Anisotropic Random Fields Research Articles

Probabilistic bearing capacity of strip footing on reinforced soil slope

The probabilistic bearing capacity of the strip footing placed on the edge of a reinforced cohesionless soil slope is computed by combining lower bound finite element limit analysis method with anisotropic random field modelling and Monte Carlo simulation technique. Friction and dilation angle of purely cohesionless soil are considered as log-normally distributed random variables. The mean value of the bearing capacity factor Nγ reduces with the increasing randomness of the soil friction angle for particular values of correlation length in the horizontal and vertical directions. At smaller correlation lengths, the mean Nγ values of unreinforced and reinforced slopes are always lower than the deterministic values. However, the difference reduces with the increasing values of correlation lengths. At a higher correlation length, the deterministic and probabilistic mean values become almost equal. With the increasing value of correlation lengths and factor of safety, the failure probability of the bearing capacity factor associated with the reinforced slope reduces. Failure mechanisms of unreinforced and reinforced slopes obtained from the deterministic and probabilistic analysis are illustrated.

Magnetism and structure in nanocomposite Fe nanoparticle/Al matrix films

We describe magnetometry measurements performed on nanocomposite films formed of Fe nanoparticles embedded in Al matrix. The samples were prepared using a flexible co-deposition technique under Ultra-High Vacuum (UHV) environments. Fe nanoparticles, made by using a gas aggregation source, were co-deposited with an atomic Al beam, made by an MBE source. The Volume Filling Fraction (VFF) of nanoparticles was varied controllably between 4% and 45%, while the mean diameter of Fe nanoparticles produced by the source was ∼ 2 nm. Fe K edge extended x-ray absorption fine structure (EXAFS) experiments show that there is a high degree of alloying between Fe nanoparticles and Al atoms. Magnetism in the embedded Fe nanoparticle samples was investigated using a SQUID magnetometer. Atomic moment of Fe in the Fe/Al nanocomposite films increases slightly when the proportion of Fe nanoparticles is increased, but is still significantly lower than the bulk Fe value. In a more detailed analysis, the magnetisation curves were fitted using the Random Anisotropy Model (RAM) which allowed the exchange field and random anisotropy field to be evaluated. The findings and issues raised by this approach are discussed.

Model-data fusion for spatial and statistical characterization of soil parameters from geophysical measurements

A recently developed PDE-constrained stochastic inverse analysis algorithm for spatial and statistical characterization of soil parameters from geophysical measurements, considering uncertainty due to limited measurements and sensor noise, is exemplified and validated. A 60m × 60 m geotechnical site in Garner Valley, CA is used as the validation testbed. Advanced geophysical test measurements – in terms of velocity waveforms at a few locations on the surface due to surface excitations using a mobile shaker – are available for the site. The algorithm inversely analyzes the available measurements to probabilistically estimate the elastic parameters of the soil at the site up to a depth of 40 m. The algorithm relies on (1) hypothesizing the soil parameters to be heterogeneous, anisotropic random fields, (2) making prior assumptions on them, (3) numerically simulating the geophysical experiment using the finite element method in conjunction with a stochastic collocation approach, and (4) fusing simulated measurements with experimental measurements using a minimum variance framework to update the prior assumptions on the soil parameter random fields. The estimated elastic parameters of the soil are presented in terms of marginal mean and marginal standard deviation profiles of the soil's P- and S-wave velocities as well as their correlation structures in the x-, y-, and z-direction. In ascertaining the accuracy of the inverse analysis algorithm, the geophysical experiment is numerically re-simulated with the estimated P- and S-wave velocity profiles and the model predicted velocity waveforms are compared against the field observations at all the measurement locations. Comments are made at appropriate places regarding several aspects of the algorithm in highlighting the lessons learned through this validation effort towards accurate stochastic full waveform inversion of geophysical measurements.

An Improved Anisotropic Markov Random Field Approach for Prestack Seismic Inversion

This letter presents an improved anisotropic Markov random field (IAMRF) approach for prestack seismic inversion. Instead of using multiple potential functions or adding extra weights on clique potentials as done in existing AMRF approaches, IAMRF removes the effects of anisotropic gradients in subsurface models by means of scaling parameters, which directly tune the model gradients right above. In particular, the scaling parameters are not only directionally varied at certain iterations but also updated by a statistical method at every iteration throughout the inversion, thereby generating more accurate and better edge-preserving inverted results. The prestack seismic inversion based on the IAMRF prior constraints is demonstrated on a field data example, which presents encouraging results.

