Page - 96 - 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

uncertainty observed at low frequencies in the dynamic orbits. Complicated higher- order or time-variable parametrizations of the reference ellipse can thus be avoided, keeping the complexity of the implementation low. Further, the reference ellipse was transformed from a parametrisation in classical Kepler elements to a parametrisation in equinoctial elements. Without changing the geometry of the ellipse, this improved the stability of the derived reference motion mainly at higher frequencies. The resulting orbits combining both of these improvements reach machine precision at frequencies above two cycles per orbital revolution. The uncertainty of the original and improved dynamic orbits were compared with the uncertainty of GRACE and GRACE-FO ll-SST ranging observations through error propagation to the range rate domain. It was shown that the improved orbits are now self-consistent to well below the expected precision of the GRACE-FO LRI instrument. This represents an improvement of several orders of magnitude over previous results achieved at IfG. Such an improvement can be important as the dynamic orbits are used as a Taylor point in the linearisation of the observation equations for gravity field recovery from GRACE, and later GRACE-FO. Any extraneous errors that originate in the processing chain, and do not arise directly from the observations, should be avoided. The largest significance of this work can be found in the reduction of the in-track variability between iterations by several orders of magnitude, as it is the component with the largest influence on the GRACE ll-SST ranging observations. It is conceivable to further reduce this error using an ensemble approach. In such an approach, the results of each iteration after convergence would be treated as a separate realisation of the dynamic orbit. An ensemble of such realisations could then be directly used to compute an orbit of best agreement, possibly reducing the integration error at each individual epoch. The assessment of the effectiveness of the equinoctial best-fit reference ellipse presented in this work is very specific to the GRACE orbital configuration and chosen arc length. For the case of GRACE precession of the orbital plane during the integration period is negligible, which ensures a consistently small Encke ratio. The method is thus directly applicable to satellites in similar orbital configurations such as the European Space Agency’s gravity field and steady-state ocean circulation explorer (GOCE) mission (Drinkwater et al., 2003) or the Swarm constellation (Friis-Christensen, Lu¨hr, and Hulot, 2006). For satellites in other orbits, or for longer arcs, the effect of nodal precession might need to be considered. The nodal precession of the orbital plane is dependent on the spacecraft’s inclination (Brouwer, 1959), and is smallest for polar orbits. The Encke ratio would thus increase at a faster rate for a satellite at any other inclination than GRACE. A better approximation of such an orbit could be made by introducing a co-precessing ellipse, which would keep the Encke ratio smaller for longer integration periods. Such an ellipse would also again allow for longer integration periods, as shown by Escobal, 1966; Kyner and Bennett, 1966. Jezewski (1983a,b) gives an analytical solution for such a reference motion, where the precession of the ellipse is due to Earth’s oblateness. Chapter7 Numerical Optimization in Orbit Integration96