A methodology has been developed for an accurate estimation of mass anomalies in the Earth system using level-2 data products from satellite gravimetry GRACE and GRACE Follow-On (GFO) missions. Its key elements are: (i) direct inversion of Spherical Harmonic Coefficients (SHCs)—or SHC trends—into a global distribution of mass anomalies (or their trends); (ii) Spatially-varying regularization that takes into account available information about the behavior of mass anomalies; and (iii) rigorous optimization of the data processing consistently with the target estimates. The methodology is applied to quantify the mass balance of the Greenland Ice Sheet and its individual Drainage Systems (DSs) in Apr. 2002–Aug. 2023 on the basis of GRACE/GFO monthly solutions from the Institute of Geodesy at Graz University of Technology (ITSG). It is found that the rate of the total mass loss in Greenland was 271±10 Gt/yr. It varied between 19±4 Gt/yr in northeast DS and 77±7 Gt/yr in southeast DS. In average, the mass balance of individual DSs is estimated with an accuracy better than 5 Gt/yr. As a consequence, the obtained estimates show a sufficiently high signal-to-noise ratio (between 5 in the northeast DS and 42 in the northwest DS). This opens the door, among other, for using GRACE/GFO data for a comparison, validation, and calibration of physical models describing mass changes in Greenland, including its surface mass balance, at the scale of individual DSs.

Gravity Recovery and Climate Experiment (GRACE) observation data processed by various institutions yields somewhat different spherical harmonic solutions, which are further used to derive terrestrial water storage (TWS) changes. Combining TWS solutions from different institutions helps to refine the effective signal while removing noise. This study investigates regularization constraints in the context of TWS fusion to enhance the resulting estimates. The considered constraints are Tikhonov regularization of different orders, as well as minimization of month-to-month year-to-year double differences (MYDD), and triple differences (MYTD). Different accuracy and signal evaluation approaches are implemented for both individual and combined solutions. Compared to individual solutions and unregularized combinations, the regularized TWS combined solutions demonstrate lower noise levels. Among them, the second-order Tikhonov regularization performs slightly better than other constraints, providing lower noise levels. This study offers a novel perspective for exploring GRACE-based TWS combination methodologies.

This paper reports on an innovative mass concentration (mascon) solution obtained with the short-arc approach, named "GCL-Mascon2024", for estimating spatially enhanced mass variations on the Earth's surface by analyzing K- and Ka-band ranging satellite-to-satellite tracking data collected by the Gravity Recovery And Climate Experiment (GRACE) mission. Compared to contemporary GRACE mascon solutions, this contribution has three notable and distinct features: first, this solution recovery process incorporates frequency-dependent data-weighting techniques to reduce the influence of low-frequency noise in observations. Second, this solution uses variably shaped mascon geometry with physical constraints such as coastline and basin boundary geometries to more accurately capture temporal gravity signals while minimizing signal leakage. Finally, we employ a solution regularization scheme that integrates climate factors and cryospheric elevation models to alleviate the ill-posed nature of the GRACE mascon inversion problem. Our research has led to the following conclusions: (a) GCL-Mascon2024 mass anomaly estimates from GRACE data show strong agreement with the (Release) RL06 versions of mascon solutions (GSFC, CSR, JPL) in both spatial and temporal domains; (b) in Greenland and global hydrologic basins, the correlation coefficients of estimated mass changes between GCL-Mascon2024 and other RL06 mascon solutions exceed 95.0 %, with comparable amplitudes, and, especially over non-humid river basins, the GCL-Mascon2024 suppresses random noise by 27.8 % compared to contemporary mascon products; and (c) in desert regions, the analysis of residuals calculated after removing the climatological components from the mass variations indicates that the GCL-Mascon2024 solution achieves noise reductions of over 29.3 % as compared to the GSFC and CSR RL06 mascon solutions. The GCL-Mascon2024 gravity field solution (Yan and Ran, 2025) is available at 10.5281/zenodo.15525467.

The Greenland ice sheet (GrIS) is at present the largest single contributor to global-mass-induced sea-level rise, primarily because of Arctic amplification on an increasingly warmer Earth^1–5. However, the processes of englacial water accumulation, storage and ultimate release remain poorly constrained. Here we show that a noticeable amount of the summertime meltwater mass is temporally buffered along the entire GrIS periphery, peaking in July and gradually reducing thereafter. Our results arise from quantifying the spatiotemporal behaviour of the total mass of water leaving the GrIS by analysing bedrock elastic deformation measured by Global Navigation Satellite System (GNSS) stations. The buffered meltwater causes a subsidence of the bedrock close to GNSS stations of at most approximately 5 mm during the melt season. Regionally, the duration of meltwater storage ranges from 4.5 weeks in the southeast to 9 weeks elsewhere. We also show that the meltwater runoff modelled from regional climate models may contain systematic errors, requiring further scaling of up to about 20% for the warmest years. These results reveal a high potential for GNSS data to constrain poorly known hydrological processes in Greenland, forming the basis for improved projections of future GrIS melt behaviour and the associated sea-level rise⁶.

