Page - 72 - 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

be unstable, due to the quotient eq. (6.5.19) becoming ill-defined for some values of Ω˜0 and s˜0. To work around this issue in ITSG-Grace2016, the zero frequency amplitude was set to the same value as that of the lowest non-zero frequency of the estimated power spectral density. 6.5.7 Summary The input to the algorithm for the estimation of the stochastic model are the observa- tionsforhl-SST, lpod,A, lpod,B, andthell-SSTobservations lsst.Further,anapproximation of the stochastic model for these observation types is needed. If no information is available, white noise can be assumed, with Sjxx=1 , j∈ [0,Nmax) and σ2m=1 , m∈ [1,M] . (6.5.62) In addition, the cofactor matrix for the geophysical process modelQgpm is needed, the scale of which can also be assumed to beσ2gpm=1. With this data in place, the determination of the stochastic model can begin. The iteration roughly follows the scheme presented in fig. 6.7, where the location of the individual steps in the following algorithm are marked with their respective Arabic numerals. To determine the stochastic model, for each arc and observation group 1. compute∆l,A, using the functional models from section 6.4 and the parametriza- tion described in table 6.2, but only up to degree and order 60. 2. compute the covariance matrix Σ using eq. (6.5.62), and store the Cholesky decompositionΣ=WTW . 3. decorrelate thereducedobservationvectorandtheDesignmatrixusingeq. (2.3.4), giving∆l¯, A¯. 4. compute and accumulate the normal equationsN and right hand sidesn. 5. compute the Cholesky decompositionN=UTU, and solve for∆xˆ. 6. create a matrix of Monte Carlo vectors Z¯, and computeU−1Z¯. 7. compute the decorrelated residuals as ˆ¯e=∆l¯−A¯∆xˆ. 8. compute R˜, e˜, and, using them, Ω˜ and s˜. 9. compute updated estimates of Sˆjxx, σˆ2m for all observation groups, as well as σˆ2gpm. 10. if thesequantitieshavenotsufficientlyconverged,computeanupdatedcovariance matrix Σˆ and its Cholesky decomposition, continue from the 2nd step. Figure 6.8 gives an example of the estimated PSDs Sˆxx and the arc-wise variance factors σ2 for the month of June 2010. Figure 6.8a shows the estimated PSD of the ll-SST observation noise, derived from the KBR residuals. The PSD exhibits a structure typical for GRACE ll-SST data. The noise spectrum shows an ascending branch above 1 ·10−2Hz, which is due to the processing of the KBR data in the range rate domain. As the range rates are derived from the observed biased ranges through differentiation, noise at higher frequencies is amplified, while noise at lower frequencies is damped. At frequencies below 1 ·10−2Hz, noise due to a combination of ACC observation errors and residual geophysical signals dominates. Both the ACC observations, and Chapter6 ITSG-Grace201672