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    (left) The Adriatic Sea bathymetry and the surrounding orography. The black-red dot shows the position of the Acqua Alta oceanographic platform (AA). (right) The platform structure and the position (white-red markers) of the two stereo cameras, looking northeastward.

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    Time series of wind and wave parameters from reference instrumentation at Acqua Alta on 10 Mar 2014. (top) Average wind speed (U10) and direction (θU, nautical convention). (bottom) Spectral significant wave height (Hm0) and peak period (Tp). In both panels, the red dashed lines limit the time frame of the WASS acquisition.

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    (right) Example of 3D reconstruction of the sea surface elevation and (left) corresponding right camera image with superimposed locations (red dot markers) of the maximum and minimum wave elevations (black dot markers in the right panel).

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    (left) Frequency–direction spectrum and (right) omnidirectional frequency spectrum estimated from stereo wave data (OBS). In the right panel, the dashed and solid black lines are reference spectral slopes.

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    Orthorectified right camera image plane within the stereo-matched area and locations (red dots) of the 23 wave records ηi(t) = η(xi, yi, t), such that max[ηi(t)] > 1.25Hs. The blue arrow shows the peak direction of wave propagation (θp) with respect to the camera axes.

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    (left) Extract of the dimensional sea surface elevation η1(t) = η(x1, y1, t) of the record B1 around max[η1(t)] and (right) dimensionless sea surface elevation for all records Bi around max[ηi(t)]. In the right panel, the mean wave profile is shown as red line, and, for graphical purposes, the time axis is centered at the moment when max[ηi(t)] occurs and extended to include 4Tp.

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    PDF of dimensionless sea surface elevations of the records Bi (OBSB) and of the space–time ensemble (OBSENS). The Gaussian distribution (G), the third- (G-C3), and fourth-order (G-C4) nonlinear corrections are displayed for reference.

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    Probability of exceedance (EDF) of (left) dimensionless crest heights and (right) wave heights of the record B1 (OBS). The empirical EDF stability band is plotted as blue solid line. Reference distributions: Rayleigh (R), Tayfun (T), Tayfun–Fedele (TF), and Boccotti (BO).

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    Probability of exceedance (EDF) of dimensionless observed extreme crest heights ηi,max = max[ηi(t)] of the records Bi (OBS). The empirical EDF stability band is plotted as blue solid line. Reference distributions of extremes: Rayleigh (R); Tayfun (T); Tayfun–Fedele (TF); Fedele (FM); second-order FM (FM2); and asymptotic Gumbel limit of FM2 (FM2-GL). The reference duration is D = 1798 s. Space–time extremes are computed over an area XY = 11.2 × 11.2 m2 = 126 m2.

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    Probability of exceedance (EDF) of dimensionless observed extreme crest heights max[η(SEj)] of the space–time subensembles SEj (OBS). The empirical EDF stability band is plotted as blue solid line. Reference distributions of extremes: Rayleigh (R); Tayfun (T); Tayfun–Fedele (TF); Fedele (FM); second-order FM (FM2); and asymptotic Gumbel limit of FM2 (FM2-GL). The reference duration is D = 450 s. Space–time extremes are computed over an area XY = 26.9 × 26.9 m2 = 723 m2.

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Observation of Extreme Sea Waves in a Space–Time Ensemble

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  • 1 Institute of Marine Sciences-Italian National Research Council (ISMAR-CNR), Venice, Italy
  • | 2 DAIS, Università Ca’ Foscari, Venice, Italy
  • | 3 Institute of Marine Sciences-Italian National Research Council (ISMAR-CNR), Venice, Italy
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Abstract

In this paper, an observational space–time ensemble of sea surface elevations is investigated in search of the highest waves of the sea state. Wave data were gathered by means of a stereo camera system, which was installed on top of a fixed oceanographic platform located in the Adriatic Sea (Italy). Waves were measured during a mature sea state with an average wind speed of 11 m s−1. By examining the space–time ensemble, the 3D wave groups have been isolated while evolving in the 2D space and grabbed “when and where” they have been close to the apex of their development, thus exhibiting large surface displacements. The authors have selected the groups displaying maximal crest height exceeding the threshold adopted to define rogue waves in a time record, that is, 1.25 times the significant wave height (Hs). The records at the spatial positions where such large crests occurred have been analyzed to derive the empirical distributions of crest and wave heights, which have been compared against standard statistical linear and nonlinear models. Here, the maximal observed wave crests have resulted to be outliers of the standard statistics, behaving as isolated members of the sample, apparently uncorrelated with other waves of the record. However, this study has found that these unexpectedly large wave crests are better approximated by a space–time model for extreme crest heights. The space–time model performance has been improved, deriving a second-order approximation of the linear model, which has provided a fair agreement with the empirical maxima. The present investigation suggests that very large waves may be more numerous than generally expected.

Corresponding author address: Alvise Benetazzo, Institute of Marine Sciences (ISMAR)-Italian National Research Council (CNR), Arsenale Tesa 104, Castello 2737/F, 30122, Venice, Italy. E-mail: alvise.benetazzo@ve.ismar.cnr.it

Abstract

In this paper, an observational space–time ensemble of sea surface elevations is investigated in search of the highest waves of the sea state. Wave data were gathered by means of a stereo camera system, which was installed on top of a fixed oceanographic platform located in the Adriatic Sea (Italy). Waves were measured during a mature sea state with an average wind speed of 11 m s−1. By examining the space–time ensemble, the 3D wave groups have been isolated while evolving in the 2D space and grabbed “when and where” they have been close to the apex of their development, thus exhibiting large surface displacements. The authors have selected the groups displaying maximal crest height exceeding the threshold adopted to define rogue waves in a time record, that is, 1.25 times the significant wave height (Hs). The records at the spatial positions where such large crests occurred have been analyzed to derive the empirical distributions of crest and wave heights, which have been compared against standard statistical linear and nonlinear models. Here, the maximal observed wave crests have resulted to be outliers of the standard statistics, behaving as isolated members of the sample, apparently uncorrelated with other waves of the record. However, this study has found that these unexpectedly large wave crests are better approximated by a space–time model for extreme crest heights. The space–time model performance has been improved, deriving a second-order approximation of the linear model, which has provided a fair agreement with the empirical maxima. The present investigation suggests that very large waves may be more numerous than generally expected.

