Page - 84 - 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

becoming large again, and in consequence to the loss of any numerical advantages attributed to the method. The separation of the reference motion and the perturbed motion is commonly quanti- fied in the Encke ratio e= ‖∆r‖ ‖r‖ , (7.2.24) the ratio of the magnitude of the Encke vector in relation to the magnitude of the position vector. A large Encke ratio indicates a relatively speaking large numerical inte- grand, and consequently the loss of the numerical precision associated with the Encke method. Lundberg, Bettadpur, and Eanes (2000) state that the general recommendation is to aim for e<1%. The general approach to treating large Encke ratios e is rectification. Rectification means that the integration is interrupted at a certain epoch and then continued from there using a newly defined reference trajectory. In essence, this implies restarting the orbit integrator with new initial values, which are defined by the last epoch of the previous integration arc. This entails possibly negative effects on precision of the orbit arc (Milani and Nobili, 1987). The new trajectory will however again have a small e, at least for some time until the deviation of the new reference trajectory from the true trajectory starts to grow again. The first efforts to reduce the Encke ratio for long arc orbit determination, or dynamic orbit integration in general, were based on the premise of considering the secular terms induced in the satellite motion by Earth’s oblateness in the reference force (Escobal, 1966; Kyner and Bennett, 1966). Closed equations exist for such a trajectory. The drift induced by Earth’s oblateness in some of the Kepler elements, notably the argument of perigee and the right ascension of the ascending node, contribute significantly to the deviation of the reference trajectory from the true trajectory, especially for longer arc lengths. Liu and Hu (1997) later focused on considering higher order terms of Earth’s potential, as well as higher-order secular terms, in the reference force. Lundberg, Bettadpur, and Eanes (2000) developed a long arc model that allows general variations in all six orbital elements, mentioning successful results with Encke ratios on the order of 10% to 20%. All of these studies have in common that they consider medium to high orbiting laser ranging satellites like the laser geodynamics satellite (LAGEOS) (Liu and Hu, 1997; Lundberg, Bettadpur, and Eanes, 2000; Lundberg, Schutz, et al., 1990) or the Satellite de Taille Adapte´e avec Re´flecteurs Laser pour les Etudes de la Terre (STARLETTE) (Lundberg, Bettadpur, and Eanes, 2000). The arc lengths considered in these works are on the order of multiple years or decades, not hours as is usual in GRACE processing. For the considered GRACE case of a low-earth orbiter with moderate arc lengths of at most 24h, a distinctly simpler and more elegant solution presents itself. Where the methods mentioned above consider ever-more precise refinements of the refer- ence force, the initial parameters are always kept to be those of an osculating ellipse for the reference epoch. For GRACE, a mean static Kepler ellipse, with no temporal Chapter7 Numerical Optimization in Orbit Integration84