Anisotropic Correlation Length Research Articles

In-situ investigation on coal mine overburden dump slope and its seismic stability considering heterogeneity

Stability of the coal mine overburden (OB) dump slopes is many times compromised owing to the lesser availability of the in-situ characteristics of the OB dump materials and their spatial heterogeneity. In this study, multi-channel analysis of surface waves (MASW) investigation was carried out for the characterization of OB dump materials of Jambad opencast coal mine of India and the variation of heterogeneous in-situ characteristics has been reported in terms of statistical parameters. Next, the stability of a two-bench OB dump slope against seismic forces was investigated through finite element limit analyses coupled with random field theory utilizing Karhunen-Loeve expansion method for the spatial heterogeneity of the OB dump material. The effects of spatial heterogeneity of the OB dump material on the probability of failure and the critical pseudostatic seismic coefficient of the slope were investigated through Monte-Carlo simulations. The influence of spatial heterogeneity of OB dump materials were investigated with both isotropic and anisotropic correlation lengths. Finally, seismic slope stability analyses were also carried out using the discrete element method for the size heterogeneity of the OB dump materials. This study summarizes the effects of heterogeneity of material property and size on the seismic stability of OB dump slope.

Numerical limit analyses on the uplift capacity of strip plate anchor in anisotropic spatially random cohesive soil

Plate anchors are used to resist uplift forces of both offshore and on-land structures. Numerous studies have been carried out to date to evaluate the ultimate uplift capacity of anchor plates. Most of them assumed the soil as homogeneous and isotropic. Naturally, the soil is heterogeneous and anisotropic. The undrained shear strength of soil may vary significantly within a seemingly homogeneous layer. This paper analyzes the impact of spatial shear strength variability of cohesive soil on the uplift capacity of strip anchors through numerical limit analyses. Repeated Monte Carlo simulations within a selected range of both isotropic and anisotropic spatial correlation length and coefficient of variation of undrained shear strength are conducted using the random field theory to evaluate the effect of probabilistic uplift capacity of shallow and deep strip anchors. The outcomes are then compared to those of a similar study in which the probabilistic uplift capacity was investigated using a displacement-based analysis employing the local averaging subdivision of the random finite element method (LAS-RFEM). The comparison depicts the requirement of a higher safety factor than the published results of RFEM. The findings suggest considering the spatial variation of soil strength for the safe design of plate anchors.

Anisotropic scaling for 3D topological models

A proposal to study topological models beyond the standard topological classification and that exhibit breakdown of Lorentz invariance is presented. The focus of the investigation relies on their anisotropic quantum critical behavior. We study anisotropic effects on three-dimensional (3D) topological models, computing their anisotropic correlation length critical exponent nu obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. A generalized Weyl semimetal model with broken time-reversal symmetry is introduced and studied using a modified Dirac equation. An approach to characterize topological surface states in topological insulators when applied to Fermi arcs allows to capture the anisotropic critical exponent theta =nu _{x}/nu _{z}. We also consider the Hopf insulator model, for which the study of the topological surface states yields unusual values for nu and for the dynamic critical exponent z. From an analysis of the energy dispersions, we propose a scaling relation nu _{bar{alpha }}z_{bar{alpha }}=2q and theta =nu _{x}/nu _{z}=z_{z}/z_{x} for nu and z that only depends on the Hopf insulator Hamiltonian parameters p and q and the axis direction bar{alpha }. An anisotropic quantum hyperscaling relation is also obtained.

Probabilistic Settlement Analysis of Strip Footing on Spatially Random Soil

The prediction of settlement in a traditional design approach usually uses a deterministic value of modulus of elasticity (E), which is estimated as an average value by testing the soil at selected locations. However, these deterministic properties of soil may not represent the actual properties of soil and site condition. Due to numerous sources of uncertainty, the properties of soil mass are spatially varying and anisotropic in the natural field condition. In this study, a random finite element method (RFEM) is used to evaluate the reliability of settlement of the strip footing on spatially random soil. Modulus of elasticity is the only considered random parameter. For this purpose, 2000 spatially random realizations of E-field are generated using Monte Carlo Simulation. Each of these realizations of heterogeneous soil profile is passed to FEM to analyze the settlement of footing. The final settlement results measured in all these realizations are then statistically evaluated and compared. The results of analysis show that the mean and standard deviation of the footing settlement are increased with increasing spatial correlation length. The large value of isotropic correlation length led to an increase in the mean settlement value by more than 25% as compared with the deterministic settlement calculations. Also, it is concluded that the rate of increase of settlements for anisotropic correlation length is lower than the one under isotropic condition.

