1 Introduction

Themonitoring of sea level is conventionally performed using tide gauges and a network of radar altimeters orbiting the Earth. Tide gauges are in situ instruments that register measurements at high frequency (often multiple measurements per hour) and are scattered irregularly along the global coastlines (Woodworth et al. 2016). Altimeters sample along satellite tracks, spanning the same area after a defined number of days depending on the chosen repeating orbit (Fu and Cazenave 2001). Efforts aimed at finding new strategies to improve the characterisation of sea level variability at sub-seasonal time scales and in the coastal and shelf seas to “reduce the gap” between altimetric and tide gauge observations are ongoing as shown in previous works such as Cipollini et al. (2017).

Since altimetry data are along-track measurements scattered in time and space, interpolating algorithms are routinely used to generate sea level maps that are regularly sampled in space and time. The European Union’s Earth observation programme, Copernicus, currently releases daily sea level maps and their along-track sources through the Copernicus Marine Service (CMEMS). The CMEMS daily maps are produced using a processing based on optimal interpolation, requiring several steps and assumptions described in Le Traon et al. (1998) and Taburet et al. (2019). The along-track data are sub-sampled and filtered twice, using variable cut-off wavelengths ranging from 200 to 65 km depending on the latitude. The optimal interpolation uses a variable number of observations in time and space, with spatial correlation scales ranging from 80 to 400 km and time correlation scales ranging from 10 to 45 days. It is based on the best linear least square estimator described by Bretherton et al. (1976), in which the covariance matrix of the observations is needed as an input. The covariance matrix is provided by means of assumptions on the errors of the different geophysical corrections applied to the along-track measurements (Pujol et al. 2016).

It has been recently argued that data-driven interpolation is able to perform better than conventional optimal interpolation schemes, whose choice of covariance priors tends to over-smooth the sea level variability (Lguensat et al. 2019). The concept behind data-driven interpolation is to exploit machine learning to provide an estimation based on patterns and statistical relations acquired from the training data, rather than from external instructions and assumptions (Zhou et al. 2017). The objective of this paper is to adapt an established machine learning technique to the problem of the estimation of daily sea level maps from along-track altimetry measurements. We use the Random Forest Regression algorithm, described by Breiman (2001), in the implementation of Pedregosa et al. (2011), which has already been successfully used to fill gaps due to missing observations of the ocean (e.g. Gregor et al. (2017) used it to interpolate sparse in situ surface CO2 observations in the Southern Ocean). In this study, we test a method based on Random Forest Regression to grid sparse along-track measurements based on neighbouring observations. The test is carried out on a regional scale for 1 year of data in the North Sea and we validated using tide gauge data and the optimally interpolated maps from CMEMS. While CMEMS daily grids have only been validated using monthly averages from tide gauges as ground truth, we adopt in this work the daily averages of the Global Extreme Sea Level Analysis (GESLA, version 3), a global archive of high-frequency tide gauge data (Woodworth et al. 2016; Haigh et al. 2021).

2 Data

In this case study, we consider the year 2004 and the extended North Sea including Skagerak/Kattegat in the east and the English Channel in the west. The available altimetry missions in this year were Jason-1, Envisat, Topex/Poseidon and Geosat Follow-on. The North Sea is an ideal testbed for our experiment, thanks to the availability of an extensive tide gauge network at high frequency, which allows for validation of a daily product. To our knowledge, previous studies involving the comparison of gridded altimetry and tide gauges have only involved monthly data, analysing trends and interannual variability (e.g. Dettmering et al. (2021)). The region of interest and its geographical coordinates are delimited by the red square in Fig. 1.

Fig. 1
figure 1

Examples of along-track observations included in spatial (left) and temporal (right) neighbourhoods associated to one particular location. The red box indicates the area of study. The latter is extended in the search for neighbouring observations, in order to allow for the estimations at the domain’s border

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To train the Random Forest Regression, we use the CMEMS Level 3 (i.e. along-track) sea level anomalies (SLA), reference number: SEALEVEL_GLO_PHY_L3_REP_ OBSERVATIONS_008_062. We recall that the SLA is defined as the sea level above the mean, corrected for atmospheric and tidal effects. A list of all applied corrections is available in Taburet et al. (2019). We compare the daily machine learning–based SLA from this study (nicknamed ML from now on) with the latest version of the CMEMS Level 4 gridded SLAs, reference number: SEALEVEL_ GLO_PHY_L4_MY_008_047. We stress the difference in the use of the data sources from CMEMS: the along-track data are the observations that are used to build the regression model; the gridded SLAs are only used for comparison with respect to the results of this study.

