We have used two tide gauge datasets: GESLA-2 (Global Extreme Sea Level Analysis, https://www.gesla.org/, last access: 13 January 2021), which has data with a maximum frequency of 1 h, and PSMSL (Permanent Service for Mean Sea Level, https://www.psmsl.org/, last access: 22 April 2021), which uses monthly data. The GESLA database was created from the need to have information on extreme events (in particular on their interannual variability), and in a first version (GESLA-1, which was not published) it collected data from two international banks: UHSLC (University of Hawaii Sea Level Center) and GLOSS 2 (Global Sea Level Observing System), as well as from several national banks. In 2016, with the aim of updating the database and extending its spatial coverage, the GESLA-2 set was published. In this new set, the geographical coverage of some regions (including the Mediterranean Sea) was improved, especially for the second half of the 20th century. GESLA-2 has allowed more globally representative analyses since about 1970 (Woodworth et al., 2016). GESLA-2 has its own quality control: the quality and possible use of each data is specified (Piccioni et al., 2019). In this work, only the measurements marked as “correct” were selected. For the period between 1980 and 2015 and for the western Mediterranean basin, 34 tide gauge records are available (Table 1). Only Cagliari tide gauge series showed an obvious datum shift; this made it necessary to visually identify the intervals with different datums and subtract their means separately in order to convert the original data into zero-mean anomalies.

The PSMSL, managed by the National Oceanography Centre in Liverpool, collects data from numerous institutions around the world. This bank has a revised local reference (RLR) series, which are referenced to a common datum and constitute approximately two-thirds of the total number of stations. All of the series used in this work are of the RLR type. Although the PSMSL has some series starting in the late 19th century, the number of stations increases considerably in the second half of the 20th century, concentrating along the most developed coastal regions, especially Europe and North America (Holgate et al., 2013). From the PSMSL, 38 tide gauges were selected from the western Mediterranean basin (Table 2). In this case the reconstruction was carried out from the time when the first tide gauge has monthly measurements, namely the Genoa tide gauge, whose series starts in 1884.

The glacial isostatic adjustment (GIA) has not been applied to the tide gauge series. This correction is rather small in the Mediterranean, except for in the Adriatic Sea (Marcos and Tsimplis, 2007a), which is not part of our reconstruction domain.

In order to obtain information on the dynamic relationships between different locations (see Sect. 3), the outputs of the numerical model managed by the Balearic Islands Coastal Observing and Forecasting System (SOCIB) have been used. SOCIB is a multi-platform observatory whose products include numerical model outputs to support operational oceanography. Through a regional configuration of the Regional Oceanic Modeling System (ROMS) model (Shchepetkin and McWilliams, 2005) for the western Mediterranean basin, SOCIB has implemented a forecasting system called WMOP that provides daily forecasts with a 3 d time horizon for temperature, salinity, sea level, and currents (Juza et al., 2016). The daily forecasts and all model outputs from the moment it was implemented can be downloaded from the SOCIB website (https://www.socib.es, last access: 8 April 2021). For sea level, the spatial resolution of the outputs varies between 1.8 and 2.2 km, and they are available every 3–4 h since August 2013 (Juza et al., 2016). The coastal points of this model grid were used to define the grid for the sea level reconstructions presented later on, as the spatial correlations on which the optimal interpolation method is based were calculated from the model outputs.

The European observation program Copernicus provides, through the Copernicus Marine Environment Monitoring Service (CMEMS), regular and systematic baseline information on the global ocean and European seas. Among other results, CMEMS delivers sea level products derived from a direct processing of altimetric observations (von Schuckmann et al., 2018). In order to compare the coastal reconstructions obtained in this work with the latest generation of altimetric data, sea level anomalies from multi-mission satellite altimetry products were downloaded from the CMEMS website (product identifier: SEALEVEL_MED_PHY_L4_REP_ OBSERVATIONS_008_051, which was last accessed on 6 May 2021 and is now included as part of the SEALEVEL_EUR_PHY_L4_MY_008_068). The processing of this dataset includes some corrections, such as the removal of high-frequency variability, implemented to avoid the aliasing that could result from the low spatiotemporal resolution of altimetric observations (Gomis et al., 2012). This correction, called DAC (dynamic atmospheric correction), removes a significant part of the sea level variability associated with the atmospheric component, and therefore had to be reversed. The DAC is produced by CLS using the Mog2D model from Legos (Carrère and Lyard, 2003) and distributed by Aviso+, with support from CNES (https://www.aviso.altimetry.fr/en/data/, last access: 9 May 2021). These DAC data were bilinearly interpolated onto the coastal points where altimetric sea level anomalies were available and added to these series.

