Analytic Properties and Covariance Functions for a New Class of Generalized Gibbs Random Fields

Abstract

Spartan Spatial Random Fields (SSRFs) are generalized Gibbs random fields, equipped with a coarse-graining kernel that acts as a low-pass filter for the fluctuations. SSRFs are defined by means of physically motivated spatial interactions and a small set of free parameters (interaction couplings). This paper focuses on the FGC-SSRF model, which is defined on the Euclidean space R

On the correlation functions associated with polynomials of the diffusion operator

“…The obtained CFs generalize earlier results of Hristopulos and Elogne (2007;hereinafter HE07) and Mirouze and Weaver (2010), and may facilitate practical design of the cost functions in variational data assimilation problems, as they give explicit relationships between the shape of the CFs and the structure of the corresponding iBEC operators in the analytic form.…”

An application of Spartan spatial random fields in environmental mapping: focus on automatic mapping capabilities

“…The SSRFs provide a new class of generalized covariance functions, which are by construction positive definite for an explicitly specified range of parameter values (Hristopulos 2003;Hristopulos and Elogne 2006a). To date only one class of SSRFs derived from a specific interaction scheme has been investigated: FGC-SSRFs have an energy functional that involves the squares of the fluctuations, the gradient and the curvature of the field.…”

“…In Hristopulos and Elogne (2006a), it is proved that the isotropic SSRF model specified by Eq. (4) provides a class of covariance functions that incorporate both nondifferentiable and differentiable covariance models.…”

On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation

“…Hristopulos (2003), Hristopulos and Elogne (2007) and Yaremchuk and Smith (2011) have studied extensively the special case M = 1 and P = 2 on R d for which the parameter settings κ 1 = ρ L 2 and κ 2 = L 4 with ρ < 0 and satisfying ρ 2 < 4 yield a family of positive-definite, oscillatory functions such as those illustrated in Figure 3 on S 2 . With all of these approaches, however, the advantages of increasing the flexibility in the correlation model have to be carefully measured against the increase in computational cost that results from the need to solve additional or more complicated large linear systems, and the difficulty of having to estimate additional parameters.…”