Reconstruction of Baltic Gridded Sea Levels from Tide Gauge and Altimetry Observations Using Spatiotemporal Statistics from Reanalysis

Abstract

A method for reconstructing weekly Baltic gridded sea levels was developed and tested. This method uses input data from tide gauge and altimetry observations. The reconstruction is based on sea level empirical orthogonal function (EOF) modes, calculated as spatiotemporal statistics from daily model reanalysis results available from the Copernicus Marine Service for the 1993–2021 period. In the semi-enclosed, tideless Baltic Sea, the four leading EOF modes cover 99% of the sea level variance. Several experiments with different combinations of input data were carried out. This method was validated against coastal tide gauges and altimetry data. The best reconstruction was obtained when both the tide gauges and altimetry data were used as inputs. An assessment of the centered root-mean-square difference (cRMSD) of the reconstruction versus the tide gauges revealed a value of 0.05 m, and a result of 0.10 m was revealed versus altimetry. The average coefficient of determination (R²) was determined to be 0.93 for the tide gauges and 0.82 for the altimetry data. In the cases where only one type of input data was used, the reconstruction worsened with respect to other data sources. The reconstruction method demonstrated its usefulness for the reconstruction of coastal sea levels in unsampled locations and the calculation of changes in sea volume.

1. Introduction

The spatiotemporal features of sea level variability in the semi-enclosed Baltic Sea are rather different from those in the ocean and tidally dominated seas. Enhanced observation-based knowledge of sea levels, including gridded data, is essential both for addressing practical issues related to sea levels and for understanding the dynamics of the sea.

1.1. Sea Level Variability in the Baltic Sea

Sea level variations are one of the main concerns of ongoing climate change, pointing to the adverse effects of global sea levels rising, including the increase in floods and other extreme events [1]. In the semi-enclosed, tideless Baltic Sea, sea level dynamics are governed by a multitude of oceanographic, atmospheric, hydrological, and geological processes, which have temporal scales ranging from minutes to millennia and geological time scales [2,3]. In the northern regions, water is seemingly retreating from the coasts, the reason for which is land uplift after the Ice Age. From the hydrodynamic point of view, sea level variations with a time scale from a few days to several years can be divided by their forcing mechanisms into external (50–80% of the total variance) and internal variability [4]. The highest pulse-like sea level elevations—floods due to storm surges—have been observed in the southwestern, eastern, and northern ends of the sea (3.4 m in Kiel (1872), 4.2 m in St. Petersburg (1824), and 2.1 m in Kemi (1984), respectively).

The longest instrumental sea level data series in the Baltic Sea originates from coastal tide gauges. In Stockholm, observations started in 1774 [5]. Tide gauges register the relative sea level with respect to the local geodetic height system. A second data source—altimetry—has provided offshore absolute sea levels along satellite tracks since the early 1990s [6]. A third data source—numerical modeling—has been used for short-term storm surge predictions since the 1950s [7], but this method only became mature enough for comprehensive sea level variability analysis in the 2000s [8,9].

On the longest secular time scales, the dominant sea level signal in the Baltic Sea is glacial isostatic adjustment (GIA), for which the direction and magnitude depend on the sub-region [10]. In the northern Fennoscandia region, the recent global mean sea level rise [11,12] has been superseded by land uplift, resulting in a relative lowering in sea level of a few millimeters per year, and the associated coastal land gain [13,14,15].

On weekly-to-multidecadal scales, the mean sea level is controlled by the changes in sea volume (i.e., filling and emptying of the sea) caused by climate forcing in terms of the mean wind and temperature, river discharge, and precipitation/evaporation. Multiyear sea level changes are connected to variations in climate indices such as the North Atlantic Oscillation (NAO) [16] and other atmospheric circulation descriptors [17]. The annual sea level cycle is well distinguished because of cyclic atmospheric forcing and river runoff [18]. For weekly scales, Svansson [19] applied a Helmholtz resonator model on a basin representing simplified geometry of the Baltic Sea. A resonant period of ~10 days was found for filling and emptying the sea basin, resulting from variable forcing from the open ocean side.

Storm surges [20] and other weather-forced sea level oscillations [21], including basin eigenoscillations (seiches), have time scales ranging from hours to several days; they redistribute the water within the sea, on top of the mean sea level in the forcing period. Extreme storm surges occur with specific preconditioning [22], such as high filling of the sea and the occurrence of selected weather patterns and storm tracks, generating high surges [23,24]. On top of the storm surge sea level elevation, the local wave setup can further elevate the water line on sloping coasts [3,25].

1.2. Overview of Sea Level Reconstruction

Extending beyond time series analysis from Baltic Sea tide gauges [5,14], spatiotemporal analysis requires gridded sea level data. Simple coast-to-coast interpolation techniques are not considered to be appropriate in the Baltic Sea, and recent studies have proposed methods that statistically combine the three abovementioned data sources. These studies differ in terms of their statistical methods, selected spectral windows of reconstruction, and preprocessing of geodetic and altimetry data.

Estimates of global sea surface variability have used gridded data since the 1990s [26], when empirical orthogonal function (EOF) reconstruction was introduced based on remote sensing data for sea surface temperature. For sea level variability, in the earliest studies, the statistical covariance patterns were calculated using early altimetry data [27,28]. For the pre-altimetry era (i.e., before 1985), only sparse tide gauge records were used as input data for reconstruction on a 1° × 1° grid with a monthly resolution. Another option is to calculate the covariance patterns from the model’s reanalysis results [29,30,31,32]. With improved model results (longer calculation period and better resolution and accuracy) becoming available, this approach has the benefit of providing more detailed and robust covariance estimates than observations. In the case of a few tide gauge observations being available, the cyclostationary empirical orthogonal function (CEOF) method has better performance than standard EOF reconstruction techniques [33].