High-frequency and correlated noise filtering is one of the important preprocessing steps for GRACE level-2 products before calculating mass anomaly. Decorrelation and denoising kernel (DDK) filters are usually considered as such optimal filters to solve this problem. In this work, a sparse DDK filter is proposed. This is achieved by replacing Tikhonov regularization in traditional DDK filters with weighted L1 norm regularization. The proposed sparse DDK filter adopts a time-varying error covariance matrix, while the equivalent signal covariance matrix is adaptively determined by the Gravity Recovery and Climate Experiment (GRACE) monthly solution. The covariance matrix of the sparse DDK filtered solution is also developed from the Bayesian and error-propagation perspectives, respectively. Furthermore, we also compare and discuss the properties of different filters. The proposed sparse DDK has all the advantages of traditional filters, such as time-varying, location inhomogeneity, and anisotropy, etc. In addition, the filtered solution is sparse; that is, some high-degree and high-order terms are strictly zeros. This sparsity is beneficial in the following sense: high-degree and high-order sparsity mean that the dominating noise in high-degree and high-order terms is completely suppressed, at a slight cost that the tiny signals of these terms are also discarded. The Center for Space Research (CSR) GRACE monthly solutions and their error covariance matrices, from January 2004 to December 2010, are used to test the performance of the proposed sparse DDK filter. The results show that the sparse DDK can effectively decorrelate and denoise these data.

A novel technique has been developed to assess noise levels in GRACE-based mass anomaly time-series when the true signal is not known. The technique is based on computing an optimal combination of analyzed time-series in the presence of a regularization. To find the optimal weights associated with individual time-series, variance component estimation is used. In this way, noise variance (and, therefore, noise standard deviation) for each time-series is estimated. To validate the developed technique, altimetry-based water level variations in several lakes are used as independent information. Those variations are compared with mass anomaly time-series extracted from eight GRACE models of time-varying Earth’s gravity field from different data processing centers. The lake tests demonstrate a good performance of the developed technique, provided that the regularization functional is properly chosen. The best results are obtained with a novel regularization functional, which can be understood as a minimization of year-to-year differences between the values of the second time-derivative of the unknown function. Finally, the GRACE models under consideration are analyzed globally. It is found that the models produced at the Institute of Geodesy at Graz University of Technology (ITSG) and at the Center of Space Research of the university of Texas at Austin (CSR) show, in general, the lowest noise levels. The aforementioned lake tests also allow the signal damping in GRACE models to be quantified. It is shown, among others, that regularized GRACE models may suffer from a noticeable signal damping (up to ∼ 15 %).

Mascon products derived from Gravity Recovery and Climate Experiment satellite gravimetry data are widely used to study the Greenland ice sheet mass balance. However, the products released by different research groups—JPL, CSR, and GSFC—show noticeable discrepancies. To understand them, we compare those mascon products with mascon solutions computed in-house using a varying regularization parameter. We show that the observed discrepancies are likely dominated by differences in the applied regularization. Furthermore, we present a numerical study aimed at an in-depth analysis of regularization-driven biases in the solutions. We demonstrate the ability of our simulations to reproduce 60%–80% of biases observed in real data, which proves that our simulations are sufficiently realistic. After that, we demonstrate that the quality of mascon-based estimates can be increased by a proper modification of the applied regularization: no correlation between mascons is assumed when they belong to different drainage systems. Using both simulations and real data analysis, we show that the improved regularization mitigates signal leakage between drainage systems by 11%–56%. Finally, we validate various mascon solutions over the SW drainage system, using trends from (i) the GOCO-06S model and (ii) the Input-Output Method as control data. In general, the in-house computed trend estimates are consistent with the trends from CSR and JPL solutions and the trends from the control data.

To monitor temporal variations of the Earth’s gravity field and mass transport in the Earth’s system, data from gravity recovery and climate experiment (GRACE) satellite mission and its successor GRACE Follow-On (GFO) are used. To fill in the temporal gap between these missions, other satellites’ kinematic orbits derived from GPS-based high-low satellite-to-satellite tracking data may be considered. However, it is well known that kinematic orbits are highly sensitive to various systematic errors. These errors are responsible for a non-stationary noise in the kinematic orbits, which is difficult to handle. As a result, the quality of the obtained gravity field solutions is reduced. In this research, we propose to apply an epoch-difference (ED) scheme in the context of the classical dynamic approach to gravity field recovery. Compared to the traditional undifferenced (UD) scheme, the ED scheme is able to mitigate constant or slowly varying systematic errors. To demonstrate the added value of the ED scheme, three sets of monthly gravity field solutions produced from 6 years of GRACE kinematic orbits are compared: two sets produced in-house (with the ED and UD scheme), and a set produced with the undifferenced scheme in the frame of the short-arc approach (Zehentner and Mayer-Gürr in J Geodesy 90(3):275–286, 2015. https://doi.org/10.1007/s00190-015-0872-7). As a reference, we use state-of-the-art ITSG-Grace2018 monthly gravity field solutions. A comparison in the spectral domain shows that the gravity field solutions suffer from a lower noise level when the ED scheme is applied, particularly at low-degree terms, with cumulative errors up to degree 20 being reduced by at least 20%. In the spatial domain, the ED scheme notably reduces noise levels in the mass anomalies recovered. In addition, the signals in terms of mean mass anomalies in selected regions become closer to those inferred from ITSG-Grace2018 solutions, while showing no evidence of any damping, when the ED scheme is used. We conclude that the proposed ED scheme is preferable for time-varying gravity field modeling, as compared to the traditional UD scheme. Our findings may facilitate, among others, bridging the gap between GRACE and GFO satellite mission.