Continuity in law of some additive functionals of bifractional Brownian motion

ABSTRACTLet be a bifractional Brownain motion with indices and . We prove the continuity in law, in some anisotropic Besov spaces, with respect to H and K. Our result generalizes those obtained by Jolis and Viles [Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theor. Probab. 20(2) (2007), pp. 133–152] of the fractional Brownian motion local time and gives a new result for the generalized fractional derivatives with kernel depending on slowly varying function of the local time of . Notice that their result was generalized by Wu and Xiao [Continuity in the Hurst index of the local times of anisotropic gaussian random fields, Stoch. Proc. Their Appl. 119 (2009), pp. 1823–1844] for wide class of anisotropic gaussian random fields satisfying some condition (A) which is not satisfied by . To prove our result, we use the decomposition in law of given by Lei and Nualart [A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett. 79 (2009), pp. 619–624]. Our result is also new in the space of continuous functions.

Reservoir Characterization Through Target-Oriented AVA-Petrophysical Inversions with Spatial Constraints

We apply three methods that use different regularization strategies to insert spatial constraints into the seismic-petrophysical inversion. The first method is what we call the structurally constrained inversion (SCI), which directly uses the structural information brought by the seismic stack image to insert geological (structural) constraints into the inversion. The second method is based on anisotropic Markov random field (AMRF) and uses the Huber energy function to reasonably model the spatial heterogeneity of petrophysical reservoir properties. Finally, the last method includes both geostatistical information (describing the lateral variability of reservoir properties) and hard data (i.e. well log data) constraints into the inversion kernel (GHDC inversion). For computationally feasibility, we apply a target-oriented inversion that uses the amplitude versus angle (AVA) responses extracted along the top reservoir reflections to infer the petrophysical properties of interest (i.e. porosity, water saturation and shaliness) for the target layer. In particular, an empirical, linear rock-physics model, properly calibrated for the investigated area, is used to rewrite the P-wave reflectivity equation as a function of the petrophysical contrasts instead of the elastic constants. This reformulation allows for a direct and a simultaneous estimation of petrophysical properties from AVA data. The implemented approaches are tested both on synthetic and field seismic data and compared against the standard method in which each AVA response is inverted independently (laterally unconstrained Bayesian inversion; LUBI). In the case of poor signal-to-noise ratio it turns out that the three considered spatially constrained methods achieve better delineations of reservoir bodies and provide more reliable results than the standard LUBI approach. More in detail, the AMRF recovers sharper geological boundaries than the SCI and GHDC algorithms. The SCI algorithms is more sensitive to data noise, whereas the key advantage of the GHDC consists in analytically providing posterior uncertainties for the model parameters. Finally, for what concerns the computational cost the GHDC and the SCI methods result the most and the least computationally demanding, respectively.

Stability Analysis of a Large Gold Mine Open-Pit Slope Using Advanced Probabilistic Method

A large gold reserve was recently discovered at Haveri district of Karnataka state of India where open-pit mining was planned to extract these deposits. Stability analysis for open-pit mine slope at this site is presented in the article. Extensive geological investigations and laboratory testing suggested high variability in geological features of discontinuities, rock mass quality and intact rock properties. Hence, it was decided to perform stability analysis of the rock slope using probabilistic approach along with deterministic approach. Deterministic analysis was carried out with average properties of rock, and reliability analysis of the rock slope was carried out using both traditional and advanced probabilistic methods. In traditional probabilistic method, rock mass strength properties were treated as random variables without considering spatial variation of rock properties and reliability index was evaluated by Monte Carlo (MC) simulation on augmented radial basis function-based response surface. In advanced probabilistic analysis, spatial variability of rock mass strength properties was considered by generating anisotropic random field using Fourier series method with spatial averaging over finite difference zones. Reliability index was then estimated by performing MC simulation using random finite difference method. A comparison was provided between the results of stability analysis of slope from all these approaches. Rock slope was found to be stable in both deterministic and probabilistic approaches; however, the degree of stability predicted was different for both methods. Deterministic approach was found to be inappropriate to analyse the stability of slope having rock mass with variable properties. Further, reliability index and expected performance level of slope were highly underestimated by traditional probabilistic method as compared to advanced probabilistic method.

