Short-Term/Range Extreme-Value Probability Distributions of Upper Bounded Space-Time Maximum Ocean Waves

There is general consensus that accurate model predictions of extreme wave events during marine storms can substantially contribute to avoiding or minimizing human losses and material damage. Reliable wave forecasts and hindcasts, together with statistical analysis of extreme conditions, are then of utmost importance for monitoring marine areas. In this study, we perform an analysis of the limitations of the available short-term/range extreme-value distributions suitable for space-time maximum wave and crest heights. In particular, we propose an improvement of the theoretical distributions by including upper bounds on the maximum heights that waves may reach. The modification of the space-time probability distributions and its impact for extreme-value assessment is discussed in the paper. We show that unbounded space-time distributions are still effective provided that the surface area included in the analysis has sides smaller than O(102 m). For wider surfaces, the use of the bounded distributions is consistent with the expected saturation of maximum heights that ocean waves attain.


Introduction
The characterization of maximum wind-wave heights during marine storms has been an active topic of research for decades because of its importance for marine safety, coastal hazards, offshore design and operations. Significant efforts have been undertaken to better understand the likelihood and amplitude of extreme events, including rogue and freak waves [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Present strategies, however, resulted sometimes ineffective in warning seafarers or avoiding structural damage to offshore facilities (see e.g., [6,15,16]). Theoretical progress has been accompanied by numerical model improvements, so that currently most of the state-of-the art phase-averaged wave models, used for forecast and hindcast numerical studies, are equipped with routines that provide estimates of wave extremes (namely maximum crest height and maximum wave height) during storms [17][18][19][20].
The condition that supports the estimate of extreme sea waves is the steady-state (in time t) and homogenous (on the two-dimensional xy-space) sea condition, and the derived statistics is referred to as short-term/range (time interval from minutes to 1 h or so and spatial distance up to a thousand meters). In the temporal domain t only, the short-term extremal probability pertains to sea surface elevation η(t) time series at a fixed point x 0 = (x 0 , y 0 ) on the water surface, where x and y are the two Cartesian horizontal axes. Starting from the Rayleigh distribution [21], it has been established that nonlinear second-order bound waves and four-wave nonlinear interactions have a profound impact on the statistics of extreme events over η(t) [2,12,22,23]. More recently, following the works of Adler [24] and Piterbarg [25] for multidimensional manifolds, the concept of excursion probability was introduced The short-term/range excursion probability Pr{·} of second-order nonlinear maximum crest heights C m belonging to wave groups crossing a two-dimensional sea surface area A = XY (X and Y are the area sides along the two orthogonal directions x and y, respectively) and time duration D was here approximated after Adler [40], and following Fedele [26], Baxevani and Rychlik [29] and Benetazzo et al. [32] as follows where the elevation threshold h is normalized with the standard deviation σ of the sea surface elevation field η(x, y, t), and is the solution of the Tayfun equation [37] that relates the second-order nonlinear dimensionless threshold (h) to its linear counterpart (h 0 ) via the steepness parameter µ > 0 [23]. In Equation (1), N 3 is the average number of waves within the space-time region Γ of three-dimensional volume V = XYD, N 2 is the average number of waves on the two-dimensional faces of Γ, and N 1 is the average number of waves on the one-dimensional edges of Γ. These numbers of waves depend on the spectral moments through the definition of the mean zero-crossing period, the mean zero-crossing wave length, the mean wave crest length, and a measure of the irregularity of the space-time sea surface elevation field [26,38]. Assuming the excursion probability in Equation (1) is continuous, its asymptotic Gumbel limit for high values of h is easily found to have a cumulative distribution function (cdf) as follows [26]: where P represents the probability that the random variable C m takes on a value less than or equal to σh. The Gumbel variable z (that incorporates location and scale parameters) is written as and ξ 0 is the dimensionless mode of the linear part of the extremal probability in Equation (1). By differentiating, the probability density function (pdf) of the Gumbel distribution of the random variable C m is given by f Cm (h) = dF Cm (h)/dh (5) and the expected value of C m is calculated as follows where E{·} denotes expectation. In summary, the extreme-value statistical model based on the Adler approach and the Tayfun theory allows finding a short-term/range extreme-value statistics for second-order weakly nonlinear crest heights C m belonging to space-time wave groups. Unlike C m or in the temporal domain [39,41], in the two-dimensional spatial and three-dimensional spatio-temporal domain, an analogous distribution for maximum crest-to-trough wave height H m does not have a closed-form solution [26,35], given that the generalization to high-dimension excursion sets of the notion of wave profile, from which to infer the crest-to-trough vertical distance, is ambiguous. Notwithstanding, we have used the linear QD model by assuming it as to be effective for each realization of a space-time wave group holding maximum waves. In this case, by imposing µ = 0, the Gumbel variable in Equation (4) loses its nonlinear part (therefore h = h 0 ) and z simplifies to [26] The Gumbel-like cumulative distribution function for space-time linear maximum crest heights can be then approximated as where "0" in the subscripts indicates linear theory prediction. The pdf of C m0 can be analytically inferred by derivation as follows f Cm0 (h) = dF Cm0 (h)/dh (9) We then made use of the linear QD theory by imposing a change of variable of the type H = αh, where the coefficient α = 2(1 − ψ * ) was purposely used to relate the maximum crest and wave heights in sea states with finite bandwidth, which is specified by the first minimum ψ * ∈ [−1, 0) of the sea surface elevation autocovariance function [39]. The pdf of maximum wave heights H m was thus obtained from that of maximum crest heights as follows The expectation of the random variable H m is given by It is worth noting that this result is consistent with the prediction of the QD theory that yields an equality for mean quantities, i.e., E{H m } = αE{C m }, as corollary of the relationship that exists between the space-time wave group of the maximum expected crest height and the wave group of the maximum expected crest-to-trough height. To complete this section, we recall, and it was used in the following analyses, that all parameters of the extreme-value probability distributions may be obtained using higher-order moments of the directional wave spectrum [26].