Corresponding author address: Alvise Benetazzo, Institute of Marine Sciences (ISMAR)-Italian National Research Council (CNR), Arsenale Tesa 104, Castello 2737/F, 30122, Venice, Italy. E-mail: alvise.benetazzo@ve.ismar.cnr.it

1. Introduction

In situ observation of waves is a challenging task; apart from the measurement of usual sea state parameters (e.g., the significant wave height), large wave elevations are often missed by classical instrumentation. Indeed, most of the operational wave measurements rely on surface-following buoys, which tend in general to underestimate the displacement of very high waves (Krogstad and Barstow 2000). In addition to this, pointlike observatory systems (buoys, wave probes, etc.) underestimate (especially in short-crested seas) the actual maximum wave displacements that can occur on sea surface areas even smaller than the wave characteristics dimensions, namely, the wavelength and the crest length (Fedele et al. 2013; Forristall 2011). In recent years, a great effort has therefore been devoted to develop instrumentation apt to capture the complete dynamics of the waves as they propagate both in time and space. In this context, the field of wave observation has lately been characterized by the advent of new instrumentation and knowledge (Liu 2013; Kharif et al. 2009). Instrumentation, like radars (Rosenthal and Lehner 2008; Lehner and Gunther 2004), laser scanners (Hwang et al. 2000a; Romero and Melville 2010), and stereo cameras (Wanek and Wu 2006; Benetazzo 2006; Kosnik and Dulov 2011; Benetazzo et al. 2012; Gallego et al. 2011), are indeed providing new samples of sea surface elevations η, for which the spatial dimensions x and y are added to the wave time records η(t), producing spatial η(x, y) or even space–time ensembles η(x, y, t) of wave data. These newly explored sets of observations led to new insights on sea waves behavior, including the speed of large wave crests (Banner et al. 2014), the frequency and wavenumber distributions of sea waves (Hwang et al. 2000b; Leckler et al. 2015; Romero and Melville 2010), the spatial statistics of waves (Romero and Melville 2011), and some properties of the spatiotemporal wave groups (Nieto Borge et al. 2013; Nieto Borge 2004).

A relevant application of space–time wave data is evident in the field of wave extreme analysis. Indeed, the description of large waves requires measurements capable of capturing as completely as possible the dynamics of the 3D groups (Onorato et al. 2013). Moreover, recent studies (Forristall 2006; Dysthe et al. 2008; Fedele et al. 2013) proved that wave crest maxima occurring over a portion of the sea surface are generally larger (especially for short-crested sea conditions) than the values expected at a single point of observation. This result has also been endorsed by the conclusions of the EU-funded project “MaxWave” (Rosenthal and Lehner 2008), which suggest that criteria for modeling extreme waves must be modified and include the contribution of the spatial domain.

In the time domain, large waves are statistically described using non-Gaussian models. When the second-order nonlinearities are dominant, the waves exhibit sharper and higher crests that are distributed according to Tayfun’s model (Tayfun 1980). Second-order nonlinearities are governed by a wave steepness parameter (Fedele and Tayfun 2009) linked to the skewness coefficient of the wave record. Deviations from Tayfun’s model appear when the free waves exchange energy via third-order nonlinearities; the probability and the expected value of large waves are enhanced (Tayfun and Fedele 2007; Janssen 2003) and related to the fourth-order cumulant of the surface elevation (Mori and Janssen 2006; Fedele 2008), which also contributes to the resonant wave interactions that characterize the Benjamin–Feir instability (Benjamin and Feir 1967; Onorato et al. 2001). The impact of the spatial dimension on wave maxima have been first modeled assuming that the sea states are well approximated by Gaussian multidimensional (2D space and time) random fields using Adler and Taylor’s Euler characteristics approach (Adler 1981; Adler and Taylor 2007; Fedele et al. 2012) and Piterbarg’s theorem (Piterbarg 1996; Krogstad et al. 2004), the latter being extended to the second order by Socquet-Juglard et al. (2005). Space–time theories model the probability of large sea elevations also accounting for the number of waves that occur within the spatial domain. In this respect, space–time models predict high probabilities for large waves, and they likely describe the extent of some extreme waves (Dysthe et al. 2008).

In our analysis, we take advantage of stereo images recorded from an oceanographic platform in the northern Adriatic Sea (Italy; see Fig. 1). Images have been processed to provide a time sequence of 2D snapshots of the sea surface elevation, which is thus a function of time t and of the horizontal coordinates x and y, namely, η = η(x, y, t). We have analyzed the space–time ensemble to derive the distribution of wave maxima occurring in different wave time records η(xi, yi, t), selected where, on the horizontal x–y plane, the 3D wave groups passing within the cameras’ fields of view are close to the apex of their development. In these conditions, most probably, the wave groups produce the highest wave elevations with respect to the conditions of the sea state (Boccotti 2000). In particular, we have chosen the crests whose height has exceeded 1.25 times the significant wave height (Hs), one of the common criteria used to define rogue waves (Dysthe et al. 2008).

Fig. 1.
Fig. 1.

(left) The Adriatic Sea bathymetry and the surrounding orography. The black-red dot shows the position of the Acqua Alta oceanographic platform (AA). (right) The platform structure and the position (white-red markers) of the two stereo cameras, looking northeastward.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

The main focus of the presented work is to verify, by means of observational data, how large the wave crest heights may be when they are assumed to be space–time maxima and how the theoretical models used to describe extreme waves fit the empirical distribution. Outcomes are presented in the form of exceedance distribution functions of crest heights in a given record and in form of extreme values distributions. Also, the Euler characteristics approach as formulated by Fedele (2012) has been here extended to the second order for wave elevations.

