A model comparison in least squares collocation

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$\tilde s$.