Burkhard Schaffrin

In regression analysis, oftentimes a linear (or linearized) Gauss-Markov Model (GMM) is used to describe the relationship between certain unknown parameters and measurements taken to learn about them. As soon as there are more than enough data collected to determine a unique solution for the parameters, an estimation technique needs to be applied such as ‘Least-Squares adjustment’, for instance, which turns out to be optimal under a wide range of criteria. In this context, the matrix connecting the parameters with the observations is considered fully known, and the parameter vector is considered fully unknown. This, however, is not always the reality. Therefore, two modifications of the GMM have been considered, in particular. First, ‘stochastic prior information’ (p. i.) was added on the parameters, thereby creating the – still linear – Random Effects Model (REM) where the optimal determination of the parameters (random effects) is based on ‘Least Squares collocation’, showing higher precision as long as the p. i. was adequate (Wallace test). Secondly, the coefficient matrix was allowed to contain observed elements, thus leading to the – now nonlinear – Errors-In-Variables (EIV) Model. If not using iterative linearization, the optimal estimates for the parameters would be obtained by ‘Total Least Squares adjustment’ and with generally lower, but perhaps more realistic precision. Here the two concepts are combined, thus leading to the (nonlinear) ’EIV-Model with p. i.’, where an optimal estimation (resp. prediction) technique is developed under the name of ‘Total Least-Squares collocation’. At this stage, however, the covariance matrix of the data matrix – in vector form – is still being assumed to show a Kronecker product structure.

In a Gauss–Markov Model (GMM) with fixed constraints, all the relevant estimators perfectly satisfy these constraints. As soon as they become stochastic, most estimators are allowed to satisfy them only approximately, thereby leaving room for nonvanishing residuals to describe the deviation from the prior information. Sometimes, however, linear estimators may be preferred that are able to perfectly reproduce the prior information in form of stochastic constraints, including their variances and covariances. As typical example may be considered the case where a geodetic network ought to be densified without changing the higher-order point coordinates that are usually introduced together with their variances and (some) covariances. Traditional estimators are based on the “Helmert” or “S-transformation,” respectively an adaptation of the fixed-constraints Least-Squares estimator. Here we show that neither approach generates the optimal reproducing estimator, which will be presented in detail and compared with the other reproducing estimators in terms of their MSE-risks.

Geodetic networks, when derived solely from observed GPS baseline vectors, have an inherent datum deficiency of dimension three due to the respective unknown translation parameters. Thus, estimated coordinates from a (weighted) least-squares adjustment will not be unique, although the adjusted baseline vectors are. Uniqueness, without affecting the adjustment as such, can be achieved by introducing a minimum number of constraints for the coordinates, or by applying an objective function on the set of least-squares solutions (LESS) that fulfill the so-called normal equations. In both alternatives, bias control for some (or all) coordinate estimates has so far been treated only as a secondary issue. Here it is shown that the recently introduced estimator of type BLIMPBE—although generally not a LESS—can indeed be expected to be superior to all other linear estimators that minimize the bias for certain coordinates. This performance is demonstrated in a GPS network that includes GPS receivers at several conti...