Page - 104 - 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

With these derivatives, the variance of the satellite orientation observation from the SCA/ACC sensor fusion can be propagated to the AOC for one spacecraft. As the AOC at one epoch only depends on the orientation at that epoch, the matrix of partials is block diagonal, with Fsα=      ∂∆ρsAOC(τ1) ∂α(τ1) 0 ... 0 ∂∆ρ s AOC(τN) ∂α(τN)      N×3N . (8.1.21) With∆ρ˙AOC= ∂∆ρAOC/∂τ the full AOC covariance matrix in the range rate domain for one spacecraft is Σs∆ρ˙AOC=DF s αΣˆ s αˆαˆ(DF s α) T , (8.1.22) withD the polynomial differentiation matrix from eq. (6.4.56). The complete AOC covariance matrix considering the influence of both GRACE-A and GRACE-B is Σ∆ρ˙AOC=Σ A ∆ρ˙AOC+Σ B ∆ρ˙AOC . (8.1.23) Figure 8.3 shows the AOC covariance matrices due to the orientation uncertainty together with the Toeplitz covariance matrix constructed from the estimated covariance function for one short arc in June 2010. In ITSG-Grace2016, this is arc 134 of the month, containing data from 13:35:45 UTC to 16:35:40 UTC on June 17, 2018. Figures 8.3b and 8.3c show that the overall magnitude of the AOC covariance matrices is much smaller than that of the Toeplitz covariance matrix, displayed in fig. 8.3a. Figures 8.3d and 8.3e show magnified views of the AOC covariance matrices. Due to blinding by the sun or the moon, there are times when only one SCA head is active and considered in the sensor fusion. These times are marked in red in the margins of the matrix plots. They correspond well to larger variances on the off-diagonal elements of the covariance matrices. During this arc, at around 15:00:00, GRACE-A was commanded to attitude hold mode. Due to the deteriorated pointing accuracy, the opening angle βA for GRACE-A starts to increase. This can be observed in the increased magnitude of the AOC covariance matrix in fig. 8.3b. The effect is much clearer when focusing only on the main diagonal elements of the covariance matrices, as displayed in fig. 8.4. Here, the relation of the increased opening angle and the increase in the propagated standard deviationσ for GRACE-A (in green) can be clearly seen. Note that the standard deviation is given on a logarithmic scale. The increase in opening angle for GRACE-A by a factor of≈2 to ≈3 leads to an increase in the AOC standard deviation for GRACE-A of approximately one order of magnitude. The main diagonal elements of the AOC covariance matrix for GRACE-A (in green) are however still smaller than those of the Toeplitz matrix (in blue). The timing of the increase in opening angle corresponds to the beginning of the increase in the arc-wise variance factors previously observed in fig. 6.8. Chapter8 Star Camera Observations and Uncertainties104