As external truth for the validation, we use high-frequency data from tide gauges available from the Global Extreme Sea Level Analysis (GESLA-3, https://www.gesla.org, Woodworth et al. (2016)). To make the tide gauge data comparable to the altimetry dataset, the following processing steps are needed. Firstly, the atmospheric component is removed using the same correction applied to obtain the SLAs, i.e. the dynamic atmosphere correction from Carrère and Lyard (2003). Secondly, the tidal variability is suppressed using a 40-h LOESS filter, which has been tested to most effectively reduce tidal variance at periods lower than 2 days by Saraceno et al. (2008). The mean of the full sea level record is computed and subtracted from each time series in order to obtain the sea level anomalies. Finally, since data are provided at hourly and sub-hourly frequencies, the obtained tide gauge sea level anomalies are averaged at a daily rate.

3 Method

The concept of our methodology is the use of along-track SLAs as truth to train the random forest regressor in the estimation of unknown SLAs (our target variable) on a set of grid points. As predictors, we use means, weighted means and standard deviations of the SLAs at different neighbourhoods in space and time. Furthermore, to better describe the evolution of the target variable in both space and time, the ratios among these predictors from the different neighbours are also used as predictors.

This methodology is inherited from Leirvik and Yuan (2021), who used spatial neighbourhoods to constrain a Random Forest Regression for the interpolation of a surface solar radiation dataset. We expand the methodology by considering the time dimension as well. The following subsections are dedicated to the details of our implementation.

3.1 Preliminary steps

All along-track data for 2004 from CMEMS are collected in the area of study, enlarged by 2.5∘ in latitude and longitude to guarantee the definition of the neighbourhoods at its borders.

The target variable ytraining to train the regressor is the field sla_unfiltered, where the 1-Hz SLAs (roughly one measurement every 7 km along the track) are stored. The CMEMS Level 4 gridded SLA uses the field sla_filtered when interpolating Level 3 data. Such a field is a smoother version of the along-track data obtained using variable filter lengths of several tens of kilometers. Our experiments have shown that the neighbourhood method proposed in this study does not need further filtering and our objective is to keep as much signal as possible. Further discussion and comparison with the CMEMS Level 4 in these regards are provided in Section 4.

We define the locations for computing the SLA, our unknown independent variable y, as the geographical coordinates of a daily unstructured grid. This grid is spaced at intervals of 0.125 degrees in both latitude and longitude, which is equivalent to the grid resolution of the CMEMS Level 4 product.

3.2 Definition of neighbourhoods

We define three spatial neighbourhoods and three temporal neighbourhoods to group the along-track altimetry observations in the proximity of the locations in ytraining and y.

The spatial neighbourhoods are concentric circles with a radius of 100 km, 200 km and 300 km from the location of the target variable. The temporal neighbourhoods contain the along-track data collected within 5, 10 and 15 days from the time set by the target variable, within a distance that does not exceed 300 km. An example of the along-track locations assembled through the neighbourhoods of one target variable is provided in Fig. 1.

The borders of the neighbourhoods are selected to be within the average global correlation scales of sea level in time and space (see for example Fig. 4 from Pujol et al. (2016)). Nevertheless, the choice for this experimental study is empirical and could be further optimised, for example by using global maps of variable correlation scales depending on the region, such as what is done in the generation of the CMEMS Level 4 grids. We anticipate that we do not observe a substantial change in the performances by slightly changing the neighbourhood definitions.

3.3 Definition of predictors

We define in this section the following classes of predictors: time and space clusters, single-neighbourhoods statistics and multi-neighbourhoods statistics.

3.3.1 Time clustering

The time cluster contains the month in which the variable of interest is defined. Given that the annual cycle is the most prominent SLA periodic signal in time series whose length cannot catch decadal variability, we expect this information to be relevant for the regression. Indeed, Fig. 2a shows the two very different probability densities (PDs) of the SLA for January (blue) and July (red) based on the full training dataset.

Fig. 2
figure 2

Probability density of the sea level anomalies associated to specific predictors from the training dataset. Panel (a): months of January and July. Panel (b): two geographical clusters. Panel (c): the mean of the sea level anomalies for the first (100-km radius) and third (300-km radius) spatial neighbourhoods. Panel (d): the mean of the sea level anomalies for the first (5 days) and third (30 days) time neighbourhoods

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3.3.2 Spatial clustering

Several choices could be done concerning spatial clustering. In this exploratory study and in order to generalise the approach, we choose the agglomerative hierarchical clustering (Ward Jr. 1963) in the implementation of Pedregosa et al. (2011). This is an unsupervised classification method that we use to separate the domain in different regions, simply based on the Euclidean distance between the locations in our case. We choose to divide our subdomain in nine clusters, an example of the different PDs of SLA from two of them is visualised in Fig. 2b. We reckon that this is a choice driven by simplicity and other oceanographic information could be used to refine the clustering, for example taking into consideration the spatial correlation with respect to tide gauges (the so-called zone of influence approach from Oelsmann et al. (2020)).

