Page - 46 - 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

6.2.2 Star Camera Observations In addition to the observed orientation of the spacecraft q, the nominal orientation of the spacecraft as computed from eq. (6.2.3) is also given as a quaternion q0. The nominal orientation serves as the Taylor point for the iterative adjustment of the spacecraft orientation. Due to the difference in the rotations∆α only being small, multiplication of the quaternions yields ∆q=q−10 ·q≈      1 0.5∆αx 0.5∆αy 0.5∆αz      . (6.2.8) The linearised observation equation for one epoch of star camera observations is thus ∆q[1:3]=0.5 ·I3×3∆α . (6.2.9) The full design matrix for the star camera observations is then ASCA=0.5 ·I3N×3N . (6.2.10) For optimal results in the data combination, a priori covariance information for the SCA observations is introduced. According to Romans (2003), one GRACE star camera head has the nominal noise characteristic Σ=σ2 ·    1 0 00 1 0 0 0 κ2    , (6.2.11) expressed in the respective star camera frame. The noise for rotations about the image plane axes is at the same level, with the noise for rotations about the boresight axis worse by a factor of κ≈ 8. The a priori error of the image plane rotations is given as σ≈ 6′′. The orientation of the star camera heads in the SRF is known through calibration and delivered in the QSA record of the GRACE sequence of events file. Further, the SCA1B data contains information on which SCA head was active for a particular epoch. This allows for variance propagation of the a priori covariance matrices to the SRF and their combination. This combined covariance matrix is then introduced as the observation weight for the SCA observation in the SCA/ACC sensor fusion. 6.2.3 Accelerometer Observations The angular acceleration observations of the accelerometer are directly linked to the Euler angles in the SRF through double differentiation ω˙(t)= ∂2α(t) ∂t2 . (6.2.12) Chapter6 ITSG-Grace201646