Effects of large-scale atmospheric circulation on the Baltic Sea wave climate: application of EOF method on multi-mission satellite altimetry data

Wave heights in the Baltic Sea in 1992–2015 have predominantly increased in the sea's western parts. The linear trends in the winter wave heights exhibit a prominent meridional pattern. Using the technique of Empirical Orthogonal Functions (EOF) applied to the multi-mission satellite altimetry data, we link a large part of this increase in the wave heights with the climatic indices of the Scandinavian mode, North Atlantic Oscillation, and Arctic Oscillation. The winter trends show a statistically signicant negative correlation (correlation coecient –0.47±0.19) with the Scandinavian pattern and a positive correlation with the North Atlantic Oscillation (0.31±0.22) and Arctic Oscillation (0.42±0.20). The meridional pattern is associated with more predominant north-westerly and westerly winds driven by the Scandinavian and North Atlantic Oscillation, respectively. All three climatic indices show a statistically signicant time-variable correlation with Baltic Sea wave climate during the winter season. When the Scandinavian pattern's inuence is strong, North Atlantic and Arctic Oscillations' effect is low and vice versa. The results are backed up by simulations using synthetic data that demonstrate that the percentage of variance retrieved using EOF analysis from the satellite-derived wave measurements is directly related to the percentage of noise in the data and the retrieved spatial patterns are insensitive to the level of noise.


Introduction
The knowledge of both long-term changes and short-term variations in the wave climate has great importance for the safety of navigation (e.g., Barbariol et al. 2019), design purposes (e.g., Hemer et al. 2013), coastal protection (e.g., Weisse et al. 2012), and sediment transport (e.g., Masselink et al. 2016). In semi-sheltered seas, such as the Baltic Sea ( Fig. 1), due to the limited size of the water body, small changes in the wind direction can lead to large spatiotemporal variations in the wave climate (e.g., Jönsson, Broman and Rahm 2002;Soomere and Räämet 2011;Kudryavtseva and Soomere 2017) and alongshore transport patterns (Soomere et al. 2015). Therefore, to minimise the coastal menaces, economic losses, provide adequate information for engineering and coastal planning in the future, it is crucial to determine the spatial and temporal variability of the wave climate in such regions.
Wave properties in different locations of the Baltic Sea vary signi cantly over the decades Suursaar and Kullas 2009;Różyński 2010). For example, wave intensity showed a quick drop in the northern part of the Baltic proper at the beginning of 1980s, while it had a small increase along the Lithuanian coast (Kelpšaite, Herrmann and Soomere 2008). The average signi cant wave height (SWH) in this basin, estimated using satellite altimetry was in the range of 0.44-1.94 m during 1991-2015 with an increasing trend of 0.005 m/yr (Kudryavtseva and Soomere 2017). The largest mean wave height of 1.2 m occurs in the Baltic proper (Nikolkina, Soomere, and Räämet 2014;Björkqvist et al., 2018) and also in the Sea of Bothnia even though the seasonal appearance of ice in the Baltic Sea affects the wave growth in this subbasin (Fig. 1). As expected, semi-enclosed areas of this basin, such as the Gulf of Finland, the Gulf of Riga, and the Bay of Bothnia are characterised by a lower range of the mean wave height of 0.5-1.0 m (Tuomi, Kahma and Pettersson 2011;Nikolkina, Soomere and Räämet 2014; Kudryavtseva and Soomere 2017).
The wave heights and periods can reach ~8 m and 10-12 s respectively, in the northern part of the Baltic proper (Tuomi, Kahma and Pettersson 2011) and in the Sea of Bothnia (Björkqvist et al., 2020), ~6 m and 8-11 s at the entrance to the Gulf of Finland, and ~4 m and 6-8 s in the eastern part of this gulf (Räämet, Soomere and Zaitseva-Pärnaste 2010). Most of the wave events with SWH more than 7 m occurred from November to January (Björkqvist et al. 2018), and the roughest wave storms were the prominent ones in November 1969 and January 2005 (Suursaar and Kullas 2009). During the wave storm in January 2005, the measured SWH reached 7.2 m in the northern Baltic proper and 4.5 m in the Gulf of Finland ).
