The workflow of COST-G is illustrated by Fig. 1. It consists of harmonization and quality control of the individual input solution of the different ACs and PCs, and the combination and validation of the resulting COST-G solution. All aspects of this process are explained and illustrated in more detail in the subsequent subsections by discussing the results of the first release of combined GRACE Level-2 products (GRAC_COSTG_BF01_0100, subsequently addressed as COST-G RL01).
3.1 Harmonization of Input Solutions
The individual input solutions of the different ACs may differ in the underlying constants (Earth’s gravity constant, Earth’s equatorial radius), tide system, and mean pole convention (see Eq. 21 in Wahr et al. (2015)). Individual solutions may thus need to be re-scaled and individual coefficients (C
21) may need to be further corrected to be compliant with the tide system and mean pole convention used by COST-G. Additional background models, such as the atmosphere and ocean dealiasing product (AOD), may also differ between groups and are consolidated before the combination to ensure a consistent signal definition.
3.2 Quality Control of Input Solutions
For quality control of the COST-G products, the signal and noise content of the harmonized AC contributions are first compared. The signal content is analyzed by computing the amplitude of seasonal variations of equivalent water height (EWH) for a large number of river basins and by computing ice mass trends in Greenland and Antarctica. An assessment of the noise is performed by analyzing anomalies, which represent the monthly variability after subtraction of a deterministic model of secular and seasonal variations. The assessment of the seasonal variations of the input solutions for COST-G RL01 is discussed below, whereas the assessment of ice mass trends will be separately discussed in Sect. 3.4.1. The noise assessment is not reproduced here as it largely corresponds to earlier results presented in Jean et al. (2018) and Meyer et al. (2019).
Figure 2 (top) shows the amplitudes, expressed in mean equivalent water height (MEWH), of the estimated seasonal variations for the 500 largest river basins as derived from all AC and PC solutions used for the COST-G RL01. The underlying river basin masks were taken from the Total Runoff Integrating Pathways (TRIP4Footnote 2) model. The corresponding formal errors of the estimation are shown in Fig. 2 (bottom). The analysis shows that no systematic signal attenuation may be observed for seasonal signals in any of the contributing gravity field time-series. The cluster of outlying larger formal errors visible in Fig. 2 (bottom) is related to regions with small seasonal signals and large non-linear trends as occurring, e.g. for regions with accelerated ice mass loss.
3.3 Combination of Input Solutions
Generally, the planned strategy for future COST-G releases is to provide a combination based on the underlying NEQs of the individual ACs according to the methodology presented in Meyer et al. (2019). As NEQs were not available from all centers, the combination was performed on the solution level for the COST-G RL01 as a weighted combination of the SH coefficients using variance component estimation (VCE) (Jean et al. 2018). The underlying AC und PC solutions are, AIUB RL02 (Meyer et al. 2016), GRGS RL04, GFZ RL06 (Dahle et al. 2019), ITSG-Grace2018 (Kvas et al. 2019), and CSR RL06 (Save et al. 2018).
Apart from AIUB RL02, which is still based on the RL02 of the GRACE Level-1B data, all input solutions are based on the most recent RL03 of the GRACE Level-1B data (GRACE Level 1B JPL Release 3.0 2018). For the GRGS solution the underlying NEQs were inverted by the ACC to obtain a solution without regularization.
Figure 3 shows the relative weights assigned to the individual input solutions as determined by VCE. The weights can be interpreted as quality indicators of the solutions as they are inversely proportional to the noise levels of the individual contributions. The highest weights are usually assigned to the ITSG-Grace2018 solutions due to their very low noise level. This is largely related to the sophisticated empirical noise modeling of the ITSG-Grace2018 solution (Kvas et al. 2019) and in accordance with analyses based on earlier ITSG releases (Jean et al. 2018). Lowest weights are usually assigned to the AIUB RL02 solutions due to the use of (meanwhile) outdated Level-1B data and, in particular, since active thermal control of the GRACE satellites was switched off in April 2011 (Tapley et al. 2015), which would have required adaptions in the accelerometer parametrization as shown in Klinger and Mayer-Gürr (2016).
Figure 4 shows median degree amplitudes of anomalies for the combined solution as well as for the input solutions that contributed to the combination for the entire GRACE time period with respect to a linear and seasonal model without applying any filtering. The analysis reveals that in the spectral domain the main gain of the combination is in the range of degrees 15–45. When truncating all gravity fields at order 29 to exclude the effect of the noisy higher-order SH coefficients, which are usually attenuated in applications by post-processing filters, e.g. Kusche (2007), the gain of the combination may even be seen up to about degree 65. The lower noise of the combined solution may also be confirmed in the spatial domain by analyzing the RMS of EWH anomalies over the oceans (not shown).
3.4 Validation of Combined Solution
Internal and external validation is performed to ensure the quality of the COST-G solutions by identifying outlying or systematically deviating input solutions.
