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

• C. C. Tscherning
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 122)

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 σon.

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 σn $$\mathop \sigma \nolimits_n^2 = (2n + 1)*\sigma _{0n}^2 *\left( {\frac{{R_0 }} {{R_1 }}} \right)\left( {2n + 2} \right)*(cons\tan t{\text{)}}$$

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 σon so that σn 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 .

Keywords

Covariance Function Distinct Point Evaluation Functional Reproduce Kernel Hilbert Space Concentric Sphere
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

• C. C. Tscherning
• 1
1. 1.Department of GeophysicsCopenhagen OeDenmark

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