# Combination of GRACE monthly gravity field solutions from different processing strategies

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## Abstract

We combine the publicly available GRACE monthly gravity field time series to produce gravity fields with reduced systematic errors. We first compare the monthly gravity fields in the spatial domain in terms of signal and noise. Then, we combine the individual gravity fields with comparable signal content, but diverse noise characteristics. We test five different weighting schemes: equal weights, non-iterative coefficient-wise, order-wise, or field-wise weights, and iterative field-wise weights applying variance component estimation (VCE). The combined solutions are evaluated in terms of signal and noise in the spectral and spatial domains. Compared to the individual contributions, they in general show lower noise. In case the noise characteristics of the individual solutions differ significantly, the weighted means are less noisy, compared to the arithmetic mean: The non-seasonal variability over the oceans is reduced by up to 7.7% and the root mean square (RMS) of the residuals of mass change estimates within Antarctic drainage basins is reduced by 18.1% on average. The field-wise weighting schemes in general show better performance, compared to the order- or coefficient-wise weighting schemes. The combination of the full set of considered time series results in lower noise levels, compared to the combination of a subset consisting of the official GRACE Science Data System gravity fields only: The RMS of coefficient-wise anomalies is smaller by up to 22.4% and the non-seasonal variability over the oceans by 25.4%. This study was performed in the frame of the European Gravity Service for Improved Emergency Management (EGSIEM; http://www.egsiem.eu) project. The gravity fields provided by the EGSIEM scientific combination service (ftp://ftp.aiub.unibe.ch/EGSIEM/) are combined, based on the weights derived by VCE as described in this article.

## Keywords

GRACE Gravity field combination Noise assessment Weighting schemes Variance component estimation Gravity field validation EGSIEM## 1 Introduction

The National Aeronautics and Space Administration (NASA)/Deutsches Zentrum für Luft- und Raumfahrt (DLR) Gravity Recovery and Climate Experiment (GRACE) mission (Tapley et al. 2004) to map the Earth’s static and time-varying global gravity field has been successfully operated from 2002 till 2017. Using the *K*-band and GPS measurements from the GRACE twin satellites (Dunn et al. 2003), gravity fields at various spatial scales and temporal resolutions have been derived. They have been used for a wide range of geoscience research such as geodesy, hydrology, oceanography, atmospheric science, and glaciology (e.g., Güntner 2008; Johnson and Chambers 2013; Steffen et al. 2009; and an overview in Wouters et al. 2014).

To acquire dense enough observational coverage to map the Earth’s gravity field up to a spatial resolution of about 400 km, thirty days of measurements are usually required (Tapley et al. 2004) in the standard case of non-repeating orbits. Monthly global gravity fields have been processed and released by the official GRACE Science Data System (SDS, Watkins et al. 2000) which consists of the Center for Space Research at the University of Texas (CSR), the German Research Center for Geosciences (GFZ), and NASA’s Jet Propulsion Laboratory (JPL). There are also additional processing centers, which produce GRACE monthly global gravity fields on a best effort basis such as the Astronomical Institute of the University of Bern (AIUB), the Groupe de recherche de Géodésie spatiale (GRGS), the Technical University of Delft (TU Delft), the Institute of Geodesy at the Graz University of Technology (ITSG), and the Tongji University (Tongji).

From individual processing centers there are up to five consecutive releases of monthly GRACE gravity fields. Different processing strategies were applied, resulting in various noise characteristics of the individual solutions. Numerous studies compare subsets of the available monthly gravity fields in various geophysical applications such as ocean bottom pressure, ice mass change, and glacial isostatic adjustment (e.g., Chambers and Bonin 2012; Johnson and Chambers 2013; Sasgen et al. 2007a; Steffen et al. 2009). Sasgen et al. (2007a) and Sakumura et al. (2014) compared and combined the three official SDS and the GRGS gravity fields.

