Nonlinear crest distribution for shallow water Stokes waves

Applied Ocean Research 57 (2016) 152–161

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Nonlinear crest distribution for shallow water Stokes waves Yingguang Wang a,b,c,∗ a

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai 200240, PR China c School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China b

a r t i c l e

i n f o

Article history: Received 24 July 2015 Received in revised form 28 February 2016 Accepted 13 March 2016 Available online 28 March 2016 Keywords: Shallow water Stokes waves Wave crest distribution Asymptotic method

a b s t r a c t This article concerns the calculation of nonlinear crest distribution for shallow water Stokes waves. The calculations have been carried out by incorporating a second order nonlinear wave model into an asymptotic analysis method. This is a new approach to the calculation of wave crest distribution, and, as all of the calculations are performed in the probability domain, avoids the need for long time-domain simulations. The accuracy and efficiency of this new approach for calculating the wave crest distribution are validated by comparing the results predicted using it with those predicted by using the Monte Carlo simulation (MCS) method, by using a previous Transformed Rayleigh method, by using some existing wave crest distribution formulas, and by using the measured surface elevation data at the Poseidon platform in the Japan Sea. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The wave crest distribution is an important factor to be considered for the design of various kinds of ships and ocean engineering structures. The wave crest distribution must be established carefully because it is used for the calculation of wave loads on a ship or an offshore platform. It is also a key input parameter for the stability analysis of ships and floating offshore platforms. Meanwhile, knowledge of the wave crest distribution is also critical to the successful design of various kinds of coastal shore protection structures. It is known that the wave crest distribution obeys the Rayleigh probability law in an ideal Gaussian random sea (see e.g. [1–6]). In the ideal Gaussian sea model the individual cosine wave trains superimpose linearly (add) without interaction, and therefore, the model is also called the linear sea model. However, it is known that when deep water waves become too steep or as the water depths decrease, the non-linearities of sea waves become more and more relevant, and the ocean surface elevation process deviates significantly from the linear Gaussian assumption (see e.g. [7]). The crests become higher and sharper than expected from a summation of cosine waves with random phase, and the troughs become

∗ Correspondence to: State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China. Tel.: +86 021 34206514; fax: +86 021 34206701. E-mail address: [email protected] http://dx.doi.org/10.1016/j.apor.2016.03.006 0141-1187/© 2016 Elsevier Ltd. All rights reserved.

shallower and flatter (see e.g. [8–11]). Consequently, in nonlinear seas, the application of the Rayleigh distribution to the wave crests will become invalid, and other more suitable methods should be applied to predict the distribution of wave crests for the nonlinear random model of the sea elevation. In the literature, there exist some theoretical and/or empirical models of wave crest height distributions of nonlinear random waves. The deep water wave crest distribution proposed by Tayfun [12] is derived by assuming that second-order water waves with a narrow-banded spectrum can be described in a form in which each realization of the surface elevation becomes an amplitudemodulated Stokes wave with a mean frequency and a random phase. Subsequent approximate crest distributions based on the narrow-band model of sea waves include those described by Dawson et al. [13], Kriebel and Dawson [14], and others. Tayfun [15] considered statistics of nonlinear wave crests in transitional water depths using an appropriate second-order representation for crest heights. In particular, theoretical expressions describing the statistics of nonlinear wave crests are posed in the form of a simple second-order transformation of well-known results on linear waves. However, the previous research work of this author’s research group has shown that such kinds of theoretical and/or empirical formulas will sometimes predict wave characteristic distributions that differ considerably from the true ones. In the literature, the inadequacies of such kinds of theoretical and/or empirical formulas have also been discussed in many other papers (see e.g. Forristall et al. [16]; Prevosto and Forristall [17]; Izadparast and Niedzwecki [18]; Latheef and Swan [19]; or Petrova and

Y. Wang / Applied Ocean Research 57 (2016) 152–161

Guedes Soares [20]). In his two previous papers [21,22] the author of this article has developed two transformed Rayleigh methods for the calculation of wave crest height distributions in nonlinear random seas. However, the transformed Rayleigh method in [21] has been specifically developed for calculating the wave crest height distributions in nonlinear mixed sea states (i.e. sea states of combined wind waves and swell) and this method involves an empirical criterion (for the approximation of the cumulant generating function) that was only tested to work fine for the bimodal spectra of the mixed sea states. The transformed Rayleigh method in [22] is essentially a semi-empirical method whose accuracy depends on how many moments are kept in the equation of the functional transformation. The more moments are kept, the more accurate the functional transformation will be. Motivated by the above-mentioned facts, in this paper, the author has proposed a new approach by incorporating a second order nonlinear wave model into an asymptotic analysis method in order to calculate the nonlinear crest distributions of shallow water Stokes waves more accurately and efficiently. In order to substantiate the proposed new approach, it will be applied for calculating the wave crest distributions of two sea states (one with a McCormick spectrum in an effective water depth of 0.8862, and another one with a McCormick spectrum in an effective water depth of 1.2607). Because all of the calculations will be performed in the probability domain, there will be no need for long time-domain simulations. The calculation results will be validated against Monte Carlo simulation results. Meanwhile, the wave crest height distributions of these two sea states obtained from using the previous transformed Rayleigh method in [22] and from using the Tayfun model [15] will also be included for comparison purpose. Finally, the proposed new approach in this article will be applied to predict the wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea, and its accuracy will be once again validated. This paper begins in Section 2 by introducing the knowledge of the shallow water Stokes waves. It continues in Section 3 by elucidating the theoretical background of the proposed asymptotic analysis method. In Section 4 the calculation examples and discussions will be provided, with conclusions summarized in Section 5. 2. The shallow water Stokes waves



 h 2 

P(Ac < h) = 1 − exp −8

Hs

For surface gravity waves above a horizontal bed, the mean depth d is a constant. Using a perturbation-series approach (Stokes expansion), the velocity potential ˚ and the free surface elevation  can be expanded as (see, e.g., [28]):



˚ = ˚(1) + ˚(2) + · · ·

˚(n+1) ˚(n)

=

(n+1) (n)

= O(ε)

(2)

