The EOF modes were calculated on the coarse grid (5 × 10 arcmin by N and E) on the basis of space-averaged results from the fine-grid (0.5 × 1 arcmin by N and E) model, running from 1 July 2010 to 30 June 2015 (Elken et al., 2019). This analysis revealed that mean distributions of modelled SST and SSS, serving as the basis for calculation of deviations in the variability studies, were close to the climatological maps calculated on the basis of observations (Janssen et al., 1999). The highest temporal variability was found in the shallow coastal areas for SST, whereas the largest SSS variations were revealed near the larger river mouths and in the NE area of the Gulf of Finland. While temporal changes strongly dominate in the variability of SST, spatial changes prevail in SSS variability.

Calculated SST and SSS covariance matrices have significant spreading of individual values over pairs of points, especially for the dominating gravest modes where big covariance values may occur over large distances. Covariance of residual fields (sum of higher EOF modes) has a decay scale of about 30 km with increasing space lag, both for SST and SSS. The first, most energetic EOF modes have nearly “flat” patterns without sign change (energy share 97.6 % for SST and 36.2 % for SSS); their amplitudes are dominated by a seasonal signal. A space-dependent mean biharmonic seasonal cycle was not removed from the model time series prior to the analysis, since special experiments revealed only a small effect of seasonality suppression on EOF mode patterns. The second EOF mode of SST (1.3 %) presents differential heating and cooling in shallow areas, compared to the deeper offshore waters. Transverse anomaly stripes near northern or southern coasts, like those due to coherent upwelling and downwelling in the region, were evident in the second SSS mode pattern (16.9 %) and third SST mode pattern (0.31 %). There is also a pattern of SSS changes in the freshwater spreading pathway near the northern coast of the Gulf of Finland (third SSS mode, 7.1 %) and longitudinal SST changes in the east–west direction (fourth SST mode, 0.14 %).

The data set used in the present DA study (Fig. 2) is rather irregular, compared to the reconstruction experiments by Elken et al. (2019). Therefore, we revisit the covariance issues and perform additional reconstruction tests, before finding in the next subsection the best options for the automatic reconstruction procedure. Spatial interrelation of observed values at a specific point to the values in the rest of the region is found from the extract of the spatial covariance matrix, which can be shown as a map. One example of SSS covariance with a frequently sampled HELCOM monitoring station BMP F3 is shown in Fig. 3. The covariance of three dominating EOF modes (Fig. 3b) comprises most of the unfiltered data covariance (Fig. 3a) at large distances. High covariance locations have clear basin-scale geographical explanations: under the similar weather and seasonal forcing, which is spatially nearly uniform, SSS changes in distant river influence areas are closely interlinked. Correlation (not shown) may exceed 0.4 at distances greater than 500 km; therefore, assumptions of fast decay of correlation with space lag (like using the Gaussian covariance approximation), adopted in offshore areas with negligible coastal influence, are not valid. Covariance of residuals to the large-scale variations are presented by higher EOF modes (Fig. 3c). Such smaller-scale variations have nearly Gaussian structure, with elliptical anisotropy stretched along the axis of the basins similar to the results by Høyer and She (2007): spatial scales in Fig. 3c are 30 and 15 km along the main axis and perpendicular to the axis, respectively. Similar regularities – physically explained high covariance at large distances, localized covariance patterns for the higher EOF modes – were found for other points of reference, both for SSS and SST fields.

The EOF reconstruction method relies on the full covariance matrix, without any approximation. Covariance is further treated using EOF modes. For the reconstruction procedure, we keep the lowest EOF modes without any approximation, and covariance from higher modes as shown in Fig. 3c is not taken into account. The large-scale features of the EOF reconstruction and associated DA exclude the possibility of creating spurious “bullseye” patterns around observation points, which may happen for instance during unfavourable selection of optimal interpolation parameters. Subsequently, our DA method handles the large-scale features and excludes the possibility to assimilate smaller-scale features, which can be described by the higher modes.

