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Bulletin géodésique

, Volume 50, Issue 2, pp 181–192 | Cite as

A model comparison in least squares collocation

  • Reiner Rummel
Article

Abstract

The well known least squares collocation model
$$\ell = Ax + \left[ {\begin{array}{*{20}c} O \\ I \\ \end{array} } \right]^T \left[ {\begin{array}{*{20}c} s \\ {s' + n} \\ \end{array} } \right]$$
(I)
is compared with the model
$$\ell = Ax + \left[ {\begin{array}{*{20}c} R \\ I \\ \end{array} } \right]^T \left[ {\begin{array}{*{20}c} s \\ n \\ \end{array} } \right]$$
(II)
The basic differences of these two models in the framework of physical geodesy are pointed out by analyzing the validity of the equation
$$s' = Rs$$
that transforms one model into the other, for different cases. For clarification purposes least squares filtering, prediction and collocation are discussed separately. In filtering problems the coefficient matrix R becomes the unit matrix and by this the two models become identical. For prediction and collocation problems the relation s′=Rs is only fulfilled in the global limit where s becomes either a continuous function on the earth or an intinite set of spherical harmonic coefficients. Applying Model (II), we see that for any finite dimension of s the operator equations of physical geodesy are approximated by a finite matrix relation whereas in Model (I) the operator equations are applied in their correct form on a continuous, approximate function\(\tilde s\).

Keywords

Gravity Anomaly Operator Equation Random Signal Altimeter Data Geoid Undulation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Bureau Central de L’Association Internationale de Géodésie 1976

Authors and Affiliations

  • Reiner Rummel
    • 1
  1. 1.Department of Geodetic ScienceOhio State UniversityColumbus

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