One of the main drivers of wind direction is the different phases of atmospheric circulation which can be explained in connection with climate change (Wang, Zwiers, and Swail 2004). The Baltic Sea is located in a transition area between the North Atlantic Ocean and the continental area of Eurasia. Consequently, the wave climate in this basin is highly in uenced by the North Atlantic Oscillation (NAO), Arctic Oscillation (AO), and Scandinavia pattern (SCAND; previously Eurasia-1; Barnston and Livezey 1987) climate indices.
The NAO pattern is one of the most prominent teleconnection patterns over the North Atlantic and European regions in all seasons (e.g., Walker and Bliss 1932). However, it has the strongest in uence over the Baltic Sea in winter (Rózyński 2010) explaining an essential part of wave height variability from December to February. This climatic index consists of the north-south dipole of anomalies with one centre located over Greenland and another in the North Atlantic at latitudes ~40°N. In terms of the wind direction, the positive phase of the NAO corresponds to stronger westerly winds. During the negative phase, westerly winds are weaker, and the winds from the east and north-east are more frequent ( Fig. 1; e.g., Trigo, Osborn and Corte-Real 2002).
The AO mode corresponds to the air circulating counterclockwise at ~55°N latitude, in the Arctic. The AO is strongly related to the NAO. When the AO is in its positive phase, a belt of strong winds circulating the North Pole con nes colder air inside the Polar Regions (e.g., Thompson and Wallace 1998). In the AO's negative phase, this belt becomes weaker, resulting in a southward movement of colder arctic air into the mid-latitudes, accompanied by increased storminess (Fig. 1). The positive phase of the AO, similarly to the analogous phase of the NAO, corresponds to strong westerly winds over the Baltic Sea region. The negative phase of the AO results in a shift of storm tracks to the south and the winds from the east are more frequent in the region. The NAO and AO are widely used to connect the regional climate variability to the sea level (Andersson, 2002, Jevrejeva et al., 2005, storm events (Surkova, Arkhipkin, and Kislov 2015), SWH (Rózyński 2010), and ice condition (Jevrejeva, Moore, and Grinsted 2003) in the Baltic Sea.
The processes driven by the NAO and AO have a time variable effect on the Baltic Sea region. A signi cant change in the frequency of cyclonic circulation occurred at the end of the 19th century, which presumably coincided with a retreat of the Little Ice Age (e.g., Omstedt et al. 2004). The water level records showed a changing effect of the NAO and AO on the Baltic Sea sea-level with a decrease in the correlation of the relevant time series with these indices in 1900-1954(Andersson 2002Jevrejeva et al. 2005), and an increase in 1980-2000 (Wakelin et al. 2003;Jevrejeva et al. 2005;Suursaar and Sooäär 2007). There are also indications of spatial dependence of the correlation between the NAO index and sea-level. The reported correlations were very low in 1961-1980 at Narva and Pärnu tide gauges (Suursaar and Sooäär 2007) but much stronger at Wismar, Kronstadt, and Cuxhaven during the same time (Jevrejeva et al. 2005). The time variability of the correlation with the NAO index is also prominent in the temperature (e.g., Jacobeit et al. 2001) and ice conditions (e.g., Jevrejeva, Moore and Grinsted 2003). However, the possibility of a time variable effect of the NAO and AO or the other climatic indices on the Baltic Sea wave climate is currently not studied.
The SCAND pattern consists of a main circulation centre over Scandinavia, with weaker centres of an opposite sign over eastern Russia and Western Europe. This pattern affects regional weather conditions and westerlies mostly in winter (Gao, Yu, and Paek 2017). During the negative phase of the SCAND, the wind rotates anticlockwise around the midpoint of the anomaly, creating strong westerly or north-westerly winds over the northern Europe, Scandinavia, and the Baltic Sea region. During the SCAND positive phase, the wind reverses its direction, rotating clockwise, which results in a strong easterly and south-easterly winds belt (Bueh and Nakamura 2007;Gao, Yu and Paek 2017). Fig. 1 shows wind directions during the positive and negative phases of the SCAND over the Baltic Sea (Bueh and Nakamura 2007). This pattern's positive phase blocks the cold air in central Eurasia with below-average temperatures across central Russia and Western Europe.
