Page - 90 - 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

1 10 100 Iteration l 10−9 10−7 10−5 10−3 10−1 101 Linear Osculating/Kepler Osculating/Equinoctial Best-fit/Kepler Best-fit/Equinoctial Figure 7.4: Simulated data. RMS of 3D coordinate difference‖∆re‖ for a complete arc after each iteration l. For the first two iterations, no difference in convergence between the configurations is apparent, with the changes in positions for all configurations quickly shrinking from tens of metres to around 1cm. The convergence behaviour starts to differ from the third iteration. Here, the configuration with no reference acceleration (brown) can be easily identified as the one with the worst convergence. The RMS of the corrections never decreases below 10µm. The two configurations using the osculating reference ellipse and Kepler elements (pink) or equinoctial elements (purple) also exhibit comparatively large position differences between iterations, never falling much below 10µm. The two configurations using a best-fit reference ellipse, either with Kepler elements (green) or equinoctial elements (orange), show the smallest differences, reaching an equilibrium at changes below 100nm. All configurations have converged after four iterations, showing no significant reduction in inter-iteration differences thereafter. 7.3.3 Absolute Differences from Simulated Data The following results are based on the corrections ∆re obtained after a generous number of 80 iterations, picked at random from the available results. The exact numeric values are only representative, as they of course change from iteration to iteration. The spectral behaviour of the differences is however rather consistent. Figure 7.5 shows the differences∆ralonge of the in-track coordinate component for one complete day. Figures 7.5a and 7.5b show the coordinate differences in the spatial domain at two different scales. Figure 7.5c shows the PSDs of the coordinate differences, allowing for interpretation of frequency-dependent patterns therein. Chapter7 Numerical Optimization in Orbit Integration90