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In this study, a modified constrained centroid Voronoi tessellation (MCCVT) algorithm was used to generate stochastic models of irregular columnar jointed rock masses (CJRMs) considering the joint characteristics within the cross section. Subsequently, the effect of the size and joint characteristics, such as the joint density (JD) and coefficient of variation (COV), on the elastic parameters of the CJRMs were analyzed. The results indicated that the MCCVT algorithm generated irregular CJRMs with a high-accuracy advantage over conventional algorithms, with JD and COV relative errors of only 0.73% and 0.90%, respectively. The minimum representative volume element (RVE) size was found to be larger than 2.0 m. With increasing JD, the elastic modulus decreased almost linearly, whereas Poisson's ratio did not change significantly. As the COV increased, the elastic modulus decreased before increasing, although this effect is not significant. Meanwhile, Poisson's ratio continued to increase. Additionally, the anisotropy ratio of the elastic modulus first increased before decreasing with increasing COV, whereas that of Poisson's ratio continued to increase. The maximum anisotropy ratios of the elastic modulus and Poisson's ratio were 1.87 and 1.35, respectively. These findings suggest that treating CJRMs as transverse isotropic media is oversimplified. Finally, fitting equations based on the numerical simulation results were proposed to estimate the elastic parameters of irregular CJRMs based on joint characteristics, which could help researchers better understand the mechanical characteristics of CJRMs. [ABSTRACT FROM AUTHOR]

The construction of a high-precision geomagnetic map is a prerequisite for geomagnetic navigation and magnetic target-detection technology. The Kriging interpolation algorithm makes use of the variogram to perform linear unbiased and optimal estimation of unknown sample points. It has strong spatial autocorrelation and is one of the important methods for geomagnetic map construction. However, in a region with a complex geomagnetic field, the sparse geomagnetic survey lines make the ratio of line-spacing resolution to in-line resolution larger, and the survey line direction differs from the geomagnetic trend, which leads to a serious effect of geometric anisotropy and thus, reduces the interpolation accuracy of the geomagnetic maps. Therefore, this paper focuses on the problem of geometric anisotropy in the process of constructing a geomagnetic map with sparse data, analyzes the influence of sparse data on geometric anisotropy, deduces the formula of geometric anisotropy correction, and proposes a modified interpolation algorithm accounting for geometric anisotropy correction of variogram for sparse geomagnetic data. The results of several sets of simulations and measured data show that the proposed method has higher interpolation accuracy than the conventional spherical variogram model in a region where the geomagnetic anomaly gradient changes sharply, which provides an effective way to build a high-precision magnetic map of the complex geomagnetic field under the condition of sparse survey lines. [ABSTRACT FROM AUTHOR]

Today, Phased Array Ultrasonic Testing (PAUT) is a suitable method for detecting defects in Non-destructive Testing (NDT). This simulation was done with a new method on a stainless steel pipe with a diameter of 500 mm and a length of 1,000 mm, and cracks were modeled on the external and internal surfaces and corrosion defects on the internal surface. Four probes with a characteristic of 2 MHz were used simultaneously, each probe having 64 elements. By increasing the number of probes and their elements, more accurate information about the defects is obtained, which makes it easier to detect the location and leads to a reduction of scattered signals and noise, which can detect even small size defects. Considering this important advantage, a new use of the method of increasing the number of probes in PAUT is reported in this study to detect defects, especially corrosion defects. The results of geometric anisotropy studies of group velocity and phase of horizontally polarized shear waves (SH-waves) for stainless steel pipe were presented. Due to the isotropic properties of the pipe material, the speed on the outer surface of the pipe in the direction of the cover is 40 m/s higher than the generatrix. [ABSTRACT FROM AUTHOR]

Today, Phased Array Ultrasonic Testing (PAUT) is a suitable method for detecting defects in Non-destructive Testing (NDT). This simulation was done with a new method on a stainless steel pipe with a diameter of 500 mm and a length of 1,000 mm, and cracks were modeled on the external and internal surfaces and corrosion defects on the internal surface. Four probes with a characteristic of 2 MHz were used simultaneously, each probe having 64 elements. By increasing the number of probes and their elements, more accurate information about the defects is obtained, which makes it easier to detect the location and leads to a reduction of scattered signals and noise, which can detect even small size defects. Considering this important advantage, a new use of the method of increasing the number of probes in PAUT is reported in this study to detect defects, especially corrosion defects. The results of geometric anisotropy studies of group velocity and phase of horizontally polarized shear waves (SH-waves) for stainless steel pipe were presented. Due to the isotropic properties of the pipe material, the speed on the outer surface of the pipe in the direction of the cover is 40 m/s higher than the generatrix.

