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.

This paper investigates the full variance–covariance (VC) matrix of two high-resolution regional quasi-geoid models, utilizing a spherical radial basis function parameterization. Model parameters were estimated using weighted least-squares techniques and variance component estimation (VCE) for data weighting. The first model, known as the “RCR model,” is computed through the remove–compute–restore method, incorporating various local gravity and radar altimeter datasets. The second model, the “combined model,” includes the GOCO05s satellite-only global geopotential model as an additional dataset with a full-noise VC matrix. Validation of the noise VC matrix scaling for each quasi-geoid model is achieved by comparing observed and formal noise standard deviations of differences between geometric and gravimetric height anomalies at GPS height markers in the Netherlands. Analysis of the noise VC matrix of height anomalies at grid nodes reveals significantly smaller formal noise standard deviations for the RCR model compared to the combined model. This difference is attributed to VCE assigning larger weights to the GOCO05s dataset, which exhibits greater noise standard deviations for the specific spatial scales used. Additionally, the formal noise standard deviations of height anomaly differences, relevant for GNSS-heighting, favor the RCR model. However, the disparity between the two models is smaller than implied by the height anomaly noise standard deviations. This is due to the combined model’s noise autocorrelation function displaying a longer correlation length (67 km) in contrast to the RCR model’s (17 km). Consequently, the combined model exhibits a greater reduction in noise variance for height anomaly differences relative to white noise compared to the RCR model.

All realizations of the European Vertical Reference System (EVRS) computed so far are solely based on geopotential differences obtained by spirit leveling/gravimetry. As such, there are no direct connections between height benchmarks separated by large water bodies. In this study, such connections are added by means of model-based hydrodynamic leveling resulting in a new, yet unofficial realization of the EVRS. The model-derived mean water levels used in computing the hydrodynamic leveling connections were obtained from the Nemo-Nordic (Baltic Sea) and 3D DCSM-FM (northwest European continental shelf) hydrodynamic models. The impact of model-based hydrodynamic leveling on the European Vertical Reference Frame is significant, especially for France and Great Britain. Compared to a solution which only uses spirit leveling/gravimetry, the differences in these countries reach tens to hundreds of kgalmm . We also observed an improved agreement with normal heights obtained by differencing GNSS and the European gravimetric quasi-geoid 2015 (EGG2015) heights. In Great Britain, the south-north slope of 48 mm deg ^{- 1} present in the solution which uses only spirit leveling/gravimetry data reduced to 2.2 mm deg ^{- 1} . In France, the improvement is confined to the southwest. The choice of the period over which water levels are averaged has an impact on the results as it determines, among others, the set of tide gauges available to establish the hydrodynamic leveling connections. When using an averaging period that can be considered as the least preferred choice based on three established criteria, the positive impact for France has gone. For Great Britain, the estimated south-north slope became 12.6 mm deg ^{- 1} . This is larger than the slope obtained using the most preferred averaging period but still substantially lower compared to the slope associated with a solution that uses only spirit leveling/gravimetry.

We present exact, closed-form expressions for the complete RTM correction and the harmonic correction to disturbing potential, gravity disturbance, gravity anomaly, and height anomaly. They need to be applied in quasi-geoid modelling whenever data points are buried inside the masses after residual terrain model (RTM) reduction and analytically downward-continued functionals of the disturbing potential at the original locations of the data points are required. Compared to recent work of the authors published in this journal, no Taylor series enter the expressions and numerical instabilities of the harmonic downward continuation from the RTM surface to the Earth’s surface are avoided as are inaccuracies in the free-air upward continuation from the Earth’s surface to the RTM surface caused by a lack of precise information about higher-order derivatives of the disturbing potential. The new expressions can easily be implemented in any existing RTM software package and do not require additional computational resources. For two test areas located in western Norway and the Auvergne in France, we compute the complete RTM correction and the harmonic correction to the afore-mentioned functionals of the disturbing potential. Overall, all harmonic corrections are non-negative with maximum values of 1.54 m ²/ s ², 263.0 μ Gal, 263.9 μ Gal, and 15.7 m (Norway) and 1.55 m ²/ s ², 263.3 μ Gal, 263.3 μ Gal, and 15.8 cm (Auvergne) for disturbing potential, gravity disturbance, gravity anomaly, and height anomaly, respectively. The medians are 0.02 m ²/ s ², 33.6 μ Gal, 33.7 μ Gal, and 0.3 cm (Norway) and 0.01 m ²/ s ², 19.2 μ Gal, 19.2 μ Gal, and 0.1 cm (Auvergne). We also show that the nth Taylor polynomials used in the recent work of the authors published in this journal may have large remainders depending on the topography in the vicinity of the evaluation point no matter how n is chosen. Finally, we show that the commonly used expression for the harmonic correction to gravity anomaly introduced in 1981 is almost exact, though it was derived along a completely different line of reasoning. The errors do not exceed 49 μ Gal in both test areas. Moreover, the errors have a negligible impact on the computed height anomalies in one-centimetre quasi-geoid modelling, as the mean error does not exceed a few μ Gal in both test areas.