Asymptotic correlation functions in the Q-state Potts model: A universal form for point group C_{4v}.

Reexamining algebraic curves found in the eight-vertex model, we propose an asymptotic form of the correlation functions for off-critical systems possessing rotational and mirror symmetries of the square lattice, i.e., the C_{4v} symmetry. In comparison with the use of the Ornstein-Zernike form, it is efficient to investigate the correlation length with its directional dependence (or anisotropy). We investigate the Q-state Potts model on the square lattice. Monte Carlo (MC) simulations are performed using the infinite-size algorithm by Evertz and von der Linden. Fitting the MC data with the asymptotic form above the critical temperature, we reproduce the exact solution of the the anisotropic correlation length (ACL) of the Ising model (Q=2) within a five-digit accuracy. For Q=3 and 4, we obtain numerical evidence that the asymptotic form is applicable to their correlation functions and the ACLs. Furthermore, we successfully apply it to the bond percolation problem which corresponds to the Q→1 limit. From the calculated ACLs, the equilibrium crystal shapes (ECSs) are derived via duality and Wulff's construction. Regarding Q as a continuous variable, we find that the ECS of the Q-state Potts model is essentially the same as those of the Ising models on the Union Jack and 4-8 lattices, which are represented in terms of a simple algebraic curve of genus 1.

Partial Up-Up-Down Order with the Continuously Distributed Order Parameter in the Triangular Antiferromagnet TmMgGaO4

Frustrated quasidoublets without time-reversal symmetry can host highly unconventional magnetic structures with continuously distributed order parameters even in a single-phase crystal. Here, we report the comprehensive thermodynamic and neutron diffraction investigation on the single crystal of TmMgGaO$_4$, which entails non-Kramers Tm$^{3+}$ ions arranged on a geometrically perfect triangular lattice. The crystal electric field (CEF) randomness caused by the site-mixing disorder of the nonmagnetic Mg$^{2+}$ and Ga$^{3+}$ ions, merges two lowest-lying CEF singlets of Tm$^{3+}$ into a ground-state (GS) quasidoublet. Well below $T_c$ $\sim$ 0.7 K, a small fraction of the antiferromagnetically coupled Tm$^{3+}$ Ising quasidoublets with small inner gaps condense into two-dimensional (2D) up-up-down magnetic structures with continuously distributed order parameters, and give rise to the \emph{columnar} magnetic neutron reflections below $\mu_0H_c$ $\sim$ 2.6 T, with highly anisotropic correlation lengths, $\xi_{ab}$ $\geq$ 250$a$ in the triangular plane and $\xi_c$ $<$ $c$/12 between the planes. The remaining fraction of the Tm$^{3+}$ ions remain nonmagnetic at 0 T and become uniformly polarized by the applied longitudinal field at low temperatures. We argue that the similar model can be generally applied to other compounds of non-Kramers rare-earth ions with correlated GS quasidoublets.

Energy spectrum and critical exponents of the free parafermion ZN spin chain

Results are given for the ground state energy and excitation spectrum of a simple N-state ZN spin chain described by free parafermions. The model is non-Hermitian for with a real ground state energy and a complex excitation spectrum. Although having a simpler Hamiltonian than the superintegrable chiral Potts model, the model is seen to share some properties with it, e.g. the specific heat exponent and the anisotropic correlation length exponents and .

Optimal channels for channelized quadratic estimators.