The construction of a high-precision geomagnetic map is a prerequisite for geomagnetic navigation and magnetic target-detection technology. The Kriging interpolation algorithm makes use of the variogram to perform linear unbiased and optimal estimation of unknown sample points. It has strong spatial autocorrelation and is one of the important methods for geomagnetic map construction. However, in a region with a complex geomagnetic field, the sparse geomagnetic survey lines make the ratio of line-spacing resolution to in-line resolution larger, and the survey line direction differs from the geomagnetic trend, which leads to a serious effect of geometric anisotropy and thus, reduces the interpolation accuracy of the geomagnetic maps. Therefore, this paper focuses on the problem of geometric anisotropy in the process of constructing a geomagnetic map with sparse data, analyzes the influence of sparse data on geometric anisotropy, deduces the formula of geometric anisotropy correction, and proposes a modified interpolation algorithm accounting for geometric anisotropy correction of variogram for sparse geomagnetic data. The results of several sets of simulations and measured data show that the proposed method has higher interpolation accuracy than the conventional spherical variogram model in a region where the geomagnetic anomaly gradient changes sharply, which provides an effective way to build a high-precision magnetic map of the complex geomagnetic field under the condition of sparse survey lines.

Paper deals with statistical classification of spatial data as a part of widely applicable statistical approach to pattern recognition. Error rates in supervised classification of Gaussian random field observation into one of two populations specified by different constant means and common stationary geometric anisotropic covariance are considered. Formula for the exact Bayesian error rate is derived. The influence of the ratio of anisotropy to the error rates is evaluated numerically for the case of complete parametric certainty.

When working with contemporary spatial ecological datasets, statistical modellers are often confronted by two major challenges: (I) the need for covariance models with the flexibility to accommodate directional patterns of anisotropy; and (II) the computational effort demanded by high-dimensional inverse and determinant problems involving the covariance matrix V → . In the case of rectangular lattice data, the spatially separable covariogram is a longstanding but underused model that can reduce arithmetic complexity by orders of magnitude. We examine a class of covariograms for stationary data that extends the separable model through affine coordinate transformations, providing a far greater flexibility for handling anisotropy than that offered by the standard approach of using geometric anisotropy to extend an isotropic model. This motivates our development of an extremely fast estimator of the orientation of the axes of range anisotropy on spatial lattice data, and a powerful visual diagnostic for nonstationarity. In a case study, we demonstrate how these tools can be used to analyze and predict forest damage patterns caused by outbreaks of the mountain pine beetle. [ABSTRACT FROM AUTHOR]

The columnar jointed basalt (CJB) in the dam site of the Baihetan hydropower station exhibits noticeable geometric anisotropy. The current constitutive relationship based on a quadrangular columnar structure fails to truly represent the natural characteristics of the CJB. To address the stability problem of the diversion tunnel at the dam foundation, the mechanics of composite materials and Goodman superposition principle were employed to establish improved three-dimensional constitutive relations for pentagonal and hexagonal CJBs. The constitutive relations of three types, including the quadrangular prism CJB, were optimized by considering the impact of transverse joints and internal staggered zones. The mechanical parameters of the Baihetan CJB were appropriately evaluated using the three constitutive relations, and the achieved results demonstrated that the determined elastic moduli of the three improved constitutive models were consistent with field results. Subsequently, the three anisotropic constitutive models were employed for numerical simulations of the diversion tunnel excavation in the Baihetan CJB stratum. Strain field tests were conducted at different locations within the tunnel, and an empirical method was proposed that combined statistical data on the shape of CJBs at the site, which can effectively predict the deformation of the surrounding rock. The theoretical and numerical results provide crucial reference values for engineering optimization design. [ABSTRACT FROM AUTHOR]

