## 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 *H*_{s} or if the crest height exceeds the threshold 1.25*H*_{s}, where *H*_{s} = 4*σ* 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 *H*_{s} = 4*σ* = 9.2 m, mean period *T*_{0} = 13.2 s, and wavelength *L*_{0} = 220 m. The Andrea crest height is *h* = 1.63*H*_{s} = 15 m, and the crest-to-trough height is *H* = 2.3*H*_{s} = 21.1 m. The sea state during which the Draupner wave occurred had a significant wave height of *H*_{s} = 11.9 m, mean period of *T*_{0} = 13.1 s, and wavelength of *L*_{0} = 250 m. The Draupner crest height is *h* = 18.5 m (*h*/*H*_{s} = 1.55), and the associated crest-to-trough height is *H* = 25.6 m (*H*/*H*_{s} = 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.

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 10^{5} waves on average. Also unexpected crest heights are more probable than unexpected wave heights as they occur on average once every 7 × 10^{4} in Gaussian seas and once every 10^{4} 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 10^{6} waves a set of 100 waves are unexpected, as they occur once every 10^{4} waves on average. However, only about 10–20 waves of the set have crest heights that are rogue, that is, larger than 1.34*H*_{s}, 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 10^{5} 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 1.34*H*_{s}. Consequently, unexpected crest heights larger than 1.34*H*_{s} 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

*x*=

*h*/

*H*

_{s}is the crest amplitude

*h*scaled by the significant wave height

*H*

_{s}= 4

*σ*, and

*x*

_{0}follows from the quadratic equation (Tayfun 1980)Here, the wave steepness

*μ*=

*λ*

_{3}/3 relates to the skewness of surface elevations (Fedele and Tayfun 2009), and the parameteris a measure of third-order nonlinearities as a function of the fourth-order cumulants

*λ*

_{nm}of the wave surface

*η*and its Hilbert transform

*λ*

_{40}bywhich 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

*λ*

_{40}is mainly affected by bound nonlinearities (Annenkov and Shrira 2014; Fedele 2015).

Consider now a time interval *N*_{w} = *T*_{m} 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 *N*_{a} + 1 successive waves, it is irrelevant what wave is the unexpected wave larger than the *N*_{a} 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 *N*_{a} − *m* waves, with *m* = 1, … *N*_{a}/2 and *p* = *N*_{a}/*m*. For instance, the last wave in the sample could be larger than the preceding (one sided) *N*_{a} waves (*m* = *N*_{a} and *p* = 1) or the central wave could extend above the preceding and following (two sided) *m* = *N*_{a}/2 waves (*p* = 2 and *N*_{a} even; see Fig. 1). Note that our definition of two-sided unexpectedness is different than that in Gemmrich and Garrett (2008), as they consider *N*_{a} waves on each side.

*p*-sided statistics, are the same if stochastic independence of successive waves holds. On this basis, the fraction of waves

*n*(

*x*;

*α*,

*N*

_{a}) that have a dimensionless crest height

*h*/

*H*

_{s}within the interval (

*x*,

*x*+

*dx*) and that are

*α*times larger than any of the surrounding

*N*

_{a}waves is given bywhere

*P*(

*x*) is the exceedance probability given in Eq. (1) andis the pdf of

*x*. Then the probability that the crest height

*ξ*is in (

*x*,

*x*+

*dx*) follows aswhere

*n*(

*α*,

*N*

_{a}) is the fraction of waves whose crest height is

*α*times larger than the surrounding

*N*

_{a}waves, namely,By definition, the unconditional return period

*R*or the average time interval between two consecutive occurrences of the unexpected wave event

*T*

_{m}is the mean wave period,

*N*

_{R}waves where

*N*

_{R}is as follows: Consider the average number of unexpected waves

*n*

_{j}(

*α*,

*N*

_{a})Δ

*x*with a crest height between

*x*

_{j}− Δ

*x*/2 and

*x*

_{j}+ Δ

*x*/2, where Δ

*x*≪ 1 is small and

*x*

_{j}are increasing amplitudes starting from

*x*

_{1}= Δ

*x*/2, that is, for

*j*= 1, …,

*x*

_{j+1}>

*x*

_{j}. Then,is the return period of an unexpected wave whose crest height is nearly

*x*

_{j}. Then, Eq. (11) is approximated aswhich reveals that

*N*

_{R}is the harmonic mean of the return periods

*N*

_{R,j}of all unexpected waves with any crest height.

