Variance Component Estimation for Combination of Terrestrial Reference Frames.
Prof. Dr.-Ing. B. Heck
Dr.- Ing. Z. Altamimi (LAREG)
- Person in Charge:Bähr, H.
Terrestrial Reference Frames (TRF) are based on heterogeneous data derived from continuous observations by different space geodetic techniques. They are established by diverse analysis centers pursuing various strategies of mathematical modelling. Combining individual solutions to frames of superior quality usually involves homogenisation by an empirical weighting scheme. For this purpose, different approaches on variance component estimation are available, the evaluation of which was subject of this study.
Combining TRFs, such as the ITRF (International Terrestrial Reference Frame), is and has ever been a principal field of activity of the Laboratoire de Recherche en Géodésie (LAREG) . It is part of the French Institut Géographique National (IGN) and situated in the facilities of the Ecole Nationale des Sciences Géographiques (ENSG) at Marne la Vallée in France. Since 1995, the software CATREF (Combination and Analysis of Terrestrial Reference Frames) has been developed at the LAREG to perform combinations.
The CATREF combination algorithm implies that the individual solutions are statistically independent populations. The analysis centers provide full covariance matrices that are scaled by individual factors (variance components) before being introduced into the combination process. These factors can be estimated, and the estimates can be reintroduced to specify the weighting of subsequent iterations.
Up to now, two methods of variance component estimation have been realised in CATREF software: an approximate least squares estimator (the "classical" estimator) and the degree of freedom method, which is a special case of an approach known as "Förstner's method". Within the scope of this study, the application of the statistically rigorous Helmert estimator has been evaluated comparatively on the basis of CATREF's mathematical model.
Firstly, it turned out that the Helmert estimator is hardly applicable for most practical purposes due to its high requirements on computation time. Nevertheless, some tests have been performed, covering two elementary types of combinations:
- Intratechnique Combination: Combining weekly SLR-solutions of consecutive weeks, derived by a unique analysis center, yielded relatively homogeneous variance components for all of the three investigated estimators. Convergence was achieved quickly after a few iterations.
- Intertechnique Combination: Three individual solutions derived from different techniques (VLBI, SLR, GPS) were combined along with 46 sets of local ties, corresponding to respective collocation sites. Unfortunately, none of the estimators succeeded in estimating 46 individual variance components for the local ties in addition to the three for the space geodetic solutions. Otherwise, fixing the weights for the local ties to empirical values resulted in slow convergence of the Helmert estimator for the space geodetic solutions.
Whereas variance component estimation turned out to be relatively unproblematic for intratechnique combinations, no appropriate approach has been found for the intertechnique case. The traditional manual choice of empirical weights for local ties is in some respects arbitrary and thus not satisfactory.