Combination of GRACE monthly gravity fields on the normal equation level
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Abstract
A large number of timeseries of monthly gravity fields derived from GRACE data provide users with a wealth of information on mass transport processes in the system Earth. The users are, however, left alone with the decision which timeseries to analyze. Following the example of other wellknown combination services provided by the geodetic community, the prototype of a combination service has been developed within the frame of the project EGSIEM (2015–2017) to combine the different timeseries with the goal to provide a unique and superior product to the user community. Four associated analysis centers (ACs) of EGSIEM, namely AIUB, GFZ, GRGS and IfG, generated monthly gravity fields which were then combined using the different normal equations (NEQs). But the relative weights determined by variance component estimation (VCE) on the NEQ level do not lead to an optimal combined product due to the different processing strategies applied by the individual ACs. We therefore resort to VCE on the solution level to derive relative weights that are representative of the noise levels of the individual solutions. These weights are then applied in the combination on the NEQ level. Prior to combination, empirical scaling factors that are based on pairwise combinations of NEQs are derived to balance the impact of the NEQs on the combined solution. We compare the processing approaches of the different ACs and introduce quality measures derived either from the differences w.r.t. the monthly means of the individual gravity fields or w.r.t. a deterministic signal model. After combination, the gravity fields are validated by comparison to the official GRACE SDS RL05 timeseries and the individual contributions of the associated ACs in the spectral and the spatial domain. While the combined gravity fields are comparable in signal strength to the individual timeseries, they stand out by their low noise level. In terms of noise, they are in 90% of all months as good or better than the best individual contribution from IfG and significantly less noisy than the official GRACE SDS RL05 timeseries.
Keywords
EGSIEM GRACE Satellite gravimetry Timevariable gravity Combination service1 Introduction
Monthly Earth gravity fields based on the observations of the Gravity Recovery And Climate Experiment [GRACE, Tapley et al. (2004)] satellite mission are an important source of information on temporal mass variations in the system Earth (Wouters et al. 2014). Monthly gravity fields are not only provided by the official GRACE Science Data System (SDS) processing centers JPL, CSR and GFZ, but also by an increasing number of independent analysis centers (ACs) worldwide.
The standard approach is to expand the gravity field in spherical harmonics and provide the weight coefficients of this expansion (L2products) that may be transformed to global grids (L3products) for easier use. Depending on the field of application, the grids are complemented by monthly mean values of the shortterm atmosphere and ocean mass variations [socalled GAXproducts, Flechtner and Dobslaw (2013)] to restore the nontidal signal content. Moreover, the L3products are usually prefiltered to reduce noise.
Web services (e.g., PO.DAAC,^{1} ISDC,^{2} or Tellus^{3} for L3products) are available to download the GRACE SDS products. Timeseries of monthly gravity fields, also from the alternative ACs, are collected and made available via the International Centre for Global Earth Models (ICGEM).^{4}
Based on these sources of information, the user has to decide which timeseries of monthly gravity fields to use. For most users, the peculiarities of the different processing approaches of the individual GRACE ACs remain unclear. Therefore, there is urgent need for a unification of gravity models like it is done for the products of other space geodetic techniques by, e.g., the International GNSS Service [IGS, Dow et al. (2009)], the International Laser Ranging Service [ILRS, Pearlman et al. (2002)], the International VLBI Service [IVS, Nothnagel et al. (2017)] or the International DORIS Service [IDS, Tavernier et al. (2005)].
Noise in the monthly gravity fields is dominated by either measurement system errors (observation noise), or temporal aliasing errors caused by imperfections in the background models [see, e.g., Flechtner et al. (2016); Seo et al. (2008)]. Due to the onedimensional observation geometry, the temporal aliasing error is manifest as north–south striping. The noise characteristics of the various solutions differ because of the different parameterizations used by the individual ACs to compensate for both error sources. In the following, we refer to the solution noise not explained by the measurement system error as analysis noise.
To fully take into account correlations between the individual gravity field parameters, but also between gravity field, orbit, instrument, stochastic or other model parameters, gravity fields have to be combined on the normal equation (NEQ) level. Up to now, a combination of gravity fields on the NEQ level was not possible because the individual ACs normally do not provide NEQs. This was only changed in the frame of the EGSIEM project (see Sect. 2).
It is a wellknown technique when combining NEQs to define relative weights iteratively by variance component estimation [VCE, Koch and Kusche (2002)]. This approach is applied, e.g., by the IVS (Böckmann et al. 2010). The technique is recapitulated in “Appendix A.1.” But applying VCE to NEQs provided by different ACs is hampered by a basic problem. Proper stochastic models of noise in the original data are not available. The individual ACs apply different noise modeling strategies and rely on different parameters to absorb background model errors, but the inversion of the individual normal matrices does not yield realistic covariance matrices of errors of model parameters. The error estimates of the unknown parameters differ considerably between the ACs, and consequently classical VCE converges to nonoptimal results. This problem is not only encountered in the combination of GRACE gravity fields. Lerch (1989), in case of the combination of data from different satellite missions, proposed a procedure to derive relative weights based on the analysis noise. Seitz et al. (2012), in the computation of the global reference frame DTRF2008, based on terrestrial and satellite data, completely replaced VCE derived by empirical weights.
We propose an alternative weighting scheme which is based on the noise levels of the individual solutions (Sect. 5). Relative weights are derived by VCE on the solution level (Jean et al. 2018) and then consequently applied to the NEQs. The main difference between the classical VCE on the NEQ level and the alternative VCE on the solution level is that in the latter case, while correlations between the unknown parameters are lost, the error assessment does not depend on the different error modeling and absorption strategies of the individual ACs, but on the differences to a (weighted) mean of the individual solutions.
Prior to weighting, the impact of the individual NEQs on the combination has to be balanced. This is achieved by empirical factors derived from the study of pairwise combinations and has to be done independently from the VCE, since the VCE is converging robustly to the same results, independent of the a priori weights used.