3.3.3 Single-neighbourhoods statistics

For the SLAs contained in every spatial and temporal neighbourhood, we compute the following statistics: mean, spatial-based weighted mean, time-based weighted mean and standard deviation. The weighted means are based on inverse distance weighting, i.e. maintaining the notation of Leirvik and Yuan (2021), the weighted means are defined as:

$$ \tilde{z}(N) = \displaystyle\sum\limits_{z_{i}\in N} \lambda_{i} z_{i} $$
(1)

where N defines the neighbourhood, zi is every SLA value within it and the weights λi are defined as:

$$ \lambda_{i} = \frac{d_{i0}^{-r}}{\displaystyle\sum\limits_{z_{i}\in N}d_{i0}^{-r}} $$
(2)

For the spatial-based weighted mean, di0 is the Euclidean distance in kilometers of every SLA within the neighbourhood and the location of the target variable. For the time-based weighted mean, di0 is the time difference in seconds between the passing time of the altimeter at the observation i and the time stamp of the target variable. Note that the time difference is multiplied by a factor 10− 4 in order to achieve similar orders of magnitude between spatial-based and time-based weighted means.

The exponent, r, expresses the relative importance of close-by observations. The highest exponent found in the literature is r = 5 from Leirvik and Yuan (2021). We tested lower values and the sensitivity of our results to the choice of r (results are reported in Section 4.2 and found the best performances for r = 2). This is not surprising, since a high exponential gives a high importance to the closest observations, while SLA is a field characterised by large spatial and temporal scales of correlation. Given three spatial neighbourhoods and three temporal neighbourhoods, we obtain therefore twenty-four single-neighbourhood predictors. An example of the different PDs of the predictors is given in Fig. 2, where the PD of the mean SLA for the first and third spatial (panel c) and temporal (panel d) neighbourhoods is provided.

3.3.4 Multiple-neighbourhoods statistics

The multiple-neighbourhoods statistics are the ratios between the single-neighbourhoods statistics of the same kind for consecutive neighbourhoods. For example, as in Leirvik and Yuan (2021), considering the mean of the SLAs, we compute the ratio of the mean SLAs between first and second neighbourhoods, \(\overline {Z}^{k1,k2}\), and the ratio of the mean SLAs between second and third neighbourhoods, \(\overline {Z}^{k2,k3}\). Considering the typical objective of the altimetry missions to achieve a 1-cm SLA accuracy at 1-Hz posting rate (Bonnefond et al. 2013), we round up (or down, for negative numbers) to the nearest centimetre the single neighbourhood statistics previously obtained before computing each ratio.

3.4 Final steps

The predictors are computed for both ytraining and y locations, generating the predictor matrices Xtraining and X, in which each row corresponds to the predictors associated with each location. Outliers in ytraining and Xtraining are identified using a 3σ criterion, where σ is the standard deviation of each variable. Observations in which the SLA or its predictors are identified as outliers are eliminated from the training dataset. Finally, the Random Forest Regression is applied on the training dataset. The obtained regressor f(⋅) is then applied to estimate the desired SLA on the grid points as y = f(X).

4 Results and discussion

4.1 Examples

To investigate the advantages and the limitations of the generated daily ML product, we first consider examples in time and space. Figure 3 shows the time series of daily averaged data from tide gauges (in green), whose locations are specified at the top of each subplot. The ML product (in blue) and the CMEMS product (in orange), corresponding to the closest location to each tide gauge, are shown for a period comprised between the 15th of January and the 15th of December. This latest choice is because we have only worked with data from 2004 and therefore the regression would generate worse results at the beginning and the end of the period investigated.

Fig. 3
figure 3

Time series estimated from satellite altimetry from this study (ML, blue) and CMEMS (orange) at the closest point to four tide gauges (green), whose coordinates are shown at the top of each panel. Also shown as text is the root mean square error (RMSE) of the altimetry dataset considering the tide gauges as ground truth

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The CMEMS time series appears significantly smooth in time, while the ML product preserves time scales that better match the ones of the tide gauges, although of course the full extent of the high-frequency variability is not captured. Despite CMEMS being smoother than the ML product, the root mean square error (RMSE), computed taking the tide gauges as the truth, is systematically lower for ML. This gives us confidence that the ML time series is not simply noisier than the CMEMS, but it is indeed more accurate.