In Eq. (2) ε is a small parameter in the expansion and it is typically proportional to the wave steepness. O(·) denotes “on the order of”. For an irregular sea state characterized by a specific wave spectrum S (ω) where ω is the angular frequency, it is straightforward to show that the first order velocity potential ˚(1) and surface elevation (1) take the following forms (see, e.g., [28]): ˚(1) (x, t) = Re

N  igcn cosh k(z + d) n=0

(1) (x, t) = Re

N 

ωn

cosh kd

exp(i(ωn t − kn x + εn ))

cn exp(i(ωn t − kn x + εn ))

(3)

(4)

n=0

as N tends to infinity. In Eqs. (3) and (4), Re denotes the real part of the complex number and i denotes the imaginary unit. For each elementary cosine wave cn denotes its complex valued amplitude which is directly related to the specific wave spectrum S (ω). In Eqs. (3) and (4), ωn is the angular frequency, kn is the wave number, and εn is the phase angle that is uniformly distributed in the range of [0, 2]. Meanwhile, in Eqs. (3) and (4) the individual frequencies, ωn and wave numbers, kn are functionally related through the linear dispersion relation: ωn2 = gkn tanh(kn d)

(5)

where g and d are the gravitational acceleration and water depth, respectively. Due to the bottom effects, shallow water wave data usually does not follow the linear Gaussian model. The linear Gaussian sea model can be corrected by including quadratic (second-order) terms. Following [29,30] the quadratic (second-order) corrections are given by: N N  

icn cm P(ωn , ωm )

n=0 m=−0

cosh(kn + km )(z + d) cosh(kn + km )d

× exp(i(ωn t − kn x + εn )) exp(i(ωm t − km x + εn )) N  cn2 gkn

+2 (1)

In Eq. (1) h is the wave crest height, Hs is the significant wave height and AC is a specific wave crest height. However, in a nonlinear random sea [23–26], the observed surface process is positively skewed with higher crests and shallower troughs than expected under the Gaussian assumption (see e.g. [7]). In the following we briefly introduce the theoretical background of the nonlinear shallow water Stokes waves. The fluid region is described using three-dimensional Cartesian coordinates (x, y, z), with x and y the horizontal coordinates, and z the vertical coordinate. The positive z-direction opposes the direction of the gravitational acceleration. Time is denoted by t. The free surface is located at z = (x, y, t), and the bottom of the fluid region is at z = −d(x,y). In many instances, the oscillatory flow in the fluid interior of surface waves can be described accurately using potential flow theory. Then, the flow velocity can be described as the gradient of a velocity potential ˚(x, y, z, t) (see, e.g. Faltinsen [27]).

where

 = (1) + (2) + · · ·

˚(2) (x, t) = 2Re

The wave crest height distributions in an ideal Gaussian random sea are considered as to obey the Rayleigh probability law (see e.g. [1,4]) as shown in Eq. (1):

153

n=0

(2) (x, t) = Re

sinh 2kn h

N N  

(6)

t

cm cn [rmn exp(i(ωm t − km x + εm + ωn t − kn x + εn ))

m=0 n=0

+ qmn exp(i(ωm t − km x + εm − ωn t + kn x − εn ))]

(7)

In Eqs. (6) and (7), P(ωn , ωm ), rmn and qmn are called quadratic (second-order) transfer functions. The quadratic transfer function P(ωn , ωm ) is given by: P(ωn , ωm ) = (1 − ı−n,m ) ×

g 2 kn km 2ωn ωm

2 +ω ω )+ − 14 (ωn2 + ωm n m

2 2 g 2 (ωn km +ωm kn ) 4 ωn ωm (ωn +ωm )

m (ωn + ωm ) − g (ωkn +k tanh((kn + km )d) +ω ) n

m

(8)

154

Y. Wang / Applied Ocean Research 57 (2016) 152–161

The Kroenecker delta (ı−n,m = 1 if n + m = 0, zero otherwise) is introduced to avoid a singular P(ωn , ωm ). The quadratic (second-order) transfer functions rmn and qmn are given by: rmn = −

1



1 4ωm ωn



˛ = 0.3536 + 0.2892S1 + 0.1060Ur

2 − k k g 2 ) + ω (ω 4 − g 2 k2 ) + ω (ω 4 − g 2 k2 ) 2(ωm + ωn )(ωn2 ωm n m n m m n m n

(16)



(ωm + ωn )2 cosh((km + kn )d) − g(km + kn ) sinh((km + kn )d)

g

× (ωm + ωn ) cosh((km + kn )d) −

1

qmn = −

for a frequency of 1/T1 , Hs is significant wave height. In the case of a second order long-crested sea (Forristall [9]):



1 4ωm ωn





1 4gωm ωn



2 (km kn g 2 − ωn2 ωm )+

1 4g

2 (ωm + ωn2 )

2 + k k g 2 ) − ω (ω4 − g 2 k2 ) + ω (ω4 − g 2 k2 ) 2(ωm − ωn )(ωn2 ωm n m n m m m n n

(9)



(ωn − ωm )2 cosh(|km − kn |d) − g|kn − km | sinh(|kn − km |d)

g

× (ωm − ωn ) cosh(|kn − km |d) −



1 4gωm ωn



2 (km kn g 2 + ωn2 ωm )+

In Eqs. (9) and (10) the wave numbers kn and frequencies ωn satisfy the same relation as in the linear case. Finally, by combining Eqs. (4) and (7) the wave surface elevations for the shallow water nonlinear Stokes waves can be written as:

1 4g

2 (ωm + ωn2 )

(10)

ˇ = 2 − 2.1597S1 + 0.0968(Ur )2

(17)

In the case of a second order short-crested sea (Forristall [9]):

 N

(x, t) = (1) (x, t) + (2) (x, t) = Re

˛ = 0.3536 + 0.2568S1 + 0.08Ur

cn exp(i(ωn t − kn x + εn ))

ˇ = 2 − 1.7912S1 − 0.5302(Ur ) + 0.284(Ur )

n=0

+ Re

N N  

cm cn [rmn exp(i(ωm t − km x + εm + ωn t − kn x + εn ))

m=0 n=0

+ qmn exp(i(ωm t − km x + εm − ωn t + kn x − εn ))]

(11)

For shallow water nonlinear Stokes waves, the bottom effects should be considered when predicting the wave crest height distributions. Obviously, the Rayleigh distribution which is good for predicting the crest heights of linear Gaussian waves will become invalid for calculating the wave crest height distributions of shallow water nonlinear Stokes waves. Tayfun [15] proposed a formula for calculating the exceedance probabilities of crest heights of nonlinear Stokes waves in transitional water depths as follows:

