Page - 150 - 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

can be observed at the first orbital resonant frequency of approximately order 16. For higherdegrees, a reduction in variabilitycan alsobeobservedfor tesseral coefficients. The ratio for the TLS solution, displayed in fig. 9.13e, shows similar features, but with a larger magnitude. The reduction in variability in the zonal and high-order tesseral Stokes coefficients is again present. In contrast to the ratios for the solution using AOC covariance matrices only, there does not appear to be an increase in variability at the first orbital resonant frequency. For medium orders, a decrease in variability can be observed between multiples of the resonant frequency, while Stokes coefficients at the resonant frequencies themselves show relatively little change. Higher degree non-zonal Stokes coefficients of all orders decrease in variability at a similar magnitude, with no obvious correlation to the resonant frequencies. It is expected that the variability should remain relatively unchanged at degrees below approximately 30, as strong temporal signals are expected at these scales. In general, this is the case for both solutions. The TLS solution however shows a strong reduction in variability for the zonal Stokes coefficients of degrees 17 to 20 (see inset in fig. 9.13e). Because of the sharp jump of variability and the strict limitation to zonal Stokes coefficients, a geophysical origin of this signal can be ruled out. A back-of-the-envelope calculation gives the temporal frequency that these Stokes coefficients correspond to for the GRACE spacecraft on their polar orbits. For a revolution period of Trev≈89min, the lower bound of the affected frequency spectrum, where the reduction in variability is smaller (c17,0), is 1/(Trev/17)≈3.2mHz. The corresponding frequency for the upper bound, formedbythedegree20zonalStokescoefficient is1/(Trev/20)≈3.7mHz.This band encompasses, again, the dominant frequency in the pitch angle variations of the GRACE spacecrafts, 3.3mHz (Bandikova, Flury, and Ko, 2012), which was previously encountered in section 8.3.3. This leads to the conclusion that the TLS estimate reduces noise in these Stokes coefficients, and that this noise originated in GRACE pointing variations at the appropriate frequencies. Figure 9.13e shows a sharp decrease in temporal variability for the TLS solution between near-zonal Stokes coefficients of degree 60 and 61. A geophysical source for this pattern is unlikely. Instead, the chosen processing strategy is the most likely origin for this artefact. The estimation of the stochastic model and the satellite orientation was only performed while co-estimating a gravity field up to D/O 60. The resulting orientations were used together with the stochastic model to compute a full D/O 120 field. As no such jump exists in the solution considering AOC covariances only (fig. 9.13c), it stands to reason that this approach is not suitable in the process of TLS orientation estimation. Rather, the iterative co-estimation of the orientation should be performed with a full D/O 120 gravity field. 9.4 Discussion This chapter described the application of the total least squares approach of least squares adjustment to the combined estimation of GRACE-derived gravity fields and Chapter9 Co-Estimation of Orientation Parameters150