Extreme-Value Distribution of Upper Bounded Maximum Heights
As we reported earlier, the pdf of unbounded space-time extremes, being based on Gaussianity, has no physical limits on the values that the surface height or crest-to-trough height can attain. In practice, zero-probability events, which are events whose probability is zero, are not allowed. This is far from being realistic, since many physical processes govern the scale of wave growth. One of the important processes driving the wind-wave evolution is the energy dissipation via breaking [31]. When breaking occurs, it produces a sudden reduction of wave height and consequently wave energy; it was estimated that a breaking wave may lose more than half of its height [42] in a space less than one wavelength [43]. Over a surface area, breaking appears randomly and the fraction of breaking waves is such that, on average, every 20th to 50th wave displays breaking [31], a fact that would favor the likelihood of encountering a breaking wave as the area increases. During the breaking onset, waves approach an instability condition that ultimately reduces their height, whether waves relax back or steepen further and collapse. After Stokes [44], the most common criterion for wave breaking in deep water employs the local steepness, by limiting the height-to-wavelength ratio H/L. This geometric criterion, originally thought for linear monochromatic waves, was confirmed as the limit for wave breaking induced by linear wave focusing [45], and for breaking due to modulation instability [46]. Further, local effects, such as the superimposed wind forcing [47] or the ocean surface current gradients [48], may influence the formation of breaking, steep waves.
The parameterization of the ultimate steepness beyond which directional waves will certainly break, and the pdf of wave steepness H/L has a sudden cut-off, is not straightforward in the context of extreme-value theories. There are two main limitations. First of all, theoretical criteria based on the maximum steepness appear rather as upper bounds, even when they are converted into equivalent formulations for nonlinear crest heights [49]. Indeed, measurements from laboratory [50] and field experiments [51] suggested that waves generally break below the Stokes limiting steepness. Secondly, those criteria require the knowledge of the local wave-by-wave length L [52] from which one can infer the maximum wave steepness H m /L (or C m /L). However, since parameters of the extreme height pdf are computed from the wave spectrum, the deterministic value of the local L for each realization of maximum waves is unknown; indeed, amplitudes and therefore steepness of waves with a specific wavenumber are not deterministically defined in the continuous-spectrum environment [31], so that a probabilistic approach was established [53,54]. This makes the computation of the individual steepness and the simulation of breaking an unviable solution to judge whether or not an individual maximum wave of a given height is breaking. We have therefore decided to give up the search of a limiting steepness, and we have loosely used normalized (with the significant wave height H s = 4σ) limiting crest and crest-to-trough heights for the maximum waves. In the following, we show how the extreme-value pdfs are modified to account for such a maximum value, while in the next section we discussed how to characterize maximum allowable heights, and we assessed the theoretical formulation against literature data.
From a statistical point of view, we aimed at restricting the unbounded sample space (0, ∞) of the two density functions f Cm and f Hm by equalling them to zero above a finite height threshold; in other words, we wish to know the probability density of the non-negative random variables C m and H m , after limiting the corresponding support to be below a threshold scaled with H s . We proceed by assuming that (i) before reaching the threshold, maximum waves are allowed to grow and decay and the random process and its extreme-value probability distribution are unaffected, and (ii) the upper bound is an accumulation point for the extreme-value statistics. This is a further simplification of the effects of wave breaking, since it forces the height reduction after breaking, by assigning a normalized height that maximum waves cannot exceed. This choice is however reasonable since we are taking into account the extreme-value pdf of maximum heights, which describe the distribution of maximum parameters over an ensemble of realizations.
Accordingly, we might first write the condition for maximum crest heights C m as follows by defining the bounded pdf g Cm as follows where B c > 0 is the maximum elevation a normalized crest height C m /H s may reach. The expectation of the bounded random variable is given by which depends on B c and it is, by construction, smaller than E{C m }, being E{C m } BC = E{C m } in the limit B c → ∞. Notice that g Cm is a density since For maximum wave heights H m , we followed a similar strategy by thresholding the values with an upper limit for the crest-to-trough vertical distance B H > B c , so that the pdf of bounded H m is defined as and may be written as The expectation of maximum wave heights H m that are constrained to be below the normalized threshold B H is evaluated as HG Hm dH (18) and, as anticipated, it is smaller than E{H m }.
In summary, we proposed a modification of the unbounded space-time pdfs of H m and C m by including an upper threshold for both maximum heights that permits a transition of the extreme-value statistics towards zero-probability events. Since in the distribution of extremes the likelihood of having the largest waves above such thresholds increases with the sample size (namely, the number of individual waves N 3 , N 2 and N 1 ), the effect of the bounds is greater when the sea surface area (or the interval duration) for the characterization of extremes is relatively large. An example of the thresholding effect is shown in Figures 1 and 2, where the bounded and unbounded pdf, cdf, expected values and errors are shown for varying sea surface area width √ XY. We note that the probability that the threshold values are larger than any wave maximum diminishes with increasing sample size (colored curves from blue to red in Figures 1 and 2). Differences between the bounded and unbounded distributions increase accordingly. With this in mind, in the following sections we focused on the meaningfulness of the threshold and its consequence on the extreme wave prediction over sea states of variable severity. of having the largest waves above such thresholds increases with the sample size (namely, the number of individual waves N3, N2 and N1), the effect of the bounds is greater when the sea surface area (or the interval duration) for the characterization of extremes is relatively large. An example of the thresholding effect is shown in Figures 1 and 2, where the bounded and unbounded pdf, cdf, expected values and errors are shown for varying sea surface area width √ . We note that the probability that the threshold values are larger than any wave maximum diminishes with increasing sample size (colored curves from blue to red in Figures 1 and 2). Differences between the bounded and unbounded distributions increase accordingly. With this in mind, in the following sections we focused on the meaningfulness of the threshold and its consequence on the extreme wave prediction over sea states of variable severity.