In order to study the relationship between the variability inferred from our coastal sea level reconstructions and large-scale atmospheric modes, climatic indices were downloaded for the four modes that are most relevant for the western Mediterranean basin: the North Atlantic Oscillation (NAO), the East Atlantic pattern (EA), the East Atlantic/Western Russian (EA/WR) pattern, and the Scandinavian pattern (SCAN). The NAO index is the main mode of variability in winter and accounts for the large-scale variation in atmospheric mass between the areas of the Azores subtropical anticyclone and the low-pressure area near Iceland. The EA index influences the freshwater flux into the northeastern Atlantic and also influences the western Mediterranean basin. The EA/WR index contributes to heat fluxes and has an impact on precipitation in the Mediterranean. The SCAN index is related to the climate of the Scandinavian Peninsula and East Asia, as well as to the precipitation in northern Europe (Bueh and Nakamura, 2007; Josey et al., 2011; Martínez-Asensio et al., 2014). All series were obtained from the NOAA Climate Prediction Centre (http://www.cpc.ncep.noaa.gov/data/teledoc/telecontents.shtml, last access: 23 September 2021) and consist of monthly data covering the period from 1950 to present.

Optimal interpolation is a type of linear statistical interpolation in which the best representation of the state vector of a system at a given point (ϕg) is obtained through the superposition of a first guess at that point (ϕgfg) and the weighting of the differences between observed data (ϕio) and those estimated by the first guess at each observation point (ϕifg) (Hasselmann et al., 1997): 1aϕg=ϕgfg+∑i=1NWgi(ϕio-ϕifg)=ϕgfg+∑i=1NWgiϕi′, where Wgi are the weights applied to the differences at each observation point i to obtain the state vector at point g, and N is the number of observation points. For a discrete set of m interpolation points, Eq. (1a) can be written as follows: 1bϕg=ϕgfg+W⋅ϕ′, where ϕg and ϕgfg are m vectors, ϕ′ is the n vector of anomalies at the n observation points, and W is the m×n weight matrix. The method is named as “optimal interpolation” because the weights are determined through the statistical minimization of the mean square error between the real and the interpolated field. The development of this minimization leads to the following expression: 2ϕg=ϕgfg+θT-1ϕ′, where θ is an m×n matrix whose elements are correlation values between series at interpolation points and at observation points and T is an n×n matrix whose elements are correlation values between series at observation points. Under the assumption of spatially uncorrelated noise, this has no effect on θ, but the diagonal of T (the correlation of observation series with themselves) includes the noise of the observations. Thus, matrix T can be expressed in terms of a correlation matrix between true values θo, the noise-to-signal coefficient (variance of the errors divided by the variance of the signal) of the observations η2 and the identity matrix I: 3T=θo+η2I, where the signal-to-noise coefficient η2 has been assumed to be the same for all observation points. This method also provides an explicit expression of the interpolation error in a statistical sense. That is, the mean value of the interpolation errors that would result if an infinite number of realizations of the observed field were interpolated in the same way (with the same observation points and the same weights). The errors at each interpolation point (εg) are given by Gomis and Pedder (2005): 4εg=σg1-DiaggθTT-1θ, where σg is the variance of the signal at point g and Diagg [⋅] denotes the element of the diagonal of matrix [⋅] corresponding to the interpolation point g.

