A model comparison in least squares collocation

The well known least squares collocation model (I) $$\ell = Ax + \left {\begin{array}{*{20}c} O \\ I \\ \end{array} } \right]^T \left {\begin{array}{*{20}c} s \\ {s' + n} \\ \end{array} } \right]$$ is compared with the model (II) $$\ell = Ax + \left {\begin{array}{*{20}c} R \\ I \\ \end{array} } \right]^T \left {\begin{array}{*{20}c} s \\ n \\ \end{array} } \right]$$ 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 \(\tilde s\) .