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.

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.

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.