⎡  2 ⎤ (1 + 2∗ h∗ ) − 1 ⎢ ⎥ E(h∗ ) = exp ⎣− ⎦ ∗2 2

(18)

(12)

2

(19)

However, many researchers (see e.g. Forristall et al. [16]; Prevosto and Forristall [17]; Izadparast and Niedzwecki [18]; Latheef and Swan [19]; or Petrova and Guedes Soares [20]) have pointed out that such kinds of theoretical and/or empirical models not always, if not almost never, show good comparison with the actual crest heights distribution of the shallow water waves. Therefore, the author of this article has proposed an asymptotic analysis method in order to calculate the wave crest height distributions of shallow water nonlinear Stokes waves more accurately and efficiently. In Section 3 of this article the theoretical background of this asymptotic analysis method will be elucidated. 3. Theoretical background of the proposed asymptotic analysis method We rewrite Eq. (4) at a specific reference location (say x = 0) as follows: (1) (x, t) = (1) (0, t) = (1) (t) = Re

N 

cn exp(i(ωn t + εn ))

n=0

where h* is a non-dimensional wave crest height scaled by (1/4)Hs , and * is a non-dimensional steepness parameter estimated from Forristall’s Weibull crest height distribution model as follows (Tayfun [15]): ˛3  = 16  ˇ ∗

3 ˇ

1 − 4



 2

(13)

In the above equation,  (·) denotes the Gamma function. The parameters ˛ and ˇ are given in terms of S1 , which is a measure of steepness and the Ursel number Ur , which is a measure of the impact of water depth on the non-linearity of waves. These quantities read (Forristall [9]): 2 Hs S1 = g T2 1 Ur =

Hs k12 d3

=

N 

(an cos(ωn t) + bn sin(ωn t))

(20)

n=0

In the above equation, ωn = 2n/T, T is the time interval, cn ’s are mutually independent of one another, and an and bn are Gaussian random variables. Meanwhile, the coefficients of the above formula have the following properties (see [29]): E[a2n ] = E[b2n ] = S (ω)dω;

E[an bm ] = 0;

(21)

E[am an ] = E[bm bn ] = 0m = / n

(22)

2 cm

(23)

=

a2m

+ b2m

 −b  m

(14)

εm = tan−1

(15)

In Eq. (21) E[·] denotes mathematical expectation, S (ω) is the wave spectrum and dω = 2/T. In the following we re-express the complex term appearing in Eq. (20) in the form (see [29]):

where T1 is the mean wave period calculated from the ratio of the first two moments of the wave spectrum, k1 is the wave number

(24)

am

cm exp(i(ωm t + εm )) =



S (ωm )dω(xm + iym )

(25)

Y. Wang / Applied Ocean Research 57 (2016) 152–161

With comparison with Eq. (20) shows that: xm ym



S (ωm )dω = am cos(ωm t) + bm sin(ωm t)

(26)

S (ωm )dω = am sin(ωm t) + bm cos(ωm t)

(27)



Because am and bm are Gaussian random variables, xm and ym will also be Gaussian random variables at any fixed time t. Further, Eqs. (21) and (22) imply that: 2 2 E[xm ] = E[ym ] = 1;

E[xm yn ] = 0

E[xm xn ] = E[ym yn ] = 0;

(28)

m= / n

(29)

We rewrite Eq. (7) at a specific reference location (say x = 0) as follows: (2) (x, t) = (2) (0, t) = (2) (t) = Re

N N   m=0 n=0

+ qmn exp(i(ωm t + εm − ωn t − εn ))]

(30)

The terms rmn and qmn are the same as shown in Eqs. (9) and (10). It can be seen that rmn = rnm ; qmn = qnm . Using Eq. (25), Eq. (11) at a specific reference location (say x = 0) can now be rewritten as: (x, t) = (0, t) = (t) = Re

N  

S (ωm )dω(xn + iyn )

m=0

+ Re

N N  

[sn sm rmn (xn + iyn )(xm + iym )

m=0 n=0

+sn sm qmn (xn + iyn )(xm + iym )]

where (i )d denotes a column vector formed by the diagonal elements of i . We know that (t) is the elevation of the sea level at a fixed point as a function of time t. We here use the so called mean up crossing wave to define a wave, i.e. the wave is considered as a part of a function between the consecutive up crossings of the mean sea level. Assume (t) crosses a mean sea level u* finite many times and denote the times of up crossings of u* by ti , 0< t1 < t2 < · · ·. The wave crest height Mi *, say, of the ith wave equals the global maximum of (t) during the time interval ti < t < ti+1 . For a specific nonlinear irregular sea state, (t) is a non-Gaussian stochastic process. Here we will use the simplest (but widely effective) model for a non-Gaussian sea where (t) is expressed as a function of a stationary zero mean Gaussian process (t) ˜ with vari˜ = 1), i.e. ance one (Var[(0)] (t) = G((t)), ˜

cm cn [rmn exp(i(ωm t + εm + ωn t + εn ))

(31)

(t) = sT x + xT [Q + R]x + yT [Q − R]y

(32)

where Q and R are real symmetric matrix whose nmth components are sn sm qmn and sn sm rmn respectively, s, x and y are vectors  whose nth components are sn , xn and yn respectively and sn = S (ωm )dω. By performing an eigenvalue decomposition, the above equation becomes (see [29]): (t) = sT x + xT P1T 1 P1 x + yT P2T 2 P2 y

(33)

where i is a diagonal matrix with the eigenvalues in the respective diagonal and Pi contains the corresponding eigenvectors per row. Introducing a new set of Gaussian random variables Zj, such that:



Z=

P1

0

0

P2



x

 (34)

y

we can write the stochastic process (t) as (this is called the KacSiegert solution): (t) =

2N 

(ˇj Zj + j Zj2 )

(35)

j=0

where Zj are independent Gaussian processes with unit variance, and ˇj and j are coefficients computed based on the information provided by the sea spectrum S (ω). Specifically, the coefficients in the above equation are computed as follows:



␤=

P1 s 0





,

␭=

(1 )

d

(2 )

d



(36)

dG > 0, d

G(0) = 0.

