An empirical noise model for the benefit of model-based hydrodynamic leveling

Abstract

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 cm$$^2$$ 2 (3D DCSM-FM) and 16.3 cm$$^2$$ 2 (DCSMv6-ZUNOv4). The variance (i.e., covariance at zero sea distance) for these two models is 15.3 cm$$^2$$ 2 and 21.7 cm$$^2$$ 2 , respectively. Averaging the water levels over three summers, lowered the variance and nugget effect for 3D DCSM-FM to 12.7 cm$$^2$$ 2 and 10.0 cm$$^2$$ 2 , 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 $$7\%$$ 7 % and $$8\%$$ 8 % , when we average the water levels over one and three summer periods, respectively.

Realizing the European Vertical Reference System using model-based hydrodynamic leveling data

“…The full noise variance-covariance matrix associated with the middle term has been obtained using the noise models developed by Afrasteh et al (2023). For the specific model being used, see Sect.…”

“…For this, there are at most N N −2 possibilities (Afrasteh et al 2021). Similar to Afrasteh et al (2021Afrasteh et al ( , 2023, we use a heuristic search method that identifies the connections one-by-one. Each connection added to the final set is the one that provided the lowest median standard deviation of the adjusted heights.…”

“…They also found increased redundancy numbers for leveling observations close to the coastlines. Afrasteh et al (2023) reassessed the potential impact of hydrodynamic leveling data on the quality of the EVRF using an empirical noise model for the model-derived SMWLs allowing to include error correlations. Their results suggest an improvement up to 25% if the water levels are averaged over three summer periods.…”

“…Both Afrasteh et al (2021) and Afrasteh et al (2023) assessed the impact of hydrodynamic leveling data on the quality of the EVRF through geodetic network analyzes, using only (assumed) stochastic information of the modelbased hydrodynamic leveling data. This study goes further because it uses real model-based hydrodynamic leveling data between tide gauges in the Baltic Sea and the northwest European continental shelf, including the North Sea and Wadden Sea.…”

“…The water levels are averaged over three successive summer periods that are the same per region. The reason for using three summer periods is that this is the maximum averaging period for which a noise model (needed to build the full noise variance-covariance matrix of the dataset) is available (Afrasteh et al 2023).…”