*α*times larger than the surrounding

*N*

_{a}waves follows from Eq. (8) as

*h*

_{n}, and

*h*

_{1/n}for crest heights (Tayfun and Fedele 2007). In particular,

*n*waves:which admits Gumbel-type asymptotic approximations (Tayfun and Fedele 2007). Further,

*h*

_{n}is the threshold exceeded by the 1/

*n*fraction of largest crest heights and it satisfieswhere

*P*(

*x*) is the unconditional nonlinear probability of exceedance for crest heights given in Eq. (1). The statistic

*h*

_{1/n}is the conditional mean

*n*fraction of largest crest heights:One can show that

*h*

_{1/n}, and they tend to be the same as

*n*increases (Tayfun and Fedele 2007).

*N*

_{h}(

*ξ*) (in number of waves) of a wave whose crest height exceeds the threshold

*h*=

*ξH*

_{s}, namely,where the exceedance probability

*P*(

*ξ*) is that in Eq. (1). From Eq. (15), the threshold

*h*

_{n}exceeded with probability 1/

*n*implies that

*N*

_{h}(

*h*

_{n}/

*H*

_{s}) =

*n*, that is, on average

*h*

_{n}is exceeded once every

*n*waves.

*y*=

*H*/

*H*

_{s}of unexpected waves follow by replacing the crest exceedance probability

*P*in Eq. (1) with the generalized Boccotti distribution (Alkhalidi and Tayfun 2013):whereand

*η*(

*t*), which is attained at

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

*x*

_{j}is only stochastically dependent on the preceding crest height

*x*

_{j−1}, that is,Since the sequence is stationary, the conditional pdf

*p*(

*x*

_{j}|

*x*

_{j−1}) is the same for any

*j*, say

*p*(

*x*

_{2}|

*x*

_{1}) =

*p*(

*x*

_{1},

*x*

_{2})/

*p*(

*x*

_{1}), where the crest

*x*

_{1}precedes

*x*

_{2}and

*p*(

*x*

_{1},

*x*

_{2}) and

*p*(

*x*

_{1}) are the associated joint and marginal pdfs.

*n*(

*x*;

*α*,

*N*

_{a}) that has a dimensionless crest height within the interval (

*x*,

*x*+

*dx*) and that is

*α*times larger than any of the surrounding

*N*

_{a}waves is given bywhereandThen, the return period

*R*(

*α*,

*N*

_{a}) of unexpected wave crests follows from Eqs. (9) and (11). Clearly, if successive waves were stochastically independent,

*p*(

*x*

_{1},

*x*

_{2}) =

*p*(

*x*

_{1})

*p*(

*x*

_{2}) and Eq. (19) reduces to Eq. (6) for the stationary case.

*I*

_{0}(

*y*) is the modified Bessel function, and the parameter

*ψ*(

*τ*) of the zero-mean random wave process (Fedele 2005). Further, the marginal pdfis the univariate Rayleigh distribution. As

*γ*> 300 and a spectral bandwidth

*ν*< 0.01. For typical oceanic seas,

*ν*~ 0.3–0.5 and

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

*h*generalizes towhere {

*b*

_{j}}

_{j=1,M}are

*M*time-varying wave parameters, for example,

*σ*,

*μ*,

*λ*

_{40},

*λ*

_{22}, and

*λ*

_{04}; the conditional pdf

*p*

_{h}(

*x*;

*α*,

*N*

_{a}|

*b*

_{1}, …,

*b*

_{M}) is the stationary pdf in Eq. (8) for given values of

*b*

_{j}; and

*p*(

*b*

_{1}, …,

*b*

_{M}) is the joint pdf of the parameters, which encodes their time variability. Equation (21) can be interpreted as the average value of

*p*

_{h}(

*x*;

*α*,

*N*

_{a}|

*b*

_{1}, …,

*b*

_{M}) with respect to the random variables

*b*

_{j}, that is,where the vector

**b**= [

*b*

_{1}, …,

*b*

_{M}] and the labeled overbar denotes the statistical average with respect to

**b**only. Taylor expanding around the mean

*T*denotes matrix transposition, the vector

**g**has entriesand the Hessian matrixTaking the averages in Eq. (22) yieldswhereare correlation coefficients, in particular

