Page - 56 - 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

The second partials in eq. (6.4.23) are ∂v ∂cA =−RA and ∂v ∂cB =RB . (6.4.26) Inserting into eq. (6.4.23) gives ∂ρKBR ∂cA =−(u+v) T ‖u+v‖RA , (6.4.27) ∂ρKBR ∂cB = (u+v)T ‖u+v‖RB , (6.4.28) which are the observation equations for the 6 ll-SST parameters per month: psst= [ cA cB ] . (6.4.29) The complete Jacobian for the ll-SST parameters is then simply Jsst= [ ∂ρKBR ∂cA ∂ρKBR ∂cB ] . (6.4.30) 6.4.3 Observation Equation System With all Jacobians available, the observation equation system for the GRACE hl-SST and ll-SST observations can be set up. The observation equations are set up for each short arc m ∈ [1,M], using the variational equations integrated for each day of a month k∈ [1,K]. The combined parameter vector for one short arc is xm= [ pgrav,d p k sat,A p k sat,B psst y k 0,A y k 0,B pgrav,m ]T . (6.4.31) It is important to note that the estimated parameters can be the same for multiple short arcs. The monthly Stokes coefficients, for example, are only estimated once per month. The parameterspksat,A,p k sat,B,y k 0,A, andy k 0,B are estimated once per day k, and are the same for all short arcs m in this day. This must be considered when accumulating the normal equations from the individual short arcs. The peculiar ordering of the parameters, with the daily Stokes coefficients at the beginning of the parameter vector, and the monthly Stokes coefficients at the end, is due to some optimizations that can then be employed in eliminating the non-target parameters, namely all butpgrav,m, from the resulting normal equation system. These optimizations are described in Kvas (2014). In ITSG-Grace2016 three observation groups are processed in the LSA for gravity field determination: The spacecraft positions from kinematic orbits as observed by the GPS, the ll-SST KBR range-rate observations, and the pseudo-observations constraining the daily gravity fields. Starting from the sought parameters and the known observations, Chapter6 ITSG-Grace201656