In the global ocean, the main interest is to detect and quantify the global sea level rise. Therefore, preprocessing of tide gauge data includes sophisticated correction of generally unknown datum shifts between the locations, as shown by Church et al. [28]. Short-term variations in winds and air pressure, as well as tides, cause unwanted noise for sea level trend estimations; their impact is corrected or suppressed.

One important field of sea level reconstruction is the study of ocean dynamics [34].

In the Baltic Sea, the comparison of observations and model results [9] revealed that in the 2000s, the model results matched observations better on shorter time scales (i.e., characteristic time of less than 2 months) than on longer scales. However, during extreme storm surges, the model errors were up to 25 cm, a few times higher than the background centered root-mean-square difference (cRMSD) of 8 cm. For the period 1993–2011, the comparison of the monthly mean tide gauge and altimetry data revealed a cRMSD of 3.4 cm, while the standard deviation (STD) was ~15 cm [35]. In this analysis, the original L3 altimetry data along the tracks were averaged during a month over 0.25°× 0.25° grid boxes. Model reanalysis showed similar performance results to the two reconstructed data sets.

Recently, the reconstruction of gridded monthly sea level data was performed for the period 1900–2014. The method used tide gauges and altimetry as input data, whereas model reanalysis was used for estimation of covariance matrices for each reconstruction grid point [36]. Over the longer period, the cRMSD at coastal tide gauges was estimated to be 3.8 cm, as in the variability study [35] for the shorter period. Another study focused on the annual sea level cycle [37]. The CEOF method was applied for the estimation of sea level background statistics, using the model reanalysis data. In this study, monthly sea levels reconstructed from either tide gauges or altimetry data demonstrated similar variability on the annual and longer time scales.

In the present study, observation-based sea level reconstruction methods were validated in the Baltic Sea for the monthly time scale as the shortest period, whereas hourly-to-weekly time scales remained unresolved.

1.3. Aims and Study Design

The present study aimed to develop and test a computationally efficient method for the reconstruction of Baltic gridded sea levels from tide gauge and altimetry observations. The method should be applicable on shorter-than-monthly time scales. Spatiotemporal sea level variability statistics, needed for the optimized reconstruction, are estimated from the model reanalysis data that cover the study domain with high resolution over a long period without gaps.

Sea level dynamics have well-defined basin-scale variation patterns, as described in Section 1.1. To suppress shorter period variability that has more complex covariance patterns, we used weekly average sea level data. By “computationally efficient”, we mean that new weekly sea level maps, necessary for the bias correction of operational numerical forecasts, could be reconstructed nearly immediately when new observations become available, without the need to conduct new, computationally demanding numerical model runs. This study was made possible by the newly available altimetry and model data sets from the Copernicus Marine Service [38], along with the continued delivery of tide gauge data.

Covariance estimates reveal that high covariance values occur over considerable distances. These basin-guided covariance patterns can be mathematically represented by EOF modes [39], and their physical interpretation is the response of the basin to the forcing fields (e.g., atmosphere, rivers, adjacent sea area) at scales comparable to or larger than the basin dimensions. The reconstruction of the large-scale gridded sea surface temperature and salinity data from sparse observations (e.g., monitoring data, FerryBox, buoy stations) proved useful [39]. The method is based on finding the values of principal components (PCs, also called EOF amplitudes) that satisfy the least-squares minimization of differences between observed values and superpositions of leading EOF modes. This method has been successfully applied in operational forecasts for the data assimilation of sea surface temperature and salinity [40].

This manuscript starts with the Introduction (Section 1), followed by the description of data and methods in Section 2. After introducing the used source data, the steps of data preparation are outlined. Section 2 also presents the EOF reconstruction framework and specifies the conducted reconstruction experiments. The results are presented in Section 3, which contains the calculated EOF modes, comparisons of reconstructed and observed sea levels, reconstruction of coastal sea levels in unsampled locations, and sample estimation of changes in sea volume. The paper ends with the Discussion (Section 4) and Conclusions (Section 5).

2. Data and Methods

2.1. Source Data from the Copernicus Marine Service

2.1.1. Reanalysis 1993–2021

Gridded data from the recent Copernicus Baltic Sea reanalysis were used for the evaluation of spatiotemporal statistics of sea level variability [38]. NEMO 4.0 was used for this reanalysis in combination with the sea ice and thermodynamic model SI3. This has a horizontal resolution of ~1 nautical mile (1′40″E by longitude and 1′N by latitude), and it has up to 56 levels along the vertical direction, with a grid step of ~1 m at the surface, but with depth, the grid step increases up to 25 m, whereas the maximum depth is 712 m. The model domain covers the Baltic Sea and North Sea; data are extracted from 9° to 30.2°E and 53° to 65.9°N on a grid of 763 by 774 points along longitude and latitude, respectively. The model was forced by atmospheric data from the ERA5 reanalysis data set available from the Copernicus Climate Service, river discharge data from the Ehype model, and open boundary conditions from the North West Shelf model. Reanalysis involved the assimilation of temperature and salinity, but not for the sea level. Data assimilation was carried out using the LESTKF filter within the PDAF (Parallel Data Assimilation Framework, [41]). All of the physical variables are presented on the Copernicus data portal for the period 1993–2021 as daily mean gridded values. In the delivered NetCDF files, the sea level variable is sla (sea level anomaly), calculated with the boundary conditions from the CMEMS North West Shelf multiyear product.

