Page - 69 - 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

Where the left-hand trace is the simple trace from before, but the right-hand trace is now a rewritten form of the Monte-Carlo trace estimator. The right-hand trace is shifted back to its original permutation, after which the traces are combined, giving sn= trace ( Σ−1Vn ) − trace ( Σ−1AU−1Z¯Z¯TU−TATΣ−TVn ) = trace (( Σ−1−Σ−1AU−1Z¯Z¯TU−TATΣ−T ) Vn ) = trace(R¯Vn) . (6.5.45) This new quantity is an estimator for the matrix of redundancies R¯=Σ−1−Σ−1AU−1Z¯Z¯TU−TATΣ−T , (6.5.46) or R¯=Σ−1−R˜R˜T (6.5.47) with R˜=Σ−1AU−1Z¯ =W−1W−TAU−1Z¯ =W−1A¯U−1Z¯ . (6.5.48) WithW ,A, andU known from previous computations, R˜ can be computed efficiently and quickly. First the triangular systemU−1Z¯ is solved, giving a P×Z temporary matrix. AsU is computed from the accumulated normal equations and is identical for all short arcs,U−1Z¯ only needs to be computed once, and can then be reused for all short arcs. After determining the product A¯U−1Z¯, only one more triangular system needs to be solved, giving the N×Z matrix R˜. In GROOPS, Z= 100, giving a very manageably small matrix R˜, which is then used to determine R¯using eq. (6.5.47). To summarize, this implementation avoids the explicit computation ofN−1 in eq. (6.5.35). Further the computation of R¯ is reduced in complexity through exploitation of sym- metries in eq. (6.5.46). Elegant application of the Monte Carlo trace estimator reduces the operations needed to compute R˜. Overall, this algorithm reduces the computational cost to determineR considerably, at the expense of a small error due to the approximation introduced by the Monte Carlo trace estimator. As the stochastic model is determined through multiple iterations, this uncertainty does however not affect the resulting model significantly. Computation of Variance Factors The computation ofΩmn and smn from eqs. (6.5.22) and (6.5.23) can be optimized by not explicitlycomputing allmatrixproducts involved.For a timelag n=0, trace(R¯mVn) is 6.5 Fit of Stochastic Model 69