The Rhine-Meuse Delta is a low-lying delta in the Netherlands that is subject to both salt intrusion events and storm surges. Typically, storm surges only temporarily cause increased salt intrusion and do not cause severe problems for freshwater availability. However, during the storm surge of December 2013, salt reached the closed southern branch of the delta and higher salinities were observed for weeks after the storm surge. The purpose of this study is to examine the mechanisms controlling salt intrusion in the Rhine-Meuse Delta during and after a severe storm surge event. A three-dimensional hydrodynamic model (Delft3D-FM) of the Rhine-Meuse Delta was developed that successfully reproduces salt intrusion for both normal and storm surge conditions. During the storm, high water levels in the northern branch caused a salt flux toward the southern branch. The southern branch of the Rhine-Meuse Delta is closed off by an estuarine dam, consequently salt was retained landward of the dam. Local stratification in the southern branch caused salt to remain in the deeper parts, limiting the effectiveness of flushing after the storm surge. In the post-storm period, salt was gradually released from the southern branch, raising salinity levels in an adjacent channel. The river discharge was only just below the yearly average, showing prolonged salt intrusion can also occur outside of dry periods.

Tidal models that incorporate satellite altimeter data have historically shown discrepancies in accuracy between shallow and deep marine environments. A recent study suggests that these differences may partly stem from neglecting the nonlinear tide-surge interactions in tidal analyses. In this study, we introduce a novel method for estimating tidal constituents from satellite altimeter data in shallow waters, leveraging a 2D hydrodynamic model that accounts for these nonlinear interactions. This approach substantially reduces the variance of unaccounted water level variability, thereby benefiting the estimation. A distinctive feature of our method is the treatment of prior model tidal constituents as stochastic, which helps manage the low temporal resolution of altimeter data by ensuring that unresolved tidal constituents are not updated. We tested our method in the data-rich northwest European continental shelf region, using the high-resolution 2D Dutch Continental Shelf Model version 7 (DCSM). Results show a substantial reduction in the standard deviations of residual water level time series in the shallow waters around Great Britain and in the German Bight, from 11 cm to 5 cm. In deep waters (>200 m), the median standard deviation decreased from 6.8 cm to 6.2 cm. When compared to state-of-the-art ocean tide and surge corrections from publicly available models, our method outperformed them in shallow waters (median standard deviation of 6.0 cm versus 7.5 cm), though the alternative products performed better in deep waters (median standard deviation of 5.5 cm versus 6.2 cm). An estimate of the accuracy at satellite crossovers resulted in an estimated total tidal error of about 1.5 cm (RSS VD). We acknowledge that comparisons in shallow waters are complicated, as alternative products do not account for nonlinear tide-surge interactions. Overall, the demonstration along-track tidal product developed in this study shows potential for improving the tidal representation in the DCSM model. In data-poor regions, the number of tidal constituents that can be reliably estimated using the method may be limited, and alternative strategies might be needed to evaluate the model’s uncertainty in representing tides.

Sea surface currents are of significant importance in various scientific and maritime applications. There are several measurement techniques available to study surface currents, however, they have limitations in spatial coverage and resolution. This study presents a proof-of-concept for a new measurement principle that relies on the difference between a ship's speed relative to water and land. The approach involves estimating the ship speed vector relative to water from optical satellite imagery of Kelvin wakes. This ship speed vector is subtracted from the ship speed over ground, which is determined from Automatic Identification System (AIS) data, to estimate the surface current. A case study in the Strait of Gibraltar was performed using two months of Sentinel-2 imagery, which yielded 81 visible Kelvin wakes over 25 images. Surface currents were estimated in directions parallel and perpendicular to the ship's sailing line for each Kelvin wake. The estimated currents were validated with respect to surface currents derived from High-Frequency Radars (HFRs) and modelled currents from the Copernicus Marine Environmental Monitoring Service (CMEMS). The uncertainty in the two surface current components was estimated using triple collocation. After removing 12 data points with large ship course variability, standard deviations of 0.14 and 0.16 m s⁻¹ were estimated for the surface currents along and across the sailing line, respectively. Despite limitations in measurement frequency due to satellite revisit times, cloud cover and Kelvin wake visibility, this new method can provide accurate estimates of sea surface currents in regions with high vessel density.

