• Alkhalidi, M. A., , and M. A. Tayfun, 2013: Generalized Boccotti distribution for nonlinear wave heights. Ocean Eng., 74, 101106, doi:10.1016/j.oceaneng.2013.09.014.

    • Search Google Scholar
    • Export Citation
  • Annenkov, S. Y., , and V. I. Shrira, 2014: Evaluation of skewness and kurtosis of wind waves parameterized by JONSWAP spectra. J. Phys. Oceanogr., 44, 1582–1594, doi:10.1175/JPO-D-13-0218.1.

  • Bitner-Gregersen, E. M., , L. Fernandez, , J. M. Lefèvre, , J. Monbaliu, , and A. Toffoli, 2014: The North Sea Andrea storm and numerical simulations. Nat. Hazards Earth Syst. Sci., 14, 14071415, doi:10.5194/nhess-14-1407-2014.

    • Search Google Scholar
    • Export Citation
  • Boccotti, P., 2000: Wave Mechanics for Ocean Engineering. Elsevier, 496 pp.

  • Borgman, L. E., 1970: Maximum wave height probabilities for a random number of random intensity storms. Proc.12th Conf. on Coastal Engineering, Vol. 12, Washington, D.C., ASCE, 53–64. [Available online at https://journals.tdl.org/icce/index.php/icce/article/view/2608.]

  • Dias, F., , J. Brennan, , S. Ponce de Leon, , C. Clancy, , and J. Dudley, 2015: Local analysis of wave fields produced from hindcasted rogue wave sea states. ASME 2015 34th Int. Conf. on Ocean, Offshore and Arctic Engineering, St. John’s, Newfoundland, Canada, American Society of Mechanical Engineers, OMAE2015-41458, doi:10.1115/OMAE2015-41458.

  • Dysthe, K. B., , H. E. Krogstad, , and P. Muller, 2008: Oceanic rogue waves. Annu. Rev. Fluid Mech., 40, 287310, doi:10.1146/annurev.fluid.40.111406.102203.

    • Search Google Scholar
    • Export Citation
  • Fedele, F., 2005: Successive wave crests in Gaussian Seas. Probab. Eng. Mech., 20, 355363, doi:10.1016/j.probengmech.2004.05.008.

  • Fedele, F., 2012: Space–time extremes in short-crested storm seas. J. Phys. Oceanogr., 42, 1601–1615, doi:10.1175/JPO-D-11-0179.1.

  • Fedele, F., 2015: On the kurtosis of ocean waves in deep water. J. Fluid Mech., 782, 2536, doi:10.1017/jfm.2015.538.

  • Fedele, F., , and M. A. Tayfun, 2009: On nonlinear wave groups and crest statistics. J. Fluid Mech., 620, 221239, doi:10.1017/S0022112008004424.

    • Search Google Scholar
    • Export Citation
  • Forristall, G. Z., 2000: Wave crest distributions: Observations and second-order theory. J. Phys. Oceanogr., 30, 1931–1943, doi:10.1175/1520-0485(2000)030<1931:WCDOAS>2.0.CO;2.

  • Forristall, G. Z., , S. F. Barstow, , H. E. Krogstad, , M. Prevosto, , P. H. Taylor, , and P. S. Tromans, 2004: Wave crest sensor intercomparison study: An overview of WACSIS. J. Offshore Mech. Arct. Eng., 126, 26–34, doi:10.1115/1.1641388.

  • Gemmrich, J., , and C. Garrett, 2008: Unexpected waves. J. Phys. Oceanogr., 38, 23302336, doi:10.1175/2008JPO3960.1.

  • Gemmrich, J., , and C. Garrett, 2010: Unexpected waves: Intermediate depth simulations and comparison with observations. Ocean Eng., 37, 262267, doi:10.1016/j.oceaneng.2009.10.007.

    • Search Google Scholar
    • Export Citation
  • Haver, S., 2001: Evidences of the existence of freak waves. Proc. Rogue Waves 2000, Brest, France, Ifremer, 129–140.

  • Haver, S., 2004: A possible freak wave event measured at the Draupner jacket January 1 1995. Proc. Rogue Waves 2004, Brest, France, Ifremer, 1–8. [Available online at http://www.ifremer.fr/web-com/stw2004/rw/fullpapers/walk_on_haver.pdf.]

  • Magnusson, K. A., , and M. A. Donelan, 2013: The Andrea wave characteristics of a measured North Sea rogue wave. J. Offshore Mech. Arct. Eng., 135, 031108, doi:10.1115/1.4023800.

  • Mori, N., , and P. A. E. M. Janssen, 2006: On kurtosis and occurrence probability of freak waves. J. Phys. Oceanogr., 36, 1471–1483, doi:10.1175/JPO2922.1.

  • Osborne, A. R., 1995: The numerical inverse scattering transform: Nonlinear Fourier analysis and nonlinear filtering of oceanic surface waves. Chaos Solitons Fractals, 5, 26232637, doi:10.1016/0960-0779(94)E0118-9.

    • Search Google Scholar
    • Export Citation
  • Prevosto, M., , and B. Bouffandeau, 2002: Probability of occurrence of a “giant” wave crest. ASME 2002 21st Int. Conf. on Offshore Mechanics and Arctic Engineering, Oslo, Norway, American Society of Mechanical Engineers, 483–490, doi:10.1115/OMAE2002-28446.

  • Tayfun, M. A., 1980: Narrow-band nonlinear sea waves. J. Geophys. Res., 85, 15481552, doi:10.1029/JC085iC03p01548.

  • Tayfun, M. A., 2006: Statistics of nonlinear wave crests and groups. Ocean Eng., 33, 15891622, doi:10.1016/j.oceaneng.2005.10.007.

  • Tayfun, M. A., , and J. Lo, 1990: Nonlinear effects on wave envelope and phase. J. Waterw. Port Coastal Ocean Eng., 116, 79100, doi:10.1061/(ASCE)0733-950X(1990)116:1(79).

    • Search Google Scholar
    • Export Citation
  • Tayfun, M. A., , and F. Fedele, 2007: Wave-height distributions and nonlinear effects. Ocean Eng., 34, 16311649, doi:10.1016/j.oceaneng.2006.11.006.

    • Search Google Scholar
    • Export Citation
  • Watson, G. S., 1954: Extreme values in samples from m-dependent stationary stochastic processes. Ann. Math. Stat., 25, 798–800, doi:10.1214/aoms/1177728670.

  • View in gallery

    WACSIS measurements: the observed largest crest height is α = 2 times larger than the crests of the one-sided (two sided) Na ~ 50 (60) waves. Wave parameters Hs = 4.16 m, Tm = 6.6 s, and depth d = 18 m (Forristall et al. 2004).

  • View in gallery

    Unexpected crest heights in broadbanded Gaussian seas. (left) Empirical one-sided (thin dashed line with squares) and two-sided (thin solid line with plus signs, Na even) unexpected wave statistics vs (solid line) predicted theoretical unconditional return period NR in the number of waves of a wave whose crest height is α times larger than the surrounding Na waves for increasing values of α = 1.5, 2, and 2.5. The 95% confidence bands are also shown. (right) Empirical one-sided (thin dashed line with squares) and two-sided (thin solid line with plus signs, Na even) unexpected wave statistics vs theoretical predictions (solid line) of the mean crest height of a wave whose crest height is α times larger than surrounding Na waves for α = 1.5, 1.75, and 2. Sea state parameters: fully developed JONSWAP spectrum (peak enhancement factor γ = 1), mean period Tm = 8.3 s, spectral bandwidth ν = 0.35, Boccotti parameters and , and simulated ~106 waves (see left panel inset). The theoretical predictions accounting for the stochastic independence and dependence of successive crest heights are practically the same as the sea state is broadbanded.