(37)

Therefore, in order to use the model it is necessary to estimate the transformation G. Here transformation G relating the nonGaussian process and the original Gaussian process will be obtained based on the equivalence of the level up-crossing rates of the two processes. Therefore, we can see that in order to calculate the wave crest height distributions of the non-Gaussian process (t), the critical task is to calculate the sea level u up-crossing rate (u) of (t). In the following, the principles of an asymptotic analysis method for calculating (u) will be outlined. Let (t) be a strictly stationary process and non-Gaussian. For a fixed level u, assuming (u) is the expected number of times, in the interval [0,1], the process (t) crosses the u level in the upward direction. We then have the following extension of Rice formula (see [31]): a.a.u

Collecting real parts, Eq. (31) can be re-written in the following matrix notation (see [29]):

155



+∞

(u) =

zf(0)(0) (u, z)dz ˙

(38)

0 a.a.u

where = means that the equality is valid for almost all u. In the ˙ and (0) ˙ is the derivative of above equation, z is a realization of (0) (0). We see that the computation of (u) requires the estimation ˙ i.e. f(0)(0) (u, z). of the joint density of (0) and (0), ˙ Although seemingly trivial, the problem of estimating the joint ˙ is actually very difficult because for most density of (0) and (0) cases of non-Gaussian processes this density does not have an explicit form. Therefore, in this article we consider a different approach to the approximation of the sea level u up-crossing rate (u), using the theory of asymptotic expansions. The asymptotic results proposed in Breitung [32] and further investigated in Hagberg [33] are employed. In Appendix A we show that the level u up-crossing rate (u) is well approximated by 2

(u) ≈

e−ˇu /2 c(ˇu ), 2␲

(39)

The explicit formula for the constant c(ˇ) as well as a justification of this approximation can be found in Appendix A. The increasing function ˇu is equal to the so-called Hasofer-Linds safety index often employed in the reliability analysis. Note that the estimation of the function ˇu is achieved by standard numerical methods. It should be noted that in the existing literature the Rice’s logic has been already extended to model distributions of nonlinear asymmetric waves, including shallow water waves (see, e. g. Machado and Rychlik [31]; Butler et al. [34])). However, the approach in the present paper for the calculation of nonlinear crest distribution is very much different from the existing approaches as used in Machado and Rychlik [31] and in Butler et al. [34]. A key procedure in the present approach is to use a deterministic and memoryless functional transformation to relate a stationary zero

156

Y. Wang / Applied Ocean Research 57 (2016) 152–161

mean variance one Gaussian process to the original non-Gaussian process. Another key procedure in the present approach is to utilize the asymptotic analysis method for calculating the up crossing rates of second order nonlinear shallow water waves. For calculating the up crossing rates of second order nonlinear water waves, Machado and Rychlik [31] and Butler et al. actually used a “saddle-point approximation method” which is based on equations entirely different from the asymptotic analysis Eqs. (A1)–(A13) in the present paper. The saddle point approximation was first introduced by Daniels [35] as a formula to approximate the probability density function from its cumulant generating function. It is for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. However, “Saddle-point approximation” is one method, and “asymptotic analysis” is another method. Saddle-point approximation method is entirely different from the asymptotic analysis method in this paper. The essence of the asymptotic analysis method in the present paper is to use the second order reliability analysis method to calculate the up-crossing rates for a noncentral Chi square process. Due to the curse of dimensionality in the probability-of-failure calculation, numerous methods are used to simplify the numerical treatment of the integration process. The Taylor series expansion is often used to linearize the limit-state. In the present approach, the second-order Taylor series expansion is used to perform the second-order approximation of the response surface at the Most Probable failure Point. This approach is actually a mathematical optimization problem for finding the point on the response surface that has the shortest distance from the origin to the surface in the standard normal space. For detailed mathematical theories of the present approach, please refer to the Appendix. After the level up-crossing rate (u) of the non-Gaussian process (t) is calculated by using Eq. (39), it can subsequently be used to obtain the inverse transformation G−1 (u) (Wang and Xia [21]):

(u) =

if u ≥ 0

(0)

  (u)  ⎪ ⎪ ⎪ if u < 0 ⎩ − −2 ln

(40)

(0)

Next, we turn to the relation between the level up-crossing rate (u) and the probability distribution of the crest height FM ∗ (u). As specified in Wang and Xia [21], we have the following two equations: P(Ac > u) = 1 − FM ∗ (u) ≤ min (u) = exp (0)

(z)

0≤z≤u (0)

 −

(G−1 (u)) 2

2

(41)

FM ∗ (u) ≥ 1 − exp

(0.25 × Hs )2 ωp

 ω (M+1) p

ω



exp

−

M+1 M

 ω M  p

ω (44)

where ωp =

2 Tp

M+1 1/M

Tp M

= Tz  M+1 M

where  () is a Gamma function. In this study we choose a standard McCormick spectrum for the simulation because the spectral formula of Eq. (44) is based on the Weibull probability formula, which is generic in nature. By comparison with measured data the spectral formula of Eq. (44) has been shown to be generic in nature in that it well-describes the energy distributions in the wind-generated seas (McCormick [36]). The motivation for developing the generic spectral formula (44) is a need for a single expression that could be used in both performance analysis and survival analyses of offshore systems (McCormick [36]). In Fig. 1 the blue curve shows a McCormick wave spectrum with a significant wave height Hs = 3.0 m, a spectral peak period Tp = 8.0 s, and a zero up-crossing period Tz = 6.5 s. For this sea state we first utilize a Newton Raphson method to find the wave number kp in the dispersion relation:



ωp2 =

2 Tp

2

= gkp tanh(kp d)

(45)

Then we obtain the effective water depth (for definition see: Katsardi et al. [37]) is: kp d = 0.8862. Fig. 2 shows our calculation results of the non-dimensional wave crest height exceedance probabilities for a sea state with a McCormick wave spectrum with Hs = 3.0 m, Tp = 8.0 s and Tz = 6.5 s for a water depth of 10 m. The solid green line in Fig. 2 represents the Monte Carlo Simulation (MCS) results of the non-dimensional wave crest height exceedance probability of this nonlinear irregular sea state (note that h in Fig. 2 denotes wave crest heights while h/Hs denotes nondimensional wave crest heights). In the process of the Monte Carlo simulation, a wave elevation time series consisting of 1,500,000 points (enough to reduce the simulation variance) were first generated based on the blue curve spectrum in Fig. 1 by summation



(G−1 (u)) − 2

Spectral density

(42)

Combining the above two equations we obtain the following relationship for the crest height distribution:



S(ω) = (M + 1)

2



(43)

for all u ≥ 0, and G−1 (u) is calculated by Eq. (40).