Multi-mission satellite altimetry data have been used to study the spatial and temporal variability in global storm surge water levels. This was done by means of a time-dependent extreme value analysis applied to the monthly maximum detided water levels. To account for the limited temporal resolution of the satellite data, the data were first stacked on a 5^∘× 5^∘ grid. Moreover, additional scaling was applied to the extreme value analysis for which the scaling factors were determined by means of a resampling method using reanalysis data. In addition to the conventional analysis using data from tide gauges, this study provides an insight in the ocean-wide storm surge properties. Nonetheless, where possible, results were compared to similar information derived from tide gauge data. Except for secular changes, the satellite-derived results are comparable to the information derived from tide gauges (correlation > 0.5), although the tide gauges show more local variability. Where limited correlation was observed for the secular change, it was suggested that the satellites may not be able to fully capture the temporal variability in the short-lived, tropical storms, as opposed to extra-tropical storms.

Arctic sea ice leads to a significant dissipation of tidal energy, necessitating its inclusion in global tidal models. However, most global tidal models either neglect or only partially incorporate the impact of sea ice on tides. This study proposes a method to model the dissipative forces exerted by sea ice on tides without directly coupling to a sea ice model, yet utilizing sea ice parameters such as thickness and concentration. Our approach involves (re)-categorizing the sea ice cover into regions dominated either by the velocity difference between sea ice and tides (Vertical Shear (VS)) or by the shear from drifting sea ice on tides (Horizontal Shear (HS)), which primarily govern the energy dissipation between tides and sea ice. The subdivision and resulting areas of these HS and VS regions are based on a nondimensional number referred to as the Friction number, which is the ratio of the internal stress of the sea ice field to the ice–water frictional stress, and directly depends on the thickness and concentration of the sea ice. The new parameterization is validated through a performance assessment comparing it to a commonly used approach of assuming all the sea ice to be stationary (landfast). The seasonal modulation of the M₂ tidal component, quantified as the March–September difference, serves as the performance metric, demonstrating that the new parameterization has better agreement with observations from altimeter- and tide gauge-derived seasonal modulation. The results indicate that the physics of ice–tide interaction is better represented with the new parameterization of sea ice-induced dissipation, making it suitable for investigating the effects of declining sea ice thickness on tides.

Deltas are home to billions of people and are often highly developed and engineered systems. Extreme weather events such as droughts are a threat to deltas worldwide. During droughts salt can intrude far inland and threaten the drinking, agricultural and industrial water supply of many people. Under climate change the frequency of extreme events is expected to increase and the threat of salt intrusion may intensify. Here we use data and models to explore salt intrusion in the Rhine-Meuse Delta (RMD) during the severe European drought in the summer of 2022. The RMD is one of the most highly managed deltas in the world, with numerous interconnected waterways and an open connection to the sea at the mouth of the Rotterdam Waterway. The outflow of the Rhine River through the Rotterdam Waterway generates the strongly stratified Rhine River plume. Under normal conditions a salt wedge intrudes about 16-18 km inland on every tide. In contrast, under drought conditions in summer 2022, observations show salt intruding over 42 km inland and the Rhine River plume diminished in size. We explore the changes in estuarine dynamics during the drought using velocity, salinity and temperature data from various field campaigns near the mouth of the Rotterdam Waterway and within the delta, together with numerical models. We also compare drought condition observations with data from prior field campaigns during normal discharge conditions. Shifts in the relative strength of the dominant mechanisms of landward salt flux throughout the drought are explored and linked to the changes in estuarine response.

