Isotropic Random Field Research Articles

On the properties of multivariate isotropic Random fields on the Ball

We consider multivariate isotropic random fields on the ball Bd. We first study their regularity properties in terms of Sobolev spaces. We further derive conditions guaranteeing the Hölder continuity of their covariance kernels and we prove the existence of sample Hölder continuous modifications for Gaussian random fields. Furthermore, we measure the error of truncated approximations of the corresponding series' representations. Moreover our developments are supported by numerical experiments. The majority of our results are new for multivariate random fields indexed over other domains, too. We express some of them for the case of the sphere.

Isotropic random tangential vector fields on spheres

In this paper we characterized isotropic random tangential vector fields on d-spheres for d≥1 by the cross-covariance, and derived their Karhunen–Loève expansion. The tangential vector field can be decomposed into a curl-free part and a divergence-free part by the Helmholtz–Hodge decomposition. We proved that the two parts can be correlated on a 2-sphere, while they must be uncorrelated on a d-sphere for d≥3. On a 3-sphere, the divergence-free part can be further decomposed into two isotropic flows.

Optical momentum distributions in monochromatic, isotropic random vector fields

Abstract We investigate the decomposition of the electromagnetic Poynting momentum density in three-dimensional random monochromatic fields into orbital and spin parts, using analytical and numerical methods. In sharp contrast with the paraxial case, the orbital and spin momenta in isotropic random fields are found to be identically distributed in magnitude, increasing the discrepancy between the Poynting and orbital pictures of energy flow. Spatial correlation functions reveal differences in the generic organization of the optical momenta in complex natural light fields, with the orbital current typically forming broad channels of unidirectional flow, and the spin current manifesting larger vorticity and changing direction over subwavelength distances. These results are extended to random fields with pure helicity, in relation to the inclusion of electric-magnetic democracy in the definition of optical momenta.

Stochastic analysis for bearing capacity computation of twin strip footings on geogrid-reinforced sands

ABSTRACT The interaction of closely-spaced footings on soils is of concern for recent decades. The inherent variability of soil makes the topic more challenging. This study investigates the behaviour of twin strip foundations on both unreinforced and geogrid-reinforced spatially random sands. The Random Finite Difference is performed by coupling Matlab and FLAC2D in each Monte Carlo simulation. Two types of sands, dense and loose, are assumed as the spatially correlated log-normal random fields and the friction angle is considered a random parameter. This study discusses how much the bearing capacity of twin footings is impressed by the heterogeneity of random sands. The unreinforced soil results show that when the uncertainty of φ is high, the homogeneous soil assumption could overestimate the interference effect by about 18% and 9% in the dense and loose sands, respectively. While soil reinforcement reduces the difference between results obtained from probabilistic analysis with those calculated with deterministic analysis. Moreover, the isotropic random fields with rapid fluctuation of φ yield the greatest interference influence in the unreinforced and reinforced dense sands. The combined impact of interference and reinforcement is greater in loose sand than in dense sand, regardless of whether the soil is heterogeneous or homogeneous. HIGHLIGHT Random heterogeneous variability of φ on the interaction of the twin foundations on both unreinforced and geogrid-reinforced sand was investigated. For twin footings on unreinforced dense sand, Scr/B changes from 1.25 to somewhere between 1.25 and 1.5 by increasing COV φ , while on the unreinforced loose sand, Scr/B = 1 remains constant, irrespective of COV φ values. For high variability of φ, the assumption of soil without variability results in overestimation of twin footings interference effect by about 18% and 9%, which means the risk acceptance yields irrevocable damages. Increasing the number of geogrid layers, N, declines the risk of estimating the analysis with homogeneous soil, especially for high variability of φ. By all means, the footing interference and geogrid effects become lower when N is added. The interference effect is usually higher for twin footings on unreinforced sand than on the same reinforced sand. Overall, the integrated influence of interference and reinforcement in loose sand is more than that in dense sand The combined influence of interference and the first geogrid layer is the most in the isotropic random field (θ x = θ y = 1 m) for both sand since the rapid variation of φ in both directions increases the influence and reinforcement effects. The deterministic analysis underestimates the interaction of twin foundations on the reinforced loose sand compared to the anisotropic random fields.

