The Climate Forecast System Reanalysis (CFSR) is a hindcast database (Saha et al., 2010). Wind and other environmental variables (humidity, pressure, etc.) are recomputed with a homogeneous model and data assimilation system to eliminate fictitious trends caused by model and data assimilation changes in real time. The main purpose of this type of treatment is to obtain consistent datasets for the study of climate.

Only 10 m wind components U10 and V10 are extracted with a degraded space–time resolution of 3 h for the time resolution and half a degree in latitude and longitude for the spatial resolution. The predictors of the proposed method are derived from these components. Their construction is detailed in Sect. 3.1.

Predicting wave climatology is particularly complex due to the random nature of the ocean, the lack of data, and the difference in scale between the phenomena. Numerical wave prediction models are powerful tools to address this problem, based on the physics of the phenomenon (Thomas and Dwarakish, 2015). WAVEWATCH III is a spectral-phase average wave model based on a discretization of the energy balance equation with finite differences in time, as follows: 1∂Ek∂t+∇⋅(cgEk)=Sink+Snlk+Sdis(k), where k is the wavenumber and Ek the energy spectrum. The left-hand side constitutes a local term and an advective term moving at the group velocity, cg. The energy supplied by the wind accounted for in the source term Sin is redistributed via the non-linear wave interaction term Snl. The dissipation term Sds addresses the energy loss due to wave breaking caused by high wave steepness (Ardhuin, 2021). As it is a phase average model, WAVEWATCH III cannot simulate diffraction phenomena.

WAVEWATCH III's model accuracy depends on the one of the forcing fields and parameterization of the source terms and on the effect of the numerical schemes (Roland and Ardhuin, 2014). In deep water, the predominant forcing is the wind, followed by currents and water height. This order is reversed in coastal areas.

Hs and other wave parameters, such as Tp and wave direction, are directly estimated from the energy spectrum Ek. For reasons of memory limitation, the spectra are only stored for a few nodes. For the remaining nodes, only the wave parameters are available in the model output.

In this study, data from two hindcast databases using the WAVEWATCH III model are exploited as follows:

HOMERE. This specific hindcast database has been developed by Ifremer (French National Institute for Ocean Science) for the “assessment of sea-state climatologies to support research and engineering activities related to the development of marine renewable energies over the Bay of Biscay and the Channel” (Boudière et al., 2013). This model takes realistic sea bed roughness mapping into account. The forcing considered includes CFSR wind, water levels, and tidal currents computed using the MARS 2D hydrodynamic model (Lazure and Dumas, 2007). The HOMERE mesh consists of an unstructured grid built from the depth and wave propagation velocity. The nodes are spaced up to 10 km offshore and only 200 m near the coast.

In situ data are difficult to use in statistical downscaling models. Indeed, deep learning models usually require a large amount of data to be fitted. In situ data often suffer from too short a recording (less than a decade and missing values). Sea state data from satellite altimetry are particularly complex to use because of their sparsity. Buoys provide high-frequency wave measurements at their location. However, buoys with a sufficiently long and continuous history are rather rare, and their spatial repartition is sparse.

The idea of a ground truth can be misleading, and there are challenges in using existing in situ wave measurements. Indeed, data from physical measurements are noisy, which may induce bias and extra variances in the models. Moreover, a buoy estimates the significant wave height from its own motion. Thus, the characteristics of the buoy, such as its size, composition, or structure, as well as the characteristics of the sensor and the processing chain, can alter the estimation of Hs (Ardhuin et al., 2019). Several studies have quantified these measurement errors. For example, some buoys overestimate wave heights by 56 % in hurricane conditions, while measurements from light vessels have a systematic bias, causing an underestimation of high waves (Anderson et al., 2016). In this study, in situ data from buoys have been chosen because they remain the most suitable for the learning method we have developed.

The predictors introduced here have been proposed in Obakrim et al. (2022) for a linear statistical downscaling model for the significant wave height. They are based on some simplified physics rules. In particular, the deep-water hypothesis (depth ≥ wavelength/2) is assumed to be true. At this depth, the energy of the wave is reduced by a factor of exp⁡π≈25 compared to the surface (Ardhuin, 2021). Under this hypothesis, the wind can be considered as the only forcing source.

At a given time, the wave field is the combination of the swells generated over long distances, often several days before, and wind sea waves generated by the local wind. It is, therefore, important that the predictors consider the complex phenomenon of wave generation and the non-instantaneous and non-local relationship between wind and waves. Therefore, two sets of predictors are needed, i.e., a local and global predictor, denoted, respectively, as Xl and Xg in the sequel. This approach has already been tested by taking the pressure field as the forcing (Camus et al., 2014). Unlike pressure, wind is represented by two components, i.e., zonal and meridional. Assuming that waves travel along a great circle path (Pérez et al., 2014), wind can be reduced to a single component by projection into the bearing of the target point. Many CFSR wind sources are then masked by continents or islands. Fig. 1 shows the remaining wind sources for the station selected in the Bay of Biscay (see Sect. 4).

