Page - 80 - 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

Use can however be made of the iterative nature of the dynamic orbit integration. An alternative quality check can be derived from the position corrections applied at each iteration l to give the new positions at iteration l+1. These are the∆re from eq. (5.2.26). Given a perfect dynamic orbit integration process consisting of a correct orbit integration implementation and error-free data, it is expected that the∆re must grow smaller with each iteration. The integrated dynamic orbit approaches the true satellite orbit, and the difference between subsequent iterations of orbits must thus vanish: lim l→∞ ∆re=0 . (7.1.1) For all practical considerations, constraints such as flawed algorithms or the limited precision of computations set a lower bound for the achievable repeatability of the orbit integration. This precludes the differences in eq. (7.1.1) from disappearing entirely. Instead, even as more iterations of computation are performed, the differences∆re stop to grow smaller after some computational threshold is reached. This convergence limit can be used as a benchmark to test the quality of orbit determination strategies both amongst different implementation and on their own merit, always given the same input data. The better the algorithm is designed and implemented, the smaller the ultimate limit of the position differences becomes. Figure 7.1 shows the convergence in terms of∆re for 100 iterations of computation, based on simulated data. The exact specifications of the simulation are given later in section 7.3. At this point the magnitude of the remaining variability between iterations is of primary interest. It can be clearly seen that even after convergence,∆re does not drop much below 10µm. As the expected ranging accuracy of the GRACE-FO LRI is expected to be smaller than 100nm (Heinzel et al., 2012), it is worth investigating avenues to improve on this result. 1 10 100 Iteration l 10−5 10−3 10−1 101 Figure 7.1: Unsatisfying convergence of dynamic orbit integration after several iterations l, expressed as the RMS of ‖∆re‖. From simulated data. Chapter7 Numerical Optimization in Orbit Integration80