Volume 8 ocean energy 8 03 – resource assessment for wave energy

Volume 8 ocean energy 8 03 – resource assessment for wave energy Volume 8 ocean energy 8 03 – resource assessment for wave energy Volume 8 ocean energy 8 03 – resource assessment for wave energy Volume 8 ocean energy 8 03 – resource assessment for wave energy Volume 8 ocean energy 8 03 – resource assessment for wave energy Volume 8 ocean energy 8 03 – resource assessment for wave energy Volume 8 ocean energy 8 03 – resource assessment for wave energy

8.03 Resource Assessment for Wave Energy EBL Mackay, GL Garrad Hassan, Bristol, UK © 2012 Elsevier Ltd All rights reserved 8.03.1 8.03.2 8.03.2.1 8.03.2.1.1 8.03.2.1.2 8.03.2.2 8.03.2.2.1 8.03.2.2.2 8.03.2.2.3 8.03.2.2.4 8.03.2.2.5 8.03.2.2.6 8.03.3 8.03.4 8.03.4.1 8.03.4.1.1 8.03.4.1.2 8.03.4.1.3 8.03.4.1.4 8.03.4.1.5 8.03.4.2 8.03.4.2.1 8.03.4.2.2 8.03.4.2.3 8.03.4.2.4 8.03.4.2.5 8.03.4.3 8.03.4.3.1 8.03.4.3.2 8.03.4.3.3 8.03.4.3.4 8.03.4.3.5 8.03.5 8.03.5.1 8.03.5.2 8.03.5.3 8.03.6 8.03.6.1 8.03.6.2 8.03.6.2.1 8.03.6.2.2 8.03.6.3 8.03.6.3.1 8.03.6.3.2 8.03.6.4 8.03.6.4.1 8.03.6.4.2 8.03.6.4.3 References Introduction Mathematical Description of Ocean Waves Regular Waves The Airy wave equations The dispersion relation, phase speed, and group speed Irregular Waves The wave spectrum Height and period parameters Directional parameters Standard shapes for the frequency spectrum Standard shapes for the directional distribution Examples of sea surface elevation for standard spectral shapes Estimating WEC Power Wave Measurements and Modeling Wave Measurements from Moored Buoys Instrumental characteristics Estimation techniques for buoy data Quality checks for buoy data Sampling variability for temporal averages Presentation of wave climate data from buoy measurements Wave Measurements from Satellite Altimeters Instrumental characteristics Quality checks for altimeter data Sampling variability for spatial averages Calibration and validation of altimeter wave measurements Mapping the wave resource Numerical Wave Models Brief introduction to spectral wave models Sources of error in wave models Qualitative description of model errors Calibration of model data against in situ measurements Uncertainties in WEC power estimated from model data Variability and Predictability of WEC Yield Sampling Variability Synoptic and Seasonal Variability Interannual and Climatic Variability Estimation of Extremes Introduction Short-Term Distributions of Wave and Crest Heights The short-term distribution of crest-to-trough wave heights The short-term distribution of crest heights Long-Term Distributions of Extreme Sea States Overview of methods for estimating extreme Hs The POT method Combining Long-Term and Short-Term Distributions The distribution of the maximum wave or crest height in a storm The equivalent triangular storm The long-term distribution of the maximum wave or crest height Comprehensive Renewable Energy, Volume doi:10.1016/B978-0-08-087872-0.00803-9 12 12 12 12 13 13 13 14 15 15 18 19 20 23 23 24 24 28 28 31 31 34 36 37 37 41 45 45 46 47 47 50 50 51 51 51 52 52 52 52 53 54 54 55 70 71 71 72 73 11 12 Resource Assessment for Wave Energy 8.03.1 Introduction The assessment of the wave energy resource is an important step in the planning of a wave energy project The process can be split into two stages The first stage is comparative: to select the best sites for development There are many factors other than the wave resource that influence site selection, such as availability of a grid connection, proximity to ports, or appropriate seabed conditions These factors can often be limiting, but without an adequate wave resource a project is not viable The second stage is quantitative: to accurately determine the resource at a given site A detailed understanding of the wave energy resource is necessary to assess the economic viability of a wave energy project Like other sources of renewable energy, ocean waves are a variable resource, impossible to predict precisely This increases the risk associated with the development of a wave energy farm, since the upfront cost is large and the return is variable and imprecisely known It is therefore necessary to calculate the average power produced, the variability in power production, and the confidence bounds on these estimates Another important issue for wave energy developers is survivability Wave energy converters (WECs) must be designed to withstand the most severe conditions expected in their lifetime Since it is not possible to predict the severity of a storm at a certain location more than a few days in advance, a probabilistic approach must be taken to determine design conditions that represent an acceptable level of risk The aim of this chapter is to provide information on the wave resource so that the reader understands • • • • • how sea states are described, how energy yield is estimated from wave data, the available sources of wave data and their characteristics, the variability and predictability of energy yield, and the methods for the estimation of extreme wave conditions 8.03.2 Mathematical Description of Ocean Waves To understand the concepts and definitions used to describe ocean waves, it is useful to be aware of the mathematics used to describe wave motion The full solution to the hydrodynamic equations describing water wave motion is quite complicated and involves nonlinear terms Fortunately, a linear approximation to the full solution, where it is assumed that the wave height is negligible