The outline of the paper is as follows: Section 2 describes the stereo cameras’ installation and processing and the environmental conditions during the experiment. The paper continues in section 3 with a description of the largest waves of the space–time ensemble. The empirical distribution of extreme wave crests is discussed in section 4, where observational data are compared against maxima derived from different statistical models. Conclusions are presented in section 5.

2. The observational space–time ensemble

For the purpose of the study, collection of sea surface elevation η data (usually gathered by systems recording η as function of time t only) was extended to the sample space (x, y) to produce a space–time ensemble η(x, y, t) of the wavy sea surface. The sequence of 3D wave elevations was collected during an experiment conducted starting at 0940 UTC 10 March 2014 with a stereo wave imaging system [namely, Wave Acquisition Stereo System (WASS); Benetazzo 2006].

a. Metocean conditions

The WASS was mounted on top of the Acqua Alta oceanographic platform (Cavaleri 1999), located in the northernmost part of the Adriatic Sea, Italy (Fig. 1), where the sea bottom is gently sloping with a local depth d ≈ 17 m. Stereo images were grabbed during a well-established east-northeast wind condition (namely, Bora wind; Benetazzo et al. 2013; see Fig. 2; Table 1), with the average wind speed (at 10-m height) U10 stable at about 11 m s−1 during the 2 h prior to the WASS acquisition. At Acqua Alta, the sea surface elevation and wave parameters are routinely measured with an acoustic surface tracking system [Nortek Acoustic Wave and Current profiler (AWAC)] at 1-Hz sampling frequency and with an accuracy of 1% of the measured value for wave elevation and 2° for wave direction. During the experiment, AWAC recorded a sea state with spectral significant wave height Hm0 = 1.36 m, zero-crossing significant wave height H1/3 = 1.33 m, spectral mean period Tm02 = 3.8 s, and spectral peak period Tp = 5.27 s. These conditions correspond to a mature sea with a wave age cp/U10 = 0.76, where cp is the wave phase speed corresponding to Tp. At Acqua Alta, the local shoaling coefficient for the dominant waves (i.e., those with period Tp) was 0.97, and the dimensionless water depth kpd was about 2.5 (kp being the dominant wavenumber), such that we can assume as negligible the shallow-water effects on wave propagation (Holthuijsen 2008). During the WASS acquisition, the oceanic currents, measured by AWAC, were weak, experiencing a depth-averaged speed of 0.16 m s−1 (and equal to 0.31 m s−1 at 1 m underneath the mean sea surface and 0.19 m s−1 at 2 m) and mostly oriented with the wind and wave direction from the sea surface through the bottom.

Fig. 2.
Fig. 2.

Time series of wind and wave parameters from reference instrumentation at Acqua Alta on 10 Mar 2014. (top) Average wind speed (U10) and direction (θU, nautical convention). (bottom) Spectral significant wave height (Hm0) and peak period (Tp). In both panels, the red dashed lines limit the time frame of the WASS acquisition.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

Table 1.

Metocean conditions at the Acqua Alta (AA) platform on 10 Mar 2014 from 0940 to 1010 UTC. Observations from reference instrumentation (AA) and WASS data. U10: average wind speed; θU: average wind direction (nautical convention); θm: spectral mean direction of wave propagation; θp: spectral peak direction of wave propagation; Tp: spectral peak wave period; Tm02: spectral mean wave period; Tz: zero-crossing wave period; H1/3: zero-crossing significant wave height; and Hm0: spectral significant wave height.

Table 1.

b. The Wave Acquisition Stereo System

The WASS setup at Acqua Alta is similar to that used in previous applications (Benetazzo et al. 2012; Leckler et al. 2015) and consisted of a pair of 5 megapixel digital cameras (with 2456 columns by 2048 rows array of 3.45-µm square active elements) and mounting 5-mm distortionless lenses, placed 2.5 m apart and 12.5 m above the mean sea level.

The cameras’ internal parameters were estimated as in Albarelli et al. (2010), whereas the spatial configuration of cameras (i.e., the extrinsic parameters) was computed using an autocalibration approach based on the photometric consistency between stereo images (as in Benetazzo et al. 2014). With the device calibrated, each image pair acquired was stereo rectified and processed by a modified version of the dense stereo algorithm proposed by Hirschmüller (2008), available in the OpenCV library (http://opencv.org) by Bradski and Kaehler (2008). The semiglobal nature of the approach has the great advantage that it can relate the photometric consistency of several matching pixels to improve the reliability of the disparity map, especially for areas with loosely distinctive features. As a consequence, we can keep a relatively small window size (13 × 13 pixels), while still obtain a precise localization of the matches. This accounts for the matching bias for points around white-capped areas described by Leckler et al. (2015) without introducing brightness equalizations or complicated pyramidal search approaches. A discussion of the errors associated with the WASS measurements is reported in the study of Benetazzo et al. (2012), from which we recall the mean absolute quantization error of the adopted WASS setup, which is on the order of 2–3 cm along the three camera axes.

For each image pair, output of the stereo process is a cloud of 3D points of the sea surface elevation. For each point cloud, after being transformed to a common earth reference frame (Benetazzo 2006), a patchwise planar surface was constructed by means of 2D Delaunay triangulation. Then, the surface was resampled over a regular grid at a resolution of 0.2 m to span the space x ∈ [−42.5 m, 42.5 m] and y ∈ [−70.0 m, −5.0 m] (Fig. 3). The trapezoidal area of the sea surface spanned by the stereo data is about A = 2893 m2; the undisturbed surface of the sea coincides with the xy plane, the y axis is turned 46° clockwise from the geographical north, and the z axis is pointed upward. The WASS sequence comprises 26 971 image pairs, grabbed at 15 Hz for a total of 1798 s (about 30 min) of data; such duration ensures reliability for waves’ characterization and stationarity of the sea state (Boccotti 2000). The resulting space–time grid consists of 72 325 (2D space) × 26 971 (time) sea surface elevation points. To limit the influence of high-frequency noise, time records η(x0, y0, t) taken at each position (x0, y0) of the 3D wave space–time ensemble were smoothed using a weighted linear least squares local regression and low-pass filtered at 2.0 Hz (Fig. 4).