2.1.2. Tide Gauges

The Copernicus Marine Service provides hourly sea level data from 105 coastal stations during 1993–2021 (Figure 1), as presented in the technical report of in situ TAC [42]. The data contain the variable slev (water surface height above a specific datum) in reference to BSCD2000. The data have undergone quality check, and each data record has a quality flag. Basic statistical (e.g., mean, trends, variability) and geodetic (e.g., coordinates, leveling) details of tide gauge observations have been presented in several research papers, with variable numbers of involved stations [2,3,4,13,15,18,21,35,36,37]. Because of the low tidal activity [21], the sea level instruments in the Baltic are often named also called “mareographs”. Data coverage over time is variable in the Copernicus Service [15,42]; however, for technical reasons, we decided not to search for other sources of data, such as the Permanent Service for Mean Sea Level, referenced by Kapsi et al. [15]. There is nearly gap-free data coverage during the study period from Gedser to Hamina (counting the stations clockwise), but only ~50% from Kronstadt to Kolka (the data records in the Copernicus service start mainly from 2005 onwards) and 30% from Gdansk to Rostock. For a basin-wide overview along the coast, we selected 21 stations from different coastal sections for detailed analysis, as shown in Figure 1 by the station names.

2.1.3. L3 Altimetry

Altimetry data, observed along the satellite tracks, were acquired from the Copernicus Marine Service multiyear observations “European Seas Along Track L3 Sea Surface Heights Reprocessed” [43,44], tailored for data assimilation into the models. The data originate from altimeter Copernicus missions (Sentinel-6A, Sentinel-3A/B) and other collaborative or opportunity missions (e.g., Jason-3, Saral[-DP]/AltiKa, Cryosat-2, OSTM/Jason2, Jason-1, Topex/Poseidon, Envisat, GFO, ERS-1/2, Haiyang-2A/B). The data from these >20 satellites are presented at various times on the initial grid, with a step of 7 km.

The data records of the altimetry product [43] contain sea level variables given in

Table 1. Sea level variables Iin the L3 altimetry product.

Variable Name	Explanation
sla_unfiltered	raw sea level anomaly measurement including noises; corrected for the dac, lwe, ocean_tide and internal_tide
sla_filtered	sea level anomaly low pass filtered for noise reduction; corrected for the dac, lwe, ocean_tide and internal_tide
mdt	mean dynamic topography (long-term mean of sea surface height 1993–2012 above geoid) that is constant for grid point during time; it is used to compute the absolute dynamic topography adt = sla + mdt
dac	dynamic atmospheric correction for the removal of high frequency variability induced by the atmospheric forcing and aliased by the altimetric measurements; the sla is already corrected for the dac
lwe	long wavelength error due to changes in satellite orbits; the sla is already corrected for the lwe; it is stored with opposite sign compared to the other corrections so if the user wants to uncorrect it or to use another correction instead, he must subtract it from the sla in the product
ocean_tide	ocean barotropic tide correction (including S1S2 signal) based on tidal model; the sla is already corrected for the ocean_tide
internal_tide	internal tide correction: coherent part of the baroclinic tide (phase-locked with barotropic tide frequency); the sla is already corrected for the internal_tide

.

2.2. Preparation of the Data

2.2.1. Selecting the Reference Levels

Although the different data sets—model reanalysis ${\eta }_M$, tide gauges ${\eta }_T$, and altimetry ${\eta }_A$—should ideally have the same mean reference field $\overline{\eta }=\overline{\eta }\left(x,y\right)$, the actual references for ${\overline{\eta }}_M$, ${\overline{\eta }}_T$, and ${\overline{\eta }}_A$ are somewhat different [45,46]. From the oceanographic viewpoint, the adopted common reference should provide realistic gradients of $\overline{\eta }$ that are consistent with known circulation. Among the three data sources, model reanalysis is the only one that produces dynamically balanced sea levels. Sea level gradients cause pressure gradients and therefore, induce water motions. On the other hand, horizontal divergence or convergence of vertically integrated water motions causes lowering or rising of the sea level due to the continuity equation. With the aim of following these dynamical balances, the model-based reference ${\overline{\eta }}_M$ (also named as mean dynamic topography, MDT_m) was adopted as the baseline reference (Figure 1). Model-based leveling was also proposed for the European Vertical Reference System [47]. Note that, in other studies, geodetically refined MDT_g has also been adopted [48].

Model reanalysis and tide gauge stations have different mean sea level values at coastal stations; therefore, this pointwise determined bias was subtracted prior to the sea level reconstruction. While comparing the reconstruction with TG observations, pointwise corrections were added again, to follow the mean local coastal reference.

Regarding altimetry data, mean dynamic topography (mdt = MDT_a) in the Copernicus product has been defined independent of the model results. The dynamic topography is variable over time due to the changes in forcing and internal circulation. The averaging periods are also different: 1993–2012 for altimetry and 1993–2021 for reanalysis. According to the rough analysis, the mean dynamic topography determined from the altimetry is ~0.20 m higher than that determined from the model reanalysis (mean sea level in Figure 1). The difference in dynamic topographies is slightly space-dependent, from 0.19 m in the Western Baltic and the northern part of the Bay of Bothnia, to 0.26 m in the southern part of the Baltic Proper. During the sea level reconstruction, the altimetry values were brought to the model reanalysis background using the space-dependent bias between the two topographies.

