Page - 40 - 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

is introduced. Taking the derivative of eq. (5.3.2) w.r.t. the initial state gives ∂ y˙(τ) ∂y0 = ∂z(τ) ∂y0 = ∂z(τ) ∂y(τ) ∂y(τ) ∂y0 . (5.3.3) Introducing the matrix Zy= ∂z(τ) ∂y(τ) (5.3.4) eq. (5.3.3) can be compactly written as Φ˙=ZyΦ , (5.3.5) the differential equation of the state transition matrix. Similarly, taking the derivative of eq. (5.3.2) w.r.t. the force model parameters gives, using first the product rule and then the chain rule, ∂ y˙(τ) ∂p = ∂z(τ) ∂p = ∂z(τ) ∂p + ∂z(τ) ∂y(τ) ∂y(τ) ∂p . (5.3.6) Note that z(τ) is a function of p both directly through f(τ,r(τ),p, . . .), as well as indirectly through the satellite statey(τ), which also depends onp. Introducing the matrix Zp= ∂z(τ) ∂p (5.3.7) eq. (5.3.6) can be compactly written as S˙=ZyS+Zp , (5.3.8) the differential equation of the parameter sensitivity matrix. Looking closer, Zp(τ)=      ∂ r˙(τ) ∂p ∂ r¨(τ) ∂p      =     0 ∂f(τ) ∂p     , (5.3.9) which can be computed for all times τ. The inhomogeneous differential equation system formed by eqs. (5.3.5) and (5.3.8) −ZyΦ+Φ˙=0 (5.3.10) −ZyS+S˙=Zp (5.3.11) can be solved forS(τ) through the approach of variation of constants, yielding S(τ)=−Φ(τ) [∫ τ 0 Φ−1(τ′)Zp(τ′)dτ′+C ] . (5.3.12) The integration constant can be fixed toC=0due to eq. (5.3.1), resulting in S(τ)=−Φ(τ) ∫ τ 0 Φ−1(τ′)Zp(τ′)dτ′ , (5.3.13) Using this equation, the parameter sensitivity matrix for a complete orbit arc can be computed through integration from a known start value and the known quantitiesΦ andZp. Chapter5 Variational Equations40