Page - 94 - 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

7.3.5 Propagation to Ranging Measurement The final integrated dynamic GRACE orbits are used at many steps in gravity field determination, as laid out in chapter 5. In eq. (6.4.58) a linearisation of the GRACE ll-SST KBR observations is computed from the dynamic orbits using eq. (4.3.9). Here, bothorbitsrA andrB forGRACE-AandGRACE-Bareused.Theimpactof thedynamic orbit noise on the term used to reduce the ranging measurement can be computed from eq. (4.3.2), ρCOM=‖u‖=‖rB−rA‖ . The norm of the position difference is ‖rB−rA‖= √ ∆x2+∆y2+∆z2 (7.3.1) regardless of the choice of reference frame. Let both rA and rB be given in the SRF of either satellite, for example arbitrarily GRACE-A. Then x is approximately the along-track, y the cross-track, and z the radial difference in position. From eq. (7.3.1), simple error propagation gives σ2ρCOM=2 [( ∆x ‖u‖ )2 σ2x+ ( ∆y ‖u‖ )2 σ2y+ ( ∆z ‖u‖ )2 σ2z ] . (7.3.2) With x being the along-track axis, the partials can be approximated with ∆x ‖u‖ ≈1 (7.3.3) and ∆y ‖u‖ ≈ ∆z ‖u‖ ≈0 . (7.3.4) This gives the standard deviation of the derived baseline as σρCOM= √ 2σx . (7.3.5) The uncertainty of the orbit in the along-track axis due to the integration algorithm is given by the PSDs displayed in fig. 7.6. After variance propagation with eq. (7.3.5), the PSDs can directly be differentiated in the frequency domain to compute the uncertainty of the reduction term in the range rate domain: σρ˙COM(f)=2pi f ·σρCOM(f) (7.3.6) The resulting PSDs are displayed in fig. 7.7. Here, the solid black line shows an error estimate for the GRACE KBR instrument. The dashed black line is an error estimate for the GRACE-FO laser ranging interferometer. All tested configurations lead to a Chapter7 Numerical Optimization in Orbit Integration94