Fig. 3.
Fig. 3.

(right) Example of 3D reconstruction of the sea surface elevation and (left) corresponding right camera image with superimposed locations (red dot markers) of the maximum and minimum wave elevations (black dot markers in the right panel).

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

Fig. 4.
Fig. 4.

(left) Frequency–direction spectrum and (right) omnidirectional frequency spectrum estimated from stereo wave data (OBS). In the right panel, the dashed and solid black lines are reference spectral slopes.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

c. Space–time wave data

Assuming the observed sea condition was stationary in time and homogenous in space, the frequency–direction wave spectrum S(f, θ) was computed using a stochastic approach, namely, the extended maximum entropy method (EMEP; Hashimoto et al. 1994). The frequency–direction spectrum was resolved with 180 equally spaced directions to cover the full circle and 1024 equally distributed frequencies from 0.05 to 2 Hz. The EMEP method has already been successfully applied to WASS data (Fedele et al. 2013; Barbariol et al. 2014) and preferred here to the 3D Fourier transform of the space–time ensemble η(x, y, t) that otherwise would have provided a small directional resolution around the spectral peak (Hwang et al. 2000b). The directional spectrum S(f, θ) shown in Fig. 4 is unimodal with Hm0 = 1.33 m, peak direction of wave propagation θp = 249°N, and peak frequency fp = 0.19 Hz. The directional spreading of S(f, θ) integrated over the frequencies, estimated according to Kuik et al. (1988), is equal to 39°, a characteristic value of wind sea states.

An estimate of the mean 〈η〉 = 〈η(x, y, t)〉 of the dataset is −0.001 m, where the angle brackets denote space–time averaging. The standard deviation σ and the coefficients of skewness λ3 and kurtosis λ4 have been estimated from the second-, third-, and fourth-order moments of the surface displacement η(x, y, t) probability density function (PDF) as follows:
e1
e2
e3
The observed values for these parameters are σ = 0.334 m, λ3 = 0.16, and λ4 = 3.22. The significant wave height Hs, taken as 4 times the standard deviation σ of the surface elevation, is 1.34 m.

In a previous study (Benetazzo et al. 2012), WASS data have already been validated through a comparison with observations from other instruments. Here, for the sake of assessment, sea state parameters estimated using the space–time ensemble have been compared against observations from the AWAC instrument. Results in Table 1 show that WASS observations are consistent with the measurements collected from AWAC, with small differences on the order of a few centimeters for wave heights and a few tenths of a second for wave periods.

3. An observatory for large waves

a. Recognition of time records with the highest waves

With the purpose of analyzing the highest waves of the space–time ensemble ENS, it is remarkable to note that the maximal sea surface elevation ηENS,max = max[η(x, y, t)] is 2.12 m, about 6.4 times larger than the standard deviation σ, and it exceeds Hs by a factor of 1.59. We have therefore examined the entire ensemble to isolate the 3D wave groups, passing through the stereo camera field of view, whose maximum elevation ηc has exceeded a specific value, which we have chosen corresponding to the rogue wave threshold for crest heights, namely, 1.25Hs (Haver 2004; Dysthe et al. 2008). Our selected threshold is somehow arbitrary and adopted with the only purpose of isolating the highest waves within the ensemble. Additionally, we have visually checked that the selected wave maxima were not contained in areas of large breaking foam, where the images tend to saturate, thus lowering the photometric consistency of the stereo pair. Also, data analysis has been limited to the central part of the 2D elevation maps to prevent reconstruction biases due to the image borders.

In this way, we have identified 23 independent 3D wave crests (e.g., the crest shown in Fig. 3 at x ≈ −3 m and y ≈ −42 m) at instants ti and spatial locations (xi, yi) shown in Fig. 5, most likely when and where the crests traveled close to the center of the 3D wave groups (Boccotti 2000). We note that the wave crest positions displayed in Fig. 5 are distributed all over the spatial coverage, fulfilling the assumption of homogeneity of the wave process. The fixed (xi, yi) positions of these maximal crests have therefore been assumed to act as virtual pointlike probes of the space–time ensemble (see Fig. 9 of Benetazzo et al. 2012). At these points, we have extracted the corresponding time series ηi(t) = η(xi, yi, t), with i = 1, 2, …, 23. As the sea state under investigation was generated by a Bora wind condition, we have labeled the time records as Bi (Table 2) and ordered them in descending order according to the maximum crest height of the record ηi,max = max[ηi(t)].

Fig. 5.
Fig. 5.

Orthorectified right camera image plane within the stereo-matched area and locations (red dots) of the 23 wave records ηi(t) = η(xi, yi, t), such that max[ηi(t)] > 1.25Hs. The blue arrow shows the peak direction of wave propagation (θp) with respect to the camera axes.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

Table 2.

Parameters of the individual maximal waves of the records Bi and average values (Avg). Wave index i (from 1 to 23) corresponds to the record number ordered with decreasing ηi,max = max[ηi(t)]. Hi,max: maximum wave height; εi,max: steepness of the maximal wave; and Hs: significant wave height.

Table 2.

