Order Statistics of Additive Uniform Exponential Distribution

In this paper we investigated the order statistics by using Additive Uniform Exponential Distribution (AUED) proposed by Venkata Subbarao Uppu (2010).The probability density functions of rth order Statistics, lth moment of the rth order Statistic, minimum, maximum order statistics, mean of the maximum and minimum order statistics, the joint density function of two order statistics were calculated and discussed in detailed . Applications and several aspects were discussed Keywords: Additive Uniform Exponential Distribution, Moments, Minimum order statistic, Maximum order statistic, Joint density of the order Statistics, complete length of service.

Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equation

In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove u ( t , x ) ∈ D ∞ u(t,x)\in \mathbb {D}^\infty for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case.

Rock joint detection from borehole imaging logs based on gray-level co-occurrence matrix and Canny edge detector

Rock joints play an important role in characterizing the rock mass quality for geo-mechanical design and stability analysis. An approach was developed to detect and characterize the rock joints from images collected by a borehole imaging system. A gray-level co-occurrence matrix was employed to locate the joint regions, allowing more focused and effective detection processing, followed by extractions of the upper and lower edges of rock using the Canny algorithm. Four basic geometrical parameters of rock joints－orientation, depth, aperture, and core length－were determined based on the fitting of sinusoids to joints’ edges. Furthermore, the joint density was determined based on the geometric parameters. To calibrate the proposed approach, a borehole in the Rumei hydropower station engineering at Lantsang River was selected as a case study. Orientation of rock joints with gentle dip angles, which was determined from borehole imaging logs, corresponded to the measurement in three horizontal tunnels. Additionally, both joint density and pressure-wave velocity revealed that jointed rock mass was observed in the depth from 100 m to 120 m, and intact rock mass was presented in the depth of 150 m to 170 m, indicating the good performance of the proposed method.

Numerical investigation of the flow characteristics and permeability of 2D irregular columnar jointed rock masses

PurposeThe objective of this paper is to provide a better understanding of the effect of irregular columnar jointed structure on the permeability and flow characteristics of rock masses.Design/methodology/approachAn efficient numerical procedure is proposed to investigate the permeability and fluid flow in columnar jointed rock masses (CJRMs), of which the columnar jointed networks are generated by a modified constrained centroid Voronoi algorithm according to the field statistical results. The fractures are represented explicitly by using the lower-dimensional zero thickness elements. And the modeling scheme is validated by a benchmark test for flow in fractured porous media. The effective permeability and representative elementary volume (REV) size of CJRMs are estimated using finite element method (FEM). The influences of joint density and variation coefficient of columnar joint structure on the permeability of the rock mass are discussed.FindingsThe simulation results indicate that the permeability is scale-dependent and tends to be stable with increase of model size. The hydraulic REV size is determined as 3.5 m for CJRMs in the present study. Moreover, the joint density is a dominant factor affecting the permeability of CJRMs. The average permeability of columnar jointed structures increases linearly with the joint density under the same REV size, while the influence from the coefficient of variation can be neglected.Originality/valueThe present paper investigates the REV size of the CJRMs and the effect of joint parameters on the permeability. The proposed method and the results obtained are useful on understanding the hydraulic characteristic of the irregular CJRMs in engineering projects.

Effect of Joint Density on Rockburst Proneness of the Elastic-Brittle-Plastic Rock Mass

The prediction of rockburst proneness is the basis of preventing and controlling rockburst disasters in rock engineering. Based on energy theory and damage mechanics, the quantitative functional relationship between joint density and energy density was derived. Then, the theoretical results were verified by numerical simulation and uniaxial compression test, and the effect of joint density on rockburst proneness of the elastic-brittle-plastic rock mass was discussed. The results show that the relationship between the joint density and the dissipated energy index of the jointed rock mass is a logarithmic function. With the same total input energy, the higher the joint density, the more the damage dissipation energy. Even in the case of high joint density, the rock mass still has limited resistance to external failure. Under the same joint density, the strength of parallel jointed rock mass is better than that of the cross-jointed rock mass, and the parallel jointed rock mass can accumulate more elastic strain energy and has higher rockburst proneness. The joint density is closely related to the ability of the rock mass to store high strain energy. The higher the joint density is, the weaker the ability to accumulate the elastic strain energy of rock mass is and the lower the rockburst proneness is. It is helpful to predict rockburst proneness by investigating and studying the properties of geological discontinuities. The research results have some theoretical and engineering guiding significance for the prediction of rockburst proneness of the jointed rock mass.

