Page - 88 - 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

Update estimate: Compute the least squares estimate∆xˆusing∆l andA. The up- dated estimate for the best-fit orbit is then ξˆe= ξe,0+∆xˆ . (7.2.37) Iterate: Until convergence is achieved. The equinoctial orbit ξˆe is then used to deter- mine the reference forcef0 in eq. (7.2.1) and as the reference orbit in eq. (7.2.4). 7.3 Results The following pages will give results from two separate sets of computations. The first results, presented in sections 7.3.1 to 7.3.3 are purely from simulations and illustrate the algorithms performance under ideal conditions. To this end, an orbit was simulated for a single spacecraft using the GOCO05s static gravitational potential (Mayer-Gu¨rr, Kvas, et al., 2015). No further conservative or non-conservative forces were considered in the orbit propagation. The static potential was expanded to degree and order 60. The simulated orbit was computed using an in-house orbit propagator based on the integration polynomials presented in section 2.7. The second set of results, given in sections 7.3.4 and 7.3.5, illustrate the performance of the algorithm for real data processing. The results were determined using real GRACE data in the context of dynamic orbit integration for the ITSG-Grace2016 gravity field solution. All orbits, both for the simulation and for real data, were determined for an arc length of 24h at a sampling of 5s resulting in N = 17280 epochs. Where applicable, the Marussi tensorTwas expanded to degree and order 10. When absolute differences between coordinates are shown, such as between iterations, they are given for the along-track axis only. In all presented results the along-track axis shows, in accordance to theory (Huang and Innanen, 1983), the largest errors. 7.3.1 Encke Ratio Figure 7.3 shows the Encke ratio for the two reference ellipses described in section 7.2.1. The first case (in pink) represents the classical Encke configuration of an osculating reference ellipse. Here, the differential initial state is∆y0=0. The ellipse is congruent to the approximate orbit at the first epoch. The second case (in green) represents the best-fit reference ellipse for the orbit arc as derived in section 7.2.3. Here, the initial state is the least squares estimate∆yˆ0. The Encke ratio for the osculating orbit is 0% at the start of the 24h arc. It grows to larger than 1% within 1h, finally reaching 20% towards the end of the arc. The Encke ratio at the final epoch corresponds to an Encke vector of 1240km. Chapter7 Numerical Optimization in Orbit Integration88