The most in uential atmospheric teleconnections in the Baltic Sea region are NAO, AO, and SCAND. In some studies, the East-Atlantic West-Russia (EAWR), the East Atlantic (EA), the Polar-Eurasia (POL), and the Atlantic Multidecadal Oscillation (AMO) are considered as the drivers of precipitation pattern (Jaagus 2009;Irannezhad, Marttila, and Kløve 2014) and coastal upwelling (Bednorz et al. 2019). However, it is still unclear what are the links between these climatic indices and the wave climate.
There are three different data collecting methods for wave climate studies: modelling, direct measurements and remote sensing. The most common wave models used in the Baltic Sea are the thirdgeneration models WAM (Cavaleri 1997;Hasselmann et al. 1988) and SWAN (Booij, Ris, and Holthuijsen 1999). Based on these models, several hindcasts of the wave climate (Soomere 2005;Cieślikiewicz and Paplińska-Swerpel 2008;Björkqvist et al., 2018;Tuomi et al. 2019) and studies of the annual and decadal changes of wave heights in the Baltic Sea (Soomere and Räämet 2014) were performed. Due to different model parameters and, in particular, depending on the wind forcing, there are mismatches between different model outputs (Tuomi, Kahma, and Pettersson 2011;Soomere and Räämet 2011). Direct measurements are the primary source in the assessment of the local wave climate. However, the relevant devices do not cover the whole area of interest, and the wave measurements are limited to the ice-free time (Tuomi, Kahma, and Pettersson 2011;Madsen, Høyer, and Tscherning 2007;Cieślikiewicz and Paplińska-Swerpel 2008). Hence, direct measurements are restricted only to a few locations and provide a limited knowledge of wave elds' spatial variability.
Signi cant advances in satellite technology made it possible to obtain homogeneous and continuous wave data over the sea's vast areas. Wave heights derived from high-resolution satellite-based techniques such as Synthetic Aperture Radar (SAR) in 2012-2017 were compared with in situ data over the Baltic Sea (Rikka et al. 2017). Although there is a good agreement between the SAR and in situ wave height data, the SAR data cover only a short period of 5 years and cannot be used to reconstruct long-term wave climate variability. Satellite altimetry becomes a powerful tool for studying the global scales' wave climate and its relation to the climatic indices (Hannachi 2004). For example, a strong relationship between the interannual variability of SWH (including trends) and both the AO and NAO reveal a connection between these climate patterns and the interannual extreme wave climate in the North Atlantic (Izaguirre et al. 2011)).
However, the use of satellite altimetry is problematic in coastal areas (e.g., Madsen, Høyer, and Tscherning 2007) and partially ice-covered (e.g., Brenner et al. 1983) water bodies like the Baltic Sea. For these reasons, the wave height data from satellites has been used scarcely in this region until recently. A signi cant effort towards validating the long-term multi-mission wave height data with the available in situ wave measurements in the Baltic Sea region was performed by Kudryavtseva and Soomere (2016).
This validation made it possible to study in detail spatial and temporal variations of the Baltic Sea wave climate during the last decades. Signi cant spatial variability is detected over this basin, which can be explained by a rotation of preferential wind directions rather than an increase in the wind speed (Kudryavtseva and Soomere 2017). However, more effort is needed to identify the main drivers behind the spatial variability of wave heights in the Baltic Sea.
To examine this question in detail, the EOF method (Hannachi 2004) is applied in this study to the wave properties retrieved from satellite altimetry. The EOFs produce an expansion of data in a series of functions that separate the spatiotemporal variations (Niroomandi et al. 2018) and facilitate variability analysis in this domain. The EOF method is a robust tool to determine correlations between the wave climate and climatic indices. It has been widely used to study major modes of climate variability such as the NAO and the El Nino indices (Lionello and Sanna 2005;Nezlin and McWilliams 2003) and to estimate monthly distributions of large-scale sea-level variability on the global scale (Church et al. 2004). The EOF analysis extracted the predominant spatial patterns of monthly averaged wave height variability in wintertime in the northern hemisphere (Shimura, Mori and Mase 2013) and determined the "climate" of seasonal directional wave energy ux in the southern hemisphere (Hemer, Church and Hunter 2010).