List of GRACE monthly gravity field solutions (as of April 2016)

Label | Solution name | Institution | Max. deg. | Processing strategy | References |
---|---|---|---|---|---|

AIUB 02 (60) | AIUB release 2 | AIUB | 60 | Celestial mechanics approach (pseudo-stochastic orbit parameters) | Meyer et al. (2016) |

AIUB 02 (90) | 90 | ||||

CSR 05 (60) | CSR release 5 | CSR (Univ. Texas) | 60 | Direct approach | Bettadpur (2012) |

CSR 05 (96) | 96 | ||||

DMT 01 (120) | DMT–1 | TU delft | 120 | Acceleration approach (pre-filtered) | Liu et al. (2010) |

GFZ 5a (90) | GFZ release 5a | GFZ | 90 | Direct approach | Dahle et al. (2012) |

GRGS 03 (80) | GRGS release 3 | GRGS | 80 | Direct approach (regularized) | Lemoine et al. (2013) |

ITSG2014 (60) | ITSG 2014 | ITSG (TU graz) | 60 | Short arc approach (empirical covariances) | Mayer-Gürr et al. (2014) |

ITSG2014 (90) | 90 | ||||

ITSG2014 (120) | 120 | ||||

JPL 05 (60) | JPL release 5 | JPL | 60 | Direct approach | Watkins and Yuan (2012) |

JPL 05 (90) | 90 | ||||

Tongji 01 (60) | Tongji release 1 | Tongji Univ. | 60 | Modified short arc approach (extended arc length) | Chen et al. (2015) |

In both combination studies, gravity fields based on variants of the direct approach (Bettadpur and McCullough 2017) were used. However, meanwhile further gravity fields from processing centers that are not members of the official SDS have become available. These additional time series are determined based on various alternative approaches such as the celestial mechanics approach (Beutler et al. 2010a, b), the short arc approach (Mayer-Gürr 2006), and the modified acceleration approach (Liu et al. 2010). The full set of these different gravity field time series has not yet been rigorously compared nor combined. A further drop of noise levels can be expected from this combination.

The combinations achieved by Sasgen et al. (2007a) and Sakumura et al. (2014) were limited to maximum degree and order 50, due to the limited resolution of the GRGS release 1 or 2 contributions. However, GRGS’s release 3 now is available up to degree 80. Moreover, some of the processing centers provide various monthly gravity fields up to degrees 60, 90, or even 120. Combined solutions up to degrees higher than 50 have not yet been investigated. In Sasgen et al. (2007a) and Sakumura et al. (2014), the combined gravity fields were evaluated in the spatial domain only. We gain further insight by investigating the different noise characteristics also in the spectral domain. Based on their reduced set of contributions, Sakumura et al. (2014) could not demonstrate weighting to be beneficial. With the larger set of gravity fields, a proper weighting scheme is expected to improve the combination, especially in case of diverse noise levels of the individual gravity fields.

In this study, we compare the full set of currently available monthly GRACE gravity fields. We combine all time series with comparable signal content, i.e., without obvious regularization. We therefore test five different weighting schemes including the arithmetic mean. Finally, we evaluate the combinations in terms of signal and noise in both the spectral and the spatial domains. The study is performed in the frame of the project European Gravity Service for Improved Emergency Management (EGSIEM^{1}) in preparation of a future combination service comparable to the International GNSS Service (IGS; Dow et al. 2009), the International VLBI Service (IVS; Schlüter and Behrend 2007), or the International Laser Ranging Service (ILRS; Pearlman et al. 2002).

## 2 Database of GRACE monthly gravity fields

^{2}), as listed in Table 1. Figure 1 (top) visualizes the availability of the individual monthly gravity fields. Gaps are related to missing observational data, mainly in the early and late mission phases, and to periods of orbit resonance. The latter can be inferred from the maximum longitudinal spacing of neighboring ground tracks per month, as shown in Fig. 1 (bottom). The density of the ground tracks of the satellites directly determines the achievable spatial and spectral resolution of the derived monthly gravity field (Weigelt et al. 2013; Klokočník et al. 2015). Extended periods of orbit resonances are encountered around September 2004 and May 2012. A period of less pronounced resonances during the years 2009 and 2010 did not impair the monthly gravity fields. Some processing centers that provide high-degree gravity fields during periods with little observation coverage or orbit resonance adopt regularization techniques (e.g., Dahle 2017). Regularized gravity fields were excluded from the combination.