The global predictor component, Xjgt;tj^,αj^=Wj2‾t-tj^-αj^:t-tj^+αj^, is defined as the mean of the squared projected wind, Wj2, on the time window [t-tj^-αj^:t-tj^+αj^] at location j.

The estimated temporal width αj^ controls the length of the time window, and tj^ estimates the wave travel time. These last two parameters are estimated for each point by maximizing the correlation between the Hs at the target location and the predictor Xjg(tjαj): tj^,αj^=corr(Hs,Xjg(tj,αj)). The parameters fields obtained by this method are shown in Fig. 2.

The local predictor Xl is defined as follows: 2Xlt={UtU2tU3tU2tFtUt-1U2t-1U3t-1U2t-1Ft-1, in which U is the speed of the local 10 m wind, and F is the fetch. The fetch is the length of the region over which the waves have been generated. The fetch can be difficult to define precisely in open sea (Ardhuin, 2021). In this study, it is defined as the minimum between the distance to the nearest coast in the wind incidence azimuth and 500 km.

Deep learning techniques are promising approaches for statistical downscaling because of their ability to learn spatial features from huge spatiotemporal datasets. The model presented in this paper is a convolutional neural network (CNN). It is inspired by the CNN-PR (pattern recognition) network, using standard topologies from pattern recognition for precipitation downscaling presented in Baño-Medina et al. (2020).

The use of CNN for the statistical downscaling of the sea state has been motivated by properties of CNN that are interesting for the prediction of wind generated waves, as follows:

Shared-weight architecture of kernels. For each filter of the convolution layer, it is the same kernel that runs through the input matrix. Thus, when a meteorological event of the same magnitude appears at one position or another on the projected and time-shifted wind map, the result of the convolution operation on the output layer will be the same. CNN are equivariant in translations (Mouton et al., 2020).

Sea state is the result of the combination of wind sea wave and swell. Thus, the model needs to take both wave systems into consideration. This is realized by implementing a hybrid CNN model, HCNN, which takes two inputs, i.e., the global and local predictor. Figure 3 shows a schematic view of the network architecture.

This neural network has been implemented with the Python deep learning library Keras (https://github.com/keras-team/keras, last access: 5 October 2021). The different layers used in the model are as follows:

ZeroPadding2D. This adds rows and columns of zeros at the top, bottom, left, and right side of the input to conserve its spatial dimensions (height and width) during the convolution operation.

The local predictor is normalized before being fed into the neural network to improve numerical stability of the model and its computational efficiency (Shanker et al., 1996). The Adam optimizer is a stochastic gradient descent algorithm in which individual adaptive learning rates are computed for different parameters from the estimates of the first and second moments of the gradients. It was chosen because its hyperparameters have an intuitive interpretation and typically require little tuning (Kingma and Ba, 2015). In addition, the Adam optimization algorithm can handle sparse gradients on noisy problems. The selected loss function is the mean squared error (MSE).

To evaluate the performance of downscaling methods, classical metrics are used. T and P denote, respectively, the value of the target Hs and the predicted one. The cardinal of the time series is n.

The root mean squared error (RMSE) is the measure of the accuracy, as follows: 3RMSE=∑T-P2/n. The Pearson's correlation coefficient is the linear correlation between two datasets, as follows: 4ρ=∑[(Ti-∑Ti/n)(Pi-∑Pi/n)]∑(Ti-∑Ti/n)2∑(Pi-∑Pi/n)2. The scatter index is the prediction error normalized by mean of observations and debiased, good metric to quantify the dispersion of the scatterplot and to compare performances at different stations, as follows: 5SI=∑T-∑Tin-P-∑Pin2/n∑Ti/n. The RMSE of errors greater than the 95th quantile of the error is as follows: 6ERRABS=ABS(T-P)7RMSEQ95=∑ERRABS[ERRABS>95∘percentile of ERRABS]2/n. In addition to these statistical indicators, the scatterplot and the quantile–quantile plot are presented. The scatterplot allows one to observe the deviation of the prediction from the identity function. The quantile–quantile plot allows one to observe the adequacy of the distribution of the predicted values with the reference dataset.

The performance of the HCNN model is evaluated through seven-fold cross-validation. To do so, the dataset is divided into seven periods of 3 consecutive years. Each of these periods is then successively taken as a validation dataset, with the rest of the data being used for learning. For each of these iterations, five models are trained, and the grand mean of each score is computed on each fold. The training of five individual models is a robust approach to evaluating the skills of deep learning models that are stochastic because of their use of randomness during training, particularly because of the initialization to a random state and the use of stochastic gradient descent algorithms.

