About approximation of 3-D random fields and statistical simulation

Soil nails used to support excavations may be inserted into soils with bedding structures at different orientations. This paper examines for the first time the influence of such cases on probabilistic margins of safety for a typical soil nail arrangement by assigning rotated anisotropic random fields of soil strength parameters that are aligned with the bedding structure of the soil domain. The results using rotated anisotropic random fields on statistical outcomes for the computed global factor of safety are compared to results using homogeneous and isotropic random fields, and deterministic analyses. Analyses using rotated anisotropic random fields show that there is a detectable worst bedding direction that gives the highest probability of failure. However, the range of factor of safety for all random field cases is small and remains in the vicinity of the deterministic value. This is ascribed to the ability of the soil nails to anchor the entire reinforced soil domain together and prevent preferential failure paths to develop through the soil. The paper includes the calculation of the confidence in the point estimate of probability of failure for different conditions, and lessons learned in selecting domain size and level of discretization using the random finite difference method.

Previous article Next article Asymptotic Distributions of Curvatures of a Homogeneous Gaussian Random Field at Points of High Local MaximaV. P. NoskoV. P. Noskohttps://doi.org/10.1137/1129110PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Yu. Belyaev, Point processes and first passage problems, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, Univ. California Press, Berkeley, Calif., 1972, 1–17, Los Angeles 53:4207 Google Scholar[2] Robert J. Adler, The geometry of random fields, John Wiley & Sons Ltd., Chichester, 1981xi+280 82h:60103 0478.60059 Google Scholar[3] A. N. Mal'kov, An investigation of the curvature of Gaussian random processes and fields at local maximum pointsInvestigations on Random Fields, MGU Press, Moscow, 1974, 103–111, (In Russian.) Google Scholar[4] P. K. Rashevskii, A Course of Differential Geometry, GITTL, Moscow—Leningrad, 1950, (In Russian.) Google Scholar[5] Yu. K. Belyaev, Random point sets and problems of the level crossing typeExcursions of Random Fields, MGU Press, Moscow, 1972, 9–37, (In Russian.) Google Scholar[6] V. P. Nosko, Local structure of Gaussian homogeneous fields in the vicinity of high-level light sources, Soviet Math. Dokl, 10 (1969), 1481–1482 0267.60036 Google Scholar[7] V. P. Nosko, The characteristics of excursions of Gaussian homogeneous fields beyond a high level, Soviet-Japan Symposium on Probability Theory (Khabarovsk, August, 1969), SO Akad Nauk SSSR, Novosibirsk, 1969, 209–215, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails On the Shape of a High Excursion of a Gaussian Stationary Process24 March 2021 | Journal of Mathematical Sciences, Vol. 254, No. 4 Cross Ref Volume 29, Issue 4| 1985Theory of Probability & Its Applications History Submitted:09 September 1982Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1129110Article page range:pp. 817-823ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

The paper deals with the theory and methods of statistical simulation of random processes and fields based on their spectral decomposition and Kotelnikov-Shennon modified interpolation sums, as well as applying these methods for environmental geophysical monitoring. Statistical simulation of multivariate random fields (those homogeneous in time and homogeneous isotropic in n other variables) are considered to be essential for seismological research into frequency characteristics of geological media. A statistical model and a numerical algorithm of simulating random fields are built on the basis of Kotelnikov-Shennon modified interpolation decomposition to generate adequate realizations of seismic noise. The paper examines real-valued random fields ξ(t,x),tϵÎR,xÎRn, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. It also considers approximation of random fields by the random fields with a bounded spectrum. There is made an analogue of the Kotelnikov–Shannon theorem for random fields with a bounded spectrum. Besides, there are obtained estimates of the mean-square approximation of random fields in the space R´Rn by a model constructed with the help of spectral decomposition and Kotelnikov–Shannon interpolation formula. The paper provides a mechanism for statistical simulation of Gaussian random fields with a bounded spectrum; namely, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. Proved have been the theorems of the mean-square approximation of random fields (those homogeneous in time and homogeneous isotropic ones relative to n- other variables) by special partial sums. A simulation method was used to formulate an algorithm of numerical simulation by means of these theorems. There are also considered ways to carry out spectral analysis of generated seismic noise realizations. Finally, there have been developed universal methods of statistical simulation (Monte Carlo methods) of multi-parameter seismology data for generating seismic noise on 2D and 3D grids of the required detail and regularity.

