Page - 60 - 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

eq. (6.4.21). With this information, the appropriate row of matrices in eq. (6.4.32) can be determined by inserting the correct partials in eq. (6.4.38). One row of the design matrix, corresponding to one point in timeτ, for the monthly gravity field parameters from ll-SST observations is A grav,m sst = ( 1 ρ u˙T− ρ˙ ρ eT )( Srgrav,B−Srgrav,A ) +eT ( Sr˙grav,B−Sr˙grav,A ) , (6.4.50) again with the time τ omitted. The design matrices for the daily Stokes coefficients can be computed herefrom according to section 6.4.1. The design matrices w.r.t. to the satellite parameters for GRACE-A and GRACE-B are Asatsst,A= ( 1 ρ u˙T− ρ˙ ρ eT )(−Srsat,A)+eT(−Sr˙sat,A) , (6.4.51) Asatsst,B= ( 1 ρ u˙T− ρ˙ ρ eT )( Srsat,B ) +eT ( Sr˙sat,B ) , (6.4.52) and equivalently for the satellite states Astatesst,A= ( 1 ρ u˙T− ρ˙ ρ eT )(−Φrsat,A)+eT(−Φr˙sat,A) , (6.4.53) Astatesst,B= ( 1 ρ u˙T− ρ˙ ρ eT )( Φrsat,B ) +eT ( Φr˙sat,B ) . (6.4.54) Only the observation equations for the ll-SST parameters remain. The equations in section 6.4.2 were computed for range observations, while the observation equations for ITSG-Grace2016 are set up for range rates. The required range rate observation equations can be computed trough ∂ρ˙ ∂psst = ∂ρ˙ ∂ρ ∂ρ ∂psst . (6.4.55) As ρ˙= ∂ρ/∂τ, eq. (6.4.55) can be written as ∂ρ˙ ∂psst =D ∂ρ ∂psst , (6.4.56) withD a polynomial derivative matrix. In ITSG-Grace2016, a polynomial of degree 2 is used. The complete design matrix for the ll-SST parameters is (with eq. (6.4.30)) Asstsst=DJsst . (6.4.57) The reduced observation vector for the KBR observations ρ˙KBR is, from the ranging equation (eq. (4.3.7)), ∆lsst= ρ˙KBR+∆ρ˙AOC+∆ρ˙TOF+∆ρ˙Iono− ρ˙COM . (6.4.58) The sum ρ˙KBR+∆ρ˙TOF+∆ρ˙Iono is given in the level 1B KBR product. ρ˙COM is com- puted according to eq. (4.3.9). The derivative∆ρ˙AOC is computed from∆ρAOC using polynomial differentiation, as above, with orientations from the SCA/ACC sensor fusion. Chapter6 ITSG-Grace201660