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  • View in gallery

    Reference frame: (left) equivalent parabolic storm and (right) equivalent triangular storm.

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    Sea storms recorded by NOAA NODC buoys. For each actual storm, the EPS models are plotted for different values of λ.

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    Comparison among the exceedance probabilities P(Hmax > H) calculated for the actual storm of the century in Fig. 2 and for the associated equivalent storms with different values of λ. The best fit of the observed distribution’s tail is attained for λ = 0.75.

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    Significant wave height time series recorded by NOAA NODC 46006 buoy located in the Pacific Ocean during the month of October 2000. The sequence of equivalent parabolic storms is also plotted (λ = 2). Dotted line shows the storm threshold hcrit = 1.5Hsm.

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    Weibull plot of the probabilities of exceedance P(Hs > h) estimated for both NOAA buoys 41008 and 46006 in the Atlantic and Pacific Oceans, respectively. Continuous lines are associated with the theoretical Weibull distributions with parameters given in Table 1.

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    NOAA 46006 buoy in the Pacific Ocean: regression of Hmax and the peak intensity a of the actual storms recorded during 2000–07.

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    NOAA buoy 46006 in the Pacific Ocean: regression (28) of the b(a) as function of the storm peak a for different values of λ from data recorded during 2000–07: (a) cubic, (b) triangular, (c) parabolic, and (d) cusp storms.

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    Mean duration bm(λ) of storms recorded by NOAA buoys 46006 and 41008 in the Pacific and Atlantic Oceans, respectively, during the period 2000–07.

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    NOOA 46006 buoy in the Pacific Ocean: (a) R(Hs > h) and (b) D(h) computed for different values of λ.

  • View in gallery

    Same as Fig. 9, except for NOOA buoy 41008 in the Atlantic Ocean.

  • View in gallery

    NOOA 46006 buoy in the Pacific Ocean: R(Hmax > H) computed for different values of λ.

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    NOAA 46006 buoy in the Pacific Ocean: conditional probability pA|F(a; H) of the intensity A of the equivalent cusp storm (λ = 0.75) given the event F = {Hmax > H}, where Hmax is the crest-to-trough height of the largest wave of the storm.

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    NOAA 46006 buoy in the Pacific Ocean: (a) the variation coefficient γ = σA|F/μA|F of the storm peak intensity A given F = (Hmax > H) and (b) the associated ratio between Hmax and the conditional mean μA|F computed for different values of λ, where Hmax is the crest-to-trough height of the largest wave of the storm.

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Long-Term Statistics and Extreme Waves of Sea Storms

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  • 1 School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, Georgia
  • | 2 Department of Mechanics and Materials, Mediterranea University, Reggio Calabria, Italy
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Abstract

A stochastic model of sea storms for describing long-term statistics of extreme wave events is presented. The formulation generalizes Boccotti’s equivalent triangular storm model by describing an actual storm history in the form of a generic power law. The latter permits the derivation of analytical solutions for the return periods of extreme wave events and associated statistical properties. Lastly, the relative validity of the new model and its predictions is assessed by analyzing wave measurements retrieved from two NOAA National Oceanographic Data Center (NODC) buoys in the Atlantic and Pacific Oceans.

Corresponding author address: Felice Arena, Department of Mechanics and Materials, Mediterranea University, Loc. Feo di Vito, 89123 Reggio Calabria, Italy. Email: arena@unirc.it

Abstract

A stochastic model of sea storms for describing long-term statistics of extreme wave events is presented. The formulation generalizes Boccotti’s equivalent triangular storm model by describing an actual storm history in the form of a generic power law. The latter permits the derivation of analytical solutions for the return periods of extreme wave events and associated statistical properties. Lastly, the relative validity of the new model and its predictions is assessed by analyzing wave measurements retrieved from two NOAA National Oceanographic Data Center (NODC) buoys in the Atlantic and Pacific Oceans.

