Page - 126 - 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

The right-hand side of the system is[ n1 n2 ] = [ AT 0 FTind I ][ Pdep Pcro PTcro Pind ][ ∆ldep ∆lind ] = [ ATPdep A TPcro FTindPdep+P T cro F T indPcro+Pind ][ ∆ldep ∆lind ] = [ ATPdep∆ldep+ATPcro∆lind( FTindPdep+P T cro ) ∆ldep+ ( FTindPcro+Pind ) ∆lind ] . (9.1.46) Parameter elimination Now the independent parameters are eliminated from the normal equation system as given by eq. (9.1.45) (see section 2.4). After some simplification, this gives NE=A TPEA . (9.1.47) with PE=Pdep− ( PdepFind+Pcro )( Pind+P T croFind+F T indPcro+F T indPdepFind )−1( FTindPdep+P T cro ) . (9.1.48) For the right-hand side as given by eq. (9.1.46), parameter elimination yields nE=A TPdep∆ldep+A TPcro∆lind − ( ATPdepFind+A TPcro )( Pind+P T croFind+F T indPcro+F T indPdepFind )−1 (( FTindPdep+P T cro ) ∆ldep+ ( FTindPcro+Pind ) ∆lind ) =ATPE∆ldep+A TPl∆lind , (9.1.49) with Pl=Pcro− ( PdepFind+Pcro )( Pind+P T croFind+F T indPcro+F T indPdepFind )−1( FTindPcro+Pind ) . (9.1.50) 9.1.3 Equivalence of Approaches Given the simplifying assumption that Σcro=0 , (9.1.51) Chapter9 Co-Estimation of Orientation Parameters126