A sea state is a statistical description of the sea surface waves generated by wind at a given time and location. The sea state is characterized by a superposition of wind seas and swells (). The local wind generates wind seas, whereas swells are generated in distant areas. Significant wave height (Hs), defined as 4 times the zeroth moment of the wave power spectrum, is commonly used to describe the sea state. Thus, Hs is an essential measure of wave height and provides information about the wave energy of a given sea state.

High-quality wave data are essential for many marine applications, such as designing coastal and offshore structures and planning marine operations. Observational, numerical, and statistical models are the methods used for sea-state characterization. Traditional in situ measurements obtained from buoys provide the most reliable data for sea-state parameters; however, they are only available for the last decades and are limited spatially . Numerical models provide simulations of spectral wave models from which sea-state parameters are extracted. They are a valuable source of data and provide decades of records, although they are computationally expensive. Statistical models constitute an alternative to numerical models for constructing the wind–waves relationship. These models are not computationally expensive, and once the statistical relationship is estimated, future predictions can be made by assuming that this relationship is stationary, meaning that the statistical relationship between the large-scale variables and the sea-state parameters does not change between the present and the future .

Various studies have compared statistical and numerical models for ocean wave parameters and other climate variables. compared these methods in terms of climatological characteristics of the present period using ERA-40 wave data. They found that the statistical models are better at reproducing the observed climate than the dynamical models. analyzed these methods for climate projections, and their study shows that statistical downscaling (SD) approaches can reproduce the present climatology and future projections. In addition, due to their low computational complexity, SD models permit the consideration of a wide range of global climate models (GCMs) and climate scenarios, which allows us to estimate the uncertainties. However, modeling the relationship between wind and sea-state parameters using statistical methods still presents some difficulties, which have been addressed by different methods in the literature, namely:

Waves depend on both local and global wind conditions. The surface wind generates wind waves. However, it is not only the local wind that defines local waves, and wind from distant regions generates swells that may reach the target point . Therefore, SD models have to consider both wind sea and swells, which is particularly challenging in swell-dominated areas . To address this issue, we use a local and a global predictor to account for wind sea and swells, respectively, as already done in and .

Wind conditions are multicollinear and multidimensional. The wind conditions are characterized by two components (zonal and meridional), which might be challenging to consider directly in a statistical model. To address the issue of multidimensionality in this study, we introduce the wind projection, which consists of retaining only the fraction of wind blowing towards the target point. The proposed preprocessing step allows the use of only one variable for each grid point, reducing the dimension of the predictor by half. Furthermore, large-scale wind variables are high-dimensional and multicollinear (strong correlations among variables) due to the strong spatiotemporal dependence of wind fields, and using them as a predictor in a statistical model might be challenging. Dimensionality-reduction methods such as principal component analysis are typically used as a preprocessing step to reduce the dimension of the large-scale variables and deal with multicollinearity . In this study, we use ridge regression , which has been proven to be beneficial for dealing with multicollinearity in various studies .

The relationship between wind and waves is not instantaneous. Wind from distant regions generates waves that may take days to reach the target point. Thus, the relationship between wind and waves is not instantaneous. Therefore, it is necessary to consider lagged wind conditions to understand the wave dynamics at a particular target location. The optimal lag at each grid point is interpreted as the travel time required for the waves to reach the target point . ESTELA (Evaluation of Source and Travel-time of wave Energy reaching a Local Area) is a method that defines the wave generation area and wave travel time at any ocean location worldwide. Using its spectral information, the method selects the fraction of energy that travels to the target point from selected source points. The ESTELA method was used in various studies to define the temporal coverage of predictors used in SD . The present study uses a statistical approach to define the wave generation area. It is based on estimating waves' travel time from each source to the target point (optimal lag) using the maximum correlation between the significant wave height and wind conditions. Therefore, this method is not computationally expensive, and only wind and Hs data at the target point are required, and unlike ESTELA, no spectral data are needed.