The reconstruction has been performed in different frequency bands. The main reason is that spatial correlations may differ for different timescales (e.g., daily variability may have associated shorter spatial scales than multidecadal changes). Thus, the splitting into frequency bands allows for a more accurate spatial interpolation. Furthermore, because high-frequency signals usually dominate and mask low frequencies, the separation in frequency bands allows for a better representation of low frequencies (e.g., interannual and decadal variability). In the following, year is abbreviated with ”y”, month is abbreviated with ”m”, and day is abbreviated with ”d”.

The monthly PSMSL data were separated into three frequency bands: a first band corresponding to periods longer than 10 years (T>10y, a second band corresponding to periods between 1 and 10 years (1y<T<10y), and a third band corresponding to periods between 1 month and 1 year (1m<T<1y). The GESLA-2 hourly data were first averaged into daily data, and then a frequency band corresponding to periods between 1 d and 1 month (1d<T<1m) was isolated. The number of stations considered for each frequency band was different (Fig. 1) and dependent on the length of the series.

For the T>10y band, five series with at least 20 years of consecutive data were selected. The frequency band was isolated by means of a low-pass filtering carried out using a Butterworth filter on the order of 10, subtracting from each series the average of a period in which all of them had data.

The implementation of the optimal interpolation required to estimate the correlations between interpolation points and observation points and also between each pair of observation points. For this purpose, two approaches were followed, one based on model data (used to interpolate the frequency bands 1y<T<10y, 1m<T<1y and 1d<T<1m) and a second based on fitting an analytical correlation function (used to interpolate the band T>10y, for which the period spanned by the model does not allow a reliable estimation of correlations). These two approaches are described in the following points.

Calculating correlations from numerical model outputs. Defining the interpolation points as the coastal points of the SOCIB model grid and approximating the location of each tide gauge to the nearest grid point (which implies a minimum error given the spatial resolution of the model) makes the calculation of all necessary correlations straightforward. The elements of matrices θ and θo appearing in Eqs. (2) and (3), which correspond to correlations between true values of the field, were computed in this way for the three frequency bands.

In order to validate this procedure, the correlations obtained for the pairs of SOCIB model series colocated with tide gauges were compared with those obtained for the original tide gauge series. The latter were found to be lower than the correlations between model series due to the presence of observational noise. In order to simulate observations with model data, a Gaussian noise was added to the model series closest to the tide gauges, with a variance adjusted to make model correlations as close as possible to tide gauge correlations. These model series with added noise were used as pseudo-observations to carry out a first test: the optimal interpolation of these pseudo-observations at all coastal points was compared with the original model series with the aim of verifying the ability of the method to reproduce coastal sea level at all locations from a discrete number of observations. For more information on this validation test, see Appendix A.

Another parameter necessary for the implementation of optimal interpolation is the signal-to-noise coefficient of observations appearing in Eq. (3). This coefficient was optimized using the golden section search, which is appropriate for finding the minimum or maximum of unimodal functions through successive reduction of the range of values within which the extreme is known to exist (Pejic and Arsic, 2019). In our case, we searched for the signal-to-noise coefficient that minimized the mean square error between the original tide gauge series and the reconstructed series through a cross-validation test that will be explained in the next section.

In order to make a diagnosis of the reconstructions, cross-validation tests were carried out. These tests use part of the available observations to fit the model, while the other part is used as a validation set (Hastie et al., 2008). In our case, the sea level series at the closest grid points to each tide gauge were obtained considering all available tide gauges as observations except the one closest to that grid point in each case, which was kept as validation series. The reconstructed series were then compared with the tide gauge series using several statistics, namely: (i) the root-mean-square error (RMSE), considered a standard metric to model errors, (ii) the percentage of the variance of observations explained by the reconstruction, and (iii) the Pearson correlation coefficient between the reconstructed series and the tide gauge series. This was done for each frequency band in which the reconstructions were carried out.

Another validation test consisted of recovering the original monthly signals from the sum of the first three frequency band reconstructions and the original daily signals from the sum of the four band reconstructions. The comparison of these unified signals with the original tide gauge series allows us to verify that the split into frequency bands has been carried out correctly and that it is possible to reconstruct the complete series at all interpolation points. From the interpolation errors calculated for the reconstructions of the different frequency bands, the theoretical interpolation error of the total reconstruction can also be obtained: assuming that the errors of the reconstructions in the different frequency bands are independent of each other, the variance of the total error is the quadratic sum of the variances of the error in each band.