  • View in gallery

    Conditional return period of large unexpected waves in Gaussian seas: (squares) empirical one-sided unexpected wave statistics vs (solid lines) predicted theoretical conditional return periods NR(ξ) in number of waves of unexpected waves whose crest height is greater than ξHs and α = 1.5 times larger than the surrounding Na waves for ξ = 0, 1.0, and 1.2. Note that NR(ξ = 0) is the unconditional return period NR. Sea state parameters: fully developed JONSWAP spectrum (peak enhancement factor γ = 1), mean period Tm = 8.3 s, spectral bandwidth ν = 0.35, Boccotti parameters and , and simulated ~106 waves. The predictions accounting for the stochastic independence and dependence of successive crest heights are practically the same as the sea state is broadbanded.

  • View in gallery

    Unexpected wave heights in Gaussian seas: (left) predicted theoretical unconditional return period NR in number of waves (solid line) vs empirical one-sided (plus signs) and two-sided (squares, Na even) statistics as a function of the number Na of surrounding waves for α = 1.5; (center) predicted mean unexpected wave height vs observations as a function of the return period NR. For comparison purposes, predicted mean wave height , conditional mean , and (right) threshold vs observations (circles) are also shown. Sea state parameters: fully developed JONSWAP spectrum (peak enhancement factor γ = 1), mean period Tm = 8.3 s, spectral bandwidth ν = 0.35, Boccotti parameter , and simulated ~106 waves.

  • View in gallery

    The role of stochastic wave dependence to the unexpectedness of crest heights in narrowband Gaussian seas: (thin dashed line with squares) empirical one-sided unexpected wave statistics vs (solid lines) predicted theoretical unconditional return periods NR in number of waves for (thin line) independent and (thick line) dependent crest heights of a wave whose crest height is α times larger than the surrounding Na waves; α = 1.5, 1.75, and 2. The 95% confidence bands are also shown. Sea state parameters: Gaussian spectrum with spectral bandwidth ν = 0.1 (similar to a JONSWAP spectrum with peak enhancement factor γ ~ 300), mean period Tm = 8.3 s, Boccotti parameters and , and simulated ~106 waves.

  • View in gallery

    Unexpected crest heights in unidirectional second-order random seas. Empirical one-sided (thin dashed lines with squares) unexpected wave statistics vs (thick solid lines) predicted theoretical unconditional return period NR in number of waves of a wave whose crest height is α times larger than the surrounding Na waves for increasing values of α = 1.5, 1.75, 2, 2.25, and 2.5. The 95% confidence bands are also shown. Sea state parameters: fully developed JONSWAP spectrum (peak enhancement factor γ = 1), mean period Tm = 8.3 s, spectral bandwidth ν = 0.35, Tayfun steepness μm = 0.06, and simulated ~106 waves. The theoretical predictions accounting for the stochastic independence and dependence of successive crest heights are practically the same as the sea state is broadbanded.

  • View in gallery

    (left) WACSIS predicted theoretical nonlinear unconditional return period NR in number of waves (solid line) of a wave whose crest height is α times larger than the surrounding Na waves and linear predictions (dash lines) and empirical one-sided observed statistics (squares) for α = 1.5 and 2. The 95% confidence bands are also shown. (right) As in (left), but for TERN measurements. Statistical parameters are taken from Tayfun (2006) and Tayfun and Fedele (2007).

  • View in gallery

    WACSIS unexpected wave crest heights: predicted theoretical nonlinear (solid line) and linear (dash line) mean heights and as a function of the number Na of surrounding waves vs empirical one-sided statistics (squares) for α = 1.5. Horizontal line denotes the observed maximum crest height 1.62Hs. Wave parameters Hs = 4.16 m, Tm = 6.6 s, and depth d = 18 m (Forristall et al. 2004). Average wave parameters are taken from Tayfun (2006) and Tayfun and Fedele (2007), in particular skewness λ3 ~ 0.23 and excess kurtosis λ40 ~ 0.11.

  • View in gallery

    WACSIS rogue wave: (left) predicted nonlinear theoretical return periods NR(ξ), in number of waves, of unexpected crest heights greater than ξHs and α = 2 times larger than the surrounding Na waves for ξ = 0, 1.0, 1.2, 1.4, 1.55, and 1.6 (solid lines) and (square) empirical one-sided unexpected wave statistics. Dashed vertical line denotes return period values at Na = 50. (center) Predicted nonlinear mean crest height , conditional mean , and average unexpected crest height vs their linear counterparts as a function of number of waves NR. Empirical conditional mean is also shown (circles). (right) Predicted (solid line) and empirical (circles) nonlinear threshold vs its linear counterpart as a function of NR. Dashed vertical lines denote values at NR = 4 × 104, 0.3 × 106, and 0.6 × 106. The horizontal line denotes the observed maximum crest height 1.62Hs. Average wave parameters are taken from Tayfun (2006) and Tayfun and Fedele (2007), in particular skewness λ3 ~ 0.23 and excess kurtosis λ40 ~ 0.11.

  • View in gallery

    Andrea rogue wave: (left) predicted nonlinear theoretical return periods NR(ξ), in number of waves, of unexpected crest heights greater than ξHs and α = 2 times larger than the surrounding Na waves for ξ = 0, 1.0, 1.2, 1.4, 1.55, and 1.6. Dashed vertical line denotes return period values at Na = 30. (center) Predicted nonlinear mean crest height , conditional mean , and average unexpected crest height vs their linear counterparts as a function of number of waves NR. Empirical conditional mean is also shown (circles). (right) Predicted (solid line) and empirical (circles) nonlinear threshold vs its linear counterpart as a function of NR. Dashed vertical lines denote values at NR = 2 × 104, 0.3 × 106, and 3 × 106. The horizontal line denotes the observed maximum crest height 1.63Hs. Wave parameters Hs = 9.2 m, Tm = 13.2 s, depth d = 70 m (Magnusson and Donelan 2013), skewness λ3 ~ 0.15, and excess kurtosis λ40 ~ 0.1 (Dias et al. 2015).

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 54 54 11
PDF Downloads 37 37 2

Are Rogue Waves Really Unexpected?

View More View Less
  • 1 School of Civil and Environmental Engineering, and School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia
© Get Permissions
Full access

Abstract

An unexpected wave is defined by Gemmrich and Garrett as a wave that is much taller than a set of neighboring waves. Their definition of “unexpected” refers to a wave that is not anticipated by a casual observer. Clearly, unexpected waves defined in this way are predictable in a statistical sense. They can occur relatively often with a small or moderate crest height, but large unexpected waves that are rogue are rare. Here, this concept is elaborated and statistically described based on a third-order nonlinear model. In particular, the conditional return period of an unexpected wave whose crest exceeds a given threshold is developed. This definition leads to greater return periods or on average less frequent occurrences of unexpected waves than those implied by the conventional return periods not conditioned on a reference threshold. Ultimately, it appears that a rogue wave that is also unexpected would have a lower occurrence frequency than that of a usual rogue wave. As specific applications, the Andrea and Wave Crest Sensor Intercomparison Study (WACSIS) rogue wave events are examined in detail. Both waves appeared without warning and their crests were nearly 2 times larger than the surrounding O(10) wave crests and thus unexpected. The two crest heights are nearly the same as the threshold ~ 1.6Hs exceeded on average once every 0.3 × 106 waves, where Hs is the significant wave height. In contrast, the Andrea and WACSIS events, as both rogue and unexpected, would occur slightly less often and on average once every 3 × 106 and 0.6 × 106 waves, respectively.

