Fast Collocation

  • Riccardo Barzaghi
  • Giampaolo Bottoni
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 106)

Abstract

The collocation theory is nowaday one of the most used techniques in Geodesy; it is indeed possible to estimate the geoid starting from eterogeneous data. However, since the most common data is the gravity anomaly, the collocation formula is frequently used with this kind of data only. Furthemore, data are often given on a regular grid. If you have regularly spaced data, the covariance matrix contained in the collocation formula has a particular structure, i.e. is of Toeplitz form. In fact, collocation formula is
$$\begin{gathered} \hat s(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} )\, = [\sum\limits_{i,\,\,j}^N {C(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} ,{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} }_i})} \,\,[\,C(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} ,{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} }_K})\, + \,\sigma _n^2\,{\delta _{1\,K}}]_{i\,j}^{ - 1}\,x({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} }_j})\, = \hfill \\ = \,{C_p}{[C\, + \,\sigma _n^2\,I]^{ - 1\,}}x\, = \,{C_p}{D^{ - 1}}x\, = \,{C_p}v \hfill \\ \end{gathered}$$
(1)
where
$$x\,({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} _i})\, = \,s\,({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} _i})\, + \,n{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} _i})\,i\, = \,1, \cdots ,N $$
(2)
are the observed values (signal plus noise), C is the covariance matrix of s and σn2 is the variance of the noise. If you consider, as an example, a one-dimensional stochastic process, i.e. Pj∈ℝ, and you have regularly distributed data (Pj = P1 + nΔP n=0,…,N−1), then C is a symmetric Toeplitz matrix.This is due to the fact that the covariance function of s, C(P,Q) is a function of |P-Q|, C(P,Q) = C(|P,-Q|). In consequence of that, also D will have the same structure.Now if you consider a two dimensional process, Pj∈ ℝ2 and the data are distributed on a regular grid, it can be shown that C is a block symmetric Toeplitz matrix and that each block is a symmetric Toeplitz matrix too (in this case the covariance function depends on the planar distance between points).

Keywords

Covariance Convolution 

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • Riccardo Barzaghi
    • 1
  • Giampaolo Bottoni
    • 2
  1. 1.Istituto di TopografiaPolitecnico di MilanoMilanoItaly
  2. 2.CILEASegrateItaly

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