Effects of lithology on bedrock channel occurrence: an examination from the Seogang River in South Korea

<div> <p>Bedrock river is rock-bound, its bed and banks are composed mainly of in-place bedrock. Bedrock channel reaches, commonly short and intermittent, often occur where transport capacity exceeds bedload sediment flux. Despite the abundant research on the typical patterns of alluvial channel reaches, the distribution of bedrock channels has not been well studied. Rock type may affect the occurrence of bedrock channels because the strength, joint density, and erosion process of bedrock vary depending on the rock type. Previous studies have viewed the bedrock channel occurrence in the aspect of the excessive sediment transport capacity, but the influence of lithology has not been considered in the literature. To understand the influence of lithology on bedrock channel occurrence in a drainage basin-scale, we investigated the distribution of bedrock channels in relation to varying lithology and unit stream power along the Seogang River in South Korea. We used satellite images with high resolution for the identification of bedrock channel reaches and then verified them through field surveys. Geological maps and 1 arc-second SRTM DEMs were used to analyze lithological effects and calculate unit stream power.  As a result of the analysis, we identified 94 bedrock channels in the studied river, varying depending on lithologies. The frequency of bedrock channels in granitic gneiss areas (0.73/km) is much higher than those in the other rock type areas (granite areas, 0.57/km; limestone areas, 0.16/km). In the more frequent granitic gneiss areas, the bedrock channels are steepened (average channel slope: 0.0074 m/m) and narrow (average channel width: 65 m) and mainly reside within steepened and narrow (average valley width: 123 m) rock-bound valleys so that their occurrence is mainly associated with high unit stream power. In contrast, the bedrock channels over the other lithologies are wider (89 m) and lower-gradient (0.0056 m/m) and occur along flat and broad valleys (391 m). Consequently, the bedrock channels in the studied river were divided into two types: confined and unconfined bedrock channels. The confined bedrock channels are within the steepened and narrow valleys composed of resistant granitic gneiss and show the evidence for recent bedrock incision processes. However, the unconfined bedrock channels are mainly within the broad and flat valleys of weak saprolites and limestone with high joint density have lower unit stream power and don't show any marker for bedrock incision. In conclusion, high-relief landscape mainly composed of more resistant rocks generates steep and narrow valleys, which leads to the formation of continuous and actively incising bedrock channels. However, low-relief landscape underlain by non-resistant rocks shows wider and lower-gradient channels, with intermittent bedrock channels due to locally more resistant rock bodies.</p> </div>

Comparison of models for missing pedigree in single-step genomic prediction

Pedigree information is often missing for some animals in a breeding program. Unknown-parent groups (UPGs) are assigned to the missing parents to avoid biased genetic evaluations. Although the use of UPGs is well established for the pedigree model, it is unclear how UPGs are integrated into the inverse of the unified relationship matrix (H-inverse) required for single-step genomic best linear unbiased prediction. A generalization of the UPG model is the metafounder (MF) model. The objectives of this study were to derive 3 H-inverses and to compare genetic trends among models with UPG and MF H-inverses using a simulated purebred population. All inverses were derived using the joint density function of the random breeding values and genetic groups. The breeding values of genotyped animals (u2) were assumed to be adjusted for UPG effects (g) using matrix Q2 as u2∗=u2+Q2g before incorporating genomic information. The Quaas–Pollak-transformed (QP) H-inverse was derived using a joint density function of u2∗ and g updated with genomic information and assuming nonzero cov(u2∗,g′). The modified QP (altered) H-inverse also assumes that the genomic information updates u2∗ and g, but cov(u2∗,g′)=0. The UPG-encapsulated (EUPG) H-inverse assumed genomic information updates the distribution of u2∗. The EUPG H-inverse had the same structure as the MF H-inverse. Fifty percent of the genotyped females in the simulation had a missing dam, and missing parents were replaced with UPGs by generation. The simulation study indicated that u2∗ and g in models using the QP and altered H-inverses may be inseparable leading to potential biases in genetic trends. Models using the EUPG and MF H-inverses showed no genetic trend biases. These 2 H-inverses yielded the same genomic EBV (GEBV). The predictive ability and inflation of GEBVs from young genotyped animals were nearly identical among models using the QP, altered, EUPG, and MF H-inverses. Although the choice of H-inverse in real applications with enough data may not result in biased genetic trends, the EUPG and MF H-inverses are to be preferred because of theoretical justification and possibility to reduce biases.