Page - 123 - 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

Normal Equation System Going beyond the derivations given by Reinking, 2008, his apparatus can be taken to completion by directly setting up the normal equation system for the sought parameters xˆ. To this end, the matrices in eq. (9.1.8) are analysed and expanded, so that they can later be substituted into eq. (9.1.18). First, note that Ax= ∂G ∂x ∣∣∣∣ x0,l0 =− ∂f ∂x ∣∣∣∣ x0,l0 =−A . (9.1.22) Further, F= [ ∂G ∂ldep ∣∣∣ x0,l0 ∂G ∂lind ∣∣∣ x0,l0 ] = [ I − ∂f∂lind ∣∣∣ x0,l0 ] = [ I −Find ] . (9.1.23) With eqs. (9.1.17) and (9.1.23), the covariance matrix of the derived observations is Σλλ= [ I −Find ][Σdep Σcro ΣTcro Σind ][ IT −FTind ] =Σdep−FindΣTcro−ΣcroFTind+FindΣindFTind , (9.1.24) giving the normal equation in the Reinking apparatus NR=A T ( Σdep−FindΣTcro−ΣcroFTind+FindΣindFTind )−1 A . (9.1.25) To compute the right-hand sidenR, eqs. (9.1.12) to (9.1.14) must be expanded. With λ=− [ I −Find ][ldep lind ] =−ldep+Findlind (9.1.26) Fl0= [ I −Find ][ldep,0 lind,0 ] = ldep,0−Findlind,0 (9.1.27) G0= ldep,0−f0 (9.1.28) the reduced derived observations are ∆λ=λ−λ0=λ−(G0−Fl0)=λ−G0+Fl0 =−ldep+Findlind−ldep,0+f0+ldep,0−Findlind,0 =−ldep+f0+Findlind−Findlind,0 =−(ldep−f0)+Find(lind−lind,0) =−((ldep−f0)−Find(lind−lind,0)) =−(∆ldep−Find∆lind) . (9.1.29) At this point it is important to note that∆ldep is the reduced observation as it appears in the classical Gauß-Markov model in section 2.2, not simply ldep−ldep,0. In fact, the 9.1 Uncertainties in Independent Variables 123