Fast Spherical Collocation



F.Sansò, DIIAR - Sezione Rilevamento, Politecnico di Milano, Piazza Leonardo da Vinci 31, I-20133 Milano, Italy. Ph: 00390223996504, Fax: 00390223996530, e-mail: f.sanso@ipmtf4.topo.polimi.it

C.C.Tscherning, Department of Geophysics,University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen Oe, Denmark, Ph: 004535320582, Fax: 004535365357, e-mail: cct@gfy.ku.dk.



Abstract:



It has long been known that a spherical harmonic analysis of gridded (and noisy) data on a sphere (with uniform error for a fixed latitude) gives rise to simple systems of equations. This has for the method of least-squares collocation (using an isotropic covariance function or reproducing kernel) been generalized. The data only needs to be at the same altitude and of the same kind for each latitude. This permits for example the combination of gravity data at the surface of the Earth and data at satellite altitude.



Suppose that data are associated with the points of a grid with N values in latitude and M values in longitude. The latitudes do not need to be spaced uniformly. Also suppose we want to determine the spherical harmonic coefficients to a maximal degree and order K. Then the method will require that we solve K systems of equations each having a symmetric positive definite matrix of size N * N, only.



Results of simulation studies using the method are described.