This study presents a statistical approach for estimating the relationship between wind conditions and ocean waves. The approach is based on weather types, which are constructed using a regression-guided clustering algorithm. These weather types are then used to link the space–time wind fields over the North Atlantic (predictors) and the significant wave height (predictand) at three locations: northwest and southwest of the French coast and the English Channel. Then, regression with ridge regularization is used to fit the relationship between wind conditions and significant wave height at each weather type. The proposed methodology considers wind sea and swells and provides additional information about the spatiotemporal relationship between wind and waves. The main contribution of this work is that it provides an entirely data-driven approach for estimating the travel time of waves from any source point to a target point, which is essential for the definition of predictors. To the best of our knowledge, the only other approach in the literature that can be used for this purpose is ESTELA , which relies on directional spectra over the spatial domain of the wave generation. These spectra are not always accessible and can be computationally costly and demand high storage capacity. Our proposed approach, however, utilizes wind fields and significant wave height at the point of interest, which are more accessible and less computationally costly. Additionally, it allows for the processing of buoy data or data from models with limited spatial coverage, as it does not require the directional spectra over the wave generation domain. Furthermore, this study proposes a relatively interpretable model with a limited number of weather types and uses ridge regularization . The regularization is used to make the parameters of the regression more interpretable and to improve the generalization capability of the model.

This paper is structured as follows. After describing the data in Sect. 2, the local predictors are defined in Sect. 3. Then, Sect. 4 describes the construction of the global predictors. Next, Sect. 5 presents the statistical model that combines the local and global predictors. Then, Sect. 6 presents the results of the SD model. Finally, the study is concluded in Sect. 7.

The atmospheric data used in this work to construct predictors are extracted from the Climate Forecast System Reanalysis (CFSR) . CFSR is a global reanalysis developed at the National Centers for Environmental Prediction (NCEP) that covers the period from 1979 to the present with an hourly time step and spatial resolution of 0.5∘ by 0.5∘. Extracted data consist of hourly 10 m zonal and meridional wind components in the North Atlantic (Fig. ).

To comprehensively evaluate the method across a range of observed sea states, we consider three different locations: Gironde (45.2∘ N, 1.6∘ W), the English Channel (49∘ N, 4.4∘ W), and the Gulf of Maine (43∘ N, 69∘ W). The Gironde location is nearshore, with the highest Hs observed at the considered period being 10.5 m, whereas in the English Channel location, which is offshore, the highest Hs observed is 11.8 m. In addition to the two eastern Atlantic locations, a location in the Gulf of Maine is situated in the western part of the Atlantic, where the maximum Hs reached 9 m. The bathymetry of the Gulf of Maine is highly complex, with a coastline dotted with numerous bays, islands, and coves .

The historical wave data used in this work for the Gironde and English Channel locations are the sea-state hindcast database HOMERE based on the WAVEWATCH III^® model forced by CFSR wind. The database covers the English Channel and the Bay of Biscay with unstructured computational mesh. It contains 37 parameters and the frequency spectra on high spatial resolution, ranging from 200 to 10 km, with a 1 h time step. For the Gulf of Maine location, we consider the IOWAGA database , which is also based on the WAVEWATCH III^® model forced by CFSR and ECMWF wind. To validate and interpret the results of the SD method, we consider the energy spectral partitioning, which identifies different wave systems. HOMERE uses the watershed algorithm to separate wind sea and different swells.

The temporal resolution of both predictors and predictand is upscaled from hourly to 3 h resolutions to facilitate the analysis. Both datasets comprise a common period of 26 years, from 1994 to 2019. The 1994–2013 period is used as the calibration period, while the 2014–2019 period is used as a validation period.

Wind speed, duration, and the fetch impact the characteristics of the wind sea . Hereafter, at time t, the variables U(t), F(t), U(t-1), and F(t-1) are considered to construct the local predictors. U(t) is the wind speed at the target point, and F(t) is the fetch length at time t, calculated as the minimum of the distance from the target point to shore in the direction from which the wind is blowing and 500 km. A minimum distance of 500 km is fixed because it is computationally expensive to calculate the distance between the target point and far away shores such as eastern American shores. Note that F(t) is not but is related to the fetch in the literature, which is defined as the distance over which waves develop . Lagged wind conditions are considered because they may provide information about the temporal variability of the wind and thus the duration of wind conditions.

