Correcting for site displacements at different levels of the Gauss-Markov model – A case study for geodetic VLBI

In a previous paper, Glomsda et al. (2020) revisited the impact of distinct parts of non-tidal loading in the analysis of geodetic Very Long Baseline Interferometry (VLBI). The loading is represented by displacements of the reference positions of the observing VLBI antennas, which are variables of a corresponding Gauss-Markov model for estimating various geodetic target parameters. These displacements were applied at two different levels of the model, the observation and the normal equation level, and quite similar results were obtained for both cases. In this paper, the authors provide a more detailed theoretical discussion of the application of site displacements at the distinct levels, which also comprises the a posteriori application at the solution level. For each case, the respective formulas and implications for the Gauss-Markov model are derived, and equations for assessing the differences between the estimated parameters are established. In this way, the authors aim to create a deeper understanding of the results of the previous paper, which show the capability of the normal equation level to approximate the application of site displacements at the observation level with less effort and prerequisites, and to provide evidence for the claims made in that paper (for VLBI only): (i) the chosen reference frame of the site displacements is basically irrelevant except for the solution level; (ii) the Jacobi matrix does not change substantially; (iii) the loss of temporal resolution of the site displacements is more important than the linear approximation of the functional model at the normal equation level; and (iv) the station coordinate estimates for all three levels strongly depend on the regularizing (datum-) conditions of the model. The theoretical results are substantiated with numerical examples, which consider site displacements generated from non-tidal atmospheric loading by the Earth-System-Modelling group of the Deutsches GeoForschungsZentrum (GFZ). However, the results are valid for site displacements of any source.