Page - 62 - 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

one month only, and can differ between any two months. Let∆t be the sampling in seconds and N the number of epochs in the arc. The covariance function for the arc can then be discretised as Cnxx=Cxx(n ·∆t) , n∈ [0,N) . (6.5.4) The observations for the month have at this point been split into short arcs l= [ l1 . . . lM ]T , (6.5.5) each of length Nm≤Nmax. For one arc of Nm POD or KBR observations, the cofactor matrix then has a Toeplitz structure Qmll=         C0xx C1xx C2xx · · · CNm−1xx C1xx C0xx C1xx C Nm−2 xx C2xx C1xx C0xx C Nm−3 xx ... ... ... CNm−1xx CNm−2xx CNm−3xx · · · C0xx         (6.5.6) with the entries determined by the time lag and the autocovariance function of the noise signal from eq. (6.5.4). This cofactor matrix is identical for all arcs of the same observation type and the same length N. All cofactor matrices are slices of the longest possiblecofactormatrixofsize Nmax×Nmax.Theelementsof thediscretisedcovariance functionCxx completely define all of these cofactor matrices. This necessitates that the covariance function is also estimated for a length of Nmax, even though some arcs do not allow for the estimation of some of the longer time lags due to their shorter length Nm<Nmax. Additionally, a variance factor is computed per observation type for each of the M short arcs of the month, giving appropriate weights to the individual arcs. These are the arc-wise variance factorsσ2m, which together with the cofactor matrix give the complete covariance matrix for the arc Σmll=σ 2 m ·Qmll . (6.5.7) A simplifying assumption is made that observations are not correlated between short arcs. This then gives the following block-diagonal covariance structure for a complete month of observations in one observation group: Σll=       Σ1ll 0 · · · 0 0 ... ... ... ... ... ... 0 0 · · · 0 ΣMll       . (6.5.8) As observations are treated as uncorrelated across observation groups and across individual short arcs, the observation equations for each arc can be computed and decorrelated independently of each other, and finally be accumulated at the normal equation level to compute a gravity field solution. Chapter6 ITSG-Grace201662