The assessment of wave conditions at sea is fruitful for many research fields in marine and atmospheric sciences and for human activities in the marine environment. In the past decades, the observational network (mostly relying on buoys, satellites and other probes) has been integrated with numerical model outputs allowing one to obtain the parameters of sea states over wider regions. Apart from the collection of wave parameters, the technique adopted to infer the wave climate at those sites is a crucial step in order to provide high-quality data and information to the community. In this context, several statistical techniques have been proposed to provide a reliable representation of the probability structure of wave parameters. While univariate and bivariate probability distribution functions (PDFs) are routinely derived, multivariate PDFs that represent the joint probability structure of more than two wave parameters are not straightforward. For individual waves, for instance, the bivariate joint PDF of wave height and period was derived by and the bivariate joint PDF of wave height and direction was obtained by . A trivariate joint PDF of wave height, wave period and direction is due to . For sea states, attempts have been made to model the joint probability structure of the integral wave parameters. For instance, a joint PDF of the significant wave height and the average zero-crossing wave period was derived by and . exploited the “copula” statistical operators to describe the dependence among several random variables, e.g., significant wave hight, storm duration, storm direction and storm interarrival time, deriving their joint probability distributions. The same approach was applied by to the significant wave height and peak water level in the context of coastal flooding.

Recently, the self-organizing map (SOM) technique has been successfully applied to represent the multivariate wave climate around the Iberian Peninsula and the South American continent . SOM is an unsupervised neural network technique that classifies multivariate input data and projects them onto a uni- or bi-dimensional output space, called map. The SOM technique was originally developed in the 1980s, and has been largely applied in various fields, including oceanography . Typical applications of SOM are vector quantization, regression and clustering. SOMs gained credit among other techniques with same applications due to its visualization capabilities that allow one to get multi-dimensional information from a two-dimensional lattice. The SOM also has the advantages of unsupervised learning; therefore, vector quantization is performed autonomously. However, the quantization is strongly driven by the input data density. Indeed, the SOM is principally forced by the most frequent conditions, while the most rare (i.e., the extreme events) are often missed. Consequently, it is highly unlike to find extremes properly represented on a SOM.

In the context of ocean waves, drawing upon the works of and , the SOM input is generally constituted by a set of wave parameters measured or simulated at a given location and evolving over the time t, e.g., the triplet composed by significant wave height Hs(t), mean wave period Tm(t) and mean wave direction θm(t), even if other variables can be added (examples of five- or six-dimensional inputs can be found in ). Several activities in the wave field could benefit from the SOM outcomes, such as selection of typical deep-water sea states for propagation towards the coast to study the longshore currents regime and coastal erosion, identification of typical sea states for wave energy resource assessment and wave farm optimization. In addition the empirical joint and marginal PDFs can be derived from SOMs. As accurately shown in , besides interesting potentials, especially in visualization, some drawbacks in using the SOM for wave analysis have emerged with respect to other classification techniques. Indeed, the largest Hs are missed by SOMs because such extreme events are both rare (few comparisons in the “competitive” stage of the SOM learning) and distant from the others in the multi-dimensional space of input data (poorly influenced during the “cooperative” stage).

Moving from this evidence, the scientific question being asked is how can we employ SOM with its visualization capabilities to improve representation of the extremes of a multivariate wave climate at a location. To answer this question we have followed three different strategies. First, we have pre-preprocessed the SOM input data using the maximum-dissimilarity algorithm (MDA) in order to reduce the redundancies of the frequent low and moderate sea states, as done by . Indeed, MDA is a technique that reduces the density of inputs by preserving only the most representative (i.e., the most distant from each other in a Euclidean sense). Doing so, the most severe sea states are expected to gain weight in the learning process. We have called this strategy MDA-SOM. Then, we have focused on the post-processing of the SOM outputs. In this context, we have applied a two-step SOM approach (herein called TSOM), by firstly running the SOM to get a reliable representation of the low/moderate (i.e., the most frequent) wave climate, and then by running a second SOM on a reduced input sample. This new sample has been obtained by taking from first-step SOM results the events exceeding a prescribed threshold (e.g., 97th percentile of Hs). To present results of two-step SOMs, we have proposed a double-sided map, showing on the left the SOM with the reliable representation of the low/moderate sea states, and on the right the map with the most severe sea states (i.e., the extremes). Then, we have applied a SOM to the peak of the storms individuated by means of a peak-over-threshold analysis (calling this strategy POT-SOM) and we have represented results using the double-sided map. An application of the proposed TSOM approach is finally reported: we have exploited the TSOM results to compute the longshore component of the wave energy flux, showing that a more proper representation of the extreme wave climate leads to an enhanced quantification of the energy approaching the shore.