An efficient PDE-constrained stochastic inverse algorithm for probabilistic geotechnical site characterization using geophysical measurements

This paper develops an efficient, PDE-constrained stochastic inverse analysis methodology to probabilistically estimate site-specific elastic parameters of soil from sparse geophysical test measurements by accounting for the uncertain spatial variability of soil deposits and any measurement uncertainty associated with the geophysical experiment. Hypothesizing the soil parameters at any site to be three-dimensional, heterogeneous, anisotropic random fields, the methodology first probabilistically simulates the geophysical experiment using the finite element method in conjunction with a stochastic collocation approach to compute statistical measures of a quantity of interest such as the soil displacement or acceleration throughout the soil domain. To this end, the random fields are discretized into finite number of random variables by utilizing a Gaussian mixture model that allows for mimicking the soil formation process. The parameters of the random fields are initially assumed based on the generic data available in the literature for the geological soil type. The stochastic collocation approach utilizes a recently developed non-product quadrature method, conjugate unscented transformation, to accurately estimate the statistical moments corresponding to the model response variables in a computationally efficient manner. The methodology, then, employs a minimum variance framework to fuse the finite element model output with sparse real measurements to update the initially assumed soil statistical parameters. The methodology is illustrated through numerical geophysical experiments at a fictitious geotechnical site and is verified with three very different true profiles of the soil modulus. Moreover, a probabilistic sensitivity analysis is carried out by varying the number and locations of sensors. It is observed that by judiciously selecting the sensor locations, following a set of information maps, obtained by exploiting the equations of the minimum variance scheme, more information may be extracted from any geophysical experiments, leading to less uncertain estimates of the soil parameters.

Exchange bias mechanism in FM/FM/AF spin valve systems in the presence of random unidirectional anisotropy field at the AF interface: The role played by the interface roughness due to randomness

We propose an atomistic model and present Monte Carlo simulation results regarding the influence of FM/AF interface structure on the hysteresis mechanism and exchange bias behavior for a spin valve type FM/FM/AF magnetic junction. We simulate perfectly flat and roughened interface structures both with uncompensated interfacial AF moments. In order to simulate rough interface effect, we introduce the concept of random exchange anisotropy field induced at the interface, and acting on the interface AF spins. Our results yield that different types of the random field distributions of anisotropy field may lead to different behavior of exchange bias.

Análisis de la historia geológica del ángulo de fricción en suelos transportados y su importancia en la capacidad portante de cimentaciones superficiales

This work shows the variability of the friction angle based on the geological history, over the bearing capacity of shallow foundations. This study was developed through the generation of random fields of friction angle using the matrix decomposition technique for a self-correlated lognormal distribution. In addition, to recreate the geological history, the anisotropic random fields were created using the rotation matrix with angles of 0, 45 and 90 degrees. Subsequently, these random fields were implemented in a finite element model of a continuous footing. A linear elastoplastic constitutive model was selected to represent stress-strain soil behavior together with the Mohr-Coulomb failure criteria. This last work was done in order to understand the influence of geological history and the variability of the friction angle in the bearing capacity of shallow foundations. Results include the weight of the variability of the friction angle in the bearing capacity and the influence of the dip over the friction angle and the bearing capacity.

Magnetic excitations in ferromagnetic glasses

We have studied theoretically (including computer simulations) dynamical magnetic properties of aggregates of ferromagnetic amorphous nanogranules in the presence of direct exchange between the neighboring granules and random anisotropy fields. It was shown that in such structures there still exist delocalized magnetic excitations similar to magnons in the bulk materials, although at much lower frequencies. The excitations at high frequencies are localized within the corresponding granules. The details of spectrum of these excitations (including structural factors) are discussed.