2.2.2. Selecting the Weekly Interval

While operational modeling provides sea level forecasts with hourly intervals, altimetry data have been used mainly for the monthly scale [35,36,37]. From a physical point of view, the weekly average sea level allows for the suppression of short-period oscillations due to local wind tilt, storm surges, Kelvin waves, and seiches that are presently not resolved by the altimetry data, but allows for keeping sea level changes related to circulation and water exchange between the sea’s sub-basins, along with the external forcing from the North Sea. For the sake of reasoning, Samuelsson and Stigebrandt [4] divided the sea level variability into short-term internal variability with a time scale of up to a few days, and longer-term variability that is forced by water exchange in the sea. Raudsepp et al. [49] estimated the temporal correlation radius of sea level oscillations in the Estonian coastal area for 10 days.

Temporal autocorrelation and spectral analysis studies over extended regions have revealed that weekly and longer period sea level changes can be effectively separated from shorter period oscillations just by taking the weekly average and residual values [50]. In this context, Lagemaa et al. [51] found that a weekly average filter length is optimal for correcting the bias of short-term sea level forecasts, using recent observations. Soomere and Pindsoo [52] have shown the importance of weekly-scale high water levels within the sea basins in forming superimposed extreme storm surges.

2.2.3. Averaging and Filtering of the Data

The sea level reanalysis data were used in the following aggregations:

• Weekly mean data on the basic grid, with the tide-dominated Kattegat and Danish Straits excluded, as shown in Figure 1. The excluded areas have higher sea level variability [36] and are not of interest for the present study.
• Weekly mean time series at model grid points corresponding to coastal tide gauge stations (Figure 1).
• Extracts of weekly mean sea levels at L3 altimetry grid points.

When calculating weekly average tide gauge observations as input and validation data for reconstruction, the weeks with more than 25% missing observations were blanked out. All of the available tide gauge data were used in the sea level reconstruction procedure. Detailed results for the 21 selected stations will be presented in Section 3.2.

Along-track altimetry data need to be converted to their weekly averaged form while also maintaining reasonable spatial averaging. With the example from an earlier study [35], where typical satellite tracks are shown, a coarse “altimetry grid” was introduced with steps of 0.25°E (~14 km) and 0.25°N (~27 km). This adopted grid for the altimetry data consisted of 89 points by longitude and 53 points by latitude. Grid cells with missing or low numbers of altimetry data were blanked. For the inclusion of weekly altimetry data, the reasonable criterion was set that non-averaged altimetry data are found at least in 25 grid cells of the fine reconstruction grid, within a 0.25°E by 0.25°N box, during the week. On average, 36 altimetry grid cells were used for reconstruction.

2.3. EOF Reconstruction Framework

As described in [39,40], we used the $M\times N$ space-time data matrix $X$, which contains time slices $x_i$ as spatial state vectors of length $M$ at time $i$, where $i=1\dots N$. The spatial vectors, containing deviations from the space-dependent temporal mean vector ${\overline{x}}_m$, are mapped from one-, two-, or three-dimensional physical coordinates into the one-dimensional vector. The covariance matrix $B=X^{T}X/N$ (averaging over time) is used to calculate the eigenvector matrix $E$ containing spatial empirical orthogonal function (EOF) vectors $e_k$. Principal components (PCs), or “amplitudes”, are found for each time $i$ from $a_i=E^T{x}_i$ when $x_i$ is known. Then, $x_i$ is decomposed as $x_i=E\Lambda a_i$, where $\Lambda$ is the diagonal matrix of eigenvalues.

When reconstructing the field of interest at time $i$ from observations $y_i$ of length $L_i$, they are then taken from different locations (usually $L<M$) from the model grid points. For the comparison of observations with the model, gridded data are transformed to the observation points by the observation operator $H_i$, using the formula $H_i{\tilde{x}}_i=H_{i}E{\tilde{a}}_i$, where ${\tilde{a}}_i$ is the “observational” PC to be found. The values ${\tilde{a}}_i$ should follow least-squares minimization of reconstruction error in relation to observations ${‖y_i-H_{i}E{\tilde{a}}_i‖}^2\Rightarrow \min$. The expressions to find observational PCs and reconstructed fields are found using matrix–vector formulation of least-squares estimators [53] in the following form:

$${\tilde{a}}_i={\left(E^T{H}_i^T{H}_{i}E\right)}^{-1}{{\rm E}}^T{H}_i^T{y}_i$$

(1)

$${\tilde{x}}_i=E{\tilde{a}}_i$$

(2)

There is also a possibility of using the time-dependence of observation data for each reconstruction time step, assuming linear time-dependence of EOF amplitudes within the calculation interval. In the present study, the dependence option was not used.

The field of interest is reconstructed by superposition of continuous model-based mode patterns multiplied by observational amplitudes that meet adopted statistical limits based on the eigenvalues of the specific mode [39].

2.4. Performed Sea Level Reconstruction Experiments

One sea level reconstruction experiment followed the flowchart presented in Figure 2. Calculation of gridded EOF modes (Section 2.3) was performed once from the reanalysis data (Section 2.1.1). For each timestep during the study period, the initial data consisted of tide gauge data (Section 2.1.2) and altimetry data (Section 2.1.3), which were brought to the selected reference (Section 2.2.1) and later filtered to the weekly values (Section 2.2.3). The observational data were merged, and in case of enough data, EOF reconstruction was performed using Formulae (1) and (2) (Section 2.3). If the calculated PC of the EOF was within acceptable limits up to the selected highest mode, the final 2D sea level grid was saved.