Upper Heights and Assessment of the Bounded Distribution
There are not many studies dealing with the ultimate normalized heights m s ⁄ and m s ⁄ that sea waves may reach, since, as we pointed out before, the individual height is not by itself a limiting factor for the wind-wave growth. In the literature, there are a number of credible accounts of giant rogue waves [3], which are among the highest and steepest waves ever recorded in the world's oceans [55,56]. A summary of parameters of some iconic and widely studied rogue waves is reported in Table 1, where individual wave and crest heights are shown after being normalized with the sea severity expressed by the vertical scale Hs, which is the standard way used by scholars and engineers to classify whether a single wave falls within the definition of rogue or not [57]. It is important to mention that the work by Fedele et al. [58] pointed out that rogue wave formation seems to result from constructive interference of elementary waves enhanced by second-order nonlinearities, which is the physical mechanism underlying the theoretical formulations for extremes used in this study.
For the selected rogue cases, maximum crest heights Cm range between 1.55Hs and 1.68Hs, and maximum wave heights Hm between 2.15Hs and 2.60Hs, the highest values pertaining the rogue event associated to the sinking of the El Faro vessel [6] that were not directly observed but instead obtained from numerical simulations. Moreover, Magnusson and Donelan [59] indicated that the conditions in the Andrea storm were more extreme for steepness and near breaking than in the Draupner case; both individual waves, however, had steepness smaller than the Stokes limit. This suggests that, excluding the very extreme El Faro case that was obtained numerically, likely observed largest values for Cm/Hs and Hm/Hs were in the ranges from 1.55 to 1.65 and from 2.45 to 2.50, respectively, the former limit being also consistent with the values found for space-time maximum elevations (AA1 and AA2