To investigate the capability of local variables to explain Hs, the polynomial regression model, 1Hs(t)=β0(ℓ)+X(ℓ)(t)β(ℓ)+ϵ(ℓ)(t), is considered. Where X(ℓ) is the local predictor, 2X(ℓ)(t)={U(t),U2(t),U3(t),U2(t)F(t),U(t-1),U2(t-1),U3(t-1),U2(t-1)F(t-1)}, β0(ℓ) and β(ℓ) are model coefficients, and ϵ(ℓ)(t) is the model error. Model () contains polynomial terms and interactions between the fetch and squared wind to consider nonlinear relationships between Hs and predictors. We considered taking into account the lagged local wind conditions up to t-4; however, the performance of the model does not change as much as taking into account only the t and t-1 wind conditions.

The model is fitted using data from 1994 to 2013 and is assessed in a validation period from 2014 to 2019 using the Pearson correlation r, root mean square error (RMSE), and bias: 3r=∑t=1n(H^s(t)-H^s‾)(Hs(t)-Hs‾)σH^sσHs4RMSE=∑t=1n(H^s(t)-Hs(t))2n5BIAS=∑t=1n(H^s(t)-Hs(t))n, where H^s(t) is the predicted Hs at time t; H^s‾ and Hs‾ are the mean of observed and predicted Hs, respectively; σH^s and σHs are the standard deviation of predicted and observed Hs, respectively; and n is the number of observations considered (58 440 for the calibration period and 17 528 for the validation period).

Results of the local model as a function of the peak period of the three considered locations are shown in Fig. . The model strongly underestimates high-period waves in all three locations and better predicts small-period waves. The high-period waves could correspond to swells that are generated far from the target point. Therefore, in order to predict Hs at these locations, it is important to consider the large-scale wind conditions that cover the swell generation as well as the local-scale wind conditions.

In order to take swells into account, a global predictor which describes wind conditions over the North Atlantic has to be considered. Wind data have two components, the zonal and meridional components. Each of the two components in space and time carries more or less information about the waves observed at the target point at a given date. However, using all of them as inputs to a statistical model is computationally challenging, given the high dimensionality of the data and may lead to hardly interpretable results due to the strong correlation between wind conditions at closed locations in space and time. This section defines the global predictor related to the spatiotemporal domain of the wave generation area.

In this section, the methodology's results are presented. As for the last section, the results of the weather types were found to be consistent across the three studied locations; therefore, only the results for Gironde will be displayed. Subsequently, the overall methodology results for all three locations will be provided (see Fig. ).

The clusters obtained in the last section seem to be interpretable and correspond to sea-state classes obtained from the energy partitioning algorithm provided by HOMERE (accuracy =0.87). However, the number of clusters K may be greater than 2; therefore, a validation analysis is done to select the optimal number of WTs. To do that, for each number of WTs (from 1 to 8), model () is fitted using the calibration period and evaluated using the validation period. Figure  illustrates the RMSE of Hs as a function of the number of WTs. The RMSE is stabilized for a number of WTs greater or equal to 5, and the RMSE decreases significantly from 1 to 5 WTs. We therefore chose the number of WTs to be 5.

Figure  shows the time series of Hs and its empirical density as a function of the five WTs. Note that the weather types were manually arranged in ascending order according to the magnitude of Hs. The resulting WTs depend on the value of Hs; for example, the first WT corresponds to small values of Hs, and the fifth corresponds to extremes. In increasing order, the other clusters (2 to 4) correspond to intermediate values of Hs. The bottom right panel of Fig.  shows the frequency of occurrence of WTs. The first WT is the most likely, and the fifth one has the smallest probability of occurrence. The transition matrix in the bottom left panel shows that the self-transition probabilities are greater than 0.9 for all WTs, meaning that the WTs are consistent in time. Note that some transition probabilities are precisely zero; for example, the transition probabilities from the first to the fourth and the fifth WT are equal to zero. This means that the probability of being in extreme sea states after being in the first WT is zero.

Figure  shows the mean of Xβ(g) at each WT where 17xβ(g)(i)=X‾β(g)(i)-X‾β(g),i=1,..,5, where X‾β(g)(i) is the mean of Xβ(g) at the ith WT, and X‾β(g) is the global mean of Xβ(g). For the first and second WT, contributions of source points far from the target points are greater than the global mean. Therefore, these two classes may correspond to swells. In the third WT, the local wind contributes more, with moderate winds, in the variance of Hs. The fourth one can be considered as a composition of wind sea and swells given that local and far source points contribute to the variance of Hs. Finally, the fifth WT corresponds to the wind sea, where the local source points contribute with the highest intensities of winds creating the highest waves.

