Page - 66 - 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

The estimated arc-wise variance factor is however not directly σˆ2m = αˆ2mσ2m. As the absolute magnitude of the covariance matrix has already been modified by adjusting the individual frequencies in the PSD, computing σˆ2m in this way would lead to a double consideration of the respective variance components. Instead, an intermediate quantity γ2m= αˆ 2 mσ 2 m (6.5.28) is determined. The arc-wise variance factor is then this intermediate value normalized by the mean over all arcs with γ¯= 1 M M ∑ m=1 γm (6.5.29) as σˆ2m= γ2m γ¯2 . (6.5.30) The mean of the arc-wise variance factors is thus 1. The arc-wise variance factors only give the relative weighting of the arcs w.r.t. each other within the observation group. The covariance function determines the overall magnitude and structure of the covariance matrix for the observations. 6.5.5 Geophysical Signal In addition to the complete stochastic model estimated for the satellite instruments, one variance factor per month is estimated for the stochastic model of the geophysical pseudo-observations. The standard apparatus for VCE given in section 2.6 is used to optimally scale the covariance matrix Σgpm= σˆ 2 gpmQgpm . (6.5.31) 6.5.6 Implementation Notes The variance component estimation as described in the previous sections is a computa- tionally expensive algorithm, especially as multiple iterations are usually necessary to achieve good convergence. In GROOPS, the software used to compute ITSG-Grace2016, some optimizations have been implemented to reduce this computational effort. The following pages will give an overview of some of the most impactful of these optimiza- tions. Further, some aspects will be shown where for practical reasons the computation strategy deviates from the strict theoretical derivations of the previous subsections. Chapter6 ITSG-Grace201666