We propose a technique to regularize a GRACE-based mass-anomaly time-series in order to (i) quantify the Standard Deviation (SD) of random noise in the data, and (ii) reduce the level of that noise. The proposed regularization functional minimizes the Month-to-month Year-to-year Double Differences (MYDD) of mass anomalies. As such, it does not introduce any bias in the linear trend and the annual component, two of the most common features in GRACE-based mass anomaly time-series. In the context of hydrological and ice sheet studies, the proposed regularization functional can be interpreted as an assumption about the stationarity of climatological conditions. The optimal regularization parameter and noise SD are obtained using Variance Component Estimation. To demonstrate the performance of the proposed technique, we apply it to both synthetic and real data. In the latter case, two geographic areas are considered: the Tonlé Sap basin in Cambodia and Greenland. We show that random noise in the data can be efficiently (1.5–2 times) mitigated in this way, whereas no noticeable bias is introduced. We also discuss various findings that can be made on the basis of the estimated noise SD. We show, among others, that knowledge of noise SD facilitates the analysis of differences between GRACE-based and alternative estimates of mass variations. Moreover, inaccuracies in the latter can also be quantified in this way. For instance, we find that noise in the surface mass anomalies in Greenland estimated using the Regional Climate Model RACMO2.3 is at the level of 2–6 cm equivalent water heights. Furthermore, we find that this noise shows a clear correlation with the amplitude of annual mass variations: it is lowest in the north-west of Greenland and largest in the south. We attribute this noise to limitations in the modelling of the meltwater accumulation and run-off.

Time-varying Stokes coefficients estimated from GRACE satellite data are routinely converted into mass anomalies at the Earth’s surface with the expression proposed for that purpose by Wahr et al. (J Geophys Res 103(B12):30,205–30,229, 1998). However, the results obtained with it represent mass transport at the spherical surface of 6378 km radius. We show that the accuracy of such conversion may be insufficient, especially if the target area is located in a polar region and the signal-to-noise ratio is high. For instance, the peak values of mean linear trends in 2003–2015 estimated over Greenland and Amundsen Sea embayment of West Antarctica may be underestimated in this way by about 15%. As a solution, we propose an updated expression for the conversion of Stokes coefficients into mass anomalies. This expression is based on the assumptions that: (i) mass transport takes place at the reference ellipsoid and (ii) at each point of interest, the ellipsoidal surface is approximated by the sphere with a radius equal to the current radial distance from the Earth’s center (“locally spherical approximation”). The updated expression is nearly as simple as the traditionally used one but reduces the inaccuracies of the conversion procedure by an order of magnitude. In addition, we remind the reader that the conversion expressions are defined in spherical (geocentric) coordinates. We demonstrate that the difference between mass anomalies computed in spherical and ellipsoidal (geodetic) coordinates may not be negligible, so that a conversion of geodetic colatitudes into geocentric ones should not be omitted.

The Gravity Recovery And Climate Experiment (GRACE) mission has achieved a quantum leap in knowledge of the Earth's gravity field. However, current gravity field solutions still cannot reach the prelaunch baseline accuracy. One of the reasons for that is the presence of colored noise in GRACE data, which is typically ignored in the classical dynamic approach to gravity field modeling. In this research, we propose to account for colored noise in the classical dynamic approach by applying the frequency-dependent data weighting (FDDW) scheme, so that enhanced estimates of gravity field solutions are produced. The monthly solutions are compared with those produced using the standard least squares adjustment without a data weighting scheme. The comparison is performed in both spectral and spatial domains, showing the positive effect of the FDDW scheme in all considered cases. For instance, the cumulative geoid height errors up to degree 96 are reduced by 18%. In the spatial domain, the FDDW scheme lowers noise level in mass changes over the oceans, Mississippi river basin, and Greenland by 20, 38, and 23%, respectively, when compared to the without a data weighting scheme. In addition, the consistency of mass changes over the Mississippi and Congo river basins with those inferred from the state-of-the-art hydrology model WaterGAP is substantially improved when the FDDW scheme is applied. These results indicate that modeling colored noise in the GRACE data allows to significantly improve the recovered monthly solutions. This finding is likely applicable also to the GRACE Follow-On mission.