Construction of anisotropic covariance functions using Riesz-representers |
| |
| Authors: | C C Tscherning |
| |
| Institution: | Department of Geophysics, Juliane Maries Vej 30, DK-2100 Copenhagen ?, Denmark e-mail: cct@gfy.ku.dk; Tel.: +45 35 32 05 82; Fax: +45 35 36 53 57, DK
|
| |
| Abstract: | A reproducing-kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R
0, having a reproducing kernel K
0(P,Q) is considered (P, Q, and later P
n
being points in the set of harmonicity). The degree variances of this kernel will be denoted σ0
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 two-dimensional surface surrounding the bounding sphere, will be linearly independent. These functions are used to
define a new N-dimensional RKHS with kernel (a
n
>0)
If the points all are located on a concentric sphere with radius R
1>R
0, and form an ε-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
Consequently, if K
N
(P,Q) is required to represent an isotropic covariance function of the Earth's gravity potential, COV(P,Q), σ0
n
can be selected so that σ
n
becomes equal to the empirical degree variances.
If the points are chosen at varying radial distances R
n
>R
0, 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
Values of anisotropic covariance functions constructed based on these ideas are calculated, and some initial ideas are presented
on how to select the points P
n
.
Received: 24 September 1998 / Accepted: 10 March 1999 |
| |
| Keywords: | , Reproducing-kernel Hilbert Space,Gravity,Anisotropic,Non-homogeneous |
| 本文献已被 SpringerLink 等数据库收录！ |
|
