Covariance Models for Divergence-Free and Curl-Free Random Vector Fields

Thousands of permanent GNSS stations sample nowadays the 3D deformation of the Earth’s surface. The temporal covariance structure of the field of GNSS station displacements is well characterized and modelled. On the other hand, there lacks a general agreed-upon model of its spatial covariance structure, in part because the theory of random vector fields on the sphere remains hardly developed.In this contribution, we show how the well-established theory of random isotropic scalar fields on the sphere generalizes to the case of vector fields. We derive in particular a spectral representation of random isotropic vector fields on the sphere in the domain of vector spherical harmonics, from which several properties of their covariance functions follow. We then present several parametric families of covariance functions which could be used to describe and model the covariance structure of vector fields on the sphere, such as GNSS station displacements.Although this presentation focuses on theoretical aspects, it is given with future practical applications in mind. A realistic spatio-temporal covariance model of GNSS station displacements could indeed benefit different problems such as the estimation of long-term GNSS station velocities, the identification and mitigation of offsets in GNSS station position time series, the filtering of spatial “common modes” to isolate local deformation, or the spatial interpolation of GNSS station displacements into global maps. Applications may also be found in other domains involving vector quantities distributed on a sphere, e.g., winds, ocean currents, magnetic anomalies, etc.

This paper is concerned with spherically invariant or elliptically contoured vector random fields in space and/or time, which are formulated as scale mixtures of vector Gaussian random fields. While a spherically invariant vector random field may or may not have second-order moments, a spherically invariant second-order vector random field is determined by its mean and covariance matrix functions, just like the Gaussian one. This paper explores basic properties of spherically invariant second-order vector random fields, and proposes an efficient approach to develop covariance matrix functions for such vector random fields.

ABSTRACTA simple but efficient approach is proposed in this paper to construct the isotropic random field in (d ⩾ 2), whose univariate marginal distributions may be taken as any infinitely divisible distribution with finite variance. The three building blocks in our building tool box are a second-order Lévy process on the real line, a d-variate random vector uniformly distributed on the unit sphere, and a positive random variable that generates a Pólya-type function. The approach extends readily to the multivariate case and results in a vector random field in with isotropic direct covariance functions and with any specified infinitely divisible marginal distributions. A characterization of the turning bands simulation feature is also derived for the covariance matrix function of a Gaussian or elliptically contoured random field that is isotropic and mean square continuous in .

This article introduces the hyperbolic vector random field whose finite-dimensional distributions are the generalized hyperbolic one, which is formulated as a scale mixture of Gaussian random fields and is thus an elliptically contoured (or spherically invariant) random field. Such a vector random field may or may not have second-order moments, while a second-order one is characterized by its mean function and its covariance matrix function, just as in a Gaussian case. Some covariance matrix structures of hyperbolic type are constructed in this paper for second-order hyperbolic vector random fields.

Strong disorder effects of a Dirac fermion with a random vector field

A study is made of interband absorption of light in a smooth random field that varies slowly in space. When the exciton energy in the random field is much less than the exciton binding energy, the influence of the random field on the discrete spectrum of the exciton is expressed through the appearance of an exponential “tail” to every exciton line. But if the exciton energy in the random field is much greater than the exciton binding energy, the exciton effects are unimportant because the random field breaks up the excitons.

Many processes in astrophysical plasmas, such as magnetic reconnection and current sheet formation, are sensitive to the topology of the magnetic field. Magnetic nulls are special in a topological sense because they are connected to an infinite number of field lines. Complicated magnetic field geometries may have many nulls and many null–null lines. This work examines the statistical properties of zeros in a random divergence-free vector field model. An analytic expression for the density of nulls is obtained as a function of the dissipation and integral length scales of the random field and the exponent of the energy spectrum of the field. For a broad class of spectra which are similar to the spectra of turbulent cascades magnetic nulls form self-similar clusters. The fractal dimension of the set of nulls is computed as a function of the power law index of the random field. Complicated magnetic topologies may possess tangled webs of nulls and null–null lines. These webs of nulls may be the sites of current sheet formation and heating, and they may modify the properties of particle transport in turbulent plasma media.