The monthly variability of WTs is shown in the left panel of Fig. . As expected, the fifth and fourth WTs occur primarily in winter (December–January–February), and the first WT, which corresponds mainly to swells, often occurs during summer. The long-term winter variability of frequency of occurrence of WTs is shown in the right panel of Fig. . The continuous black line corresponds to the mean annual winter of NAO index from 1994 to 2019. The horizontal black line indicates when NAO is greater or less than zero. Figure  suggests that there is a correlation between the long-term variability of weather types and the North Atlantic Oscillation (NAO) index. For instance, the winter of 2010 exhibited a lower frequency of extreme waves and corresponded with a low NAO index, while the most extreme sea states were observed in 2014, which coincided with a high NAO index. This correlation is consistent with previous research, such as in , which reported that winter waves tend to have larger significant wave heights (Hs) during the positive phase of the NAO and smaller Hs during the negative phase of the NAO.

The results of the model described in Eq. () during the validation period for the three studied locations are presented in Figs.  and  for Gironde. The model performs well in predicting Hs at Gironde and the English Channel. However, it exhibits less accuracy in the Gulf of Maine, which can be attributed to the complex wave climate in this area due to factors such as bathymetry, islands, breaking waves, and frequent storm activity . Comparing these results with those of the local model in Fig. , it appears that considering the global predictor is essential to explain the variability of Hs. Figure  illustrates the performance of the downscaling model at each weather type in the validation period. It can be seen that the model in WT 1, 2, and 4 explains less of the variability of Hs compared with the model in WT 3 and 5. This might be explained by the fact that in these WTs (1, 2, and 4), the model has to consider source points that cover the swell generation, as seen in Fig. , which means that numerous source points contributes to the variability of Hs. In contrast, in WT 3 and 5, waves are mainly generated by local wind (Fig. ); therefore, the model considers mainly local source points.

This study proposes a method that describes the spatiotemporal relationship between wind and the significant wave height (Hs). At first, the local model, based on a linear regression between the local wind and Hs, is constructed. However, the model poorly explains the variability of Hs given that the model does not consider the swell generation. Therefore, the global predictor was defined to account for both wind sea and swells. The global predictor is based on the target-projected wind, which is the wind that goes from source points to the target point in a great circle path. After wind projection, the spatial coverage of the predictor is defined based on the assumption that waves travel along a great circle path. Then its temporal coverage is defined based on two parameters, the travel time of waves and the temporal width. Both parameters exhibit spatial structure and increase as the distance between the source and target points increases.

The statistical downscaling model combines the local and global predictors to predict Hs using a weather-types-based model. The weather types were constructed using a regression-guided clustering algorithm. The comparison between the HOMERE sea-state classes (wind sea and swell) and two clusters obtained by the clustering algorithm shows a significant resemblance. The predictive model consists of fitting linear regression with ridge penalty between the predictors and the predictand in each WT, and the validation analysis shows that the optimal number of WTs is five. The obtained weather types are interpretable and correspond to different wave systems. The results of the downscaling model show its skill in predicting Hs, even for large values, which are often important for operational purposes. The proposed method can be used for various operational applications depending on the availability and quality of wind data. These applications include hindcasting, short-term forecasting, and climate projections. In the case of climate projections, the method can use bias-corrected output from global climate models (GCMs). The statistical method presented in this study is validated at three locations, and its ability to accurately predict Hs is demonstrated; the method may be therefore generalizable and can be extended to other locations. For locations nearshore, it may be necessary to take into account other local characteristics such as bathymetry and currents, as these factors can significantly impact wave behavior. In addition, the assumption of great circle wave propagation may not be valid for these locations, and alternative wave propagation models may be considered. Another limitation of the proposed methodology is its limited scope in predicting only significant wave height (Hs). To fully characterize a sea state, other parameters such as wave period and direction should also be taken into account. Future research should investigate the generalizability of the methodology to other sea-state parameters.

In this paper, we introduced a methodology based on observed weather types, constructed prior to the regression problem using a clustering algorithm. For future research, these weather types could be treated as latent variables within a mixture regression framework, which can be estimated using the expectation maximization (EM) algorithm. This approach would evaluate variables according to Hs predictions, which can yield to statistically optimal estimates.