Traditionally, nadir-looking satellite radar altimeters provide water levels of rivers only at intersections with the satellite's ground track, called virtual stations. These observations have limited spatial coverage because such cross-overs are sparse, depending on the altimeter's orbit. In this work, we introduce the novel concept of Polygon-Informed Cross-Track Altimetry (PICTA), enabling accurate estimation of water levels at cross-track distances — for as long as the target's signal is recorded in the altimeter's range window. Using fully-focused SAR data from the Sentinel-6 altimetry mission, we demonstrate how the new approach can provide detailed river water level profiles over a ground swath of about 14 km cross-track width and with an along-track resolution as fine as 10 m. On the one hand, this marks a drastic improvement in the number of available measurements when compared to the virtual station approach, on the other hand, for the first time, water surface slopes and level variations along the river, caused by rapids, dams, and sluices, can be directly observed using a nadir radar altimeter. The validation over two river segments in France reveals biases as low as ±4 cm and random errors on the order of 3–8 cm at 30 m along-track resolution. The new PICTA concept can potentially be generalized to other targets, such as lakes or even coastlines.

Traditionally, nadir-looking satellite radar altimeters provide water surface elevations of rivers only at intersections with the satellite’s ground track, called virtual stations. These observations have limited spatial coverage because such cross-overs are sparse, depending on the altimeter's orbit. In this work, we introduce the novel concept of Polygon-Informed Cross-Track Altimetry (PICTA), enabling precise measurement of water surface elevations at cross-track distances---for as long as the target's signal is recorded in the altimeter's range window. Using fully-focused SAR data from the Sentinel-6 altimetry mission, we demonstrate how the new approach can provide detailed river elevation profiles over a ground swath of about 14 km cross-track width and with an along-track resolution as fine as 10 m. On the one hand, this marks a drastic improvement in the number of available measurements when compared to the virtual station approach, on the other hand, for the first time, water surface slopes and level variations along the river, caused by rapids, dams, and sluices, can be directly observed using a nadir radar altimeter. The validation over two river segments in France reveals biases as low as [[EQUATION]]4 cm and random errors on the order of 3 to 8 cm at 30 m along-track resolution. The new PICTA concept can potentially be generalized to other targets, such as lakes or even coastlines.

With the continued rise in global mean sea level, operational predictions of tidal height and total water levels have become crucial for accurate estimations and understanding of sea level processes. The Dutch Continental Shelf Model in Delft3D Flexible Mesh (DCSM-FM) is developed at Deltares to operationally estimate the total water levels to help trigger early warning systems to mitigate against these extreme events. In this study, a regional version of the Empirical Ocean Tide model for the Northwest European Continental Sea (EOT-NECS) is developed with the aim to apply better tidal forcing along the boundary of the regional DCSM-FM. EOT-NECS is developed at DGFI-TUM by using 30 years of multi-mission along-track satellite altimetry to derive tidal constituents which are estimated both empirically and semi-empirically. Compared to the global model, EOT20, EOT-NECS showed a reduction in the root-square-sum error for the eight major tidal constituents of 0.68 cm compared to in situ tide gauges. When applying constituents from EOT-NECS at the boundaries of DCSM-FM, an overall improvement of 0.29 cm was seen in the root-mean-square error of tidal height estimations made by DCSM-FM, with some regions exceeding a 1 cm improvement. Furthermore, of the fourteen constituents tested, eleven showed a reduction of RMS when included at the boundary of DCSM-FM from EOT-NECS. The results demonstrate the importance of using the appropriate tide model(s) as boundary forcings, and in this study, the use of EOT-NECS has a positive impact on the total water level estimations made in the northwest European continental seas.