Corresponding author address: Francesco Fedele, Georgia Institute of Technology, 790 Atlantic Drive NW, Atlanta, GA 30332. E-mail: fedele@gatech.edu

Abstract

An unexpected wave is defined by Gemmrich and Garrett as a wave that is much taller than a set of neighboring waves. Their definition of “unexpected” refers to a wave that is not anticipated by a casual observer. Clearly, unexpected waves defined in this way are predictable in a statistical sense. They can occur relatively often with a small or moderate crest height, but large unexpected waves that are rogue are rare. Here, this concept is elaborated and statistically described based on a third-order nonlinear model. In particular, the conditional return period of an unexpected wave whose crest exceeds a given threshold is developed. This definition leads to greater return periods or on average less frequent occurrences of unexpected waves than those implied by the conventional return periods not conditioned on a reference threshold. Ultimately, it appears that a rogue wave that is also unexpected would have a lower occurrence frequency than that of a usual rogue wave. As specific applications, the Andrea and Wave Crest Sensor Intercomparison Study (WACSIS) rogue wave events are examined in detail. Both waves appeared without warning and their crests were nearly 2 times larger than the surrounding O(10) wave crests and thus unexpected. The two crest heights are nearly the same as the threshold ~ 1.6Hs exceeded on average once every 0.3 × 106 waves, where Hs is the significant wave height. In contrast, the Andrea and WACSIS events, as both rogue and unexpected, would occur slightly less often and on average once every 3 × 106 and 0.6 × 106 waves, respectively.

Corresponding author address: Francesco Fedele, Georgia Institute of Technology, 790 Atlantic Drive NW, Atlanta, GA 30332. E-mail: fedele@gatech.edu

1. Introduction

A rogue wave is defined as such if the crest-to-trough height is at least 2.2 times the significant wave height Hs or if the crest height exceeds the threshold 1.25Hs, where Hs = 4σ and σ is the standard deviation of surface elevations (Dysthe et al. 2008). Evidences given for the occurrence of such waves in nature include the Draupner and Andrea events. In particular, the Andrea wave was measured on 9 November 2007 by a system of four Optech lasers placed on a square array (LASAR) mounted on the Ekofisk platform in the North Sea in a water depth of d = 74 m (Magnusson and Donelan 2013). The Draupner freak wave was measured by Statoil at a nearby platform in January 1995 (Haver 2001; Prevosto and Bouffandeau 2002). In the last decade, the properties of the Draupner and Andrea waves have been extensively studied (Dysthe et al. 2008; Osborne 1995; Magnusson and Donelan 2013; Bitner-Gregersen et al. 2014; Dias et al. 2015; and references therein). The Andrea wave occurred during a sea state with significant wave height Hs = 4σ = 9.2 m, mean period T0 = 13.2 s, and wavelength L0 = 220 m. The Andrea crest height is h = 1.63Hs = 15 m, and the crest-to-trough height is H = 2.3Hs = 21.1 m. The sea state during which the Draupner wave occurred had a significant wave height of Hs = 11.9 m, mean period of T0 = 13.1 s, and wavelength of L0 = 250 m. The Draupner crest height is h = 18.5 m (h/Hs = 1.55), and the associated crest-to-trough height is H = 25.6 m (H/Hs = 2.15; Haver 2004; Magnusson and Donelan 2013). Observations of such large extreme waves show that they tend to extend above the surrounding smaller waves without warning and thus unexpectedly. Further, both waves were twice as high as the immediately preceding as well as following groups of waves.

In describing the unexpectedness of ocean waves, Gemmrich and Garrett (2008) define as unexpected a wave that is α times larger than a set of one-sided (preceding) waves or two-sided (preceding and following) waves (see Fig. 1). Note that their definition of unexpectedness refers to the time interval of apparent calm before or during which a wave is much taller than the neighboring waves. Hereinafter, the term “unexpected” refers to a wave that is not anticipated by a casual observer as emphasized by Gemmrich and Garrett (2010). Clearly, unexpected waves defined in this way are predictable in a statistical sense as one can estimate the associated return period or frequency of occurrence.

Fig. 1.
Fig. 1.

WACSIS measurements: the observed largest crest height is α = 2 times larger than the crests of the one-sided (two sided) Na ~ 50 (60) waves. Wave parameters Hs = 4.16 m, Tm = 6.6 s, and depth d = 18 m (Forristall et al. 2004).

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

In particular, unexpected waves occur often with a small or average wave height, but they are rarely the largest waves in a record or rogue waves (Gemmrich and Garrett 2010). In this regard, Gemmrich and Garrett (2008) performed Monte Carlo simulations of second-order nonlinear seas characterized with the typical Joint North Sea Wave Project (JONSWAP) ocean spectrum and initial homogeneous random conditions. They estimated that a wave with height at least twice that of any of the preceding 30 waves occurs once every 105 waves on average. Also unexpected crest heights are more probable than unexpected wave heights as they occur on average once every 7 × 104 in Gaussian seas and once every 104 waves in second-order nonlinear seas (see Fig. 2 in Gemmrich and Garrett 2008). Thus, their numerical predictions indicate that in weakly nonlinear seas unexpected waves occur frequently and more often than in Gaussian seas.

Further, Gemmrich and Garrett (2008) noted in their simulations that among the unexpected waves 2 times larger than the surrounding 30 waves, only about q = 10%–20% were rogues. With reference to second-order crest heights, this means that in a sample population of 106 waves a set of 100 waves are unexpected, as they occur once every 104 waves on average. However, only about 10–20 waves of the set have crest heights that are rogue, that is, larger than 1.34Hs, as according to the rogue threshold adopted by Gemmrich and Garrett (2008). This implies that unexpected wave crests that are rogue would occur less often, that is, once every 105 waves on average. Further, the percentage q of rogue occurrences can be interpreted as the probability that the crest of an unexpected wave exceeds the threshold 1.34Hs. Consequently, unexpected crest heights larger than 1.34Hs would occur rarely.

The preceding results provide the principal motivation here to consider a statistical model for describing unexpected waves and their “rogueness”. We will show that Gemmrich and Garrett’s (2008) definition of return period is unconditional. In particular, it is the harmonic mean of the return periods of all unexpected waves with any amplitude. Thus, unexpected waves of moderate amplitude occur relatively often. However, unexpected waves that are rogue have a lower occurrence frequency, and this is in agreement with Gemmrich and Garrett’s (2008) numerical predictions.

The remainder of the paper is structured as follows: First, we introduce a new theoretical model for the statistics of unexpected waves that accounts for both second- and third-order nonlinearities. We also study the effects of nonstationarity and stochastic dependence among successive waves. In particular, we present analytical solutions for the return period of unexpected waves and associated unconditional and conditional averages for crest and wave heights. Then, the conceptual framework is validated by way of Monte Carlo simulations and the theoretical predictions are compared to oceanic measurements. As a specific application here, we capitalize on the numerical simulations of the Andrea sea state (Bitner-Gregersen et al. 2014; Dias et al. 2015) and examine the unexpectedness of the Andrea wave in detail. Summary and conclusions follow subsequently.

2. Statistics of unexpected waves

Consider the exceedance probability distribution of wave crests characterized by third-order nonlinearities and described by Tayfun and Fedele (2007):
e1
where x = h/Hs is the crest amplitude h scaled by the significant wave height Hs = 4σ, and x0 follows from the quadratic equation (Tayfun 1980)
e2
Here, the wave steepness μ = λ3/3 relates to the skewness of surface elevations (Fedele and Tayfun 2009), and the parameter
e3
is a measure of third-order nonlinearities as a function of the fourth-order cumulants λnm of the wave surface η and its Hilbert transform (Tayfun and Fedele 2007). Mori and Janssen (2006) assume the following relations between cumulants:
e4
which, to date, have been proven to hold for second-order narrowband waves only (Tayfun and Lo 1990). Then, Λ in Eq. (3) is approximated in terms of the excess kurtosis λ40 by
e5
which will be used in this work. Then, Eq. (1) reduces to a modified Edgeworth–Rayleigh (MER) distribution (Mori and Janssen 2006). For realistic oceanic seas, the kurtosis λ40 is mainly affected by bound nonlinearities (Annenkov and Shrira 2014; Fedele 2015).