Corresponding author address: Felice Arena, Department of Mechanics and Materials, Mediterranea University, Loc. Feo di Vito, 89123 Reggio Calabria, Italy. Email: arena@unirc.it

1. Introduction

Stochastic modeling of time series of the significant wave height Hs recorded at a given ocean site is the principal focus of statistical methods employed in the long-term prediction of extreme wave events during sea storms (Krogstad 1985; Prevosto et al. 2000; Boccotti 2000). The reviews of several methods used for this can be found in the work of Isaacson and Mackenzie (1981), Guedes Soares (1989), and Goda (1999). In these methods, the effects of the sea state on the short-term scales of Ts ∼ 1–3 h are accumulated to predict the wave conditions on the long-term time scales of Tl ∼ yr. In doing so, it is reasonable to assume that with Ts, the sea state is a homogenous and stationary stochastic field whose properties are fully characterized by the directional spectrum of the sea surface and associated moments. As a result, wave parameters such as Hs, mean periods, and mean wavelengths can be easily estimated from either the time series of surface elevations or the associated spectrum. With Tl, we then have a succession of storms where each storm, according to Boccotti (2000), is identified as a nonstationary sequence of sea states in which Hs exceeds 1.5 times the mean annual significant wave height, say, Hsm at a given site, and it does not fall below that level during an interval of time longer than 12 h (see also Arena 2004).

Given a succession of storm events in time, Boccotti (1986, 2000) proposed the equivalent triangular storm (ETS) model to predict the return period of extreme wave events. In this model, a storm is described in time as a triangle of height a indicative of storm intensity and base b as a measure of duration. The statistical equivalence is achieved by requiring that a equal the actual maximum Hs in the storm, and b is chosen so that the maximum expected wave height during the storm is the same as that of the triangular storm (Borgman 1970, 1973). It is then assumed that a and b are realizations of two random variables, say, A and B, respectively. Then, the storm peak probability density function (pdf) pA(a) = Pr[A ∈ (a, a + da)] is not fitted directly to the observed storm peak data via ad hoc regressions, but it follows analytically by requiring that the average times spent by the equivalent and actual storm sequences above any threshold h be identical. So, the significant wave height history or the actual storm sequence is stochastically equivalent to a succession of random triangle storms. This type of equivalence defines the probabilistic structure of the ETS model, which depends on wave data only via the observed significant wave height exceedance P(h) = Pr(Hs > h) and the conditional average duration b(a) = B|A = a, both estimated via regression. Then, the estimates of wave extremes and their associated statistics simply follow from the density pA with no need for data fitting.

In particular, Boccotti (2000) derived an analytical solution for the return period R(Hs > h) of a storm during which the maximum Hs exceeds h as that of a triangle storm whose a is above h. Arena (2004) extended this result to account for seasonal effects and wave directionality. Boccotti (1986, 2000) also derived the return period R(Hmax > H) of a storm where the maximum individual wave height Hmax exceeds a fixed threshold H. Recently, Arena (2004) and Arena and Pavone (2006) exploited the ETS model to define the return period R(Cmax > C) of a storm during which the largest nonlinear crest height Cmax exceeds a fixed threshold C. Further, Arena and Pavone (2009) derived the solution of the return periods RN(RN) of a storm during which exactly N waves (at least N waves) exceed a given threshold. These analytical results have been compared against buoy measurements, demonstrating their relevance to the design of coastal and offshore structures (Arena and Pavone 2006, 2009).