We present a new method for computing optimized channels for estimation tasks that is feasible for high-dimensional image data. Maximum-likelihood (ML) parameter estimates are challenging to compute from high-dimensional likelihoods. The dimensionality reduction from M measurements to L channels is a critical advantage of channelized quadratic estimators (CQEs), since estimating likelihood moments from channelized data requires smaller sample sizes and inverting a smaller covariance matrix is easier. The channelized likelihood is then used to form ML estimates of the parameter(s). In this work we choose an imaging example in which the second-order statistics of the image data depend upon the parameter of interest: the correlation length. Correlation lengths are used to approximate background textures in many imaging applications, and in these cases an estimate of the correlation length is useful for pre-whitening. In a simulation study we compare the estimation performance, as measured by the root-mean-squared error (RMSE), of correlation length estimates from CQE and power spectral density (PSD) distribution fitting. To abide by the assumptions of the PSD method we simulate an ergodic, isotropic, stationary, and zero-mean random process. These assumptions are not part of the CQE formalism. The CQE method assumes a Gaussian channelized likelihood that can be a valid for non-Gaussian image data, since the channel outputs are formed from weighted sums of the image elements. We have shown that, for three or more channels, the RMSE of CQE estimates of correlation length is lower than conventional PSD estimates. We also show that computing CQE by using a standard nonlinear optimization method produces channels that yield RMSE within 2% of the analytic optimum. CQE estimates of anisotropic correlation length estimation are reported to demonstrate this technique on a two-parameter estimation problem.

Parameterization of Centimeter-Scale Sea Ice Surface Roughness Using Terrestrial LiDAR

Microwave scattering from sea ice is partially controlled by the ice surface roughness. In this paper, we propose a technique for calculating 2-D centimeter-scale surface roughness parameters, including the rms height, correlation length, and form of autocorrelation function, from 3-D terrestrial light detection and ranging data. We demonstrate that a single scale of roughness can be extracted from complex sea ice surfaces, incorporating multiple scales of topography, after sophisticated 2-D detrending, and calculate roughness parameters for a wide range of artificial and natural sea ice surface types. The 2-D technique is shown to be considerably more precise than standard 1-D profiling techniques and can therefore characterize surface roughness as a stationary single-scale process, which a 1-D technique typically cannot do. Sea ice surfaces are generally found to have strongly anisotropic correlation lengths, indicating that microwave scattering models for sea ice should include surface spectra that vary as a function of the azimuthal angle of incident radiation. However, our results demonstrate that there is no fundamental relationship between the rms height and correlation length for sea ice surfaces if the sampling area is above a threshold minimum size.

Spatial correlation length of normalized cone data in sand: case study in the north of Denmark

The main topic of this study is to assess the anisotropic spatial correlation lengths of a sand layer deposit based on cone penetration testing with pore pressure measurement (CPTu) data. Spatial correlation length can be an important factor in reliability analysis of geotechnical systems, yet it is rarely estimated during routine site investigations. Results from two different sites in the north of Denmark are reported in this paper, indicating quite strong anisotropy due to the depositional process, with significantly shorter spatial correlation lengths in the vertical direction. It is observed that the normalized cone resistance is a better estimator of spatial trends than the normalized friction ratio.

Neutron scattering measurements of spatially anisotropic magnetic exchange interactions in semiconducting K0.85 Fe1.54Se2 (TN = 280 K.

We use neutron scattering to study the spin excitations associated with the stripe antiferromagnetic order in semiconducting K(0.85)Fe(1.54)Se(2) (T(N) = 280 K). We show that the spin-wave spectra can be accurately described by an effective Heisenberg Hamiltonian with highly anisotropic inplane couplings at T = 5 K. At high temperature (T = 300 K) above T(N), short-range magnetic correlation with anisotropic correlation lengths are observed. Our results suggest that, despite the dramatic difference in the Fermi surface topology, the inplane anisotropic magnetic couplings are a fundamental property of the iron-based compounds; this implies that their antiferromagnetism may originate from local strong correlation effects rather than weak coupling Fermi surface nesting.

Helical glasses near ferromagnetic quantum criticality

We study the effects of quenched charge disorder on the phase reconstruction near itinerant ferromagnetic quantum critical points in three spatial dimensions. Combining a Replica-disorder average with a fermionic version of the quantum order-by-disorder mechanism, we show that weak disorder destabilizes the ferromagnetic state and enhances the susceptibility towards incommensurate, spiral magnetic ordering. The Goldstone modes of the spiral phase are governed by a $3d$-$XY$ model. The induced disorder in the pitch of the spiral generates a random anisotropy for the Goldstone modes, inducing vortex lines in the phase of the helical order and rendering the magnetic correlations short ranged with a strongly anisotropic correlation length.

