Page - 119 - 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

Co-Estimation of Orientation Parameters 9 The full orientation covariance matrix from the SCA/ACC sensor fusion not only allows for the disentanglement of the AOC residuals from the stationary ll-SST resid- uals. Having a complete stochastic model of both observation types — ll-SST KBR observations as well as the orientation observations — there is no longer a need to consider the orientation of the spacecraft as fixed. This chapter will describe the process of co-estimating improved orientations of the GRACE satellites at each epoch, together with the Stokes coefficients, in one least squares adjustment. Where the previous chapter only focused on the effects of the ori- entation uncertainty on the derived antenna offset correction, the following pages will outline a strategy of directly targeting improvements in the original noisy observations from the SCA/ACC sensor fusion. The assumption of a fixed, non-stochastic, perfectly observed orientation must of course introduce errors into the recovered gravity field. An attempt was made to model these errors in chapter 8 by describing the uncertainty in the AOC. Taking this thought to its logical conclusion, estimation of an improved “best-fit” satellite orientation will further allow for the computation of an improved AOC, reducing the resulting error in the recovered gravity field. The complete ll-SST observable and the estimated Stokes coefficients depend on the AOC. The ultimate independent variable for the AOC is the spacecraft orientation. To properly model this dependency, an algorithm must be employed that allows for variations in both the dependent and independent variables. Many such approaches exist, known by several names. Amongst them are total least squares, error-in-variables, the generalized case of adjustment theory, mixed model, or Gauß-Helmert model (see e.g. Amiri-Simkooei and Jazaeri, 2012; Golub and van Loan, 1980; Koch, 1997; L. Lenzmann and E. Lenzmann, 2004; Niemeier, 2008; Reinking, 2008; Schaffrin, 2007; Snow, 2012). In essence, these algorithms describe the same approach: the linearisation of the functional relationship is not only computed about the Taylor point for the unknownsx0, but also about approximate values for both dependent and independent observables l0. Both parameters and observables are then improved iteratively. This chapter will start by presenting the theoretical basis of one such algorithm, the total least squares (TLS) algorithm as outlined by Reinking, 2008. This approach will be contrasted to a formulation of the problem in the classical Gauß-Markov apparatus. The practical constraints in implementing a TLS algorithm for GRACE gravity field recovery will be enumerated, leading to a summary of the strategy employed to reprocess the GRACE time series. 119