Page - 8 - 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

Multiple observation groups lk for the same set of parametersx can be combined at the normal equation level by summation N= n ∑ k=0 Nk n= n ∑ k=0 nk . (2.2.10) The estimated covariance matrix of the adjusted parameters is the inverse of the normal equation Σˆxˆxˆ=N −1 . (2.2.11) 2.3 Decorrelation For many problems, the stochastic model for the observations is given or estimated as a covariance matrixΣll, and not as the weight matrixP needed to compute eq. (2.2.4). As direct inversion ofΣll is expensive and, depending on the covariance structure, can be numerically unstable (Bjo¨rck, 1996), it is desirable to avoid this operation. If the given covariance matrix is positive definite, the Cholesky decomposition Σll=W TW , (2.3.1) exists, withW an upper triangular matrix. With P= ( WTW )−1 =W−1W−T (2.3.2) eq. (2.2.4) is xˆ= ( ATW−1W−TA )−1 ATW−1W−Tl . (2.3.3) With the transformation A¯=W−TA , l¯=W−Tl (2.3.4) this is xˆ= ( A¯TA¯ )−1 A¯Tl¯ . (2.3.5) Due to the upper triangular shape ofW , the decorrelated observations l¯and the decorre- lated observation equations A¯ can be computed without knowing the inverseW−1 by solving the system WTx¯=x (2.3.6) through back-substitution. The normal equation and the right hand side are computed from the decorrelated matrices as N= A¯TA¯ , n= A¯Tl¯ . (2.3.7) Chapter2 Mathematical Theory and Notation8