Finally, the total reconstructed series, as well as the reconstructed series of each frequency band, were compared with the last generation of altimetric products and checked against the original tide gauge series, in order to determine the goodness of each approximation.

In general, the statistics of the cross-validation test described in Sect. 3.3 gave good results when applied to the reconstructions of the four frequency bands (Figs. 2, 3 and 4): most reconstructions explain a high percentage of the variance of the original series and also show good correlations with the original series. The lowest frequency band (T>10y) displays the best results: root-mean-square (rms) differences are below 2 cm for all tide gauges, and correlations range from a minimum value of 0.74 in Ceuta and a maximum value of 0.99 in Marseille. For this band, the reconstruction explains more than 75 % of the variance of the original series at all stations except in Ceuta, where it only explains 39 %. These results show that a few stations (only four in the present case) are enough to reconstruct the low-frequency (decadal) variability, since this is mainly associated with large-scale spatial structures (Woodworth et al., 2019). Considering the observations from the 2 years before and after each interpolation time has also helped to obtain good estimates.

For the interannual to decadal frequency band (1y<T<10y), the reconstructions explain in general smaller percentages of the tide gauge variance than for the other frequency bands. The best results are obtained at the stations located on the east coast of the Iberian Peninsula, southern France and northern Italy, where the explained variance is at least 50 % (>75 % for the Imperia, L'Estartit, and Port Vendres tide gauges), rms differences are below 4 cm, and correlations are higher than 0.7. However, for the stations close to the Strait of Gibraltar the statistics are much poorer, meaning that the reconstructions can explain almost none of the variance of the original series. The results for Ibiza and Cagliari are not good either. In order to check whether the origin of the poor statistics of this band was due to a bad representation of the correlation matrix elements used for the optimal interpolation, we also tested analytical correlation functions with different correlation characteristic lengths; however, no improvement in the results was achieved. This suggests that the stations showing poor statistics would have a sea level variability spatially decoupled from the others at this frequency band and/or that there were problems with the observations.

In the intra-annual frequency band (1m<T<1y), the statistics are better than for the previous band, with the worst values corresponding to the same stations: those located near the strait of Gibraltar, Ibiza, and Cagliari. For the other stations, rms differences are below 3 cm, correlations are above 0.8, and explained variances are above 70 % in almost all cases.

In order to visualize the bad results given by the cross-validation test near the strait of Gibraltar, we show a comparison between the original Tarifa tide gauge series and its reconstructions given by the cross-validation test for the four frequency bands (Fig. 5). Tarifa station was chosen because its series has been used for the reconstruction of all four frequency bands. The difficulties in reconstructing the original tide gauge series for the intra-annual and inter-annual frequency bands are well apparent. For these frequency bands, the differences between the original and the reconstructed series are larger than the statistical interpolation error given by the optimal interpolation formulation (Eq. 4); this suggests that for some stations the correlation elements of the optimal interpolation matrices are not correctly represented. The plots also suggest that in some frequency bands the observations may have some problems that were not identified by the quality control. For instance, in the 1 day to 1 month range the explained variances are lowered by the spikes in the observations, which look unrealistic. For comparison, Fig. 6 shows the same information as Fig. 5 but for the Genoa station, which generally shows good statistics. As for Fig. 5, results are worse for the inter-annual and intra-annual bands (though notably better than in the case of Tarifa). This suggests that some of the coastal processes driving these frequency bands cannot be correctly interpolated by the reconstruction method. On the other hand, in the daily to monthly frequency band the reconstruction is able to explain more than 88 % of the variance of the original series.

The cross-validation test has allowed for some insight into the capabilities of the method. The next step was to obtain the main result of this work: the reconstructions for each frequency band, now considering all stations, i.e., without withdrawing any station, as for the cross-validation test. Following this, the reconstructions of the different bands were merged to evaluate the extent to which total sea level can be recovered and hence to prove that the separation into frequency bands has been carried out correctly.