Consider now a time interval during which a stationary sequence of Nw = /Tm consecutive waves occurs on average. We assume that neighboring waves are stochastically independent. This assumption is convenient for the theoretical development of a probabilistic model. Furthermore, Borgman (1970, p. 54) argues, “It seems reasonable to assume that a wave height is at most interdependent with the first several wave heights occurring before and after it and essentially independent with waves further back into the past or forward into the future.” We will show later that this is justified as long as the sea state is broadbanded so that the covariance function decays sufficiently rapidly to zero after a few wave periods and successive wave peaks decorrelate faster. Thus, in a sample of Na + 1 successive waves, it is irrelevant what wave is the unexpected wave larger than the Na surrounding waves. Indeed, any wave in the sample could be “p-sided” unexpected, that is, α times larger than the previous m waves and following Nam waves, with m = 1, … Na/2 and p = Na/m. For instance, the last wave in the sample could be larger than the preceding (one sided) Na waves (m = Na and p = 1) or the central wave could extend above the preceding and following (two sided) m = Na/2 waves (p = 2 and Na even; see Fig. 1). Note that our definition of two-sided unexpectedness is different than that in Gemmrich and Garrett (2008), as they consider Na waves on each side.

Clearly, the statistics of one- and two-sided unexpected waves, or more generally the p-sided statistics, are the same if stochastic independence of successive waves holds. On this basis, the fraction of waves n(x; α, Na) that have a dimensionless crest height h/Hs within the interval (x, x + dx) and that are α times larger than any of the surrounding Na waves is given by
e6
where P(x) is the exceedance probability given in Eq. (1) and
e7
is the pdf of x. Then the probability that the crest height ξ is in (x, x + dx) follows as
e8
where n(α, Na) is the fraction of waves whose crest height is α times larger than the surrounding Na waves, namely,
e9
By definition, the unconditional return period R or the average time interval between two consecutive occurrences of the unexpected wave event is
e10
Since Tm is the mean wave period, occurs on average once every NR waves where
e11
Another statistical interpretation of the unconditional return period NR is as follows: Consider the average number of unexpected waves nj(α, Nax with a crest height between xj − Δx/2 and xj + Δx/2, where Δx ≪ 1 is small and xj are increasing amplitudes starting from x1 = Δx/2, that is, for j = 1, …, xj+1 > xj. Then,
eq1
is the return period of an unexpected wave whose crest height is nearly xj. Then, Eq. (11) is approximated as
e12
which reveals that NR is the harmonic mean of the return periods NR,j of all unexpected waves with any crest height.
The associated mean crest height of a wave α times larger than the surrounding Na waves follows from Eq. (8) as
e13
For comparison purposes, we also consider the standard statistics , hn, and h1/n for crest heights (Tayfun and Fedele 2007). In particular, is the mean maximum crest height of a sample of n waves:
e14
which admits Gumbel-type asymptotic approximations (Tayfun and Fedele 2007). Further, hn is the threshold exceeded by the 1/n fraction of largest crest heights and it satisfies
e15
where P(x) is the unconditional nonlinear probability of exceedance for crest heights given in Eq. (1). The statistic h1/n is the conditional mean , namely, the average of the 1/n fraction of largest crest heights:
e16
One can show that is always smaller than h1/n, and they tend to be the same as n increases (Tayfun and Fedele 2007).
We also consider the standard conditional return period Nh(ξ) (in number of waves) of a wave whose crest height exceeds the threshold h = ξHs, namely,
e17
where the exceedance probability P(ξ) is that in Eq. (1). From Eq. (15), the threshold hn exceeded with probability 1/n implies that Nh(hn/Hs) = n, that is, on average hn is exceeded once every n waves.
Similar statistics for the crest-to-trough height y = H/Hs of unexpected waves follow by replacing the crest exceedance probability P in Eq. (1) with the generalized Boccotti distribution (Alkhalidi and Tayfun 2013):
e18
where
eq2
and is the absolute value of the first minimum of the normalized covariance function of the zero-mean random wave process η(t), which is attained at , and is the corresponding second derivative (Boccotti 2000).

The corresponding linear statistics of unexpected wave crests follow by setting μ = 0 and Λ = 0 in Eq. (1) or Λ = 0 in Eq. (18) for wave heights. Hereinafter these will be differentiated with the superscript L. In the following, we will not dwell that much on unexpected wave heights, but our main focus will be the statistics of unexpected crests in typical oceanic sea states.

Finally, we point out that our present theory of unexpected waves can be generalized to space–time extremes drawing on Fedele (2012), but this is beyond the scope of this paper.

a. Stochastic dependence of successive waves

The statistics of unexpected waves presented so far does not take into account the stochastic dependence of neighboring waves or wave groupness. Clearly, for large crest heights, as argued by Borgman (1970), one expects that only few neighboring crests are more or less correlated (Watson 1954). To quantify this, we draw on Fedele (2005) and model a stationary sequence of wave crests as a one-step memory Markov chain, where each crest height xj is only stochastically dependent on the preceding crest height xj−1, that is,
eq3
Since the sequence is stationary, the conditional pdf p(xj|xj−1) is the same for any j, say p(x2|x1) = p(x1, x2)/p(x1), where the crest x1 precedes x2 and p(x1, x2) and p(x1) are the associated joint and marginal pdfs.
On these assumptions, following Fedele (2005), the fraction of waves n(x; α, Na) that has a dimensionless crest height within the interval (x, x + dx) and that is α times larger than any of the surrounding Na waves is given by
e19
where
eq4
and
eq5
Then, the return period R(α, Na) of unexpected wave crests follows from Eqs. (9) and (11). Clearly, if successive waves were stochastically independent, p(x1, x2) = p(x1) p(x2) and Eq. (19) reduces to Eq. (6) for the stationary case.
The theoretical probability structure of two consecutive wave crests is known for Gaussian processes, and it is given by the bivariate Rayleigh distribution (Fedele 2005):
e20
where I0(y) is the modified Bessel function, and the parameter with as the abscissa of the second absolute maximum of the normalized covariance function ψ(τ) of the zero-mean random wave process (Fedele 2005). Further, the marginal pdf
eq6
is the univariate Rayleigh distribution. As tends to zero, successive crests become stochastically independent and the sea state tends to be broadbanded. Thus, we expect that stochastic dependence of waves is dominant in very narrowband sea states, where . In particular, our numerical simulations discussed later in section 4 suggest that the dependence of consecutive crests in Gaussian seas is dominant when . This condition corresponds to unrealistic oceanic sea states characterized by a very narrowband JONSWAP spectrum with a peak enhancement factor γ > 300 and a spectral bandwidth ν < 0.01. For typical oceanic seas, ν ~ 0.3–0.5 and ~ 0.2–0.5, and successive waves can be assumed as stochastically independent.

Note that the joint pdf of consecutive Gaussian wave crests in Eq. (20) can be generalized to account for second-order bound nonlinearities following Fedele and Tayfun (2009), but this is beyond the scope of this work. Since second-order bound harmonics are phase locked to the Fourier components of the linear free surface, we expect the classical Tayfun’s (1980) enhancement of successive linear crest amplitudes, but their dependence should be unaffected by second-order nonlinearities.

b. Nonstationarity

The statistics of unexpected waves formulated so far is valid for stationary sea states. In nonstationary seas, as those during storms, our present theory can be formalized as follows: From Eq. (8), the pdf of an unexpected wave crest height h generalizes to
e21
where {bj}j=1,M are M time-varying wave parameters, for example, σ, μ, λ40, λ22, and λ04; the conditional pdf ph(x; α, Na | b1, …, bM) is the stationary pdf in Eq. (8) for given values of bj; and p(b1, …, bM) is the joint pdf of the parameters, which encodes their time variability. Equation (21) can be interpreted as the average value of ph(x; α, Na | b1, …, bM) with respect to the random variables bj, that is,
eq7
where the vector b = [b1, …, bM] and the labeled overbar denotes the statistical average with respect to b only. Taylor expanding around the mean , up to second order, yields
e22
where the superscript T denotes matrix transposition, the vector g has entries
eq8
and the Hessian matrix
eq9
Taking the averages in Eq. (22) yields
e23
where
eq10
are correlation coefficients, in particular is the variance of br. These can be easily estimated from the nonstationary time series. Thus, ph is the sum of (i) the pdf in Eq. (8) evaluated using the mean parameters and (ii) additional terms that account for the spreading of the parameters from their mean. A similar formula can be obtained for n(α, Na) in Eq. (6). The statistical moments of ph can then be obtained by integrating Eq. (23), and the nonstationary return period follows from Eq. (11).