: LONG TERM GOALS. To formulate constitutive laws and solution methods for ice fields viewed as heterogeneous multiscale materials from the standpoint of micromechanics of random media. To develop a stochastic-computational model for mechanics of pack ice treated as a spatially random granular material of Mohr-Coulomb type. APPROACH. A spatially random granular Mohr-Coulomb material is obtained from the classical, homogeneous continuum model by taking the internal friction angle rho and the resistance in tension H as a vector random field. These two constitutive coefficients specify a meso-continuum approximation on a meso-scale corresponding to the actual choice of spacing of a finite difference net of characteristics (Fig. 1) employed in the solution of a given boundary value problem.

ABSTRACTA stochastic representation is derived for a vector random field that is stationary, isotropic, and mean square continuous on a sphere or unit circle. The established stochastic representation is an infinite series involving the sequence of ultraspherical or Gegenbauer's polynomials, looks like a mimic of the series representation of the covariance matrix function of the isotropic vector random field, but differs from the spectral representation in terms of the ordinary spherical harmonics. It is also shown in this paper that some isotropic and continuous covariance matrix functions on the real line or , if they are compactly supported, can be adopted as covariance matrix functions on the unit circle or .

This presentation overviews sound propagation and scattering in turbulent media and pertinent acoustic remote sensing of turbulence. Although sound propagation through a turbulent atmosphere is mainly considered, the presented results apply to other media such as oceanic turbulence. Formulations are provided for various statistical characteristics of acoustic signals such as the scattering cross section, the variances of the phase and log-amplitude fluctuations, the spatial, temporal, and cross-frequency coherences, and the probability density functions. Monte Carlo simulation of sound propagation in a random medium is explained. Specifics of acoustic signals in turbulent media as compared to electromagnetic waves are highlighted. For example, sound waves are scattered by both the sound speed and medium velocity fluctuations, which are scalar and vector random fields, respectively, with different statistical properties. Also, fluctuations in the acoustic refractive index are much stronger than those in electromagnetic propagation, and sound can be scattered by humidity (or salinity) fluctuations. Finally, the Markov approximation, which is widely used for electromagnetic waves, might not be applicable for some statistical characteristics of acoustic signals. It is explained how these specifics of sound propagation are addressed in atmospheric acoustics.

The set IM of Neher’s classes of tripotents in an arbitrary JB*-triple Z is considered and a natural complex-analytic Banach manifold structure is defined on it. The relationship between IM and the Grassmann manifold of all complemented principal inner ideals in Z is studied in detail and the smooth complete vector fields on IM are characterized as smooth complete equivariant vector fields on the manifold M of tripotents of Z.

Assessing the impact of uncertainties in ice-sheet models is a major and challenging issue that needs to be faced by the ice-sheet community to provide more robust and reliable model-based projections of ice-sheet mass balance. In recent years, uncertainty quantification (UQ) has been increasingly used to characterize and explore uncertainty in ice-sheet models and improve the robustness of their projections. A typical UQ analysis involves first the (probabilistic) characterization of the sources of uncertainty followed by the propagation and sensitivity analysis of these sources of uncertainty. Previous studies concerned with UQ in ice-sheet models have generally focused on the last two steps but paid relatively little attention to the preliminary and critical step of the characterization of uncertainty. Sources of uncertainty in ice-sheet models, like uncertainties in ice-sheet geometry or surface mass balance, typically vary in space and potentially in time. For that reason, they are more adequately described as spatio(-temporal) random fields, which account naturally for spatial (and temporal) correlation. As a means of improving the characterization of the sources of uncertainties in ice-sheet models, we propose in this paper to represent them as Gaussian random fields with Matérn covariance function. The class of Matérn covariance functions provides a flexible model able to capture statistical dependence between locations with different degrees of spatial correlation or smoothness properties. Samples from a Gaussian random field with Matérn covariance function can be generated efficiently by solving a certain stochastic partial differential equation. Discretization of this stochastic partial differential equation by the finite element method results in a sparse approximation known as a Gaussian Markov random field. We solve this equation efficiently using the finite element method within the Ice-sheet and Sea-level System Model (ISSM). In addition, spatio-temporal samples can be generated by combining an autoregressive temporal model and the Matérn field. The implementation is tested on a set of synthetic experiments to verify that it captures well the desired spatial and temporal correlations. Finally, we demonstrate the interest of this sampling capability in an illustration concerned with assessing the impact of various sources of uncertainties on the Pine Island Glacier, West Antarctica. We find that both larger spatial and temporal correlations lengths will likely result in increased uncertainty in the projections.