Influence of spatial variability of soil Young's modulus on tunnel convergence in soft soils

Field data have shown the fact that soil spatial variability could aggravate the uncertainty of tunnel convergence ΔD (a key indicator for serviceability and safety of tunnels). This paper presents a detailed numerical analysis to investigate the probabilistic response of tunnel convergence in spatially varied soft soils. The soil Young's modulus Es is highlighted and modeled with isotropic and horizontally stratified anisotropic random fields, respectively. The influence of scale of fluctuation (SOF) δ of the Es on convergence ΔD is discussed in detail with respect to different directions, i.e., the vertical and horizontal directions both for δ and ΔD. It is observed that ignoring the spatial variability of Es, i.e., disregarding the possibility of unfavorable soft soil (low stiffness soil) locally around tunnel, can underestimate the mean value of ΔD. The horizontally stratified anisotropic random field is more appropriate than isotropic random field in the sense of an accurate prediction, especially when extreme tunnel convergence occurs. In horizontally stratified anisotropic random fields, the influence of horizontal and vertical SOF is different on tunnel convergence. The surrounding soils near tunnel crown and invert or across tunnel horizontal diameter are very critical to the tunnel convergence. In addition, the effect of horizontal SOF δx on failure probability of the calculated ΔD exceeding the specified allowable ΔDlim is limited when the δx is larger than 4.84 times of tunnel outer diameter.

Clustering of high values in random fields

The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with \(\mathbb {Z}^{2}\), and that they satisfy stationarity and isotropy conditions. Here we extend the existing theory, concerning the asymptotic behavior of the maximum and the extremal index, to non-stationary and anisotropic random fields, defined over discrete subsets of \(\mathbb {R}^{2}\). We show that, under a suitable coordinatewise mixing condition, the maximum may be regarded as the maximum of an approximately independent sequence of submaxima, although there may be high local dependence leading to clustering of high values. Under restrictions on the local path behavior of high values, criteria are given for the existence and value of the spatial extremal index which plays a key role in determining the cluster sizes and quantifying the strength of dependence between exceedances of high levels. The general theory is applied to the class of max-stable random fields, for which the extremal index is obtained as a function of well-known tail dependence measures found in the literature, leading to a simple estimation method for this parameter. The results are illustrated with non-stationary Gaussian and 1-dependent random fields. For the latter, a simulation and estimation study is performed.

On features of magnetization self-organization in 1D stochastic ferromagnetic systems

The magnetic structure of a polycrystalline nanowire at the weak or missing magnetostatic interaction exhibits the special self-organization of magnetization. As is known, the magnetization structure forming in a random crystallographic anisotropy field has a characteristic length range, which involves tens and hundreds of crystallites. This leads to the occurrence of stochastic domains. The induced uniform anisotropy of magnetostatic nature or the texture co-directed with the crystallite anisotropy axes masks the picture of stochastic domains. Nevertheless, as we show, the information on stochastic domains remains in the magnetization structure. The experimental techniques for obtaining information on the magnetic properties of stochastic domains are proposed.

Influence of spatially random soil on lateral thrust and failure surface in earth retaining walls

ABSTRACTThe present study investigates the influence of spatial variability of soil properties on the lateral thrust and failure surface of a 6 m high frictionless rigid earth retaining wall in active condition. The drained friction angle of the soil is modelled as a log-normal anisotropic random field in two dimensions, using the Cholesky decomposition technique. The effect of horizontal and vertical scale of fluctuation is observed on the lateral thrust. The failure surface in the backfill is found to be more or less similar for different combinations of spatial variability in vertical and horizontal directions. Monte-Carlo simulation technique is used to compute the probability of failure and to obtain the worst-case spatial variability configuration.

Effects of the random single-ion anisotropy and random magnetic field in the spin-3/2 Blume–Capel model

We have studied the effects of the random single-ion anisotropy and random magnetic field in the phase diagram and in the thermodynamic properties of the spin-3/2 Blume–Capel model via Curie–Weiss mean-field approximation. The phase diagrams were built in the planes temperature versus single-ion anisotropy, temperature versus magnetic field, temperature versus random parameters and the dependencies of magnetization were plotted versus temperature and single-ion anisotropy. These diagrams show that, in the space (D/J−T/J), the type (first- or second-order) of the phase transition between the ferromagnetic and paramagnetic phases is dependent on the random parameters. Therefore, within these conditions the model presents tricritical behavior. For large values, and a certain critical value of the random parameters, the phase transition is only of second-order, but it is of first-order within the ordered phase, between the phase with m=1/2 and m=3/2, which ends in a terminal critical point.

