Page - 129 - 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

the weight matrix of the equation system is PR= ( Σsst+DF A αΣ A αˆαˆ ( DFAα )T +DFBαΣ B αˆαˆ ( DFBα )T)−1 = ( Σsst+Σ A ∆ρ˙AOC+Σ B ∆ρ˙AOC )−1 . (9.2.6) Here,Σsst is the estimated stationary covariance matrix for the ll-SST observations, andΣA∆ρ˙AOC andΣ B ∆ρ˙AOC are the AOC covariance matrices from chapter 8. The obser- vation groups are uncorrelated. The covariance matrix of the derived observations is thus identical to the complete arc-wise covariance matrix of the improved stochastic model described in eq. (8.2.1). Also identical to the approach of chapter 8, a complete set of variance factors, one for each time lag in the stationary covariance function, one for each short arc, and one for each spacecraft’s AOC covariance per month, are co-estimated. The reduced derived observation from eq. (9.1.29) is then ∆λ=− ( ρ˙KBR−f0(x0,α0)−DFAα (αA−α0A)−DFBα (αB−α0B) ) , (9.2.7) with the Taylor point for the orientation the observations thereof in the first iteration, and the estimated orientation thereafter. With this information, the normal equa- tion system can be formed, and∆xˆ and eˆλ are determined in the usual way. Using eqs. (9.1.20) and (9.1.21), the estimated additions to the satellite orientation are ∆αˆA= Σˆ A αˆαˆ ( DFAα )T PReˆλ and ∆αˆB= Σˆ B αˆαˆ ( DFBα )T PReˆλ , (9.2.8) with the final updated estimated orientations αˆA=αA−∆αˆA and αˆB=αB−∆αˆB . (9.2.9) Practical Considerations The algorithm used to co-estimate orientation parameters only affects the covariance matrix and reduced observations for the ll-SST observables. The hl-SST component of the equation system remains unaffected. In GROOPS, the co-estimation of the satellite orientation has been implemented in parallel to the iterative estimation of the stochastic model with a degree and order 60 gravity field solution. This means that the updated orientation is not used to re-integrate the dynamic orbits, as this step is already completed at this point. Further, the updated estimates for the KBR antenna phase centre vectors are not used in the linearisation of the observation equations or for computing the updated AOC. Under regular observation conditions the opening angle β is small for both spacecraft. This is an unfavourable configuration for a stable estimate of the APC vectors. Until convergence of the stochastic model and the spacecraft orientation is achieved, estimates of the APC vectors fluctuate wildly. Allowing these vectors to vary at this point prohibits convergence of the system and leads to chaotic results. 9.2 Application to GRACE 129