We exploit the normal matrices provided together with solutions themselves.

Ideally, the inverse of the normal matrix is the full error variance–covariance matrix of model parameters. In reality, ACs do not guarantee a proper scaling of normal matrices. Therefore, we estimate the weight factors to be used in combining the individual NEQs.

The most obvious technique to estimate optimal weights is VCE. Unfortunately, a direct application of this technique, as it is presented in Appendix A.1, leads to suboptimal results, because the inversion of the NEQs provided by the ACs does not yield realistic covariance matrices of errors in model parameters.

In order to solve this problem, we apply VCE on the solution level (described in Sect. 5.3). In this case, the errors in model parameters are assumed to be uncorrelated and the error variances of all the parameters are assumed to be the same. Unfortunately, the weights estimates cannot be applied to the NEQs just like that, because the inversion of any of the NEQs results in an error covariance matrix that suffers, among others, from an unknown scaling factor.

We therefore compute and apply additional empirical factors, which so to say equalize the available normal matrices. Those factors are estimated such that all the individual solutions contribute equally to the combined solution, independently of their quality. (This procedure is described in Sect. 5.2.)
2 The EGSIEM combination service for monthly gravity fields
In the frame of the Horizon 2020 project European Gravity Service for Improved Emergency Management [EGSIEM, Jäggi et al. (2019)], the prototype of a scientific combination service for timevariable gravity fields was established. The goal of this service is to provide consistent, reliable and validated monthly gravity fields, which are combined on the NEQ level from standardized NEQs of all associated ACs. EGSIEM ACs contributing to the combination are the Astronomical Institute of the University of Bern (AIUB), the Helmholtz Centre Potsdam, German Research Centre for Geosciences (GFZ), the Groupe de Recherche de Géodésie Spatiale (GRGS) and the Institute of Geodesy of the Technical University of Graz (IfG, former Institute for Theoretical and Satellite Geodesy, ITSG).
To guarantee consistency between the individual contributions, EGSIEM standards were defined for reference frame, Earth rotation and antenna reference points on the GRACE satellites, as well as for the relativistic effects and for thirdbody perturbations. The EGSIEM ACs were free to use their specific processing approaches and the background force models of their choice for the static gravity field of the Earth and for tidal mass variations. Neither were the dealiasing products for shortterm atmosphere and ocean mass variations [AOD, Flechtner and Dobslaw (2013)] harmonized, because background models and dealiasing products are not free of errors. In the combination, errors in the individual models may be reduced; therefore, a wide variety of models is beneficial.
EGSIEMcombined gravity fields are provided in spherical harmonic representation (L2products) and as global grids (L3products). To generate L3products, degree 1 terms derived from Satellite Laser Ranging (SLR) are added to transform between a center of mass and a center of figure frame. Then, the monthly mean of AOD is restored to achieve full (nontidal) signal content. The AOD correction is combined from the individual monthly means provided by the ACs using the same relative weights as in the combination of the gravity fields (Jäggi et al. 2019).
For hydrological applications, monthly means of atmosphere [GAA, Flechtner and Dobslaw (2013)], ocean [GAB, Flechtner and Dobslaw (2013)] and the global isostatic adjustment (GIA) model LM17.3^{5} are subtracted. For oceanographic applications, monthly means of the atmosphere, the terrestrial water storage modeled by the WaterGAP Global Hydrological Model [WGHM; Döll et al. (2003)] and GIA (evaluated at the epochs of the monthly gravity fields) are subtracted.