The data set employed for the SOM analysis consists of wave time series gathered at the Acqua Alta oceanographic tower, owned and operated by the Italian National Research Council – Institute of Marine Sciences (CNR-ISMAR). Acqua Alta is located in the northern Adriatic Sea (Italy, northern Mediterranean Sea), approximately 15 km off the Venice coast at 17 m depth (Fig. ) and is a preferential site for marine observations (wind, wave, tide, physical and biogeochemical water properties are routinely retrieved), with a multi-parameter-measuring structure on board upgraded over the years. For this study, we have relied on a 30-year (1979–2008) data set of 3-hourly significant wave height Hs, mean wave period Tm and mean wave direction of propagation θm (measured clockwise from the geographical north), observed using pressure transducers. Preliminarily, data have been preprocessed in order to remove occasional spikes. To this end, at first the time series have been treated with an ad hoc despiking algorithm . The complete data set is therefore constituted of three variables and 50 503 sea states.

Basic statistics of the time series (Table ) point out that sea states at Acqua Alta have on average low intensity (〈Hs〉=0.62 m, where 〈-〉 denotes mean), though occasionally they can reach severe levels: the most intense event (Hs=5.23 m, Tm=5.36 s, θm=242∘ N) occurred on 9 December 1992 during a storm forced by winds coming from north-east. Such severe events are not frequent, as confirmed by the 99th percentile of Hs, which is 2.68 m. Nevertheless they populate the wave time series at Acqua Alta and constitute the most interesting part of the sample, for instance for extreme analysis. Mean wave period is on average 4.1 s, while mean wave direction is 260∘ N indeed most of the waves propagate towards the western quadrants.

This is represented more in detail by the histogram representing the PDF of θm (Fig. , bottom panel), which shows that the most frequent directions of propagation are indeed in the range 180 < θm < 360∘ N (western quadrants), with peaks at 247.5 and 315∘ N. Directions associated with the most intense sea states (Hs>4.5 m) can be obtained from the bivariate histogram (Hs-θm) representing the joint PDF of Hs and θm (Fig. , top panel): 247.5, 270 and 315∘ N. Mild sea states and calms (Hs<1.5〈Hs〉, following ) are the most frequent conditions at Acqua Alta, with 80 % of occurrence during the 30 years of observations. They mainly propagate towards the western quadrants too, though the principal propagation directions of such seas states is north-west. In this context, the most frequent sea states at Acqua Alta are represented by {Hs,θm}= {0.25 m, 315∘ N}. Storms in the area (denoted as sea states with Hs≥1.5〈Hs〉) are generated by the dominant winds, i.e., the so-called Bora and Sirocco winds . Bora is a gusty katabatic and fetch-limited wind that blows from north-east; it generates intense storms along the Italian coast of Adriatic Sea characterized by relatively short and steep waves. Sirocco is a wet wind that blows from south-east; it is not fetch limited and it generates longer and less steep waves than Bora, which come from the southern part of the basin. Denoted conventionally as Bora the events with 180≤θm≤270∘ N, and as Sirocco the events with 270<θm≤360∘ N, it follows that Bora storms have an occurrence of 12 % and Sirocco storms an occurrence of 8 %. The most frequent {Hs,Tm}, which occurred in the Bora and Sirocco quadrants, are shown in the bivariate (Hs-Tm) histogram (Fig. ) are {0.15 m, 3.6 s} and {0.35 m, 3.8 s}, respectively, Sirocco being the most frequent among the two. The associated marginal histogram (Fig. ) point out that Sirocco winds are responsible for most of the calms, in particular for sea states with Hs<1 m, while Bora for the most energetic sea states. Nevertheless, the histogram of Hs shows that Sirocco events with Hs in the range of 4–5 m can occur as well as Bora events. Bora is also associated with the shortest period waves observed: indeed, the histograms of Tm almost coincide for waves shorter than 5.5 s, while for longer waves the probability level of Bora mean periods abruptly drops to values much smaller than those of Sirocco (which remains to non-negligible levels until 9 s). The consequence of shorter and higher Bora waves, with respect to Sirocco, is steeper waves (3 % against 2 % on average, respectively).

In this section, results of the standard SOM approach (applied one time, hence called single-step SOM) and results of the different strategies proposed to improve extremes representation are presented. The performances of a single-step SOM, MDA-SOM and TSOM are assessed by comparing the wave parameters time series and their empirical marginal PDFs to the time series reconstructed from the results of the different strategies and relative PDFs, respectively. POT-SOM is treated separately because a direct comparison with the other strategies using the described methods is not possible.