Gaussian random fields (GRF) are a fundamental stochastic model for spatiotemporal data analysis. An essential ingredient of GRF is the covariance function that characterizes the joint Gaussian distribution of the field. Commonly used covariance functions give rise to fully dense and unstructured covariance matrices, for which required calculations are notoriously expensive to carry out for large data. In this work, we propose a construction of covariance functions that result in matrices with a hierarchical structure. Empowered by matrix algorithms that scale linearly with the matrix dimension, the hierarchical structure is proved to be efficient for a variety of random field computations, including sampling, kriging, and likelihood evaluation. Specifically, with n scattered sites, sampling and likelihood evaluation has an O(n) cost and kriging has an cost after preprocessing, particularly favorable for the kriging of an extremely large number of sites (e.g., predicting on more sites than observed). We demonstrate comprehensive numerical experiments to show the use of the constructed covariance functions and their appealing computation time. Numerical examples on a laptop include simulated data of size up to one million, as well as a climate data product with over two million observations.

I will present three projects that are related to the modeling of covariance structures on the Euclidean space and on sphere. Firstly, we propose a method to model isotropic random field on sphere, where a tapered Matern covariance function is used to capture the local behavior while a nonparametric expansion controls the behavior at large distances. A model selection procedure based on residual sum of squares with penalization is used to reduce over fitting. Secondly, we address the issue of modeling axially symmetric spatial random fields on sphere with a kernel convolution approach. The observed random field is generated by convolving a latent uncorrelated random field with a class of Matern type kernel functions. By allowing the parameters in the kernel functions to vary with locations, we are able to generate a flexible class of covariance functions and capture the nonstationary properties. We use pre-computation tables to speed up the computation. For regular grid data on sphere, the block circulant property of the covariance matrix enables us to use Fast Fourier Transform (FFT) to get the determinant and the inverse efficiently. Thirdly, we proposed a semiparametric variogram estimating method through its spectral representation to model the intrinsically stationary random on R. The low frequency part of the spectral density is estimated by solving a regularized inverse problem through quadratic programming. The behavior at high frequencies, however, is modeled via a parametric tail in the form of a power decaying function. The power parameter in the tail is estimated by a log likelihood method. All proposed methodologies are supplemented with simulation studies and real data analyses.

In studies involving environmental risk assessment, Gaussian random field generators are often used to yield realizations of a Gaussian random field, and then realizations of the non-Gaussian target random field are obtained by an inverse-normal transformation. Such simulation process requires a set of observed data for estimation of the empirical cumulative distribution function (ECDF) and covariance function of the random field under investigation. However, if realizations of a non-Gaussian random field with specific probability density and covariance function are needed, such observed-data-based simulation process will not work when no observed data are available. In this paper we present details of a gamma random field simulation approach which does not require a set of observed data. A key element of the approach lies on the theoretical relationship between the covariance functions of a gamma random field and its corresponding standard normal random field. Through a set of devised simulation scenarios, the proposed technique is shown to be capable of generating realizations of the given gamma random fields.