Page - 110 - 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

Correlation with Opening Angle The presented results from June 2010 and December 2005 suggest that the arc-wise variance factors are indeed correlated with periods of larger opening angles. This hypothesis can be tested by computing Pearson’s product-moment correlation coeffi- cient r between the arc-wise variance factors and the mean opening angles for each month. The distribution of all correlation coefficients, one for each month, is then studied. The mean of the opening angle for both satellites for one arc m with Nm epochs is β¯m= 1 2Nm Nm ∑ n=1 ( βAn+β B n ) . (8.3.1) Figure 8.7 shows the distribution of the monthly correlation coefficients for the pro- cessed GRACE time series. If the observation geometry, represented through the mean opening angle, had no impact on the arc-wise variance factors, the correlation coefficients would be expected to scatter around 0. Due to the distribution of both the mean opening angle and the arc-wise variance factor estimates, some spurious correlation is to be expected. The probability distribution for the old model considering only the stationary covariance function (in brown) is not centred at 0. The shift of the distribution towards positive correlations indicates that arc-wise variance factors and the mean opening angles are positively correlated. Introducing the AOC covariance information largely eliminates this shift (in blue). This indicates that the correlation of observation geometry and arc-wise variance factors is largely removed by use of the improved stochastic model. Numerically, themeanof thecorrelationcoefficients is reducedfrom0.15 in theoriginal processing to 0.03 when considering the AOC covariance matrices. −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 Correlation r [unitless] 0 1 2 3 4 Oldmodel Incl. AOC Figure 8.7: Distribution of correlation coefficients r between arc-wise variance factors σˆ2m and mean opening angles β¯m. One correlation coefficient was estimated per month. Chapter8 Star Camera Observations and Uncertainties110