Several considerations can be drawn from the wave data of records Bi. First, records have been analyzed via zero-crossing analysis to isolate single crest heights and wave heights as crest-to-trough vertical distance. The average (H1/3) of the highest one-third of the waves from all records Bi is 1.32 m; therefore, the average is comparable (albeit smaller) to Hs. In the following analysis, however, we will retain Hs = 4σ instead of H1/3, as a conservative upper bound to estimate dimensionless crest and wave heights. We have therefore examined each record in proximity to the maximal crest height ηi,max. In this respect, the time portrait of the highest wave of the ensemble (belonging to the record B1 and having ηc = ηENS,max) is displayed in Fig. 6, where the nature of this large wave is visible within the wave group passing through the position (x1, y1) from 610 to 640 s. In the record B1, increasing high waves do not precede the highest wave that seems to appear out of nowhere, whereas the spatial continuity is fulfilled (see for example the crest in Fig. 3). The time profile of the wave is asymmetric and skewed around the peak with a maximum crest-to-trough height H1,max = 2.96 m, about 2.22 times larger than Hs. The plot of the individual highest waves of all records Bi around ηi,max is shown in Fig. 6. These large waves have sharper and narrower crests than the other waves of the group and tend to assume a deterministic profile (Boccotti 2000; Tayfun and Fedele 2007), which was proved to be in very good agreement with experimental data (Fedele et al. 2013). The average period of the largest waves of the records is 4.7 s, 13% smaller than the peak period. The waves adjacent (next or previous) to the maximal ones have a broader time profile with crest heights generally one-third (~0.4Hs compared to ~1.4Hs) of the highest wave of the group, as also displayed by the well-known Draupner and Andrea rogue waves (Magnusson and Donelan 2013). As discussed hereinafter, even if the smaller crests of the group have heights that are well fitted by the theoretical distributions for wave crest heights, the highest one is a real outlier from the standard statistics.

Fig. 6.
Fig. 6.

(left) Extract of the dimensional sea surface elevation η1(t) = η(x1, y1, t) of the record B1 around max[η1(t)] and (right) dimensionless sea surface elevation for all records Bi around max[ηi(t)]. In the right panel, the mean wave profile is shown as red line, and, for graphical purposes, the time axis is centered at the moment when max[ηi(t)] occurs and extended to include 4Tp.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

Some parameters describing the largest waves of the records Bi are reported in Table 2. Within the data presented, it is worth noting that 6 out of 23 records have a maximal wave height Hi,max exceeding Hs by a factor of 2.2 and 16 exceed Hs by a factor of 2.0. The average maximal crest and wave heights are 1.38Hs and 2.08Hs, respectively. For reference, the abnormal Draupner wave (Magnusson and Donelan 2013; Haver 2004) had ηmax = 1.55Hs and Hmax = 2.15Hs. The local steepness of the highest wave of the records, defined as εi,max = ηi,maxki,max (where ki,max is the wavenumber corresponding to the period of the wave with ηc = ηi,max), is given in Table 2. The steepness is 0.34 on average and 0.45 at most for record B22, where, from visual inspection of the stereo images, the maximal crest was at the breaking onset. We also note that not all the highest waves of the records were in incipient breaking conditions.

The PDF of the zero-mean and dimensionless wave surface elevations ηi(t) for the 23 Bi records aggregated (for a total of about 620 000 sea elevation data) is provided in Fig. 7 (OBSB), with the Gaussian distribution and the nonlinear third- and fourth-order corrections (Longuet-Higgins 1963), respectively given by
e4
plotted for reference. For the nonlinear corrections, the skewness [(2)] and kurtosis [(3)] estimated from the ensemble of the space–time data have been adopted. The deviation of the positive tail of the empirical PDF from the theoretical distributions (that otherwise fit well the observations) starts at elevations of about 4σ (=Hs). Even the corrections of the Gaussian model cannot reproduce the extreme wave crests observed in the large elevation region. On the contrary, once the probabilities are computed using the whole space–time ensemble (OBSENS in Fig. 7), the G-C4 approximation fits the empirical PDF well.
Fig. 7.
Fig. 7.

PDF of dimensionless sea surface elevations of the records Bi (OBSB) and of the space–time ensemble (OBSENS). The Gaussian distribution (G), the third- (G-C3), and fourth-order (G-C4) nonlinear corrections are displayed for reference.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

To summarize, the ensemble of the wave records Bi comprises a sample of time-varying sea surface displacements observed at fixed spatial locations, chosen where the 3D wave groups have exhibited a maximal elevation. Hence, the (xi, yi) position of each record is a preferential site for wave observation, as the surface area retrieved by the cameras allows the observer to follow the wave groups in their spatial evolution and to freeze the sea surface when the wave groups are close to the apex of their development and display large amplitude waves. So constrained, wave records Bi hold the largest crest heights among all those observable within the space–time ensemble.

b. Distribution of crest and wave heights of the record B1

As stated in section 3a, the maximal crest height of each record Bi corresponds to a very large displacement compared to the variance of the sea state. In this respect, a zero-crossing analysis of single waves of the record B1 reveals some insights into the probability of occurrence of the largest wave. The exceedance distribution functions (EDF; hereinafter represented as P{}) of the crest ηc and wave heights H of record B1 are presented in Fig. 8, where the estimates and the stability band of EDF have been derived from mean and standard deviation [Eq. (16) of Tayfun and Fedele 2007] of the jth value of the order statistics. It is worth noting that two single waves of the record display large ηc and H, with exceedance levels that deviate strongly from those of the remaining waves. A question arises whether existing theoretical models predict these probability levels.

Fig. 8.
Fig. 8.

Probability of exceedance (EDF) of (left) dimensionless crest heights and (right) wave heights of the record B1 (OBS). The empirical EDF stability band is plotted as blue solid line. Reference distributions: Rayleigh (R), Tayfun (T), Tayfun–Fedele (TF), and Boccotti (BO).