On fractional spherically restricted hyperbolic diffusion random field

The paper investigates solutions of the fractional hyperbolic diffusion equation in its most general form with two fractional derivatives of distinct orders. The solutions are given as spatial–temporal homogeneous and isotropic random fields and their spherical restrictions are studied. The spectral representations of these fields are derived and the associated angular spectrum is analysed. The obtained mathematical results are illustrated by numerical examples. In addition, the numerical investigations assess the dependence of the covariance structure and other properties of these fields on the orders of fractional derivatives.

Group sparse optimization for inpainting of random fields on the sphere

Abstract We propose a group sparse optimization model for inpainting of a square-integrable isotropic random field on the unit sphere, where the field is represented by spherical harmonics with random complex coefficients. In the proposed optimization model, the variable is an infinite-dimensional complex vector and the objective function is a real-valued function defined by a hybrid of the $\ell _2$ norm and non-Lipschitz $\ell _p (0<p<1)$ norm that preserves rotational invariance property and group structure of the random complex coefficients. We show that the infinite-dimensional optimization problem is equivalent to a convexly-constrained finite-dimensional optimization problem. Moreover, we propose a smoothing penalty algorithm to solve the finite-dimensional problem via unconstrained optimization problems. We provide an approximation error bound of the inpainted random field defined by a scaled Karush–Kuhn–Tucker (KKT) point of the constrained optimization problem in the square-integrable space on the sphere with probability measure. Finally, we conduct numerical experiments on band-limited random fields on the sphere and images from Cosmic Microwave Background (CMB) data to show the promising performance of the smoothing penalty algorithm for inpainting of random fields on the sphere.

Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields

AbstractThe Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, whose k-point correlation functions (kth-order cumulants) have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-squared random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.

Pitfalls of Climate Network Construction—A Statistical Perspective

Abstract Network-based analyses of dynamical systems have become increasingly popular in climate science. Here, we address network construction from a statistical perspective and highlight the often-ignored fact that the calculated correlation values are only empirical estimates. To measure spurious behavior as deviation from a ground truth network, we simulate time-dependent isotropic random fields on the sphere and apply common network-construction techniques. We find several ways in which the uncertainty stemming from the estimation procedure has a major impact on network characteristics. When the data have a locally coherent correlation structure, spurious link bundle teleconnections and spurious high-degree clusters have to be expected. Anisotropic estimation variance can also induce severe biases into empirical networks. We validate our findings with ERA5 data. Moreover, we explain why commonly applied resampling procedures are inappropriate for significance evaluation and propose a statistically more meaningful ensemble construction framework. By communicating which difficulties arise in estimation from scarce data and by presenting which design decisions increase robustness, we hope to contribute to more reliable climate network construction in the future. Significance Statement Network-based approaches have gained renewed attention regarding the prediction of climate phenomena such as El Niño events, extreme regional precipitation patterns, anomalous polar vortex dynamics, and regarding understanding the Earth system. Even though climate networks are constructed from a limited amount of noisy data, they typically are not studied from a statistical perspective. However, such an approach is crucial: due to sampling uncertainty, climate networks unavoidably contain false and missing edges. We analyze how sampling artifacts impact the conclusions drawn from the networks and present both pitfalls and statistically robust procedures of network construction and evaluation. We aim to contribute to understanding the limitations and fully leveraging the potentials of network methods in climate and Earth system science.

Series Expansions of Fractional Brownian Motions and Strong Local Nondeterminism of Bifractional Brownian Motions on Balls and Spheres

This paper provides series expansions for fractional Brownian motions on the unit ball and the unit sphere by means of ultraspherical polynomials and spherical harmonics. It establishes the property of strong local nondeterminism of isotropic Gaussian random fields on the unit sphere and that of fractional and bifractional Brownian motions on the unit ball and the unit sphere.

