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The combination service for time-variable gravity fields (COST-G) provides the full time-series of monthly GRACE gravity fields: COST-G GRACE RL01, combined in reprocessing mode, and a steadily growing time-series of monthly GRACE-FO gravity fields: COST-G GRACE-FO RL01 OP, combined on an operational basis. Both time-series are currently considered for re-combination. In case of GRACE, new high-quality time-series from Chinese analysis centers are available for combination. In case of GRACE-FO, a revision of the weighting scheme, developed in the frame of the Horizon2020 project Global Gravity-based Groundwater Product (G3P), and the availability of reprocessed GRACE-FO time-series from AIUB, CSR, GFZ, and JPL, lead to a significant improvement of the combined gravity fields. We present the preliminary re-combined GRACE and GRACE-FO time-series and quantify the differences with respect to the COST-G RL01 series in terms of signal and noise content.

Historically, there have been two fundamental views, timewise and spacewise, to recover the Earth’s gravity field using satellite observations. This has resulted in different temporal gravity field solutions using the Gravity Recovery and Climate Experiment (GRACE) observations. In this chapter, we compare timewise batch processor algorithm for solving variational equations with spacewise energy balance approach using simulated GRACE observations. When using error free simulated observations, both approaches perform similarly well. Energy balance approach has the advantage of using less data storage and less computational time. With error contaminated observations, energy balance approach performs worse than variational equations. Because the noise in orbital velocity corrupts the potential difference observables, and respectively the estimate of the gravity field. Although, variational equations perform better, it is important that both positions and range rates are combined and they are properly weighted in solving normal equations.

The Gravity Recovery and Climate Experiment (GRACE) mission has yielded data on the Earth's gravity field to monitor temporal changes for more than 15 years. The GRACE twin satellites use microwave ranging with micrometre precision to measure the distance variations between two satellites caused by the Earth's global gravitational field. GRACE Follow-on (GRACE-FO) will be the first satellite mission to use inter-satellite laser interferometry in space. The laser ranging instrument (LRI) will provide two additional measurements compared to the GRACE mission: Interferometric inter-satellite ranging with nanometre precision and inter-satellite pointing information. We have designed a set of simulated GRACE-FO data, which include LRI measurements, apart from all other GRACE instrument data needed for the Earth's gravity field recovery. The simulated data files are publicly available via https://doi.org/10.22027/AMDC2 and can be used to derive gravity field solutions like from GRACE data. This paper describes the scientific basis and technical approaches used to simulate the GRACE-FO instrument data.data. This paper describes the scientific basis and technical approaches used to simulate the GRACE-FO instrument data.

This paper reviews a field that is herein termed spatial ‘non-stationarity’, which is specifically concerned with non-stationarity in the geodetic theory of least-squares collocation (LSC). In practice, many geodesists rely on stationary assumptions in LSC, i.e. using a constant mean and isotropic and spatially invariant covariance for estimation and prediction of geodetic quantities. However, new theories in spatial statistics and geostatistics allow for better statistical methodologies to be used in geodesy. The aim of this paper is to introduce these methodologies and adapt them for dealing with non-stationarity in LSC.

This paper addresses implementation issues in order to apply non-stationary least-squares collocation (LSC) to a practical geodetic problem: fitting a gravimetric quasigeoid to discrete geometric quasigeoid heights at a local scale. This yields a surface that is useful for direct GPS heighting. Non-stationary covariance functions and a non-stationary model of the mean were applied to residual gravimetric quasigeoid determination by planar LSC in the Perth region of Western Australia. The non-stationary model of the mean did not change the LSC results significantly. However, elliptical kernels in non-stationary covariance functions were used successfully to create an iterative optimisation loop to decrease the difference between the gravimetric quasigeoid and geometric quasigeoid at 99 GPS-levelling points to a user-prescribed tolerance.

Standard least-squares collocation (LSC) assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the observed data. However, the assumption that the spatial dependence is constant throughout the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing of, e.g., the gravity field in mountains and under-smoothing in great plains. We introduce the kernel convolution method from spatial statistics for non-stationary covariance structures, and demonstrate its advantage for dealing with non-stationarity in geodetic data. We then compared stationary and non- stationary covariance functions in 2D LSC to the empirical example of gravity anomaly interpolation near the Darling Fault, Western Australia, where the field is anisotropic and non-stationary. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation against data not used in the interpolation, demonstrating that the use of non-stationary covariance functions can improve upon standard (stationary) LSC.