Page - 9 - 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

2.4 Parameter Elimination Given a normal equation systemNxˆ=nwhere only some of the parameters xˆ1 are of interest and the remaining parameters xˆ2 are only necessary for proper modelling of the system, these non-target parameters can be eliminated from the system. Let the normal equation system be partitioned as[ N11 N12 NT12 N22 ][ xˆ1 xˆ2 ] = [ n1 n2 ] . (2.4.1) Solving the second matrix equation of eq. (2.4.1) for xˆ2 gives xˆ2=N −1 22 ( n2−NT12xˆ1 ) , (2.4.2) assuming thatN22 is invertible. Equation (2.4.2) can then be inserted into the first equation of eq. (2.4.1), giving N11xˆ1+ N12 ( N−122 ( n2−NT12xˆ1 )) =n1 N11xˆ1+N12N −1 22n2−N12N−122NT12xˆ1=n1( N11−N12N−122NT12 ) xˆ1=n1−N12N−122n2 . (2.4.3) More compactly, this isN′xˆ1=n′, with N′=N11−N12N−122NT12 and n′=n1−N12N−122n2 . (2.4.4) The system eq. (2.4.4) gives the same solution for xˆ1 as eq. (2.4.1). Solving for xˆ1 in this manner can be advantageous, depending on the structure and size of the initial normal equation system. 2.5 Variance Propagation Given a functional relationship between some dependent variables y and a set of parameters xwith given covariance Σxx, the covariance matrix of the dependent variables is (e.g. Niemeier, 2008) Σyy=BΣxxB T (2.5.1) with B= ∂y ∂x . (2.5.2) 2.4 Parameter Elimination 9