Stepwise Solutions to Random Field Prediction Problems

  • M. Reguzzoni
  • N. Tselfes
  • G. Venuti
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 132)


The approximation of functionals of second order random fields from observations is a method widely used in gravity field modelling. This procedure is known as collocation or Wiener – Kolmogorov technique. A drawback of this theory is the need to invert matrices (or solve systems) as large as the number of observations. In order to overcome this difficulty, it is common practice to decimate the data or to average them or (as in the case of this study) to produce more manageable gridded values. Gridding the data has sometimes the great advantage of stabilizing the solution.

Once such a procedure is envisaged, several questions arise: how much information is lost in this operation? Is rigorous covariance propagation necessary in order to obtain consistent estimates? How do different approximation methods compare? Such questions find a clear formulation in the paper: some of them (the simplest ones) are answered from the theoretical point of view, while others are investigated numerically.

From this study it results that no information is lost if the intermediate grid has the same dimension of the original data, or if the functionals to be predicted can be expressed as linear combinations of the gridded data. In the case of a band-limited signal, these linear relations can be exploited to obtain the final estimates from the gridded values without a second step of collocation. A similar result can be obtained even in the case of non-band limited signal, due to the low pass filtering of the signal, along with the noise, performed by the intermediate gridding.


Wiener-Kolmogorov principle local gridding collocation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Colombo, O. L. (1981). Numerical Methods for Harmonic Analysis on the Sphere. Report No. 310, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio.Google Scholar
  2. Driscoll, J. R., and D. M. Healy (1994). Computing Fourier transforms and convolutions on the 2-sphere. Advances in applied mathematics, 15, pp. 202–250.CrossRefGoogle Scholar
  3. ESA (1999). Gravity Field and Steady-State Ocean Circulation Mission. ESA SP-1233 (1). ESA Publication Division, c/o ESTEC, Noordwijk, The Netherlands.Google Scholar
  4. Jekeli, C. (1999). The determination of gravitational potential differences from satellite-to-satellite tracking. Celestial Mechanics and Dynamical Astronomy, 75, pp. 85–101.CrossRefGoogle Scholar
  5. Migliaccio, F., and F. Sansò (1989). Data processing for the Aristoteles mission. In: Proc. of the Italian Workshop on the European Solid-Earth Mission Aristoteles. Trevi, Italy, May 30–31 1989, pp. 91–123.Google Scholar
  6. Migliaccio, F., M. Reguzzoni and F. Sansò (2004). Space-wise approach to satellite gravity field determination in the presence of coloured noise. Journal of Geodesy, 78, pp. 304–313.CrossRefGoogle Scholar
  7. Migliaccio, F., M. Reguzzoni and N. Tselfes (2006). GOCE: a full-gradient solution in the space-wise approach. In: International Association of Geodesy Symposia, “Dynamic Planet”, Proc. of the IAG Scientific Assembly, 22–26 August 2005, Cairns, Australia, P. Tregoning and C. Rizos (eds), Vol. 130, Springer-Verlag, Berlin, pp. 383–390.Google Scholar
  8. Moritz, H. (1980). Advanced Physical Geodesy, Wichmann Verlag, Karlsruhe.Google Scholar
  9. Sansò, F., and C. C. Tscherning (2003). Fast Spherical Collocation: theory and examples. Journal of Geodesy, 77, pp. 101–112.CrossRefGoogle Scholar
  10. Visser, P. N. A. M., N. Sneeuw and C. Gerlach (2003). Energy integral method for gravity field determination from satellite orbit coordinates. Journal of Geodesy, 77,pp. 207–216.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • M. Reguzzoni
    • 1
  • N. Tselfes
    • 2
  • G. Venuti
    • 2
  1. 1.Italian National Institute of Oceanography and Applied Geophysics (OGS), c/o Politecnico di Milano, Polo Regionale di ComoItaly
  2. 2.DIIAR, Politecnico di Milano, Polo Regionale di ComoItaly

Personalised recommendations