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

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

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

The ocean load in glacial isostatic adjustment (GIA) modelling is represented by the so-called sea level equation (SLE). The SLE describes the mass redistribution of water between ice sheets and oceans on a deforming Earth. Despite various teams independently investigating GIA, there has been no systematic intercomparison among the numerical solvers of the SLE through which the methods may be validated. The goal of this paper is to present a series of synthetic examples designed for testing and comparing the numerical implementations of the SLE in GIA modelling. The 10 numerical codes tested combine various temporal and spatial parametrizations. The time-domain or Laplace-domain discretizations are used to solve the SLE through time, while spherical harmonics, finite differences or finite elements parametrize the GIA-related field variables spatially. The surface ice-water load and solid Earth's topography are represented spatially either on an equiangular grid, a Gauss-Legendre or an equiarea grid with icosahedron-shaped spherical pixels. Comparisons aremade in a series of five benchmark examples with an increasing degree of complexity. Due to the complexity of the SLE, there is no analytical solution to it. The accuracy of the numerical implementations is therefore assessed by the differences of the individual solutions with respect to a reference solution. While the benchmark study does not result in GIA predictions for a realistic loading scenario, we establish a set of agreed-upon results that can be extended in the future by including more complex case studies, such as solutions with realistic loading scenarios, the rotational feedback in the linear-momentum equation, and by considering a 3-D viscosity structure of the Earth's mantle. The test computations performed so far show very good agreement between the individual results and their ability to capture the main features of sea-surface variation and the surface vertical displacement. The differences found can often be attributed to the different approximations inherent in the various algorithms. This shows the accuracy that can be expected from different implementations of the SLE, which helps to assess differences noted in the literature between predictions for realistic loading cases.

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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

Annual cycles in the geocenter motion time series are primarily driven by mass changes in the Earth’s hydrologic system, which includes land hydrology, atmosphere, and oceans. Seasonal variations of the geocenter motion have been reliably determined according to Sun et al. (J Geophys Res Solid Earth 121(11):8352–8370, 2016) by combining the Gravity Recovery And Climate Experiment (GRACE) data with an ocean model output. In this study, we reconstructed the observed seasonal geocenter motion with geophysical model predictions of mass variations in the polar ice sheets, continental glaciers, terrestrial water storage (TWS), and atmosphere and dynamic ocean (AO). The reconstructed geocenter motion time series is shown to be in close agreement with the solution based on GRACE data supporting with an ocean bottom pressure model. Over 85% of the observed geocenter motion time series, variance can be explained by the reconstructed solution, which allows a further investigation of the driving mechanisms. We then demonstrated that AO component accounts for 54, 62, and 25% of the observed geocenter motion variances in the X, Y, and Z directions, respectively. The TWS component alone explains 42, 32, and 39% of the observed variances. The net mass changes over oceans together with self-attraction and loading effects also contribute significantly (about 30%) to the seasonal geocenter motion in the X and Z directions. Other contributing sources, on the other hand, have marginal (less than 10%) impact on the seasonal variations but introduce a linear trend in the time series.

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We present a high resolution model of the linear trend in the Earth’s mass variations based on DMT-2 (Delft Mass Transport model, release 2). DMT-2 was produced primarily from K-Band Ranging (KBR) data of the Gravity Recovery And Climate Experiment (GRACE). It comprises a time series of monthly solutions complete to spherical harmonic degree 120. A novel feature in its production was the accurate computation and incorporation of stochastic properties of coloured noise when processing KBR data. The unconstrained DMT-2 monthly solutions are used to estimate the linear trend together with a bias, as well as annual and semi-annual sinusoidal terms. The linear term is further processed with an anisotropic Wiener filter, which uses full noise and signal covariance matrices. Given the fact that noise in an unconstrained model of the trend is reduced substantially as compared to monthly solutions, the Wiener filter associated with the trend is much less aggressive compared to a Wiener filter applied to monthly solutions. Consequently, the trend estimate shows an enhanced spatial resolution. It allows signals in relatively small water bodies, such as Aral sea and Ladoga lake, to be detected. Over the ice sheets, it allows for a clear identification of signals associated with some outlet glaciers or their groups. We compare the obtained trend estimate with the ones from the CSR-RL05 model using (i) the same approach based on monthly noise covariance matrices and (ii) a commonly-used approach based on the DDK-filtered monthly solutions. We use satellite altimetry data as independent control data. The comparison demonstrates a high spatial resolution of the DMT-2 linear trend. We link this to the usage of high-accuracy monthly noise covariance matrices, which is due to an accurate computation and incorporation of coloured noise when processing KBR data. A preliminary comparison of the linear trend based on DMT-2 with that computed from GSFC global mascons v01 reveals, among other, a high concentration of the signal along the coast for both models in areas like the ice sheets, Gulf of Alaska, and Iceland.@en