We consider a continuous model for transverse magnetization of spins diffusing in a homogeneous Gaussian random longitudinal field \(\{\lambda V(x);\, x \in {\Bbb R}^{d} \} \), where \(\lambda \) is the coupling constant giving the intensity of the random field. In this setting, the transverse magnetization is given by the formula \(M(t)={\Bbb E} \exp \{ -{\lambda}^{2} \int_{0}^{t} \int_{0}^{t} K(B_r-B_s) \; ds dr \} \), where \(\{B_t;\,t\ge 0\}\) is the standard process of Brownian motion and \(K(x)\) is the covariance function of the original random field \(V(x)\). We use large deviation techniques to show that the limit \(S(\lambda)=\lim_{t \rightarrow \infty} \frac{1}{t} \ln M(t)\) exists. We also determine the small \(\lambda\) behavior of the rate \(S(\lambda)\) and show that it is indeed decaying as conjectured in the physics literature.

The paper tackles the problem of simulating isotropic vector-valued Gaussian random fields defined over the unit two-dimensional sphere embedded in the three-dimensional Euclidean space. Such random fields are used in different disciplines of the natural sciences to model observations located on the Earth or in the sky, or direction-dependent subsoil properties measured along borehole core samples. The simulation is obtained through a weighted sum of finitely many spherical harmonics with random degrees and orders, which allows accurately reproducing the desired multivariate covariance structure, a construction that can actually be generalized to the simulation of isotropic vector random fields on the d-dimensional sphere. The proposed algorithm is illustrated with the simulation of bivariate random fields whose covariances belong to the $${{{\mathcal{F}}}}$$ , spectral Matern and negative binomial classes of covariance functions on the two-dimensional sphere.

For perfect surface roughness description is not enough to know characteristics of surface profile. It is necessary to use topography methods, so called microtopography. Thereby, surface roughness in microtopographycal understanding must be described with three coordinates, whose in Cartesian coordinates system compose point under consideration height h, abscissa and ordinate, determines point position in the plane. Most efficient methods in irregular surface roughness research are random function theory methods. Therefore, microtopography, analogically to profile, may consider as random function, but two dimensional function, i.e. two variable x and y random field h(x,y). From analogy with random process, random field can be normal – ordinates are distributed by normal (Gaussian) distribution. Moreover, random field can be homogeneous and heterogeneous. Random field is deemed homogeneous if its mean value is discretionary and correlation function depends only from distance between surface points. Important characteristic of random field is correlation function, whose depends of two variables t1 and t2 – orthogonal Cartesian coordinates of vector t. Random field is homogeneous and isotropic when its characteristics are equivalent in any direction. There are three types of surface anisotropy: • General event of surface anisotropy. Characteristics of this event roughness parameters are depend of surface split direction. • Surface roughness with direct anisotropy. Those surfaces are with typical traces of tool and they proper two mutually perpendicular surface roughness directions. • Extended anisotropy area – special event of anisotropy roughness. Of analytical opinion, gainfully anisotropy roughness see as extended occasional isotropy area. This let easy cross from anisotropy surface to isotropy and contrariwise, thereby embrace amount class of surface roughness. Let’s formulate microtopography model of rough surface [1]. Surface roughness is described with homogeneous normal random field h(x,y) that has uninterrupted correlation function and uninterrupted deriviates. We may consider that E{h(x,y)}=0. The mean random field value is plane called mean plane. For describing random field we must know mathematical expectation and field correlation function, what in fact reduces on determining dispersion and rationed correlation function r(t1, t2). Homogeneous random field dispersion D{h} doesn’t depends of direction and can be founded in any surface split. Given model of rough surface let inspect surfaces produced by abrasive instruments and friction surfaces after wear-in period.