^{3}/s

^{2}. Both values are consistent with the IERS Conventions 2010 (Petit and Luzum 2010). When necessary, the individual gravity fields’ fully normalized spherical harmonic coefficients \(\bar{C}_{lm}\) and \(\bar{S}_{lm}\) of degree

*l*and order

*m*are rescaled (Hofmann-Wellenhof and Moritz 2006):

*orig*and

*ref*indicate the original and reference values.

Each time series of the individual monthly gravity field solutions is screened separately. Outliers are determined on the basis of the RMS of the non-seasonal variability over the oceans, which is a good indicator for the noise level of a gravity field (see Sect. 3.2). As threshold three times the median absolute deviation^{3} is used. Up to five monthly gravity fields per time series, mostly in the early mission phase before 2004 and around the resonance period in May 2012, are excluded by the screening. Finally, we group the individual time series according to their maximum degrees 60, 90, or 120. The degree 80 GRGS gravity fields are cut to degree 60 and the degree 96 CSR gravity fields are cut to degree 90, to be comparable to the other time series.

## 3 Comparison of individual time series

The preprocessed time series are compared in the spatial domain in terms of signal and noise. This step is necessary to exclude obviously regularized time series that may bias the combination. We study the mean equivalent water height (MEWH) within selected river basins to assess the signal content and the RMS of the non-seasonal variability over the oceans to assess the noise levels of the monthly gravity fields. The \(C_{20}\) coefficient is excluded from the analysis because it is degraded in most of the GRACE gravity fields (Chen et al. 2005) and normally replaced by SLR-derived values (e.g., Chambers and Bonin 2012; King et al. 2012 and Velicogna and Wahr 2013).

### 3.1 Signal content

We compare the amplitudes of annual variations of MEWH within selected river basins. Therefore, the unitless spherical harmonic coefficients of the individual gravity fields are transformed to equivalent water heights (EWH, Wahr et al. 1998) and the \(3^{\circ }\times 3^{\circ }\) grids on the Earth’s surface are synthesized. The grid cells are weighted by the sine of their colatitude. The MEWH within a river basin is computed as the normalized sum over all basin grid cells (e.g., Eq. (8) in Zhao et al. 2011). For this signal evaluation, all gravity fields were truncated at degree 60 and no filtering was applied.

### 3.2 Noise level

*i*, and \(e_i\) is the corresponding EWH anomaly at time

*t*.

Figure 4 shows the monthly weighted RMS over the oceans for the individual degree 60 and 90 gravity fields, without (left) or with smoothing (right) by a 300-km Gaussian filter which reduces noise, especially in the high-degree coefficients (Jekeli 1981; Wahr et al. 1998). The GRGS time series exhibits a distinctly low noise level and also the least fluctuations. This is due to a regularization during the data processing (Lemoine et al. 2013). Consequently, the GRGS time series was not considered for combination. The other contributions have comparable noise levels and show similar fluctuations, such as increased noise around periods of orbit resonance. Neglecting the GRGS gravity fields, the ITSG time series has the lowest noise levels in both the degree 60 and the degree 90 cases. This is most probably due to the empirical noise modeling applied in their approach (Mayer-Gürr et al. 2014).

