Page - 71 - 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

Arc-wise Variance Factors As described in section 6.5.4, the arc-wise variance factors are normalized to remove a double-consideration of the amplitude determination. To this end, the mean of the arc-wise variance factors is determined in eq. (6.5.29). In the implementation in GROOPS, this mean is not the direct arithmetic mean of all variance factors. Instead, the mean is computed robustly by only employing the central half of the sorted values, ignoring outliers at either tail of the distribution. With theγm sorted according to their magnitude, the mean is then γ¯= 2 M 3 ·M/4 ∑ m=M/4 γm . (6.5.57) Variance Factor for Geophysical Pseudo-Observations The variance factor for the geophysical pseudo-observations σˆ2gpm from eq. (6.5.31) is computed using eq. (2.6.12). As the pseudo-observations lgpm are0, the residuals are directly eˆgpm=−Agpmxˆgpm . (6.5.58) Dropping the index for the geophysical process, the square sum of residuals is Ω= eˆTΣ−1eˆ= xˆTATΣ−1Axˆ= xˆTNgpmxˆ . (6.5.59) The redundancy is s=Ngpm− trace ( NgpmN −1 ) . (6.5.60) Here, the Monte Carlo trace estimator from eq. (6.5.41) can be employed again, with trace ( NgpmN −1 ) = trace ( NgpmU −1U−T ) = trace ( U−TNgpmU−1 ) = trace ( Z¯TU−TNgpmU−1Z¯ ) (6.5.61) Importantly,U−1Z¯ is the same as in eq. (6.5.48), and can be reused directly. Estimation of the Zero-Frequency For a zero-mean random process, the theoretical expected value of the zero frequency of the PSD is well defined. Through discretisation and approximation effects, the actual zero frequency amplitude for the PSD of such a time series can be close, or equal to, zero. In practice, this means that the estimation of the zero frequency amplitude can 6.5 Fit of Stochastic Model 71