*b*

_{r}. These can be easily estimated from the nonstationary time series. Thus,

*p*

_{h}is the sum of (i) the pdf in Eq. (8) evaluated using the mean parameters

*n*(

*α*,

*N*

_{a}) in Eq. (6). The statistical moments of

*p*

_{h}can then be obtained by integrating Eq. (23), and the nonstationary return period

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 *B*_{rs} 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 1.25*H*_{s} (Dysthe et al. 2008). For example, observations indicate that the Andrea rogue wave appeared without warning suddenly, attained a crest height *h*_{obs} = 1.62*H*_{s}, 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 *N*_{a} = 30 waves occurs on average once every *N*_{R} ~ 10^{4} 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 ^{6}) fraction of largest crests. Equation (17) also suggests that the Andrea wave is likely a rare event as the crest threshold 1.6*H*_{s} is exceeded once every *N*_{h} = 0.3 × 10^{6} 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 *N*_{R} ~ 10^{4} waves.

The difference in occurrence rates is explained by first noting that the return period *N*_{R} is the average time interval between two consecutive waves whose crest height *h*, of any possible amplitude, is *α* times larger than the surrounding *N*_{a} wave crests. In other words, Eq. (12) reveals that *N*_{R} 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).

*N*

_{R}(

*ξ*,

*α*,

*N*

_{a}) of an unexpected wave whose crest height

*h*exceeds the threshold

*ξH*

_{s}and is

*α*times larger than the surrounding

*N*

_{a}wave crests. This is given bywhereis the exceedance probability of the unexpected crest height

*h*from Eq. (8). Clearly, for given

*α*and

*N*

_{a}, the conditional return period

*N*

_{R}(

*ξ*) is always greater than the unconditional

*N*

_{R}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

*ξH*

_{s}= 1.6

*H*

_{s}would occur rarely, that is, on average once every

*N*

_{R}(

*ξ*= 1.6) ~ 6 × 10

^{6}. Instead, unexpected waves of any amplitude occur more often and on average once every

*N*

_{R}~ 10

^{4}.

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 1.25*H*_{s} (Dysthe et al. 2008). What is the occurrence frequency of such a bivariate event in comparison to being only rogue as an univariate event?

*n*(

*x*;

*α*,

*N*

_{a}) ≤

*p*(

*x*) from Eq. (6). Here,

*N*

_{h}(

*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 = ξH*

_{s}. Thus, a wave whose crest is both larger than

*ξH*

_{s}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 *h*_{n} = 1.6*H*_{s} is *N*_{h}(*h*_{n}) = 0.3 × 10^{6}. This is smaller than the return period *N*_{R} of an unexpected wave exceeding the same threshold, that is, *N*_{R}(*ξ* = 1.6) ~ 6 × 10^{6} [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 *T*_{m} = 8.3 s, peak period *T*_{p} = 10 s, spectral bandwidth *ν* ~ 0.35, and Boccotti parameters ^{6} 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.

The left panel of Fig. 2 shows the empirical return period *N*_{R} =*R*/*T*_{m} 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 *N*_{a} waves for different values of *α* (*N*_{a} 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 *N*_{a} 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 *N*_{R}(*ξ*) (solid lines) of an unexpected wave whose crest height is greater than *ξH*_{s} for *ξ* = 0, 1.0, and 1.2 (*α* = 1.5). Note that *N*_{R}(*ξ* = 0) is the unconditional return period *N*_{R}. We find a fair agreement with the empirical one-sided unexpected wave statistics (squares). For *α* = 2 and *N*_{a} = 30, our predicted return period is *N*_{R} ~ 6 × 10^{4} and in fair agreement with the linear predictions (~7 × 10^{4}) 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.

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 *N*_{R} 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).

### 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* = *λ*_{3}/3 ~ 0.06, where *λ*_{3} 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 *N*_{R} = *R*/*T*_{m} 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 *N*_{a} waves for different values of *α*. For *α* = 2 and *N*_{a} = 30, our predicted second-order return period *N*_{R} ~ 2 × 10^{4} is shorter than the linear counterpart (~6 × 10^{4}) 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 *N*_{R} ~ 10^{4} for *α* = 2 and *N*_{a} = 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 *N*_{R}. In contrast, our simulated sea states are based on the exact second-order solution for unidirectional broadband waves in deep water (Tayfun 1980).