Upper Heights and Assessment of the Bounded Distribution
There are not many studies dealing with the ultimate normalized heights C m /H s and H m /H s that sea waves may reach, since, as we pointed out before, the individual height is not by itself a limiting factor for the wind-wave growth. In the literature, there are a number of credible accounts of giant rogue waves [3], which are among the highest and steepest waves ever recorded in the world's oceans [55,56]. A summary of parameters of some iconic and widely studied rogue waves is reported in Table 1, where individual wave and crest heights are shown after being normalized with the sea severity expressed by the vertical scale H s , which is the standard way used by scholars and engineers to classify whether a single wave falls within the definition of rogue or not [57]. It is important to mention that the work by Fedele et al. [58] pointed out that rogue wave formation seems to result from constructive interference of elementary waves enhanced by second-order nonlinearities, which is the physical mechanism underlying the theoretical formulations for extremes used in this study. Table 1. Crest and crest-to-trough maximum wave height parameters of iconic rogue waves (namely Draupner, Andrea, Killard, and El Faro; [6,8]) and of maximum waves AA1 and AA2 gathered within space-time fields of sea elevation [5]. For the selected rogue cases, maximum crest heights C m range between 1.55H s and 1.68H s , and maximum wave heights H m between 2.15H s and 2.60H s , the highest values pertaining the rogue event associated to the sinking of the El Faro vessel [6] that were not directly observed but instead obtained from numerical simulations. Moreover, Magnusson and Donelan [59] indicated that the conditions in the Andrea storm were more extreme for steepness and near breaking than in the Draupner case; both individual waves, however, had steepness smaller than the Stokes limit. This suggests that, excluding the very extreme El Faro case that was obtained numerically, likely observed largest values for C m /H s and H m /H s were in the ranges from 1.55 to 1.65 and from 2.45 to 2.50, respectively, the former limit being also consistent with the values found for space-time maximum elevations (AA1 and AA2 cases in Table 1), and the latter with rogue wave observations off the US West coast [60] for which H m /H s peaked at 2.57.

