Page - 64 - 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

specific time lag. LetVn be the Nmax×Nmax identity matrix for n= 0, and a zero matrixwithonlythe n-th lowerandupperdiagonalonesotherwise,e.g. for Nmax=5 V2=         0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0         . (6.5.16) The cofactor matrix eq. (6.5.6) is then Qmll= Nm−1 ∑ n=0 Cnxx ·Vn . (6.5.17) Each value of the discretised covariance function corresponds to one specific covariance componentVn, and as such they can be separated using VCE. The arc-wise variance factors, the variance factor determining the weight of one individual short arc of one observation type w.r.t. all other arcs of that observation type, depend on the sum of all individual cofactor matrices associated with all time lags: Σmll=σ 2 m · Nm−1 ∑ n=0 Cnxx ·Vn (6.5.18) To fit the stochastic model to the observation noise, the amplitude of the covariance function for each time lag Cnxx must be determined. If the covariance function were estimated directly in the time domain, the value for one specific time lag Cnxx could however never change its sign, as the appropriate variance factor is always estimated as its square (compare eq. (2.6.8)). As the PSD is by definition always positive, this restriction does not impact its estimation. Negative values of the covariance function can then occur in the inverse DCT that give the covariance function from the estimated PSD. Thus, in ITSG-Grace2016, the PSD is estimated instead of the covariance function, even though the two are functionally interchangeable. This can be achieved in the spectral domain by determining the amplitude of the PSD at each frequency Sjxx. Using VCE, the estimated amplitude for each frequency j∈ [0,N) can be written as Sˆjxx= αˆ2jS j xx , (6.5.19) with Sjxx an a priori value for the amplitude of the PSD at that frequency, and αˆ2j = Ω˜j s˜j . (6.5.20) As the time lag of the covariance function, or conversely the frequency of the PSD, can appear in any of the covariance matrices Σmll of the M short arcs, depending Chapter6 ITSG-Grace201664