1.8 1.6 1.4 S(w) [m 2 s / rad]

G

−1

⎧  (u)  ⎪ ⎪ −2 ln ⎪ ⎨

a spectral peak period Tp , and a zero up- crossing period Tz is as follows (McCormick [36]):

1.2 1 0.8 0.6 0.4

4. Calculation examples and discussions In this section we first utilize the asymptotic analysis method to calculate the shallow water wave crest height exceedance probabilities based on a standard McCormick spectrum corresponding to a water depth of 10 m. The mathematical expression of the McCormick frequency spectrum with a significant wave height Hs ,

0.2 0 0

0.5

1

1.5

2 Frequency [rad/s]

2.5

3

3.5

4

Fig. 1. McCormick wave spectrum with Hs = 3.0 m, Tp = 8.0 s and Tz = 6.5 s for a water depth of 10 m (d = 10 m). For this sea state the effective water depth is: kp d = 0.8862. (For interpretation of the references to colour in the citation of this figure, the reader is referred to the web version of this article.)

Y. Wang / Applied Ocean Research 57 (2016) 152–161

157

Fig. 2. The non-dimensional wave crest height exceedance probabilities for a sea state with a McCormick wave spectrum with a significant wave height Hs = 3.0 m, Tp = 8.0 s and Tz = 6.5 s for a water depth of 10 m. For this sea state the effective water depth is: kp d = 0.8862. (For interpretation of the references to colour in the citation of this figure, the reader is referred to the web version of this article.)

Fig. 3. The non-dimensional wave crest height exceedance probabilities for a sea state with a McCormick wave spectrum with a significant wave height Hs = 3.0 m, Tp = 7.5 s and Tz = 6.1 s for a water depth of 15 m. For this sea state the effective water depth is: kp d = 1.2607. (For interpretation of the references to colour in the citation of this figure, the reader is referred to the web version of this article.)

of sine functions with random phase angles uniformly distributed in the range of [0,2]. The wave crest height time series were subsequently extracted from these 1,500,000 wave elevation points. After extracting these crest height time series, exact Epanechnikov kernel density estimates were carried out for obtaining the probability density function of the non-dimensional wave crest heights. Next, cumulative trapezoidal numerical integration was performed on the above-mentioned probability density function for getting the probability distribution (F) of the non-dimensional wave crest heights. Finally, the non-dimensional wave crest height exceedance probabilities were obtained based on the probability distribution by utilizing the formula P = 1 − F. The above Monte Carlo simulation results are used as the benchmark against which the accuracy of the results from the asymptotic analysis method and from the theoretical and/or empirical crest distribution models is checked. In Fig. 2, the red dashed line represents the results of the nondimensional wave crest height exceedance probabilities obtained from utilizing the proposed asymptotic analysis method. In applying the asymptotic analysis method, the mean and standard deviation of the surface elevation were first calculated based on the blue curve spectrum in Fig. 1. Then the G−1 function was calculated by using the procedures as listed in Section 3, and the non-dimensional wave crest height distribution was subsequently calculated using Eq. (43) in Section 3. Finally, the non-dimensional wave crest height exceedance probabilities were obtained based on the probability distribution by utilizing the formula P = 1 − F. We can see that the results from the asymptotic analysis method fit the Monte Carlo simulation results very well. In Fig. 2, the small blue “+” represents a non-dimensional wave crest height exceedance probability predicted by using the Tayfun [15] model (i.e. by using Eqs. (12)–(17) with the parameters T1 = 7.0984 s, S1 = 0.0382, Ur = 0.4693, ˛ = 0.4144, ˇ = 1.9389 and * = 0.2086). Please note that during the calculations the value of Hs was taken to be four times the standard deviation ( ) of the surface elevation based on the blue curve spectrum in Fig. 1. We can find that the Tayfun [15] model performs poorly for crests larger than √ the rms value represented on the abscissa as 0.4 Hs = 2 than the asymptotic analysis method does. In Fig. 3, the small red “•” represents a non-dimensional wave crest height exceedance probability predicted by using the Tayfun [15] model (i.e. by using Eqs. (12)–(15) and Eqs. (18) and (19) with the parameters T1 = 7.0984 s, S1 = 0.0382, Ur = 0.4693, ˛ = 0.4010, ˇ = 1.7453 and * = 0.2258). We can notice that in Fig. 2, the small red “•” distribution deviates a little bit more from the Monte Carlo simulation results than the small blue “+” distribution does. However, the two distributions from the Tayfun [15] model are almost indistinguishable.

In Fig. 2, the small blue “*” represents a wave crest height exceedance probability predicted by using the Transformed Rayleigh method (as proposed in Wang [22]) that differs from the present asymptotic analysis method by that the functional transformation is calculated without using the information of the level up-crossing rates of the non-Gaussian process. The functional transformation in the Transformed Rayleigh method (as proposed in Wang [22]) is calculated based on the notion of a “Hermite moment”. We can find that the Transformed Rayleigh method as proposed in Wang [22] gives poorer predictions in the nondimensional wave crest height region of about [0.7, 1.05] than the asymptotic analysis method does. In Fig. 2, the solid blue line represents the non-dimensional wave crest height exceedance probabilities calculated by using a Theoretical Rayleigh model (i.e. by using Eq. (1)). We notice that the Theoretical Rayleigh method underestimates the exceedance probabilities in the nondimensional wave crest height region of about [0.4, 1.05]. Finally, it is mentioned that in this case, the time series simulation with 1,500,000 wave elevation points and the post statistical processing for getting the solid green line in Fig. 2 took about 139 s on a desktop computer (Dell OPTIPLEX 360, Intel(R) Core(TM)2 Duo CPU [email protected] GHz). By comparison, it took less than 3 s to obtain the results of the non-dimensional wave crest height exceedance probabilities represented by the red dashed line in Fig. 2 by applying the asymptotic analysis method. The accuracy and efficiency of the asymptotic analysis method for calculating the crest height distributions of shallow water Stokes waves can thus be validated. We next utilize the asymptotic analysis method to calculate the shallow water non-dimensional wave crest height exceedance probabilities based on another McCormick spectrum (with a significant wave height Hs = 3.0 m, a spectral peak period Tp = 7.5 s, and a zero-down crossing period Tz = 6.1 s) corresponding to a water depth of 15 m. The effective water depth kp d for this sea state was obtained to be 1.2607. Fig. 3 shows our calculation results for this specific sea state. In Fig. 3, the solid green line represents the Monte Carlo Simulation (MCS) results of the non-dimensional wave crest height exceedance probability of this specific nonlinear irregular sea state. In the process of the Monte Carlo simulation, a wave elevation time series consisting of 1,500,000 points were first generated based on this specific McCormick spectrum. In Fig. 3, the red dashed line represents the results of the non-dimensional wave crest height exceedance probabilities obtained from utilizing the proposed asymptotic analysis method. In Fig. 3, the solid blue line represents the non-dimensional wave crest height exceedance probabilities calculated by using a Theoretical Rayleigh model. In

