Different radial basis functions and their applicability for regional gravity field representation on the sphere
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Different radial basis functions and their applicability for regional gravity field representation on the sphere Katrin Bentel · Michael Schmidt · Christian Gerlach
Received: 17 September 2012 / Accepted: 12 December 2012 / Published online: 3 February 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract Global gravity field solutions are commonly modelled in spherical harmonic basis functions. Additionally, radial basis functions on the sphere with quasi-local support are used to model regional refinements of gravity fields. However, these functions are usually not orthogonal on a sphere, which makes the modelling process more complex. In this paper we study and compare different radial basis functions and their performance in regional gravity field modelling on the sphere by making use of simulated data. In addition to the type of radial basis function also the size of the study area on the sphere, the point grid, the margins and the method which is used to solve the singular system have to be taken into account. The synthetic signal, which we use in our simulation, is a residual signal in a bandwidth which corresponds to the bandwidth of GOCE satellite gravity observations. Keywords Radial basis function · Regional gravity field modelling · Singular problem Mathematics Subject Classification
86A30 · 86A22
K. Bentel (B) Department of Mathematical Sciences and Technology, IMT, Norwegian University of Life Sciences, Postboks 5003, 1432 Ås, Norway e-mail: [email protected] M. Schmidt Deutsches Geodätisches Forschungsinstitut, Munich, Germany C. Gerlach Bavarian Academy of Sciences and Humanities, Commission of Geodesy and Glaciology, Munich, Germany
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Int J Geomath (2013) 4:67–96
1 Introduction Global Earth gravity models are commonly represented in spherical harmonic basis functions. These global models can be regionally refined by additional satellite, airborne, or terrestrial gravity measurements. Spherical harmonic basis functions, which are globally oscillating functions, provide no spatial localization and therefore regional refinements of the gravity field on the sphere cannot be represented in an optimal way. If the signal varies in only one point on the sphere, in such a representation, the whole global solution has to change, since they are globally-optimized best-fit solutions. Small spatial details are difficult to represent and can even be masked in the solutions. In contrast to spherical harmonics, spherical radial basis functions provide quasicompact support. Their influence decreases rapidly with distance from their center. Thus, they are very appropriate to model regional detailed refinements to global gravity models. The synthetic example signal we use to test different modelling scenarios in spherical radial basis functions is such a regional detailed gravity signal, from spherical harmonic degrees 150–250, which corresponds to the sensitive bandwidth of gravity measurements taken by the GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) satellite gravity mission,
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