## 4 Combination of gravity fields

*X*(either

*C*or

*S*) of degree

*l*and order

*m*, in gravity field

*i*at time

*t*. \(N_\mathrm{sol}\) is the number of contributions. The formal error of the weighted mean is

Weighting schemes

Type | Weight | Computed monthly | Formula |
---|---|---|---|

Reference | Identical | Per order, degree | \( w_{l,m}^{i,t}=1 \) |

Non-iterative | Coefficient-wise | Per order, degree | \( w_{l,m}^{i,t}=\Bigg [\left( X_{l,m}^{i,t}-\bar{X}_{l,m}^{t}\right) ^{2}\Bigg ]^{-1} \) |

Order-wise | Per order | \( w_{m}^{i,t}=\Bigg [\frac{1}{l_\mathrm{max}-m+1}\sum _{l=m}^{l_\mathrm{max}}\left( X_{l,m}^{i,t}-\bar{X}_{l,m}^{t}\right) ^{2}\Bigg ]^{-1}\) | |

Field-wise | Per field | \( w_{}^{i,t}=\Bigg [\frac{1}{N_\mathrm{coef}}\sum _{l=2}^{l_\mathrm{max}}\sum _{m=0}^{l}\left( X_{l,m}^{t}-\bar{X}_{l,m}^{t}\right) ^{2}\Bigg ]^{-1}\) | |

Iterative | Field-wise (VCE) | Per field | \( w^{i,t,(k)}=\Bigg [\frac{\sum _{l=2}^{l_\mathrm{max}}\sum _{m=0}^{l}\left( X_{l,m}^{t}-\hat{X}_{l,m}^{t,(k-1)}\right) ^{2}}{N_\mathrm{coef}\cdot \left( 1-\frac{w^{i,t,(k-1)}}{\sum _{i=1}^{N_\mathrm{sol}}w^{i,t,(k-1)}} \right) }\Bigg ]^{-1}\) |

### 4.1 Arithmetic mean

The arithmetic mean is the simplest way to compute combined solutions. Each gravity field coefficient enters the combination with equal weight.

### 4.2 Coefficient-wise weighting

Weights are computed per gravity field coefficient by the inverse of the squared difference to the arithmetic mean (Table 2). This weighting scheme corresponds to the inverse of the squared variance which is commonly used in statistics. Figure 5 shows the normalized coefficient-wise weights per time series (averaged over all monthly combinations for this illustration). AIUB, CSR, and ITSG in general obtain higher weights than GFZ, JPL, and Tongji. The AIUB contribution obtains the highest weights, indicating that the AIUB gravity fields are closest to the arithmetic mean, i.e., are least affected by biases. However, its coefficients at resonance orders (15, 31, 46, 61, 76) seem to be deteriorated. The CSR gravity fields obtain high weights at their high-degree low-order coefficients. This corresponds to the observation that their noise level in the corresponding spectral range is very low.

*K*-Band correction. Its effect on the zonal coefficients could be reproduced at AIUB for the test month January 2007, replacing the original GRACE Level 1B data with ITSG’s sensor fusion data (Fig. 6, left). Note that, compared to the general noise levels of the gravity fields (see Fig. 11), the effect of the sensor fusion data on the zonal coefficients is quite small, but systematic.

The main difference between the original L1B and the sensor fusion data is a significant reduction in high-frequency noise. We therefore made one additional test and simply smoothed the original geometric *K*-band correction using a low-pass Savitzky–Golay filter (Savitzky and Golay 1964) of polynomial order 3 and a half window length of 60 seconds. The effect on the zonal coefficients (Fig. 6, right) closely resembles the true sensor fusion results. We conclude that by smoothing the geometric *K*-band correction the zonal coefficients can be improved. This example reveals a weakness of all weighting schemes based on comparison to a mean: A superior contribution cannot be distinguished from a degraded one if it systematically differs from the bulk of contributions.

The Tongji time series in contrast to AIUB, CSR, and ITSG obtains lower weights at low orders. JPL has problems at resonance orders, but profits for sectorial coefficients. GFZ in contrary obtains rather high weights around resonance orders in response to the degradation at these orders in the AIUB and JPL contributions.

### 4.3 Order-wise weighting

Spherical harmonic coefficients of the same order and parity are correlated, as it becomes evident applying the concept of lumped coefficients (Sneeuw 1992). This correlation results in comparable noise characteristics per order and motivated an order-wise weighting scheme, drastically reducing the number of weights. The order-wise weights can be derived from the coefficient-wise differences according to Table 2.