compared with the wave length, is a good model for ocean waves in many situations In particular, most of the terminology used for resource assessment can be understood in terms of linear theory Nonlinear aspects become important for steep waves and shallow water and are essential for understanding the evolution of the wave spectrum as waves are generated, propagate, and dissipate However, even when nonlinear aspects cannot be considered insignificant, much of the terminology based on linear theory used to describe the sea state is still applicable The section begins by discussing the equations that describe the motion of regular, low-amplitude waves and then goes on to discuss how a sea state can be described as a linear superposition of a large number of regular sinusoidal components using the concept of the wave spectrum 8.03.2.1 8.03.2.1.1 Regular Waves The Airy wave equations The linear solutions to the hydrodynamic equations that describe ocean wave motion were first presented by Sir George Biddell Airy in 1845 They are often referred to simply as the Airy wave equations It is easiest to work in two dimensions to begin with, with x as the horizontal coordinate and z as the vertical coordinate, positive upward and with the origin at the mean sea level Hence, in water of depth h, z = −h at the seabed Then, the Airy wave equations can be presented as follows Let χ and ζ be the horizontal and vertical displacement of a water particle from its rest position (x, z), respectively At time t, ẳa cosh kz ỵ hị coskx t ị sinh kh ẳa sinh kz ỵ hị sinkx t ị sinh kh ẵ1 ẵ1 where a is the wave amplitude; , the phase; ω = 2πf, the angular frequency of the water particles; f, the frequency; k = 2π/λ, the wave number; and λ, the wavelength In deep water, where h → ∞, the equations reduce to ẳ a expkzị coskx t ị ẳ a expkzị sinkx t ị ẵ2 ẵ2 Figure shows the orbits of a water particle described by eqn [1] The water particles move forward in the wave crests and backward in the troughs The orbits become increasingly elliptical with depth, until at seabed the motion is purely oscillatory In deep water, the orbits are circular at any depth, with the size of the orbit decreasing exponentially with depth Deep water can be considered as depths for which h/λ is greater than ½ 13 Resource Assessment for Wave Energy Direction of wave propagation Direction of water particle orbits Water surface Seabed Figure Motion of water particles in an Airy wave in finite depth Ellipses show complete orbits and lines show displacement from rest positions 8.03.2.1.2 The dispersion relation, phase speed, and group speed The equation that governs the relationship between wavelength and period is called the dispersion relation It is given by ẳ gk kh ẵ3 where g is the acceleration due to gravity For deepwater waves, kh → and we have ω2 ¼ gk ½4Š Substituting ω = 2π/T and k = 2π/λ, we see that for deep water gT ½5Š 2π For finite values of water depth, h, it is not possible to solve eqn [3] analytically for k in terms of ω, and numerical methods must be used to find the wavelength for a given period and water depth The speed at which wave crests pass a fixed point is called the phase speed and is denoted cp It is given by deepị ẳ cp ẳ ẳ T k ẵ6 Substituting eqn [3] gives cp ẳ g k kh For deep water, kh → 1, and the phase speed is given by cp deepị ẳ 1=2 rffiffiffiffi g k For very shallow water, kh → kh and the phase speed is given by pffiffiffiffiffi cp shallowị gh ẵ7 ẵ8 ẵ9 The speed at which the energy propagates is known as the group speed and is denoted cg It is given by cg ¼ Substituting eqn [3] and rearranging gives dω dk   2kh cg ẳ cp ỵ sinh 2kh ẵ10 ẵ11 For deep water, this reduces to cp ½12Š Equation [3] is called the dispersion relation because it governs how waves of different periods and wavelengths disperse from a fixed point From eqns [6] and [11], it is clear that the group speed increases with wavelength From eqn [9], we see that in very shallow water, the phase speed is no longer dependent on wavelength In this case, we say that the waves are nondispersive cg deepị ẳ 8.03.2.2 8.03.2.2.1 Irregular Waves The wave spectrum Waves in the ocean generally look very different from the monochromatic sinusoidal form shown in Figure For many purposes, we can think of the sea surface elevation, η, as a linear superposition of a large number of sine wave components with different amplitudes, periods, and directions: 14 Resource Assessment for Wave Energy ηðx; y; tị ẳ X nẳ1 an sinẵkn x cos n þ y sin θn Þ − ωn t þ n Š ½13Š where θn is the direction of the nth component It is normally assumed that phases are distributed randomly over [0 2π] with a uniform probability density Under these assumptions, the sea surface elevation follows a Gaussian distribution The directional variance spectrum S(f, θ) describes how the energy in the wave field is distributed with frequency and direction For small f and , we have fX ỵ f X ỵ f a ẳ Sf; ịf δθ n ½14Š That is, the spectral density is the sum of the variances of the individual sinusoidal components over a given frequency and directional range The directional spectrum can be decomposed into two functions, one representing the total energy at each frequency and the other describing how the energy at each frequency is distributed with direction: Sðf; θÞ ẳ Sf ịDf; ị ẵ15 S(f ) is called the omnidirectional spectrum or frequency spectrum and is related to the directional spectrum by Sf ị ẳ Sf; ịd ½16Š D(f, θ) is the directional spreading function or directional