A variant of the DDKfilter (Kusche 2007) making use of the full, monthly covariance information is applied to filter the different versions (still in spherical harmonics representation). Therefore, the calibrated error covariances of the ITSG gravity fields were used and the characteristics of the expected hydrological or oceanographic signals were taken into account. Finally, the spherical harmonic coefficients (SHC) were transformed to global grids with \(1^\circ \)resolution.
All EGSIEM products are representative of the time span within a given month defined by the start and end day. Short GRACE data gaps are ignored when computing the monthly means of the AOD products, assuming that users in general do not thin out their observation database according to the availability of the GRACE observation data either.
EGSIEMcombined L2 and L3 products can be downloaded from the “Data” section of the EGSIEM homepage.^{6} Furthermore, mass variations derived from individual timeseries as well as from the combined gravity fields can be visualized by the EGSIEM plotter.^{7}
The prototype combination service continues after the completion of the EGSIEM project in the frame of the Combination Service for Timevariable Gravity field models (COSTG) as a product center of the International Gravity Field Service (IGFS) under the umbrella of the International Association of Geodesy (IAG).
3 Noise assessment
We need to assess the noise levels of the individual and the combined gravity fields for quality control, and we have to independently define relative weights to consider the different noise levels in the models to be combined. Prior to combination, the monthly gravity fields provided by the individual ACs undergo strict quality control based on their signal and noise content in the spectral and spatial domains. While noise levels may vary between ACs and are taken into account in the combination by noisebased relative weights, the signal content is expected to be the same in all gravity field timeseries accepted for combination. Gravity field solutions with attenuated temporal variation due to intended or accidental regularization are excluded from the combination to avoid damaging the signal content.
The signal content is evaluated by the comparison of the amplitudes of seasonal mass variations in a large number of river basins and by the study of mass trends in polar regions. The tests of the signal content are described in detail by Jean et al. (2018).

Comparison with the monthly mean of different gravity fields, assuming that all gravity fields contain the same signal, but are different in noise. The noise ideally is greatly reduced in the averaging process, while the signal content remains unchanged.

Comparison with a signal model. As our knowledge of mass transport in the system Earth is limited, we refer to a deterministic model of mass variation containing bias, trend, annual and semiannual variations fitted by a leastsquares process to the monthly mean values of the individual gravity fields. The residuals with respect to this model are called anomalies.
Figure 2 shows the rootmeansquare (RMS) in geoid heights over all monthly gravity fields 2004–2010 of the degree amplitudes of differences to the mean and anomalies of the EGSIEMAIUB contribution. The differences to the mean values are in general smaller than the anomalies, because our signal model is incomplete and does not represent nonsecular, nonseasonal variations, and because the differences to the mean do not reflect the errors that are shared by all the models.
Due to the polar orbits of the GRACE satellites and due to the related sparse observation sampling in crosstrack direction, high spherical harmonic orders are especially noisy and often removed by filtering (Kusche 2007). We therefore also show degree amplitudes computed from orders \(0,\ldots ,29\) only to focus on the geophysically most meaningful part of the spectrum. The spikes visible at degrees 15, 31, 46, 61 and around 76 are related to orbit resonances. The GRACE satellites circle the Earth approximately 15.3 times per day. Spherical harmonic orders at integer multiples of 15.3 are called resonant orders and suffer from aliasing by longperiodic signal of geophysical origin (Seo et al. 2008). Whenever the degree amplitudes include a new resonant order, this causes a jump in the noise level.

Differences to the mean are significantly smaller than anomalies and only show a small latitude dependence.