In our applications (see section 4), time wave measurements at a point are subdivided in a sequence of optimal 30-min intervals during which the sea state can be assumed to be stationary. We observed that shorter time intervals lead to unstable estimates of higher-order moments, whereas longer intervals violate the stationarity assumption. The variability of the standard deviation σ was taken into account by normalizing the surface height measurements in each 30-min interval by the respective observed σ. In our data analysis, wave parameters are estimated as the average values over the available time record. Then, the statistics of unexpected waves can be based on Eq. (23), where the Brs terms accounting for nonstationarity can be neglected.

3. Are rogue waves really unexpected waves?

Our interest is to describe statistically the occurrence of rogue waves with crest heights larger than 1.25Hs (Dysthe et al. 2008). For example, observations indicate that the Andrea rogue wave appeared without warning suddenly, attained a crest height hobs = 1.62Hs, and it was as nearly 2 times larger than the surrounding O(30) waves (Magnusson and Donelan 2013). Thus, the Andrea wave is unexpected in accordance with the definition of Gemmrich and Garrett (2008). However, it will be discussed later in section 6, an application of our present theory using Eq. (11) predicts that a wave with a crest height at least twice as that of any of the surrounding Na = 30 waves occurs on average once every NR ~ 104 waves. This is clearly observed in the left panel of Fig. 10 (shown below). Further, the right panel of the same figure shows that the actual Andrea crest height is nearly the same as the threshold exceeded by the 1/(0.3 × 106) fraction of largest crests. Equation (17) also suggests that the Andrea wave is likely a rare event as the crest threshold 1.6Hs is exceeded once every Nh = 0.3 × 106 waves on average. In contrast, our present theory predicts that the Andrea event would occur relatively often as an unexpected wave, that is, on average once every NR ~ 104 waves.

The difference in occurrence rates is explained by first noting that the return period NR is the average time interval between two consecutive waves whose crest height h, of any possible amplitude, is α times larger than the surrounding Na wave crests. In other words, Eq. (12) reveals that NR is the harmonic mean of the return periods of all unexpected waves of any crest amplitude, and it is smaller than the return period of large (rare) unexpected waves.

In summary, unexpected waves occur relatively often with small or moderate amplitude. However, unexpected waves that are rogue are rare, in agreement with the numerical predictions by Gemmrich and Garrett (2008; see also Gemmrich and Garrett 2010).

To quantify the difference in occurrence frequencies of small and large unexpected waves, it is natural to define the conditional return period NR(ξ, α, Na) of an unexpected wave whose crest height h exceeds the threshold ξHs and is α times larger than the surrounding Na wave crests. This is given by
e24
where
e25
is the exceedance probability of the unexpected crest height h from Eq. (8). Clearly, for given α and Na, the conditional return period NR(ξ) is always greater than the unconditional NR for any ξ > 0, and they are the same if ξ = 0. The left panel of Fig. 10 (below) shows that the Andrea rogue wave as an unexpected wave that exceeds ξHs = 1.6Hs would occur rarely, that is, on average once every NR(ξ = 1.6) ~ 6 × 106. Instead, unexpected waves of any amplitude occur more often and on average once every NR ~ 104.

Clearly, the Andrea wave is both rogue and unexpected, that is, its crest is larger than the crests of surrounding waves and it exceeds the threshold 1.25Hs (Dysthe et al. 2008). What is the occurrence frequency of such a bivariate event in comparison to being only rogue as an univariate event?

From Eq. (24), the following inequality holds:
e26
where we have used n(x; α, Na) ≤ p(x) from Eq. (6). Here, Nh(ξ) is defined in Eq. (17) as the standard conditional return period (in number of waves) of a wave whose crest exceeds the threshold h = ξHs. Thus, a wave whose crest is both larger than ξHs and unexpected (as being larger than the surrounding waves) has a lower occurrence frequency than a wave whose crest is just larger than the same threshold.

The preceding results imply that a rogue wave that is also unexpected has a lower occurrence frequency than just being rogue. For example, for the Andrea sea state the return period of a crest larger than hn = 1.6Hs is Nh(hn) = 0.3 × 106. This is smaller than the return period NR of an unexpected wave exceeding the same threshold, that is, NR(ξ = 1.6) ~ 6 × 106 [see left panel of Fig. 10 (below)]. Similar conclusions hold for the Wave Crest Sensor Intercomparison Study (WACSIS) rogue wave (see section 4).

4. Verification and comparisons

a. Monte Carlo simulations of Gaussian seas

Drawing on Gemmrich and Garrett (2008), we performed Monte Carlo simulations of a Gaussian sea described by the average JONSWAP spectrum with a peak enhancement factor of γ = 1. The sea state is broadbanded with mean period Tm = 8.3 s, peak period Tp = 10 s, spectral bandwidth ν ~ 0.35, and Boccotti parameters and (see covariance function in the panel inset of Fig. 2). A long time series of wave surface displacements was randomly generated containing a total of ~106 waves, from which unexpected waves were sampled. As the sea state is broadbanded, our theoretical predictions can be based on Eqs. (9) and (11), assuming the stochastic independence of successive crest heights.

Fig. 2.
Fig. 2.

Unexpected crest heights in broadbanded Gaussian seas. (left) Empirical one-sided (thin dashed line with squares) and two-sided (thin solid line with plus signs, Na even) unexpected wave statistics vs (solid line) predicted theoretical unconditional return period NR in the number of waves of a wave whose crest height is α times larger than the surrounding Na waves for increasing values of α = 1.5, 2, and 2.5. The 95% confidence bands are also shown. (right) Empirical one-sided (thin dashed line with squares) and two-sided (thin solid line with plus signs, Na even) unexpected wave statistics vs theoretical predictions (solid line) of the mean crest height of a wave whose crest height is α times larger than surrounding Na waves for α = 1.5, 1.75, and 2. Sea state parameters: fully developed JONSWAP spectrum (peak enhancement factor γ = 1), mean period Tm = 8.3 s, spectral bandwidth ν = 0.35, Boccotti parameters and , and simulated ~106 waves (see left panel inset). The theoretical predictions accounting for the stochastic independence and dependence of successive crest heights are practically the same as the sea state is broadbanded.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

The left panel of Fig. 2 shows the empirical return period NR =R/Tm in the number of waves of both one-sided (thin dashed line) and two-sided (thin solid line) unexpected wave crests as a function of the surrounding Na waves for different values of α (Na even for the two-sided statistics). The two statistics are roughly the same with two-sided unexpected waves slightly less frequent than the one-sided waves. Note that for the two-sided unexpectedness, Gemmrich and Garrett (2008) consider Na waves on each side; thus, their two-sided return period is larger than ours. Shown in the right panel of Fig. 2 are also the empirical statistics of mean crest heights in comparison to our theoretical predictions for stochastically independent waves. In particular, we note that the mean crest height of two-sided unexpected waves is slightly smaller than that of one-sided waves, especially as α increases. Further, in Fig. 3 there are shown the predicted conditional return periods NR(ξ) (solid lines) of an unexpected wave whose crest height is greater than ξHs for ξ = 0, 1.0, and 1.2 (α = 1.5). Note that NR(ξ = 0) is the unconditional return period NR. We find a fair agreement with the empirical one-sided unexpected wave statistics (squares). For α = 2 and Na = 30, our predicted return period is NR ~ 6 × 104 and in fair agreement with the linear predictions (~7 × 104) by Gemmrich and Garrett (2008) as shown in their Fig. 2. In regards to unexpected crest-to-trough heights, our theoretical model fairly predicts the empirical wave height statistics from simulations as clearly seen in Fig. 4.