distribution and satisfies two properties: 2π ð Dðf; θÞdθ ẳ 1 ẵ17 Df; ị over ½0 2πŠ ½18Š It is a slight abuse of notation to use S to denote both directional and omnidirectional spectral density, but it is usually clear from the context which function S denotes 8.03.2.2.2 Height and period parameters The wave spectrum can be summarized to a reasonable accuracy using a small number of parameters The most important are a measure of average wave height and period, followed by descriptors of directional properties Wave height and period parameters are defined in terms of moments of the omnidirectional spectrum The nth moment of the spectrum is defined as ð∞ mn ¼ f n S f ịdf ẵ19 Wave height and period parameters are defined as follows: Significant wave height pffiffiffiffiffiffi Hs ¼ m0 ½20Š m −1 Energy period Te ¼ m0 ẵ21 m0 m1 r m0 Zero-crossing period Tz ẳ m2 ẵ22 Mean period Tm ẳ Peak period Tp ẳ fp ½23Š ½24Š where fp is the peak frequency, that is, the frequency at which S(f ) takes its maximum value These definitions have a natural interpretation: m0 is the variance of the sea surface elevation, and hence Hs as defined in eqn [20] is times the root mean square (RMS) displacement of the sea surface The factor arises for historical reasons The term significant wave height was originally introduced to correspond to the visual estimate made by a ‘trained observer’ and was defined as the average height of the highest 1/3 zero up- or down-cross waves In narrow-band seas, the height of the average highest 1/3 of the Resource Assessment for Wave Energy 15 pffiffiffiffiffiffi waves is equal to 4:01 m0 For neatness, the 0.01 has been dropped from the definition of Hs Some authors use the symbols H1/3 and Hm0 to distinguish between the two definitions of significant wave height In this chapter, Hs always denotes the spectral definition given in eqn [20] The mean period is simply the reciprocal of the mean frequency of the spectrum The zero-crossing period is approximately equal to the average time between waves crossing the mean sea level in an upward direction This was first shown by Rice [1, 2] for the case of a random Gaussian signal, which is a good approximation for ocean waves The definition of the energy period stems from the formula for wave power in deep water For a unidirectional wave system, the power transported forward per meter of crest length is ð∞ P ẳ g cg f ịSf ịdf ẵ25 where ρ is the density of seawater, usually taken as 1025 kg m−3 In deep water, using eqns [5], [8], and [12], we have Pdeep ð∞ ρg2 −1 ¼ f Sðf ịdf ẵ26 g ẳ m So from the definition of energy period Pdeep ¼ 8.03.2.2.3 ρg 2 H Te 64π s ½27Š Directional parameters Directional parameters are defined as follows The mean direction, θm(f ), at each frequency is given by 22π 2π m f ị ẳ ATAN2 Df; ị sinịd; D f; ị cosịd5 ẵ28 where ATAN2(y, x) is the four-quadrant inverse tangent function, which uses logic on the signs of x and y to resolve the 180° ambiguity in direction There are two commonly used definitions of the spread of energy about the mean direction at each frequency, defined in terms of either line moments or circular moments (denoted with the subscripts ‘l’ and ‘c’, respectively): 31=2 22π ð Dðf ; θÞ ðθ − m ị d5 l f ị ẳ ẵ29 31=2 22π  !2 ð θ − θ m dθ5 c f ị ẳ Df; ị 2sin ½30Š The circular moment definition approximates the line moment definition for narrow directional bandwidths since 2sin(x/2) ≈ x for small values of x Moreover, as will be discussed in Section 8.03.4.1.2(ii), σc can be calculated directly from measured data without the need to estimate the directional spreading function D(f, θ) An average direction and spread over the whole spectrum can be defined as follows: 2∞ ð ð∞ MDIR ¼ ATAN24 Sðf Þsinðθm ðf ÞÞdf ; Sðf Þcosðθm ðf ịịdf ẵ31 0 SDIR ẳ Sf Þσðf Þdf m0 ½32Š 8.03.2.2.4 Standard shapes for the frequency spectrum For calculations, it is often useful to assume a standard form for the shape of the frequency spectrum and directional distribution In deep water, the shape of the spectrum is controlled by the balance between the wind input, dissipation from whitecapping (the breaking of wave crests due to wind forcing), and nonlinear interactions between wave components During active wave growth, when the waves are relatively steep, nonlinear interactions play a central role in controlling the shape of the spectrum, forcing it toward ‘standard’ unimodal shapes and smoothing local deviations (see Reference 3) The part of the wave spectrum under active input from the local wind is known as the wind sea 16 Resource Assessment for Wave Energy If the wind drops or the waves propagate away from their generation area, they are referred to as swell Without input from the wind, the steepness of the waves will decrease (due to frequency and direction dispersion) and the nonlinear interactions will decrease accordingly In this case, the spectral shape will depend upon the history of the individual wave components, and a ‘standard’ shape will not necessarily be applicable This can result in spectra with multiple peaks, from one or more swells possibly together with a local wind sea In these cases, parametric descriptions of the frequency spectrum can be formed as the sum of two or more standard unimodal spectra The most commonly used forms of unimodal spectra for deepwater applications belong to the family given by Sðf Þ ¼ αf − r expð−βf − s Þγδ ðf Þ for α; β; r; s > and γ ≥ ẵ33 where f ị ẳ exp  f − fp σf p 2 ! ½34Š and it is usually assumed that & σ¼ 0:07 0:09 for f < f p for f ≥ fp ½35Š