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

First, for a sea state such that the surface elevation is Gaussian distributed and surface waves are narrowbanded in frequency, the wave crest heights gathered at a fixed point have an EDF given by the Rayleigh formula (hereinafter R; Longuet-Higgins 1952)
e5
that underpredicts the empirical probabilities of crest heights (Fig. 8) because of nonlinear effects occurring in realistic active sea states.
The second-order correction in the Stokes expansion of the linear model affects the probability distribution of ηc, which, here, is expressed in the Tayfun form (hereinafter T; Tayfun 1980), namely,
e6
where ξ is the second-order elevation. In the T model, the variable z satisfies the quadratic equation
e7
where μ is the wave steepness. Following Fedele and Tayfun (2009), the wave steepness has been evaluated as a statistically stable estimate from the zeroth- (m000), first- (m001), and second-order (m002) moments of the directional spectrum given by
e8
where k = (kx, ky) is the wavenumber vector associated with the frequency f (through the linear dispersion relation for sea waves) and direction θ. The steepness is expressed as
e9
where μm = σ(2πm001/m000)2/g is an integral measure of the wave steepness (Fedele and Tayfun 2009), corrected with the spectral bandwidth (Longuet-Higgins 1975). The observed values for these parameters are μ = 0.06 and ν = 0.50. In Fig. 8, the T model tends to underestimate the experimental data for crest heights larger than 2σ, similar to what was found by Fedele (2008) for steepness of about 0.07.
At third-order correction, interactions between free waves produce a deviation from Gaussian and second-order theories (e.g., Janssen 2003). In this respect, the Tayfun–Fedele approximation (hereinafter TF; Tayfun and Fedele 2007) gives the EDF of dimensionless wave crest heights as
e10
where the variable Λ is a function of the fourth-order joint cumulants of the sea surface elevation and its conjugate [Eq. (42) of Tayfun and Fedele 2007], and for the record B1 it takes the value 1.09. The TF model increases the probability of large waves over that given by the R and T models and well reproduces the empirical distribution of wave crests in Fig. 8, except the two extreme waves that occur on the tail of the empirical EDF.
As far as the observed wave height EDF is concerned (right panel of Fig. 8), for linear waves the Rayleigh model overestimates the probability levels as it does not account for finite bandwidth effects (Tayfun 1981; Boccotti 2000; Forristall 1978), which are included in the asymptotic linear distribution derived by Boccotti (2000; hereinafter BO):
e11
where Ψ* is the absolute value of the ratio between the global minimum and the global maximum of the autocovariance function of the record. For the record B1 the parameter Ψ* = 0.66 (belonging to the typical range for wind waves; Boccotti 2000). Second-order nonlinearities have negligible effects to the crest-to-through wave heights (Tayfun and Fedele 2007). At a higher order of approximation, the TF model for wave height is expressed as
e12
which improves the prediction of the BO model in Fig. 8. For narrowbanded sea states, Λ ≈ 8/3(λ4 − 3), showing a direct dependence of the probability levels on the kurtosis coefficient (Tayfun and Fedele 2007; Mori and Janssen 2006). In this respect, the kurtosis [(3)] of the space–time data (λ4 = 3.22) is different from the corresponding value of the wave record B1 (λ4,B1 = 3.52). The higher kurtosis of the record B1 is a direct consequence of the fact that significantly large displacements occur in the proximity of the highest wave. In fact, removing this single wave from the series and recalculating the kurtosis of the remaining record, we have found a kurtosis of 3.21, very close to the ensemble value.

Crest and wave heights distributions displayed in Fig. 8 for the record B1 are typical of each record Bi, where the highest waves are real outliers of the standard statistics. These unusual waves will be discussed in section 4 in the context of extreme values statistics, also including space–time effects.

4. Statistics of extremes

a. Empirical distribution

The extreme crest heights ηi,max are members of an ordered sample of the largest sea surface elevations within the space–time ensemble. In this respect, some studies (Fedele 2012; Fedele et al. 2013; Forristall 2011; Krogstad et al. 2004; Dysthe et al. 2008) documented that wave maximal crests occurring over an area are generally larger than those expected (on average) at a given point inside that area, especially for short-crested sea waves. Distributions of such space–time extremes were derived from theories developed to describe multidimensional random fields, in which the wave crests are labeled as maxima exceeding a given threshold (Adler 1981; Adler and Taylor 2007; Piterbarg 1996). In fact, the concept of a single 3D “wave” is per se ambiguous, as a generalization of the zero-crossing procedure to higher dimensions is far from being trivial (Fedele et al. 2012; Worsley 1996).

In the present study, the empirical distribution of wave extremes relies on the maximal crest height ηi,max of each record Bi (Table 2): (η1,max, η2,max, …, η23,max). These extremes, homogeneously distributed within the stereo camera field of view, are assumed to pertain to, on average, a sea surface area A/23 = 126 m2, equivalent to a square with sides X = Y = 11.2 m. The duration D of the sea state is 1798 s, such that the average number of waves in time is N = D/Tm02 = 498. The set of ηi,max variables defines a stochastic process, where each ηi,max is assumed to be an independent realization of a large wave crest (Boccotti 2000). The empirical EDF of the maximal crests is plotted in Fig. 9; the expected maximum 〈ηi,max〉 is 1.85 m (equivalent to 5.52σ and 1.38Hs) with a standard deviation of 0.12 m.

Fig. 9.
Fig. 9.

Probability of exceedance (EDF) of dimensionless observed extreme crest heights ηi,max = max[ηi(t)] of the records Bi (OBS). The empirical EDF stability band is plotted as blue solid line. Reference distributions of extremes: Rayleigh (R); Tayfun (T); Tayfun–Fedele (TF); Fedele (FM); second-order FM (FM2); and asymptotic Gumbel limit of FM2 (FM2-GL). The reference duration is D = 1798 s. Space–time extremes are computed over an area XY = 11.2 × 11.2 m2 = 126 m2.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