Draupner Andrea Killard AA1 AA2 El Faro
Additionally, it is worth noting, that the two above-mentioned intervals are consistent with each other in the framework of the QD model, supporting the choice of upper bounds that lie within the suggested intervals. Indeed, if we assume for second-order nonlinear crest heights the upper bound B C = 1.55 and the steepness parameter µ = 0.06, the linear threshold for crest height, after [37], equals 1.34H s ; then, using the wave height bound B H = 2.45, the coefficient α = 2(1 − ψ * ) would equal 1.83 and, as a consequence, ψ * = −0.67, which is well within the range from −0.75 to −0.65 of typical values for wind-generated waves [39]. The other way around, assuming B H = 2.45, µ = 0.06, and the mid-value ψ * = −0.70, the upper bound for crest height would be B C = 1.54.
To assess the meaningfulness of the limit B C for nonlinear crest heights and its impact on the extreme-value distribution, we used the results presented by Fedele et al. [6], who show that the variation with the surface area of the space-time maximum crest heights suggests statistical similarity and universal law (Figure 3). Those authors used a combination of stereo observations (OBS in Figure 3) and Higher-Order pseudo-Spectral numerical simulations (HOS in Figure 3) to assess the growth of space-time maxima (expected values E{·} with subscript ST in Figure 3) over purely temporal maxima (expected values E{·} with subscript T in Figure 3).
Aiming at reproducing the results of Fedele et al. with the unbounded distribution, we have adopted theoretical expectations that were obtained from a sample of one hundred directional wave spectra computed at hourly interval in the Mediterranean Sea, south of the Gulf of Lion (point of coordinates 4.65 • E, 42.06 • N), using the wave model WAVEWATCH III ® (https://polar.ncep.noaa. gov/waves/) forced with ERA5 reanalysis wind fields at 0.25 • resolution (https://www.ecmwf.int/en/ forecasts/datasets/reanalysis-datasets/era5). The Mediterranean Sea wave model setup is based to cover the whole basin with 0.05 • uniform resolution in longitude and latitude and with a spectral grid composed of 36 evenly spaced directions and 32 frequencies exponentially spaced from 0.0500 to 0.9597 Hz at an increment of 10%. WAVEWATCH III ® was formulated using the ST4 source term configuration [61], but with adjusted coefficients β max = 1.55 and z 0,max = 0.002. To make the analysis reliable, we used a large variety of sea conditions, with the significant wave height H s ranging between 1.3 and 6.7 m, the mean wavelength L x between 24 and 85 m, the zero-crossing mean period T z between 4.4 and 7.9 s, and the mean steepness µ between 0.035 and 0.074. For the extreme-value analysis, the square sea surface region of area XY was considered around the selected point.
We observe in Figure 3 that normalized values of unbounded E{C m } and bounded E{C m } BC extreme-value crest heights tend to part when √ XY is above~2L x , and the maximum difference between E{C m } BC and El Faro HOS simulations is about +6% (+9%) of the mean temporal maximum when the limiting threshold B c = 1.55 (B c = 1.60) is adopted. Then, over larger areas, the differences between numerical outputs and theoretical expectations E{C m } BC reduce, and they tend to reconcile for width √ XY >~20L x by setting B c = 1.55. The use of a higher limit such as B c = 1.60 seems to produce a poorer matching against HOS numerical outputs, at least within the spatial range for which they are available (blue line in Figure 3). In conclusion, we first acknowledge that a universal law for space-time wave extremes is not yet fully validated and needs further understanding. Nevertheless, these results show that the use of an upper bounded distribution (green markers in Figure 3) both reduces the overestimation over large areas of space time maximum crest heights and produces a more realistic saturation of surface heights in comparison to an unbounded distribution (red markers in Figure 3). J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 9 of 14 more realistic saturation of surface heights in comparison to an unbounded distribution (red markers in Figure 3).

Impact of the Upper Bounds on Space-Time Extreme Waves
The impact of the upper bound on space-time extreme waves was examined by evaluating the theoretical difference between the expected values of the unbounded and bounded maximum heights when the sea state characteristics and sea surface areas were varied. Expected maximum crest and wave heights were obtained via numerical integration of the corresponding pdf. We considered two time intervals: D = 1200 s, which is the wave buoy record standard length that has been used in previous assessments of space-time extremes from model data [33], and D = 3600 s, which is the generally accepted maximum time interval for a sea state to be considered stationary [63]. We assume the two upper limits c = 1.55 and H = 2.45, and we vary the area width √ by keeping X = Y. Results are presented, at first, by assessing the error (i.e., overestimation) committed in using the unbounded distribution as a function of the unbounded expected heights; then, we considered how this error changes with the area size and the sea severity. Theoretical results were obtained using the same set of directional spectra as in the previous section. Figure 4 shows that the overestimation increases with increasing extreme height values E{ m } s ⁄ and E{ m } s ⁄ . When the normalized expected values of the unbounded pdf equal the two thresholds c = 1.55 and H = 2.45, the difference (unbounded -bounded) is however smaller than +4% of Hs, while for higher values the errors tend to cluster around a steady growth. The two time intervals produce different results (the shorter one, 1200 s, allows for smaller errors), but for both cases these errors remain quite small below the threshold (< +4%). Since the expected values of extremes depend on the sea severity (i.e., Hs) of the sea state [64], overestimations do vary with Hs, as is shown in Figures 5 and 6, which provide the relative errors ∆(Cm) = (E{Cm} -E{Cm}BC)/Hs and ∆(Hm) = (E{Hm} -E{Hm}BH)/Hs as a function of Hs, √ , and D. For both interval durations, errors are smaller for more energetic sea states (i.e., large Hs) and grow with the sea surface area (i.e., sample size).