Geocenter motion and changes in the Earth’s dynamic oblateness (J2) are of great importance in many applications. Among others, they are critical indicators of largescale mass redistributions, which is invaluable to understand ongoing global climate change. The revolutionary Gravity Recovery and Climate Experiment (GRACE) satellite mission enables a constant monitoring of redistributing masses within the Earth’s system. However, it still cannot provide reliable time variations in degree-1 coefficients and degree-2 zonal coefficients, which are directly related to geocenter motion and J2 variations.@en

The HY-2A satellite is the first ocean dynamic environment monitoring satellite of China. Centimeter-level radial accuracy is a fundamental requirement for its scientific research and applications. To achieve this goal, we designed the strategies of precise orbit determination (POD) in detail. To achieve the relative optimal orbit for HY-2A, we carried out POD using DORIS-only, SLR-only, and DORIS + SLR tracking data, respectively. POD tests demonstrated that the consistency level of DORIS-only and SLR-only orbits with respect to the CNES orbits were about 1.81 cm and 3.34 cm in radial direction in the dynamic sense, respectively. We designed 6 cases of different weight combinations for DORIS and SLR data, and found that the optimal relative weight group was 0.2 mm/s for DORIS and 15.0 cm for SLR, and RMS of orbit differences with respect to the CNES orbits in radial direction and three-dimensional (3D) were 1.37 cm and 5.87 cm, respectively. These tests indicated that the relative radial and 3D accuracies computed using DORIS + SLR data with the optimal relative weight set were obviously higher than those computed using DORIS-only and SLR-only data, and satisfied the requirement of designed precision. The POD for HY-2A will provide the invaluable experience for the following HY-2B, HY-2C, and HY-2D satellites.

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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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Uncertainty in the geocenter position and its subsequent motion affects positioning estimates on the surface of the Earth and downstream products such as site velocities, particularly the vertical component. The current version of the International Terrestrial Reference Frame, ITRF2014, derives its origin as the long-term averaged center of mass as sensed by satellite laser ranging (SLR), and by definition, it adopts only linear motion of the origin with uncertainty determined using a white noise process. We compare weekly SLR translations relative to the ITRF2014 origin, with network translations estimated from station displacements from surface mass transport models. We find that the proportion of variance explained in SLR translations by the model-derived translations is on average less than 10%. Time-correlated noise and nonlinear rates, particularly evident in the Y and Z components of the SLR translations with respect to the ITRF2014 origin, are not fully replicated by the model-derived translations. This suggests that translation-related uncertainties are underestimated when a white noise model is adopted and that substantial systematic errors remain in the data defining the ITRF origin. When using a white noise model, we find uncertainties in the rate of SLR X, Y, and Z translations of ±0.03, ±0.03, and ±0.06, respectively, increasing to ±0.13, ±0.17, and ±0.33 (mm/yr, 1 sigma) when a power law and white noise model is adopted.