The parameters r and s control the shape of the spectrum, α is the scale parameter, β is the location parameter (in terms of frequency), and γ is known as the peak enhancement factor The peak frequency of the spectrum is given by  fp ẳ s r 1=s ẵ36 The high-frequency (HF) tail of the spectrum is proportional to f −r There is some debate on whether the spectral tail follows an f −4 shape or an f−5 shape Most recent theoretical and empirical evidence suggests that an f −4 shape is more appropriate (see Reference for a review) However, the most commonly used spectra in ocean engineering have an f −5 tail For practical purposes though, the difference appears to be small The family of spectra given by eqn [33] has five free parameters To describe the sea state with fewer variables, some of these parameters can be fixed, whereas the others are left free The most commonly used families of spectra with one, two, and three free parameters are summarized in Table Fixing r = 5, s = 4, and γ = gives the commonly used form proposed by Bretschneider [5] A special form of the Bretschneider spectrum for ‘fully developed’ seas was proposed by Pierson and Moskowitz [6], where α is fixed and the energy in the spectrum depends on the value of β only (equivalently Hs is in a fixed ratio to Tp) The Joint North Sea Wave Project (JONSWAP) form [7] was a further generalization of Bretschneider spectra, which accounted for the more peaked spectral shapes observed in fetch-limited wind seas The term ‘Ochi spectra’ has been used here for the case where s = 4, γ = 1, and r is a free parameter, after the use of this type of spectrum by Ochi and Hubble [8] Finally, the term ‘Gamma spectra’ is used by some authors to describe the form where γ = 1, r is a free parameter, and s = r − Obviously, for this type of spectrum, it could have been written equivalently that s is free and r is fixed as s + For shallow water applications, the commonly used spectral forms include the TMA spectrum [9] or the form proposed by Young and Babanin [10] Examples of the JONSWAP, Ochi, and Gamma families are shown in Figure for fixed Hs and fp and a range of the third free parameter In each case, the Bretschneider spectrum is a special case and is indicated with a bold line For the JONSWAP family, the Bretschneider spectrum is the limiting form, corresponding to the most broad-banded member, whereas both Gamma and Ochi can take more broad-banded forms When γ = 1, the moments of the spectra defined by eqn [33] can be expressed as explicit functions of α, β, r, and s:   α r n1 mn ẳ n r ỵ Þ =s Γ s s for n < r−1 ½37Š Table Free and fixed parameters for families of unimodal spectra given by eqn [33] Name α β r s γ Pierson–Moskowitz Bretschneider JONSWAP Ochi Gamma 5.0  10−4 Free Free Free Free Free Free Free Free Free 5 Free Free 4 4 r−1 1 Free 1 Resource Assessment for Wave Energy Ochi Gamma 3 2.5 2.5 2.5 1.5 0.5 0 fp 2fp 3fp Normalized spectral density Normalized spectral density Normalized spectral density JONSWAP 1.5 0.5 0 2fp fp 17 1.5 0.5 3fp fp 2fp 3fp Figure Examples of JONSWAP spectra for γ = 1–5 and Ochi and Gamma spectra for r = 2–10 Spectral densities have been normalized by the peak spectral density of a Bretschneider spectrum with the same Hs and fp In each plot, the Bretschneider spectrum is shown in bold where Γ is the Gamma function, defined by zị ẳ t z e t dt ½38Š If γ ≠ 1, then numerical integration must be used to compute the moments and hence the relationships between the spectral parameters Functions of γ can be defined so that the parameters α and β can be expressed in terms of height and period parameters as follows: ẳ gh ịHs2 ẳ Tp4 Tp ẳ ge ịTe ẳ gm ịTm ẳ gz ịTz ẵ39 Note that the relation between and Tp is independent of the value of γ The values of gh, ge, gm, and gz for ≤ γ ≤ 10 are shown in Table (the values for γ = correspond to the Bretschneider spectrum) The mean value of γ found in the JONSWAP spectrum was 3.3 This is sometimes referred to as the ‘standard’ JONSWAP spectrum and is often used to model extreme sea states The most commonly used multimodal spectral forms are formulated as the summation of JONSWAP, Gamma, or Ochi spectra Ochi and Hubble [8] proposed a six-parameter spectrum formed as the sum of two Ochi spectra However, each of the six free parameters was given as a function of Hs (together with 10 spectra representing a 95% confidence interval), so in essence this is a one-parameter spectrum Guedes Soares [11] proposed a bimodal spectrum formed as the sum of two JONSWAP spectra, but with γ fixed as for both components, resulting in a four-parameter spectrum Torsethaugen [12] and Torsethaugen and Haver [13] have proposed a form consisting of two JONSWAP spectra However, the values of the parameters of each spectrum are determined by the values of Hs and fp, so the number of free parameters is reduced from to Finally, Boukhanovsky and Guedes Soares [14] modeled multimodal spectra as the summation of Gamma spectra, imposing no restrictions on the parameter values, resulting in a true six-parameter spectrum Table Values of the height and period ratio functions for the JONSWAP spectrum γ gh(γ) ge(γ) gm(γ) gz(γ) 3.3 10 0.250 0.201 0.171 0.164 0.150 0.135 0.122 0.112 0.104 0.097 0.091 1.167 1.132 1.112 1.107 1.098 1.088 1.080 1.074 1.069 1.064 1.061 1.296 1.240 1.206 1.198 1.183 1.165 1.151 1.140 1.130 1.122 1.115 1.405 1.338 1.295 1.285 1.264 1.240 1.221 1.206 1.193 1.181 1.172 18 Resource Assessment for Wave Energy 8.03.2.2.5 Standard shapes for the directional