3.4.1 Internal Validation
Ice mass loss in the polar regions is of enormous societal relevance (Shepherd et al. 2012). Evaluating GRACE mass change time series for the ice sheets in Antarctica and Greenland, as derived from the individual input solutions and the combined COST-G solution, is thus an essential task of the COST-G product evaluation group to detect potential inconsistencies between the input solutions.
126.96.36.199 Greenland Ice Sheet
GRACE-derived mass change time series are derived from all input solutions that contributed to the combination for the entire GRACE time period and from the combined COST-G solution for all drainage basins of the Greenland Ice Sheet (GISFootnote 3) and for the entire Greenland Ice Sheet by adopting the sensitivity kernel approach (Groh et al. 2019 and references in there).
The estimated linear trends of ice mass change for all drainage basins and for the entire Greenland ice sheet agree very well for the different solutions (not shown). Most notably the COST-G time series is characterized by a very favorable noise behavior. Figure 5 shows the noise levels of the mass change time series available over the entire time span in terms of the scaled standard deviation of the derived noise time series for all Greenland Ice Sheet drainage basins and the entire ice sheet following the method of Groh et al. (2019). For the majority of the basins the COST-G time series shows indeed the lowest noise of all solutions used for the combination.
188.8.131.52 Antarctic Ice Sheet
A similar analysis is performed for the drainage basins of the Antarctic Ice Sheet (AIS, see footnote 3), selected aggregations, and for the entire Antarctic ice sheet. Whereas most of the results of this analysis confirm the level of agreement found for the Greenland Ice Sheet, the linear trends resulting for different gravity field solutions for some drainage basins deserve special attention. Figure 6 compiles for the solutions available over the entire time span the estimated linear trends and error bars in ice mass change for selected drainage basins, the Antarctic Peninsula, East Antarctica, West Antarctica, and for the entire ice sheet. The displayed error bars account for the formal errors of the estimated trends, as well as for leakage errors and errors in the applied model reductions.
Figure 6 shows that the trends resulting from all solutions agree fairly well for West Antarctica. It also reveals, however, that discrepancies are occurring for East Antarctica. Whereas the trends resulting from the CSR RL06 and the ITSG-Grace2018 solutions are in almost perfect agreement, the GFZ RL06 solution suggests a different trend for this entire region. However, all trend estimates agree within their error estimates, since these are clearly dominated by the error of the reduced glacial isostatic adjustment model. As a consequence of the weighting scheme used for the combined solution, the trend of the COST-G solution is in-between the two concurring trends. Due to the larger weights assigned to the CSR / ITSG solution (cf. Fig. 3), it is closer to these solutions. Figure 6 also shows that the different trends over East Antarctica influence the trends resulting for the entire ice sheet. Trend analyses based on shorter time periods common to all solutions show again different trends for the AIUB RL02 and GRGS RL04 solutions (not shown).
3.4.2 External Validation
Currently two methods are realized within COST-G to assess the quality of the solutions by independent data sets. More external validations may follow in the future.
184.108.40.206 Comparison to Altimetry
Currently two test areas (Caspian Sea, Black Sea) are selected within COST-G for an independent signal assessment. The time series of the time-variable gravity field solutions are compared with the time series of altimetric heights, derived from Hydroweb for the Caspian Sea and AVISO+ for the Black Sea. One bias and one scale factor are adjusted to perform the comparison.
Table 1 exemplarily shows the correlation coefficients over the Caspian Sea when filtering the gravity field solutions with different DDK filters (Kusche 2007). It can be seen that the COST-G solution presents in this metric a slight improvement with respect to the input solutions.
220.127.116.11 Orbital Fits
The long to medium wavelength accuracy of gravity field models can be evaluated through dynamic orbit computations as commonly done for the evaluation of static gravity fields, e.g. Gruber et al. (2011). In the frame of COST-G the same concept is adopted to the time-variable gravity field solutions from the individual ACs and thereof derived combined solution. In order to not suffer from large omission errors of the time-variable GRACE solutions, which are generally only available up to degree and order 90, all solutions are filled up to degree and order 240 with the SH coefficients of the GOCE static model DIR-6 (Förste et al. 2019). For the COST-G RL01 dynamic GOCE orbits with an arc length of 1.25 days were fitted to the GOCE kinematic precise science orbits (Bock et al. 2014) serving as pseudo-observations. The gravitational forces were modeled according to the different gravity field models under consideration, whereas non-gravitational accelerations were described by the high-quality GOCE accelerometer data. For each arc three common mode acceleration biases are estimated in addition to the initial state vector. The scale factors of the common mode accelerations were fixed to one (Gruber et al. 2011).
Table 2 shows the RMS of orbit fits for the different test cases, derived as mean values from the 3D residuals from the 30/31 individual arcs under consideration for each month. It can be seen that the COST-G RL01 solution is also assessed in this metric of very good quality, but there is potential for further improvements as the solution is not yet best for each of the tested months. Note that the orbit test does not primarily validate C20 of the individual solutions. Only marginal differences would result if C20 is replaced in all solutions with one and the same value (not shown).