Tidal river plumes dominate many shelf seas, transporting freshwater, sediment, nutrients, pollutants and larvae downstream. The Rhine River Plume is one of the largest in Europe, under typical discharge conditions it is dominated by tidal plume fronts in the near to mid-field plume and by tidal straining in the mid- to far field plume. Moreover, in agreement with other tidal river plumes discharging onto the shelf, internal waves generated ahead of tidal plume fronts are an important source of mixing in the river plume. We compare field data collected downstream of the mouth of the Rhine River in 2013 and 2014 under typical discharge conditions, with data collected in the near field plume during 2022 during a major drought. Together with numerical models we explore how extreme variations in freshwater discharge impact both tidal straining and the formation and strength of tidal plume fronts. Furthermore we explore how in turn, this influences the structure and mixing of the near to far-field Rhine River Plume. We use a 3D hydrostatic model of the Rhine River Plume and a potential energy anomaly analysis to explore changes in the mixing. We explore how the river plume adjusts to extremely low discharge conditions and discuss the possible impact on the transport of freshwater, tracers, larvae and fine sediment.

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.

Earlier work has empirically demonstrated some advantages of an increased posting rate of Synthetic Aperture Radar (SAR) altimeters beyond the expected ground resolution of about 320 m in Delay-Doppler (unfocused SAR, UFSAR) processing, corresponding to ∼20 Hz sampling. Higher posting rates of 40–80 Hz were shown to prevent spectral aliasing of the signal, enable to measure swell wave related signal distortions and may lead to a reduced root mean square error of 1 Hz estimates of Sea Surface Heights (SSH), radar cross section (sigma0) and Significant Wave Heights (SWH) from current SAR altimeters. These improvements were explained by the narrow noise autocovariance function of the waveform signal's power speckle noise in along-track direction on one hand, and frequency doubling by power detection (squaring of the signal) on the other. It has not been explained, however, why the power speckle noise decorrelates faster than anticipated by the predicted Doppler resolution, and whether this decorrelation depends on the altimeter and processing configuration. Also, it has not been shown explicitly that the estimates of SSH, SWH and sigma0 decorrelate in the same way. Describing the noise autocovariance function – or equivalently the noise power spectral density via the Wiener-Kintchin theorem – is necessary on two counts: Knowing the noise autocovariance allows to apply optimal filtering strategies that maximize precision on one hand, while the noise power spectral density predicts the frequencies contained in the noise (and signals), which in turn determines the required sampling frequency according to the Nyquist theorem. Using a newly derived analytic noise autocovariance model for UFSAR-processed altimeter data, we show that the swift signal decorrelation is mainly due to the observation geometry. Furthermore, our results demonstrate that the noise autocovariance functions of power speckle, SSH, SWH and sigma0 estimates in along-track direction are different and depend on the sea state. On top of that, the noise autocovariance functions are strongly dependent on the number of Doppler beams used for multilooking, the used retracker, and the processing choices such as antenna gain pattern compensation and windowing within the UFSAR processing (Level-1b). We validated our noise autocovariance model with segments of 42 Sentinel-3B overpasses. Our findings are in accordance to all earlier work, but indicate that the reported precision improvements with respect to 20 Hz may have been too optimistic and that the SSH, SWH and sigma0 generally decorrelate slower than the power speckle noise. We found that the required posting rate is always higher or equal to 40 Hz. Our results will potentially enable improved spectral analysis and optimal filtering of any UFSAR altimetry data. More importantly, our results can be used to trade off different aspects for determining an optimal posting rate in UFSAR altimeter processing in different sea states and with changing processing parameters, which is necessary in view of strict precision requirements of existing and future SAR altimetry missions.

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 the coming decades, coastal flooding will become more frequent due to sea-level rise and potential changes in storms. To produce global storm surge projections from 1950 to 2050, we force the Global Tide and Surge Model with a ∼25-km resolution climate model ensemble from the Coupled Model Intercomparison Project Phase 6 High Resolution Model Intercomparison Project (HighResMIP). This is the first time that such a high-resolution ensemble is used to assess changes in future storm surges across the globe. We validate the present epoch (1985–2014) against the ERA5 climate reanalysis, which shows a good overall agreement. However, there is a clear spatial bias with generally a positive bias in coastal areas along semi-enclosed seas and negative bias in equatorial regions. Comparing the future epoch (2021–2050) against the historical epoch (1951–1980), we project ensemble-median changes up to 0.1 (or 20%) in the 1 in 10-year storm surge levels. These changes are not uniform across the globe with decreases along the coast of Mediterranean and northern Africa and southern Australia and increases along the south coast of Australia and Alaska. There are also increases along (parts) of the coasts of northern Caribbean, eastern Africa, China and the Korean peninsula, but with less agreement among the HighResMIP ensemble. Information resulting from this study can be used to inform broad-scale assessment of coastal impacts under future climate change.