distribution There are several commonly used forms of the directional distribution One of the most popular forms is due to Cartwright [15], who suggested using Dðθ; f ị ẳ Fsịcos 2s m ị ẵ40 where F(s) is a factor to satisfy condition of eqn [17] and is given by s ỵ 1ị Fsị ẳ p s ỵ =2 ị ẵ41 The circular moment definition of directional spread (eqn [30]) is related to the index s by σ 2c ẳ 1ỵs ẵ42 Another commonly used formulation is the wrapped normal distribution: "   # ∞ X θ − θm ðf Þ − 2πk p D; f ị ẳ exp l f ị l f ị k ẳ ∞ ½43Š This formulation directly includes the line moment spread parameter σl (eqn [29]) The summation over k in eqn [43] is to ensure that energy outside the interval [0 2π] is added back in In practice, the summation can be taken over the range k = − 2,…,2 For σ < 30, the ‘cosine-2s’ and wrapped normal distributions have very similar shapes For fetch-limited sea states, the directional distribution is bimodal at frequencies greater than about twice the peak frequency (see, e.g., References 16–18) Ewans [16] has proposed the use of a double Gaussian distribution to model this bimodality It can be written as ( "  "  #  #) ∞ X θ − θ2 ðf Þ − 2πk 1 θ − θ1 ðf Þ − 2πk pffiffiffiffiffiffi Dðf; θÞ ¼ exp − ỵ exp ẵ44 f ị f ị f ị k ẳ where f ị ẳ m ỵ f ị 2 f ị ¼ θm − Δθðf Þ and Δθ is the separation between the peaks of the two modes Note that the parameter σ in eqn [44] no longer corresponds to the directional spread The values of Δθ and σ are given as functions of frequency: ẵ45 ẳ 14:93 for f < f p "  f Δθ ¼ exp 5:453 − 2:750 fp − #  − 7:929 f ẳ 11:38 ỵ 5:357 fp   f σ ¼ 32:13 − 15:39 fp for f ≥ f p for f < f p for f ≥ f p ½46Š ½47Š ½48Š The resulting distribution is unimodal for f < 2fp and becomes bimodal at higher frequencies This formulation results in a directional distribution that is qualitatively the same as in earlier studies (e.g., References 19–21) in the way that the spread varies with frequency However, earlier studies made the a priori assumption that the distribution was unimodal Mitsuyasu et al [21] and Hasselmann et al [20] also suggested that the distribution was dependent on the wave age (a function of the wind speed and phase speed of the waves), whereas no such dependence was noted in later studies The shape of the distribution is shown in Figure The directional distribution of swell was examined by Ewans [22] Less evidence of bimodality in the directional distribution was found than for wind seas The use of the wrapped normal distribution (eqn [43]) was therefore proposed, with  l f ị ẳ ỵ f fp − for f < fp ½49Š Resource Assessment for Wave Energy 19 0.9 0.8 0.7 0.6 f/fp 0.5 0.4 0.3 0.2 0.5 θm − π 0.1 θm θm + π θ (rad) Figure Directional distribution specified by Ewans [16] Levels have been normalized to have a maximum value of at each frequency Figure Comparison of directional spreading with frequency for the swell and wind sea directional distributions proposed by Ewans [16, 22]  − 0:3 f σ l ðf ị ẳ 36 ỵ 46 fp for f f p ½50Š A comparison of the directional spread (circular moment definition) as a function of frequency for the wind sea and swell distributions proposed by Ewans [16, 22] is shown in Figure 8.03.2.2.6 Examples of sea surface elevation for standard spectral shapes It is useful to visualize how the standard spectral shapes discussed in the previous sections relate to waves that would be observed in the ocean, by simulating the sea surface elevation for various theoretical spectra The examples in this section are chosen for comparison with the measured spectra presented in Section 8.03.4.1.5 Figure shows the polar plot of a typical swell spectrum The polar plot shows how the energy is distributed with frequency and direction In this case, the frequency increases radially from the center, the directions are those from which the energy is coming, and color denotes the spectral density in square meter per hertz per degree The spectrum shown is a JONSWAP spectrum with a wrapped normal directional distribution for Hs = 1.25 m, Tp = 13 s, γ = 1.5, θm = 270°, and the directional distribution for swell proposed by Ewans [22] Figure also shows a simulation of the instantaneous sea surface elevation from this spectrum over an area of 2.5 km  2.5 km The sea surface elevation has been simulated from the spectrum using eqns [13] and [14] and assigning a random phase to each sine wave component Note the relatively large spatial scale of the waves shown here, with the crests of the larger waves extending for over 500 m in the y-direction 20 Resource Assessment for Wave Energy Figure Example of a swell wave system (see text for details) Left: Polar spectral density plot Right: Simulated sea surface elevation Figure Example of a wind sea (see text for details) Left: Polar spectral density plot Right: Simulated sea surface elevation The wavelength corresponding to the peak period is λp = gTp2/2π = 264 m, but the higher-frequency waves with greater directional spread are also visible as a shorter-scale roughness over the larger waves Figure shows a spectrum and simulated sea surface elevation typical of a wind sea This time, a JONSWAP frequency spectrum with Hs = m, Tp = s, and γ = 1.5 has been used, together with the Ewans [16] directional distribution with θm = 145° The same color scale has been used as in Figure In this case, the wavelengths are much shorter than the swell spectrum shown in Figure 5, with λp = 25.0 m A mixed sea, which is the sum of the swell and wind sea spectra illustrated in Figures and 6, respectively, is