As an example of the results, Fig. 7 shows the following results for the interpolation point closest to the Barcelona tide gauge: (i) the reconstructions in the four frequency bands, (ii) a comparison between the original monthly series of Barcelona tide gauge and the merging of the three bands that correspond to periods T>1m, and (iii) a comparison between the original daily series of Barcelona tide gauge and the merging of the four frequency bands, which corresponds to periods T>1d. In both cases there is a high coincidence between the original and the merging of the reconstructed series, showing a correlation of 0.97 for the monthly case and 0.99 for the daily case.

Figure 8 shows the average interpolation errors of the merged series for the monthly case. Although it has been shown that actual errors can be higher than the theoretical error estimate at some stations, the latter can be useful to reflect the spatial distribution of the interpolation accuracy. The quoted values are an average of the interpolation errors over the period from 1884 to 2019, since errors depend on the number of available stations and this varies with time. The interpolation errors of the daily merged series are also quoted, in this case averaged over the period from 1980 to 2015. Maximum values of 5.26 cm are obtained for the monthly case and of 7.05 cm for the daily case. The magnitude of the interpolation errors depends not only on the number of available stations but also on their location with respect to the considered interpolation point. For this reason, the spatial pattern of the errors clearly shows higher values in regions where no observations are available, such as the North African coast, or where tide gauge series are recent (and hence the average involves time periods when these stations were not available), such as in the Balearic archipelago for the monthly merging.

With the aim of characterizing coastal sea level variability, some indicators were estimated from the obtained reconstructions. First, sea level trends were estimated for (i) the merged reconstructed series during the period covered by altimetry (1993–2019), (ii) tide gauge series spanning at least 80 % of that period, and (iii) altimetry series for the grid points closest to the coast. Trends were also calculated for the period 1884–2019. Figure 9 and Table 3 show all these trend values.

For the period covered by altimetry, the trends of the reconstructions and of the altimetry series are similar in magnitude, with a basin-wide mean value of 2.70 ± 0.32 and 2.45 ± 0.49 mm yr-1, respectively. However, the values along the coast show a smoother continuity for the reconstruction than for altimetry. The heterogeneous coastal trend pattern obtained from altimetry does not seem to make physical sense and could be explained by the limitations of altimetry in coastal areas. When compared with tide gauge trends computed for the period common to the three datasets (Table 3), a good agreement between tide gauges and the reconstruction is obtained, except for at Algeciras, Barcelona, and Tarifa. The trends computed for these stations also show discrepancies with altimetric trends and even with the trends of nearby tide gauges. The lack of coherence between tide gauge trends in the Strait of Gibraltar has already been reported by different authors (e.g., Marcos and Tsimplis, 2008; Ross et al., 2000).

The main advantage of the reconstruction over tide gauges is that it allows for the estimation of sea level trends along the entire coastline. Moreover, it does not have data gaps, which are responsible for significantly increasing the uncertainty of the trends estimated from tide gauge series. The advantages of the reconstruction over altimetry are that it spans a longer period, provides more accurate results, and smooths out eventual local anomalous trend values.

For the total reconstruction period (1884–2019) the trends range from less than 1 mm yr-1 on the African coast close to Gibraltar to about 1.5 mm yr-1 in the Gulf of Lion, with a regional mean value of 1.20 ± 0.14 mm yr-1. This result is consistent with the Mediterranean sea level trend computed from the three stations with the longest series, which is estimated to be between 1.1 and 1.3 mm yr-1 (Gomis et al., 2012), as well as with the global rate of sea level rise for the 20th century, estimated through various reconstructions between 1.3 and 2 mm yr-1 (Dangendorf et al., 2017). The trends show a rapid increase from the 1990s, coinciding with the period covered by altimetry with values ranging from 1.89 to 3.16 mm yr-1. For that period the trends estimated from the reconstruction are coherent with those estimated by other authors (e.g., Bonaduce et al., 2016, who obtained an average value for the whole Mediterranean basin of 2.44 ± 0.5 mm yr-1 for the period 1993–2012).

