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Journal of Geodesy

, Volume 81, Issue 1, pp 39–51 | Cite as

Local Solutions to Inverse Problems in Geodesy

The Impact of the Noise Covariance Structure upon the Accuracy of Estimation
  • Frank Bauer
  • Peter Mathé
  • Sergei Pereverzev
Original Article

Abstract

In many geoscientific applications, one needs to recover the quantities of interest from indirect observations blurred by colored noise. Such quantities often correspond to the values of bounded linear functionals acting on the solution of some observation equation. For example, various quantities are derived from harmonic coefficients of the Earth’s gravity potential. Each such coefficient is the value of the corresponding linear functional. The goal of this paper is to discuss new means to use information about the noise covariance structure, which allows order-optimal estimation of the functionals of interest and does not involve a covariance operator directly in the estimation process. It is done on the basis of a balancing principle for the choice of the regularization parameter, which is new in geoscientific applications. A number of tests demonstrate its applicability. In particular, we could find appropriate regularization parameters by knowing a small part of the gravitational field on the Earth’s surface with high precision and reconstructing the rest globally by downward continuation from satellite data.

Keywords

Regularization by local data Ill-posed inverse problems Gaussian random noise Satellite gravity gradiometry 

Mathematics Subject Classification (2000)

47A52 65J22 49J55 93E14 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institute for Mathematical Stochastics, Department of MathematicsUniversity of GöttingenGöttingenGermany
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  3. 3.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria

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