shown in Figure Even though the total energy in each component is similar, the swell shows a much higher peak in spectral density, since the energy is more focused in both frequency and direction Due to the large difference in the wavelengths of the swell and wind sea components, the wind sea is clearly discernable over the swell 8.03.3 Estimating WEC Power The response of a WEC is dependent on the full directional spectrum However, for the purposes of estimating the energy yield, it is useful to describe the response in terms of a small number of parameters Currently, few manufacturers of WECs publish details of the response of their device, partly for commercial reasons and partly because many devices are still at the developmental stage For Resource Assessment for Wave Energy 69 Figure 40 Ratio of density function estimated from iterative deconvolution procedure, fi, to true density function, f If we assume that storms occur at an average rate of once every days, we can calculate the effect that the errors in the model distribution can have on the estimated return values The 50- and 100-year return values for the true distribution are 17.47 and 18.93 m, respectively (rather higher than the return values estimated in Section 8.03.6.3 for this location, which emphasizes the drawbacks with the ID method) The bias in the return values as a function of the number of iterations is shown in Figure 41, with the zeroth iteration representing the initial model distribution The initial biases in the 50- and 100-year return values are 2.93 and 3.30 m, respectively It is perhaps worth reiterating here that this bias was caused by the effect of a zero-mean random error (although with a standard error of 25%, which is somewhat larger than would be expected in practice) The first iterative estimate of the true density function produces return values that are too low, since the effect of the convolution is overestimated at first However, after 20 iterations, the bias is less than cm for both the 50- and 100-year return values In practice, where the true return values are not known, the iterations can be terminated when the return values converge to a desired precision Figure 41 Bias in return value for iterative estimates of the density function Solid line is bias in 50-year return value and dashed line is bias in 100-year return value 70 Resource Assessment for Wave Energy In the example presented above, we have assumed that the model distribution is fitted over the entire range of the data In practice, however, the GPD is only fitted for exceedances over a high threshold and we not model the distribution of the bulk of the data This can be overcome by fitting a distribution to the bulk of the data and adjusting it so that the density function matches the GPD at the threshold value The choice of distribution for the bulk of the data is not important, as the far right tail is only affected by the convolution with the GPD part of the tail However, to avoid numerical instabilities in the deconvolution procedure, it is necessary to ensure that the density function has smooth transition between the distribution of the bulk of the data and the GPD A method to achieve this was proposed by Mackay [178] and it produced reasonable results The algorithm converged in all trials with error STDs less than 15%, but diverged in around 20% of the trials with an error STD of 20% and diverged in over half of the trials for error STDs of 25% Despite the convergence problems, the algorithm still reduced the bias to less than 1% in trials with error STDs up to 20% and around 2% in trials with an error STD of 25% For error STDs less than about 5%, it does not seem worthy of using the deconvolution procedure as the bias is not significant and may indeed be offset if an estimator with negative bias in estimates of high quantiles is used such as the LM or ML estimators (although note that estimators such as the PWM and moment estimators give positive bias for small sample sizes) However, the use of a deconvolution procedure can significantly reduce the bias when the hindcast has a high error STD 8.03.6.4 Combining Long-Term and Short-Term Distributions After the short-term and long-term distributions have been estimated, they can be combined to estimate the distribution of the maximum wave or crest height in a given period The problem of how to this correctly is rather subtle and the answer you get depends on the question you ask The problem is asking the correct question For the design of wave farms, the most relevant question is likely to be: “What is the probability that a storm will occur in a given period in which at least one wave exceeds a given size?” The ‘classic’ method for combining long-term and short-term distributions involves modeling the long-term distribution of sea states and then calculating the distribution of individual wave or crest heights for the most severe r-hour sea state (where r is typically or 6) in an N-year period This neglects the probability that the highest wave will occur in a lower sea state Carter and Challenor [179] showed that this can lead to a significant underestimation in estimates of extremes, since there will be several r-hour periods with Hs close to the most severe value, either within the same storm or in separate storms Forristall [180] has compared the various methods that address the shortcoming of the ‘classic’ method Battjes [181] proposed a method (subsequently refined by Tucker [182]) to calculate the distribution of the maximum