Y. Wang / Applied Ocean Research 57 (2016) 152–161

Fig. 3, the small blue “*” represents a wave crest height exceedance probability predicted by using the Transformed Rayleigh method as proposed in Wang [22]. We can find that the Transformed Rayleigh method as proposed in Wang [22] gives poorer predictions in the non-dimensional wave crest height region of about [0.8, 1.05] than the asymptotic analysis method does. We can notice that the tendencies of the above mentioned four lines are almost identical to those of the corresponding four lines in Fig. 2, and the accuracy and efficiency of the asymptotic analysis method for calculating the crest height distributions of shallow water waves in this specific sea state has also been validated. In Fig. 3, the small blue “+” represents a non-dimensional wave crest height exceedance probability predicted by using the Tayfun [15] model (i.e. by using Eqs. (12)–(17) with the parameters T1 = 6.6548 s, S1 = 0.0434, Ur = 0.1074, ˛ = 0.3775, ˇ = 1.9073 and * = 0.0889). In Fig. 3, the small red “•” represents a nondimensional wave crest height exceedance probability predicted by using the Tayfun [15] model (i.e. by using Eqs. (12)–(15) and Eqs. (18) and (19) with the parameters T1 = 6.6548 s, S1 = 0.0434, Ur = 0.1074, ˛ = 0.3733, ˇ = 1.8685 and * = 0.0851). We can notice that in Fig. 3, the small blue “+” distribution deviates a little bit more from the Monte Carlo simulation results than the small red “•” distribution does. However, the two distributions from the Tayfun [15] model are almost indistinguishable. We can also notice that in this case (kp d = 1.2607) both distributions from the Tayfun [15] model have provided predictions of the wave crest height exceedance probabilities more closer to the Monte Carlo Simulation results. Finally, our proposed asymptotic analysis method has been applied to calculate the non-dimensional wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea. Poseidon platform was located 3 km off the coast of Yura in the Yamagata prefecture, in the Japan Sea during the measurements. Fig. 4 shows the geographical position of the coast of Yura. As shown in Fig. 4 Yura is to the east of the Japan Sea. The latitude of Yura is 38.3◦ N (N means north). The longitude of Yura is 139.5◦ E (E means east).

48 oN

China 44 oN

40 oN

4 Elevation from mean water level (m)

158

3 2 1 0 -1 -2 -3 0

50

100

Time (s)

150

200

250

300

Fig. 5. The first 300 s wave elevation points time series measured with the 2# ultrasonic wave gauge at the Poseidon platform.

The surface elevation data at the Poseidon platform were measured with three ultrasonic wave gauges located at the sea floor from 8:12 a.m. on November 24, 1987 to 7:57 a.m. on November 25, 1987 and three sets of data each consisting of 85,547 wave elevation points had been obtained during the measurement. The sampling rate is 1 Hz. The water depth at the Poseidon platform is 41 m. Fig. 5 shows the first 300 wave elevation points time series measured with the 2# ultrasonic wave gauge at the Poseidon platform. It can be noticed that the crests are higher and sharper and the troughs are shallower. Therefore, these waves are nonlinear. Because a sea state remains stationary only within a short time period of less than three hours, we should divide each of the above three sets of data of 85,547 wave elevation points into eight short records respectively when we calculate the wave crest height distributions. In this study we have chosen a three-hour record of wave elevation points time series measured with the 2# ultrasonic wave gauge (Data source is from [38]. Please note that these data are used for academic research only and not for commercial purposes [38]). In Fig. 6 the power spectrum corresponding to this specific three-hour wave record is shown. The spectral peak frequency of this specific spectrum has been calculated to be 0.5062 rad/s. After using a Newton Raphson method to find the wave number kp in the dispersion relation we have obtained that for this specific sea state the effective water depth is: kp d = 1.2589. Fig. 7 shows our calculated non-dimensional wave crest height exceedance probabilities for this measured specific three-hour sea state. The solid green line in Fig. 7 represents the results of the non-dimensional wave crest height exceedance probabilities directly obtained from the measured three-hour wave elevation points at the Poseidon platform. For obtaining these results, the wave crest height time series were first extracted from these three-hour wave elevation points. Then exact Epanechnikov kernel Spectral density

Japan Sea

5

Yura(139.5oE, 38.3oN)

S(w) [m2 s / rad]

4

36 oN

Tokyo Japan

32 oN

3

2

1

0

o

132 E

o

136 E

o

140 E

o

144 E

Fig. 4. The geographical position of the coast of Yura.

o

148 E

0.4

0.6

0.8

1 Frequency [rad/s]

1.2

1.4

1.6

Fig. 6. The spectrum corresponding to the measured (2# gauge) three-hour data at the Poseidon platform.