Both empirical and assimilative global ocean tidal models are significantly more accurate in the deep ocean than in shelf and coastal waters. In this study, we answered whether this is due to the quality of the models used to reduce tide and surge or the general approach to treat tide and surge as two separate components of the water level obtained from stand-alone models, which ignores the nonlinear tide–surge interaction. In doing so, we used tide gauge observations as partially synthetic altimeter time series, tide–surge water-level time series obtained with the 2D Dutch Continental Shelf Model–Flexible Mesh (DCSM), and tide and surge water-level time series obtained using the DCSM, FES2014 (FES) and the Dynamic Atmospheric Correction (DAC) product. Expressed in the root-sum-square (RSS) of the eight main tidal constituents, we obtained a reduction (Formula presented.) % when removing the DCSM tide–surge water levels compared to when we removed the sum of the DCSM tide and DCSM surge water levels. The RSS obtained in the latter case was only 3.3% lower than with FES and DAC. We conclude that the lower tidal estimates accuracy in shelf-coastal waters derives from the missing nonlinear tide–surge interactions.

The main objective of this study is to develop and analyze an empirical noise model for model-derived coastal summer mean water levels (SMWLs) and use that to obtain a more realistic quality impact of combining hydrodynamic leveling and Unified European Leveling Network (UELN) data in realizing the European Vertical Reference System (EVRS). We considered three state-of-the-art hydrodynamic models for the Northeast Atlantic Ocean, including the North Sea and Wadden Sea; AMM7, DCSMv6-ZUNOv4, and 3D DCSM-FM. Moreover, we assess the spatiotemporal performance of these three models in representing coastal SMWLs. The empirical noise models are determined from the differences between observation- and model-derived SMWLs at coastal tide gauges. All three noise models show that the model noise is indeed correlated over sea distances up to hundreds of kilometers. At the same time, they all show a relatively large discontinuity at the origin (i.e., nugget effect); between 12.1 cm² (3D DCSM-FM) and 16.3 cm² (DCSMv6-ZUNOv4). The variance (i.e., covariance at zero sea distance) for these two models is 15.3 cm² and 21.7 cm², respectively. Averaging the water levels over three summers, lowered the variance and nugget effect for 3D DCSM-FM to 12.7 cm² and 10.0 cm², respectively. Our analysis also showed that between 30 and 50% of the variance has to be attributed to errors in the vertical referencing of the tide gauges. We lacked the information to assess what proportion of the observed noise covariances should be attributed to these errors. The performance assessments revealed significant variations over both space and time as well as among the three hydrodynamic models. The results suggest that there is still room for model improvement. In the final experiments, we used the noise model of the best overall performing model (i.e., 3D DCSM-FM) to reassess the quality impact of combining hydrodynamic leveling and UELN data in realizing the EVRS. The results suggest that not including the noise covariances leads to an overestimation of the total quality impact by 7 % and 8 % , when we average the water levels over one and three summer periods, respectively.