Anomalies include nonsecular, nonseasonal signals, which are concentrated over land regions with strong mass variability.
4 Individual timeseries

their use of either the original GPS observations or kinematic satellite orbits derived thereof,

the relative weighting or sampling of observables,

the noise model or the parameters estimated to absorb the noise and

the background models used for signal separation.
Types and numbers of GRACE observations used by the different ACs
KRR  GPS code per satellite  GPS phase per satellite  Kin. Pos. per satellite  Relative weight monthly sum of obs.  

AIUB  
Observation sampling  5 s  –  –  30 s  
Observation weight  \(1/(3\times 10^{7}\frac{\mathrm{m}}{\mathrm{s}})^2\)  –  –  \(1/(15\times 2\times 10^{3}\,\hbox {m})^2\)  \(1\times 10^{10}\frac{1}{s^2}\) 
Observations per month  535.680  0  0  267.840  Max. 1.071.360 
Observations in 2006  810.000–1.070.000  
GFZ  
Observation sampling  5 s  30 s  30 s  –  
Observation weight  \(1/(1\times 10^{7}\frac{\mathrm{m}}{\mathrm{s}})^2\)  1 / 1.0 m\(^2\)  \(L3: 1/(1\times 10^{2}\) m)\(^2\)  –  \(1\times 10^{10}\frac{1}{s^2}\) 
Observations per month  535.680  892.800  892.800  0  Max. 4.106.880 
Observations in 2006  2.200.000–2.700.000  
GRGS  
Observation sampling  5 s  30 s  30 s  –  
Observation weight  \(1/(1\times 10^{7}\frac{\mathrm{m}}{\mathrm{s}})^2\)  \(1/1.0\,\)m\(^2\)  \(L1: 1/(2\times 10^{3}\)m)\(^2\)  –  \(0.25\times 10^{10}\frac{1}{s^2}\) 
Observations per month  535.680  892.800  892.800  0  Max. 4.106.880 
Observations in 2006  2.400.000–3.000.000  
ITSG  
Observation sampling  5 s  –  –  300 s  
Observation weight  VCE  –  –  VCE  VCE 
Observations per month  535.680  0  0  26.784  Max. 589.248 
Observations in 2006  510.000–580.000 
The weights of the observables are generally based on the RMS of the corresponding residuals, i.e., 0.7 m for GPS code, 0.2 cm for GPS carrier phase (L1), 0.7 cm for the ionospherefree linear combination (L3) of carrier phases (L1 and L2) and 0.1–0.3 \(\upmu \)m/s for KRR. In the case of ITSG, the relative weights of the different observables are determined by VCE.
All ACs observe inconsistencies between GPS and KRR observations leading to increased noise in the gravity field solutions. This problem seems to be even more serious, if kinematic orbits are used as pseudoobservations instead of the original GPS observables. Both, GFZ and GRGS, downweight the GPS code observable, and GFZ in addition downweights the GPS phases (see Table 1). GRGS moreover limits the resolution of the gravity field contribution determined by GPS to degree and order 40. AIUB downweights the kinematic positions by an empirically determined factor of \(15^2\), and ITSG downsamples the pseudoobservations by a factor of 10. The reason for the inconsistencies between GPS and KRR is still under investigation.
Types and numbers of nuisance parameters estimated by the different ACs
Orbit (per satellite)  ACC (per satellite)  Others  Monthly sum  