In this paper, we extend and generalize Boccotti’s triangular storm model to include other possible and plausibly more realistic descriptions of the temporal history of the significant wave heights observed at a fixed point during a sea storm. In particular, we describe an actual storm in time t in the form of a generic power law |tt0|λ, where λ is a shape parameter and t0 is the time of the storm peak. Boccotti’s ETS model is recovered for λ = 1. For the “generalized” model, the associated storm peak density pA stems from the analytical solution of a Volterra integral equation of the first kind. We then derive new analytical expressions for both R(Hs > h) and R(Hmax > H) as function of pA and perform a sensitivity analysis with respect to λ to assess the deviations from the ETS predictions. Further, we present some statistical properties of the largest waves in storms. Finally, we apply the generalized model to the wave data gathered by two National Oceanic and Atmospheric Administration (NOAA) National Oceanographic Data Center (NODC) buoys moored off the Georgia and California coasts.

2. Equivalent power storm (EPS) model

Consider a time interval τ during which N(τ) storm events occur at a site. The exceedance probability of the Hmax in a particular storm is approximated by (Borgman 1973)
i1520-0485-40-5-1106-e1
where D is the storm duration, h(t) is the significant wave height history, T is the mean zero upcrossing wave period (Boccotti 2000), and P(H|Hs = h) is the exceedance probability of the crest-to-trough wave height H, given Hs = h. The latter is of the form (Boccotti 1981, 1997, 2000)
i1520-0485-40-5-1106-e2
where ψ* = |ψ(T*)|/ψ(0), with T* representing the abscissa of the first absolute minimum of the surface elevation covariance ψ(T). In the narrowband limit, ψ* → 1 and (2) reduce to the Rayleigh law (Rice 1944, 1945; Longuet-Higgins 1952). The maximum wave height expected during the actual storm follows from
i1520-0485-40-5-1106-e3
Now, assume that each actual storm can be described as an EPS whose significant wave height h varies in t according to the power law
i1520-0485-40-5-1106-e4
where b is the storm duration, a is the peak amplitude at t0, and λ (0 < λ < ∞) is a shape parameter (see Fig. 1). The EPS model has one degree of freedom in λ to better represent the actual storm peak. Given (4), the probability that Hmax > H follows from (1) as
i1520-0485-40-5-1106-e5
Then, the expected Hmax of the equivalent storm can be computed from (3). The estimation of the EPS parameters follows easily by equating a to the maximum significant wave height and b corresponds to the expected Hmax of the actual storm.

As an example, Fig. 2 shows the equivalent model (4) estimated for some actual storms for different values of λ. It is seen that the EPS approximation has smooth peaks for λ ≥ 1 and sharp cusps if 0 < λ < 1. Peaks become smoothly sharper as λ decreases. For λ = 1, the ETS model of Boccotti with linear cusps is recovered. The analysis of the data suggests the presence of a correlation trend between duration b and peak a. Small values of b—say, 20–40 h—in the ETS model are typical of brief and very severe storms, whereas large values of b—say, 100–200 h—for the same model tend to characterize storms with either a persistent single peak or multiple peaks.

Note that it is not generally guaranteed a priori that the P(Hmax > H) in an actual storm is the same as that of the associated EPS, as we require just the first-order statistical moments of the two distributions to be identical. In practice they are so and also slightly sensitive to the parameter λ, as seen in Fig. 3 for the “storm of the century” from Fig. 2a (Cardone et al. 1996). The same figure also shows that the ETS model slightly overestimates the observed P(Hmax > H) of the actual storm in the range of low probabilities. More significantly, the EPS model allows us to improve on the ETS predictions by choosing the parameter λ that will best fit the tail of the exceedance distribution describing the maximum wave height in a storm. For the storms analyzed in this paper, the optimal λ turns out to be roughly 0.75, as shown in Fig. 3. The analysis of oceanic data later on will show that an EPS model with an optimized λ provides long-term predictions of wave extremes slightly more conservative than those based on ETS models.