To reveal the meso-mechanical mechanism of shear behaviors in the graded crushed rocks (GCRs), which are heterogeneous in geometry and widely used in the flexible pavement, a numerical technique was proposed to simulate the geometric anisotropy of the GCRs using randomly generated models. Then, studies on meso-mechanical responses during the loading were performed through the biaxial shear test and California bearing ratio (CBR) test. The results showed that the changes of both friction and anisotropy had a similar trend with the particle size enlarging in the dense assembly, which maintain stable when the size is less than 3.0 mm. In addition, the breaking sieve (BS) between coarse and fine aggregates was six times of the base diameter (0.8 mm). The predicted shear properties in macrostructure were in good agreement with those laboratory measurements. For the shear behaviors, the fractures and force chains were mainly distributed in the area of action that was limited by force size and side boundary. [ABSTRACT FROM AUTHOR]

We introduce a Gibbs–Markov random field for spatial data on Cartesian grids, which is based on the generalized planar rotator (GPR) model. The GPR model generalizes the recently proposed modified planar rotator (MPR) model by including in the Hamiltonian additional terms that better capture realistic features of spatial data, such as smoothness, non-Gaussianity, and geometric anisotropy. In particular, the GPR model can include an infinite number of higher-order harmonics with exponentially vanishing interaction strength, directional dependence of the bilinear interaction term between nearest neighbors, longer-distance neighbor interactions, and two types of an external bias field. Hence, in contrast with the single-parameter MPR, the GPR model features five additional parameters: the number n of higher-order terms and the decay-rate parameter α , the exchange anisotropy parameter J nn , the further-neighbor interaction coupling J fn , and the external field (bias) parameters K (or K ′). We present numerical tests on various synthetic and real data which demonstrate the effects of the respective terms on the model's prediction performance, and we discuss these results in connection with the data properties. • We introduce Gibbs–Markov random fields for spatial data prediction on regular grids. • We generalize a simple MPR model (GPR) by including additional terms in Hamiltonian. • GPR model thus features terms controlling non-Gaussianity, anisotropy and smoothness. • Tests on synthetic and real data show their benefits on prediction performance. • We discuss these results in connection with the data properties. [ABSTRACT FROM AUTHOR]

Anisotropic models are often used in spatial statistics to analyze spatially referenced data. Within a Bayesian framework we develop default priors for the anisotropic Gaussian random field model with and without including a nugget parameter accounting for the effects of microscale variations and measurement errors. We present Jeffreys priors and a reference prior and study their posterior propriety. Moreover, we obtain that the predictive distributions at ungauged locations have finite variance. We also show that the seemingly uninformative uniform prior for the anisotropy parameters, ratio and angle, yields an improper posterior. Finally, we find that the proposed priors have good frequentist properties and we illustrate our approach by analyzing two data sets for which we discuss model choice as well as predictions and uncertainty estimates. [ABSTRACT FROM AUTHOR]

model for computing thermal stress inside a crystal with facets is presented. Using a systematical perturbation analysis, a semi-analytical thermal-stress solution is obtained for constrained directional growth with small lateral heat flux. This solution can be applied to crystals grown by various growth techniques such as the Czochralski method. The semi-analytical nature of the solution leads to a much more efficient approach for computing thermal stress in crystals with facets, compared to a full 3D simulation. Examples are given for crystals pulled in a variety of seed orientations. [ABSTRACT FROM AUTHOR]

Frequently an observed spatial variable may have anisotropic variability which varies signi­ficantly with direction. ln this article overview of anisotropy forms for spatial variograms is presented. Different forms of variograms anisotropy were obtained by fitting salinity data collected at the coastal zone of Baltic Sea. Analysis was carried out using R, a system for statistical computation and graphics.

In monitoring the environment one often wishes to detect the temporal trend in a variable that varies across a region. A useful executive summary is then the temporal trend in the spatial mean. In this article, the best linear unbiased predictor of the spatial mean temporal trend and its variance are derived under a universal kriging model. Five different, spatially explicit sampling patterns are compared in terms of this variance. For small spatial ranges, the time-separation pattern is optimal, regardless of the temporal range. For larger spatial ranges, the best pattern depends on the temporal range. If the temporal range is small (less than ca. 1/20 of the spatial range), then the always-revisit pattern is best. For larger temporal ranges (between ca. 1/20 and ca. 1/4 of the spatial range) the serially alternating pattern is best, whereas the time-separation pattern is best for larger temporal ranges. The gain in using the serially alternating pattern or time-separation pattern instead of the always-revisit pattern can be substantial. The potential loss with respect to the always-revisit pattern is only minor for the serially alternating pattern, but can be substantial for the time-separation pattern. Unless one has good knowledge of the spatial and temporal range, the serially alternating pattern is thus a good choice. This work extends that by Urquhart and Kincaid and gives further support to their plea for the serially alternating pattern. [ABSTRACT FROM AUTHOR]