The order-wise weights per gravity field are shown in Fig. 7 for the whole period. As expected from the previous results, AIUB, CSR, and ITSG again obtain generally higher weights. In case of the Tongji time series, it becomes obvious that the low weights at low orders are mainly caused by a degradation at these orders toward the end of the period. GFZ correspondingly shows degraded performance before 2006 and from 2012 to 2014. This agrees well with periods of increased noise as shown in Fig. 4 and corresponds to periods of increased solar and correspondingly ionosphere activity.

### 4.4 Field-wise weighting

### 4.5 Iteration by variance component estimation

*k*) indicates the iteration depth.

## 5 Evaluation of the combined solutions

We evaluate the combined solutions in both the spectral and the spatial domains. We focus on the noise assessment because the signal content of the combined solutions is expected to be similar to that of the individual contributions. In the spectral domain, we evaluate the combined solutions, investigating the RMS and the degree amplitudes of the coefficient-wise anomalies. Additionally, we examine the significance of annual variations per coefficient applying a statistical *F*-test. In the spatial domain, we use the weighted RMS of anomalies over the oceans to assess the noise. We also estimate secular mass change in Antarctica and compare the size of the RMS of residuals.

### 5.1 Spectral domain

#### 5.1.1 Coefficient-wise anomalies

The coefficient-wise anomalies are defined as the residuals after subtraction of deterministic models consisting of bias, trend, annual, and semiannual variations from the time series of spherical harmonic coefficients (compare to Sect. 3.2). Figure 11 shows the coefficient-wise RMS of anomalies of the individual contributions and of the combined solutions of the degree 60 (top) or 90 (bottom) gravity fields. We focus on the comparison of the simple arithmetic mean and the weighted mean using VCE because they perform best. The coefficient-wise weighting scheme is obviously impaired by the small sample size. It is the same for the order-wise weighting scheme in case of high orders.

In Fig. 11, we can distinguish three different sections of spherical harmonic coefficients: low degrees, high orders, and the central part of the triangles of coefficients. The central part containing the low- to medium-order coefficients generally has smaller anomalies than the corners containing the low-degree or high-order coefficients. The low- to medium-order coefficients usually contain less noise than the high-order coefficients. The relatively large anomalies in the low-degree coefficients are most probably caused by unmodeled signal, whereas the larger anomalies in the high-order coefficients are dominated by colored noise. For the assessment of the noise, we focus on the high-order coefficients. The degree 60 individual contributions are more similar than the degree 90 individual contributions because the latter contain more noisy high-order coefficients.

In both the degree 60 and 90 cases, the combined solutions have smaller coefficient-wise RMS than the individual contributions. This indicates that the coefficients of the combined solutions are less noisy and fit the modeled signals better than the individual contributions. However, the zonal coefficients of the degree 60 and 90 ITSG time series have smaller anomalies than the corresponding combined solutions. This effect is related to the use of the sensor fusion attitude data by ITSG (see Sect. 4.2). The high quality of the ITSG zonal coefficients is not adequately represented by the different weighting schemes and the combination does not profit.

The weighted mean has smaller anomalies than the arithmetic mean in both the degree 60 and 90 cases. As expected, the difference is more pronounced in the degree 90 case: The improvement in the weighted mean with respect to the arithmetic mean is 0.9% in the degree 60 case and 1.6% in the degree 90 case. The diverse degree 90 gravity fields profit more from the weights than the uniform degree 60 gravity fields.

#### 5.1.2 Degree amplitudes of anomalies

The coefficient-wise anomalies may be condensed to degree amplitudes to give a clearer view of the noise characteristics of the spherical harmonic spectrum. Figure 13 shows the medians of the degree amplitudes of the anomalies per time series in the degree 90 case. First, only coefficients up to order 29 are considered to focus on the geophysically most meaningful part of the spectrum (left), then all coefficients are included (right). The motivation for truncation at order 29 is to exclude the noisy coefficients around the second resonant order 31.