A meshfree-Galerkin method in modelling and synthesizing spatially varying soil properties

We develop multispectral random field image models for use in image processing applications. The simultaneous autoregressive and Markov random field (MRF) models have been widely used in modeling intensity images. In this work we extend these models to include the more general multispectral case where images are represented by multiple intensity planes. In particular, we focus on the obvious application to color texture modeling using the RGB color model. For each model type we present the model equations, develop methods for synthesizing images based on these models and procedures for estimating the model parameters. In addition, the conditions necessary to ensure model validity are identified. We also provide experimental results which, substantiate the validity of these results. Color images synthesized from these models are shown to have the statistical characteristics implied by the model equations and parameters estimated from these images are very close to the known values from which the images were generated. In further experiments color random field models were fitted to natural texture samples. Images synthesized from these models are observed to be visually similar to the original images.

Generalizing the Affine Framework to HJM and Random Field Models

A homogeneous isotropic Gaussian random field is accepted as a statistical model of objective functions, aiming to construct global optimization algorithms. The asymptotic of the conditional mean and variance is considered, assuming that the random field values are known at the vertices of a simplex, and that the latter is contracting. The obtained result theoretically substantiates the construction of the recently proposed bi-variate global optimization algorithm, which arouses interest due to good performance in testing experiments and the established convergence rate. The obtained result also enhances motivation to extend the aforementioned algorithm to higher dimensions.

Needlet approximation for isotropic random fields on the sphere

A random field model of external metal-loss corrosion on buried pipelines

In the scanning magnetic domain by using the conventional magnetic force microscopy (MFM), a laser beam reflection is used to detect the static magnetic force between probe and sample. Therefore, for the MFM, it is a challenge to directly detect the dynamic magnetic force between probe and sample under an external alternating-current (AC) magnetic field. In this study, it is proved that in an alternating magnetic force microscopy (A-MFM) a sensitive Co-GdO<sub><i>x</i></sub> superparamagnetic probe can be usedto detect the dynamic magnetic force under an external AC magnetic field (frequency <i>ω</i><sub>m</sub>). In the present method, the magnetization of Co-GdO<sub><i>x</i></sub> probe is modulated by an external AC magnetic field. Collecting <i>ω</i><sub>m</sub> and 2<i>ω</i><sub>m</sub> signals by using the combination of phase-locked loop (PLL) and lock in amplifiers can accurately represent the static (DC, which stands for direct current) magnetic field areas (the external AC magnetic field has no effect on the magnetized status of the sample) and dynamic (AC) magnetic field areas (the external AC magnetic field changes the magnetized status of the sample) of an anisotropic Sr ferrite sintered magnet at the same time, respectively. The Sr ferrite sample is a single-domain-type magnet where magnetization mainly changes via magnetic rotation. The A-MFM method can measure the strength and identify the polarities of the static magnetic field of sample with a DC demagnetized state. By modifying the traditional tapping-lift mode into a tapping-multiply lift mode, the A-MFM by using superparamagnetic tips can measure the static and dynamic magnetic field distribution in three-dimensional (3D) space. It is proved that the static and dynamic magnetic field as a function of the distance <i>z</i> between probe and sample are both expressed as <i>H<sub>z</sub></i>(<i>z</i>) = <i>H<sub>z</sub></i>(0)·exp(–<i>kz</i>). The experimental data are consistent with the previous theoretical calculations. The A-MFM can be used to study the dynamic magnetization process and to evaluate the magnetic homogeneity (microstructural homogeneity) of magnetic materials.

Influence of soil spatial variability on the response of strip footing on geocell-reinforced slope

In this paper, the properties of a new class of causal Markov random fields, named symmetric Markov mesh random field, are initially discussed. It is shown that the symmetric Markov mesh random fields from the upper corners are equivalent to the symmetric Markov mesh random fields from the lower corners. Based on this new random field, a symmetric, corner-independent, and isotropic image model is then derived which incorporates the dependency of a pixel on all its neighbors. The introduced image model comprises the product of several local 1D density and 2D joint density functions of pixels in an image thus making it computationally tractable and practically feasible by allowing the use of histogram and joint histogram approximations to estimate the model parameters. An image restoration application is also presented to confirm the effectiveness of the model developed. The experimental results demonstrate that this new model provides an improved tool for image modeling purposes compared to the conventional Markov random field models.