In this paper, we derive improved expressions for the harmonic correction to gravity and, for the first time, expressions for the harmonic correction to potential and height anomaly. They need to be applied at stations buried inside the masses to transform internal values into harmonically downward continued values, which are then input to local quasi-geoid modelling using least-squares collocation or least-squares techniques in combination with the remove-compute-restore approach. Harmonic corrections to potential and height anomaly were assumed to be negligible so far resulting in yet unknown quasi-geoid model errors. The improved expressions for the harmonic correction to gravity, and the new expressions for the harmonic correction to potential and height anomaly are used to quantify the approximation errors of the commonly used harmonic correction to gravity and to quantify the magnitude of the harmonic correction to potential and height anomaly. This is done for two test areas with different topographic regimes. One comprises parts of Norway and the North Atlantic where the presence of deep, long, and narrow fjords suggest extreme values for the harmonic correction to potential and height anomaly and corresponding large errors of the commonly used approximation of the harmonic correction to gravity. The other one is located in the Auvergne test area with a moderate topography comprising both flat and hilly areas and therefore may be representative for many areas around the world. For both test areas, two RTM surfaces with different smoothness are computed simulating the use of a medium-resolution and an ultra-high-resolution reference gravity field, respectively. We show that the errors of the commonly used harmonic correction to gravity may be as large as the harmonic correction itself and attain peak values in areas of strong topographic variations of about 100 mGal. Moreover, we show that this correction may introduce long-wavelength biases in the computed quasi-geoid model. Furthermore, we show that the harmonic correction to height anomaly can attain values on the order of a decimetre at some points. Overall, however, the harmonic correction to height anomaly needs to be applied only in areas of strong topographic variations. In flat or hilly areas, it is mostly smaller than one centimetre. Finally, we show that the harmonic corrections increase with increasing smoothness of the RTM surface, which suggests to use a RTM surface with a spatial resolution comparable to the finest scales which can be resolved by the data rather than depending on the resolution of the global geopotential model used to reduce the data.

In this study, we compare six commonly used methods for the downward continuation of airborne gravity data. We consider exact and noisy simulated data on grids and along flight trajectories and real data from the GRAV-D airborne campaign. We use simulated and real surface gravity data for validation. The methods comprise spherical harmonic analysis, least-squares collocation, residual least-squares collocation, least-squares radial basis functions, the inverse Poisson method and Moritz’s analytical downward continuation method. We show that all the methods perform similar in terms of surface gravity values. For real data, the downward continued airborne gravity values are used to compute a geoid model using a Stokes-integral-based approach. The quality of the computed geoid model is validated using high-quality GSVS17 GPS-levelling data. We show that the geoid model quality is similar for all the methods. However, the least-squares collocation approach appears to be more flexible and easier to use than the other methods provided that the optimal covariance function is found. We recommend it for the downward continuation of GRAV-D data, and other methods for second check.

The first objective of this paper is to assess by means of geodetic network analyses the impact of adding model-based hydrodynamic leveling data to the Unified European Leveling Network (UELN) data on the precision and reliability of the European Vertical Reference Frame (EVRF). In doing so, we used variance information from the latest UELN adjustment. The model-based hydrodynamic leveling data are assumed to be obtained from not-yet existing hydrodynamic models covering either all European seas surrounding the European mainland or parts of it that provide the required mean water level with uniform precision. A heuristic search algorithm was implemented to identify the set of hydrodynamic leveling connections that provide the lowest median of the propagated height standard deviations. In the scenario in which we only allow for connections between tide gauges located in the same sea basin, all having a precision of 3 cm, the median of the propagated height standard deviations improved by 38 % compared to the spirit leveling-only solution. Except for the countries around the Black Sea, coastal countries benefit the most with a maximum improvement of 60 % for Great Britain. We also found decreased redundancy numbers for the observations in the coastal areas and over the entire Great Britain. Allowing for connections between tide gauges among all European seas increased the impact to 42%. Lowering the precision of the hydrodynamic leveling data lowers the impact. The results show, however, that even in case the assumed precision is 5 cm, the overall improvement is still 29%. The second objective is to identify which tide gauges are most profitable in terms of impact. Our results show that these are the ones located in Sweden in which most height markers are located. The impact, however, hardly depends on the geographic location of the tide gauges within a country.

Joint effects of the dynamic sea-level rise projected changes in the large-scale atmosphere/ocean circulation, and wave climate on hurricane-induced extreme water levels in the Caribbean region are assessed. We use the 2D-depth integrated ADCIRC + SWAN wave-ocean model, baroclinically coupled to an ocean-eddying version of the Community Earth System Model, to compare impacts of the September 2017 hurricanes with projected impacts of similar hypothetical tropical storms occurring in the future. The model predicts only minor changes in the hurricane-induced extreme water levels for those Caribbean islands which were severely devastated by the 2017 tropical storms (Irma and Maria). That is, provided that the hurricane intensity remains at the present-day level, the global mean sea-level rise is the main future coastal flood risk factor.

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.

