Journal of Geodesy

, Volume 81, Issue 1, pp 53–66 | Cite as

Integration of the Monte Carlo Covariance Estimation Strategy into Tailored Solution Procedures for Large-Scale Least Squares Problems

  • H. Alkhatib
  • W. -D. SchuhEmail author
Original Article


Large-scale least squares problems require tailored numerical techniques to overcome the computational burden. For these types of problems, iterative strategies are suitable because of their flexibility and effectiveness. The only shortcoming of iterative strategies in least squares estimation is that the inverse of the normal equation matrix as the carrier of the covariance information is either unavailable or very expensive to compute. This paper presents algorithms based on Monte Carlo integration, which can be incorporated very efficiently into iterative solvers and which are demonstrated to close the aforementioned gap. Tailored strategies for different types of solution techniques with respect to normal equations, observation equations, and combined models are treated. Finally, the paper presents new criteria to define confidence regions for the estimated covariance matrix of the parameters, as well as for all additional derived quantities. In a case study these techniques are applied to simulated GOCE data, where satellite gravity gradiometry and satellite-to-satellite tracking information are combined for reconstructing the gravity field. The problem of deriving the covariance matrix of gravity fields with high spatial resolution by combined iterative estimation processes, unsolved until now, is treated.


Covariance estimation Variance propagation Monte Carlo integration Inverse problems GOCE gravity field processing 


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institute of Theoretical Geodesy (ITG)University of BonnBonnGermany

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