a. Distribution of storm peak

The stochastic modeling of the EPS approximation for an actual sea storm sequence, as that shown in Fig. 4, proceeds by assuming that the height a and base b of the equivalent storm are values of the random variables A and B, respectively. Thus, we introduce the joint pdf pA,B(a, b) = pA(a)pB|A(b|a) of A and B and define pA,B(a, b) db da as the fraction of equivalent storms having a duration in (b, b + db) and peak amplitude in (a, a + da). The pdf of A follows from
i1520-0485-40-5-1106-e6
and the conditional average duration of B, given A = a, is
i1520-0485-40-5-1106-e7
This and the exceedance probability P(h) = P(Hs > h) are the only two quantities in the EPS model that are estimated from data via regression, as it will be shown later on. The storm peak density pA(a) is not fitted directly from the observed storm peak data via ad hoc regressions, but it is stemmed in an analytical form as function of both P(h) and b(a) by invoking the stochastic equivalence between the sequence of actual storms and that of the equivalent storms (see Fig. 4). This is accomplished by imposing the condition that the average time during which Hs is above h is the same in both the actual and equivalent storm sequences. For the actual storm sequence, the average time TR within the interval τ during which Hs stays above h is given by
i1520-0485-40-5-1106-e8
To derive the average time TEPS during which Hs is above h in the equivalent storm sequence, we first consider
i1520-0485-40-5-1106-e9
as the number of equivalent storms in the random sequence having a duration B in (b, b + db) and peak amplitude A in (a, a + da), and we recall that N(τ) is the total number of storms in τ. It follows then that
i1520-0485-40-5-1106-e10
where, from (4)
i1520-0485-40-5-1106-e11
is the time interval during which Hs stays above h in an equivalent storm with a and b (see Fig. 1). Thus, using (7), (9), and (11), (10) can be simplified further as
i1520-0485-40-5-1106-e12
We can now require that TEPS(h) = TR(h) for any h, and this will lead to the following Volterra integral equation of the first kind:
i1520-0485-40-5-1106-e13
Solving (13) for pA yields (see appendix)
i1520-0485-40-5-1106-e14
In the applications to follow, we will assume that P(h) is given by the Weibull distribution (cf. Battjes 1972; Isaacson and Mackenzie 1981; Ochi 1998)
i1520-0485-40-5-1106-e15
where the parameters hl, w, and u can be estimated iteratively (Goda 1999).

In general, (15) does not describe the observed overall distributions of Hs well, but it does fit the tails fairly accurately (Ferreira and Guedes Soares 2000). Thus, it is preferable to fit (15) to relatively large values of the observational data in the low-probability region (Guedes Soares 1989; Boccotti 2000) and to use a lognormal fit for the high-probability region, as suggested by Haver (1985).

We point out that the EPS model depends on the measured data only via the observed P(h) and b(a), as in the ETS formulation. Then, the density pA satisfies the Volterra’s integral equation of first kind (13), which imposes the stochastic equivalence between the equivalent and actual storm sequences for an arbitrary λ > 0. As a result, the EPS model is defined in a probabilistic setting and no more data fitting is required to estimate wave extremes and develop their associated statistics. Indeed, in the following we shall derive analytical expressions of the return periods R(Hs > h) and R(Hmax > H) explicitly as a function of the peak distribution (14) as well as investigate some statistical properties of large waves in storms using probabilistic principles.

b. Return period R(Hs > h)

The return period R(Hs > h) of an actual storm where the maximum Hs exceeds h is the same as the return period of an equivalent power storm whose peak A exceeds h. Thus,
i1520-0485-40-5-1106-e16
where N(τ ; A > h) represents the average number of equivalent storms whose peak A exceeds h during τ. Integrating (9) over all the possible b and a > h yields
i1520-0485-40-5-1106-e17
and together with (14), (16) reduces to
i1520-0485-40-5-1106-e18
The mean persistence of Hs above h is given by the general expression (Boccotti 2000)
i1520-0485-40-5-1106-e19
For λ = 1, (18) reduces to the expressions valid for ETS models, as to be expected (cf. Boccotti 2000).

c. Return period R(Hmax > H)