While performing the experiments, unified data and mode selection procedures were introduced to handle cases with few or spatially unfavorably distributed data. In case of too few observations being available—i.e., fewer than 25—reconstruction was not performed for this time step.

The number of EOF modes included in the reconstruction was determined for each time interval. For this, amplitude limits were set for the reconstructed fields, based on the EOF statistics from the model reanalysis. Based on [39], a reasonable limit is double the standard deviation of the PC time series of reanalysis. When the amplitude of a particular mode exceeded its limit, then this mode and higher modes were considered to be too noisy, and they were trimmed from the reconstruction. The success of reconstruction depends on the number and spatial configuration of observations [39]. This is usually considered undesirable since variable number of included EOF modes creates changes in feature resolution; however, by our experience in this study, it has less effect on the accuracy than inclusion of modes with outlier amplitudes, which may create completely distorted patterns. In most cases, 6 to 10 leading modes was found to be acceptable; the mean number of modes was 7.

The skill of sea level reconstruction was evaluated with five experiments using different combinations of tide gauge and altimetry data (

Table 2. Main features of the performed sea level reconstruction experiments. Altimetry sea level data were taken according to Table 1.

Name	Tide Gauges	Altimetry	Description
rec1	Y	N	Input from tide gauge data only; altimetry comparison with rec4.
rec2	Y	Y	Input from tide gauge and altimetry sla_filtered; sea level anomalies accounted for on top of model-based dynamic topography.
rec3	Y	Y	Input from tide gauge and altimetry ADT = sla_filtered + mdt; sea level anomalies accounted for on top of altimetry-based dynamic topography.
rec4	Y	Y	Input from tide gauge and altimetry ADT_c0 = sla_filtered + dac + ocean_tide + iw − lwe + mdt; same as rec3, but corrections to sla_filtered removed.
rec5	N	Y	Same as rec3, but tide gauge data not included.

). Regarding altimetry, it is not yet clear which of the several data options would be best for the semi-enclosed non-tidal Baltic Sea, where weather forcing and seiches play important roles. Therefore, we selected three options, from rec2 to rec4, with different altimetry data.

3. Results

3.1. Calculated EOF Modes of Sea Level

The full covariance matrix $B=X^{T}X/N$ contains $M^2$ pairs of second-order statistical moments, whereas spatial lag between the correlated points amounts from the model grid step ~2 km to the basin scale ~1500 km. When conducting widespread optimal interpolation [54], it is assumed that the spatial correlation function (derived from covariance) follows a spatial decay that depends on the distance between the correlated points that can be fitted by an exponential, Gaussian “bell curve”, or some other function. It is also often assumed that statistical properties such as variance are not changing between the spatial locations (i.e., homogeneity assumption). In our case, sea level covariance calculated from the model reanalysis has a complex two-dimensional histogram that is dependent on the spatial lag between the points over the whole sea domain (Figure 3). Covariance estimates cover the pairs of points within and between different basins of the sea (Figure 1); multiple maxima occur in covariance histogram extracts at fixed lags. Therefore, it is justified to use the empirical orthogonal function (EOF) framework, which considers the full covariance matrix but allows it to be reduced to a few of the most energetic EOF modes.

The EOF modes represent the perturbations to the mean sea level shown in Figure 1. The first mode, which contains 94.6% of the overall variance, is nearly flat, meaning that it represents filling and emptying (predominantly through the Danish Straits) of the whole sea basin. The modes from two to four represent tilting of the sea level in different directions, forced mainly by variable weather patterns. In the PC time series (Figure 4c), the dominant first mode is rather variable. Although in the seasonal average there are high PC values (and mean sea level) in winter and low PC values in summer, this is not evident from the data of individual years. Exceptionally high winter PC values were noted for the years 1996, 2003, 2010, 2013, and 2018.

The reconstruction sums the mean sea level and contributions of different modes, whose superposition weights are calculated by the reconstruction procedure based on the location of observations and observed values. The first 10 modes were calculated; however, the first four modes, presented in Figure 4, already cover 99% of the total sea level variance during 1993–2021 over the entire sea.

3.2. Comparison of Reconstructed and Observed Sea Levels

Weekly mean sea levels vary in a range of more than a meter. Examples of the sea level time series, extracted for 2007, are given for Helsinki coastal station (Figure 5). Data from experiments rec1 to rec4, which used tide gauge data for sea level reconstruction, show similar results, close to the observed values. Data from rec5, which used only altimetry data and did not use tide gauge data for reconstruction, are also close to the tide gauge observations, although some deviations are visible.

Model data were used to set up the long-term (1993–2021) datum (mean reference level) for observations; however, the variability in modeled sea levels at coastal points (an example is shown in Figure 5) shows greater differences from the observed values than the reconstructions. Specifically, the centered root-mean-square difference (cRMSD) of the model results from the tide gauge observations is 0.085 m, while experiments from rec1 rec4 show cRMSD ranging from 0.034 to 0.046 m, and rec5 has the highest value of 0.106 m. Pearson’s coefficient of determination (R²) is higher than 0.90 for rec1 to rec4, but only 0.66 for rec5. The standard deviation (STD) of the model results at coastal points is 0.222 m, 22% higher than the STD of the observations (0.182 m). Reconstruction experiments rec1 to rec4 reveal STD from 0.176 to 0.198 m, while rec5 provides a smaller STD of 0.163 m.

