Seite - 128 - 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

9.2 Application to GRACE GRACE KBR range-rate observations are described by the ranging equation (compare eq. (4.3.7)) f(x,α)= ρ˙KBR= ρ˙COM(x)−∆ρ˙AOC(x,α)−∆ρ˙TOF−∆ρ˙Iono+e . (9.2.1) Here, the dependent observable is the ll-SST KBR range rate ρ˙KBR. The independent observables are the satellite orientationαs for both GRACE spacecraft. The sought parameters x are those described in section 6.4: The force model parameters, the satellite parameters, and the ll-SST parameters. The observation equations for these parameters, as summarized in eq. (6.4.32), are unchanged. They are collected and linearised in the Taylor point for the parameters and the satellite orientation f0(x0,α0)= ρ˙COM(x0)−∆ρ˙AOC(x0,α0)−∆ρ˙TOF−∆ρ˙Iono . (9.2.2) The Taylor point for the force model parameters is given by the background models and the dynamic orbits integrated therein. The Taylor point for the satellite orientation is simply the observed satellite orientation as obtained from the SCA/ACC sensor fusion. The influence of the linearisation about the Taylor point on the time-of-flight correction and the ionospheric correction is omitted. The design matrix for the ll-SST observationsA is unchanged from the equations laid out in section 6.4, with the exception of the entries for the APC coordinates in eq. (6.4.30). The observation equations for the APC must now be linearised about the current Taylor point for the orientation at every iteration. The expanded observation vector is l=    ρ˙KBRαA αB    . (9.2.3) TheJacobianof theAOCintherangeratedomainw.r.t. theorientationofonespacecraft was previously computed in chapter 8. WithD the polynomial differentiation matrix andFsα= ∂∆ρ s AOC/∂αs from eq. (8.1.21), the full Jacobian for both the ll-SST and the orientation observations in the Reinking model (compare eq. (9.1.23)) is F= [ I −DFAα −DFBα ] . (9.2.4) With the full covariance matrix Σll=    Σsst 0 00 ΣAαˆαˆ 0 0 0 ΣBαˆαˆ    (9.2.5) Chapter9 Co-Estimation of Orientation Parameters128