Y. Wang / Applied Ocean Research 57 (2016) 152–161

159

In Fig. 7, the solid blue line represents the non-dimensional wave crest height exceedance probabilities calculated by using a Theoretical Rayleigh model. We notice that the Theoretical Rayleigh method underestimates the exceedance probabilities in the nondimensional wave crest height region of about [0.4, 1.1]. Through comparison of the calculation results shown in Fig. 7, the accuracy of the asymptotic analysis method for calculating the crest height distributions of shallow water Stokes waves has once again been validated. 5. Conclusions

Fig. 7. The non-dimensional wave crest height exceedance probabilities for a sea state with the measured Poseidon data. For this sea state the effective water depth is: kp d = 1.2589. (For interpretation of the references to colour in the citation of this figure, the reader is referred to the web version of this article.)

density estimates were carried out for obtaining the probability density function of the non-dimensional wave crest heights. Next, cumulative trapezoidal numerical integration was performed on the above-mentioned probability density function for getting the probability distribution (F) of the non-dimensional wave crest height. Finally, the non-dimensional wave crest height exceedance probabilities were obtained by using the formula P = 1 − F. These exceedance probabilities results are used as the benchmark against which the accuracy of the results from the asymptotic analysis method and from the theoretical and/or empirical crest distribution models is checked. In Fig. 7, the red dashed line represents the results of the nondimensional wave crest height exceedance probabilities obtained from utilizing the proposed asymptotic analysis method. In applying the asymptotic analysis method, the mean and standard deviation of the surface elevation were first calculated based on the measured specific three-hour data. Then the G−1 function was calculated by using the procedures as listed in Section 3, and the non-dimensional wave crest height distribution was subsequently calculated using Eq. (43) in Section 3. Finally, the non-dimensional wave crest height exceedance probabilities were obtained based on the probability distribution by utilizing the formula P = 1 − F. We can see that the results from the asymptotic analysis method fit the results from the measured Poseidon data very well except in a small region (the non-dimensional wave crest height region of about [0.7, 0.81]). In Fig. 7, the small blue “*” represents a wave crest height exceedance probability predicted by using the Transformed Rayleigh method as proposed in Wang [22]. We can find that the Transformed Rayleigh method as proposed in Wang [22] gives slightly poorer predictions in the non-dimensional wave crest height region of about [0.9, 1.05] than the asymptotic analysis method does. In Fig. 7, the small blue “+” represents a non-dimensional wave crest height exceedance probability predicted by using the Tayfun [15] model (i.e. by using Eqs. (12)–(17) with the parameters T1 = 8.1543 s, S1 = 0.0477, Ur = 0.0196, ˛ = 0.3695, ˇ = 1.8969 and * = 0.0660). In Fig. 7, the small red “•” represents a nondimensional wave crest height exceedance probability predicted by using the Tayfun [15] model (i.e. by using Eqs. (12)–(15) and Eqs. (18) and (19) with the parameters T1 = 8.1543 s, S1 = 0.0477, Ur = 0.0196, ˛ = 0.3674, ˇ = 1.9042 and * = 0.0658). We can notice that in Fig. 7, the small blue “+” distribution is actually indistinguishable from the small red “•” distribution. We can also find that the Tayfun [15] model (either the * = 0.0660 case or the * = 0.0658 case) gives slightly poorer predictions than the asymptotic analysis method does.

The detailed mathematical procedures and formulas of an asymptotic analysis method for calculating the wave crest height probability distributions of shallow water Stokes waves have been elucidated in this article, and the shallow water nonlinear Stokes wave model has been integrated into this asymptotic analysis method. The essence of the asymptotic analysis method in the present paper is to use the second order reliability analysis method to calculate the up-crossing rates for a noncentral Chi square process. Due to the curse of dimensionality in the probability-of-failure calculation, numerous methods are used to simplify the numerical treatment of the integration process. The Taylor series expansion is often used to linearize the limit-state. In the present approach, the second-order Taylor series expansion is used to perform the second-order approximation of the response surface at the Most Probable failure Point. This approach is actually a mathematical optimization problem for finding the point on the response surface that has the shortest distance from the origin to the surface in the standard normal space. The proposed asymptotic analysis method has been applied for calculating the non-dimensional wave crest height exceedance probabilities of two sea states (one with a McCormick wave spectrum in an effective water depth of 0.8862, and another one with a McCormick wave spectrum in an effective water depth of 1.2607). It is demonstrated in these two cases that the asymptotic analysis method can offer better predictions than those from using the existing wave crest height distribution models and from using a previous Transformed Rayleigh method. The accuracy and efficiency of the proposed asymptotic analysis method have also been validated by using the results from Monte Carlo simulations. Finally, our proposed asymptotic analysis method has been applied to calculate the non-dimensional wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea, and its accuracy has been once again substantiated. The research findings gained from this study demonstrate the suitability of utilizing the proposed asymptotic analysis method for engineering purposes. Acknowledgments The funding from an independent research project from the Chinese State Key Laboratory of Ocean Engineering (Grant No. GKZD010038) that supports this research work is gratefully acknowledged. Appendix A. In this paper we are interested in measurements of the sea surface elevation at a fixed origin point, hence we may write (t) = X(t).

(A1)

The following derivation process follows that in Baxevani et al. [39,40]:

160

Y. Wang / Applied Ocean Research 57 (2016) 152–161

With comparison with Eq. (35) we may write the process X(t) in the form (see e.g. Baxevani et al. [39,40]): X(t) =

2N 

(ˇj Zj (t) + j Zj (t)2 )

(A2)

j=0



=



I

12 21

(A3)

22

cess with a spectrum S (ω) and variance Var(Xl (t)) = 0

2N 2 ˇ j=0 j

≈

S (ω)dω = 0 in which the equality is for N going to infinity.

2N  Z (t)2 has mean zero j=1 j j 2N 2 2j . By the independence and variance obtained as Var(Xq ) = j=1 for fixed t of the different Zj (t) and because Zj (t) and Zj (t)2 are uncor-

The quadratic correction term Xq (t) =

related, the variance of X(t) is the sum of the variances of the terms in Eq. (A2) (see Baxevani et al. [39,40]): Var(X(t)) =

2N 

ˇj2

+

j=0

2N 

2

(A4)

2j .

j=1

The following generalization of Breitung’s approximation (Breitung [32]) can be found in Baxevani et al. [39,40]): Let g: Rn → R be a function such that the surface S = {x = (x1 , x2 , . . . xn ), g(x) = 0} has a point x0 such that ||x0 ||=1 and ||x|| > 1 for all other x ∈ S. By x we denote both the vector (x1 , x2 , . . . xn ) and the n × 1 column matrix. Suppose Z(t) is an n-dimensional, station˙ ary, differentiable, Gaussian vector process, and let Z(t) denote its ˙ is denoted by derivative. The correlation of the vector (Z(t), Z(t)) ,





=



I

12 21

(A5)

22

For a family of processes g(Z(t)/ˇ), ˇ > 0, under some mild technical assumptions, the intensity of zero upcrossings is given by (for detailed derivation process, please refer to: Breitung [32]; Hagberg [33]):



2

+ (0) ˇ

e−ˇ /2 = (c + O(ˇ−2 )), 2

c=

x0 T (

22

−

G 21 0

det(I + P0 G0 P0 )

12

)x0

,

(A6) as ˇ tends to infinity, where



G0 :=

(A9)

2



∂ g 1 (x0 ) |∇ g(x0 )| ∂xi ∂xj

and

P0 :=I − x0 xT0 .