Under the influence of climate change, estuaries around the world are increasingly exposed to more extreme weather conditions. In recent years droughts specifically have been occurring more frequently and for prolonged periods. During a period of drought, salt intrusion is exacerbated, impacting the availability and quality of water resources and the estuarine ecosystem. As the impacts of droughts can be severe, assessment of droughts and their influence on the estuarine system is of great importance.

Using a high-resolution 3D coupled ocean-delta model we investigate the influence of the record-breaking European drought of the summer of 2022 on the Rhine-Meuse Delta and compare this to the estuarine response under average discharge conditions, putting the drought’s influence into perspective. Spatial patterns of stratification, mixing, and straining and their evolution throughout the drought period are studied by a salinity variance analysis. The progression of the salt wedge and retreat of the tidal plume fronts are examined and related to the changing strength of the individual estuarine processes influencing stratification. We show that as the tidal plume fronts retreat during the drought, we see a corresponding change in the structure of the salt wedge, demonstrating the importance of the coupling between the tidal plume fronts and the estuarine dynamics.

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.

Strongly forced salt wedge estuaries are known to demonstrate significant variability in stratification, currents, and mixing within a tidal cycle. Such estuaries also show strongly spatially varying patterns of stratification and mixing, and the processes causing this spatial and temporal variability are known to interact. As a consequence, the dynamics of these estuaries are distinctly different from well-mixed and partially mixed estuaries, and intratidal processes may play a more central role in the evolution of stratification.

The Rhine-Meuse estuary is an example of a strongly forced salt wedge estuary. It is very dynamic, mesotidal, and stably stratified. In this study, we investigate how individual intratidal estuarine processes contribute to the evolution of stratification in the Rhine-Meuse estuary. Data from recent shipboard measurements are used to assess their relative influence and highlight the potential importance of interactions between these processes.

From measurements in the Rhine-Meuse estuary we find that the availability of salt is determined by tidal advection of the salt wedge. Additionally, exchange flows transfer salt from high-density to low-density regions such as harbour basins and side branches while the salt wedge is advected through the estuary. The combination of the barotropic tidal asymmetry imposed at the river mouth and turbulence damping at the pycnocline results in strong shear and subsequent formation of mid-depth jets at the onset of flood. These mid-depth jets contribute to the transfer of salt by transporting salt from regions of higher momentum to regions of lower momentum. Furthermore, the measurements suggest that several bathymetric transitions locally generate internal wave activity, although the resulting turbulent mixing is not strong enough to erode the persistent salt wedge structure.

These findings underline the importance of interactions between intratidal processes on different spatial scales and their effect on the evolution of stratification in the Rhine-Meuse estuary. As an extension to our findings, measurements in the Rhine region of influence (ROFI) are used to further examine the role of the seaside forcing on the individual physical processes and the resulting intratidal variability of stratification in the estuary.