As one option for the comparison of reconstructed and observed sea levels, along-coast changes in the statistical characteristics of the reconstruction experiments versus selected coastal tide gauge observations (locations in Figure 1) are presented in Figure 6. In the Bothnian Sea and Bothnian Bay, the mean observed sea levels (Figure 6a) on the Swedish coastal section from Forsmark to Ratan are ~0.1 m higher than on the opposite Finnish coastal section from Kemi to Hanko. This coast-to-coast difference is considered further in the Discussion. On the Swedish coastal section in the Baltic Proper, from Simrishamn to Landsort, the mean observed sea levels are higher as well. The reconstruction experiments give similar mean sea levels to the model results, except for rec5, which produces higher mean sea levels in the eastern part of the sea, from Hamina to Kolka.

Regarding the standard deviation over the study period (Figure 6b), the inclusion of altimetry data reduces the STD from rec1 (closest results to the observations; altimetry data not included in the reconstruction) to rec5. Comparison of rec1 to rec5 with tide gauge observations, using cRMSD and R², reveals the closest match (i.e., smallest cRMSD and highest R²) for rec1 and the worst reproduction for rec5. Along the coast, larger reconstruction uncertainties are found in the Swedish section of the Bothnian Sea and Bothnian Bay (Forsmark to Ratan), as well as on the Polish and German sections of the western Baltic (Gdansk to Rostock).

The numerical results of the statistical comparison of the reconstructed data with the tide gauge and altimetry data are presented in

Table 3. Statistical sea level characteristics of reconstruction experiments rec1 to rec5 compared to the observations at tide gauges and altimetry observations. Tide gauge and altimetry data were shifted to the model reanalysis reference during the reconstruction.

	rec1	rec2	rec3	rec4	rec5
Tide gauges at coastal stations
Mean, tide gauges (m)	0.085
Mean, reconstruction (m)	0.089	0.106	0.101	0.095	0.117
STD, tide gauges (m)	0.182
STD, reconstruction (m)	0.198	0.176	0.176	0.190	0.163
cRMSD, reconstruction to tide gauges (m)	0.034	0.046	0.046	0.040	0.106
R2 coefficient of determination, reconstruction to tide gauges	0.952	0.935	0.935	0.948	0.659
Altimetry points
Mean, altimetry (m)	0.101	0.143	0.117	0.101	0.117
Mean, reconstruction (m)	0.103	0.126	0.112	0.103	0.117
STD, altimetry (m)	0.206	0.168	0.167	0.206	0.167
STD, reconstruction (m)	0.201	0.175	0.174	0.193	0.162
cRMSD, reconstruction to altimetry (m)	0.112	0.060	0.066	0.086	0.094
R2 coefficient of determination, reconstruction to altimetry	0.723	0.881	0.855	0.827	0.930

. It can be concluded that combination of data from coastal tide gauges and altimetry (rec2 to rec4) give the best results in terms of the minimum cRMSD and maximum R² in both the coastal (i.e., tide gauges) and offshore (i.e., altimetry) observation domains. Omitting the altimetry input data (rec1) improves the accuracy at the coastal stations insignificantly (the cRMSD of 0.034 m in rec1 is slightly smaller than the 0.046 m in rec2 and rec3, and 0.040 m in rec4) but reduces the offshore accuracy (the cRMSD of 0.112 m in rec1 is larger than the 0.060 m and 0.066 m in rec2 and rec3, and 0.086 m in rec4). The other specific case—input from altimetry data only (rec5)—provides roughly the same offshore accuracy as other altimetry-based reconstruction experiments, but the coastal accuracy is reduced (the cRMSD of 0.106 in rec5 is higher than the 0.046 in rec2 and rec3). The experiment rec4, with input from tide gauge and altimetry ADT_c0 (Table 2), yields higher accuracy in relation to tide gauges compared to rec2 and rec3, but it provides slightly worse offshore accuracy.

Regarding the altimetry options from rec2 to rec4, the cRMSD and R² values are rather similar, and within this study, it was not possible to determine which of the altimetry options is better suited for the Baltic Sea’s conditions.

The two-dimensional histograms of the observed versus reconstructed sea levels, presented in Figure 7 for rec1, rec3, and rec5, illustrate that although the reconstruction procedure is of good quality, the most accurate reconstruction in both the coastal stations and offshore regions was obtained for rec3 when both the tide gauge and altimetry data were involved.

3.3. Reconstruction of Coastal Sea Levels in Unsampled Locations

Spatial coverage of coastal tide gauges has increased over time. Still, it remains a challenging task to obtain reliable sea level characteristics for the coastal locations where observations are missing. Therefore, an experiment rec1b, modified from rec1 (Table 2), was conducted with a reduced number of input tide gauge locations. The calculations were made using input data from 21 “named” coastal tide gauges, as shown by the station names in Figure 1. The evaluation was carried out over all 105 stations. Some of the data were considered to be missing and were not used in the reconstruction, but after the reconstruction the full data set was used for skill estimates. It should be noted that the statistical characteristics of reconstruction at these stations, based on full input data, are given in Figure 6.

