2001 | OriginalPaper | Buchkapitel
Construction of An-isotropic Covariance-Functions Using Sums of Riesz-Representers
verfasst von : C. C. Tscherning
Erschienen in: IV Hotine-Marussi Symposium on Mathematical Geodesy
Verlag: Springer Berlin Heidelberg
Enthalten in: Professional Book Archive
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We regard 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). (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 Pn,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 R1 > 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, CONN(P,Q)=CON(P,Q)+KN(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 .