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 %).

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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.

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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.

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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.

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We present a new approach to estimate time variations in J2. Those variations are represented as the sum of contributions from individual sources. This approach uses solely Gravity Recovery And Climate Experiment (GRACE) data and the geoid fingerprints of mass redistributions that take place both at the surface and in the interior of the solid Earth. The results agree remarkably well with those based on satellite laser ranging, while estimates of the sources explain the observed variations in J2. Seasonal variations are dominated by terrestrial water storage and by mass redistribution in the atmosphere and ocean. Trends, however, are primarily controlled by the Greenland and Antarctic ice sheets and by glacial isostatic adjustment. The positive trend from surface mass variations is larger than the negative trend due to glacial isostatic adjustment and leads to an overall rising trend during the GRACE period (2002–2017).@en

Satellite gravimetry data acquired by the Gravity Recovery and Climate Experiment (GRACE) allows to derive the temporal evolution in ice mass for both the Antarctic Ice Sheet (AIS) and the Greenland Ice Sheet (GIS). Various algorithms have been used in a wide range of studies to generate Gravimetric Mass Balance (GMB) products. Results from different studies may be affected by substantial differences in the processing, including the applied algorithm, the utilised background models and the time period under consideration. This study gives a detailed description of an assessment of the performance of GMB algorithms using actual GRACE monthly solutions for a prescribed period as well as synthetic data sets. The inter-comparison exercise was conducted in the scope of the European Space Agency’s Climate Change Initiative (CCI) project for the AIS and GIS, and was, for the first time, open to everyone. GMB products generated by different groups could be evaluated and directly compared against each other. For the period from 2003-02 to 2013-12, estimated linear trends in ice mass vary between −99 Gt/yr and −108 Gt/yr for the AIS and between −252 Gt/yr and −274 Gt/yr for the GIS, respectively. The spread between the solutions is larger if smaller drainage basins or gridded GMB products are considered. Finally, findings from the exercise formed the basis to select the algorithms used for the GMB product generation within the AIS and GIS CCI project.@en

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.

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The mascon approach is a well-known technique to estimate mass anomalies in Greenland using GRACE satellite gravity data. It partitions the area of interest into laterally-homogeneous patches (mascons). An important aspect of the mascon approach is the chosen geometry of mascons. So far, its impact has not been fully understood. In this study, we use a full-scale numerical study and real data analysis to identify the optimal strategy (primarily, the size of the mascons) for the extrac- tion of mass anomalies over the Greenland drainage systems from GRACE monthly solutions. We use ordinary and weighted least-squares techniques to estimate the mascon parameters. The weighted least-squares estimator uses the full noise covari- ance matrices of monthly GRACE models, i.e. is designed to suppress random noise in the estimates. In addition, the zero-order Tikhonov regularization is applied. Four types of quantities of interest, which are associated with di
erent temporal scales, are investigated in this study: monthly mass anomalies, mean mass anomalies per calendar month, inter-annual mass variations, and long-term linear trends. We show that the dominant error sources are random errors and parameterization (model) errors, as well as the bias introduced by the regularization. Errors in long-term lin- ear trend estimates are dominated by parameterization errors, whereas the role of random errors increases with the decreasing temporal scale. The best solutions are Downloaded from https://academic.oup.com/gji/advance-article-abstract/doi/10.1093/gji/ggy242/5040767 by guest on 05 July 2018 2 J. Ran, P. Ditmar and R. Klees obtained when the territory of Greenland is split into at least 23 mascons (the area of each one being 90; 000 km2). The usage of smaller mascons does not worsen the solutions in most cases, which is explained by the application of the regularization. Usage of larger mascons leads in most cases to inferior results due to the impact of parameterization errors. The application of the weighted least-squares estimator noticeably improves the quality of the solutions, with the exception of long-term linear trends estimated at the drainage system scale. In addition, we considered the long-term linear trend estimates integrated over entire Greenland. It is shown that the best results are obtained in that case when no regularization is applied. The results of real GRACE data processing are consistent with those obtained in the numerical study.
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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.

