Page - 58 - 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

Observation Equations for hl-SST The observable in hl-SST POD is the spacecraft position. The equations describing the dependence of the spacecraft position at timeτon the sought parametersxare exactly the variational equations. In ITSG-Grace2016, only the spacecraft position is used as an observable for POD. The velocity is not used, as it is only derived from the POD positions, not determined directly. The linearised observation difference is then the POD positionrpod reduced by the dynamic orbit positionrdyn: ∆lpod=rpod−rdyn (6.4.33) The design matrix for the monthly gravity field parameters for each spacecraft i is simply the appropriate parameter sensitivity matrix A grav,m pod,i =S r grav,i . (6.4.34) The design matrices for the daily Stokes coefficients can be computed herefrom accord- ing to section 6.4.1. The design matrices for the satellite parameters are similarly Asatpod,i=S r sat,i . (6.4.35) The design matrices for the satellite states are the state transition matrices Astatepod,i=Φ r i . (6.4.36) Observation Equations for ll-SST In ITSG-Grace2016, ll-SST observations are processed at the range rate level. The range rate for the satellites COMs is given in eq. (4.3.8) as ρ˙COM(τ)= 〈e(τ),u˙(τ)〉=eT(τ)u˙(τ)= 1‖u(τ)‖u T(τ)u˙(τ) . (6.4.37) The range rate depends on both the positions of the spacecraft, as components ofu, and their velocities, as components of u˙. For brevity, the index for the COM is omitted in the following derivations, while understanding that all ranges and range rates refer to the spacecraft COM. Similarly, the time index is also dropped. The derivative of the range rate w.r.t. to some parametersx is ∂ρ˙ ∂x = ∂ρ˙ ∂u ∂u ∂x + ∂ρ˙ ∂u˙ ∂u˙ ∂x . (6.4.38) The derivative of the range rate w.r.t. the satellite baseline is ∂ρ˙ ∂u = ∂ ∂u 〈e,u˙〉= ∂ ∂u eTu˙= ∂ ∂u ( 1 ‖u‖u Tu˙ ) (6.4.39) Chapter6 ITSG-Grace201658