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. @en

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.

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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.

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Estimating the three geophysical variables significant wave height (SWH), sea surface height, and wind speed from satellite altimetry continues to be challenging in the coastal zone because the received radar echoes exhibit significant interference from strongly reflective targets such as sandbanks, sheltered bays, ships etc. Fully focused SAR (FF-SAR) processing exhibits a theoretical along-track resolution of up to less than half a metre. This suggests that the application of FF-SAR altimetry might give potential gains over unfocused SAR (UF-SAR) altimetry to resolve and mitigate small-scale interferers in the along-track direction to improve the accuracy and precision of the geophysical estimates. The objective of this study is to assess the applicability of FF-SAR-processed Sentinel-6 Michael Freilich (S6-MF) coastal altimetry data to obtain SWH estimates as close as possible to the coast. We have developed a multi-mission FF-SAR processor and applied the coastal retracking algorithm CORALv2 to estimate SWH. We assess different FF-SAR and UF-SAR processing configurations, as well as the baseline Level-2 product from EUMETSAT, by comparison with the coastal, high-resolution SWAN-Kuststrook wave model from the Deltares RWsOS North Sea operational forecasting system. This includes the evaluation of the correlation, the median offset, and the percentage of cycles with high correlation as a function of distance to the nearest coastline. Moreover, we analyse the number of valid records and the L2 noise of the records. The case study comprises five coastal crossings of S6-MF that are located along the Dutch coast and the German coast along the East Frisian Islands in the North Sea. We observe that accurate and precise SWH records can be estimated in the nearshore zone within 1–3 km from the coast using satellite SAR altimetry. We find that the FF-SAR-processed dataset with a Level-1b posting rate of 140 Hz shows the greatest similarity with the wave model. We achieve a correlation of ∼0.8 at 80% of valid records and a gain in precision of up to 29% of FF-SAR vs UF-SAR for 1–3 km from the coast. FF-SAR shows, for all cycles, a high correlation of greater than or equal to 0.8 for 1–3 km from the coast. We estimate the decay of SWH from offshore at 30 km to up to 1 km from the coast to amount to 26.4% ± 3.1%.

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We demonstrate in this work how we can take advantage of known unfocused SAR (UF-SAR) retracking methods (e.g. the physical SAMOSA model) for retracking of fully-focused SAR (FF-SAR) waveforms. Our insights are an important step towards consistent observations of sea surface height, significant wave height and backscatter coefficient (wind speed) with both UF-SAR and FF-SAR. This is of particular interest for SAR altimetry in the coastal zone, since coastal clutter may be filtered out more efficiently in the high-resolution FF-SAR waveform data, which has the potential to improve data quality. We implemented a multi-mission FF-SAR altimetry processor for Sentinel-3 (S3) and Sentinel-6 Michael Freilich (S6), using a back-projection algorithm, and analysed ocean waveform statistics compared to multilooked UF-SAR. We find for Sentinel-3 that the averaged power waveforms of UF-SAR and FF-SAR over ocean are virtually identical, while for Sentinel-6 the FF-SAR power waveforms better resemble the UF-SAR zero-Doppler beam. We can explain and model the similarities and differences in the data via theoretical considerations of the waveform integrals. These findings suggest to use the existing UF-SAR SAMOSA model for retracking S3 FF-SAR waveforms but the SAMOSA zero-Doppler beam model for S6 FF-SAR waveforms, instead. Testing the outlined approach over short track segments, we obtain range biases between UF-SAR and FF-SAR lower than 2 mm and significant wave height biases lower than 5 cm.

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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.

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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.

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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.

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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.

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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.

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Intertidal wetlands, such as mangroves in the tropics, are increasingly recognized for their role in nature-based mitigation of coastal flood risks. Yet it is still poorly understood how effective they are at attenuating the propagation of extreme sea levels through large (order of 100 km2) estuarine or deltaic systems, with complex geometry formed by networks of branching channels intertwined with mangrove and intertidal flat areas. Here, we present a delta-scale hydrodynamic modelling study, aiming to explicitly account for these complex landforms, for the case of the Guayas delta (Ecuador), the largest estuarine system on the Pacific coast of Latin America. Despite coping with data scarcity, our model accurately reproduces the observed propagation of high water levels during a spring tide. Further, based on a model sensitivity analysis, we show that high water levels are most sensitive to the mangrove platform elevation and degree of channelization but to a much lesser extent to vegetation-induced friction. Mangroves with a lower surface elevation, lower vegetation density, and higher degree of channelization all favour a more efficient flooding of the mangroves and therefore more effectively attenuate the high water levels in the deltaic channels. Our findings indicate that vast areas of channelized mangrove forests, rather than densely vegetated forests, are most effective for nature-based flood risk mitigation in a river delta.@en