Several recent studies have addressed the options for its use in different regions. For example, EOFs helped establish an annual cycle and validate the Mediterranean Sea's spatial wave distribution (Lionello and Sanna 2005;Cañellas et al. 2010;Sartini, Besio and Cassola 2017) and to represent interseasonal variations in the Bengal region (Patra and Bhaskaran 2016). It has also been used to de ne long-term trends and spatiotemporal variability of wave heights in the Chesapeake Bay (Niroomandi et al. 2018). In the Baltic Sea, this method was applied to the daily maximum values of the total SWH and corresponding windsea and swell heights by using a 5-year hindcast dataset from 1988 to 1993 (Mietus and von Storch 1997). The rst EOF of the wave properties revealed a signi cant anomaly of wave heights in the eastern part of the Baltic Sea. The second and the third EOFs described an anomaly of wave heights from the southwest to the northeast and an oscillation between the eastern and western parts of the basin, respectively (Mietus and von Storch 1997).
In this study, multi-mission satellite altimetry SWH dataset is used, which was previously validated against all available in situ wave measurements for the Baltic Sea basin (Kudryavtseva and Soomere 2016). Based on these data, spatial and temporal variations of monthly averages of the SWH in the Baltic Sea are analysed by applying the EOF method, and a correlation analysis is performed between the EOF modes and various climatic indices.
The manuscript is described as follows. First, a description of the data and the methodology is introduced in Section 2. In Section 3, the EOF technique is used to retrieve the SWH pattern in the Baltic Sea. Discussion and conclusions are presented Section 4.
Several features of the Baltic Sea such as the presence of ice (Tuomi, Kahma and Pettersson 2011), complicated geometry and the presence of extensive archipelago areas in this relatively small basin (Madsen, Høyer and Tscherning 2007) can signi cantly affect the quality of satellite data (Kudryavtseva and Soomere 2016). Therefore, data with backscatter coe cient >13.5 cdb, large errors in SWH normalised standard deviation, data closer than 0.2° to the coast, and measured over the sea areas with more than 30% ice concentration were eliminated from further analysis and all measurements were corrected for biases (Kudryavtseva and Soomere 2016). Finally, the data were thoroughly checked for consistency and possible erroneous measurements and validated against the records of in situ data (Kudryavtseva and Soomere 2016).
The resulting dataset (Fig. 2) showed very good quality and is well suitable for the analysis of the Baltic Sea wave climate (Kudryavtseva and Soomere 2017). More than two decades of satellite altimetry data with a large number of measurements provide excellent spatial and temporal coverage over the Baltic Sea. The wave data exhibits intricate variability patterns in both temporal and spatial domain (e.g., Jönsson, Broman and Rahm 2002;Soomere and Räämet 2011;Kudryavtseva and Soomere 2017). To extract these patterns, an EOF analysis (Mietus and von Storch 1997), which separates individual modes of variability, was performed on monthly-averaged SWHs. The analysis was implemented by using the software package "spacetime" (version 1.2-1) in R (version 3.4.4).

Data gridding
Each satellite altimetry mission measures continuous data only along a speci c line. Merging the measurements from different missions into a combined dataset provides much better spatial and temporal coverage but can be prone to gaps. Due to differences in orbits of various missions, the gaps are not regular, which signi cantly complicates further analysis.
A monthly averaged dataset with a regular rectangular grid was created. The latitudes and longitudes ranged from 54.11°N to 65.57°N and from 10°E to 28.83°E, respectively. Different grid sizes and their effect on the results were thoroughly tested, and the cell size of 0.8°´0.8° showed the optimal performance. Additionally, the grid size effect on the percentage of temporal variability retrieved by EOF was studied. It did not show any dependency on the grid resolution. Despite an increase in resolution, no signi cant change in the magnitude of reconstructed temporal variability was found.
Due to the nature of the satellite altimetry data, each grid cell has a different number of measurements (Fig. 2). A small number of observations in a particular cell can lead to errors in the EOF results. Hence, the cells with less than a certain number of measurements should be omitted from the analysis. To test the optimal lower cut-off value, the distribution of the number of measurements per grid cell was examined. Interestingly, it is very close to a uniform distribution. To calculate the minimum number of measurements in a single grid cell that does not cause errors in the EOF results, the cut-off values varied in the range of 2%-iles -5%iles and the EOF analysis results were compared. This test was performed for the whole dataset and several subsets containing measurements during various seasons. The 2nd percentile cut-off provided consistent results and was used in further analysis. As a result, the grid cells that have less than 645 (whole dataset), 181 (winter), 141 (spring), 145 (summer), and 180 (fall) measurements are omitted, leaving more than 99.1% of the data available for the analysis. Most of the omitted data were excluded from the Gulf of Finland, a narrow gulf with a rigged coastline. The spatial distribution of the other omitted data points appeared to be random and, therefore, does not affect the retrieved wave climate spatial patterns.

