Page - 32 - 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

and the unknown force model parameters p gives the variational equationsΩ. The variational equations Ω(τ)= [ S(τ) Φ(τ) ] (5.1.11) are divided into the parameter sensitivity matrix S(τ)= ∂y(τ) ∂p =      ∂r(τ) ∂p ∂r˙(τ) ∂p      (5.1.12) and the state transition matrix Φ(τ)= ∂y(τ) ∂y0 =      ∂r(τ) ∂r0 ∂r(τ) ∂r˙0 ∂r˙(τ) ∂r0 ∂r˙(τ) ∂r˙0      . (5.1.13) The parameter sensitive matrixSdescribes the influence of a change δp in the force model parameterspon the satellite orbit. This could for example be due to a change in Earth’s gravitational potential, or due to a change in the relative position of a celestial body and a resulting disturbance to its tidal potential. The state transition matrixΦ in turn describes the influence of changes in the initial state δy0 on each epoch of the satellite orbit. The linearisation of the satellite statey about these parameters δp and δy0 is then y¯(τ)= y(τ)|p,y0+ [ S(τ) Φ(τ) ][δp δy0 ] . (5.1.14) The dynamic orbit is the zero order term of this expansion. The dynamic orbit for the entire arc τ ∈ (0,1] can thus be fixed by setting a set of initial conditions y0 and parameters p. Interestingly, the parameter sensitivity matricesS(τ), τ ∈ (0,1] can be computed through integration having knowledge of its initial stateS0 as well as of the full state transition matrixΦ for all epochs τ. This process is explained later in section 5.3. The full parameter sensitivity matrix can then be used to set up the observation equations for GRACE hl-SST and ll-SST observations, as described in section 6.4. 5.2 Orbit Integration and State Transition Matrix Inorder tosolve for thesoughtgravityfieldparameters in theparametersensitivityma- trixS, the state transition matrix for the complete orbit arc, and thus the dynamic orbit, must be known. These two quantities are computed together through integration. Chapter5 Variational Equations32