A full covariance matrix can be implemented in optimal interpolation as well. While the EOF method needs to limit the number of included modes, smoothing in such a way smaller-scale variability and observational errors, optimal interpolation needs to include observational error variance (“nugget effect” in terms of Kriging method, equivalent to optimal interpolation); otherwise the system of underlying linear equations may become close to singular and the result may become unrealistically spiky. In some examples (not shown), EOF reconstruction and optimal interpolation based on full covariance produced similar results, but these relations need further studies. When observed values were close to the model-computed climatological background, visual similarity was caused mainly by the dominance of spatial gradients of mean SSS over the spatio-temporal variability. Optimal interpolation with Gaussian approximation to the covariance produced realistic results in the neighbourhood of observation points but gave unrealistic patterns and values in the distant SW extrapolation area.

Multiple checks performed on our data set suggested that only the three leading modes were included in the EOF reconstruction. In order to find the best options for reconstruction, experiments were made with different intervals (time window) tR around the reference time ti; including the observations within time window from ti-tR/2 to ti+tR/2. The results were evaluated to fulfil the following goals:

a small RMSD between the observed values and the reconstructed fields;

As a first step in all the experiments with variable time windows, the EOF amplitudes of the mode k were checked for the limit a^i,k<2λk=2σãk , where σ denotes standard deviation. DA data for the days with higher amplitudes were left blank since these reconstruction results most frequently became unrealistic. In addition, when the number of observations was less than six, reconstruction was not performed and the DA step using Eq. (4) was skipped.

The time windows tR for experiments (a) and (b) were selected to be 10, 20 and 30 d. Elken et al. (2019) have found that the correlation timescales (e-folding drop, correlation value 0.368) of EOF SST amplitudes were 65 d for the seasonal first (overall heating/cooling) and second (faster heating/cooling in shallow coastal areas) modes, and 15 d for the third “upwelling” mode. Timescales of the SSS modes were 65 d for the second and third modes, representing the large-scale gradients, and 110 d for the first mode describing long-term variations of mean salinity.

Methods of time-fixed (a) and time-dependent (b) reconstructions revealed similar statistical results during the study period in 2015, whereas RMSD between observed and reconstructed values of (a) was by 5 % larger than that of (b). By increasing the time window, RMSD of reconstruction slightly increases due to the stronger smoothing. The smoothing effect can be seen from the reconstruction examples given in Fig. 4. It should be noted that the reconstruction is designed to yield the best approximation to the observations over the entire region; therefore, it does not need to present the local best fit at individual points.

A network of observations, available during the study period, appeared favourable for the reconstruction, although observations were missing in the southern part of the Gulf of Riga and eastern part of the Gulf of Finland. With a time window of 30 d, there were no reconstruction gaps identified during the study period, determined for both of the methods by the above-described amplitude limit criteria. Smaller time windows yielded some gaps in 2015. During the longer period from 2010–2018, gaps were found in most of the years (except our study period), whereas shorter time windows result in more reconstruction gaps. Detailed comparison of the time-fixed (a) and time-dependent (b) methods revealed that time-fixed reconstruction might create spurious “jumps” when there is a gap in observations which has a length close to the time window. In that case, a backward average is taken before the gap and forward average after the gap, which may result in “jumpy” results. Time-dependent reconstruction, which also accounts for the temporal changes within the time window, handled such situations more smoothly.

There was an interesting oceanographic situation in the beginning of August, when a moderate but extensive upwelling SST pattern at the northern coasts of the basins (Fig. 5), with some effects on SSS (Fig. 6), was combined with fast heating of the thin (6–9 m) surface layer (Fig. 7). Since the middle of July, moderate winds with speeds from 4 to 6 m s-1, which had a westerly zonal component (favouring upwelling at the northern coasts of the basins), were blowing above the Gulf of Finland. After 3 August 2015 (the maps in Figs. 5 and 6 are taken on this date), wind ceased and air temperatures increased by 10 August across the study area up to 25–27 ∘C in the Gulf of Finland and up to 31 ∘C in the southern Gulf of Riga, creating a thin layer of warm surface water. Heating of surface waters was favoured by high night-time air temperatures, higher than SST. Vertical profiles (not shown) in the Gulf of Finland revealed a deep thermocline at 40 m depth near the southern (downwelling) coast and a shallower thermocline near the northern coast; the warm surface water column near Tallinn was 2 to 3 times thicker than near Helsinki. From the end of July to 10 August, warming resulted in an increase in SST (Fig. 7) near Tallinn from 16.5 to 18.5 ∘C and near Helsinki from 14.5 to 18 ∘C.

