Page - 127 - 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 two approaches from sections 9.1.1 and 9.1.2 are identical. The proof is obtained by comparing the normal equation systems for both approaches under the condition thatΣcro=0. For the approach due to Reinking, the simplified normal equation system is obtained directly from eqs. (9.1.25) and (9.1.31) as NR=A TPRA and nR=ATPR ( ∆ldep−Find∆lind ) (9.1.52) with the weight matrix in the Reinking system PR= ( Σdep+FindΣindF T ind )−1 . (9.1.53) For the normal equation system derived using parameter elimination with NE=A TPEA and nE=ATPE∆ldep+A TPl∆lind , (9.1.54) lettingΣcro=0 results in the weights from eqs. (9.1.41) to (9.1.44) simplifying to Pdep=Σ −1 dep , Pind=Σ −1 ind and Pcro=0 . (9.1.55) Again using the matrix inversion lemma, this gives a simplified version of eq. (9.1.48). Thefirstweightmatrixfor thenormalequationsystemduetotheparameterelimination algorithm is PE=Σ −1 dep−Σ−1depFind ( Σ−1ind+FTindΣ −1 depFind )−1 FTindΣ −1 dep = ( Σdep+FindΣindF T ind )−1 =PR . (9.1.56) WithPE≡PR, it directly follows thatNE≡NR. It remains to show that the right-hand side of the systems are identical as well. When lettingΣcro=0 the weight matrixPl from eq. (9.1.50) is Pl=−Σ−1depFind ( Σ−1ind+FTindΣ −1 depFind )−1 Σ−1ind =− ( Σdep+FindΣindF T ind ) FindΣindΣ −1 ind =− ( Σdep+FindΣindF T ind ) Find =−PRFind (9.1.57) Inserting eqs. (9.1.56) and (9.1.57) into eq. (9.1.49) gives nE=A TPR∆ldep+A T(−PRFind)∆lind =ATPR∆ldep−ATPRFind∆lind =ATPR ( ∆ldep−Find∆lind ) =nR (9.1.58) WithNR≡NE andnR≡nE, it is proven that both approaches give the same results. It stands to reason that this equivalence should also hold forΣcro 6=0. Numerical tests support this hypothesis, but a rigorous proof for this conjecture is not known to the author. 9.1 Uncertainties in Independent Variables 127