Page - 92 - 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

In fig. 7.5a, the configuration using no reference acceleration (brown) can be identified as the one with the largest coordinate differences. This matches the observations from fig. 7.4. The configurations using the osculating reference ellipse (pink and purple) also show comparatively large errors. For all configurations the magnitude of the differences is smallest at around 8h and 20h, not at the beginning of the arc as might be expected. This is due to the re-estimation of the initial state from eqs. (5.2.40) and (7.2.19), fitting the integrated orbit to the approximate positionsre. The corrections applied to the best-fit configurations (green and orange) can not be seen at this scale. Figure 7.5b shows a magnification of only the best-fit cases. Here it becomes clear that the corrections for the best-fit configuration using equinoctial elements (orange) are smaller than those for the best-fit configuration using Kepler elements (green). The differences and similarities between the five configurations become most clear not in the spatial domain but when observing the PSDs of the corrections∆ralonge . The PSDs were computed using Welch’s method with a segment length of 6h, and are displayed in fig. 7.5c. The vertical grey lines in fig. 7.5c denote multiples of the orbital frequency, starting at approximately one cycle per 89min for the leftmost line. The best-performing configuration of a best-fit reference ellipse parametrised in equinoctial elements (orange) shows white noise behaviour at frequencies higher than two cycles per revolution. Significantly, the magnitude of the corrections in the high-frequency part of the spectrum is at the level of the numerical resolution of a double precision floating point number at orbital altitude. For this configuration, machine precision is completely exhausted here. This is not the case for the best-fit ellipse parametrised in Kepler elements (green), which shows a consistently higher power for all frequencies above two cycles per revolution. This clearly illustrates the advantages of the equinoctial parametrisation over the Kepler parametrisation. Both configurations show some residual error at very long wavelengths. At low frequencies, the configurations based on the osculating reference orbit (pink and purple) show much larger deviations. The corrections for these orbits at one cycle per revolution are two orders of magnitude larger than those of the two best-fit ellipse configurations (green and orange). At higher frequencies, the corrections of the osculating configuration using Kepler elements (pink) asymptotically approach those of the best-fit configuration also using Kepler elements (green). The same holds true for the two equinoctial ellipses, with the osculating configuration (purple) almost reaching the level of the best-fit ellipses (orange) at the Nyquist frequency. Also at higher frequencies, the osculating configuration employing equinoctial elements (purple) displays corrections smaller than those of the best-fit orbit with Kepler elements (green). At very high frequencies close to the Nyquist frequency, the configurations employing Kepler elements (pink and green) show no significant improvements over using no reference acceleration at all (brown). Not shown in fig. 7.5, the coordinate differences∆re in the cross-track and radial axes show similar spectral behaviour. The magnitude of the differences is however smaller by approximately two orders of magnitude. Chapter7 Numerical Optimization in Orbit Integration92