The SST maps presented in Fig. 5 include control run, reconstructed in situ observations, one experiment with DA (the other experiment yielded similar results) and satellite observations. When warm waters with SST above 17 ∘C dominated the study area, all the maps revealed moderate upwelling near the northern coasts of the basins. However, the minimum temperatures and the spatial extent of the colder waters were different. The warmest “cold” waters were observed on satellite images. While satellites measure SST of a thin surface layer, FerryBox and models acquire temperature over a much thicker layer. It is known that in the Gulf of Finland satellite and FerryBox can have similar SST values in cases of winds stronger than 5 m s-1 (Uiboupin and Laanemets, 2015); at smaller wind speeds the SST bias can be 1–3 ∘C in reference to FerryBox observations. Within these accuracy limitations, satellite observations presented in Fig. 5d confirm the model patterns to some extent. The control run (Fig. 5a) was characterized by SST contrasts that are too high, compared to the satellite data (Fig. 5d; for the data source see Sect. 2.2). From the earlier study by Zujev and Elken (2018), it is known that the free model without DA forecasts faster heating and cooling of shallow coastal areas and slower heat dynamics in offshore areas. Data assimilation (Fig. 5c), made using the reconstructed FerryBox data (Fig. 5b), reduced discrepancies with satellite observation. The major large-scale differences between the satellite data (Fig. 5d) and the best DA02 (Fig. 5c) can be outlined as follows: (1) the colder upwelling water extended on the satellite image further to the east, (2) warmer waters were found on the satellite images in the southern Gulf of Riga, near the Daugava river and in the shallow areas between the Estonian islands, and (3) in the Gulf of Riga, a strip of colder waters was modelled along the western coast, while satellite observations revealed warmer waters near this coast.

There were also numerous mesoscale features evident on SST (Fig. 5) and SSS (Fig. 6) maps, like colder upwelling filaments along the northern coasts of the Gulf of Finland and the Gulf of Riga, and decaying anticyclonic warm-core eddies near the southern coast of the Gulf of Finland. The Irbe Front (Lilover et al., 1998; Raudsepp and Elken, 1999), formed by the salinity difference between the Gulf of Riga and the Baltic Proper, was found by the SSS maps in the outward position, stretching from the strait towards the open sea. This salinity structure was also repeated in the SST patterns; the satellite observations confirmed the predicted outward position during the taken snapshot. The model predicted that in the Gulf of Riga the Daugava river waters were spreading by narrow coastal strips of lower salinity in both the NE and NW directions (Fig. 6).

Locations with dense observations allow us to validate the model and visually evaluate assimilation quality. We compared SST and SSS data of the control run (FR) and DA options DA01 and DA02 with FerryBox data (FB) at two points near Tallinn and Helsinki (Fig. 7). While SST followed the seasonal cycle, with weather-dependent deviations, then SSS behaviour was more irregular. In the given variation scales of SST and SSS (16 ∘C and 2 g kg-1 respectively), all the compared SST data sources were more similar to each other than that of SSS. Still, most of the time the assimilation curve (DA02) was closer to the FerryBox observations than the control run, for both SST and SSS.

Warm conditions in the beginning of August (Sect. 3.2.1) are clearly visible on SST time series (Fig. 7a, c). Comparing the values near Tallinn and Helsinki, the southern part of the Gulf of Finland was roughly 2 ∘C warmer than the northern part, whereas the northern part had an unstable day-to-day pattern, possibly due to the fluctuations of the upwelling pattern. This is consistent with the spatial maps given in Fig. 5. Near the southern coast, an upwelling event occurred in September, reducing SST during a few days by nearly 4 ∘C (Fig. 7a). A larger SST drop during the southern coast upwelling (at easterly winds), compared to the northern coast upwelling (at westerly winds of the same magnitude), is explained by the steeper topography slopes in the southern part of the Gulf of Finland (Laanemets at al., 2009). This upwelling event was properly resolved by all the data sets, with DA02 being closest to observations. In general, a free model without DA expected warming at a lower rate during summer and was more precise in autumn, while both assimilation experiments properly corrected the SST and SSS values. However, in some cases, assimilated temperature was somewhat higher than observed and modelled SST.