Inverse Random Potential Scattering for Elastic Waves

This paper is concerned with the inverse elastic scattering problem for a random potential in three dimensions. Interpreted as a distribution, the potential is assumed to be a microlocally isotropic Gaussian random field whose covariance operator is a classical pseudodifferential operator. Given the potential, the direct scattering problem is shown to be well-posed in the sense of distributions by studying the equivalent Lippmann–Schwinger integral equation. For the inverse scattering problem, we demonstrate that the microlocal strength of the random potential can be uniquely determined with probability one by a single realization of the high frequency limit of the averaged compressional or shear backscattered far-field pattern of the scattered wave. The analysis employs the integral operator theory, the Born approximation in the high frequency regime, the microlocal analysis for the Fourier integral operators, and the ergodicity of the wave field.

Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces

This paper is concerned with sample path properties of real-valued isotropic Gaussian fields on compact two-point homogeneous spaces. In particular, we establish the property of strong local nondeterminism of an isotropic Gaussian field and then exploit this result to establish an exact uniform modulus of continuity for its sample paths.

Tensor- and spinor-valued random fields with applications to continuum physics and cosmology

In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes the physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp., rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp., its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic. For the convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.

THE STATISTICAL SIMULATION OF RANDOM FIELDS WITH THE GAUSSIAN TYPE CORRELATION FUNCTION BY THE INVESTIGATION OF THE MAGNETOMETRY DATA

In the article, universal methods of statistical modeling (Monte Carlo methods) of geophysical data using the Gaussian correlation function have been developed, which make it possible to solve the problems of generating adequate realizations of random fields on a grid in three-dimensional space of required regularity and detail. Since in geophysics, most of the results of object research are presented in digital form, the accuracy of which depends on various random influences, the problem of the condition of the maps arises in the case when the data cannot be obtained with the specified detail in some observation areas. It is proposed to apply statistical simulation of random fields methods, to solve the problems of conditional maps, supplement the required detail of research results with additional data, to achieve the required accuracy of observations, and other similar problems in geophysics. An algorithm for numerical modeling of realizations of homogeneous isotropic random fields in three-dimensional space with a Gaussian correlation function is formulated on the basis of the theorem on estimation of the mean-square approximation of such random fields by the partial sum of the "spectral decomposition" series. Using the example of data from aeromagnetic surveying in the area of the Ovruch depression, the proposed algorithm for statistical modeling of random fields is implemented in solving the problems of map fitness by supplementing the data with simulated adequate implementations to the required level of detail. When analyzing data by profiles, they are divided into deterministic (trend) and random components. The trend is proposed to approximate by cubic splines and the homogeneous isotropic random component is proposed to modeling on the basis "spectral decomposition" of random fields on 3-D space in the Ovruch depression. According to the algorithm, authors received random component implementations on the study area with twice detail for each profile. When checking their adequacy, authors made the conclusions that the relevant random components histogram has Gaussian distribution. The built variogram of these implementations has the best approximation by theoretical variogram which is connected to the Gaussian type correlation function. As a result of superimposing the simulated array of the random component on the spline approximation of the real data, a more detailed implementation was obtained for the data of geomagnetic observations in the selected area. A comparative analysis of the results of modeling realizations random fields with the Gaussian correlation function with other correlation functions is carried out. Therefore, the method of statistical modeling of realizations of random fields in three-dimensional space with the Gaussian correlation function makes it possible to supplement the results of measurements of the full magnetic field intensity vector with data with a given detail as much as possible.

A mesostructure-informed cohesion-based numerical method for fracture behavior of slate with foliation structure

The foliation structure and spatial variability are important factors affecting the fracture behavior of slate. According to the statistical analysis of mesostructural information, random slate specimens with the foliation structure are physically reconstructed by coupling a one-dimensional random field with a two-dimensional isotropic random field. A cohesion-based numerical method incorporating the foliation structure is developed to simulate the fracture behavior of slate-like materials. The elastic and fracture parameters are evaluated by direct tensile tests, and the numerical model is validated to be self-consistency. It has been confirmed by numerical simulations and tests that the elastic modulus and tensile fracture increase with the foliation angle, and there is a critical foliation angle from the tensile fracture to the shear fracture for the slate under uniaxial tension. Affected by the spatial variability of the foliation structure, slate fractures in the dense and thick area of the foliation plane, and the mechanical properties follow lognormal distribution.

