Satellite gravity gradiometry as tensorial inverse problem
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Satellite gravity gradiometry as tensorial inverse problem Willi Freeden · Helga Nutz
Received: 12 August 2011 / Accepted: 31 August 2011 / Published online: 22 September 2011 © Springer-Verlag 2011
Abstract In the context of inverse problems of mathematical geodesy, the calculation of the gravitational potential at the Earth’s surface from its Hesse tensor at satellite’s height turns out to be exponentially ill-posed. In fact, it requires specific tensorial procedures for its solution. This paper proposes a wavelet-based regularization method to overcome the calamities of the ill-posedness, thereby providing a “zooming-in” technique of modeling the gravitational potential from global to local scale. As a particularly remarkable ingredient the paper offers a new procedure of multiscale regularization by use of locally adapted regularization parameters. Keywords
Satellite gravity gradiometry · Multiscale Runge regularization
Mathematics Subject Classification (2000)
66F22 · 65R32 · 86A22
1 Introduction The launch of the satellite Gravity field and steady-state Ocean Circulation Explorer (GOCE) by the European Space Agency (ESA) on March 17th 2009 was the initial point for realizing the concept of Satellite Gravity Gradiometry (SGG) under realistic conditions. The principle of SGG can be characterized as follows: The satellite carries a set of ultra-modern high-sensitive accelerometers which measure the components of the gravity field along all three axes (see Fig. 1) and ensure a coverage of the entire Earth with gravity measurements (apart from gaps at the polar regions), however, at orbital W. Freeden · H. Nutz (B) Geomathematics Group, University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany e-mail: [email protected]; [email protected] W. Freeden e-mail: [email protected]
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Fig. 1 The concept of gradiometry: proof masses in the orbit providing the second derivatives εi (x+x)−ε k (x−x) = 2x ∂2 ∂ xi ∂ xk F(x), i, k ∈ {1, 2, 3} (cf.
Freeden and Schreiner 2010)
altitude. The test masses located in the low orbiting satellite are subjected to local changes of the gravity field. This effect leads to different gravity forces (cf. Groten 1979, 1980; Heiskanen and Moritz 1967) and, therefore, different accelerations are detectable. The accelerometers measure the relative accelerations between two test masses and thus provide information about the second order partial derivatives of the gravitational potential at satellite’s height (cf. Fig. 1). Since the distances in the gradiometer itself are small, these differences may numerically be identified with differentials and, thus, all second order partial derivatives of the gravitational potential, i.e., the full Hesse matrix (in an ideal case), are available. In order to realize the GOCE mission, ESA and its partners (see, e.g., Laur and Liebig 2010; Rummel 2010) had to overcome an impressive technical challenge by designing a Low Earth Orbiter (LEO) at an altitude of about 265 km to
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