### 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 *B*_{rs} 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).

In regards to WACSIS measurements, the left panel of Fig. 7 compares the theoretical nonlinear return period *N*_{R} (solid line) of unexpected wave crests *α* times larger than the surrounding *N*_{a} 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 *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

We observe that the empirical statistics tend to deviate from the theoretical predictions for large values of *α* and *N*_{a}. In particular, for both TERN and WACSIS we could not produce statistically stable estimates of extreme values for *N*_{a} > 10 when *α* > 1.5 because of the limited number of waves in the time series [*O*(10^{3}) waves in comparison to the 10^{6} 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 *h*_{obs} is 1.62*H*_{s}. Figure 1 shows that the WACSIS rogue wave is also unexpected as it is *α* = 2 times larger than the surrounding *N*_{a} ~ 50 waves. According to our statistical model, such unexpected wave would occur often and on average once every *N*_{R} = 4 × 10^{4} waves, as seen in the left panel of Fig. 9. Here, we report the theoretical predictions of the unconditional nonlinear return period *N*_{R} as a function of *N*_{a} using Eqs. (9) and (11). Further, from the center panel of Fig. 9, it is seen that the associated, average, nonlinear, unexpected crest height *H*_{s} and smaller than the conditional mean *N*_{R} = 4 × 10^{4} waves. Note that these average values underestimate the actual maximum crest amplitude *h*_{obs} ~ 1.62*H*_{s} observed. In contrast, the right panel of Fig. 9 shows that *h*_{obs} is nearly the same as the threshold *H*_{s} exceeded on average once every *N*_{h} = 0.3 × 10^{6} 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 *N*_{R}(*ξ*) of an unexpected wave whose crest height is larger than *ξH*_{s} [see Eq. (24)]. In particular, the left panel of Fig. 9 depicts plots of *N*_{R}(*ξ*) as a function of *N*_{a} for increasing values of *ξ* = 1, 1.2, 1.4, 1.55, and 1.6 (*α* = 2). For *ξ* = 1.6*H*_{s}, we find that an unexpected wave exceeding this threshold and standing above *N*_{a} = 50 waves would occur rarely and once every *N*_{R}(*ξ* = 1.6) ~ 0.6 × 10^{6}, in contrast to the smaller unconditional value *N*_{R} ~ 4 × 10^{4}.

In summary, the WACSIS wave crest as both unexpected and rogue, that is, 2 times larger than *N*_{a} = 50 surrounding waves and exceeding the 1.6*H*_{s}, would occur once every *N*_{R} = 0.6 × 10^{6} waves on average. In contrast, the WACSIS wave as a rogue event has a crest height that is nearly the same as the threshold *H*_{s} exceeded on average once every *N*_{h} = 0.3 × 10^{6} waves [see Eq. (17)]. Thus, the WACSIS rogue wave has a slightly greater occurrence frequency than being both rogue and unexpected since *N*_{h} < *N*_{R} = 0.6 × 10^{6}. This implies that the threshold *N*_{R} fraction of the largest crests is larger than 1.6*H*_{s} and nearly the same as 1.65*H*_{s}.

## 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 *h*_{obs} is 1.63*H*_{s} 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 *N*_{R} and *N*_{R}(*ξ*) as a function of *N*_{a}. 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 *N*_{a} = 30 waves occurs on average once every *N*_{R} = 2 × 10^{4} waves irrespective of its crest amplitude. In contrast, an unexpected wave whose crest height exceeds the threshold 1.6*H*_{s} occurs less often since our predicted conditional return period *N*_{R}(*ξ* = 1.6) ~ 3 × 10^{6} is greater than the unconditional counterpart *N*_{R} = 2 × 10^{4}, as seen in the left panel of Fig. 10. Furthermore, the crest height 1.6*H*_{s} is nearly the same as the threshold *N*_{h} = 0.3 × 10^{6} 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 *N*_{h} < *N*_{R} = 3 × 10^{6}, 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 *H*_{s} exceeded on average once every 0.3 × 10^{6} waves. In contrast, the Andrea and WACSIS events would occur less often being both unexpected and rogue, that is, on average once every 3 × 10^{6} and 0.6 × 10^{6}, 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.

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