Isotropic random spin weighted functions on 𝑆² vs isotropic random fields on 𝑆³

We show that an isotropic random field on S U ( 2 ) SU(2) is not necessarily isotropic as a random field on S 3 S^3 , although the two spaces can be identified. The ambiguity is due to the fact that the notion of isotropy on a group and on a sphere are different, the latter being much stronger. We show that any isotropic random field on S 3 S^3 is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree d d is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range { − d 2 , … , d 2 } \bigl \{-\frac {d}{2},\dots ,\frac {d}{2}\bigr \} , each of which is isotropic in the sense of S U ( 2 ) SU(2) . Moreover, for a random field of fixed degree, each spin weight appears with the same magnitude, in a sense to be specified. In addition we will give an overview of the theory of spin weighted functions and Wigner D D -matrices, with the purpose of gathering together many different points of view and adding ours. As a byproduct of this survey we will prove some new properties of the Wigner matrices and a formula relating the operators ð ð ¯ \eth \overline {\eth } and the horizontal Laplacian of the Hopf fibration S 3 → S 2 S^3\to S^2 , in the sense of [Bérard Bergery and Bourguignon, Illinois J. Math. 26 (1982), no. 2, 181–200.]

On multifractionality of spherical random fields with cosmological applications

This paper investigates spatial data on the unit sphere. Traditionally, isotropic Gaussian random fields are considered as the underlying mathematical model of the cosmic microwave background (CMB) data. We discuss the generalized multifractional Brownian motion and its pointwise Hölder exponent on the sphere. The multifractional approach is used to investigate the CMB data from the Planck mission. These data consist of CMB radiation measurements at narrow angles of the sky sphere. The results obtained suggest that the estimated Hölder exponents for different CMB regions do change from location to location. Therefore, the CMB temperature intensities are multifractional. The methodology developed is used to suggest two approaches for detecting regions with anomalies in the cleaned CMB maps. doi:10.1017/S1446181122000104

Polyadic random fields

The paper considers mean-square continuous, wide-sense homogeneous, and isotropic random fields taking values in a linear space of polyadics. We find a set of such fields whose values are symmetric and positive-definite dyadics, and outline a strategy for their simulation.

Multilevel representations of isotropic Gaussian random fields on the sphere

Abstract Series expansions of isotropic Gaussian random fields on $\mathbb {S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations with multilevel localized structure provide an alternative to the standard Karhunen–Loève expansions of isotropic random fields in terms of spherical harmonics. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field. In addition, numerical illustrations are given and an application to random elliptic PDEs on the sphere is analysed.

An Inverse Random Source Problem for the Biharmonic Wave Equation

This paper is concerned with an inverse source problem for the stochastic biharmonic wave equation. The driven source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The well-posedness of the direct problem is examined in the distribution sense, and the regularity of the solution is discussed for the given rough source. For the inverse problem, the strength of the random source, involved in the principal symbol of its covariance operator, is shown to be uniquely determined by a single realization of the magnitude of the wave field averaged over the frequency band with probability one. Numerical experiments are presented to illustrate the validity and effectiveness of the proposed method for the case that the random source is white noise.

Seismic responses of rectangular tunnels in liquefiable soil considering spatial variability of soil properties

The objective of this paper was to investigate the dynamic response of a tunnel-soil system in the soil with spatial variability of shear modulus with various coefficients of variation and scales of fluctuation. The soil shear modulus was highlighted and modeled as an isotropic random field using the spectral representation method, then mapped into the finite difference model. The dynamic stochastic responses of the excess pore water pressure ratio, the liquefied zone, the ground displacement, and the uplift displacement of the tunnel were discussed. It was observed that the effect of coefficient of variation was pronounced in which an increase of the coefficient of variation brought out a reduction in the corresponding dynamic stochastic responses. However, the scale of fluctuation was found to be insignificant in the tunnel soil system.