Construction of anisotropic covariance functions using Riesz-representers: Fid-Bau Portal

Construction of anisotropic covariance functions using Riesz-representers

Tscherning, C. C.

Abstract. A reproducing-kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R0, having a reproducing kernel K0(P,Q) is considered (P, Q, and later Pn being points in the set of harmonicity). The degree variances of this kernel will be denoted $ σ_{0} $n. The set of Riesz representers associated with the evaluation functionals (or gravity functionals) related to distinct points Pn,n = 1,…,N, on a two-dimensional surface surrounding the bounding sphere, will be linearly independent. These functions are used to define a new N-dimensional RKHS with kernel (an>0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} If the points all are located on a concentric sphere with radius R1>R0, and form an ε-net covering the sphere, and an are suitable area elements (depending on N), then this kernel will converge towards an isotropic kernel with degree variances\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} Consequently, if KN(P,Q) is required to represent an isotropic covariance function of the Earth's gravity potential, COV(P,Q), $ σ_{0} $n can be selected so that $ σ_{n} $ becomes equal to the empirical degree variances. If the points are chosen at varying radial distances Rn>R0, then an anisotropic kernel, or equivalent covariance function representation, can be constructed. If the points are located in a bounded region, the kernel may be used to modify the original kernel\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} Values of anisotropic covariance functions constructed based on these ideas are calculated, and some initial ideas are presented on how to select the points Pn.