Page - 55 - 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

Integration of the Parameter Sensitivity Matrix The parameter sensitivity matrix is integrated for each spacecraft i = A,B using eq. (5.3.13) with Zp,i(τ)= [ 0 0 Jgrav,i(τ) Jsat,i(τ) ] (6.4.19) and an integration polynomial of degree 7, yielding — for each satellite — the complete variational equations Ωi= [ Sgrav,i Ssat,i Φi ] . (6.4.20) The variational equations are now available for all epochs, containing both the deriva- tives w.r.t. the position and the derivatives w.r.t. the velocity: Ωi(τ)= [ Srgrav,i(τ) S r sat,i(τ) Φ r i (τ) Sr˙grav,i(τ) S r˙ sat,i(τ) Φ r˙ i (τ) ] . (6.4.21) 6.4.2 Low-Low SST Parameters In ITSG-Grace2016, the KBR antenna phase centre coordinates for each spacecraft are estimated as constants, once per month. The observation equations are computed by taking the derivative of the ranging equation eq. (4.3.7) w.r.t. the KBR APC coordi- natescA andcB. As the APCs only occur in the antenna offset correction, the derivative for an APCci is ∂ρKBR ∂ci = ∂ ‖u+v‖ ∂ci . (6.4.22) The chain rule applies, with ∂ ‖u+v‖ ∂ci = ∂ ‖u+v‖ ∂v ∂v ∂ci . (6.4.23) Writing ‖u+v‖= √ 〈u,u〉+〈v,v〉+2 · 〈u,v〉 , (6.4.24) the first partial in eq. (6.4.23) is ∂ ‖u+v‖ ∂v = 1 2 ·‖u+v‖ ∂ ∂v (〈u,u〉+〈v,v〉+2 · 〈u,v〉) = 1 ‖u+v‖ (u+v) T . (6.4.25) 6.4 Functional Models 55