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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.

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Global Glacial Isostatic Adjustment (GIA) processes are usually represented by numerical models that simultaneously solve for glacial evolution and Earth rheology, being mainly constrained by the geological evidences of local ice extent and global sea level, as well as geodetic observations of Earth rotation. In recent years, GPS and GRACE observations have often been used to improve those models, especially in the context of regional studies, where the largest effects of lateral heterogeneities in the Earth structure are expected. However, regional models are intrinsically limited when it comes to answering questions from global scale geodesy. Examples are the closure of the sea level budget, the explanation of observed changes in Earth rotation, and the determination of the origin of the Earth reference frame. Furthermore, the issue of the consistency of regional models with each other is typically ignored. We consider this as a drawback, since such a consistency may not only be considered as an additional constraint, but is also important if the regional models are to be integrated into a global one. Here, we present a global empirical model of present-day mass changes driven by GIA, solely based on GRACE data and on geoid fingerprints of regional GIA combined with surface mass redistribution. We will show how the use of observations from a single space-borne platform allows us to tackle the questions from global scale geodesy mentioned above, and at the same time obtain a self-consistent GIA model at regional scales. @en

A methodology has been developed to estimate the accuracy of GRACE monthly solutions as functions of space or time without using independent geophysical models. An application of the methodology to several commonly-used solution time-series reveals that the ITSG-Grace2016 solutions show the highest accuracy among those considered. Furthermore, it is found that the accuracy of background models exploited to produce GRACE RL05 solutions was likely insufficient to describe adequately mass re-distribution in the oceans at both short (< 1 month) and long (>1 month) time scales. Insufficiently accurate modelling of ocean signals may reduce the accuracy of mass anomaly estimates not only over oceans, but also in the coastal areas of continents.@en

The Greenland Ice Sheet (GrIS) is currently losing ice mass as the result of changes in the complex ice-climate interactions that have been driven by global climate change. In order to accurately predict future sea level rise, the mechanisms driving the observed mass loss must be better understood. Here, we combine data from the satellite gravimetry mission GRACE, surface mass balance (SMB) output of RACMO 2.3, and ice discharge estimates to analyze the mass budget of Greenland at various temporal and spatial scales. Firstly, in agreement with previous estimates, we find that the rate of mass loss from Greenland observed by GRACE was between −277 and −269 Gt/yr in 2003–2012. This estimate is consistent with the sum of individual contributions: surface mass balance (SMB, around 216 ± 122 Gt/yr) and ice discharge (520 ± 31 Gt/yr), indicating a good performance of the regional climate model. Secondly, we examine the average accelerations of mass anomalies in Greenland over 2003–2012, suggesting that the SMB (−23.3 ± 2.7 Gt/yr2) contributes 75 % to the total acceleration observed by GRACE. The remaining contributions to the mass loss acceleration for entire Greenland are statistically insignificant. Finally and most importantly, this study suggests the existence of a substantial meltwater storage during summer, with a peak value of 80–120 Gt in July. The robustness of this estimate is demonstrated by using both different GRACE-based solutions and different meltwater runoff estimates (namely, RACMO 2.3 and SNOWPACK). Meltwater storage in the ice sheet occurs primarily due to storage in the high-accumulation regions of the southeast (SE) and northwest (NW) parts of Greenland. Analysis of seasonal variations in outlet glacier discharge shows that the contribution of ice discharge to the observed signal is minor (at the level of only a few Gt) and does not explain the intra-annual differences between the total mass and SMB signals.@en

