Construction of An-isotropic Covariance-Functions Using Sums of Riesz-Representers

Abstract

We regard a reproducing kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R0, having a reproducing kernel K ₀ (P,Q). (P, Q, and later P _n being points in the set of harmonicity). The degree-variances of this kernel will be denoted σo_n.

The set of Riesz-representers associated with the evaluation functionals (or gravity functionals) related to distinct points P _n ,n = 1,..., N, on a 2D-surface surrounding the bounding sphere will be linear independent. These functions are used to define a new N-dimensional RKHS with. kernel {fy1|90-1}

If the points all are located on a concentric sphere with radius R ¹ > Po, and form an є-net covering the sphere, and a _n are suitable area-elements (depending on N) then this kernel will converge towards an isotropic kernel with degree-variances σ_n $\mathop \sigma \nolimits_n^2 = (2n + 1)*\sigma _{0n}^2 *\left( {\frac{{R_0 }} {{R_1 }}} \right)\left( {2n + 2} \right)*(cons\tan t{\text{)}}$

Consequently, if we want K _N (P,Q) to represent an isotropic covariance function of the Earth’s gravity potential, COV(P,Q), we can select σo_n so that σ_n becomes equal to the empirical degree-variances.

If the points are chosen at varying radial distances R _n > R _o then we have constructed an anisotropic kernel, or equivalent covariance function representation.

If the points are located in a bounded region, the kernel may be used to modify the original kernel, CON _N (P,Q)=CON(P,Q)+K _N (P,Q)

Values of an-isotropic covariance functions constructed based on these ideas have been calculated, and some first ideas are presented on how to select the points P _n .