Regularized collocation for spherical harmonics gravitational field modeling
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Regularized collocation for spherical harmonics gravitational field modeling Valeriya Naumova · Sergei V. Pereverzyev · Pavlo Tkachenko
Received: 5 September 2013 / Accepted: 30 October 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract Motivated by the problem of satellite gravity gradiometry, which is the reconstruction of the Earth gravity potential from the satellite data provided in the form of the second-order partial derivatives of the gravity potential at a satellite altitude, we discuss a special regularization technique for solving this severely ill-posed problem in a spherical framework. We are especially interested in the regularized collocation method. As a core ingredient we present an a posteriori parameter choice rule, namely the weighted discrepancy principle, and prove its order optimality. Finally, we illustrate our theoretical findings by numerical results for the computation of the Fourier coefficients of the gravitational potential directly from the noisy synthetic data. Keywords Ill-posed problem · Collocation method · Regularization · Discrepancy principle · Satellite gravity modeling · Spherical harmonics Mathematics Subject Classification (2000)
65J20 · 47A52 · 86A30
1 Introduction Satellite missions, Gravity recovery and climate experiment (GRACE) [see, e.g., Tapley et al. (2005)] and Gravity field and steady-state Ocean Circulation Explorer (GOCE) [see, e.g., Rebhan et al. (2000)] launched in 2005 and 2009 respectively, are dedicated to measuring the Earth’s gravity field and modeling the geoid that allow us
Dedicated to Willi Freeden’s 65th Birthday. V. Naumova · S. V. Pereverzyev · P. Tkachenko (B) Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria e-mail: [email protected]
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Int J Geomath
to increase our knowledge and reveal many fascinating things in studying dynamic processes in the Earth’s interior, ocean circulation, etc. After collecting data from a satellite orbit the following problem naturally arises: “How to transform the satellite data into parameters of the gravitational field model?” At this point it is worth to mention that, on the one hand, in the existing models, such as Earth Gravity Model (EGM2008) (Pavlis et al. 2008), for example, the gravitational potential is parametrized by the Fourier coefficients with respect to the spherical harmonics up to some degree M. On the other hand, the satellite data collected during a mission such as GOCE are given as the values of the second-order partial derivatives of the gravitational potential calculated at the satellite orbit. Of particular interest, from the mathematical point of view, is the use of the second-order radial derivatives, which indeed can be found from the above mentioned values. In the spherical framework, using the second-order radial derivatives on the orbital sphere Ωρ , one can relate the satellite data and the parameters of the gravity model by means of the so-called gravity gradiometry equation wi
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