Page - 61 - 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

6.5 Fit of Stochastic Model To compute an optimal least squares solution of the daily and monthly Stokes coef- ficients, the stochastic characteristics of the observation data must be considered. In ITSG-Grace2016, four observation groups are present: the hl-SST POD observations for both GRACE-A and GRACE-B, the ll-SST KBR observations, and the pseudo- observations used to constrain the daily Stokes coefficients. Each observation group has associated residuals     ∆lpod,A ∆lpod,B ∆lsst lgpm      =Ax+      epod,A epod,B esst egpm      (6.5.1) which are distributed according to their respective covariance matrices     epod,A epod,B esst egpm      ∼N      0,      ΣpodA 0 0 0 0 ΣpodB 0 0 0 0 Σsst 0 0 0 0 Σgpm           . (6.5.2) In ITSG-Grace2016, it is assumed that the observation groups are not cross-correlated. Only the structure of the covariance matrix due to the geophysical modelQgpm is known a priori, the remaining covariance matrices are determined in ITSG-Grace2016 through variance component estimation. Thestochasticmodel isre-estimatedforeachgeneratedmonthlysolutioninamulti-step process. To this end, a GRACE monthly gravity field solution of decreased degree and order of only 60 is computed while iteratively adjusting the weights of the observation groups and refining the covariance structure of each observation group. The gravity field solution is computed at a reduced fidelity to speed up the time-consuming iteration of the computation. 6.5.1 Toeplitz Covariance Structure The noise of the POD and KBR observations is assumed to be the result of a wide-sense stationary process, meaning that the autocovariance function of the noise signal does not vary with time, or in the case of ITSG-Grace2016, within one month. Such a process can be fully described by its autocovariance function (Etten, 2006) Cxx(t1,t2)=Cxx(t2− t1)=Cxx(∆t) . (6.5.3) This covariance function is estimated once per observation type and month. The re- estimation for every month implies that the estimated noise model is stationary for 6.5 Fit of Stochastic Model 61