A parameter choice strategy for the inversion of multiple observations
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A parameter choice strategy for the inversion of multiple observations Christian Gerhards1 · Sergiy Pereverzyev Jr.2 · Pavlo Tkachenko3
Received: 7 July 2015 / Accepted: 14 July 2016 © Springer Science+Business Media New York 2016
Abstract In many geoscientific applications, multiple noisy observations of different origin need to be combined to improve the reconstruction of a common underlying quantity. This naturally leads to multi-parameter models for which adequate strategies are required to choose a set of ‘good’ parameters. In this study, we present a fairly general method for choosing such a set of parameters, provided that discrete direct, but maybe noisy, measurements of the underlying quantity are included in the observation data, and the inner product of the reconstruction space can be accurately estimated by the inner product of the discretization space. Then the proposed parameter choice method gives an accuracy that only by an absolute constant multiplier differs from the noise level and the accuracy of the best approximant in the reconstruction and in the discretization spaces. Keywords Parameter choice · Multiple observations · Spherical approximation Communicated by: Jan Hesthaven. Pavlo Tkachenko
[email protected] Christian Gerhards [email protected] Sergiy Pereverzyev Jr. [email protected] 1
Computational Science Center, University of Vienna, Oskar Morgenstern Platz 1, 1090 Vienna, Austria
2
Department of Mathematics, University of Innsbruck, Technikerstraβe 13, 6020 Innsbruck, Austria
3
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria
C. Gerhards et al.
1 Introduction Satellite missions like CHAMP, GRACE, GOCE, or Swarm (e.g., [4, 6, 10, 12]) provide highly accurate data of the Earth’s gravity and magnetic field, e.g., by giving information on the first- or second-order radial derivative of the gravitational potential or measurements of the vectorial geomagnetic field, which, once certain ionoand magnetospheric contributions have been filtered out, can be expressed as the gradient of a harmonic potential. Drawing conclusions from such satellite measurements on the gravitational potential or the magnetic field at or near the Earth’s surface is a classical exponentially ill-posed problem (see, e.g., [9, 18, 21]). Measurements at or near the Earth’s surface (which we simply denote as ground measurements), on the other hand, do not suffer from this ill-posedness but are typically only available in restricted regions (e.g., aeromagnetic surveying [22]). Combining both data sets becomes necessary when aiming at local high resolution models that also take global trends into account. This is a classical setting for multiparameter modeling (e.g., [3, 16–18]) that involves the regularization of an ill-posed inverse problem (downward continuation of satellite data) and the weighting of the satellite data against the ground data. An exemplary situation that we also use for
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