Linear theory of random textures of 3He-A in an aerogel

Spacial variation of the orbital part of the order parameter of $^3$He-A in aerogel is represented as a random walk of the unit vector $\mathbf{l}$ in a field of random anisotropy produced by the strands of aerogel. For a range of distances, where variation of $\mathbf{l}$ is small in comparison with its absolute value correlation function of directions of $\mathbf{l}(\mathbf{r})$ is expressed in terms of the correlation function of the random anisotropy field. With simplifying assumptions about this correlation function a spatial dependence of the average variation $\langle\delta\mathbf{l}^2\rangle$ is found analytically for isotropic and axially anisotropic aerogels. Average projections of $\mathbf{l}$ on the axes of anisotropy are expressed in terms of characteristic parameters of the problem. Within the "model of random cylinders" numerical estimations of characteristic length for disruption of the long-range order and of the critical anisotropy for restoration of this order are made and compared with other estimations .

New constraints on modelling the random magnetic field of the MW

We extend the description of the isotropic and anisotropic random component of the small-scale magnetic field within the existing magnetic field model of the Milky Way from Jansson & Farrar, by including random realizations of the small-scale component. Using a magnetic-field power spectrum with Gaussian random fields, the NE2001 model for the thermal electrons and the Galactic cosmic-ray electron distribution from the current GALPROP model we derive full-sky maps for the total and polarized synchrotron intensity as well as the Faraday rotation-measure distribution. While previous work assumed that small-scale fluctuations average out along the line-of-sight or which only computed ensemble averages of random fields, we show that these fluctuations need to be carefully taken into account. Comparing with observational data we obtain not only good agreement with 408 MHz total and WMAP7 22 GHz polarized intensity emission maps, but also an improved agreement with Galactic foreground rotation-measure maps and power spectra, whose amplitude and shape strongly depend on the parameters of the random field. We demonstrate that a correlation length of 0≈22 pc (05 pc being a 5σ lower limit) is needed to match the slope of the observed power spectrum of Galactic foreground rotation-measure maps. Using multiple realizations allows us also to infer errors on individual observables. We find that previously-used amplitudes for random and anisotropic random magnetic field components need to be rescaled by factors of ≈0.3 and 0.6 to account for the new small-scale contributions. Our model predicts a rotation measure of −2.8±7.1 rad/m2 and 04.4±11. rad/m2 for the north and south Galactic poles respectively, in good agreement with observations. Applying our model to deflections of ultra-high-energy cosmic rays we infer a mean deflection of ≈3.5±1.1 degree for 60 EeV protons arriving from CenA.

Theory and generation of conditional, scalable sub‐Gaussian random fields

AbstractMany earth and environmental (as well as a host of other) variables, Y, and their spatial (or temporal) increments, ΔY, exhibit non‐Gaussian statistical scaling. Previously we were able to capture key aspects of such non‐Gaussian scaling by treating Y and/or ΔY as sub‐Gaussian random fields (or processes). This however left unaddressed the empirical finding that whereas sample frequency distributions of Y tend to display relatively mild non‐Gaussian peaks and tails, those of ΔY often reveal peaks that grow sharper and tails that become heavier with decreasing separation distance or lag. Recently we proposed a generalized sub‐Gaussian model (GSG) which resolves this apparent inconsistency between the statistical scaling behaviors of observed variables and their increments. We presented an algorithm to generate unconditional random realizations of statistically isotropic or anisotropic GSG functions and illustrated it in two dimensions. Most importantly, we demonstrated the feasibility of estimating all parameters of a GSG model underlying a single realization of Y by analyzing jointly spatial moments of Y data and corresponding increments, ΔY. Here, we extend our GSG model to account for noisy measurements of Y at a discrete set of points in space (or time), present an algorithm to generate conditional realizations of corresponding isotropic or anisotropic random fields, introduce two approximate versions of this algorithm to reduce CPU time, and explore them on one and two‐dimensional synthetic test cases.

Hitting probabilities and the Hausdorff dimension of the inverse images of a class of anisotropic random fields

Let X = {X(t): t ∈ RN} be an anisotropic random field with values in Rd. Under certain conditions on X, we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel–Riesz capacity. We also obtain the Hausdorff dimension of its inverse image, and the Hausdorff and packing dimensions of its level sets. These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.