The modified experiment rec1b revealed very similar statistical characteristics to those presented in Table 3 and Figure 6. Specifically, the mean sea level calculated over all of the stations during 1993–2021 differed by less than 0.001 m, where the mean level was 0.089 m. The overall STD of all weekly sea levels was 0.198 m for rec1 and 0.196 m for the modified rec1b (a STD map is presented in Figure 8a). It should be noted that sea level variability is higher in the northern and eastern regions (i.e., Bothnian Sea, Gulf of Finland, Gulf of Riga) than in the central Baltic Proper. The change in the reconstructed sea level patterns from rec1 to rec1b is visualized by their cRMSD map shown in Figure 8b. The main differences between the experiments became evident in the southwestern region of the sea, close to the Danish Straits, with the tidal influence from the North Sea. In the coastal areas, the highest weekly sea level cRMSD was 0.03 m, or 15% of the standard deviation. Greater differences, up to 0.07 m, were found in the deep basins of the Baltic Proper, which can be interpreted as a loss of reconstruction quality due to the reduction in the input data. It should be noted that, in both the rec1 and rec1b experiments, altimetry input data were not used for reconstruction, in order to imitate earlier time periods when only limited tide gauge data were available. Pearson’s coefficient of determination (R²) revealed a close match with the reconstructed sea levels; the R² of reconstruction with observations was higher than 0.95 for both the main and the modified experiments.

The results show that weekly sea levels can be reliably estimated in unsampled locations.

3.4. Changes in Sea Volume

One of the indicators of sea level variability is sea volume, which is of primary interest for water budget calculations [55]. When calculating the changes in sea volume from gridded data, it is convenient to convert the results to the mean sea level. Since the western sea area was excluded from the reconstruction domain (Figure 1), we calculated the domain area of 379,100 km² from the gridded data. From the weekly reconstructed time series, annual and seasonal courses (Figure 9) were calculated. The reconstruction results of experiments rec1 to rec4 revealed a close match. Experiment rec5, which had only altimetry data as inputs, gave higher mean sea levels than the other experiments, especially starting from 2011, when the surplus amounted to between 0.04 and 0.08 m. The annual mean sea levels shown in Figure 9a correlate well (R² = 0.93) with the earlier results reported by Madsen et al. [36].

On the seasonal scale (Figure 9b), the known maximum sea volume in winter and minimum sea volume in spring [56] were well reproduced. The annual spatial mean sea level amplitude is ~0.18 m, as reported by previous altimetry data.

4. Discussion

4.1. Reconstruction Skill

The sea level height variation is practically the most important on the coastline, where the location of the moving waterline affects many aspects of human life; it is also important for navigation in harbors and shallow ship passages.

The reconstruction skill is estimated by differences between two data sets (estimated/modeled and observed), calculating the cRMSD and Pearson’s coefficient of determination (R²) using standard formulae [36,57]. Bias, when presented, is only considered indicative, due to the uncertainty of the sea level reference. As benchmarks for the present study, previous skill assessments of gridded sea level data are presented in

Table 4. Skill estimates for the gridded sea level data sets.

Source	Compared Variables	Period	Time Interval	cRMSD (m)	R2
Madsen et al., 2019 [36]	Reconstruction—tide gauges	1900–2014	Month	0.04	0.92
Madsen et al., 2019 [36]	Reconstruction—altimetry in Baltic Proper	1900–2014	Month	0.06	0.81
Present study	Reconstruction—tide gauges	1993–2021	Week	0.05	0.93
Present study	Reconstruction—altimetry	1993–2021	Week	0.10	0.82
Copernicus Reanalysis, 2023 [38]	Reanalysis—tide gauges	1993–2018	Day	0.10	0.77
Kärna et al., 2021 [57]	Operational forecast—tide gauges	2014–2016	Hour	0.10	0.90
Hordoir et al., 2019 [58]	Research model—tide gauges	2011–2012	Hour	0.06	0.90
Jahanmard et al., 2022 [50]	Bias-corrected research model—tide gauges	2017–2020	Hour	0.03	0.90
Mostafavi et al., 2023 [52]	Bias-corrected research model—altimetry	2017–2019	Hour	0.09	na

, together with the results from the present study. For comparison, gridded data were extracted at the times and locations of the observations. The same approach was followed in the present study.

Copernicus reanalysis data (fifth line in the table) were used in the present study for spatial covariance estimates. Since the reanalysis has already been evaluated versus tide gauge data [38], comparison of the reanalysis results with sea level reconstruction was performed only in certain cases. Lower R² values with tide gauges for reanalysis [38] than for the operational model [57] might be related to the specifics of atmospheric forcing; the reanalysis was forced by the ERA5 atmospheric reanalysis with a resolution of 0.25°E by 0.25°N, while the operational forecast used the 2.5 km MetCoOP HARMONIE model [57].

The reconstructed weekly sea level products in our study were compared with coastal tide gauge data, with cRMSD less than 0.05 m, and altimetry data, with cRMSD less than 0.1 m, when both of the data sources were included as input data. The average coefficient of determination was 0.93 when comparing the sea level reconstruction with coastal tide gauges and 0.82 in comparison with altimetry. The reconstructed sea levels at the coastal stations were closer to the observations than the reanalysis data set, as also indicated by Madsen et al. [36]. Reconstruction on shorter time scales—hours or days—was not performed in our initial study since the spatiotemporal coverage of altimetry data becomes more irregular for shorter time intervals. Operational forecasting [57] and bias-corrected research modeling [50,52,57] have revealed an hourly cRMSD of less than 0.1 m at most of the coastal stations.

