Page - 87 - 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

needed. This is the state transition matrix of the equinoctial elements Φe= [ Φe,r Φe,r˙ ] =     ∂r ∂ξe ∂ r˙ ∂ξe     . (7.2.32) Similarly to the state transition matrixΦ introduced in chapter 5,Φe describes the change in the position and velocity of a spacecraft due to a change in the equinoctial el- ements describing its orbit. This state transition matrix contains exactly the observation equations needed to determine a best-fit equinoctial orbit ξˆe that satisfies eq. (7.2.25) in a linearised least squares adjustment. Danielson et al. (1995) gives the derivatives needed to compute eq. (7.2.32) in a clear and concise formalism. Beware however of wrong partial derivatives of the equinoctial element a with regard to Cartesian position and velocity as given by Danielson et al. (1995, section 2.1.6, eqs. 2 and 4). Comparison with Broucke and Cefola (1972) gives the correct partials in Danielson et al.’s notation. These are ∂r ∂a = 1 a · ( r− r˙3t 2 ) and ∂r˙ ∂a =− 1 2a · ( r˙−GM 3r‖r‖3 · t ) . (7.2.33) In this work, one equinoctial best-fit orbit is determined for each 24h variational orbit arc. It has proven unnecessary to introduce positions and velocities from all 17280 epochs in the orbit arc as observations. Instead, only positions from up to 100 epochs are used. The algorithm to determine the best-fit orbit is: Select observations: Select N = 100 epochs from the orbit arc. Start with the first epoch, then select epochs spaced at regular intervals from the remaining arc, giving even coverage of the observations. These positions are inserted into the observation vector l= [ r(τ1) T · · · r(τN)T ]T (7.2.34) Compute approximate solution: The initial guess for the best-fit equinoctial elements ξe,0 is taken to be the osculating orbit at the first epoch. Reduced observations: Compute the unperturbed equinoctial orbit l0= [ rref(τ1) T · · · rref(τN)T ]T (7.2.35) and then∆l= l−l0. Observation equations: The observation equation system is A=          ∂r(τ1) ∂ξe ... ∂r(τN) ∂ξe          =     Φe,r(τ1) ... Φe,r(τN)     (7.2.36) 7.2 Improved Algorithm 87