Statistics – Computation
Scientific paper
Dec 2005
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2005agufmin23a1198s&link_type=abstract
American Geophysical Union, Fall Meeting 2005, abstract #IN23A-1198
Statistics
Computation
1214 Geopotential Theory And Determination (0903), 1241 Satellite Geodesy: Technical Issues (6994, 7969), 3255 Spectral Analysis (3205, 3280), 3265 Stochastic Processes (3235, 4468, 4475, 7857)
Scientific paper
Spherical Slepian functions provide a natural solution to the geodetic problem of the polar gap arising from incomplete satellite coverage of planetary gravitational or magnetic fields. The orthogonal Slepian basis consists of bandlimited functions whose energy is optimally concentrated within a closed domain on the unit sphere. In geodesy the region in question is a circularly symmetric polar cap, a pair of antipodal caps, or their complement, a latitudinal belt. The spherical Slepian functions solve an algebraic eigenvalue problem in the spectral domain. However, constructing the spatiospectral localization kernel and diagonalizing it is computationally cumbersome and often numerically unstable. Here we show that the spatiospectral localization operator bandlimiting a field on the unit sphere and projecting it to a single or double polar cap commutes with a Sturm-Liouville operator. Its eigenfunctions can be computed extremely accurately and efficiently by diagonalizing a tridiagonal matrix with analytically prescribed elements. The gains in ease, speed, and accuracy thus achieved will make the usage of Slepian functions in earth and planetary geodesy practical, as we show by example. We derive analytical formulas that characterize the estimation error of the determination of stochastic potential fields from noisy and incomplete observations at an altitude above their source. We do this in the spherical harmonic basis, as well as in the Slepian basis, and present a truncated Slepian basis method that alleviates the drawbacks of alternative methods. The problems we solve are not limited to geodesy and observations made from a satellite. In geomagnetism, our observation level may be the Earth's surface, and the source level at or near the core-mantle boundary. In cosmology, the unit sphere constituting the sky is observed from the inside out, and the galactic plane masking the measurement is akin to a latitudinal belt.
Dahlen F. A.
Simons Frederik J.
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