Page - 38 - 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

below the main block diagonal. Some values appear above the main diagonal due to the evaluation of the integration polynomial at the central support. The matrix of Marussi tensors T is block-diagonal. Each block, one per epoch, has size 3×3. Overall, the MatrixI−KrT is asymmetric and only populated on or below the main block diagonal. An iterative solver, such as the biconjugate gradient stabilized method (BiCGSTAB) (van der Vorst, 1992), can be used to solve the equation system epoch by epoch. Due to the ability to programmatically exploit the special structure of the inverse in a tailored implementation of the solver, this is magnitudes faster and more efficient than direct inversion. This however comes at the cost of a loss of generality in the solver implementation. The complete state transition matrix for the velocityΦr˙ can be arrived at by similarly taking the derivative of eq. (5.1.7) with regard to the initial state. Again using the chain rule, this is ∂r˙(τ) ∂y0 = ∂r˙0 ∂y0 +T ∫ τ 0 ∂f(τ) ∂y0 dτ′ = ∂r˙0 ∂y0 +T ∫ τ 0 ∂f(τ) ∂r(τ) ∂r(τ) ∂y0 dτ′ . (5.2.32) Here, one can identify Φr˙= Φ¯r˙+T ∫ T 0 T(τ′)Φr(τ′)dτ′ = Φ¯r˙+Kr˙TΦr = Φ¯r˙+Kr˙Φr¨ , (5.2.33) directlyyieldingthedesiredresult fromthepreviouslycomputedstate transitionmatrix for thepositionsΦr. In thisequation, thestate transitionmatrix for theaccelerationsΦr¨ appears. This matrix can, in analogy to the previous steps, also be derived by taking the derivative of eq. (5.1.6) with regard to the initial state: ∂r¨(τ) ∂y0 = ∂f(τ) ∂y0 . (5.2.34) Application of the chain rule to the derivative of the force function yields ∂r¨(τ) ∂y0 = ∂f(τ) ∂r(τ) ∂r(τ) ∂y0 (5.2.35) in which one can identify the state transition matrix for the accelerations Φr¨=TΦr . (5.2.36) 5.2.4 Final Estimate All integrated positions and velocities up to this point were integrated from forces evaluated at the approximate positionsf(re), as determined in eq. (5.2.1). An updated Chapter5 Variational Equations38