Impact of the Upper Bounds on Space-Time Extreme Waves
The impact of the upper bound on space-time extreme waves was examined by evaluating the theoretical difference between the expected values of the unbounded and bounded maximum heights when the sea state characteristics and sea surface areas were varied. Expected maximum crest and wave heights were obtained via numerical integration of the corresponding pdf. We considered two time intervals: D = 1200 s, which is the wave buoy record standard length that has been used in previous assessments of space-time extremes from model data [33], and D = 3600 s, which is the generally accepted maximum time interval for a sea state to be considered stationary [63]. We assume the two upper limits B c = 1.55 and B H = 2.45, and we vary the area width √ XY by keeping X = Y. Results are presented, at first, by assessing the error (i.e., overestimation) committed in using the unbounded distribution as a function of the unbounded expected heights; then, we considered how this error changes with the area size and the sea severity. Theoretical results were obtained using the same set of directional spectra as in the previous section. Figure 4 shows that the overestimation increases with increasing extreme height values E{C m }/H s and E{H m }/H s . When the normalized expected values of the unbounded pdf equal the two thresholds B c = 1.55 and B H = 2.45, the difference (unbounded -bounded) is however smaller than +4% of H s , while for higher values the errors tend to cluster around a steady growth. The two time intervals produce different results (the shorter one, 1200 s, allows for smaller errors), but for both cases these errors remain quite small below the threshold (<+4%). Since the expected values of extremes depend on the sea severity (i.e., H s ) of the sea state [64], overestimations do vary with H s , as is shown in Figures 5 and 6, which provide the relative errors ∆(C m ) = (E{C m } − E{C m } BC )/H s and ∆(H m ) = (E{H m } − E{H m } BH )/H s as a function of H s , √ XY, and D. For both interval durations, errors are smaller for more energetic sea states (i.e., large H s ) and grow with the sea surface area (i.e., sample size). Moreover, in order to keep errors as low as 5% of H s for the more severe sea conditions, area sides must remain smaller than about 300 m. Moreover, in order to keep errors as low as 5% of Hs for the more severe sea conditions, area sides must remain smaller than about 300 m.

Concluding Remarks
In this study we have proposed and tested an improvement of the extreme-value pdf of space-time maximum waves in order to include an upper bound for nonlinear crest heights and crest-to-trough wave heights. Indeed, the pdf of unbounded extremes, being based on Gaussianity, has no physical limits on the maximum values that the surface height or crest-to-trough height could attain. This poses a limitation on the probability functions based on asymptotic expansion, which require large heights in order to be effective. A practical trade-off was suggested in this study. The solution was based on a thresholded Gumbel-like distribution of short-term/range space-time extreme waves, but its general approach might be applied to time extreme waves as well. Fundamental in the thresholding is the selection of the upper bounds that we have fixed as a multiple of the significant wave height, stemming from the highest values reached by rogue waves. A preliminary assessment of the bounded pdf was made using reference values from previous research studies. Finally, the overestimation induced by the unbounded pdf over large areas was evaluated. Main conclusions of the study may be summarized as follows: • The extreme-value bounded distribution alleviates the overestimation of the unbounded distribution over large areas. The use of limiting heights allows a smooth transition towards a realistic saturation of crest and wave heights with increasing sample size. Although the proposed pdf used a simplified measure of the limit for wave growth, it improves the performance of the space-time extreme pdf, while leaving its skill for small areas unchanged. • Primary in the proper assessment of the bounded pdf is the definition of the upper limits. Here we have used 1.55H s and 2.45H s for the maximum crest and wave heights, respectively, which were derived from historical rogue wave parameters. However, more validation studies are needed to improve the knowledge on the confidence limits for varying sea state characteristics. Numerical studies, using for instance HOS simulations, seem to be promising for this purpose, allowing for investigation of nonlinear wave groups crossing sea surface regions with a different area.

•
Our analysis has shown that the unbounded pdfs are reliable over surface areas with a side smaller than O(10 2 m) for all sea states and time interval shorter than one hour. More energetic (and potentially damaging) sea conditions however are less influenced by the inclusion of the bounds, since, for a given area, they provide smaller sample sizes.

•
The proposed formulations are suitable for being integrated into phase-averaged spectral wave models to expand their range of applicability for a proper characterization of extreme wave parameters.