Averaging errors
To create a more complete dataset, a monthly average wave height was calculated for each grid cell before applying the EOF method. Due to the natural variability of wave heights on timescales shorter than a month, each monthly average has an uncertainty. This uncertainty σ is estimated as an error of the mean: where SWH i is a set of SWH measured for one month in one grid cell, and N is the number of the data points in each grid cell for that month. A characteristic error of the mean for the whole dataset is estimated as a median of σ. The typical error of the mean in the dataset of interest is ~9%.

Simulated data sets
To characterise the effect of spatial and temporal gaps (that are inherent in the satellite altimetry data) on the resulting EOFs, simulated data are created and thoroughly tested. The gaps in the data appear due to the variable ice cover, proximity to the coast and satellite tracks. Synthetic data with distinct zonal and meridional patterns are generated and classi ed as: Scenario A: A simulated dataset with a distinct zonal (north-south) variation of wave heights, where the wave heights in the northern part (to the north of 59°N) follow a linear trend of 0.005 m/yr and the SWH in the southern part have no distinct long-term variability (Fig. 3). The northern part includes the Bothnian Bay, the Bothnian Sea and the Gulf of Finland and the southern part comprises the Gotland Basin, the Gulf of Riga, the Bornholm Basin and the Arkona Basin.
Scenario B: A simulated dataset with a prominent meridional (west-east) variation of wave heights. The SWH in the western part (to the west of 19.4°E) follow a linear trend of 0.005 m/yr, and in the eastern part, the wave heights have no long-term variability (Fig. 4). The eastern part covers the Bothnian Bay, eastern Bothnian Sea, the Gulf of Finland, the Gulf of Riga and the Eastern Gotland Basin.
These created datasets emulated precisely the same timings and locations as the real multi-mission satellite measurements. The trend of 0.005 m/yr was assigned as a measured wave height trend in the Baltic Sea (Kudryavtseva and Soomere 2017). Gaussian random noise was added to all the simulated data using a Gaussian distribution of a particular width. To test the effect of noise level on the retrieval of the wave height spatial distribution, the Gaussian noise distribution width was varied from 5% to 30% of the total average of the simulated wave heights.
The EOF analysis results on the simulated data are shown in Fig. 3 for the zonal and Fig. 4 for the meridional spatial patterns. The rst EOF pattern between northern and southern parts as well as eastern and western parts of the Baltic Sea is visible when the noise level is less than 30%. However, at the noise level higher than 30%, the spatial pattern disappears. These features indicate that (i) spatial and temporal gaps of the satellite altimetry data do not signi cantly affect the EOF results and (ii) the results of the EOF analysis are susceptible to the noise level.
The percentage of retrieved variance explained by the rst EOF mode diminishes with an increase in the noise level (Fig. 5). The low level of introduced noise (5%) results in high variance of ~90% for the simulated datasets in both spatial patterns. However, the highest level of introduced uncertainty (30%) leads to only 18% and 29% of retrieved temporal variabilities for zonal and meridional patterns, respectively. Therefore, this indicates that the temporal signal's strength obtained with the EOF method's application highly depends on the noise level. In particular, the noise in the data can substantially reduce the rst EOFs percentage of variability, even for clearly visible and de ned spatial patterns. For example, in Fig. 3 (d, e) and Fig. 4 (d, e) the observed patterns are quite prominent, but the percentage of the recovered variability is less than 30% and 40% for zonal and meridional patterns, respectively. This indicates that even when the retrieved percentage of variability by the EOF method is relatively low, the method can detect correct patterns.
Considering that no other pattern was introduced in the simulated data, all the retrieved temporal variabilities by the 2nd and 3rd EOFs modes should be not real and caused by fake patterns picked up in the random noise. The second and third EOF modes for both scenarios retrieve less than 10% of the variance. This percentage then can be used as an indication of the reliability of the EOF method. Therefore, any EOF mode with a percentage of less than 10% variability is caused purely by noise and is unreliable.