Consider the number Nw(H) of equivalent storms where the largest wave occurs with a crest-to-trough height Hmax greater than H. Then, the return period R(Hmax > H) of an actual storm is defined as that of an equivalent storm whose maximum Hmax exceeds H. Thus,
i1520-0485-40-5-1106-e20
where
i1520-0485-40-5-1106-e21
In the preceding, dNw(H, a, b) is the number of equivalent storms in the sequence of random storms with peak amplitude A in (a, a + da) and duration B in (b, b + db), and whose maximum wave occurs with a height Hmax in (H, H + dH). More explicitly,
i1520-0485-40-5-1106-e22
where p(Hmax = H; a, b) is the pdf of Hmax that follows from (5) and dN is given by (9). Thus, (20) reduces to
i1520-0485-40-5-1106-e23
Using (14), this is simplified further to
i1520-0485-40-5-1106-e24
which generalizes the solution appropriate to the ETS model (Boccotti 2000; Arena and Pavone 2006, 2009).

3. Extreme waves in sea storms

Given λ and R, consider now the wave with the largest crest-to-trough height Hmax greater than H that follows from (24). What is the most probable value of the significant wave height peak A of the storm during which that wave occurred?

This question has some practical interest, and it can be addressed in the context of EPS models without data fitting, by simply using probabilistic principles as follows. We first integrate (22) over b to obtain the number dNw(H, a) of equivalent storms whose largest wave height Hmax is greater than H, and the peak intensity A in [a, a + da] as
i1520-0485-40-5-1106-e25
Then, given F = {Hmax > H}, the conditional probability that the extreme event occurs in an equivalent storm whose peak intensity A is in [a, a + da] follows from (25) as
i1520-0485-40-5-1106-e26
By virtue of (22), the preceding can be rewritten as
i1520-0485-40-5-1106-e27
This pdf is characterized by its conditional mean μA|F(H) and standard deviation σA|F(H), which are both functions of the given height H. If the coefficient of variation γ = σA|F/μA|F ≪ 1, then an exceptional wave event most probably occurs during a storm whose maximum significant wave height, that is, the storm peak A, is very close to μA|F. In the applications, we will show that such theoretical predictions based on EPS models are approximately satisfied in actual storm data.

4. Analysis of storm data

Hereafter, we will apply the EPS model to elaborate some wave measurements retrieved by the NOOA buoys 41008 and 46006 moored off the Georgia and California coasts, respectively. Operational since 1997, buoy 41008 is at 40 n mi southeast of Savannah, Georgia, in a water depth of 18 m. Buoy 46006 has been operational since 1977 at 600 n mi west of Eureka, California, in a water depth of 4023 m.

Given that a sequence of actual storms occurred at either buoy locations, the long-term wave statistics are uniquely defined by the following:
  1. the distribution P(Hs > h) of Hs at the site,
  2. the conditional average base b(a), and
  3. the pdf pA(a) of A.
The first two items above are readily estimated from data, whereas pA follows from the analytical solution (14) of the Volterra integral Eq. (13). For example, Fig. 5 shows that P(Hs > h) is well represented by the Weibull law (15) at both buoy locations. The corresponding distributional parameters u, w, and hl are given in Table 1. For each storm, we also computed the Hmax as function of a, as in Fig. 6. Further, for λ given, the conditional duration b(a) of a location is described by
i1520-0485-40-5-1106-e28
where K1 and K2 are regression parameters and a′ is set equal to 1 m. In particular, consider the significant wave height series recorded by buoy 46006 during the period 2000–07. Figure 7 shows the regression (28) estimated from the values of a and b observed during the actual storms for different shapes of the equivalent storm, including λ = 0.5 (cusp), 1 (triangular), 2 (parabolic), and 3 (cubic). The corresponding regression parameters are given in Table 2. It is seen that b(a) tends to increase as λ decreases. In particular, the duration of a parabolic-type storm (λ > 1) is in general smaller than that of a cusp-type storm (λ < 1). This trend is confirmed in Fig. 8, where we plotted for each buoy the mean duration
i1520-0485-40-5-1106-e29
estimated from the actual storms recorded as a function of λ. Note that bm follows the same trend at both buoys despite that one buoy is located in the Pacific Ocean and the other in the Atlantic Ocean. Further, bm slightly changes for values of λ above λc = 0.7. For λ values smaller than λc, bm increases sharply toward unrealistic durations of 300 h and longer, causing the long-term predictions of wave extremes to be underestimated significantly, as we will show later.

