Page - 11 - 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

whereR is a symmetric matrix R=Σ−1ll −Σ−1llAN−1ATΣ−Tll . (2.6.8) The estimated variance factors of this iteration σˆ2i = αˆ 2 iσ 2 i (2.6.9) are used as the initial valueσ2i for the next iteration. When theα 2 i converge to 1, the σ2i are the desired variance factors for the cofactor matricesQi. Iteration is needed due to a simplifying assumption in eq. (2.6.5) that the individualσ2i can be estimated independently. It can be shown that after convergence, the computedσ2i are identical to those resulting from a strict solution (Fo¨rstner, 1979). The algorithm of variance component estimation can also be employed to determine the optimal respective weights of several uncorrelated observation groups lk of nk observations each:    l0 ... lm     =     A0 ... Am     x+     e0 ... em     (2.6.10) Here, each observation group has a cofactor matrixQk with an unknown variance factorσ2k . The full covariance matrix of all observations is Σll=       σ20Q0 0 · · · 0 0 ... ... ... ... ... ... 0 0 · · · 0 σ2mQm       . (2.6.11) As the observation groups are assumed to be uncorrelated, all variables pertaining to the other observation groups disappear when inserting into eq. (2.6.5), except for the combined normal equationN: αˆ2k= eˆTkΣ −1 k eˆk nk− trace(NkN−1) (2.6.12) Then, the estimated variance factor for the observation group is σˆ2k = αˆ 2 kσ 2 k . (2.6.13) 2.7 Interpolation, Numerical Integration, and Numerical Differentiation Techniques for numerical differentiation, integration, and interpolation of discrete time series of data find many applications in the computation of GRACE gravity field 2.7 Interpolation, Numerical Integration, and Numerical Differentiation 11