Page - 120 - 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

Results of the performed reanalysis will be presented, focusing on time series of estimated Stokes coefficients, their formal errors, and derived quantities such as equivalent water heights. Monthly estimates for the GRACE antenna phase centre vectors will be given and contrasted with those from the ITSG-Grace2016 processing chain. The sections regarding Stokes coefficient and APC vectors will include results from the processing chain including AOC covariance matrices only, as described in chapter 8. 9.1 Uncertainties in Independent Variables This section will outline two formalisms that can be used to treat the problem of inde- pendent variables of a stochastic nature. Their outlines will be given in a generalized notation, before the application to GRACE processing is discussed in section 9.2. For both formulations, there shall be two sets of observations, a set of independent obser- vations lind, and a set of associated dependent observations ldep. These observations are used to estimate some parameters xˆ. The classical example for such a configuration is the estimation of a straight line, with the independent variable the ordinate of an observed point xn, the dependent variable the coordinate of the point yn, and the sought parameters the intercept and slope of the line. The functional model connecting the observation groups and the parameter vector is ldep=f (lind,x)+edep , (9.1.1) which in the classical linearisation in the Gauß-Markov model according to eq. (2.2.1) would lead to the observation equations ldep−f (lind,x0)= ∂f (lind,x)∂x ∣∣∣∣ x0 (x−x0)+edep (9.1.2) or ∆ldep=A∆x+edep . (9.1.3) The residuals in eq. (9.1.3) are attributed exclusively to a misfit in the dependent observations ldep. The uncertainty in the independent variables is not considered in this formalism. The fundamental idea behind the approach commonly termed TLS is to introduce a second matrix of residualsEA as ∆ldep=(A+EA)∆x+edep . (9.1.4) The norm of bothedep andEA is then minimized together (see e.g. Golub and van Loan, 1980). Here,EA allows for variations inAdue to the uncertainty in lind. Chapter9 Co-Estimation of Orientation Parameters120