Advertisement

How to handle colored noise in large least-squares problems in the presence of data gaps?

  • R. Klees
  • P. Ditmar
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 127)

Abstract

An approach to handle stationary colored noise in least-squares problems in the presence of data gaps is presented. The presence of colored noise implies that a non-diagonal covariance matrix has to be inverted. The problem is reduced to the solution of systems of linear equations with the covariance matrix as equation matrix. Each system is solved using a pre-conditioned conjugate gradient method. An ARMA representation of the colored noise is used both to design an efficient pre-conditioner and to compute the product of the covariance matrix with a vector. This results in an algorithm that has a numerical complexity of O(N) operations, where N is the number of observations. This makes the algorithm particularly suited for large least-squares problems. It is shown that the approach works perfectly in the sense that no edge effects show up and no observations have to be thrown away.

Keywords

Colored noise ARMA filters data gaps satellite gravity gradiometry (SGG) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bottoni GP, Barzaghi R (1993) Fast collocation. Bulletin Géodesique 67: 119–126.CrossRefGoogle Scholar
  2. Brockwell PJ, Davis RA (1991) Time series: theory and methods. Second Edition, Springer Series in Statistics, New York.CrossRefGoogle Scholar
  3. Colombo OL (1979) Optimal estimation from data regularly sampled on a sphere with applications in geodesy. Reports of the Department of Geodetic Science, Report No. 291, The Ohio State University, Columbus, Ohio, USA.Google Scholar
  4. Eren K (1980) Spectral analysis of GEOS-3 altimeter data and frequency domain collocation. Reports of the Department of Geodetic Science, Report No. 297, The Ohio State University, Columbus, Ohio, USA.Google Scholar
  5. Eren K (1982) Toeplitz matrices and frequency domain collocation. Manuscripta Geodaetica 7: 85–118.Google Scholar
  6. ESA (1999) Gravity field and steady-state ocean circulation mission. Reports for Mission Selection - The Four Candidate Earth Explorer Core Missions, ESA SP-1233 ( 1 ), Noordwijk, The Netherlands.Google Scholar
  7. Ditmar P, Klees R (2002) A method to compute the earth’s gravity field from SGG/SST data to be acquired by the GOCE satellite. Delft University Press (DUP) Science, Delft, The Netherlands, 64 pages.Google Scholar
  8. Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards 49: 409–436.CrossRefGoogle Scholar
  9. Klees R, Ditmar P, Broersen P (2003) How to handle colored observations noise in large least-squares problems? Journal of Geodesy 76: 629–640.CrossRefGoogle Scholar
  10. Moritz H (1980) Geodetic reference system 1980. Bulletin Géodesique 54: 395–405.Google Scholar
  11. Rapp R, Wang Y, Pavlis N (1991) The Ohio State 1991 geopotential and sea surface topography harmonic coefficient models. Rep 410, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, USA.Google Scholar
  12. Schuh WD (1996) Tailored numerical solution strategies for the global determination of the earth’s gravity field. Mitteilungen der geodätischen Institute der Technischen Universität Graz, Folge 81, Graz, Austria.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • R. Klees
    • 1
  • P. Ditmar
    • 1
  1. 1.Department of Geodesy, Physical, Geometric and Space GeodesyDelft University of TechnologyDelftThe Netherlands

Personalised recommendations