The seasonal cycle is one of the main components of sea level variability. Seasonal sea level changes are mainly caused by changes in the heat content of the upper layers of the ocean and by changes in the atmospheric pressure field and winds that modulate the inflow of water from the Atlantic. In the Mediterranean Sea, the seasonal cycle is estimated to account for, on average, 20 % of the variance of tide gauge series and shows significant interannual variability. In addition, seasonal cycle variations in coastal areas may differ significantly from those reported in the open ocean (Gomis et al., 2012; Woodworth et al., 2019). The seasonal cycle of the coastal reconstructions, estimated from monthly mean values, accounts for 24 % of the coastal sea level variance on average.

Figure 10 shows the patterns of the variability in different frequency bands quantified in terms of the standard deviation. For the seasonal cycle the variability ranges between 2.92 and 4.97 cm, with a smooth variation along the coast. The largest standard deviations are found at the eastern coast of the Iberian Peninsula, in agreement with previous authors that located the maximum annual sea level cycle of the western Mediterranean in Alicante (Marcos and Tsimplis, 2007b) and the maximum annual cycle of the atmospheric contribution to sea level in the Alborán Sea (Gomis et al., 2008). The subtraction of the seasonal cycle does not lead to a large reduction in the standard deviation of the reconstruction, which is 1 cm lower without the seasonal cycle on average. The deseasoned reconstruction has an average standard deviation of 3.77 cm for periods T>1y, 4.91 cm for periods 1m<T<1y, and 4.27 cm for periods 1d<T<1m. In all frequency bands, the highest standard deviations of the deseasoned series are obtained in the Gulf of Lion.

The four main atmospheric modes driving western Mediterranean sea level variability are NAO, EA, EA/WR, and SCAN. Their influence has already been studied by (among others) Martínez-Asensio et al. (2014), who used long tide gauge series from the whole basin, as well as altimetry products. Our coastal reconstructions allow us to complement the study of the influence of these climate modes in two ways: by covering the entire coastal region (i.e., not only where tide gauge records are available) and by avoiding the use of coastal altimetry. In addition, the sea level series of our reconstructions cover a longer period than altimetric products, thus enabling the analysis to extend to the whole period covered by climate indices (since 1950).

Figure 11 shows the correlation patterns between the monthly coastal reconstruction and the four climate indices. Correlations have been calculated for both the complete series and the seasonal mean values of the reconstruction and the indices, with winter accounting for January–March, spring for April–June, summer for July–September, and autumn for October–December.

For the total series, the NAO index is anti-correlated with sea level since the western Mediterranean is part of the subtropical high pressure, and hence sea level lowers when the NAO index is in a positive phase. Conversely, the EA and SCAN indices show a positive correlation. In winter, two indices dominate sea level variability: the NAO index, with correlation values below -0.6 at some points and showing significant correlations along the entire basin coastline, and the SCAN index, with positive correlations (the NAO and SCAN indices are interdependent and anti–correlated with each other). The obtained correlation patterns are similar to those obtained from open-ocean altimetry for the period 1993–2010 (Martínez-Asensio et al., 2014), although the values obtained for the coastal reconstructions are somewhat weaker for the two dominant indices. In spring, the EA index clearly dominates over the others, being the only one with significant correlations throughout the basin, with positive correlation values above 0.3.

In summer, the correlation patterns obtained from our reconstructions slightly differ from those obtained from tide gauge series in the western Mediterranean by Martínez-Asensio et al. (2014). Our results show that during summer the EA is the dominating index, with basin-wide positive correlations up to 0.5, while the EA/WR index shows negative correlations of around -0.3. Martínez-Asensio et al. (2014) also obtain positive correlations between the tide gauges and the EA index, but these are always lower than 0.5 and often not significant; for the EA/WR index they obtain non-significant negative correlations. Overall, the sea level reconstruction suggests a greater influence of the EA and EA/WR indices (mainly related to freshwater and heat fluxes) on western Mediterranean sea level variability in summer than what is obtained from pointwise observations. Finally, in autumn it is the NAO index that seems to dominate the variability, with correlations around -0.5, followed by EA, with values up to 0.4.