wave in each r-hour period Another method was proposed by Krogstad [183], where the distribution of all wave heights in an N-year period is given by integrating the short-term distribution over the probability density function of significant wave height Forristall [180] demonstrated that the Krogstad method gives significantly higher estimates of return values for individual waves than the Battjes method He explains that the reason for this discrepancy is that when a storm occurs in which the Hs exceeds the N-year return value, there can be many very large individual waves during that interval All of these individual waves go into the distribution given by Krogstad [183], whereas the Battjes method only counts the highest wave in each r-hour interval Forristall goes on to note that the Battjes method gives higher estimates than methods that only consider the highest wave in each storm, since there can be several high waves in separate r-hour periods within each storm Or as Arena and Pavone [184] put it, “the number of crests exceeding a fixed threshold η is greater than the number of storms in which the maximum crest height exceeds η” As noted above, it seems more important for engineering purposes to know the probability that a single storm occurs in which a wave exceeds a given size If such a storm occurs, usually it does not matter if that wave height is exceeded more than once within that storm Therefore, in this section, we consider methods to calculate the distribution of the maximum wave in a random storm (Although more recently, Arena and Pavone [185] have given formulas for the probability of occurrence of a storm in which k waves exceed a threshold.) The first approach to estimating this distribution may be to perform a POT analysis on the maximum wave in each storm However, records of individual wave heights that are long enough to estimate the long-term distribution not exist The most probable maximum in each storm can be calculated using the method described in Section 8.03.6.4.1, but fitting the GPD to the most probable highest wave in each storm neglects the chance that the highest wave may exceed the most probable Forristall [180] demonstrated that the method of Tromans and Vanderschuren [161] gives the correct solution to this problem and recommends that their method should be used to combine long-term and short-term distributions The method of Tromans and Vanderschuren [161] combines the distribution of most probable maximum wave heights in a storm, with the distribution of the maximum wave height, conditional on the most probable wave height The method presented in the remainder of the section is based on the equivalent triangular storm (ETS) method of Boccotti [186], later simplified by Arena and Pavone [184] It is similar to Tromans and Vanderschuren’s approach and ends up at the same answer Both methods make some approximations to end up at a solution, but it is easier to understand how the approximations come about in the ETS method and to demonstrate that the approximations not affect the end result We start in Section 8.03.6.4.1 by showing how the distribution of the maximum wave in a storm is calculated In Section 8.03.6.4.2, we show how this distribution is closely matched by an ETS Finally, in Section 8.03.6.4.3, we show how the distribution of storm peak wave heights can be combined with the short-term distribution of wave or crest heights, using the concept of the ETS Throughout this section, we will use notation referring to wave heights rather than crest heights, but the theory is applicable to either of these Resource Assessment for Wave Energy 8.03.6.4.1 71 The distribution of the maximum wave or crest height in a storm If it is assumed that individual wave or crest heights are independent, then the probability that the maximum does not exceed a level h in an interval Δt is simply the product of the probability that each individual wave does not exceed h In the interval Δt, there are Δt/Tz waves, so PrfHmax ≤ h j Hs gẳẵPrfH h j Hs g t=Tz ½155Š In reality, consecutive wave and crest heights are correlated, with the largest waves occurring in groups However, the assumption of independence in eqn [155] is not restrictive Krogstad and Barstow [187] note that for Gaussian processes, there is an analytical theory for the distribution of the maximum value during a given period of time, and that this theory gives identical results to eqn [155] They argue that this observation makes it reasonable to use eqn [155] in the case of nonlinear waves, where in general no analytical theory exists During a storm, the value of Hs will vary over time Battjes [181] and Borgman [188] developed a method to calculate the distribution of the maximum wave in a storm with time-varying Hs at roughly the same time, but the method is now usually referred to as the Borgman integral Given a storm s with time-varying significant wave height Hs(t), the probability that the maximum wave in the storm does not exceed a value h is the product of the probabilities that it does not exceed h within each interval ti, i = 1,…, k, during the storm: k PrfHmax h j sgẳ ẵPrfH h j Hs ti ịg t=Tz ti ị iẳ1 ! k X t lnẵPrfH h j Hs ti ịg T t ị iẳ1 z i ẳ exp ẵ156 As t 0, eqn [156] can be expressed as an integral 0D ð PrfHmax ≤ h j sg ¼ exp@ 1 lnẵPrfH h j Hs tịgdt A Tz tị ½157Š where D is the duration of the storm The integral in eqn [157] needs to be evaluated over the interval where Hs(t) ≥ 0.8max (Hs(t)) as values outside this range not affect the distribution of