Page - 36 - 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

Taking the Taylor expansion of the acting forces f(r) up to the linear term, and evaluated at the approximate positionre as the Taylor point, gives f(r)=f(re)+∇f|re ·(r−re) . (5.2.18) Inserting eq. (5.2.18) into eq. (5.2.17) gives rdyn−rdyne =Kr [ f(re)+∇f|re ·(r−re)−f(re) ] =Kr [ ∇f|re ·(r−re) ] (5.2.19) Here,∇f is the Marussi tensor, or gravity tensor. The Marussi tensor is populated with gravity gradients, the second derivative of the force-generating potential. With a Matrix T=     ∇f(re(τ1)) 0 ... 0 ∇f(re(τn)))     , (5.2.20) containing the Marussi tensors for all epochs of the orbit arc, eq. (5.2.19) can be written as rdyn−rdyne =KrT(r−re) . (5.2.21) Given a correct implementation, the estimated position can be seen as an approxi- mation for the true position, or in other terms rdyn !=r. Inserting the approximate estimaterdyne of the position from eq. (5.2.14) into eq. (5.2.21) gives r−Φ¯ry0−Krr¨e=KrT(r−re) . (5.2.22) Reducing both sides of this equation by the approximate positionre and some reorder- ing gives r−re−KrT(r−re)= Φ¯ry0+Krr¨e−re , (5.2.23) whichcanbesolvedfor thecoordinatedifferencebetweenthe trueandtheapproximate positions∆re=r−re: [I−KrT](r−re)= Φ¯ry0+Krr¨e−re (5.2.24) ∆re=(r−re)= [I−KrT]−1 [Φ¯ry0+Krr¨e−re] . (5.2.25) ∆re is an estimate of the linearisation error made in the dynamic orbit integration due to the initial evaluation of the accelerations from background models atre instead of the true positionr. Using this estimate, the position of the spacecraft along the arc can then be updated: r=re+∆re (5.2.26) Chapter5 Variational Equations36