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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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A new method is proposed to connect the orthometric height datum of island far from mainland with measurements of both ship-borne gravimetry and GNSS along the ship route connecting these datums. The methodology of route height datum connection is based on the astronomical leveling principle. The effect of the deflection of the vertical (DOV) error and the ellipsoidal height difference error are major factors on the precision of the orthometric height connection. To improve the computational accuracy of DOVs along the ship route, the collinear adjustment is used to improve the accuracy of the gravity measurements and GNSS ellipsoidal heights. The remove-restore technique in conjunction with the measured DOVs on the mainland and island is used to improve the estimation accuracy of the DOVs along the ship route. The least squares collocation method is used to estimate the residual DOVs. The covariance of gravity anomalies and cross-covariance function between gravity anomalies and DOVs are derived from the disturbing potential spherical harmonic expansion. The optimal partition number is also studied for the high precision orthometric height connection across sea. Finally, we use the proposed method to connect the orthometric height datum point on Qingdao coast in Shandong Province of China with the datum point on Caoyu coast in Fujian Province of China.@en

. In this study, for the first time, an attempt is made to close the sea level budget on a sub-basin scale in terms of trend and amplitude of the annual cycle. We also compare the residual time series after removing the trend, the semiannual and the annual signals. To obtain errors for altimetry and Argo, full variance–covariance matrices are computed using correlation functions and their errors are fully propagated. For altimetry, we apply a geographically dependent intermission bias (Ablain et al., 2015), which leads to differences in trends up to 0.8 mm yr−1 . Since Argo float measurements are non-homogeneously spaced, steric sea levels are first objectively interpolated onto a grid before averaging. For the Gravity Recovery And Climate Experiment (GRACE), gravity fields full variance–covariance matrices are used to propagate errors and statistically filter the gravity fields. We use four different filtered gravity field solutions and determine which post-processing strategy is best for budget closure. As a reference, the standard 96 degree Dense Decorrelation Kernel-5 (DDK5)-filtered Center for Space Research (CSR) solution is used to compute the mass component (MC). A comparison is made with two anisotropic Wiener- filtered CSR solutions up to degree and order 60 and 96 and a Wiener-filtered 90 degree ITSG solution. Budgets are computed for 10 polygons in the North Atlantic Ocean, defined in a way that the error on the trend of the MC plus steric sea level remains within 1 mm yr−1 . Using the anisotropic Wiener filter on CSR gravity fields expanded up to spherical harmonic degree 96, it is possible to close the sea level budget in 9 of 10 sub-basins in terms of trend. Wiener-filtered Institute of Theoretical geodesy and Satellite Geodesy (ITSG) and the standard DDK5-filtered CSR solutions also close the trend budget if a glacial isostatic adjustment (GIA) correction error of 10–20 % is applied; however, the performance of the DDK5-filtered solution strongly depends on the orientation of the polygon due to residual striping. In 7 of 10 sub-basins, the budget of the annual cycle is closed, using the DDK5-filtered CSR or the Wiener-filtered ITSG solutions. The Wiener-filtered 60 and 96 degree CSR solutions, in combination with Argo, lack amplitude and suffer from what appears to be hydrological leakage in the Amazon and Sahel regions. After reducing the trend, the semiannual and the annual signals, 24–53 % of the residual variance in altimetryderived sea level time series is explained by the combination of Argo steric sea levels and the Wiener-filtered ITSG MC. Based on this, we believe that the best overall solution for the MC of the sub-basin-scale budgets is the Wiener-filtered ITSG gravity fields. The interannual variability is primarily a steric signal in the North Atlantic Ocean, so for this the choice of filter and gravity field solution is not really signifi- cant. @en

The focus of the study is optimizing the technique for estimating geocenter motion and variations in J2 by combining data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission with output from an Ocean Bottom Pressure model and a Glacial Isostatic Adjustment (GIA) model. First, we conduct an end-to-end numerical simulation study. We generate input time-variable gravity field observations by perturbing a synthetic Earth model with realistically simulated errors. We show that it is important to avoid large errors at short wavelengths and signal leakage from land to ocean, as well as to account for self-attraction and loading effects. Second, the optimal implementation strategy is applied to real GRACE data. We show that the estimates of annual amplitude in geocenter motion are in line with estimates from other techniques, such as satellite laser ranging (SLR) and global GPS inversion. At the same time, annual amplitudes of C10 and C11 are increased by about 50% and 20%, respectively, compared to estimates based on Swenson et al. (2008). Estimates of J2 variations are by about 15% larger than SLR results in terms of annual amplitude. Linear trend estimates are dependent on the adopted GIA model but still comparable to some SLR results.@en