Page - 29 - 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

Variational Equations 5 Attribution This chapter, as well as chapter 7 of this thesis focusing on orbit integration, are an extended version of a previous publication by the author: Ellmer and Mayer-Gu¨rr, 2017. Specifically, this chapter reproduces and expands on sections 1 to 3.5 of Ellmer and Mayer-Gu¨rr, 2017. These are the sections giving background on the state-of-the-art of dynamic orbit integration at IfG, as imple- mented mainly by Torsten Mayer-Gu¨rr. Section 5.3 does not appear in Ellmer and Mayer-Gu¨rr, 2017, and is first published in this work. Determining Stokes coefficients from GRACE data requires a functional model con- necting the target parameters to the satellite observations. In ITSG-Grace2016 and this work, variational equations (Beutler and Mervart, 2010; Montenbruck and Gill, 2000) are employed to set up the functional model. This approach combines the integration of the spacecraft’s dynamic orbit with the set-up of observation equations for Stokes coefficients or other sought parameters in a numerically efficient procedure. Where kinematic orbits represent discrete epoch-wise position solutions for the space- craft, dynamic orbits are computed for a complete orbit arc through integration of the accelerations acting on the spacecraft. Positions at a later epoch thus implicitly depend on the spacecraft position at an earlier epoch. Direct position observations of the spacecraft, such as from GPS, may be used to evaluate background force models to yield the accelerations to be integrated, but are not used directly in the computation of the dynamic orbit. Due to the integral nature of the orbit, the spacecraft trajectory is smooth. On account of unavoidable approximation errors, such as from evaluating flawed background models, this smooth trajectory does usually not directly coincide with the true spacecraft position. The divergence increases with growing arc lengths. The smooth dynamic orbit should thus also be fitted to more accurate but less precise observations, such as from GPS (Zehentner and Mayer-Gu¨rr, 2016), to ensure that it is localized correctly. This determination of dynamic orbits in an integrate-and-fit procedure is an essential component in computing gravity field solutions from GRACE satellite-to-satellite tracking observations. Theaccelerationsthatare integratedtoyieldadynamicorbitcanbebroadlycategorized into the two groups described in chapter 3: Accelerations due to conservative forces and accelerations due to non-conservative forces. As conservative forces act on all masses of the spacecraft equivalently, they can not be measured by the spacecraft directly. They must be derived from a background model using some approximate position information. This approximation introduces an error in the integrated orbit which can be treated by iterating the integration procedure, using the resulting orbit 29