The combined solutions have smaller degree amplitudes than the individual contributions up to degree 60. Beyond degree 60, the ITSG contribution performs as good as or even better than the combinations. But small anomalies could also indicate attenuated signal. In a simulation study based on white noise, a comparable behavior could be reproduced including an individual contribution containing attenuated signal (Jean et al. 2016). However, in our case we could not prove any signal attenuation in the ITSG contribution. Moreover, the small anomalies of the ITSG gravity fields are observed at high degrees where the signal content of the anomalies is rather small, compared to the noise. Therefore, we conclude that the ITSG solution contains obviously less noise in the high-degree coefficients than even the combined solutions.

Overall, the different weighting schemes perform similarly. The weighted means outperform the arithmetic mean in the higher degree coefficients approximately beyond degree 60. This is the part of the spectrum where the individual contributions are most diverse and the greatest benefit due to the weighting has to be expected. Among the weighted means, the combination based on VCE excels. Among the non-iterative combinations, the coefficient-wise weighting unexpectedly performs worst. This is most probably caused by the small sample size of only four (in the degree 60 case) or five (in the degree 90 case) time series available for combination.

#### 5.1.3 Significance of annual variations

In addition to the study of anomalies, we perform a statistical *F*-test to assess the signal content within the different time series of monthly gravity fields. We focus on the annual variations that are mainly caused by the hydrological cycle. To test the significance of the signal under question, we fit deterministic models with or without the corresponding model parameter to time series of the individual gravity field coefficients. The *F*-test evaluates the ratio of the residuals of the complete and the reduced model (e.g., Davis et al. 2008 or Meyer et al. 2012). Our complete model consists of offset, trend, annual, and semiannual periodic signals.

### 5.2 Spatial domain

#### 5.2.1 Weighted RMS of anomalies over the oceans

In the spatial domain, we first focus on the noise levels as indicated by the RMS of the anomalies over the oceans, weighted by the sine of the colatitude of the grid cells (see Sect. 3.2). As shown by Fig. 15, the combined solutions in general have smaller anomalies over the oceans than the individual contributions, except for the unfiltered degree 90 case where the ITSG contribution sometimes even outperforms the combinations. After smoothing, the combined solutions outperform all individual contributions in both the degree 60 and 90 cases.

The five different weighting schemes result in very similar noise levels. In the degree 60 case, no clear advantage for one of the weighting schemes can be found: The reduction in noise relative to the arithmetic mean is at the level of 1.3% only. In the unsmoothed degree 90 case, order-wise weights or VCE produces the best results: The reduction in noise amounts to 7.7% with respect to the arithmetic mean. This difference is ironed out by smoothing.

We conclude that weighting only is worth the effort in case of degree 90 monthly gravity fields where the noise content in the high-degree and high-order coefficients differs significantly between the individual contributions. This is in line with Sakumura et al. (2014) who truncated all gravity fields at degree 50 prior to combination and concluded that in this case no weights are needed. Moreover, the benefit of weighting becomes obvious in the presence of outliers, as the case in January 2007 in the degree 90 case, where all weighted combinations clearly outperform the arithmetic mean.

Finally, we again compute subset combinations based only on the official SDS time series and compare them to the combinations of the full set of contributions. The subset combinations have significantly larger anomalies (Fig. 16) than the regular combinations. The improvement taking all contributions into account amounts to 25.4% in terms of the median of the weighted RMS of the anomalies over the oceans.

#### 5.2.2 Mass trends in Antarctica

We fit deterministic models consisting of bias, trend, annual, and semiannual periodic signal to the time series of MEWH within the 27 major glacial basins of Antarctica (Fig. 17), considering the arithmetic mean, the combination based on VCE, and the individual contributions. The estimated linear trends, estimated for the period from February 2003 to November 2011 from the unfiltered degree 60 or 90 time series, are shown in Fig. 18 (top). To derive residuals representative for the noise (Fig. 18, bottom), we further reduce quadratic trends. We compute the mass in each drainage basin simply from the basins’ MEWH without further corrections. This is justified because the purpose of this investigation is the comparison of the combined solutions with respect to the individual contributions rather than the derivation of ice mass change in Antarctica.