The sea level trends have not been considered in our 29-year study, although they may also have some effect in such an interval. Tide gauge data were adopted from the Copernicus Marine Service without modification. In further implementations of the EOF reconstruction, the trend and reference level corrections can be easily introduced, following the example from oceanic [28] and Baltic Sea [36] applications.

Mostafavi et al. [52] have shown along-track altimetry sea level variations of up to 0.2 m that can be dynamically attributed to meandering unsteady frontal systems and mesoscale eddies. The appropriate resolution, reconstruction, and/or assimilation of such features would be a challenge for the significant improvement of physical circulation models.

4.2. Reference Level Problem for Flow Calculation

Geodetic height systems around the Baltic Sea have recently been updated [51]. The reported tide gauge observations reveal that the mean sea levels in the Gulf of Bothnia (including the Bothnian Sea and Bothnian Bay) drop from 0.10 to 0.15 m from the Swedish to the Finnish coast (Figure 6a). This drop is insignificant for coastal engineering and management, but it creates problems in the water budget calculations. The mean sea level drop between Ratan and Pietarsaari, on the opposite coasts of the Bothnian Sea, is 0.003 m (from 0.158 m in Ratan to 0.155 m in Pietarsaari) in the model reanalysis, but the observations with local datums give a difference of 0.114 m (from 0.154 m in Ratan to 0.040 m in Pietarsaari). Regarding the circulation and water budget, where sea level slope is an important driver of water motions, this difference between the modeled and observed sea level drops is an item of concern.

Let us consider simplified flow calculations. Over a long period, the stationary river discharge to the Gulf of Bothnia is $Q$ = 193 km³ year⁻¹ [59], which equals to 6100 m³ s⁻¹. Geostrophic transport due to the sea level slope can be estimated in the first approach as $Q_{\eta }=gH\Delta \eta /f$ [60], where $g$ = 9.81 m s⁻² is the gravity acceleration, $H$ = 80 m is the approximate depth of the Bothnian Bay, $\Delta \eta$ is the sea level drop across the Bothnian Bay, and $f$ = 0.00013 s⁻¹ is the Coriolis parameter at a latitude of 65°N. For the modeled $\Delta \eta$ = 0.003 m, the estimated $Q_{\eta }$ = 17,800 m³ s⁻¹, while for $\Delta \eta$ = 0.114 m, the estimated $Q_{\eta }$ = 678,700 m³ s⁻¹. The volume transport estimated by the modeled sea level drop can be considered realistic, since in addition to the river discharge, there is also the inflow of water from the adjacent sea area, which is usually a few times greater than the river discharge [61].

It is anticipated that assimilating the geodetic sea level differences in the order $\Delta \eta$ = 0.12 m into the circulation model may distort the water exchange between the basins of the Baltic Sea. In addition to the freshwater budget, in estuarine systems such as the Baltic Sea, the circulation calculated by the models should follow the salinity budget [61].

4.3. Applicability of EOF Reconstruction

Reconstruction is a computationally fast procedure, since the only operations it contains are Equations (1) and (2), with matrices whose maximum size is the number of grid points. Therefore, the sea level maps can be updated nearly immediately when new observations become available. This feature can be used for bias correction in various applications. For example, Lagemaa et al. [47] introduced interval pointwise bias correction at each forecast step based on a backward weekly average model to observe differences using recent observations. Spatial reconstruction allows the bias to be corrected over the whole forecast domain. It has been shown that when bias correction has been applied to the sea level operational model forecast, the cRMSD in the Gulf of Finland has been reduced [50].

Reliable sea level estimates all over the coastline can be obtained using the presented reconstruction method from only a limited number of observations. This means that, based on tide gauge observations, available from the 19th century, it is possible to reconstruct historical weekly sea levels to estimate changes in sea volume and water budget variations for specific Baltic sub-regions, such as larger gulfs and smaller bays. It is also possible to evaluate high and low sea level cases and associated events (storm surges, coastal erosion events, etc.) in the areas of missing tide gauge observations. This is particularly important for navigation in harbors and shallow ship passages.

Estimates of sea level trends, including vertical land motions, can be refined from the present point-wise estimates, but this requires more harmonization of the reference levels from different data sets. The reconstructed gridded sea levels can be used for the quantification of changes in sea volume. It is also interesting to consider variations in the Baltic Sea’s circulation that affect the ecosystem state. However, it is still a challenge to obtain reliable circulation indices.

5. Conclusions

A new method for the reconstruction of Baltic gridded sea levels from tide gauge and altimetry observations was developed and tested. The method is based on sea level EOF modes, calculated as spatiotemporal statistics from weekly model reanalysis results. The leading EOF modes converge well, and the first four modes already cover 99% of sea level variance. The reconstruction uses observations at coastal tide gauges and altimetry as input data. Because of the limitations of data coverage, weekly averages were considered as the shortest time interval to be resolved.

Several experiments were carried out with different combinations of input data. The method was validated against coastal tide gauges and altimetry data. The best reconstruction skill was obtained when both the tide gauges and altimetry data were taken into account. The cRMSD of the reconstruction was determined to be 0.05 m versus tide gauges and 0.10 m versus altimetry. The coefficient of determination R² averaged 0.93 for the tide gauges and 0.82 for the altimetry data. In the cases where only one type of input data was used, the reconstruction became worse with respect to other data sources. For example, using only altimetry data for the input improved the R² for altimetry data to 0.93, but it worsened the R² for tide gauge data to 0.66.

The reconstruction method proved useful for the reconstruction of coastal sea levels in unsampled locations and for the calculation of changes in sea volume.