Fig. 3.
Fig. 3.

Conditional return period of large unexpected waves in Gaussian seas: (squares) empirical one-sided unexpected wave statistics vs (solid lines) predicted theoretical conditional return periods NR(ξ) in number of waves of unexpected waves whose crest height is greater than ξHs and α = 1.5 times larger than the surrounding Na waves for ξ = 0, 1.0, and 1.2. Note that NR(ξ = 0) is the unconditional return period NR. Sea state parameters: fully developed JONSWAP spectrum (peak enhancement factor γ = 1), mean period Tm = 8.3 s, spectral bandwidth ν = 0.35, Boccotti parameters and , and simulated ~106 waves. The predictions accounting for the stochastic independence and dependence of successive crest heights are practically the same as the sea state is broadbanded.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

Fig. 4.
Fig. 4.

Unexpected wave heights in Gaussian seas: (left) predicted theoretical unconditional return period NR in number of waves (solid line) vs empirical one-sided (plus signs) and two-sided (squares, Na even) statistics as a function of the number Na of surrounding waves for α = 1.5; (center) predicted mean unexpected wave height vs observations as a function of the return period NR. For comparison purposes, predicted mean wave height , conditional mean , and (right) threshold vs observations (circles) are also shown. Sea state parameters: fully developed JONSWAP spectrum (peak enhancement factor γ = 1), mean period Tm = 8.3 s, spectral bandwidth ν = 0.35, Boccotti parameter , and simulated ~106 waves.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

In the above comparisons, the fair agreement with our theoretical predictions indicates that the stochastic independence of waves holds approximately as the sea state is broadbanded. However, in very narrowband seas the stochastic dependence of neighboring waves cannot be neglected. Indeed, consider a linear sea state characterized with a Gaussian spectrum with spectral bandwidth ν = 0.1. This is similar to an unrealistic JONSWAP spectrum with a peak enhancement factor γ ~ 300. From the panel inset of Fig. 5, the Boccotti parameters are and , indicating a strong correlation between consecutive waves. Indeed, from the same figure the empirical one-sided (square) unexpected wave statistics tends to agree with our predicted theoretical return period NR for dependent waves (thick solid line) computed using Eqs. (19) and (11). Instead, our predictions for independent waves (thin solid line) are less conservative, where we use Eqs. (9) and (11).

Fig. 5.
Fig. 5.

The role of stochastic wave dependence to the unexpectedness of crest heights in narrowband Gaussian seas: (thin dashed line with squares) empirical one-sided unexpected wave statistics vs (solid lines) predicted theoretical unconditional return periods NR in number of waves for (thin line) independent and (thick line) dependent crest heights of a wave whose crest height is α times larger than the surrounding Na waves; α = 1.5, 1.75, and 2. The 95% confidence bands are also shown. Sea state parameters: Gaussian spectrum with spectral bandwidth ν = 0.1 (similar to a JONSWAP spectrum with peak enhancement factor γ ~ 300), mean period Tm = 8.3 s, Boccotti parameters and , and simulated ~106 waves.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

b. Monte Carlo simulations of second-order random seas

Drawing on Tayfun and Fedele (2007), we performed Monte Carlo simulations of unidirectional second-order broadband random seas in deep water described by the same average JONSWAP spectrum introduced in the previous section for simulating Gaussian seas. The associated wave steepness μ = λ3/3 ~ 0.06, where λ3 is the skewness of surface elevations (Fedele and Tayfun 2009). Our theoretical predictions are based on Eqs. (9) and (11) and assume the stochastic independence of successive crest heights as the sea state is broadbanded.

In Fig. 6, the comparison between the empirical return period NR = R/Tm in the number of waves of one-sided (squares) unexpected wave crests and theoretical predictions from our model is shown as a function of the surrounding Na waves for different values of α. For α = 2 and Na = 30, our predicted second-order return period NR ~ 2 × 104 is shorter than the linear counterpart (~6 × 104) for Gaussian seas (see Fig. 2) as nonlinearities enhance crest heights (Tayfun and Fedele 2007; Fedele and Tayfun 2009). Further, our second-order predictions fairly agree with those by Gemmrich and Garrett (2008) in their Fig. 2. For example, they predict a slightly shorter nonlinear period NR ~ 104 for α = 2 and Na = 30. This is because their second-order correction for crest heights is based on the narrowband assumption of the sea state. This yields a slight overestimation of crest heights shortening NR. In contrast, our simulated sea states are based on the exact second-order solution for unidirectional broadband waves in deep water (Tayfun 1980).

Fig. 6.
Fig. 6.

Unexpected crest heights in unidirectional second-order random seas. Empirical one-sided (thin dashed lines with squares) unexpected wave statistics vs (thick solid lines) predicted theoretical unconditional return period NR in number of waves of a wave whose crest height is α times larger than the surrounding Na waves for increasing values of α = 1.5, 1.75, 2, 2.25, and 2.5. The 95% confidence bands are also shown. Sea state parameters: fully developed JONSWAP spectrum (peak enhancement factor γ = 1), mean period Tm = 8.3 s, spectral bandwidth ν = 0.35, Tayfun steepness μm = 0.06, and simulated ~106 waves. The theoretical predictions accounting for the stochastic independence and dependence of successive crest heights are practically the same as the sea state is broadbanded.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

c. Oceanic observations

We will analyze two datasets. The first comprises 9 h of measurements gathered during a severe storm in January 1993 with a Marex radar from the Tern platform located in the northern North Sea in a water depth of d = 167 m. We refer to Forristall (2000) for further details on the dataset, hereinafter referred to as TERN. The second dataset is from WACSIS (Forristall et al. 2004). It consists of 5 h of measurements gathered in January 1998 with a Baylor wave staff from Meetpost Noordwijk in the southern North Sea (average water depth d = 18 m). Tayfun (2006) and Tayfun and Fedele (2007) elaborated both datasets and provided accurate estimates of statistical parameters, especially skewness and fourth-order cumulants that will be used in this work. The data analysis indicates that the statistics of unexpected waves can be based on Eq. (23), where the Brs terms accounting for nonstationarity are neglected. Further, successive waves can be assumed as stochastically independent as both sea states are broadbanded, as indicated by their estimated covariance functions (see panel insets in Fig. 7).

Fig. 7.
Fig. 7.

(left) WACSIS predicted theoretical nonlinear unconditional return period NR in number of waves (solid line) of a wave whose crest height is α times larger than the surrounding Na waves and linear predictions (dash lines) and empirical one-sided observed statistics (squares) for α = 1.5 and 2. The 95% confidence bands are also shown. (right) As in (left), but for TERN measurements. Statistical parameters are taken from Tayfun (2006) and Tayfun and Fedele (2007).

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

In regards to WACSIS measurements, the left panel of Fig. 7 compares the theoretical nonlinear return period NR (solid line) of unexpected wave crests α times larger than the surrounding Na waves, the respective linear predictions (dashed line) and the WACSIS empirical one-sided statistics for α = 1.5, 2 (dashed line with squares). The right panel of the same figure shows similar comparisons for TERN. The observed occurrence rates are close to the theoretical predictions, indicating that the assumption of stochastic independence of waves holds approximately. It is noticed that nonlinearities tend to reduce the return period of unexpected waves and increase their mean crest amplitudes. In particular, in the left panel of Fig. 8 we compare our predicted nonlinear (solid line) and linear (dashed line) mean crest heights and versus the WACSIS empirical one-sided statistics (squares) for α = 1.5. Clearly, our linear predictions underestimate the observed crest amplitudes, as expected. Indeed, it is well established that nonlinearities must be accounted for to obtain reliable wave crest statistics (Tayfun 1980; Forristall 2000; Tayfun and Fedele 2007; Fedele and Tayfun 2009). Similar trend is also observed for the WACSIS rogue wave as evident from the center panel of Fig. 9 (shown below). Here, our nonlinear predicted mean crest height , conditional mean , and mean unexpected crest height versus their linear counterparts are shown. The right panel of the same figure depicts the nonlinear threshold in comparison to its linear counterpart. The nonlinear and linear predictions for the Andrea rogue wave are also shown in Fig. 10 below.

