Seite - 34 - 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

conservative forces at each epoch: r¨conse (τ)=f(τ,re(τ),p) (5.2.1) As GRACE also provides direct accelerometer observations of accelerations due to non-conservative forces, these are added to the accelerations from conservative forces to give the complete approximate accelerations r¨e r¨e(τ)= r¨ cons e (τ)+ r¨ ACC(τ) . (5.2.2) Next, the definite integrals from eqs. (5.1.7) and (5.1.8) are computed for the complete arc using integration polynomials as introduced in section 2.7. This gives the integrated velocities and positions r˙inte (τ)=T ∫ τ 0 r¨e(τ ′)dτ′ (5.2.3) rinte (τ)=T 2 ∫ τ 0 (τ−τ′)r¨e(τ′)dτ′ . (5.2.4) Back-substituting these quantities into eqs. (5.1.7) and (5.1.8) yields the integrated equations of motion r˙ dyn e (τ)= r˙0+ r˙ int e (τ) (5.2.5) r dyn e (τ)=r0+τT·r˙0+rinte (τ) , (5.2.6) whererdyne and r˙ dyn e are the dynamic orbit computed from the initial approximation of the orbitre and r˙e. The dynamic orbit must now be fixed in space by determining its initial state. To this end, the approximate state transition matrix is computed as Φ¯(τ)= [ Φ¯r(τ) Φ¯r˙(τ) ] = [ 1 τT 0 1 ] (5.2.7) by taking the partial derivative of eqs. (5.2.5) and (5.2.6) with regard to the initial statey0. Although strictly speaking both the position and velocity components must be introduced as observations in the determination of a rigorous least-squares estimate of the initial state yˆ0, empirical tests show that it proves sufficient to use positions only at this point. This allows one to neglect the velocity components of the approximate and dynamic orbit in the observation equation system, reducing the complexity of the problem by a small margin. Rearranging eq. (5.2.6) and settingrdyne ! =re gives re−rinte = Φ¯ry0 , (5.2.8) which can directly be used to compute an estimate yˆ0 for the initial state that best fits the approximate positionsre in a standard least squares adjustment. The first complete approximate dynamic orbit is then ˆ¯y= Φ¯yˆ0+y int e . (5.2.9) This orbit is smooth due to its shape being defined by the integrated accelerations. Its absolute position in space is at this point fixed to be close to the initial approximate orbit as a result of adjusting the initial state yˆ0 with eq. (5.2.8). Chapter5 Variational Equations34