In Fig. 4, we show a snapshot of ML and CMEMS SLAs for the 24th of April 2004. While the large-scale gradients are similar in both products, the CMEMS map has more defined contours identifying mesoscale variability. The higher variability of ML is the counterpart in space of what has been seen in time in the previous example. The objectives of ML and CMEMS are indeed different: the CMEMS optimal interpolation scheme is dedicated to the retrieval of mesoscale structures (Taburet et al. 2019), while with ML we attempt to achieve a better compromise to observe local sea level variability. The latest statement is quantified and verified for this case study in Sections 4.2 and 4.3.

Fig. 4
figure 4

The daily maps of sea level anomalies (SLA) from this study (ML) and CMEMS estimated for the 24th of April 2004

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The spatial resolution of the ML grid appears degraded compared to CMEMS, in which eddy-like structures can be recognised. Still, it is important to recall that dedicated efforts to assess CMEMS effective spatial resolution concluded that it is not better than 100 km wavelength, which is the resolution reached at the highest latitudes (Ballarotta et al. 2019). Here, we further notice that the CMEMS map is affected by unrealistic extremes of SLA in single pixels in particularly challenging areas such as the English Channel. This is remarkable, considering that the input along-track data of ML and CMEMS are exactly the same, except for the along-track filtering applied by the latter.

4.2 Validation against tide gauges

We assess the general performances of ML and CMEMS computing Pearson’s correlation coefficient (CORR) and the RMS between the time series obtained from altimetry and the daily means of the tide gauge data at the closest grid point. Figure 5 shows in the upper panels the RMS and the CORR for ML and in the lower panels the difference with respect to the same statistics computed using CMEMS. The colourbar of the latter is adjusted to show that when the colour is red, an improvement is seen when using the ML product with respect to CMEMS. Good performances (CORR0.7) are reached along the coasts facing a large open ocean area at the centre of the domain, such as the eastern coast of the United Kingdom (UK). Notably, good performances are also seen in much more enclosed areas situated at the periphery of the domain, such as the Kattegat Sea between Denmark and Sweden (the easternmost part of the domain). This advocates for the robustness of the neighbourhood strategy previously presented. Lowest performances are reached in some enclosed bays and on both sides of the Channel between UK, France and Belgium (the southernmost part of the domain). Here the quality of the SLAs, also in terms of the geophysical corrections used to extract them, plays a dominant role as shown in previous studies at different temporal scales, such as Dettmering et al. (2021) using monthly time series. The most remarkable result of the validation is that in almost all of the domain (29 tide gauges out of 32) ML performs better than CMEMS. In more than half of the domain, there is at least a 5% improvement in both CORR and RMS considering the tide gauges as ground truth. The average improvement in correlation is 9.98% (6.99% considering RMS), with peaks over 30% that include some of the most problematic areas for satellite altimetry such as the Channel. Concerning the sensitivity of these results to the choice of the weighting factor r, explained in Section 3.3, we tested integer values of r from 1 to 5. The differences between the best and worse average performances are 0.36% in terms of correlation improvement and 1.2% in terms of RMS improvement with respect to CMEMS. In no case among the ones considered did the choice of r determine worse results of ML compared to CMEMS.

Fig. 5
figure 5

Results of the validation of daily sea level anomaly maps coupled with tide gauges at the closest point. Root mean square (RMS, panel a) and Pearson’s correlation coefficient (CORR, panel b) between the product of this study (ML) and the time series from the tide gauges (panel a). Panels c and d: Difference between these statistics and the equivalent computed using the CMEMS product, in which the red colour palette indicates an improvement using ML

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In order to understand whether this result depends on the choice of using unfiltered SLAs as training dataset, we also repeated the same experiment starting from the target variable sla_filtered (see Section 3.1 for description). We find that using sla_filtered produces marginal improvements in the statistics (the average improvement in correlation is 10.17%, 7.45% considering RMS). This corroborates the finding as a result of the ML approach described, with scarce dependence on the smoothing applied to the along-track data. The unfiltered approach is still kept as the basic approach of this study, since our objective is to avoid as much as possible the suppression of the physical signal.

