Page - 35 - 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

5.2.2 Refinement In the determination of the first approximate dynamic orbit, the acting forces due to the background models were evaluated at the approximate positions re, not the true positions of the satelliter. This flaw leads to the derived accelerations deviating from the true accelerations by some amount. In turn, the computed positions rdyne also deviate from the true positions. Again evaluating the accelerations at the computed positions must thus lead to accelerations different from those first evaluated at the original approximate positions — the orbit is self-consistent neither in positions nor in accelerations. Mayer-Gu¨rr (2006, section 4.2.4.3) describes a strategy for treating this problem through an iterative approach, but in the context of phrasing the dynamic orbit integration as a boundary value problem. The same approach can be applied to the formulation as an initial value problem used here, with the equivalent apparatus outlined in the following paragraphs. Two operators for the definite integrals used in the integration of both the spacecraft velocities and positions are introduced as κr˙(τ)=T ∫ τ 0 (·)dτ′ (5.2.10) κr(τ)=T2 ∫ τ 0 (τ−τ′)(·)dτ′ . (5.2.11) Phrasing the integrals in terms of polynomial integration, as introduced in section 2.7, the operators κr˙(τ) and κr(τ) can be discretised and written as linear operator matri- cesKr,Kr˙. With these integral operator matrices, eqs. (5.2.3) and (5.2.4) can be written as r˙inte =Kr˙r¨e (5.2.12) rinte =Krr¨e , (5.2.13) with r¨e a vector of all accelerations along the orbit arc and rinte and r˙inte the integrated positions and velocities. Symbolically, the difference between a hypothetical perfect and the actual computed dynamic orbit can be determined by writing eq. (5.2.6) twice, once with the (unknown) true position r as input, and once with the approximate positionsre: r dyn e = Φ¯ry0+Krr¨e (5.2.14) rdyn= Φ¯ry0+Krr¨ (5.2.15) Taking the difference of eq. (5.2.14) and eq. (5.2.15) yields rdyn−rdyne =Kr(r¨− r¨e) . (5.2.16) The equation of motion eq. (5.1.5) states that the accelerations acting on the spacecraft are a function of the force f(r). Making this substitution, eq. (5.2.16) can also be written as rdyn−rdyne =Kr [f(r)−f(re)] . (5.2.17) 5.2 Orbit Integration and State Transition Matrix 35