To test whether there is any preference in detecting the meridional pattern better than the zonal one, the variance retrieved by the EOF method for both scenarios was compared. The difference of retrieved variance for scenarios A and B varies between 10% and 21% (Fig. 5). At low noise levels, the difference is marginal (10%). It increases with the introduced uncertainty until the noise level reaches 20% when the difference starts to decrease. Overall, the method exhibited a slight preference for detecting the meridional pattern compared to the zonal one. This feature re ects a higher density of satellite measurements in the east-west direction at the Baltic Sea latitudes.

Spatial distribution of wave heights
The rst EOF mode of the Baltic Sea wave heights over the whole period (Fig. 6a) shows high values in the Baltic proper, both in the southern and eastern parts. The lower values are observed in the Gulf of Finland, the Gulf of Riga, the Danish straits, and the Gulf of Bothnia. This spatial pattern in the central Baltic Sea qualitatively matches the results of the EOF analysis of the modelled wave climate (Mietus and von Storch 1997), where the rst EOF mode also showed higher values in the eastern part of the Baltic proper.
To study the seasonality of wave patterns, two approaches are used. The rst one uses the classi cation of the seasons into "stormy" period, which includes the months from January to March and from August to December and the "calm" period, which lasts from April to July (Soomere and Pindsoo 2016;Männikus et al. 2020). The EOFs are also considered for regular seasons, such as winter (DJF, Fig. 7a), spring (MAM, Fig. 6b), summer (JJA, Fig. 6c), and fall (SON, Fig. 6d). The spring and fall seasons reveal a northsouth structure, the spring showing higher values of the rst EOF in the northern parts of the Baltic Sea, in the Bay of Bothnia (Fig. 6b), whereas the fall exhibit an EOF structure with the maxima in the southern Baltic and the Baltic proper (Fig. 6d). The EOFs for the summer season (Fig. 6c) revealed strong patchiness without a prominent large-scale pattern. The fall and spring have the most and the least number of satellite altimetry snapshots, respectively (Fig. 8). The percentages of retrieved variance with the EOF analysis are shown for different seasons and the whole year in Table 1. The total temporal variability, explained by the rst three EOF functions, ranges from 38% for the calm period up to 44% in winter.

Seasons
First EOF (%) Second EOF (%) Third EOF (%) Total variability (%) The winter season showed the strongest signal in the rst EOF. Additionally, this season revealed a prominent meridional pattern (Fig. 7a). The rst EOF mode is negative in the eastern part of the Baltic proper and the eastern part of the Gulf of Bothnia, whereas it shows positive values in the western parts of these water bodies. The meridional pattern is remarkably similar to the spatial distribution of linear trends (Fig. 7b) tted to the winter season yearly averages for 1996-2015 period (the detailed description of trends is discussed in Kudryavtseva and Soomere 2017).
The linear trends are shown in Fig. 7b are tted using the same grid like the one used for the EOF analysis (Section 2.2) for better comparability of the results. However, instead of the monthly averaging, the seasonal averaging was performed. The trends were tted for each grid cell, and only the trends with the statistical signi cance >95% are shown in Fig. 7b. The probability of the coincidence that statistically signi cant linear trends are detected in the same cell as the rst EOF with the values higher than a particular cut-off value was thoroughly tested. The selection of values above the cut-off value of -0.07 provided the highest probability of coincidence. This cut-off value results in 77% of grid cells with the detected linear trends coinciding with the rst winter EOF pattern. This means that 77% of grid cells with the signi cant linear trend are located precisely at the same locations as the grid cells with winter EOF above -0.07. Therefore, the statistically signi cant trends' location is remarkably similar to the rst EOF function, meaning that the rst winter EOF is associated with the linear trends in SWH.

Reliability of spatial patterns
In this study, the variance explained by the rst EOF is only ~27% for the winter season. The model-based studies of wave climate with the EOF method revealed a much stronger signal in the Baltic Sea (Mietus and von Storch 1997;Shimura, Mori and Mase 2013). However, similar satellite-based studies also reported a low percentage of the retrieved variance of the rst EOF. For example, Hemer, Church and Hunter (2010) using multi-mission altimeter data, found that the rst EOFs of the wave climate explain only 17-19% of the wave climate variability. Similarly, Woolf, Challenor and Cotton (2002) found that for the North Atlantic satellite-derived wave climate the rst EOF explains 41% of the variance, while the other EOFs explain <19% of temporal variability each. The retrieved variance's dependence on the gaps and noise in the satellite altimetry data are not widely discussed.