Most of the degree 60 or 90 individual or combined solutions result in similar trend estimate within thresholds of 3 times the median absolute deviations (Fig. 18, top). Only the Tongji and JPL time series show larger deviations. The RMS of the residuals (Fig. 18, bottom) is the lowest for the combinations and the ITSG time series. In the degree 60 case, the combined solutions are least noisy in most of the glacial basins, especially in the small basins in Western Antarctica, numbered from 20 to 22, where the mass loss is most prominent. In the degree 90 case, the ITSG time series is often less noisy than even the combined solutions, as it was also found in the case of anomalies over the oceans in the previous section.

## 6 Conclusion

We compared all currently available monthly GRACE gravity field time series in the spectral and the spatial domains. Most of the individual time series contain comparable mass transport signals and noise levels except for the DMT time series that has been pre-filtered in the data processing and the GRGS 03 time series that is regularized. Some of the alternative time series that are not official GRACE SDS products perform very well in terms of signal-to-noise ratio.

We further created combinations of the individual gravity field time series, excluding pre-filtered or regularized gravity fields. Compared to previous studies, the combination is extended to maximum spherical harmonic degree 90 and five different weighting schemes are explored. Degree 60 or 90 gravity fields are studied and combined separately.

For noise assessment in the spectral domain, the coefficient-wise RMS and the degree amplitudes of anomalies are computed. The signal content is evaluated applying a statistical *F*-test to the annual variation per coefficient. In the spatial domain, the weighted RMS of EWH anomalies over the oceans and mass trend estimates in Antarctica are studied.

All experiments yield consistent results: By combination the noise in general is reduced, while the signal content of the individual contributions is preserved. However, the degree 90 ITSG time series, which was generated applying empirical noise modeling, performs comparably well to the combined solutions.

Subset combination of only the SDS time series leads to significantly increased noise levels: The coefficient-wise RMS of anomalies is increased by up to 22.4%, while the weighted RMS of anomalies over the oceans is increased by 25.4%. In general, the weighted combinations perform better than the arithmetic mean in both the spectral and spatial domains. This is especially true for the unsmoothed degree 90 gravity fields that exhibit rather diverse noise characteristics in their high-degree and order coefficients: The weighted RMS of anomalies over the oceans is reduced by 7.7% in the weighted combination compared to the arithmetic mean, and residuals after trend estimation in Antarctic glacial basins are reduced by 18.1% in average over 27 major glacial basins.

We conclude that well-designed weighting schemes are necessary to produce optimal combined solutions when the individual solutions have diverse noise characteristics. With a small sample size of 4 to 5 time series considered for combination, a field-wise weighting scheme performs better than coefficient- or order-wise weighting schemes. Iteration by VCE was found to be helpful in general.

To further improve the combination results, advanced noise modeling strategies in the individual contributions would be beneficial, as shown by ITSG. With a larger number of time series, coefficient-wise weighting schemes that take into account the noise characteristics of the individual gravity field coefficients are expected to perform better than order- or field-wise weighting schemes. External comparison with hydrological, glaciological, or oceanographic observational data would strengthen the validation, but is beyond the scope of this study.

The GRACE gravity field combinations produced by the EGSIEM project make use of the VCE weighting scheme investigated in this study. They are available at ftp://ftp.aiub.unibe.ch/EGSIEM/.

## Footnotes

## Notes

### Acknowledgements

This research was supported by the European Union’s *Horizon 2020 research and innovation program* under the Grant Agreement No. 637010. The authors are thankful to ITSG at the Graz University of Technology for providing their sensor fusion data. The authors would also like to thank the anonymous reviewers, the responsible editor of this manuscript, and the editor-in-chief of the Journal of Geodesy for the valuable advice to improve this article.

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