b. Theoretical distributions

The empirical distribution of maxima is compared against theoretical EDFs of extreme crest heights, which stem from initial distributions of individual crest heights in a given record. In the linear and narrowband approximation, the probability that in a time record of N waves the maximal crest height ηmax exceeds a given threshold z is given in the general form (Gumbel 1958)
e13
where PR follows from the Rayleigh distribution (5). The asymptotic Gumbel limit of (13) permits an estimation of the expected value of the maximum crest height in N waves as (Cartwright and Longuet-Higgins 1956)
e14
where hR is the most probable value (viz., the mode) and αR is the intensity function, both depending on the initial distribution and the sample size. The parameter γ ≈ 0.5772 is the Euler–Mascheroni constant. According to Gumbel (1958), the standard deviation of the largest values is . From the Rayleigh model, the expected maximum crest height for N = 498 waves is ER,max ≈ 1.23 m (equivalent to 3.68σ and 0.92Hs) ± 0.12 m, which is smaller than the observed value (Fig. 9). Accounting for second-order nonlinearities, the EDF of crest maxima derived from the Tayfun model (6) is
e15
The mean of the maximal crests is approximated as (Tayfun and Fedele 2007)
e16
which results equal to 1.37 m (equivalent to 4.10σ or 1.03Hs) ± 0.18 m and underestimates the observations of about 1.5σ. At the next order of approximation, the TF model [(10)] provides the initial distribution for the EDF of maximal wave crests in trains of N waves
e17
for which the expected value is written in the general form as
e18
The modal value hTF satisfies
e19
and the intensity function αTF is given by
e20
The mean value of Λ over the wave records Bi is 0.83. The TF model improves the prediction of the R and T models (Fig. 9), even though the average expected maximum ETF,max ≈ 1.56 m (equivalent to 4.66σ and 1.17Hs) ± 0.18 m is still smaller than the observed value of about σ.
The theoretical R, T, and TF models assume the sea surface elevation recorded in time at a given position; within the time record, single waves are isolated using a zero-crossing procedure. As records Bi are extracted from a space–time ensemble, we have also compared the empirical EDF against those derived from stochastic distributions of maxima over multidimensional fields. The extension of the linear R model to space–time ensembles requires the generalization of the concept of waves in a multidimensional domain, which in this study spans two spatial dimensions (over a sea surface area of sides X and Y) and a time interval D. We refer here to the studies of Piterbarg (1996) and Adler and Taylor (2007) who computed the probability of maxima in generic random fields. Drawing upon the results of Fedele (2012; hereinafter FM) derived from Adler and Taylor (2007) for a stationary and homogeneous Gaussian wave field η(x, y, t) bounded by a space–time volume V = XYD (see Fig. 1 of Fedele 2012), one has to first compute the average number of waves within V(NV), on the boundary surfaces S of V(NS), and along the perimeter B of S(NB), which are given by
e21
The mean zero-crossing period (Tm = Tm02), the mean zero-crossing wavelength and wave crest length (Lx = Lxm02 and Ly = Lym02, respectively), and the irregularity parameters of the sea state (αxt, αyt, and αxy, all ranging between −1 and +1) are computed from the moments [(8)] of the directional spectrum S(f, θ) as (Baxevani and Rychlik 2006; Fedele 2012)
e22
e23
The irregularity parameters control the number of waves [(21)] and play an important role in the distribution of the highest crests. For instance, assuming x as the mean direction of wave propagation, αxt ≈ 1 in narrowband sea states, such that the numbers of waves NS and NV decrease to preserve stochastic independence among waves (Baxevani and Rychlik 2006).
Drawing upon Adler and Taylor (2007), the exceedance probabilities that one of the 3D wave crest heights (normalized by σ) exceeds the threshold z within the volume V, on the boundary surfaces S and along the perimeter B are expressed, respectively, as
e24
where PR is the Rayleigh distribution [(5)]. For large thresholds (z ≫ 1), the probability that the global surface maximum ηmax exceeds z over the space–time volume is expressed in FM as
e25
The dimensionless most probable extreme value (hFM = ηmax/σ) expected to occur over the domain XYD is obtained as the value of h that satisfies the implicit equation
e26
The expected value of ηmax is found via the asymptotic distribution of the largest values as
e27
in agreement with Fedele (2012), where, however, the maxima were normalized by Hs.

Once the x axis is aligned to the mean direction of wave propagation, the spectral mean wave and crest lengths are Lx = 13.6 m and Ly = 14.6 m, which produce a short-crestedness parameter (Baxevani et al. 2003) Lx/Ly = 0.93, characteristic of short-crested sea states. Moreover, the irregularity parameter αxt = 0.35 indicates that the sea state is (on average) confused along the peak direction of propagation. The small value (0.004) of the parameter αyt denotes that there is no organized and preferential wave movement along the y axis. These conditions support large probabilities for extreme crests’ occurrence, as visible in Fig. 9 where the FM results exceed the Rayleigh predictions of about σ. The expected maximum crest EFM,max ≈ 1.60 m (equivalent to 4.78σ or 1.19Hs) ± 0.10 m is still smaller than the observations.

Aiming at predicting the probabilities of the highest waves within the space–time ensemble, we have extended the FM model, including the second-order contribution for each free wave mode (hereinafter FM2). In accordance with Fedele and Tayfun (2009), the dimensionless largest amplitude of the nonlinear stochastic wave group can be expressed in the Tayfun form given by (7). Following the study of Socquet-Juglard et al. (2005) that reports, in the context of the Piterbarg’s theorem, a distribution for second-order space–time maximal crests (viz., the Piterbarg–Tayfun distribution), here the EDF of nonlinear extreme crest heights is derived [as done in Fedele et al. (2013) for spatial extremes] applying the transformation [(7)] in the FM model [(25)], namely,
e28
The FM2 model provides the second-order expected value of the maximal crest over the domain XYD as
e29
which is 1.83 m (equivalent to 5.46σ or 1.37Hs) ± 0.13 m and well approximates the mean of the observed maxima; as well, the probability levels [(28)] match the EDF of the observed extremes (Fig. 9).

c. Discussion

Results presented in Fig. 9 deserve additional comments. The extreme value distributions derived from nonlinear T and TF models underestimate the empirical probabilities and the corresponding expected values. Nevertheless, each extreme is an independent realization of the maximal crest height in a series of N ≈ 500 consecutive waves, and, in this respect, the time models should provide reliable probability levels. The differences may be explained by the fact that extremes have been recorded at specific spatial locations of the sea surface, that is, where wave groups maximal elevations have exceeded a specific threshold (i.e., ηi,max > 1.25Hs).

Differently, the FM and FM2 models, which prescribe the probability of extremes for multidimensional random fields, provide a better approximation for the empirical data. In these models, the leading term of the functions (25) and (28) is the volume contribution NVz2 that keeps the probability at high levels by virtue of the number NV = 1820 and the term z2 that multiply the Rayleigh distribution. Therefore, the space–time models use a larger number of waves compared to the time models. Along the same line, the number N in the time models could be arbitrarily increased to derive higher probabilities levels, but in doing so the definition of N (i.e., the average number of waves within a given duration) would lose its meaning. On the contrary, the number of waves adopted in FM and FM2 is consistent with the hypothesis underlying the theories for space–time maxima, and the inclusion of the irregularity parameters in (21) ensures the statistical independence between waves taken in time and space domain; in other words, the waves are not counted twice.