The Matérn correlation function provides great flexibility for modeling spatially correlated random processes in two dimensions, in particular via a smoothness parameter, whose estimation allows data to determine the degree of smoothness of a spatial process. The extension to include anisotropy provides a very general and flexible class of spatial covariance functions that can be used in a model-based approach to geostatistics, in which parameter estimation is achieved via REML and prediction is within the E-BLUP framework. In this article we develop a general class of linear mixed models using an anisotropic Matérn class with an extended metric. The approach is illustrated by application to soil salinity data in a rice-growing field in Australia, and to fine-scale soil pH data. It is found that anisotropy is an important aspect of both datasets, emphasizing the value of a straightforward and accessible approach to modeling anisotropy. [ABSTRACT FROM AUTHOR]

Abstract: Behaviour of fundamental and higher harmonics of ac susceptibility is studied for cylindrical geometry subjected to an applied ac field B = B accos(ωt) along any arbitrary direction with respect to the cylinder axis. Variations of the parallel and two perpendicular in-phase (χ′) and out-of-phase (χ″) ac susceptibility components are investigated in details with respect to the change in the applied field geometry (field orientations and amplitudes). The effects of geometrical anisotropy of the sample on the ac susceptibility measurements are also discussed. Some of our selected results compares well with the experimental studies. [Copyright &y& Elsevier]

The anisotropy in particular environmental phenomena is detected when behavior of a physical process differs in different directions. In this paper geometric and zonal anisotropies are considered. Various methods of geostatistical analysis, also isotropic and geometrical anisotropic variogram models are compared and applied for the Curonian lagoon depth data. The results demonstrate that after robust estimation, i.e. elimination of outliers and after elimination of geometric anisotropy the precision of prediction and adequacy of models are much better. All computations have been performed by means of gstat, base and spacial packages of R system. Prediction results are compared with the results of research where outliers and geometric anisotropy were not eliminated. [ABSTRACT FROM AUTHOR]

Geostatistical models are appropriate for spatially distributed data measured at irregularly spaced locations. We propose an efficient Markov chain Monte Carlo (MCMC) algorithm for fitting Bayesian geostatistical models with substantial numbers of unknown parameters to sizable data sets. The algorithm facilitates use of MCMC sampler output for computing Bayes factors for model selection. In our illustrative analysis of an environmental data set, model selection is critical because the inference of primary interest changes depending on whether the model includes or ignores spatial correlation. [Copyright &y& Elsevier]

Legislation passed in 1990 reducing the allowable sulfur dioxide emission levels in the United States is expected to reduce acidity in the Adirondack region of New York State. The number of fish species in a lake (species richness) depends on a number of physical and chemical factors, including the area, elevation, and acidity of the lake. Data on these and other factors are available for 1,166 Adirondack lakes. The data are analyzed with the goal of quantifying the effects of acid deposition on species richness after controlling for other important factors. A plot of the residuals from a standard multiple regression model against spatial location reveals a strong nonlinear relationship between species richness and lake location. With only one realization of the data, it is difficult, if not impossible. to tell whether this dependence should be modeled as deterministic spatial trend or as a spatial covariance structure. Two models are thus considered. The first is a multiple linear regression model with spatially correlated errors. As is common in geostatistics, it is assumed that the correlation between two errors is a function only of the vector separating the corresponding lakes. This is accomplished by modeling the correlation among the errors with a parametric variogram model. The parameters of the variogram are estimated via restricted maximum likelihood, after which the regression parameters are estimated using generalized least squares. The second model considered is a partial linear (or semiparametric) model containing a nonlinear term in spatial location. Locally weighted quadratic regression is used in conjunction with an estimation technique described by Speckman to estimate simultaneously the regression parameters and the “smooth” function of longitude and latitude. Finally, the ultimate goal of the 1990 legislation is a 50% reduction in sulfate deposition. Both models are used to form prediction intervals for species richness under that scenario. Extrapolation is avoided by using the criteria of Weisberg. The results suggest that a 50% reduction in sulfate deposition may result in a substantial increase in the fish species richness of many Adirondack lakes, particularly those with pH values between 4.5 and 6.5. [ABSTRACT FROM AUTHOR]