Polarization modulation of electromagnetic waves scattered from a quasi-homogeneous anisotropic medium

We show that, within the accuracy of the first-order Born approximation, the spatially uniform spectral degree of polarization (sDOP) of electromagnetic waves (EW) generally converts into a spatially depolarized distribution once the EW scatters from a quasi-homogeneous (QH) anisotropic medium. The influence of the anisotropic correlation lengths of the QH medium and that of the initial polarization of the EW incident on the sDOP of the scattered field are analyzed, separately. The results obtained may be of essential significance to the polarization imaging of anisotropic media, e.g. for obtaining knowledge of biological tissues or the determination of spatially anisotropic particles.

Effect of strong heterogeneity on the onset of convection in a porous medium: Importance of spatial dimensionality and geologic controls

The effect of strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated heterogeneous porous medium governed by Darcy's law is investigated. A computer package has been developed to study the applicability of an average Rayleigh number as a criterion for the onset of convection in strongly heterogeneous geologic media. The heterogeneous geologic media have been described using random spatial functions for the permeability field which are lognormally distributed and spatially correlated. Both isotropic and anisotropic correlation lengths within the geologic structure are considered. This paper presents the first 3D theoretical treatment of the conditions for the onset of convection (Rayleigh stability criteria) in strongly heterogeneous porous media. We elucidate the critical role that spatial dimensionality (2D versus 3D flow) plays in controlling convection processes and stability criteria. Our results quantitatively demonstrate for the first time that spatial dimensionality is a dominant control on the onset of convection in a strongly heterogeneous geologic medium. Unbounded Rayleigh number behavior is observed in 3D. This leads to the important new conclusion that a Rayleigh number (based on mean quantities) is unlikely to be a valid predictor for the onset of convection in 3D strongly heterogeneous porous media. Furthermore, we systematically and quantitatively demonstrate that the onset of convection in a heterogeneous geologic medium is highly sensitive to changes in the standard deviation of the lognormal permeability field, moderately sensitive to changes in the level of correlation length, and relatively insensitive to the anisotropy of correlation length.

Numerical Simulation on the Oil Recovery from an Areally Heterogeneous Reservoir by Polymer Augmented Waterflood

The influence of water-soluble polymer solution on the flow behavior and oil recovery in areally heterogeneous reservoirs with adsorption and crossflow was investigated by numerical simulation. For the reservoirs of low viscosity oil, polymer floods had a relatively small improvement on the recovery of oil. Performances of waterflood and polymer flood were not so sensitive to change in permeability variation up to σ(ln k) = 1.0. Variations in permeability fields have a pronounced effect on the tertiary oil recovery efficiency for reservoirs with viscous oil due to additional dispersion caused by the randomly heterogeneous reservoir. Oil recovery increases as the correlation length in the permeability field increases for both water and polymer floods. Anisotropic correlation length causes sudden changes in oil recovery and water-oil ratio and results in a significant decrease in oil recovery and increase in water-oil ratio over the production life.

Anisotropic spatially heterogeneous dynamics on theαandβrelaxation time scales studied via a four-point correlation function

We examine the anisotropy of a four-point correlation function G4(k[over ],r[over ];t) and its associated structure factor S4(k[over ],q[over ];t) calculated using Brownian dynamics computer simulations of a model glass forming system. These correlation functions measure the spatial correlations of the relaxation of different particles. We examine the time and temperature dependences of the anisotropy in both functions. We find that the anisotropy is strongest at nearest-neighbor distances at time scales corresponding to the peak of the non-Gaussian parameter alpha_{2}(t)=3deltar;{4}(t)/[5deltar;{2}(t);{2}]-1 but is still pronounced around the alpha relaxation time. We find that the structure factor S4(k[over ],q[over ];t) is anisotropic even for the lowest wave vector accessible in our simulation, suggesting that our system (and other systems commonly used in computer simulations) may be too small to extract the q[over ]-->0 limit of the structure factor. We find that the determination of a dynamic correlation length from S4(k[over ],q[over ];t) is influenced by the anisotropy. We extract an effective anisotropic dynamic correlation length from the small q behavior of S4(k[over ],q[over ];t) .