A summary of the performances of the different SOM strategies is given in Table . There the single-step SOM, MDA-SOM with 80 % reduction and the TSOM with Hs threshold at 97th percentile are compared in their capabilities of representing the wave climate at Acqua Alta by means of the cells. The POT-SOM is not directly comparable to the other strategies since the data set used for the second map is composed of the storm peaks only. As done in the previous sections, the performances are assessed by comparing the reconstructed time series from each strategy with the original ones, and the resulting marginal PDFs with PDFs of the original data. However, here the performances are quantified by statistical parameters (see caption of Table  for nomenclature). Generally, the reconstructed time series are in agreement with the original ones, as shown by the high rav (over 0.98) and rSD (over 0.89), as well as high CC (over 0.95) and low RMSE (below 0.19 m for Hs, 0.37 s for Tm and 23∘ for θm). Nevertheless, the highest ratios and correlation coefficients, and the lowest RMSE pertain to TSOMs. Similar conclusions can be drawn for the PDFs, which are reproduced with very high CC (over 0.95) and RMSEPDF (below 0.04) by all the approaches, but to a greater extent by TSOMs. As expected, the most wide range variability among the different strategies concerns Hs. With the only exception of θm, whose widest range is provided by MDA-SOM, TSOM turned out to be the most efficient in providing the most complete representation among the tested strategies.

We verified that a higher size single-step SOM (e.g., 25 × 25, not shown here) can produce a wider range of extremes with respect to that used in the study (i.e., 13 × 13): the units' maximum Hs is 3.56 m instead of 2.75 m. In the same map configuration (i.e., 25 × 25), MDA preselection can further widen this range towards extremes: 3.63 m, the units' maximum Hs obtained with an 80 % reduction of the sample (using MDA); 3.66 m, the units' maximum Hs with a 40 % reduction. This has the effect of reducing the absolute error on 99th percentile of Hs (1 % with 80 % reduction and 11 % with 40 % reduction). However, the most extreme sea states are still far from being properly represented (we recall that the most extreme sea state observed had Hs=5.23 m). In addition and most importantly, if a larger number of elements in the map can improve the SOM performance shown in the paper, it will certainly worsen the readability of the map and the possibility of extracting quantitative information from the map. Indeed, considering, for instance, the 25 × 25 map, sea states at a site would be represented by 625 typical sea states: a huge number that is hardly manageable for a practical classification of the wave conditions.

An application of the TSOM is proposed to show that a more detailed representation of the extreme wave climate can enhance the quantification of the longshore component of the shallow-water wave energy flux P (per unit shore length), expressed as P=Ecgsin⁡αcos⁡α, where E=ρgHs2/16 is the wave energy per unit crest length (being ρ the water density), cg is the group celerity and α is the mean wave propagation direction measured counterclockwise from the normal to the shoreline. P is a driving factor for the potential longshore transport, and its dependence upon the wave energy E (which in turn depends on the square of Hs) suggests that an accurate representation of Hs is crucial. As the shoreline in front of Acqua Alta tower is almost parallel to the 20∘ N direction (i.e., orthogonal to the 290∘ N direction), the longshore transport is directed towards southwest when P is positive, and directed towards northeast when P is negative. Given the wave energy flux Ecg, P is maximized when α=±45∘ N, which correspond to θm=245∘ N and θm=335∘ N, respectively.

In order to obtain the shallow-water values of wave parameters, following , we propagated the Acqua Alta sea state resulting from the TSOM (see maps in Fig. ) from 17 to 5 m depth (a typical closure depth in the region), approximately accounting for the wave transformations, i.e., shoaling, refraction and wave breaking. In doing so, we assumed straight and parallel bottom contour lines, we neglected wave energy dissipation prior to wave breaking, and we allowed Hs to reach the 80 % of the water depth at most (depth-induced wave breaking criterion). Roughly, shoaling mostly affects the Sirocco sea states that are typically associated with longer wavelengths with respect to Bora sea states. In shallow water, refraction tends to reduce the difference between Bora and Sirocco directions with respect to Acqua Alta, as the normal direction to the shoreline, which waves tend to align to, is very close to the boundary (i.e., 270∘ N), which we assumed in order to discriminate between the two conditions. Sea states forced by land winds (20∘ N < θm<200∘ N) were excluded from the analysis.

