Page - 48 - 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

3d 3N ncal Acal DACC Acal DACC ASCA Figure 6.3: Structure of the normal equation system of the improved SCA/ACC sensor fusion, with contributing design matrices marked. For performance reasons, the observation equations are set up and decorrelated per epoch and then combined directly at the normal equation level. The normal equation system has a kite structure. Depending on the arc length N and the chosen polynomial degree d, it can be quite sparse. The structure of the normal equation system is illustrated in fig. 6.3. The initial orientations are improved with the adjusted∆αˆ to yield qˆ, which is used asq0 for the next iteration. The estimated calibration parameters∆xˆcal are applied to the angular acceleration observations, giving the new ω˙0. After convergence, the qˆgive the adjusted orientation of the satellite, the result of the SCA/ACC sensor fusion. A further product of this approach is the covariance matrix of the adjusted parameters. This matrix is divided along the same lines as the normal equation system set up with eq. (6.2.17), and has the entries Σˆxˆxˆ= [ Σˆαˆαˆ Σˆαˆxˆcal Σˆ T αˆxˆcal Σˆxˆcalxˆcal ] . (6.2.18) Σˆαˆαˆ is the fully populated complete variance-covariance matrix for the satellite orien- tation along the orbit arc, given in the SRF. The 3×3 blocks on the main diagonal are the covariance matrices of the orientation at that specific epoch. 6.3 Orbit Integration For ITSG-Grace2016, the GRACE dynamic orbits are integrated using the algorithm described in chapter 5 with the improvements laid out in chapter 7. The orbits are Chapter6 ITSG-Grace201648