Page - 89 - 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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time [h] 0.001 0.01 0.1 1 10 100 Osculating Best-fit Figure 7.3: Encke ratio over one orbit arc for osculating reference orbit (in pink) and best-fit orbit (in green). The Encke ratio for the best-fit orbit is in general much smaller, never increasing much beyond 0.1%. The upper bound of 1% given by Lundberg, Bettadpur, and Eanes (2000) is never reached. The Encke ratio shows approximately time-symmetric behaviour, with the lowest ratio at the centre of the orbit arc, and the largest ratios at the beginning and end. In absolute terms, the largest observed deviation of the best-fit ellipse from the perturbed motion is 8.2km, a reduction of 99.3% from the osculating case. 7.3.2 Convergence As outlined in section 7.1, the convergence of the dynamic orbit solution can be taken as a benchmark indicating the correctness of the algorithm. To determine the number of iterations necessary for convergence to occur, the simulated orbit was deteriorated with Gaussian white noise. The standard deviation for the position component was set toσr=50m, that for the velocity component toσr˙=0.5m/s. The deteriorated orbit was inserted as the first approximate positionre into the orbit integration routine. Then, 100 iterations of integration and correction were computed for five configurations which differ in the choice and parametrisation of the refer- ence trajectory. The first is a configuration with no reference motion, or f0 = 0 in brown. The second pair consists of osculating reference ellipses, defined by∆y0=0, parametrised in either Kepler elements (pink) or equinoctial elements (purple). The last pair represents the best-fit ellipses with∆y0=∆yˆ0, also parametrised in either Kepler elements (green) or equinoctial elements (orange). Figure 7.4 shows the root mean square (RMS) of the corrections∆re applied for these five configurations in each iteration cycle. 7.3 Results 89