(A7)

i,j=1,2,...,n

In order to perform calculations for the general case of a second order sea, we have to construct somewhat artificial asymptotics. The idea is as follows. Fix the level u, assumed to be large, and let: p(x):=bT x + xT x,

g(x):=1 −

1 (ˇu bT x + ˇu2 xT x) u

(A10)

The process g(Z(t)/ˇu ) crosses the level 0 when the process p(Z(t)) crosses the level u. Hence, with x0 = xu /ˇu , for the specific level u, we have + (0) equal to the u-upcrossing rates for the proˇ u

cess p(Z(t)). Therefore, it is reasonable to believe that if ˇu is large, then the term O(ˇ−2 ) for ˇ = ˇu is small. Hence the approximation of omitting the term O(ˇ−2 ) in Eq. (A6) is suitable. (0) = We turn now to the computation of + ˇ u

while I is the identity matrix (note that the matrices 12 , 22 need not be identity matrices). Note that Z0 (t) = X0 (t)/ˇ0 , where X0 (t) is the Gaussian part of the process X(t) and ˇ0 2 = Var(X0 (0)), is independent of the processes Zj (t), j = 1, 2 . . ., N (obviously  0 = 0  j = 0). The linear part Xl (t) = ˇj Zj (t) is a Gaussian proand

!∞

{x ∈ Rn ; p(x) = u} of minimal distance to the origin, and define ˇu := ||xu ||. Let

where Z(t) = (Z0 (t), . . ., Z2N (t)) is a vector-valued stationary Gaussian process, such that for each t, Zj (t) ∈ N(0, 1) and the variables Zj (t), Zk (t), are independent (j and k are two different integers 0 ≤ j ≤ 2N, ˙ 0 ≤ k ≤ 2N). Let us also denote by Z(t) = (Z˙ 1 (t), . . ., Z˙ 2N (t)), the derivative of vector Z(t). Then the joint density of the vectors Z(t) ˙ ˙ and Z(t) is normal with (Z(t), Z(t)) ∈ N(0, ), where (see Baxevani et al. [39,40])



where b is a column vector containing ˇj and  = diag[ 1 ,  2 ,  3 , "n . . .  n ] is the diagonal matrix, with i=1  i = / 0. Assume that there is only one point xu on the surface

(A8)

e

−ˇ2 /2 u

2

c(ˇu ). Note

that, here we indicate c’s dependence of ˇu . Evaluating the terms gives

#

∇ g(x)T #x=x

u /ˇu



= −

# # # #

ˇu 2ˇu2 b+ x u u

=− x=xu /ˇu

−2ˇu 1  and P0 = I − 2 xu xTu . G0 = ||(b + 2xu )|| ˇu

ˇu (b + 2xu ), u

(A11)

Now we can use the following approximation: 2

(u) ≈ where: c(ˇu ) =

e−ˇu /2 c(ˇu ), 2



xTu (

22

−

(A12)

G 21 0

det(I + P0 G0 P0 )

21

)xu

(A13)

It should be noted that the point of minimum norm, xu can be found by standard optimization methods. After finishing the critical task of calculating the level up-crossing rate (u) using Eq. (A12), we can then calculate the wave crest height distributions of the non-Gaussian process (t). References [1] Longuet-Higgins MS. On the statistical distribution of the heights of sea waves. J Mar Res 1952;11:45–266. [2] Lindgren G. Some properties of a normal process near a local maximum. Ann Math Stat 1970;4(6):1870–83. [3] Lindgren G. Local maxima of Gaussian fields. Ark Mat 1972;10:195–218. [4] Chakrabarti SK. Hydrodynamics of offshore structures. Computational Mechanics Publications Inc.; 1987. [5] Lindgren G, Rychlik I. Slepian models and regression approximations in crossing and extreme value theory. Int Stat Rev 1991;59(2):195–225. [6] Boccotti P. Wave mechanics for ocean engineering. Elsevier Science; 2000. [7] Ochi MK. Ocean waves the stochastic approach. Cambridge University Press; 2005. [8] Baldock TE, Swan C. Extreme waves in shallow and intermediate water depths. Coast Eng 1996;27:21–46. [9] Forristall GZ. Wave crest distributions: observations and second-order theory. J Phys Oceanogr 2000;30(8):1931–43. [10] Fedele F, Tayfun AM. On nonlinear wave groups and crest statistics. J Fluid Mech 2009;620:221–39. [11] Cherneva Z, Petrova PG, Andreeva N, Guedes Soares C. Probability distributions of peaks, troughs and heights of wind waves measured in the Black Sea coastal zone. Coast Eng 2005;52(7):599–615. [12] Tayfun MA. Narrow-band nonlinear sea waves. J Geophys Res 1980;85:1548–52. [13] Dawson TH, Kriebel DL, Wallendorf LA. Breaking waves in laboratory-generated JONSWAP seas. Appl Ocean Res 1993;15:85–93. [14] Kriebel DL, Dawson TH. Distribution of crest amplitudes in severe seas with breaking. J Offshore Mech Arct Eng 1993;115:9–15. [15] Tayfun MA. Statistics of nonlinear wave crests and groups. Ocean Eng 2006;33:1589–622. [16] Forristall GZ, Krogstad HE, Taylor PH, Barstow SS, Prevosto M, Tromans P. Wave crest sensor intercomparison study: an overview of WACSIS. In: Proceedings, 21st International Conference on Offshore Mechanics and Arctic Engineering,

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