The detailed analysis of the simulated datasets performed in Section 2.4 demonstrates that the noise level in the satellite data signi cantly affects the percentage of retrieved variance and that the shape of the retrieved spatial patterns is practically not affected (Fig. 5). With the considerable noise level, the amount of temporal variability explained by the rst EOF can become as low as 20% even for a very distinctive and robust pattern in simulated data. However, even with high noise levels and the low percentage of variability, the EOF method can still reconstruct the correct spatial pattern (Fig. 3e, Fig. 4e). Therefore, the lower variance retrieved with the EOF method from satellite altimetry data (e.g., Hemer, Church and Hunter 2010) is most likely caused by the data's substantial uncertainty. Hence, it is likely that the observed meridional pattern in the rst EOF (Fig. 7a) does represent a real pattern of the wave climate, even if it has a relatively low percentage of retrieved variance.

Relation between SWH trends and large-scale atmospheric teleconnections
To study whether the Baltic Sea wave climate and trends in wave height are driven by the changes in the large-scale atmospheric circulation, a correlation analysis was performed between the rst three EOFs, signifying the spatio-temporal modes of the Baltic Sea wave climate, and various climatic indices. To match the satellite altimetry data's monthly timescales, the monthly time series of the climatic indices were used. The climatic records were obtained from the NOAA Center for Weather and Climate Prediction.
The strongest correlation between the rst EOF for the winter season and the Scandinavia (SCAND) climatic index is negative, with a correlation coe cient of -0.47±0.19 (Table 2). The SCAND mode exhibits substantial seasonal variability. Interestingly, the correlation of the rst EOF mode with the SCAND also shows variability over different seasons. It is the lowest throughout the spring season (0.24±0.22). During the summer and fall it is positive, 0.38±0.20 and 0.36±0.21 respectively, re ecting a prevailing (south-)easterly wind direction during these seasons.
The uncertainty of the correlation, evaluated in terms of the 95% con dence intervals, was used as a measure of reliability. Only three studied climatic indices exhibit a correlation larger than the uncertainty intervals for the winter season, namely the SCAND, the AO, and the NAO indices. Contrary to the SCAND index, the AO and NAO show positive correlations, with the coe cients of 0.42±0.20 (AO) and 0.31±0.22 (NAO). Both the AO and NAO have a signi cant correlation with the rst EOF mode during the winter season. For all other seasons, this mode reveals no correlations larger than uncertainty. The correlation analysis performed between the other climatic indices, namely AMO, EA, EAWR, POLEUR and the rst three EOFs resulted in no signi cant correlation. The EAWR index showed a correlation coe cient of 0.26±0.22 with the rst winter EOF, which is slightly higher than the associated uncertainty.
The estimated values of correlation coe cients are close to similar values from correlation analysis between the annual number of storms in the Baltic Sea with the wave heights larger than 2 m and various teleconnections reported by Myslenkov et al. (2018) and Medvedeva et al. (2016) where they found a correlation of -0.59 with the SCAND, 0.32 with the AO, and 0.12 with the NAO index based on the wave modelling results and NCAR reanalysis. The similarity of the found correlation coe cients indicates that the rst winter EOF and the wave heights trends are affected by the same processes as the storminess. Slightly stronger positive correlations with the NAO and AO indices and a weaker negative correlation with the SCAND index were obtained for signi cant wave heights (SWH) >4 m using wave hindcast for a more extended period of 1950-2010 (Surkova, Arkhipkin and Kislov 2015). A stronger positive correlation is most likely caused by the rapid reaction of wave elds to the time variability of the main teleconnection patterns' in uence over the Baltic Sea region.