According to Fig. 5, HCNN gives results of the same order of magnitude over each validation period. The variations observed are justified by the differences in climatology specific to each period. The standard deviation of the HOMERE Hs over the validation period (see Fig. 6) is indeed correlated with the RMSE (ρ: 0.65).

Since the performances of the HCNN model are homogeneous on each of the folds, only the last one is kept to conduct an in-depth study of the performances of the model. The selected training and validation datasets are as follows:

Train – 13 January 1994 to 31 December 2013 (size of 58 341)

The results over the test period are satisfactory. The RMSE is 27 cm, which is 14 % of the mean value of the target Hs.

The plots are consistent with the indicators previously outlined. The scatterplot shows a rather weak dispersion around the diagonal line. Moreover, there is no outlier (point very far from this line). The quantile–quantile diagram shows that the distribution of Hs is well reproduced.

The significant wave height derived from the energy spectrum gives an overall description of the sea state. However, it has been highlighted that a sea state is generally made of wind waves caused by local winds and several swell trains propagated over large distances. To better understand the performance of the method, it is interesting to know how much of these different components are predicted. Energy spectrum partitioning methods have been developed to identify these components. In the output of HOMERE, the partitioning is obtained using the WaveSEP (Wave Spectrum Energy Partitioning) method, which is based on a watershed algorithm (Tracy et al., 2007).

The developed neural network is unable to determine this partitioning. Yet, it is possible to use the classification of wave systems described in Maisondieu (2017) to analyze the performance of the HCNN model. The following six possible combinations of sea states are identified in increasing order of complexity: wind sea wave, swell, wind sea wave + 1 swell, 2 swells, wind sea wave + 2 swells, and at least 3 swells. The following table describes the proportion of each sea state during the test period defined above.

The following bar plot shows the value of the four statistical indicators previously introduced, plus the bias, for each of the six possible combination of sea states.

According to Fig. 8, RMSE and RMSEQ95 are higher for wind sea waves and are almost identical for the other sea states. However, the correlation decreases and the scatter index, which gives similar information, increases with the complexity of the sea states. There are two hypotheses that can explain this result. On the one hand, the chosen cost function (MSE) penalizes the errors indifferently for low and high Hs. However, as the complexity of the sea state increases, the average Hs decreases, as shown in Table 2. Since the value of SI is normalized by this average value, it is therefore normal that it increases. On the other hand, this evolution of the error may also be explained by the non-linear interactions between the different wave systems; the more complex the sea state, the more difficult the prediction will be. It is also interesting to observe the bias value which shows an overestimation of complex sea states and an underestimation of unimodal sea states characterized by higher Hs. Finally, RMSEQ95 shows that the most significant errors are associated with wind sea waves. This is probably due to the important simplification made on the calculation of the fetch, which is an important parameter for the generation of wind sea waves.

The previous results were obtained using all available history and all points that are not masked by an island or continent. However, it is reasonable to assume that, beyond a certain distance, the waves generated by the wind do not reach the target location (or only a small fraction of their energy) and that the model does not need such an amount of data. To determine the width of the spatial and temporal window, several models are trained by gradually widening them. The two RMSE indicators are used to quantify the evolution of the prediction error. Due to the geometry of the global predictor (see Fig. 1), the search for the spatial window is simplified to the search for the optimal range of longitudes, which is hereafter referred to as Δ longitude. Since France is a natural barrier east of the Bay of Biscay, only the western longitude limit has been adjusted.

The plot in Fig. 9 (top left panel) shows that the accuracy of the prediction does not improve by taking winds located at more than 50∘ longitude, which corresponds approximately to the longitude of Saint Pierre and Miquelon. The error increases even after this distance; points that are too far away bring more noise than information, which degrades the prediction.

In the second figure, one can observe that the prediction improves by increasing the learning period. However, a few years are enough to obtain sufficiently accurate predictions (RMSE less than 30 cm, with 5 years of training). Moreover, the slope of the RMSE curve becomes less steep after about 10 years. The considered learning period must also be long enough to cover the variations in the ocean–atmosphere regime over the region. Concerning the Bay of Biscay, 10 years seems to be a sufficient period to observe positive or negative North Atlantic Oscillation (NAO) phases.

The two methods used for the comparison were developed by our team and use the previously introduced predictors.

Both models were evaluated on the station presented in the previous section. The results of the three statistical downscaling methods are summarized in Table 2.

The HCNN method gives better results for the four selected indicators. Next comes the linear regression and then the analog method. It should be noted, however, that while the HCNN method outperforms the linear model, it is much less interpretable. There is a tradeoff between interpretability and performance. The analog method, even if it is the conceptually simplest of the three, obtains performances quite close to the linear model.