Accurate parameter estimation for the Global Tide and Surge Model (GTSM) benefits from observations with long time-series. However, increasing the number of measurements leads to a large computation demand and increased memory requirements, especially for the ensemble-based methods that assimilate the measurements at one batch. In this study, a memory-efficient parameter estimation scheme using model order reduction in time patterns is developed for a high-resolution global tide model. We propose using projection onto empirical time-patterns to reduce the model output time-series to a much smaller linear subspace. Then, to further improve the estimation accuracy, we introduce an outer-loop, similar to Incremental 4D-VAR, to evaluate model-increments at a lower resolution and subsequently reduce the computational cost. The inner-loop optimizes parameters using the lower-resolution model and an iterative least-squares estimation algorithm called DUD. The outer-loop updates the initial output from the high-resolution model with updated parameters from the converged inner-loop and then restarts the inner-loop. We performed experiments to adjust the bathymetry with observations from the FES2014 dataset. Results show that the time patterns of the tide series can be successfully projected to a lower dimensional subspace, and memory requirements are reduced by a factor of 22 for our experiments. The estimation is converged after three outer iterations in our experiment, and tide representation is significantly improved, achieving a 34.5% reduction of error. The model's improvement is not only shown for the calibration dataset, but also for several validation datasets consisting of one year of time-series from FES2014 and UHSLC tide gauges.

Satellite observations can help in the retrieval of constituents in shallow waters. Noise contamination, however, makes smaller constituents irretrievable and large sources of error. Throughout shallow areas, the constituent’s relevancy changes. For example, near an amphidromic point where M2 relevance drops, so does the potential of satellite contribution for improving its accuracy. Moreover, shallow waters are generally influenced by many constituents (>100). Accurately retrieving all these constituents with satellite radar altimeter data alone is not possible. Series length requirements imposed by the Rayleigh criteria to separate constituents are still unavailable.
Removing unwanted signals from satellite observations improves least-squares-based harmonic estimates, given an inversion matrix with the same condition number. This variance reduction is the core of the remove compute restore approach commonly used. First, residual harmonic sets are computed with the difference between observations and model background estimates through conventional or weighted least-squares. Then, the residual harmonics are added to the background model estimates.
Here we implemented a method that extends the typical approach by including model background estimate and error covariance in the least-squares step. This inclusion helps to weigh between constituents well represented in the model and those that must be updated.
To test the method, we designed a semi-synthetic experiment. First, we used tide gauge data to generate a satellite equivalent dataset and compared estimations between the two methods listed above and the model estimate. Next, we applied the method to compute tidal estimates along satellite radar altimeter tracks (T/P Jason) in the 2D Dutch Coastal Shelf Model (DCSMv6) domain.
Results from the synthetic experiment show that the second method produces consistently better estimates reducing RSS consistently through temporal cross-validation. In addition, it provides an effective way of keeping as many constituents estimates as the model series can resolve, adding the benefits of satellite observations. Finally, results from the North Sea implementation show the new estimates increase the variance reduction of satellite residuals across the whole domain relative to background tidal estimates. The range of improvements varies between 0 and 3cm, which is significant given already very accurate model background estimates. The benefited areas include the English Channel, the Irish Sea, the English North-Sea Coast, the Bay of Biscay, the German Bight, and the North Atlantic region close to the upper boundary of the model domain.

Previous studies have demonstrated that tides are subject to considerable changes on secular time scales. However, these studies rely on sea level observations from tide gauges that are predominantly located in coastal and shelf regions and therefore, the large-scale patterns remain uncertain. Now, for the first time, satellite radar altimetry (TOPEX/Poseidon & Jason series) has been used to study worldwide linear trends in tidal harmonic constants of four major tides (M₂, S₂, O₁, and K₁). This study demonstrates both the potential and challenges of using satellite data for the quantification of such long-term changes. Two alternative methods were implemented. In the first method, tidal harmonic constants were estimated for consecutive 4-year periods, from which the linear change was then estimated. In the second method, the estimation of linear trends in the tidal constants of the four tides was integrated in the harmonic analysis. First, both methods were assessed by application to tide gauge data that were sub-sampled to the sampling scheme of the satellites. Thereafter the methods were applied to the real satellite data. Results show both statistically significant decreases and increases in amplitude up to 1 mm/year and significant phase changes up to ∼0.1 deg/year. The level of agreement between altimeter-derived trends and estimates from tide gauge data differs per region and per tide.