AIUB  Initial state: 6  Scale: 1 per axis (X, Y, Z)  15 min stoch. acc.:  18,786 
Bias: 1 per axis (R,W)  96 per axis (R,S,W)  Partly constrained  
3. Order polynomial:  and satellite  
4 per axis (W)  Constrained  
GFZ  Initial state: 6  3 h bias: 9 per axis (X, Y, Z)  Emp. KRR  5208 
3 h scale: 9 per axis (X, Y, Z)  90 min bias: 16  
90 min drift: 16  
180 min 1/rev: 16  
GRGS  Initial state: 6  45 min bias:  14,446  
90 min 1/rev:  32 per axis (X, Y, Z)  Partly constrained  
32 per axis (S, W)  Scale: 1 per axis (X, Y, Z)  
90 min 2/rev:  
32 per axis (S,W)  
ITSG  Initial state: 6  3. Order polynomial:  Daily AOD(40): 1677  53,475 
4 per axis (X, Y, Z)  Constrained  Partly constrained  
\(3\times 3\) sym. scale matrix: 6 
AIUB applies a more conservative instrument parameterization, but estimates socalled pseudostochastic accelerations in the three axes—radial (R), alongtrack (S) and crosstrack (W)—of the corotating orbital frame every 15 min. The pseudostochastic accelerations are estimated to compensate for not only instrument noise, but also all kinds of model deficiencies. They are constrained to zero with uncertainties of \(\sigma =3\times 10^{9}\) m/s\(^2\) to prevent absorbing timevariable gravity signal (Meyer et al. 2016).
While all other ACs apply very simple noise models (diagonal weight matrices with uniform weight per observable), ITSG applies empirical noise modeling techniques to take correlations between observations over 3 h arcs into account (Ellmer 2018). Consequently, IFG has to deal with fully populated weight matrices. But a realistic noise model can only be achieved by a careful separation between signal and noise. Therefore, ITSG determines constrained daily variations up to a spherical harmonics degree of 40. The monthly mean of the daily estimates is restored in the monthly solution not to impair the signal content. On top of that, ITSG estimates fully populated (symmetric) \(3\times 3\) ACC scale factor matrices for each day. This measure drastically reduces the artifacts with a period of 161 days that impair the \(C_{20}\) estimate (Klinger and MayerGürr 2016).
To simplify the combination on the NEQ level, all but the gravity field parameters are preeliminated by the ACs and the individual NEQs are normalized (i.e., each observable is weighted according to Table 1). Despite all these measures, the differences in the choice of observables and in the observation sampling cause huge differences in the number of observations entering the daily normal equations, and the numbers of the preeliminated parameters differ significantly, too. Moreover, the various noise modeling strategies cause very different magnitudes of the formal errors. In Sect. 5, a robust combination strategy is introduced.
5 Combination of normal equations
5.1 Transformation to common geophysical constants, tide system and a priori gravity model
5.2 Empirical scaling to balance the impact of NEQs on the combination
We know, on the other hand, that each NEQ basically contains the same information representative of the same time span of GRACE observations, and the individual solutions only differ in analysis noise. The latter will be taken into account by the weights derived from the individual solutions in Sect. 5.3.
5.3 Relative weights based on solution noise
According to the argumentation in Sect. 1, we define relative weights representative of the noise content on the solution level. This can be done simply by comparing the individual solutions to their arithmetic mean. An alternative procedure based on VCE, proposed by Jean et al. (2018), is more robust against outliers. The same authors also study different weighting schemes, e.g., coefficientwise, orderwise or fieldwise weights, and conclude that monthly fieldwise weights determined by VCE on the solution level are best suited for the combination. We therefore determine fieldwise weights.

applying no weights at all,

defining the relative weights iteratively by standard VCE on the NEQ level, i.e., without empirical factors (labeled “NEQVCE” in Fig. 11),

applying empirical factors to balance the impact of the individual contributions (arithmetic mean on the NEQ level),

