# Long-Term and Decadal Sea-Level Trends of the Baltic Sea Using Along-Track Satellite Altimetry
## 2 Methodology  
Conventions of the used DT data need to be considered in the development of the sea-level trend estimation methodology from the TG and SA observations. Modern TGs contain observation sensors, which automatically record the height of the coastal water level with high precision and temporal resolution (e.g., hourly). Contrastingly, the quality of historical TG records could be quite heterogeneous, due to the usage of mechanical mareographs or visual staff readings (e.g., twice a day). The TG zeros usually coincide with (or can be converted to) the zero of the vertical datum, which nowadays is often a geoid-based one [52]. Also, TG data could refer to different charts or national vertical data. Since the land-bound TG observations yield the RSL, which is the movement of the sea relative to land, to obtain ASL, the VLM effect needs to be removed from the TG observations (Figure 1). The SA-derived SSH is computed by subtracting the distance between the satellite altitude (determined in a global geocentric coordinate system) and the instantaneous sea surface. The SA data, therefore, are not VLM-affected, thus reflecting the true sea-level rise. Also, SA data are referenced to ITRF (e.g., realizations of 2008, 2014, etc. ), while the regional/national geoid model could refer to another coordinate system, e.g., the ETRF (European Terrestrial Reference Frame). Hence, the aforementioned differences are necessary to be accounted for to achieve consistency in comparisons that contain SA and TG data. To obtain the ASL-associated \(DT_{TG}(\varphi_{TG},\lambda_{TG},t_{i})\) at the location of the TG \((\varphi_{TG},\lambda_{TG})\), the TG observations \((DT_{obs})\) need to account for the VLM and geoid change (\(\Delta\)N) corrections:  
Figure 1: Inter-relations between the participating datasets (TG, and SA), geoid model, and different reference ellipsoids (\(\Delta\)E) and relevant coordinate systems. The geoid change (\(\Delta\)N) and vertical land motion (VLM) at the time instants (t) need to be considered for TG readings according to the reference time epoch (\(t_{0}\)). TGs could refer to different national/pan-continental datums (e.g., ETRF) while SA data refer to different global reference frames (e.g., ITRF). For the rest of the used abbreviations, see the main body of the text. \[DT_{TG}(\varphi_{TG},\lambda_{TG},t_{i})=DT_{obs}(\varphi_{TG},\lambda_{TG},t_{i})+ VLM(\varphi_{TG},\lambda_{TG}).\ (t_{i}-t_{0})+\Delta N(\varphi_{TG},\lambda_{TG}).\ (t_{i}-t_{0}) \tag{1}\]  
where the selected reference time epoch is denoted by \(t_{0}\) and the TG observation time instant by \(t_{i}\). The VLM value (denoted by \(VLM(\varphi_{TG},\lambda_{TG})\)) and geoid changes (denoted by \(\Delta N(\varphi_{TG},\lambda_{TG})\)) can be obtained from an appropriate VLM model. To obtain the TG-derived sea level, \(DT_{TG\in SA\ cycle}\), for the SA overly instant, \(t_{cycle}\), the \(DT_{TG}\) data need to be selected within a predefined time window, \(w\) (before and after \(t_{cycle}\)). The matching TG data to the time instant, \(t_{cycle}\), is estimated by the following linear (timewise) interpolation:  
\[DT_{TG\in SA\ cycle}\Big{(}\varphi_{TG},\lambda_{TG},t_{cycle}\Big{)}=interp \Big{(}DT_{TG}\Big{(}\varphi_{TG},\lambda_{TG},t_{cycle}\pm w\Big{)}\Big{)} \tag{2}\]  
The instantaneous SA-derived along-track DT data point \(DT_{SA}\Big{(}\varphi,\lambda,t_{cycle}\Big{)}\) at an arbitrary location \((\varphi,\lambda)\), determine by subtracting the geoidal height (\(N\)) from the \(SSH\) using the following expression:  
\[DT_{SA}\Big{(}\varphi,\lambda,t_{cycle}\Big{)}=SSH\Big{(}\varphi,\lambda,t_{ cycle}\Big{)}-\Delta\text{E}(\varphi)-N(\varphi,\lambda) \tag{3}\]  
where \(SSH\Big{(}\varphi,\lambda,t_{cycle}\Big{)}\) is the SA-derived instantaneous SSH data point by  
\[SSH\Big{(}\varphi,\lambda,t_{cycle}\Big{)}=h_{SA}-(R+\Delta R) \tag{4}\]  
where \(\Delta R\) are the necessary geophysical corrections that need to be applied to the SA range, \(R\).