Page - 125 - 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

More compactly, this is[ ∆ldep ∆lind ] = [ A Find 0 I ][ ∆x ∆lind ] + [ edep eind ] . (9.1.39) It is important to note that the two occurrences of∆lind in eq. (9.1.39) refer to two separate variables. The first is the linearisation of the independent variable about the Taylor point,∆lind= lind−g(lind,0). The second is the parameter to be estimated in the LSA. The independent variable does not depend on the parametersx. The covariance matrix of the combined observation vector is the same as in the previous model, given in eq. (9.1.16). The weight matrix for the combined observation vector is then P=Σ−1ll = [ Σdep Σcro ΣTcro Σind ]−1 = [ Pdep Pcro PTcro Pind ] . (9.1.40) Using the matrix inversion lemma (see e.g. Bernstein, 2009, Proposition 2.8.7), the elements ofP are: Pdep=Σ −1 dep+Σ −1 depΣcro ( Σind−ΣTcroΣ−1depΣcro )−1 ΣTcroΣ −1 dep = ( Σdep−ΣcroΣ−1indΣTcro )−1 (9.1.41) Pcro=−Σ−1depΣcro ( Σind−ΣTcroΣ−1depΣcro )−1 =− ( Σdep−ΣcroΣ−1indΣTcro )−1 ΣcroΣ −1 ind (9.1.42) PTcro=− ( Σind−ΣTcroΣ−1depΣcro )−1 ΣTcroΣ −1 dep =−Σ−1indΣTcro ( Σdep−ΣcroΣ−1indΣTcro )−1 (9.1.43) Pind= ( Σind−ΣTcroΣ−1depΣcro )−1 =Σ−1ind+Σ−1indΣTcro ( Σdep−ΣcroΣ−1indΣTcro )−1 ΣcroΣ −1 ind (9.1.44) Normal Equation System The normal equation for this Gauß-Markov model is[ N11 N12 NT12 N22 ] = [ AT 0 FTind I ][ Pdep Pcro PTcro Pind ][ A Find 0 I ] = [ ATPdep A TPcro FTindPdep+P T cro F T indPcro+Pind ][ A Find 0 I ] = [ ATPdepA A TPdepFind+A TPcro FTindPdepA+P T croA Pind+P T croFind+F T indPcro+F T indPdepFind ] . (9.1.45) 9.1 Uncertainties in Independent Variables 125