Impact of tone errors in future satellite gravimetry missions

One of the main limiting factors to observe variations of the very low degrees and orders of the spherical harmonic (SH) spectrum of the Earth's gravity field with satellite gravimetry missions like GRACE and GRACE-FO are the so-called tone errors. They are deterministic errors occurring periodically at the orbital frequency of the spacecraft (one cycle-per-revolution, 1 CPR) and its multiples. Tone errors are generated by external perturbations acting on the satellite at the spacecraft environment and by spacecraft-internal processes. In this study, we investigate the impact of tone errors on the resulting gravity field model and their mitigation by numerical simulations for selected mission concepts. We start with a GRACE/GRACE-FO-like single polar pair mission concept and extend the simulations to a so-called Bender double pair constellation by adding an inclined (70°) satellite pair. Within our gravity-field simulation approach, we consider realistic instrument noise assumptions for the accelerometers and the inter-satellite ranging instrument, leading to instrument-only simulation scenarios. Tone error contributions are modeled at so-called orbital harmonics at 1, 2, and 3 CPR and incorporated into the instrumental noise time series. Three selected sets of low, moderate and large tone amplitudes and the occurrence of a single tone amplitude on either 1, 2, or 3 CPR are considered to analyze the effects on gravity field retrieval. Simulation results show, that for instrument-only scenarios, tone errors significantly affect single polar pair solutions over the complete SH spectrum by amplifying resonance orders, whereas double pair solutions are less affected. Since the tone amplitudes and occurrences are known, the applied stochastic modeling based on the instrumental behavior is extended by additional notch filters to mitigate the impact of tone errors. This approach has been selected to identify its performance and applicability for gravity field determination. Applying the adapted stochastic model, we can conclude that for both satellite constellations, the erroneous effect of tone errors in the higher SH spectrum can be mitigated at the cost of increased errors in the low degrees. The behavior, as seen in the instrument-only scenarios, cannot be confirmed in additional, more realistic simulations, including temporal gravity field contributions, called full-noise scenarios. Temporal gravity field signals are, in general, larger than the erroneous signal caused by tone errors. The under-sampling of high-frequency mass signals from atmosphere, ocean and ocean tides, causing temporal aliasing, dominates the gravity field solution errors for single and double pair constellations and is up to one order of magnitude larger than the tone errors impact considering low and moderate tone amplitudes. Only with large tone amplitudes the tone error effect exceeds temporal aliasing in the case of a single polar pair. In the presence of temporal aliasing applying the adapted stochastic modeling is disadvantageous since the down-weighting of specific frequencies via notch filters also affects the temporal gravity field solutions, in particular the single polar pair. Other suitable mitigation approaches to be applied for real data processing are identified as possible options.