Page - 65 - 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

on their length, the estimator for the variance factor must also consider all of these contributions. Therefore, Ω˜j= M ∑ m=1 Ω˜mj and s˜j= M ∑ m=1 s˜mj . (6.5.21) The contributions to each frequency can be determined in two ways. The first is to transform the residuals and covariance componentsVn to the spectral domain and then directly compute the Ω˜j and s˜j. The second option is to compute theΩn and sn in the temporal domain, and then transform these to the spectral domain afterwards. The second approach is chosen, as it enables the exploitation of some time-domain symmetries in the computation, as is later explained in section 6.5.6. To this end, the contributors for each time lag n∈ [0,N)must be computed. With the covariance matrix for one time lag from eq. (6.5.18) and eq. (2.6.6), the square sum of residuals for one time lag in one arc is Ωmn = { σ2mCnxx · ( eˆTΣ−Tll VnΣ−1ll eˆ ) if n<Nm , 0 otherwise, (6.5.22) while the redundancy is smn = { σ2mCnxx · trace(RVn) if n<Nm , 0 otherwise. (6.5.23) In both eqs. (6.5.22) and (6.5.23), it is important to note that eˆ = eˆm, Σll = Σmll, andR=Rm. This index has only been omitted for clarity. With Ωm= [ Ωm0 . . . Ω m N−1 ]T and sm= [ sm0 . . . s m N−1 ]T , (6.5.24) the contributors to the individual frequencies of the PSD to be inserted into eq. (6.5.21) are the entries of the N×1 vectors Ω˜ m =XΩm (6.5.25) s˜m=Xsm (6.5.26) 6.5.4 Estimation of Arc-wise Variance Factors The arc-wise variance factors σ2m (m∈ [1,M]) can be estimated from the equations given in section 6.5.3. Specifically, where the covariance function was determined by estimating the amplitude of a specific frequency over all arcs, the arc-wise variance factors are determined by estimating the cumulative amplitude over all frequencies for one specific arc. The estimated variance factor for one arc is then αˆ2m= ∑N−1j=0 Ω˜ m j ∑N−1j=0 s˜ m j . (6.5.27) 6.5 Fit of Stochastic Model 65