Assimilation resulted in one major SSS improvement in early summer when the model predicted upwelling with salinity near Helsinki that is too high. Nevertheless, in some cases DA made minor corrections at one of the locations, ignoring observations and sticking to the control run (e.g. late July to early August near Tallinn, and the middle of October near Helsinki). When the model overshoots at both locations, DA properly corrects temperature and salinity values. This implies that DA of surface observations tends to correct the mean values better than the cross-gulf gradients, for which 3D circulation (presently not assimilated) has a significant impact.

In the salinity time series, a “freshwater event” with reduced salinity was observed in the Gulf of Finland at the end of September and beginning of October. In the daily SSS data (Fig. 7b, d) the event was spiky, possibly due to the mesoscale features not assimilated in the present study: without DA, the eddies tend to have a random phase, and the spikes in the time series of different model options and observations do not need to be coherent. However, in the weekly averaged data (not shown) the mesoscale activity was suppressed and the fresh event appeared simultaneously in all the data within the central and western part of the Gulf of Finland.

Increasing assimilation weight in Eq. (4) two times did not make assimilation results two times closer to the observations. As can be seen from Fig. 7, the results of assimilation experiments DA01 and DA02, with relaxation times of 10 and 5 d respectively, were not placed between the free run and the observations proportionally to the corresponding weights 0.1 and 0.2. They diverged as the study region experienced a temperature drop or daily trend change. Both options of assimilated SST could either coincide for a long time or go in parallel, but DA02 was systematically closer to the FerryBox observations. Salinity fluctuations had larger amplitudes in the free run without assimilation, but both DA options, with a “thumb” rule – the bigger the weight, the bigger the change – had properly corrected them. Still, in December DA01 showed better results, being closer to the FerryBox salinity than assimilation DA02.

We have chosen to compare assimilation with the best results (DA02) to the control run without data assimilation (FR) and track the continuous time–latitude changes of SST and SSS (Fig. 8) in two sub-basins – the Gulf of Finland and Gulf of Riga – along the coast-to-coast transects given in Fig. 2a. Using DA, temperature was corrected approximately by 1–2 ∘C, and salinity by less than 1 g kg-1. Major systematic change (in the Gulf of Finland this was validated as improvement; see Sect. 3.2.4 for further details) was seen near the coasts and in the spring and autumn periods, while summer temperatures underwent minor corrections. Salinity corrections had a more uniform distribution and smooth drifting pattern – DA consistently increased SSS values with time in both of the sub-basins.

Data assimilation had increased SST in the Gulf of Finland in open waters during the warming period and in late autumn all across the gulf and had decreased in the coastal areas during the warming period, whereas near the northern coast this decrease continued until September. In the Gulf of Riga, the SST increase dominated throughout the study period, but it was interrupted occasionally by basin-wide events when DA had decreased the temperature compared to the results from FR. The largest corrections of both SST and SSS were evident in the coastal waters. Salinity was increased by DA in most of the cases in the Gulf of Finland, except for May–July near Tallinn. The largest increase in SSS occurred in November and December, when control run results dropped compared to the earlier period.

Some unusual basin-wide events can be found on the difference charts in Fig. 9. For example, abrupt warming of the surface around 10 August 2015 (Sect. 3.2.1) was correctly predicted by the free run model (Fig. 7c), but it was over-smoothed by the data assimilation. A similar line in December on both charts denotes the occurrence of fronts of cold and saline water due to strong winds and storms.

As there are not enough observations available in the Gulf of Riga for validation, we cannot definitely say whether DA improved the situation in the region and to what extent.

Ocean model performance (e.g. Stow et al., 2009; Golbeck et al., 2015; Placke et al., 2018) is usually evaluated by the differences between the observations and the model results, transferred to the times and locations of observations so that they can be directly compared. The overall mean difference (over time and space) is termed bias and the standard deviation of differences at all the observation points is denoted as RMSD (centred root-mean-square difference). The forecast skill is usually non-dimensional, with the RMSD of the studied option (in our case, DA) scaled to reference data (FR in our case) as skill = function of [RMSD(DA,FB) / RMSD(FR,FB)].