Despite the short time series considered in this experiment, we also compute the magnitude squared coherence (as defined, for example, in Thomson and Emery (2014)) to investigate the agreement between the tide gauges and the altimetry time series at different frequencies. We show the mean coherence using ML and CMEMS obtained considering all the available tide gauges and periods below 90 days, in order to have at least 4 time windows to consider out of 1 year of data. The results, displayed in Fig. 6, show that a clear improvement in the coherence is obtained with the ML approach from periods higher than 10 days. Lower periods are dominated by noise, which is expected given that the data are corrected for the dynamic atmosphere correction, which largely suppresses the oceanic variability at these frequencies that are badly sampled by the altimetry constellation (Carrère and Lyard 2003).

Fig. 6
figure 6

Results of the validation of daily sea level anomaly maps coupled with tide gauges (TG) at the closest point: mean magnitude squared coherence for periods lower than 90 days expressing similarity between the time series from TG and CMEMS (orange), and from TG and ML (blue)

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4.3 SLA variability

Finally, we assess how realistic the variability of the sea level from the daily grids is. For this purpose, we compute the interquartile range (IQR) of the time series at every grid point and every tide gauge. The IQR is an index of variability computed by taking the difference between the 75th and the 25th percentile of the data, and it is typically used instead of standard deviation or variance because of its robustness. It is commonly used in sea level studies comparing in situ and satellite time series (for example Wöppelmann and Marcos (2016)) and proves fit for our purposes, given that we only assess 1 year of data.

Figure 7a displays the results on the map, showing a consistently increasing variability of ML towards the southeastern part of the domain, which is confirmed by the tide gauge records. In Fig. 7b, the IQR at tide gauges is compared with the variability observed by ML and CMEMS at the closest point. To evaluate this comparison and considering the tide gauges as the ground truth, we compute an index of the average misrepresentation of the sea level variability:

$$ {Err}_{var}=\frac{\displaystyle\sum\limits_{i=1}^{N} \frac{({IQR}_{alti} - {IQR}_{TG})}{{IQR}_{TG}}\cdot100 }{N} $$
(3)

where N is the number of tide gauges and IQRalti is the IQR of the altimetric time series at the closest point to each tide gauge. The best results are obtained by ML with Errvar = 4.4%, while when using CMEMS Errvar = 7.6% is achieved. However, it is also noticed that ML underestimates the variability in the two stations with the highest IQR.

Fig. 7
figure 7

Panel a: Variability of sea level anomaly (SLA) estimated using the interquartile range of the time series at each grid point estimated in this study (ML) and from the tide gauges (circles). Panel b compares the same statistics at the tide gauges (TG) with the closest grid point from ML and CMEMS products

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5 Conclusions

This study has analysed the potential of using a data-driven approach to produce daily maps of SLAs starting from along-track observations from satellite altimeters. This approach allows for circumventing several hypotheses needed to characterise the covariance of the observations and their errors in the optimal interpolation. Building on the existing literature, we have tested a Random Forest Regression that uses statistics extracted from spatial and temporal neighbourhoods. By doing so, we have obtained 1 year of daily sea level maps that are on average 10% more correlated to the observations from tide gauge stations in the North Sea, compared to CMEMS data.

We believe that the main heritage of this study is the idea that along-track SLA data can be used to train machine learning routines aimed at generating gridded maps. The latter appear less smoothed in space than their CMEMS counterpart and will therefore need further filtering to be used for the identification of mesoscale features such as eddies. Nevertheless, the method presented allows for a more realistic representation of the sea level variability, as verified by the comparison against coastal in situ data. Such comparison has been conducted using high-frequency tide gauges, which is in our opinion a much more realistic external validation than the use of monthly means, if the objective is to assess the capability of the altimetry constellation to observe sea level at short time scales.

Since this is an exploratory study, we have to acknowledge both potential and limitations. To speed up the experiments, we have chosen one single year of data (2004), in which four altimeters were in orbit, and a specific region (the North Sea). Extending this methodology to a longer time series will allow to perform coherence studies and distinguish therefore the performances at different time frequencies. We have used one single regressor, because clusters based on time and geographical locations of the observations were part of the predictors. Nevertheless, the feasibility of this choice will need to be assessed for studies involving more years and a wider area, also in terms of computing time.

The validation against tide gauges shows the strong potential of machine learning to improve the characterisation of coastal sea level at a time in which the altimetry community has recognised the possibilities to improve the quality of sea level data close to the coast (Benveniste et al. 2020). We expect therefore further improvements by using SLAs whose estimation is optimised for the coastal zone (Passaro et al. 2021; Birol et al. 2021), which will nevertheless require significant post-processing of the along-track data, in order not to decrease the quality of the training dataset.