It is interesting to verify the sensitivity of the agreement between the experimental data and the FM2 predictions to the threshold used to select the highest waves of the space–time ensemble. To do so, we have modified the threshold toward values larger than 1.25Hs, thus reducing the extreme crest population size. In these cases too (not shown here), FM2 well reproduces the empirical exceedance probabilities for the entire range of observed maxima.

We note also that the FM and FM2 models provide exceedance probabilities larger than 1 for small thresholds (Fedele et al. 2013). To ensure probabilities smaller than 1 for all thresholds, a sound approximation is the Gumbel asymptotic limit of the space–time distributions given by
e30
e31
which well fit the exact probabilities (see FM2-GL in Fig. 9).

To strengthen the results presented, we have adopted a different strategy to capture the extremes of the space–time ensemble. Following Fedele et al. (2013), who used different portions of the sea surface to demonstrate the dependence of the crest height upon the observed area, we have divided the sea surface framed by the stereo system into subsets spanning approximately A/4 = 723 m2 of the whole surface. The xy plane shown in Fig. 3 has therefore been divided into four quadrants, each bounded by two axes, corresponding to the lines x = 0.0 m and y = −46.4 m. The same approach has been repeated along the time axis to split the entire sequence into four nonoverlapping subrecords approximately 450 s long. For each subrecord, the average number of waves is N = 125. As a consequence, the entire space–time ensemble has been divided into 16 space–time subensembles (SEj; j = 1, 2, …, 16) whose spatial sizes X and Y used in the FM and FM2 models have been chosen approximating each subset with a square of equivalent area and sides X = Y = 26.9 m. Within each SEj the maximal surface elevation max[η(SEj)] has been collected and considered a member of the extreme crest population. Wave extreme positions have been checked to prevent maxima being taken on the same wave crest. Results of this analysis, shown in Fig. 10, confirm that the FM2 model well predicts the probabilities of maximal wave crests.

Fig. 10.
Fig. 10.

Probability of exceedance (EDF) of dimensionless observed extreme crest heights max[η(SEj)] of the space–time subensembles SEj (OBS). The empirical EDF stability band is plotted as blue solid line. Reference distributions of extremes: Rayleigh (R); Tayfun (T); Tayfun–Fedele (TF); Fedele (FM); second-order FM (FM2); and asymptotic Gumbel limit of FM2 (FM2-GL). The reference duration is D = 450 s. Space–time extremes are computed over an area XY = 26.9 × 26.9 m2 = 723 m2.

Citation: Journal of Physical Oceanography 45, 9; 10.1175/JPO-D-15-0017.1

Additionally, we have gathered the maximum crest height η0,max = max[η(x0, y0, t)] at each fixed position (x0, y0) of the sea surface elevation map shown in Fig. 3. The ensemble average over 72 325 data is 〈η0,max〉 = 1.46 m, in line with the predictions of the T and TF models for wave crest maxima. This agreement is not surprising and states that the theoretical models T and TF used to describe nonlinear wave crest distributions over time records hold for most of the observational points, where it is likely that a pointlike instrument gathers the waves passing through. There are, however, some specific fixed positions, those of records Bi, where the crest height maxima are outliers of the aforementioned models and yet are well described by the space–time distributions.

5. Conclusions

In this study, we have analyzed a space–time ensemble of sea surface elevations η(x, y, t) in search of the largest crest heights. Wave data were acquired using a stereo camera system mounted on a fixed oceanographic platform and processed to produce a sequence of 3D elevation maps of the sea surface during a mild (Hs = 1.34 m, Tp = 5.4 s, and steepness of 0.06) and mature (cp/U10 = 0.76) sea state. From the ensemble η(x, y, t) we have estimated skewness and kurtosis coefficients equal to 0.16 and 3.22, respectively.

Within the space–time ensemble, we have isolated the 3D wave groups and selected 23 independent wave crests with height ηc exceeding the rogue wave threshold for crest heights (i.e., ηc > 1.25Hs). Analysis of the time records at the positions (xi, yi) of these maximal crests revealed that the highest elevations have an occurrence in excess of that derived by the standard statistics. This is true even when nonlinearities (for which the departure from Gaussian statistics was expressed as function of skewness and kurtosis) are taken into account, raising the issue of which probability function might be used to fit these extreme data.

Distribution of the extreme crests of the wave records has showed that the statistical models holding for time-varying nonlinear sea surface systems (viz., the Tayfun and Tayfun–Fedele models) underestimate the empirical maxima. In other words, there are points on the sea surface where extreme elevations may occur and for which all the standard distributions fail to describe their probability. More precisely, we claim that the apparent low probability of extreme elevations given by standard models may not be related to the rarity of such events but rather to the narrow spatial localization of high crests that reduces the probability of observing such phenomena if no preconditioning on the sampling point is introduced. This is possible only if space–time wave data are available. Indeed, the Tayfun and Tayfun–Fedele models well perform for most of the observational points, whereas the highest crests’ distribution derived from the records is well fitted by a second-order correction of the Fedele (2012) model for space–time maxima, which occur where the spatial wave groups are close to the apex of their development.

Therefore, even if the sea state analyzed was not extreme in terms of Hs, our results suggest that, if observational points are suitably chosen within a sample space, large waves are more numerous than expected by the standard time statistics. This should be a remarkable consideration for short-crested wave conditions, typical of stormy seas.

Acknowledgments

The authors gratefully acknowledge the funding from the Flagship Project RITMARE—The Italian Research for the Sea—coordinated by the Italian National Research Council and funded by the Italian Ministry of Education, University and Research within the National Research Program 2011–15. Authors wish to thank Prof. Mike Banner and Dr. Luigi “Gigi” Cavaleri for their helpful comments.

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