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The modeling of real-valued random fields indexed by spherical coordinates arises in different disciplines of the natural sciences, especially in environmental, atmospheric and earth sciences. However, there is currently a lack of parametric models allowing a flexible representation of the spatial correlation structure of multivariate data located on a spherical surface. To bridge this gap, we provide analytical expressions of twenty-two parametric families of isotropic p-variate covariance kernels on the d-dimensional sphere, defined for any integers p > 0 and d > 1 , together with their respective spectral representations (Schoenberg matrices) and sufficient validity conditions on the covariance parameters. These families include multiquadric, sine power, exponential, Bessel and hypergeometric kernels, and provide covariances exhibiting varied shapes, short-scale and large-scale behaviors. Our construction relies on the so-called multivariate parametric adaptation approach, where a matrix-valued covariance kernel is defined on the basis of a scalar covariance to which matrix-valued parameters are applied, coupled with matrix manipulations. [ABSTRACT FROM AUTHOR]

The conditional simulation of Gaussian random vectors is widely used in geostatistical applications to quantify uncertainty in regionalized phenomena that have been observed at finitely many sampling locations. Two iterative algorithms are presented to deal with such a simulation. The first one is a variation of the propagative version of the Gibbs sampler aimed at simulating the random vector without any conditioning data. The novelty of the presented algorithm stems from the introduction of a relaxation parameter that, if adequately chosen, allows quickening the rates of convergence and mixing of the sampler. The second algorithm is meant to convert the non-conditional simulation into a conditional one, based on the successive over-relaxation method. Again, a relaxation parameter allows quickening the convergence in distribution to the desired conditional random vector. Both algorithms are applicable in a very general setting and avoid the pivoting, inversion, square rooting or decomposition of the variance-covariance matrix of the vector to be simulated, thus reduce the computation costs and memory requirements with respect to other discrete geostatistical simulation approaches. [ABSTRACT FROM AUTHOR]

Intrinsic random fields of order k, defined as random fields whose high-order increments (generalized increments of order k) are second-order stationary, are used in spatial statistics to model regionalized variables exhibiting spatial trends, a feature that is common in earth and environmental sciences applications. A continuous spectral algorithm is proposed to simulate such random fields in a d-dimensional Euclidean space, with given generalized covariance structure and with Gaussian generalized increments of order k. The only condition needed to run the algorithm is to know the spectral measure associated with the generalized covariance function (case of a scalar random field) or with the matrix of generalized direct and cross-covariances (case of a vector random field). The algorithm is applied to synthetic examples to simulate intrinsic random fields with power generalized direct and cross-covariances, as well as an intrinsic random field with power and spline generalized direct covariances and Matérn generalized cross-covariance. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for simulating Gaussian random fields with zero mean and non-stationary covariance functions. The simulated field is obtained as a weighted sum of cosine waves with random frequencies and random phases, with weights that depend on the location-specific spectral density associated with the target non-stationary covariance. The applicability and accuracy of the algorithm are illustrated through synthetic examples, in which scalar and vector random fields with non-stationary Gaussian, exponential, Matérn or compactly-supported covariance models are simulated. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of simulating multivariate random fields with stationary Gaussian increments in a d-dimensional Euclidean space. To this end, one considers a spectral turning-bands algorithm, in which the simulated field is a mixture of basic random fields made of weighted cosine waves associated with random frequencies and random phases. The weights depend on the spectral density of the direct and cross variogram matrices of the desired random field for the specified frequencies. The algorithm is applied to synthetic examples corresponding to different spatial correlation models. The properties of these models and of the algorithm are discussed, highlighting its computational efficiency, accuracy and versatility. [ABSTRACT FROM AUTHOR]

We propose a spectral turning-bands approach for the simulation of second-order stationary vector Gaussian random fields. The approach improves existing spectral methods through coupling with importance sampling techniques. A notable insight is that one can simulate any vector random field whose direct and cross-covariance functions are continuous and absolutely integrable, provided that one knows the analytical expression of their spectral densities, without the need for these spectral densities to have a bounded support. The simulation algorithm is computationally faster than circulant-embedding techniques, lends itself to parallel computing and has a low memory storage requirement. Numerical examples with varied spatial correlation structures are presented to demonstrate the accuracy and versatility of the proposal. [ABSTRACT FROM AUTHOR]