The relationship between the wave height variability and the climatic indices was also studied at a few wave measurement locations. Wave records at Vilsandi at the West Estonian Archipelago showed a correlation of the SWH with the NAO indices at the level of 0.7 during 1966-2006(Suursaar and Kullas 2009. However, only a mild relation between the NAO index and wave heights was found in the southern Baltic (e.g., at Lubiatowo in January (Różyński 2010)

Time variable correlation with climatic indices
To study possible time variability of the in uence of the climatic indices on the wave climate of the Baltic Sea, a sliding correlation analysis is performed. A Pearson's product-moment correlation implemented in the "stats" package (R version 3.4.4) is used for the correlation analysis between the rst three dominant climatic indices (SCAND, AO, and NAO) during the winter months and the rst EOF in winter. Kendall and Spearman's correlations are also calculated and lead to the same results. Different window sizes from 2 to 9 years are tested (with one-year step), and the window with a length of 5 years is selected as it provides the optimal noise level.
For all three studied climatic indices, the sliding correlation coe cients (called correlations below for brevity) with the rst winter EOF exhibit signi cant time variability (Fig. 9). The SCAND index revealed the highest values (between -0.5 and -0.8) from 2000 to 2009. The correlation with the SCAND index was statistically signi cant at a >95% level during these years.

Discussion And Conclusions
In order to interpret the appearance of various modes and parameters of the EOF modes of the Baltic Sea wave climate derived from satellite altimetry, it was demonstrated that (i) the percentage of variability in the retrieved wave elds is directly related to the percentage of noise in the data and that (ii) the retrieved spatial patterns are practically not affected by the noise. Low variance is generally observed in the EOF analysis of the wave climate derived from the satellite altimetry (e.g., Woolf, Challenor and Cotton, 2002;Hemer, Church and Hunter 2010). However, the reasons behind the low retrieved variance of EOFs were not studied in detail. In this study, a thorough analysis was performed on how the measurements' noise affects the retrieve variance of EOF modes.
The connection between the NAO index and trends in the wave heights for the North Atlantic and the North Sea is well-established and widely discussed (e.g., Woolf, Challenor and Cotton 2002;Wolf and Woolf 2006;Hemer, Church and Hunter 2010). The phenomena behind the NAO index are driving processes in those bodies of water. For example, the long-term positive trend in the wave heights in the North Atlantic (Bertin, Prouteau and Letetrel 2013) is related to the corresponding trend in the NAO index (e.g., Wolf and Woolf 2006). However, it was not clear if there is a connection between the wave climate trends and climatic indices in the Baltic Sea region. The results of our study show a strong anticorrelation between the linear trends in the Baltic Sea wave climate and the SCAND climatic index (-0.47±0.19) and a positive correlation with the NAO (0.31±0.22) and AO (0.42±0.20) indices. During the NAO and AO's positive phase, strong westerly winds predominate in the Baltic Sea area (Fig. 7b). This negative correlation with the SCAND index and a positive correlation with the NAO and AO indices agrees well with the explanation that the trends in the Baltic Sea wave climate are due to the interplay of northwesterly (driven by the SCAND teleconnection) and westerly (driven by the NAO and AO) winds over the region.
Most importantly, using the running correlation analysis, we demonstrated for the rst time that the Baltic Sea wave climate has a signi cant time variable correlation with the SCAND, NAO, and AO indices. The phenomena that govern the SCAND mode had a strong in uence on the Baltic Sea waves between 2000 and 2009 when the effect of the NAO and AO indices was weak. Interestingly, after the year 2009, the correlation of wave properties with the SCAND index signi cantly diminished, and the NAO and AO indices exhibit a strong correlation with the winter wave climate. This feature can be interpreted as an indication that the Baltic Sea wave climate is driven alternately by several sets of processes that are characterised by the listed three indices. The correlation of wave properties and a single index dramatically varies in time. When north-westerly winds are stronger during the negative phase of the SCAND index, the effect of the processes that drive the NAO index is weaker. Alternatively, when the correlation with the NAO index is more substantial, the westerly winds are stronger, and the effect of phenomena that make up the SCAND index is weaker.

Declarations
Funding. The research was co-supported by the institutional nancing by the Estonian Ministry of Education and Research (grant IUT33-3), the Flag-ERA project FuturICT2.0.
Availability of data and material. The RADS satellite altimetry data are available at http://rads.tudelft.nl/rads/rads.shtml.
Code availability. From the authors on request.
Competing interests: The authors declare that they have no con ict of interest.