basing the relative weights on the solution noise by multiplying the empirical factors by weights determined by VCE on the solution level (labeled “EGSIEMCOMB” in Fig. 11).
The arithmetic mean on the NEQ level (based on the empirical factors to balance the impact of the individual NEQs) performs slightly worse than the “EGSIEMCOMB.” This result differs from the conclusion of Sakumura et al. (2014) (studying gravity field combinations on the solution level) that the arithmetic mean of the gravity fields of different ACs performs best. Contrary to Sakumura et al. (2014), who combined timeseries of very homogeneous quality, we are confronted with more diverse noise levels and therefore, as already mentioned by Jean et al. (2018), the benefit of relative weights becomes apparent.
Note that due to the different orbit parameterizations, the combined monthly gravity fields do not correspond to one and the same satellite orbit valid for all ACs. While the ACspecific parameters are preeliminated prior to combination and the correlations between the local and the gravity field parameters are kept, a solution of the preeliminated parameters by resubstitution of the combined gravity field coefficients would lead to, e.g., different initial state vectors and increased residuals compared to the individual solutions. As long as we have to deal with diverse parameterizations, there exists nothing like an optimal common set of orbit parameters.
6 Evaluation of combined monthly solutions
As long as no signal biases impede the combination, the fieldwise weights derived by VCE on the solution level provide a robust quality indicator for the monthly gravity fields provided by the EGSIEM ACs. Together with quality indicators based on the anomalies, noise levels can be characterized and signal attenuation due to regularization can be detected.
Within the EGSIEM project, originally only monthly gravity fields for the two years 2006–2007 were reprocessed according to the EGSIEM standards and combined on the NEQ level. For this time span, 24 monthly gravity fields are available from the EGSIEM ACs AIUB, GFZ, GRGS and ITSG. Toward the end of the project, it was decided to extend the time span to the years 2004–2010 to enable the evaluation by users inside and outside the EGSIEM project. Only contributions from AIUB, GRGS and ITSG were accessible for the extended time span.
The empirical factors are determined relative to a reference contribution, which is identified by a constant weight of 1 in the figures. During 2006–2007 GFZ, the only member of the SDS in the EGSIEM consortium was selected as reference, and for the years without GFZ contribution, GRGS was selected. As explained in Sect. 4, the empirical factors balance the effect of different types of observations, different parameterizations and different noise models in the individual timeseries. Consequently, the final weights cannot be interpreted as quality indicators anymore.
The weights derived by VCE on the NEQ level differ significantly from the final weights used for the EGSIEM combination. The fundamental difference is the low weight assigned to the ITSG contribution. This is explained by the empirical noise model applied by ITSG that leads to realistic, i.e., significantly larger formal errors compared to the other timeseries. It remains unclear why the weights derived by VCE on the NEQ level are much less favorable for GFZ than for GRGS. Both ACs base their processing on the original GPS phase observations, and their formal errors are comparable, at least for the dominating medium to highdegree SHC. We conclude that VCE on the NEQ level does not necessarily produce optimal weights if NEQs stemming from different analysis approaches have to be combined.
In the presence of signal biases, the differences between the individual contributions and their mean values include the signal biases and the weights derived by VCE on the solution level for a biased contribution are smaller than expected from noise only. In this case, the relative weights are no longer representative of the different noise levels. Consequently, small VCEderived weights together with small noise, as illustrated, e.g., by small anomalies over the oceans, indicate signal biases. In our case, all contributions of the EGSIEM ACs passed the quality control and no signal biases could be detected.
The noise level of the combined solution is also independently evaluated by means of anomalies in the spherical harmonic and the spatial domain (as proposed in Sect. 3). The deterministic signal model used to define the anomalies (see Sect. 3) was derived from the monthly arithmetic mean values of all the timeseries available at ICGEM having passed quality control according to Jean et al. (2018).