Given P(Hs > h) and b(a), we can now compute pA(a) by using (14) and make predictions for both the return period R(Hs > h) and the mean persistence D(h) from (18) and (19), respectively. Figure 9 shows the latter predictions based on the data of buoy 46006 for various values of λ. The results in Fig. 9a suggest that parabolic-type storms yield predictions consistent with those of the ETS model (λ = 1). However, for values of λ below the threshold λc = 0.7, the EPS model drastically underestimates the extreme significant wave heights in comparison to the ETS predictions. Figure 8 suggests that these are related to equivalent storms with durations longer than 400–500 h, which are unrealistic. A similar trend is also observed in the predictions of the mean persistence D(h) of buoy 46006 plotted in Fig. 9b. Figure 10 illustrates the predictions for R(Hs > h) and D(h) computed from the data of the buoy 41008 for the triangular (λ = 1), parabolic (λ = 2), and cusp (λ = 0.75) models. For the same values of λ, the predictions of the return period R(Hmax > H) are computed using (24) for buoy 46006 and shown in Fig. 11.

Our analysis reveals that if a cusp model is adopted (λ < 1), then λ should be greater than λc = 0.7 to avoid unrealistic predictions. Moreover, when λ > λc, the predictions based on the EPS model are somewhat insensitive to changes in the shape parameter λ, and they tend to be robust. Particularly for the optimal model that best fits the distribution’s tail of the maximum wave height observed in a given storm (λ = 0.75; see Fig. 3), the EPS predictions are slightly more conservative than those from the ETS model (λ = 1), but the predicted levels differ by less than 1.5%.

Figure 12 shows the conditional pdf computed from (27) for a cusp model (λ = 0.75) for various values of Hmax for buoy 46006. As Hmax increases, the conditional pdf (27) tends to become symmetric relative to its mean μA|F. Lastly, Fig. 13a shows the coefficient of variation γ = σA|F/μA|F as function of Hmax. Note that γ is slightly sensitive to λ, and it tends to decrease for larger wave heights. For Hmax > 25 m, γ is very small and nearly 0.11, irrespective of λ. This implies that an exceptional wave event most probably occurs during a storm whose maximum significant wave height A is very close to μA|F. Further, the ratio Hmax/μA|F is also slightly sensitive to λ, and the maximum wave height does not exceed twice the value of its expected storm peak intensity, as clearly seen in Fig. 13b.

5. Conclusions

We have presented a generalization of the ETS model of Boccotti (2000). To introduce flexibility in modeling the significant wave height history locally at storm peaks, we define a sequence of random equivalent storms with parabolic (λ > 1) or cusped (λ < 1) shapes, with λ as a free positive parameter. This approximation relies on the measured data, specifically, on the exceedance distribution P(h) of the significant wave heights observed and the conditional base b(a), both of which can be estimated via regression. The storm-peak density pA follows from the solution of a Volterra integral equation of the first kind by requiring that the average time interval during which the significant wave height lingers above a given threshold is identical in both the equivalent storm sequence and actual storms. No data fitting is then required in describing wave extremes and associated statistics. Hence, we were able to derive the return periods R(Hs > h) and R(Hmax > H) as explicit functions of pA and to describe the statistical properties of the largest waves in storms based solely on probabilistic principles.

