Page - 124 - in Contributions to GRACE Gravity Field Recovery - Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations

Taylor point for the dependent observations ldep,0 completely cancels from eq. (9.1.29). In the equation for the reduced derived observations, the term concerning the inde- pendent observations can be considered as a correction term for the reduced dependent observations, or ∆∆ldep=Find∆lind . (9.1.30) With eqs. (9.1.22), (9.1.24) and (9.1.29) the right-hand side of the normal equation system is nR=A T ( Σdep−FindΣTcro−ΣcroFTind+FindΣindFTind )−1( ∆ldep−Find∆lind ) . (9.1.31) 9.1.2 Total Least Squares as Gauß-Markov Model with Eliminated Observations In this section, a different formulation for the treatment of uncertainties in the depen- dent variables is proposed. Where Reinking, 2008 started from an extended observation vector (eq. (9.1.5)), this formulation, perhaps equivalently, will begin with introducing the independent variables into an extended parameter vector ξ= [ x lind ] . (9.1.32) This formulation pursues the goal of not only estimating the adjusted parameters xˆ and the adjusted dependent variable lˆdep, but to also determine a least squares estimate for the independent variable lˆind. The functional model eq. (9.1.1) is extended with a second relationship describing the observations of the independent variable ldep=f (lind,x)+edep (9.1.33) lind=g(lind)+eind . (9.1.34) This second relationship is of course simply g(lind)= lind . (9.1.35) This results in an extended observation vector identical to that of Reinking, l= [ ldep lind ] . (9.1.36) Linearising eqs. (9.1.33) and (9.1.34) at the Taylor point of the approximate values for the parameters and the independent variable yields ldep=f (lind,0,x0)+ ∂f ∂x ∣∣∣∣ x0,lind,0 ∆x+ ∂f ∂lind ∣∣∣∣ x0,lind,0 ∆lind+edep , (9.1.37) lind=g(lind,0) + ∂g ∂x ∣∣∣∣ x0,lind,0 ∆x+ ∂g ∂lind ∣∣∣∣ x0,lind,0 ∆lind+eind . (9.1.38) Chapter9 Co-Estimation of Orientation Parameters124