Page - 39 - 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

estimate for these accelerations can be computed using the estimated coordinate differ- ence∆re from eq. (5.2.25) and the linearisation of the force function from eq. (5.2.18). These are the corrected accelerations r¨c= r¨e+T∆re . (5.2.37) These corrected accelerations can then be integrated to corrected velocity and position components using the previously defined integral kernels with r˙intc =Kr˙r¨c (5.2.38) rintc =Krr¨c . (5.2.39) A new estimate for the initial state can be determined by repeating the steps from eq. (5.2.8), but now making use of the knowledge of the complete state transition matrix for the positionsΦr. One arrives at the system re−rintc =Φry0 (5.2.40) from which yˆ0 can be computed. This newly estimated initial state is then used to fix the positions and velocities of the spacecraft from the reintegrated accelerations, giving the final dynamic orbit: r˙=Φr˙yˆ0+ r˙ int c (5.2.41) r=Φryˆ0+r int c . (5.2.42) As this algorithm relies heavily on linearisations, all steps from eq. (5.2.1) to eq. (5.2.42) must be repeated. For this re-computation the final dynamic orbit from eqs. (5.2.41) and (5.2.42) is now re and r˙e, used to evaluate the background model and fit the initial state vectors. This is repeated until convergence is achieved. Dynamic orbit computation following this schema is thus an inherently iterative process. 5.3 Parameter Sensitivity Matrix WithΦnow known, the parameter sensitivity matrixS can be determined. With the P force model parametersp, e.g. the Stokes coefficients of a monthly mean potential,S describes the sensitivity of the dynamic orbit to changes in those coefficients δp. Looking at just one orbit arc, a change of the force model parameters only influences all epochsτ>0, but not the initial state of the satellitey0, as the integrals in eqs. (5.1.7) and (5.1.8) are 0 for τ= 0. The initial value of the parameter sensitivity matrix is simply a matrix of zeros S(0)=06×P . (5.3.1) To determine the value ofS for the remaining epochs, the derivative of the satellite state y˙(τ)=z(τ) (5.3.2) 5.3 Parameter Sensitivity Matrix 39