Page - 111 - 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

8.3.3 Covariance Function When only considering a stationary covariance function in the determination of the stochastic model, this covariance function will be estimated to best fit all power in the residuals, both from stationary and non-stationary noise processes. The non-stationary effects thus alias into the stationary covariance function, as there is no avenue to model them in this framework. The newly introduced information about some of the non-stationary noise in the form of the AOC covariance matrices should model some of this observed power in the residuals, reducing aliasing into the stationary covariance function. Figure 8.8a shows the estimated covariance functions, displayed as PSDs, for the previously discussed month of December 2005. In the PSD estimated according to the oldnoisemodel (inbrown), suchaliasing isclearlyvisible in theshadedarea.The lower bound of the shaded area is set to 3.3mHz, which is the dominant frequency in the GRACE pointing variations after February 2004 for GRACE-A and after January 2005 for GRACE-B (Bandikova, 2015). The upper bound of the shaded area is at 20mHz, where noise in the SCA observations at harmonics of the orbital frequency of the spacecraft starts to be dominated by purely stochastic effects (Ina´cio et al., 2015). This result can be compared to a month of “good” data without such abnormally large opening angles, e.g. April 2008 (cf. fig. 8.8b). Here, the PSD for the old model is virtually identical to that of the new model determined using the AOC covariance matrices. During this normal operation, the effect due to the orientation uncertainty is small enough to be dominated by other noise sources. It could be argued that the strict modelling of the AOC uncertainty is not necessary, as the arcs most affected by these errors are down-weighted in the VCE through application of large arc-wise variance factors. Comparison of the PSDs in fig. 8.8a shows that this is however not correct. The aliasing in the PSD estimated in the old model is not only present in the arcs affected by large opening angles, but in all arcs of the month. While arcs with large opening angles are downweighted, and thus lose influence on the monthly solution, all remaining arcs are decorrelated with a clearly wrong covariance function, introducing systematic effects into the solution. It has been demonstrated here that correct modelling of the AOC covariance mitigates this undesired effect of aliasing into the stationary covariance function. The overall magnitude of the average change in the PSD can be observed in the mean of all monthly PSDs for the processed time series. The mean of the PSDs is computed once per frequency, per processing strategy. These PSDs are displayed in fig. 8.9. The lower noise level in the highlighted band is clearly visible in the mean PSD obtained when using AOC covariance matrices (in brown). It is also interesting to note that the PSD computed under consideration of the AOC covariance information (in blue) follows the expected linear progression of the KBR-branch of the noise spectrum to a lower frequency, down to≈10mHz, as opposed to≈20mHz for the old model (in brown). 8.3 Results 111