The present ocean model has a fine resolution of about 0.5 nautical miles (930 m) (Sect. 2.1); therefore for comparison with observations we used a simplified approach and took averages of observations over the model grid cells over a daily time span (Sect. 2.2). Such a compressed fine-resolution observational data set, still having about 110 000 points for SST and SSS, originated mainly from the FerryBox (FB) lines (Fig. 2), and it covered central and western parts of the Gulf of Finland and the neighbouring part of the Baltic Proper. Areas with lower salinity in the eastern Gulf of Finland and in the Gulf of Riga had only a small number of observations.

Data from the DA experiments DA01 and DA02 were compared to the same compressed observational FB data as the data from the control run without assimilation (FR). Hernandez et al. (2015), who reviewed the problems of performance evaluations of operational ocean models, noted that most available observations are used to adjust models and reduce analysis errors. Therefore, a widespread approach is withholding part of the data set for statistical quantification of errors. In our study, the option of withholding the observations was performed: an evaluation was made of how much the DA result will change if DA is performed using 50 % of the available data (Gregg et al., 2009). The present implementation of EOF DA used about 13 000 observational averages over a coarse grid of about 5 nautical miles. The reconstruction procedure by Eqs. (1)–(2) has no direct connection to the ongoing modelling (although it includes statistical results from longer model runs), and the fields of ψo in Eqs. (3)–(4) are the only link where observations enter the DA process. The experiments which took every second available observation “box” into account (this resulted in a mean sampling interval along ship tracks about 20 km instead of 10 km) revealed that performing DA during the study period with a reduced data set (6.5 000 averaged observation data instead of 13 000) changed RMSD of SST by only 1 % and of SSS by 2 %, whereas the RMSD values were 0.05 ∘C for SST and 0.027 g kg-1 for SSS. An evaluation was made over the full time span and domain using 182 000 coarse grid cells; correlation between the data sets was higher than 0.999. We have also checked reconstruction results with FerryBox data only, excluding the data from shipborne monitoring stations. Compared with the full data set, the largest (but still minor) differences with RMSD of SSS up to 0.03 g kg-1 were found in the Gulf of Riga and the eastern Gulf of Finland, where FB data were missing. Consequently, for our large-scale approach DA results are robust to the reasonable variation of data amount, and we used FB data for reference in the performance evaluations.

Evaluated forecast performance metrics are presented in Table 2. Only those fine-grid points which had respective value of FerryBox observations on the same day were used for metrics calculation. Wet points of the model without corresponding observation value were left out from the procedure.

The statistical properties presented in Table 2 reflect that DA improves the model performance significantly: RMSD of SST was reduced by 22 % and SSS by 34 %, compared to the control run. From DA01 to DA02, a slight improvement of DA performance was observed; therefore we adopted DA02 as the major result. The spatial pattern of RMSD change between the DA and FR (Fig. 9) reveals that larger reduction rates (up to 50 %), for both SST and SSS, were found in the observation-covered areas in the Gulf of Finland. Overly cold waters produced by FR near the northern coast of the Gulf of Finland were effectively corrected by DA (see also Fig. 5); therefore highest improvement percentage scores were detected in this region. Near the western open boundary, non-assimilated SST and SSS values of the larger model were advected into the area, and therefore RMSD reduction was small, or even negative for SSS.

The applied EOF DA method does not assimilate mesoscale variability. Applying the weekly average statistics like Zujev and Elken (2018) further reduced RMSD by 13 % for SST and 9 % for SSS, compared to the daily data in Table 2. Weekly statistics suppresses the mesoscale variability and reveals a better match between the DA and the observations. DA decreased the bias, especially for SSS. At the same time, the correlation of SSS between DA and observations increased considerably. We may conclude that DA made major improvements in the forecasting of SSS. Still, the forecast RMSD in reference to the observations is 62 % of the observed standard deviations, which suggests that there may be further room for improvement. Modelling of SST is already more accurate than SSS without DA: RMSD of the control run (FR) makes 18 % of the standard deviation of observations for SST and 94 % for SSS.