As applications, we have examined the statistics of extreme waves in numerous storms recorded by two NOAA buoys located in the Atlantic and Pacific Oceans. Our analysis reveals that λ should be greater than the critical value λc = 0.7 to avoid unrealistic predictions. For λ > λc, the EPS model yields robust predictions, being slightly sensitive to changes in the shape parameter. We also observed that for a given storm, the ETS model slightly overestimates the exceedance probability P(Hmax > H) of the maximum wave height observed in the storms. The EPS model is then exploited to improve the ETS predictions by choosing the optimal value of the shape parameter λ that best fits the tail of the maximum wave height distribution observed in a storm. For the storms analyzed in this study, we find that the optimal λ roughly equals 0.75. For these optimal models, the prediction of extremes is about 1.5% larger and thus more conservative than those of the ETS model.

REFERENCES

  • Arena, F., 2004: On the prediction of extreme sea waves. Environmental Sciences and Environmental Computing, Vol. 2, P. Zannetti, Ed., EnviroComp Institute, 1–50. [Available online at http://www.envirocomp.org/esecII/esecII_flyer.pdf].

    • Search Google Scholar
    • Export Citation
  • Arena, F., , and D. Pavone, 2006: Return period of nonlinear high wave crests. J. Geophys. Res., 111 , C08004. doi:10.1029/2005JC003407.

  • Arena, F., , and D. Pavone, 2009: A generalized approach for the long-term modelling of extreme sea waves. Ocean Modell., 26 , 217225.

  • Battjes, J. A., 1972: Long-term wave height distributions at seven stations around the British Isles. Deutsch. Hydrogr. Z., 25 , 179189.

    • Search Google Scholar
    • Export Citation
  • Boccotti, P., 1981: On the highest waves in a stationary Gaussian process. Atti Accad. Ligure Sci. Lett., Genoa, 38 , 271302.

  • Boccotti, P., 1986: On coastal and offshore structure risk analysis. Excerpta Ital. Contrib. Field Hydraul. Eng., 1 , 1936.

  • Boccotti, P., 1997: A general theory of three-dimensional wave groups. Ocean Eng., 24 , 265300.

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APPENDIX

Solutions for Probability Density Function pA(a)

In (13), define
i1520-0485-40-5-1106-ea1
to obtain an integral Volterra equation of first kind for G, namely,
i1520-0485-40-5-1106-ea2
The solution of this equation varies depending on if λ > 1, λ = 1, or 0 < λ < 1, namely,
i1520-0485-40-5-1106-ea3
with (integer) n > 1 and 0 < μ < 1. If λ = 1/n is rational, that is, μ = 0, then from (A3)
i1520-0485-40-5-1106-ea4
In applications, G(λ, a) is computed via numerical integration if λ ≠ 1 or μ ≠ 0. In the following we will present the formal derivation of the Volterra integral Eq. (A2) that leads to (A3).

Solution for λ = 1

In this case, (A2) reduces to
i1520-0485-40-5-1106-ea5
The solution for G proceeds by differentiating both members of (A5) twice with respect to h and setting h = a. This yields
i1520-0485-40-5-1106-ea6

Solution for λ > 1

Consider a solution of (A2) of the form
i1520-0485-40-5-1106-ea7
where g(z ≥ 0) and H(za) are arbitrary functions. Substituting (A7) into (A2) yields
i1520-0485-40-5-1106-ea8
For z given, we first integrate with respect to a and then over z. On this basis, (A8) can be rewritten as
i1520-0485-40-5-1106-ea9
where
i1520-0485-40-5-1106-ea10
Now, (A9) can be easily solved for g if the arbitrary function H in (A10) is chosen to yield K ∼ (zh) from (A10). To do so, we first make a change of variables
i1520-0485-40-5-1106-ea11
to rewrite (A10) as
i1520-0485-40-5-1106-ea12
On the assumption that
i1520-0485-40-5-1106-ea13
it will follow that for λ > 1,
i1520-0485-40-5-1106-ea14
Thus, (A9) is simplified to
i1520-0485-40-5-1106-ea15
This is the same type of Volterra equation as in case (a). We thus solve for g by differentiating both sides of (A15) twice with respect to h and then set h = z. This yields
i1520-0485-40-5-1106-ea16
From (A13) and (A16), (A7) leads to the solution
i1520-0485-40-5-1106-ea17