Figure 14 compares the RMS of degree amplitudes of EWH anomalies in the spherical harmonic domain to the three official RL05 timeseries (evaluated for the time span 2004–2010) of the GRACE SDS ACs. Beyond degree 30, the degree amplitudes of the anomalies in general are dominated by noise [see e.g., Jean et al. (2018)]. The RMS of the anomalies of the EGSIEMcombined solutions is smaller than that of the GRACE SDS RL05 timeseries.
Figure 16 compares the combined gravity fields to the individual EGSIEM ACs’ timeseries. The monthly RMS values of anomalies over the oceans are computed in order to assess the noise levels of individual solutions. Note that for 2006–2007 the combined solutions include the GFZ contribution. The ITSG contribution is clearly less noisy than the other individual ACs’ timeseries in this evaluation. But with the exception of very few months, the noise level of the combined gravity fields is as small as or even smaller than that of ITSG. Note that poor quality of the solutions in January 2004 is caused by data gaps and in December 2005 by the satellite swap maneuver. From August to October 2004 (gray box in Fig. 16), the quality of the monthly gravity fields is impaired by the orbit resonances.
7 Conclusions and outlook
We presented the prototype of a combination service for monthly gravity fields, which was implemented in the frame of the EGSIEM project. The monthly gravity fields provided by the associated ACs show different noise levels due to different processing approaches: The number of observations used per month varies between 500,000 and 3,000,000, the number of estimated parameters between 5000 and 50,000. Moreover, the noise modeling techniques and parameter types differ substantially.
The combination is performed on the NEQ level to correctly take into account correlations between parameters. Relative weights, representative of the different noise levels, are derived by VCE on the solution level, i.e., by iterative comparison of the individual gravity fields to their weighted mean. The intrinsic weights of the individual NEQs are removed by a robust empirical procedure balancing the impact of the individual NEQs on the pairwise combinations.
Combined gravity fields were computed from three or four ACs for the time span between 2004 and 2010. An independent evaluation of the noise levels indicates that the quality of the best individual contribution (ITSG) is achieved or even topped by the combinations in 90% of the monthly solutions. Outliers can be identified with data problems. Compared to the official GRACE SDS monthly gravity fields, the anomalies of the EGSIEM combinations that are derived to assess the noise level are smaller. The original goal to provide consistent, reliable and validated gravity fields therefore is met.
The noise level differences of the individual timeseries are striking. With a more homogeneous quality of the input series, the combinations should improve substantially as well. First, experiments with the new GRACE SDS RL06 timeseries indicate a big step forward in this direction. With the availability of the new GRACE L1BRL03 observational data and the SDS RL06 gravity fields, now a final combination of all GRACE timeseries becomes feasible.
The EGSIEM initiative for gravity field combination is continuing with COSTG under the umbrella of the IAG. Since it cannot be expected that the GRACE SDS ACs will reprocess the whole GRACE timeseries to be in accordance with the EGSIEM standards, the COSTG standards will be adapted to only specify the signal content of the monthly gravity fields which should include nontidal oceanographic, hydrological, glaciological and GIA signal to the full extent.
Footnotes
Notes
Acknowledgements
This research was supported by the European Union’s Horizon 2020 research and innovation program under the Grant Agreement No. 637010. Our thanks also go to the two anonymous reviewers for their knowledgeable comments.
References
 Böckmann S, Artz T, Nothnagel A, Tesmer V (2010) International VLBI service for geodesy and astrometry: earth orientation parameter combination methodology and quality of the combined products. J Geophys Res 115:B04404. https://doi.org/10.1029/2009JB006465 CrossRefGoogle Scholar
 Brockmann E (1997) Combination of solutions for geodetic and geodynamic applications of the global positioning system (GPS). In: GeodätischGeophysikalische Arbeiten in der Schweiz, vol 55, Schweizerische Geodätische Kommission, Zürich, SwitzerlandGoogle Scholar
 Bruinsma S, Lemoine JM, Biancale R, Valès N (2010) CNES/GRGS 10day gravity field models (release 2) and their evaluation. Adv Space Res 45:587–601. https://doi.org/10.1016/j.asr.2009.10.012 CrossRefGoogle Scholar
 Dahle C, Flechtner F, Gruber C, König D, König R, Michalak G, Neumayer KH (2012) GFZ GRACE level2 processing standards document for level2 product release 0005. Scientific technical report STR12/02—Data, Revised Edition, January 2013, Potsdam. https://doi.org/10.2312/GFZ.b103120225
 Döll P, Kaspar F, Lehner B (2003) A global hydrological model for deriving water availability indicators: model tuning and validation. J Hydrol 270(1–2):105–134. https://doi.org/10.1016/S00221694(02)002834 CrossRefGoogle Scholar