ELECTRONIC TRANSPORT IN ANISOTROPIC DISORDERED QUANTUM WIRE

We have investigated the transport properties of anisotropic disordered quantum wire by the method of energy level statistics and transfer matrix method. We have found a general average conductance relation in terms of the anisotropy parameter t, disorder strength W, and the cross-sectional area, A, of the wire. From the available numerical data, we have also shown that the anisotropic correlation length ξtcan be related to the isotropic correlation length ξ as ξt≃ t2ξ, for L ≃ A and ξt≃ tξ, for L ≫ A. Here, L is the length of the wire. In addition, anisotropic dependence of conductance fluctuation and conductance distribution have been studied to a certain extent.

Developing a high-resolution climatology for the Central California coastal region

This work presents a procedure for developing a high-resolution, regional climatology estimate, named RClimo, off the coast of central California. This high-resolution climatology may provide an alternative way to initialize numerical nowcast/forecast exercises in coastal regions. The methodology includes two primary steps: (1) averaging available data on a high-resolution grid and (2) objective interpolating the resulting average profiles onto a regular grid. The first step involves the computation of averages over density layers in the vertical and allowing for data gaps in the horizontal if data are unavailable at a high resolution. The OA in the second step uses anisotropic correlation length scales derived from the data themselves and an averaging radius to preserve the scales and variability of the synoptic fields. The dataset used to compute this climatology includes the archived CalCOFI dataset, the Autonomous Ocean Sampling Network (AOSN) 2003 experiments near Monterey Bay, and many other previously undocumented profiles from various sources. As part of the climatology product, associated uncertainty is also generated through density averaging and employing the Gauss–Markov minimum error variance during Objective Analysis. The final climatology estimate is hence subject to greater error for larger mapping-grid size or lower data density, suggesting uncertainties that vary in space. The maximum value of the resulting error distribution for the RClimo estimate from the 50-km bins is less than 11% of the temperature estimate and 1% of the salinity estimate, whereas those for the 20-km-bin RClimo are 11% and 0.9%, respectively. We have conducted comparisons between the RClimo and 1/4° Levitus climatology fields via numerical simulations initialized with each field. Simulations were performed using Regional Ocean Modeling System (ROMS) and for the month of August, a peak period of upwelling-favorable wind, with and without the atmospheric forcing. The RClimo simulations produced reasonably realistic features like filaments, fronts, meanders, eddies, alongshore currents (∼50 cm s −1) and inshore undercurrents (∼15 cm s −1) under typical upwelling-favorable winds. In contrast, the Levitus simulations showed insignificant evolution over most of the domain, with or without winds. In the future, one could apply this methodology for other months and seasons in Monterey Bay, as well as develop similar high-resolution climatology for other coastal regions.

Reply to Comment on Equilibrium crystal shape of the Potts model at the first-order transition point

The eight-vertex model is defined on the square lattice rotated through an arbitrary angle with respect to the coordinate axes. We re-examine the analysis of the anisotropic correlation length in a previous paper (Fujimoto M 1996 Physica A 233 485–502). We point out that the asymptotic form of the correlation function is expressed by the use of differential forms on a Riemann surface of genus 1. Combined with the symmetry of the square lattice, this fact explains that the anisotropic correlation length is represented in terms of simple algebraic curves. The argument is applicable to a wide class of lattice models (including unsolvable ones).

Numerical Simulations of Heat Transfer in Porous Media with Effect of Heterogeneities

Numerical simulations are employed to investigate the heat transfer and fluid flow in an inhomogeneous porous medium. The permeability heterogeneity, characterized by a correlation length and variance, are seen to have strong effect on both the flow and heat transfer. The velocity fluctuation generated by permeability heterogeneity has the tendency to alter the flow features and orientation predominantly at the scale of the correlation length. For an isotropic permeability distribution, the most pronounced effect of flow alternation and fluctuation is observed at intermediate correlation length, which agrees with previous findings, thus leads to the best performance of convective heat transfer. In a permeability field with anisotropic correlation length, larger correlation length along the stream-wise direction makes the velocity fluctuation oriented more likely parallel to the main averaged velocity, and weakens the transverse transport effects. Therefore, better heat transfer performance is found if larger correlation length is aligned along the transverse direction. The simulations also show better heat transfer effect as the variance of the permeability field or global flow rate increases.