With the launch of the Gravity Recovery and Climate Experiment (GRACE) satellite mission in 2002 (http://www.csr.utexas.edu/grace), Satellite Gravimetry has become a unique tool to estimate hydrological water balance and mass balance of ice sheets, as well as to monitor mass re-distribution in the oceans and the solid Earth. However, satellite gravimetry still suffers from a poor estimation of temporal variations in the spherical harmonic coefficient C20 (which is associated with the Earth's dynamic oblateness). Therefore, these variations are typically extracted from Satellite Laser Ranging (SLR) data. Furthermore, satellite gravimetry is not sensitive to variations of degree-1 spherical harmonic coefficients (i.e., C10, C11, and S11), which are associated with the geocentre motion. Swenson et al (2008) proposed to restore those coefficients using as a reference an area where the mass anomalies are known. Such an area was chosen as the entire world ocean; mass anomalies there were defined as variations of the Ocean Bottom Pressure based on an ocean circulation model. The Glacial Isostatic Adjustment signal was corrected for by applying a remove-restore approach. Sun et al (2016) further developed the technique by Swenson et al (2008). First, the Self-Attraction and Loading (SAL) effects were additionally modelled in order to estimate water re-distribution in the ocean more accurately. Second, a buffer zone around the continents was excluded from the reference area in order to suppress the effect of “signal leakage” caused by a limited spatial resolution of satellite gravimetry. It was shown that the modified technique allows for an accurate estimation of both degree-1 and C20 variations.@en

Global Glacial Isostatic Adjustment (GIA) processes are usually represented by numerical models that simultaneously solve for glacial evolution and Earth rheology, being mainly constrained by the geological evidences of local ice extent and global sea level, as well as geodetic observations of Earth rotation. In recent years, GPS and GRACE observations have often been used to improve those models, especially in the context of regional studies, where the largest effects of lateral heterogeneities in the Earth structure are expected. However, regional models are intrinsically limited when it comes to answering questions from global scale geodesy. Examples are the closure of the sea level budget, the explanation of observed changes in Earth rotation, and the determination of the origin of the Earth reference frame. Furthermore, the issue of the consistency of regional models with each other is typically ignored. We consider this as a drawback, since such a consistency may not only be considered as an additional constraint, but is also important if the regional models are to be integrated into a global one. Here, we present a global empirical model of present-day mass changes driven by GIA, solely based on GRACE data and on geoid fingerprints of regional GIA combined with surface mass redistribution. We will show how the use of observations from a single space-borne platform allows us to tackle the questions from global scale geodesy mentioned above, and at the same time obtain a self-consistent GIA model at regional scales. @en

Transient meltwater accumulation in Greenland spans the entire ice layer, down to the ice bed. As such, this process may have a large impact on Greenland ice dynamics and on the future ice sheet evolution. GrIS subglacial hydrology is an area of active research. Unfortunately, it is difficult to validate and calibrate existing subglacial hydrology models because of intrinsic data limitations. Data collected in situ or from aircraft have a very limited spatial or/and temporal coverage, whereas space-borne electromagnetic sensors (both passive and active) are capable of observing only the near-surface part of the ice layer. With our study, we present the first direct observations of transient meltwater accumulation in Greenland with satellite gravimetry. We estimate total mass anomalies using GRACE satellite mission data and subtract from them the contributions associated with the Surface Mass Balance (SMB) and the Ice Discharge (ID). The SMB estimates are provided by the Regional Atmospheric Climate Model v. 2.3 (RACMO 2.3). The signal related to ID is approximated by a linear function fitting the “Total minus SMB” residuals in spring and autumn months. An analysis of seasonal variations in ice flow at 55 outlet glaciers in northwest and southeast Greenland shows that the deviations of ID-related mass anomalies from a linear trend are negligible. By taking the average of seasonal mass variations in 2003–2013, we observe substantial meltwater accumulation in Greenland during summer, with a peak value of 80-120 Gt in July. At the regional scale, the largest accumulation is observed in the southeast and northwest parts of Greenland: up to about 40 Gt in each region. The extracted signal is not altered substantially when using an alternative snow and firn model called SNOWPACK, which partitions meltwater into refreezing and runoff differently, as compared to RACMO 2.3. Furthermore, the meltwater accumulation signal is present in all of the considered GRACE data product variants and processing schemes, though with somewhat different magnitude and timing. Our study shows that GRACE data are capable of sensing transient meltwater accumulation not only at the scale of entire GrIS, but also at the scale of individual drainage systems. A continuation of research efforts is envisioned in order to further improve the accuracy of the obtained estimates.@en