The intensity of major storm events generated within the Atlantic Basin is projected to rise with the warming of the oceans, which is likely to exacerbate coastal erosion. Nature-based flood defence has been proposed as a sustainable and effective solution to protect coastlines. However, the ability of natural ecosystems to withstand major storms like tropical hurricanes has yet to be thoroughly tested. Seagrass meadows both stabilise sediment and attenuate waves, providing effective coastal protection services for sandy beaches. To examine the tolerance of Caribbean seagrass meadows to extreme storm events, and to investigate the extent of protection they deliver to beaches, we employed a combination of field surveys, biomechanical measurements and wave modelling simulations. Field surveys of seagrass meadows before and after a direct hit by the category 5 Hurricane Irma documented that established seagrass meadows of Thalassia testudinum remained unaltered after the extreme storm event. The flexible leaves and thalli of seagrass and calcifying macroalgae inhabiting the meadows were shown to sustain the wave forces that they are likely to experience during hurricanes. In addition, the seagrass canopy and the complex biogeomorphic landscape built by the seagrass meadows combine to significantly dissipate extreme wave forces, ensuring that erosion is minimised within sandy beach foreshores. The persistence of the Caribbean seagrass meadows and their coastal protection services during extreme storm events ensures that a stable coastal ecosystem and beach foreshore is maintained in tropical regions.

The backbone of the Amsterdam Ordnance Datum (NAP) is a network of about 400 primary subsurface markers. Relative movements between the primary subsurface markers are measured with spirit levelling once in 10-20 years. However, little is known about absolute vertical movements of the primary network. This information is indispensable for the interpretation of water level measurements at the tide gauges along the Dutch coast. It may be provided by gravity measurements. Here we present a time-series analysis of more than twenty years of gravity measurements at the stations Westerbork, Epen, Zundert, and Radio Kootwijk. It reveals that only station Epen shows a statistically significant movement of -0:252±0:066 μGal yr-1, which corresponds to an uplift of 1:3±0:5mmyr-1. For the other stations, the trends are statistically not different from zero at a significance level of 0.05. Corrections for water table variations are found to be indispensable; peak-to-peak amplitudes range from 4 μGal (Westerbork) to 28 μGal (Radio Kootwijk). Depsite some fundamental objections, corrections for instrumental offsets reduce the data scatter. First experiments with 7 years of soil moisture data acquired at station Radio Kootwijk reveal that the gravity signal of soil moisture variations has a standard deviation of 2:2 μGal, which is comparable to the noise standard deviation of measured gravity.

The mesoscale variability in the Caribbean Sea is dominated by anticyclonic eddies that are formed in the eastern part of the basin. These anticyclones intensify on their path westward while they pass the coastal upwelling region along the Venezuelan and Colombian coast. In this study, we used a regional model to show that this westward intensification of Caribbean anticyclones is steered by the advection of cold upwelling filaments. Following the thermal wind balance, the increased horizontal density gradients result in an increase in the vertical shear of the anticyclones and in their westward intensification. To assess the impact of variations in upwelling on the anticyclones, several simulations were performed in which the northward Ekman transport (and thus the upwelling strength) is altered. As expected, stronger (weaker) upwelling is associated with stronger (weaker) offshore cooling and a stronger (weaker) westward intensification of the anticyclones. Moreover, the simulations with weaker upwelling show farther advection of the Amazon and Orinoco River plumes into the basin. As a result, in these simulations the horizontal density gradients were predominantly set by horizontal salinity gradients. The importance of the horizontal density gradients driven by temperature, which are associated with the upwelling, increased with increasing upwelling strength. The results of this study highlight that both upwelling and the advection of the river plumes affect the life cycle of mesoscale eddies in the Caribbean Sea.

Global oceanic pH is lowering, which is causing great concern for the natural functioning of marine ecosystems. Current pH predictions are based on open ocean models; however, coastal zones are dynamic systems with seawater pH fluctuating temporally and spatially. To understand how coastal ecosystems will respond in the future, we first need to quantify the extent that local processes influence the pH of coastal zones. With this study, we show that over a single diurnal cycle, the total pH can fluctuate up to 0.2 units in a shallow seagrass-dominated bay, driven by the photosynthesis and respiration of the vegetation. However, these biologically controlled pH fluctuations vary significantly over small distances. Monitoring conducted at neighboring sites with contrasting hydrodynamic regimes highlights how water motion controls the extent that the local pH is altered by the metabolism of vegetation. The interactive effects of hydrodynamics and vegetation were further investigated with an in situ experiment, where the hydrodynamics were constrained and thus the local water residence time was increased, displaying the counteractive effect of hydrodynamics on the pH change caused by vegetation. With this research, we provide detailed in situ evidence of the spatial variation of pH within marine ecosystems, highlighting the need to include hydrodynamic conditions when assessing the pH-effects of vegetation, and identifying potential high-pH refuges in a future low pH ocean.