Solution for 0 < λ < 1

In this case, we seek an integer n > 1 and a real number 0 < μ < 1 such that
i1520-0485-40-5-1106-ea18
On this basis, (A2) becomes
i1520-0485-40-5-1106-ea19
Differentiate both sides of the preceding expression n times to get
i1520-0485-40-5-1106-ea20
This is the same type of integral equation as in case (b). Thus,
i1520-0485-40-5-1106-ea21

Fig. 1.
Fig. 1.

Reference frame: (left) equivalent parabolic storm and (right) equivalent triangular storm.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 2.
Fig. 2.

Sea storms recorded by NOAA NODC buoys. For each actual storm, the EPS models are plotted for different values of λ.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 3.
Fig. 3.

Comparison among the exceedance probabilities P(Hmax > H) calculated for the actual storm of the century in Fig. 2 and for the associated equivalent storms with different values of λ. The best fit of the observed distribution’s tail is attained for λ = 0.75.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 4.
Fig. 4.

Significant wave height time series recorded by NOAA NODC 46006 buoy located in the Pacific Ocean during the month of October 2000. The sequence of equivalent parabolic storms is also plotted (λ = 2). Dotted line shows the storm threshold hcrit = 1.5Hsm.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 5.
Fig. 5.

Weibull plot of the probabilities of exceedance P(Hs > h) estimated for both NOAA buoys 41008 and 46006 in the Atlantic and Pacific Oceans, respectively. Continuous lines are associated with the theoretical Weibull distributions with parameters given in Table 1.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 6.
Fig. 6.

NOAA 46006 buoy in the Pacific Ocean: regression of Hmax and the peak intensity a of the actual storms recorded during 2000–07.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 7.
Fig. 7.

NOAA buoy 46006 in the Pacific Ocean: regression (28) of the b(a) as function of the storm peak a for different values of λ from data recorded during 2000–07: (a) cubic, (b) triangular, (c) parabolic, and (d) cusp storms.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 8.
Fig. 8.

Mean duration bm(λ) of storms recorded by NOAA buoys 46006 and 41008 in the Pacific and Atlantic Oceans, respectively, during the period 2000–07.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 9.
Fig. 9.

NOOA 46006 buoy in the Pacific Ocean: (a) R(Hs > h) and (b) D(h) computed for different values of λ.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 10.
Fig. 10.

Same as Fig. 9, except for NOOA buoy 41008 in the Atlantic Ocean.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 11.
Fig. 11.

NOOA 46006 buoy in the Pacific Ocean: R(Hmax > H) computed for different values of λ.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 12.
Fig. 12.

NOAA 46006 buoy in the Pacific Ocean: conditional probability pA|F(a; H) of the intensity A of the equivalent cusp storm (λ = 0.75) given the event F = {Hmax > H}, where Hmax is the crest-to-trough height of the largest wave of the storm.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Fig. 13.
Fig. 13.

NOAA 46006 buoy in the Pacific Ocean: (a) the variation coefficient γ = σA|F/μA|F of the storm peak intensity A given F = (Hmax > H) and (b) the associated ratio between Hmax and the conditional mean μA|F computed for different values of λ, where Hmax is the crest-to-trough height of the largest wave of the storm.

Citation: Journal of Physical Oceanography 40, 5; 10.1175/2009JPO4335.1

Table 1.

Parameters of the Weibull distribution (15).

Table 1.
Table 2.

Parameters K1 and K2 (h) of the base height regression b(a) for NOAA NODC buoys for different values of λ.

Table 2.
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