Page - 67 - 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

Solution of Normal Equation System After setting up and decorrelating the observation equations for each short arc, these are accumulated into the normal equation systemNxˆ=n (see eq. (2.2.7)). At this point,N contains the full normal equation system for all parametersx from eq. (6.4.32). This includes the monthly Stokes coefficients, the daily Stokes coefficients for all days of the month, as well as the satellite and state parameters. Similarly to the algorithm for decorrelation described in section 2.3, the inversion ofN can be avoided by computing its Cholesky factorization N=UTU (6.5.32) giving UTUxˆ=n . (6.5.33) The least squares solution to the equation system can then be computed by sequentially solving two triangular systems: xˆ=U−1U−Tn=U−1n¯ . (6.5.34) In computing the Cholesky factorization eq. (6.5.32) use can be made of the special blocked structure ofN (see e.g. Higham, 2002). Matrix of Redundancies In the process of determining the stochastic model, the matrix of redundanciesR from eq. (2.6.8) must be computed for each short arc. Here,N is the completely accumulated normal equation system from all arcs, all other matrices refer to the m-th arc: R=Σ−1ll −Σ−1llAN−1ATΣ−Tll (6.5.35) Especially the productAN−1AT is expensive in this expression, as it involves the in- verse of a large P×P normal equation, as well as two products with N×P matrices. AsR is never needed directly, but only in the form of the trace of the product ofR with some covariance matrixVn sn= trace(RVn) = trace (( Σ−1−Σ−1AN−1ATΣ−T ) Vn ) = trace ( Σ−1Vn−Σ−1AN−1ATΣ−TVn ) , (6.5.36) some optimizations can be introduced in the computation. First, note that the trace of a sum of matrices is equal to the sum of the traces, giving sn= trace ( Σ−1Vn ) − trace ( Σ−1AN−1ATΣ−TVn ) . (6.5.37) 6.5 Fit of Stochastic Model 67