Our awareness of ice caps' and mountain glaciers' sensitivity to climate change has driven major advances in the application of remote sensing techniques during the past decade. Regarding ESA's SARIn altimeter CryoSat-2, processing the full waveform to generate swaths of elevation estimates has become standard practice in regions of complex topographies. This technique provides information on areas where we would be blind otherwise. In this article, we discuss systematic errors and analyze their impact on surface elevation measurements and change rates of two test areas. In particular, we focus on periodically occurring errors in elevation swaths, caused by the superposition of coherent signals from range-ambiguous surfaces. They can lead to measurement errors in excess of 10 m, affect most measurements in mountainous regions, are difficult to exclude with established post-processing techniques, and occur repeatedly for satellite revisits introducing a 369-day periodicity-difficult to distinguish from the annual cycle. We show a correlation between derived elevation swaths and the sensor view angle and explore the influence of common data exclusion choices on higher level products. Our results indicate that these systematic errors hold a substantial share of the error budget and that the choice of thresholds impacts higher level products. We conclude that error correlations need to be considered to characterize the data accuracy. With the established data editing strategies, systematic errors prevent resolving seasonal mass changes of single mountain glacier basins and impact aggregates over larger areas or longer periods.

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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.

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Satellite radar altimetry has been an important tool for cryospheric applications such as measuring ice-sheet height or assessing anomalies in snow and ice properties (e.g. the extensive melt in Greenland in 2012). Although accurate height measurements are key for such applications, slope-induced errors due to undulating topography within the kilometre-wide beam-limited footprint can cause multi-metre errors. Two main correction methods that have been developed (referred to as the slope- and point-based methods) neglect either the actual topography or the actual footprint that can be estimated by a combination of the leading edge and topography. Therefore, a leading edge point-based (LEPTA) method is presented that corrects for the slope-induced error by including the leading edge information of the radar waveform to determine the impact point. The principle of the method is that only the points on the ground that are within the range determined by the beginning and end of the leading edge are used to determine the impact point. Benchmarking of the LEPTA method against the slope- and point-based methods based on CryoSat-2 Low Resolution Mode (LRM) acquisitions over Greenland in 2019 shows that, when compared to ICESat-2 observations, the LEPTA method has a stable performance both in the flat, interior regions of Greenland and in regions with more complex topography. The median difference between the slope-corrected CryoSat-2 heights using LEPTA and the ICESat-2 heights is at the millimetre level, whereas the slope and point-based methods can have a 0.21 and 0.48 m difference, respectively, and the Level-2I (L2I) data provided by ESA have a 0.01 m difference. The median absolute deviation of height differences between CryoSat-2 and ICESat-2, which we use as an indicator of the variation in errors, is also the lowest for LEPTA (0.09 m) in comparison to the aforementioned methods (0.19 m for slope method and 0.10 m for point-based method) and ESA Level-2 data (0.14 m). Although ESA Level-2 products and the point-based method have good performance in either the median or the median absolute deviation, LEPTA shows a good performance in both metrics. Based on that, we recommend considering LEPTA for obtaining accurate height measurements with radar altimetry data, especially towards the margins of the LRM coverage where the surface slopes increase.@en

One of the major challenges facing global hydrodynamic tidal models is the modelling of the interaction between sea ice and tides in high-latitude waters. Recent studies have shown strong seasonal correlation between sea ice and tides. Hence, it is important to accurately model the effect of sea ice in a tidal model. Presence of sea ice leads to a frictional dissipation of tides. Most models either completely ignore sea ice or partly include it by assuming a fixed sea ice cover (landfast ice). However, sea ice can also be drifting and the nature of dissipation between drifting sea ice and tides is partly unknown. We assess the dissipation of tides due to free drift sea ice. In the absence of wind, this is negligible in the deeper and open ocean. For the shallow water regions, however, this dissipation is unknown. Here, we evaluate this dissipation for the Spitzbergen Shelf region using a beacon dataset showing strong free drift subdaily sea ice oscillations and a physics based point ice model. Two analyses were done which compared the model and observed motion. The analyses showed that for winds speeds below 8m/s and with low subdaily signals, the subdaily free drift sea ice motion is strongly connected to the tides and that the frictional dissipation is low. In the context of global tide and storm surge models, the dissipation from free drift sea ice on tides should be evaluated based on the region (deep ocean or shallow water) and existing wind conditions. In the presence of strong winds the dissipation between free drift sea ice and air can be significant on a subdaily scale even if there are no subdaily signals in the wind itself.

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This article shows the first spectral analysis of fully-focused Synthetic Aperture Radar (FFSAR) altimetry data with the objective of studying backscatter modulations caused by swells. Swell waves distort the backscatter in altimetry radargrams by means of velocity and range bunching. These swell signatures are visible in the tail of the waveform. By locally normalizing the backscatter and projecting the waveforms on an along-/cross-track grid, satellite altimetry can be exploited to retrieve swell information. The analysis of FFSAR spectra is supported by buoy-derived swell-wave spectra of the National Oceanic and Atmospheric Administration network. Using cases with varying wave characteristics, we discuss the altimetry-derived spectra and relate them to what is known from side-looking SAR imaging systems. Besides having a vast amount of additional data for swell-wave analysis, altimeter data can also help us to better understand the side-looking SAR spectra.

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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.

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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.

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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.

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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. @en