρ(z) − ˆρ(z) is examined for “runs” of adjacent positive or negative values. The probability density function (PDF) of the run length is compared with the corresponding PDF of random noise. This yields a threshold value for rms run length within individual reordering regions that must be exceeded for adequate resolution of overturns, taking into account both CTD characteristics and local hydrographic properties. • Temperature and salinity covariations with respect to density are screened for systematic CTD errors ...

Conditional sampling, bursting, and the intermittent structure of sensible heat flux

Sensible heat flux measurements were carried out to examine the characteristics related to frequency and duration of extreme flux events above a desert surface. First, the flux bursts were identified by a modified hyperbolic hole “quadrant analysis” in which the threshold criterion depends on the mean and standard deviation of the flux time statistics. Second, the frequency distribution of the extreme flux burst length was examined. It was found that the frequency distribution of the burst length is well approximated by two‐parameter power laws which account for two distinct types of bursting phenomena: one is organized, coherent, and has a relatively long duration, while the other is short lived, less organized, and has a relatively short duration. Linear regression analysis was used to determine the two power exponents of the burst length probability density function. Both exponents compare favorably with other reported measurements on concentration bursting in dispersing plumes from an elevated source. The cutoff burst length from one power law to the next also matches other studies. A strong correlation between bursting frequency and flux integral time scales was observed under a variety of atmospheric stability and surface wetness conditions. This strong correlation suggested a relationship between atmospheric stability and the mean time interval between bursts. It was found that the variation in the interval between bursts for the dynamic and dynamic convective sublayers is strongly related to the variation in stability. This was not the case for near‐convective and stable atmospheric conditions.