Fig. 8.
Fig. 8.

WACSIS unexpected wave crest heights: predicted theoretical nonlinear (solid line) and linear (dash line) mean heights and as a function of the number Na of surrounding waves vs empirical one-sided statistics (squares) for α = 1.5. Horizontal line denotes the observed maximum crest height 1.62Hs. Wave parameters Hs = 4.16 m, Tm = 6.6 s, and depth d = 18 m (Forristall et al. 2004). Average wave parameters are taken from Tayfun (2006) and Tayfun and Fedele (2007), in particular skewness λ3 ~ 0.23 and excess kurtosis λ40 ~ 0.11.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

Fig. 9.
Fig. 9.

WACSIS rogue wave: (left) predicted nonlinear theoretical return periods NR(ξ), in number of waves, of unexpected crest heights greater than ξHs and α = 2 times larger than the surrounding Na waves for ξ = 0, 1.0, 1.2, 1.4, 1.55, and 1.6 (solid lines) and (square) empirical one-sided unexpected wave statistics. Dashed vertical line denotes return period values at Na = 50. (center) Predicted nonlinear mean crest height , conditional mean , and average unexpected crest height vs their linear counterparts as a function of number of waves NR. Empirical conditional mean is also shown (circles). (right) Predicted (solid line) and empirical (circles) nonlinear threshold vs its linear counterpart as a function of NR. Dashed vertical lines denote values at NR = 4 × 104, 0.3 × 106, and 0.6 × 106. The horizontal line denotes the observed maximum crest height 1.62Hs. Average wave parameters are taken from Tayfun (2006) and Tayfun and Fedele (2007), in particular skewness λ3 ~ 0.23 and excess kurtosis λ40 ~ 0.11.

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

Fig. 10.
Fig. 10.

Andrea rogue wave: (left) predicted nonlinear theoretical return periods NR(ξ), in number of waves, of unexpected crest heights greater than ξHs and α = 2 times larger than the surrounding Na waves for ξ = 0, 1.0, 1.2, 1.4, 1.55, and 1.6. Dashed vertical line denotes return period values at Na = 30. (center) Predicted nonlinear mean crest height , conditional mean , and average unexpected crest height vs their linear counterparts as a function of number of waves NR. Empirical conditional mean is also shown (circles). (right) Predicted (solid line) and empirical (circles) nonlinear threshold vs its linear counterpart as a function of NR. Dashed vertical lines denote values at NR = 2 × 104, 0.3 × 106, and 3 × 106. The horizontal line denotes the observed maximum crest height 1.63Hs. Wave parameters Hs = 9.2 m, Tm = 13.2 s, depth d = 70 m (Magnusson and Donelan 2013), skewness λ3 ~ 0.15, and excess kurtosis λ40 ~ 0.1 (Dias et al. 2015).

Citation: Journal of Physical Oceanography 46, 5; 10.1175/JPO-D-15-0137.1

We observe that the empirical statistics tend to deviate from the theoretical predictions for large values of α and Na. In particular, for both TERN and WACSIS we could not produce statistically stable estimates of extreme values for Na > 10 when α > 1.5 because of the limited number of waves in the time series [O(103) waves in comparison to the 106 waves of the simulated Gaussian seas]. Nevertheless, the agreement between our present theory and observations is satisfactory, and it also provides evidence that successive waves in the samples are approximately stochastically independent.

5. How rogue are unexpected waves?

WACSIS observations indicate that the actual largest crest hobs is 1.62Hs. Figure 1 shows that the WACSIS rogue wave is also unexpected as it is α = 2 times larger than the surrounding Na ~ 50 waves. According to our statistical model, such unexpected wave would occur often and on average once every NR = 4 × 104 waves, as seen in the left panel of Fig. 9. Here, we report the theoretical predictions of the unconditional nonlinear return period NR as a function of Na using Eqs. (9) and (11). Further, from the center panel of Fig. 9, it is seen that the associated, average, nonlinear, unexpected crest height is about 1.35Hs and smaller than the conditional mean , which is slightly larger than the mean maximum crest height of NR = 4 × 104 waves. Note that these average values underestimate the actual maximum crest amplitude hobs ~ 1.62Hs observed. In contrast, the right panel of Fig. 9 shows that hobs is nearly the same as the threshold = 1.6Hs exceeded on average once every Nh = 0.3 × 106 waves [see Eq. (17)].

We have seen that a correct statistical interpretation of the WACSIS rogue wave as an unexpected event requires considering the conditional return period NR(ξ) of an unexpected wave whose crest height is larger than ξHs [see Eq. (24)]. In particular, the left panel of Fig. 9 depicts plots of NR(ξ) as a function of Na for increasing values of ξ = 1, 1.2, 1.4, 1.55, and 1.6 (α = 2). For ξ = 1.6Hs, we find that an unexpected wave exceeding this threshold and standing above Na = 50 waves would occur rarely and once every NR(ξ = 1.6) ~ 0.6 × 106, in contrast to the smaller unconditional value NR ~ 4 × 104.

In summary, the WACSIS wave crest as both unexpected and rogue, that is, 2 times larger than Na = 50 surrounding waves and exceeding the 1.6Hs, would occur once every NR = 0.6 × 106 waves on average. In contrast, the WACSIS wave as a rogue event has a crest height that is nearly the same as the threshold = 1.6Hs exceeded on average once every Nh = 0.3 × 106 waves [see Eq. (17)]. Thus, the WACSIS rogue wave has a slightly greater occurrence frequency than being both rogue and unexpected since Nh < NR = 0.6 × 106. This implies that the threshold exceeded by the 1/NR fraction of the largest crests is larger than 1.6Hs and nearly the same as 1.65Hs.

6. The Andrea rogue wave and its unexpectedness

As a specific application of the present theoretical framework, the unexpected wave statistics of the 2007 Andrea rogue wave event is examined. The actual largest crest height hobs is 1.63Hs and nearly 2 times larger than the surrounding O(30) waves (see Fig. 12 in Magnusson and Donelan 2013). For the hindcast Andrea sea state, the left panel of Fig. 10 shows the unconditional and conditional nonlinear return periods NR and NR(ξ) as a function of Na. In particular, according to our statistical model, the theoretical predictions indicate that a wave with a crest height at least twice that of any of the surrounding Na = 30 waves occurs on average once every NR = 2 × 104 waves irrespective of its crest amplitude. In contrast, an unexpected wave whose crest height exceeds the threshold 1.6Hs occurs less often since our predicted conditional return period NR(ξ = 1.6) ~ 3 × 106 is greater than the unconditional counterpart NR = 2 × 104, as seen in the left panel of Fig. 10. Furthermore, the crest height 1.6Hs is nearly the same as the threshold exceeded on average once every Nh = 0.3 × 106 waves [see Eq. (17)], as indicated in the right panel of the same figure. Thus, the Andrea wave has a greater occurrence rate than being both rogue and unexpected since Nh < NR = 3 × 106, implying the larger threshold .

7. Concluding remarks

We have presented a third-order nonlinear model for the statistics of unexpected waves. Gemmrich and Garrett (2008) define a wave that is taller than a set of neighboring waves as unexpected. The term unexpected refers to a wave that is not foreseen by a casual observer (Gemmrich and Garrett 2010). Clearly, unexpected waves are predictable in a statistical sense. Indeed, they can occur relatively often with a small or moderate crest height. However, unexpected waves that are rogue are rare. This difference in occurrence frequencies is quantified by introducing the conditional return period of an unexpected wave that exceeds a given threshold crest height. The associated unconditional return period is smaller than the conditional counterpart as it refers to the harmonic mean of the return periods of unexpected waves of any crest amplitude.