 Dow JM, Neilan RE, Rizos C (2009) The international GNSS service in a changing landscape of global navigation satellite systems. J Geod 83(3):191–198. https://doi.org/10.1007/s0019000803003 CrossRefGoogle Scholar
 Ellmer M (2018) Contributions to GRACE gravity field recovery. Dissertation, Monograph series TU Graz: geodesy, vol 1, Verlag der Technischen Universität Graz, doy. https://doi.org/10.3217/9783851256468
 Flechtner F, Dobslaw H (2013) AOD1B product description document for product release 05. GRACE project document JPL 327750, version 4.0. http://isdc.gfzpotsdam.de/grace
 Flechtner F, Neumayer KH, Dahle C, Dobslaw H, Fagiolini E, Raimondo JC, Güntner A (2016) What can be expected from the GRACEFO laser ranging interferometer for earth science applications. Surv Geophys 37(2):453–470. https://doi.org/10.1007/s107120159338y CrossRefGoogle Scholar
 Jäggi A, Weigelt M, Flechtner F, Güntner A, MayerGürr T, Martinis S, Bruinsma S, Flury J, Bourgogne S, Meyer U, Jean Y, Sušnik A, Grahsl A, CannGuthauser K, Dach R, Li Z, Chen Q, van Dam T, Gruber C, Poropat L, Gouwleeuw B, Kvas A, Klinger B, Lemoine JM, Biancale R, Zwenzner H, Bandikova T, Shabanloui A (2019) European gravity service for improved emergency management (EGSIEM): from concept to implementation. Geophys J Int 218:1572–1590. https://doi.org/10.1093/gji/ggz238 CrossRefGoogle Scholar
 Jean Y, Meyer U, Jäggi A (2018) Combination of GRACE monthly gravity field solutions from different processing strategies. J Geodesy. https://doi.org/10.1007/s0019001811235 CrossRefGoogle Scholar
 Jekeli C, Rapp R (1980) Accuracy of the determination of mean anomalies and mean geoid undulations from a satellite gravity mapping mission. Report no. 307, Department of Geodetic Science, Ohio State University, Columbus, OH, USAGoogle Scholar
 Kim J (2000) Simulation study of a lowlow satellitetosatellite tracking mission. Technical report, University of Texas at Austin, TX, USAGoogle Scholar
 Klinger B, MayerGürr T (2016) The role of accelerometer data calibration within GRACE gravity field recovery: Results from ITSGGrace2016. Adv Space Res 58:1597–1609. https://doi.org/10.1016/j.asr.2016.08.007 CrossRefGoogle Scholar
 Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. J Geod 76:259–268. https://doi.org/10.1007/s001900020245x CrossRefGoogle Scholar
 Kusche J (2007) Approximate decorrelation and nonisotropic smoothing of timevariable GRACEtype gravity field models. J Geodesy 81:733–749. https://doi.org/10.1007/s0019000701433 CrossRefGoogle Scholar
 Lerch FJ (1989) Optimum data weighting and error calibration for estimation of gravitational parameters. NASA technical memorandum 100737, Goddard Space Flight Center, Greenbelt, MarylandGoogle Scholar
 Meyer U, Jäggi A, Jean Y, Beutler G (2016) AIUBRL02: an improved timeseries of monthly gravity fields from GRACE data. Geophys J Int 205:1196–1207. https://doi.org/10.1093/gji/ggw081 CrossRefGoogle Scholar
 Nothnagel A, Artz T, Behrend D, Malkin Z (2017) International VLBI service for geodesy and astrometry. J Geod 91:711–721. https://doi.org/10.1007/s0019001609505 CrossRefGoogle Scholar
 Pearlman MR, Degnan JJ, Bosworth JM (2002) The international laser ranging service. Adv Space Res 30(2):135–143. https://doi.org/10.1016/S02741177(02)002776 CrossRefGoogle Scholar
 Sakumura C, Bettadpur S, Bruinsma S (2014) Ensemble prediction and intercomparison analysis of GRACE timevariable gravity field models. Geophys Res Lett 41:1389–1397. https://doi.org/10.1002/2013GL058632 CrossRefGoogle Scholar
 Seitz M, Angermann D, Bloßfeld M, Drewes H, Gerstl M (2012) The 2008 DGFI realization of the ITRS: DTRF2008. J Geod 86:1097–1123. https://doi.org/10.1007/s0019001205672 CrossRefGoogle Scholar
 Seo KW, Wilson CR, Chen J, Waliser DE (2008) GRACE’s spatial aliasing error. Geophys J Int 172:41–48. https://doi.org/10.1111/j.1365246X.2007.03611.x CrossRefGoogle Scholar
 Sneeuw N (2000) A semianalytical approach to gravity field analysis from satellite observations. Deutsche Geodätische Kommission, Reihe C, Heft 527Google Scholar
 Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins M (2004) GRACE measurements of mass variability in the Earth system. Science 305(5683):503–505CrossRefGoogle Scholar
 Tavernier G, Fagard H, FeisselVernier M, Lemoine F, Noll C, Ries J, Soudarin L, Wilis P (2005) The international DORIS service. Adv Space Res 36:333–341. https://doi.org/10.1016/j.asr.2005.03.102 CrossRefGoogle Scholar
 Wahr J, Molenaar M, Bryan F (1998) Time variability of the Earth’s gravity field: hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103:30205–30230. https://doi.org/10.1029/98JB02844 CrossRefGoogle Scholar
 Wouters B, Bonin JA, Chambers DP, Riva REM, Wahr J (2014) GRACE, timevarying gravity, Earth system dynamics and climate change. Rep Prog Phys 77:116801. https://doi.org/10.1088/00344885/77/11/116801 CrossRefGoogle Scholar
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