In this study, we develop a methodology to estimate monthly variations in degree-1 andC₂₀ coefficients by combing Gravity Recovery and Climate Experiment (GRACE) data withoceanic mass anomalies (combination approach).With respect to the method by Swenson et al.,the proposed approach exploits noise covariance information of both input data sets and thusproduces stochastically optimal solutions supplied with realistic error information. Numericalsimulations show that the quality of degree-1 and -2 coefficients may be increased in this wayby about 30 per cent in terms of RMS error.We also proved that the proposed approach can bereduced to the approach of Sun et al. provided that the GRACE data are noise-free and noise inoceanic data is white. Subsequently, we evaluate the quality of the resulting degree-1 and C₂₀coefficients by estimating mass anomaly time-series within carefully selected validation areas,where mass transport is small. Our validation shows that, compared to selected Satellite LaserRanging (SLR) and joint inversion degree-1 solutions, the proposed combination approachbetter complementsGRACE solutions. The annual amplitude of the SLR-based C₁₀ is probablyoverestimated by about 1 mm. The performance of the C₂₀ coefficients, on the other hand, issimilar to that of traditionally used solution from the SLR technique.

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An accurate estimation of water resources dynamics is crucial for proper management of both agriculture and the local ecology, particularly in semi-arid regions. Imperfections in model physics, uncertainties in model land parameters and meteorological data, as well as the human impact on land changes often limit the accuracy of hydrological models in estimating water storages. To mitigate this problem, this study investigated the assimilation of terrestrial water storage variation (TWSV) estimates derived from the Gravity Recovery And Climate Experiment (GRACE) data using an ensemble Kalman filter (EnKF) approach. The region considered was the Hexi Corridor in northern China. The hydrological model used for the analysis was PCR-GLOBWB, driven by satellite-based forcing data from April 2002 to December 2010. The impact of the GRACE data assimilation (DA) scheme was evaluated in terms of the TWSV, as well as the variation of individual hydrological storage estimates. The capability of GRACE DA to adjust the storage level was apparent not only for the entire TWSV but also for the groundwater component. In this study, spatially correlated errors in GRACE data were taken into account, utilizing the full error variance-covariance matrices provided as a part of the GRACE data product. The benefits of this approach were demonstrated by comparing the EnKF results obtained with and without taking into account error correlations. The results were validated against in situ groundwater data from five well sites. On average, the experiments showed that GRACE DA improved the accuracy of groundwater storage estimates by as much as 25 %. The inclusion of error correlations provided an equal or greater improvement in the estimates. In contrast, a validation against in situ streamflow data from two river gauges showed no significant benefits of GRACE DA. This is likely due to the limited spatial and temporal resolution of GRACE observations. Finally, results of the GRACE DA study were used to assess the status of water resources over the Hexi Corridor over the considered 9-year time interval. Areally averaged values revealed that TWS, soil moisture, and groundwater storages over the region decreased with an average rate of approximately 0.2, 0.1, and 0.1 cm yr^-1 in terms of equivalent water heights, respectively. A particularly rapid decline in TWS (approximately -0.4 cm yr^-1) was seen over the Shiyang River basin located in the southeastern part of Hexi Corridor. The reduction mostly occurred in the groundwater layer. An investigation of the relationship between water resources and agricultural activities suggested that groundwater consumption required to maintain crop yield in the growing season for this specific basin was likely the cause of the groundwater depletion.

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