Furthermore, our analysis indicates that a wave that is both rogue and unexpected has a lower occurrence frequency than just being rogue. This is proven both analytically and verified by way of an analysis of the Andrea and WACSIS rogue wave events. Both waves appeared without warning and their crests were nearly 2 times larger than the surrounding O(10) wave crests and thus unexpected. The two crest heights are nearly the same as the threshold ~ 1.6Hs exceeded on average once every 0.3 × 106 waves. In contrast, the Andrea and WACSIS events would occur less often being both unexpected and rogue, that is, on average once every 3 × 106 and 0.6 × 106, respectively.

Finally, we point out that our statistical model for unexpected waves supports and goes beyond the analysis by Gemmrich and Garrett (2008) based on Monte Carlo simulations. In particular, our statistical approach can be used in operational wave forecast models to predict the unexpectedness of ocean waves.

Acknowledgments

FF is grateful to George Z. Forristall and M. Aziz Tayfun for sharing the wave measurements utilized in this study. FF thanks Michael Banner, George Forristall, Peter A. E. M. Janssen, Victor Shrira, and M. Aziz Tayfun for discussions on nonlinear wave statistics. FF also thanks M. Aziz Tayfun for sharing his numerical solver for simulating second-order nonlinear waves. Further, FF thanks Michael Banner and M. Aziz Tayfun for revising an early draft of the manuscript as well as Guillermo Gallego for his support with LaTeX. The comments of an anonymous referee are acknowledged gratefully. FF also acknowledges partial support from NSF Grant CCF-1347191.

REFERENCES

  • Alkhalidi, M. A., , and M. A. Tayfun, 2013: Generalized Boccotti distribution for nonlinear wave heights. Ocean Eng., 74, 101106, doi:10.1016/j.oceaneng.2013.09.014.

    • Search Google Scholar
    • Export Citation
  • Annenkov, S. Y., , and V. I. Shrira, 2014: Evaluation of skewness and kurtosis of wind waves parameterized by JONSWAP spectra. J. Phys. Oceanogr., 44, 1582–1594, doi:10.1175/JPO-D-13-0218.1.

  • Bitner-Gregersen, E. M., , L. Fernandez, , J. M. Lefèvre, , J. Monbaliu, , and A. Toffoli, 2014: The North Sea Andrea storm and numerical simulations. Nat. Hazards Earth Syst. Sci., 14, 14071415, doi:10.5194/nhess-14-1407-2014.

    • Search Google Scholar
    • Export Citation
  • Boccotti, P., 2000: Wave Mechanics for Ocean Engineering. Elsevier, 496 pp.

  • Borgman, L. E., 1970: Maximum wave height probabilities for a random number of random intensity storms. Proc.12th Conf. on Coastal Engineering, Vol. 12, Washington, D.C., ASCE, 53–64. [Available online at https://journals.tdl.org/icce/index.php/icce/article/view/2608.]

  • Dias, F., , J. Brennan, , S. Ponce de Leon, , C. Clancy, , and J. Dudley, 2015: Local analysis of wave fields produced from hindcasted rogue wave sea states. ASME 2015 34th Int. Conf. on Ocean, Offshore and Arctic Engineering, St. John’s, Newfoundland, Canada, American Society of Mechanical Engineers, OMAE2015-41458, doi:10.1115/OMAE2015-41458.

  • Dysthe, K. B., , H. E. Krogstad, , and P. Muller, 2008: Oceanic rogue waves. Annu. Rev. Fluid Mech., 40, 287310, doi:10.1146/annurev.fluid.40.111406.102203.

    • Search Google Scholar
    • Export Citation
  • Fedele, F., 2005: Successive wave crests in Gaussian Seas. Probab. Eng. Mech., 20, 355363, doi:10.1016/j.probengmech.2004.05.008.

  • Fedele, F., 2012: Space–time extremes in short-crested storm seas. J. Phys. Oceanogr., 42, 1601–1615, doi:10.1175/JPO-D-11-0179.1.

  • Fedele, F., 2015: On the kurtosis of ocean waves in deep water. J. Fluid Mech., 782, 2536, doi:10.1017/jfm.2015.538.

  • Fedele, F., , and M. A. Tayfun, 2009: On nonlinear wave groups and crest statistics. J. Fluid Mech., 620, 221239, doi:10.1017/S0022112008004424.

    • Search Google Scholar
    • Export Citation
  • Forristall, G. Z., 2000: Wave crest distributions: Observations and second-order theory. J. Phys. Oceanogr., 30, 1931–1943, doi:10.1175/1520-0485(2000)030<1931:WCDOAS>2.0.CO;2.

  • Forristall, G. Z., , S. F. Barstow, , H. E. Krogstad, , M. Prevosto, , P. H. Taylor, , and P. S. Tromans, 2004: Wave crest sensor intercomparison study: An overview of WACSIS. J. Offshore Mech. Arct. Eng., 126, 26–34, doi:10.1115/1.1641388.

  • Gemmrich, J., , and C. Garrett, 2008: Unexpected waves. J. Phys. Oceanogr., 38, 23302336, doi:10.1175/2008JPO3960.1.

  • Gemmrich, J., , and C. Garrett, 2010: Unexpected waves: Intermediate depth simulations and comparison with observations. Ocean Eng., 37, 262267, doi:10.1016/j.oceaneng.2009.10.007.

    • Search Google Scholar
    • Export Citation
  • Haver, S., 2001: Evidences of the existence of freak waves. Proc. Rogue Waves 2000, Brest, France, Ifremer, 129–140.

  • Haver, S., 2004: A possible freak wave event measured at the Draupner jacket January 1 1995. Proc. Rogue Waves 2004, Brest, France, Ifremer, 1–8. [Available online at http://www.ifremer.fr/web-com/stw2004/rw/fullpapers/walk_on_haver.pdf.]

  • Magnusson, K. A., , and M. A. Donelan, 2013: The Andrea wave characteristics of a measured North Sea rogue wave. J. Offshore Mech. Arct. Eng., 135, 031108, doi:10.1115/1.4023800.

  • Mori, N., , and P. A. E. M. Janssen, 2006: On kurtosis and occurrence probability of freak waves. J. Phys. Oceanogr., 36, 1471–1483, doi:10.1175/JPO2922.1.

  • Osborne, A. R., 1995: The numerical inverse scattering transform: Nonlinear Fourier analysis and nonlinear filtering of oceanic surface waves. Chaos Solitons Fractals, 5, 26232637, doi:10.1016/0960-0779(94)E0118-9.

    • Search Google Scholar
    • Export Citation
  • Prevosto, M., , and B. Bouffandeau, 2002: Probability of occurrence of a “giant” wave crest. ASME 2002 21st Int. Conf. on Offshore Mechanics and Arctic Engineering, Oslo, Norway, American Society of Mechanical Engineers, 483–490, doi:10.1115/OMAE2002-28446.

  • Tayfun, M. A., 1980: Narrow-band nonlinear sea waves. J. Geophys. Res., 85, 15481552, doi:10.1029/JC085iC03p01548.

  • Tayfun, M. A., 2006: Statistics of nonlinear wave crests and groups. Ocean Eng., 33, 15891622, doi:10.1016/j.oceaneng.2005.10.007.

  • Tayfun, M. A., , and J. Lo, 1990: Nonlinear effects on wave envelope and phase. J. Waterw. Port Coastal Ocean Eng., 116, 79100, doi:10.1061/(ASCE)0733-950X(1990)116:1(79).

    • Search Google Scholar
    • Export Citation
  • Tayfun, M. A., , and F. Fedele, 2007: Wave-height distributions and nonlinear effects. Ocean Eng., 34, 16311649, doi:10.1016/j.oceaneng.2006.11.006.

    • Search Google Scholar
    • Export Citation
  • Watson, G. S., 1954: Extreme values in samples from m-dependent stationary stochastic processes. Ann. Math. Stat., 25, 798–800, doi:10.1214/aoms/1177728670.

Save