## Least-squares Geopotential Approximation by Windowed Fourier and Wavelet Transform

• Two possible substitutes of the Fourier transform in geopotential determination are windowed Fourier transform (WFT) and wavelet transform (WT). In this paper we introduce harmonic WFT and WT and show how it can be used to give information about the geopotential simultaneously in the space domain and the frequency (angular momentum) domain. The counterparts of the inverse Fourier transform are derived, which allow us to reconstruct the geopotential from its WFT and WT, respectively. Moreover, we derive a necessary and sufficient condition that an otherwise arbitrary function of space and frequency has to satisfy to be the WFT or WT of a potential. Finally, least - squares approximation and minimum norm (i.e. least - energy) representation, which will play a particular role in geodetic applications of both WFT and WT, are discussed in more detail.

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Author: Willi Freeden, Volker Michel urn:nbn:de:hbz:386-kluedo-8075 Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (216) Preprint English 1999 1999 Technische Universität Kaiserslautern 1999/09/08 arbitrary function ; geopotential determination ; harmonic WFT ; inverse Fourier transform ; squares; wavelet transform ; windowed Fourier transform Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

$Rev: 13581$