The longshore component of the wave energy flux P at 5 m depth is shown in Fig. . It is worth noting that the left map represents the longshore component of the wave energy flux resulting from the single-step SOM technique (e.g., the left panel of Fig. ). Here, P ranges between -2 and 8 kW m-1, and the highest values are mainly due to Bora events that are responsible for potential longshore transport towards southwest (even if few Sirocco events with θm close to 270∘ N have the same effect). According to the left map, the transport towards northeast is due to Sirocco events that, however, cause less intense potential transport. The highest P values are associated with the highest Hs events, clustered on the cells at the top of the Fig.  left map. The right map of Fig.  describes the longshore flux component due to the Acqua Alta sea states represented by the SOM cells exceeding the 97th percentile Hs threshold (i.e., the six cells bounded by the black line in the left map). The range of P variation widens considerably when the extreme sea states are considered, with values ranging from -20 to 20 kW m-1. As observed in the right map of Fig. , the sea states exceeding the 97th percentile threshold on Hs are Bora and Sirocco events. The Bora events in the top-left part of the map (except for two cells in the bottom-right corner) contribute to positive, i.e., south-westward, transport, while Sirocco events in the bottom-right part contribute to negative, i.e., north-eastward, transport. The most intense transport is associated with the highest Hs cells at the bottom-left, bottom-right and top-right corners of the Fig.  right map. The major difference with respect to a single-step SOM estimate concerns the Sirocco sea states, associated with negative P, that had the most intense value extended from -2 to -20 kW m-1.

The mean longshore wave energy flux in shallow-water P‾, i.e., the average of P weighted on the frequencies of occurrence F over the 30 years of observations, was obtained by taking the absolute value of P from the maps of Fig.  and is 0.57 kW m-1 (Table ). In order to support this estimate, we compared the 1.71 kW m-1 estimate of the mean wave energy flux Ecg at Acqua Alta against the 1.5 kW m-1 value obtained at the same site over 1996–2011 by . The contributions to P‾ from Bora (P‾+) and Sirocco (P‾-) are 0.45 and -0.12 kW m-1, respectively, pointing out the predominant effect of Bora on the longshore transport over the western side of the Gulf of Venice. For comparison, P‾ was also computed using single-step SOM results (see Table ): in this case, P‾ is 0.52 kW m-1, P‾+ is 0.41 kW m-1 and P‾- is -0.11 kW m-1. Hence, with respect to the TSOM, the estimate of the mean longshore energy flux is 9.0 % lower for P‾, 7.5 % lower for P‾+ and 16.5 % lower for P‾-.

In this paper, we have tested different strategies aimed at improving the characterization of multivariate wave climate using SOM. Indeed, we have verified that besides a satisfactory description of the low/moderate wave climate (in agreement with usual uni- and bivariate histograms), the single-step SOM approach misses the most severe sea states, which are hidden in SOM cells with Hs even considerably smaller than the extreme ones.

For our purpose, we used the 1979–2008 trivariate wave climate {Hs, Tm, and θm} recorded at Acqua Alta tower, and we showed that, for instance, the single-step SOM assigned most of the sea states with Hs>2.75 m to the {2.75 m, 5.9 s, 270∘ N} class. Hence, the most interesting part of the wave climate was condensed within a few cells of the map, also hindering the distinction between Bora and Sirocco events, i.e., the prevailing meteorological conditions in the northern Adriatic Sea. To increase the weight of the most severe and rare events in SOM classification, we tested a strategy based on the pre-processing of the input data set (i.e., MDA-SOM) and a strategy based on the post-processing of the SOM outputs (i.e., TSOM). Results presented in the study showed that the post-processing technique is more effective than the pre-processing one. Indeed, a TSOM allowed a more accurate and complete representation of the sea states with respect to the one furnished by MDA-SOM, because it provided a wider range of the wave parameters (particularly Hs), and more reliable a posteriori reconstructions of time series and empirical marginal PDFs. Nevertheless, some deviations from original PDFs were observed and the range of θm was not complete, such that sea states traveling towards the north were not properly described. This requires further studies to improve SOM applications to wave analysis, which are rather promising, thanks to the well recognized visualization capabilities of SOMs. In this context, we proposed a double-sided map representation, which provides (on the left) a description of the whole wave climate that is particularly reliable for the low/moderate events and is completed (on the right) by the description of the extreme wave climate. This novel representation was also employed to provide a SOM classification of the storms peaks, based on the peak-over-threshold approach, on the right (POT-SOM).

Finally, a TSOM was applied for the assessment of the potential longshore wave energy flux to show how practical oceanographic and engineering applications can benefit from this novel SOM strategy. Indeed, the mean flux in front of the Venice coast was found to be 9 % higher if evaluated after a TSOM instead of a SOM.