the maximum Tucker and Pitt [26] showed that eqn [157] is 16 times more sensitive to changes in the peak Hs than Tz This means that Tz(t) can be replaced by a mean value for a given Hs, denoted T" z ðHs Þ, without affecting the calculation In the absence of measured data, it is usually assumed that the significant steepness of the pffiffiffiffiffiffi storm, defined as s = 2πHs/gTz2, is 1/18, which gives Tz ≈ 3:40 Hs 8.03.6.4.2 The equivalent triangular storm When evaluating the summation in eqn [156], the order in which each value of Hs occurs does not matter, but only the duration at each level of Hs matters Therefore, the intervals ti could be reordered to give an equivalent storm, symmetric about the peak Hs, in which Hs(ti) increases monotonically to the peak value This reordered shape can be approximated by a triangular storm, for which the expected maximum wave or crest height is equal to the value from eqn [156] This is known as the ETS More recently, Fedele and Arena [189] have proposed a generalization of this method, known as the equivalent power storm, where Hs is assumed to vary as a power law of time, rather than the linear assumption made in the ETS method The reader is referred to this article for details The distribution of the maximum wave in a triangular storm can be calculated using the Borgman integral Consider an isosceles triangle of height a and base b The Hs in this storm is given by Hs(t) = 2at/b for < t ≤ b/2 and Hs(t) = 2a − 2at/b for b/2 < t ≤ b Noting that |dHs/dt| = 2a/b, the Borgman integral for this storm can be written as b PrfH max ≤h j a; bg¼exp @ a ða 1 ln½Pr fH ≤ h j Hs gdHs A T"z Hs ị ẵ158 The value of a for the ETS is simply the peak Hs in the measured storm, but the value of b must be found by numerical methods First, the expected value of the maximum wave in the measured storm, denoted E(Hmax), is computed as d EHmax ị ẳ h ẵPrfHmax h j sgdh dh ẵ159 Now, if we define Qhị ¼ a ða ln½Pr fH ≤ h j Hs gdHs T" z Hs ị ẵ160 72 Resource Assessment for Wave Energy Figure 42 (a) Time series of Hs measured by NDBC buoy 46002 (bold line) and ETS to storm occurring on 05/01/08 (thin line) (b) Distribution of the maximum wave in the storm calculated from measured Hs and ETS then b is found by solving ð∞ d ẳ gbị ẳ EHmax ị h ẵexpb Qhịịdh dh ½161Š It is clear that g(b) is a smooth monotonic function, with a unique solution in (0, ∞), and thus readily amenable to numerical solution Moreover, the function Q(h) needs to be computed only once Typically, it is sufficient to compute the integral over the range a ≤ h ≤ 3a An example of the ETS to a storm measured by NDBC buoy 46002 on January 2008 is shown in Figure 42(a) The storm has a peak Hs of 10.67 m (after taking a h moving average) and the ETS has a duration of b = 82.0 h The distribution of wave heights has been computed using eqn [108] with ψ = 0.75 To illustrate that eqn [157] is not sensitive to the precise value of Tz used, we have computed the distribution of the maximum wave in the storm using the measured values of Tz and the distribution for the ETS using pffiffiffiffiffiffi Tz ¼ 3:40 Hs In this case, the measured Tz at the peak of the storm was 10.8 s and the peak value used in the ETS calculation is 11.1 s, so the difference is quite small A comparison of the two distributions is shown in Figure 42(b) A double-logarithmic scale is used on the y-axis to emphasize the fit at very low exceedance probabilities The distribution from the ETS is a good match to the distribution calculated from the measured data, over the entire range The distribution of the maximum wave from the ETS usually provides a good fit to the distribution from the measured storm, even if the ‘reordered’ shape of the storm is not exactly triangular Further examples of the performance of the ETS model are presented in the work of Boccotti [186] and Arena and Pavone [184] Under the double-logarithmic scale used in Figure 42(b), a Gumbel distribution would appear as a straight line It can be seen that the distribution of the maximum wave height in the storm is very close to a Gumbel distribution Borgman [188] and Tromans and Vanderschuren [161] proposed the use of a Gumbel distribution to model the distribution of the maximum wave height in a storm The example presented here lends support to the assertion made in the introduction to this section that the ETS method is in close agreement with Tromans and Vanderschuren’s method However, it is easier to understand the motivation for the ETS method, since it does not require asymptotic arguments Moreover, it is simple to verify that the approximations made in the ETS method not affect the results 8.03.6.4.3 The long-term distribution of the maximum wave or crest height To calculate the distribution of the maximum wave in N years, we first need to calculate the distribution of the maximum wave in a random storm This can be calculated by integrating the distribution of the maxima from the ETS over the density functions of the triangle heights and bases: ð∞ ð∞ PrfHmax ≤ h j r:s:g¼ pA ðaÞpB ðb j aÞPrfHmax ≤ h j a; bgda db ½162Š 0 where pA(a) is the density function of the triangle heights and pB(b | a) is the density function of the triangle bases, conditional on a, and r.s denotes a random storm The effect of changes in the ETS duration b is actually very small and the above equation can be calculated using an approximation where b is constant for a given value of a, "bðaÞ ð∞ ẩ ẫ PrfHmax h j r:s:g ẳ pA aịPr Hmax h j a; "baị da ẵ163 Resource Assessment for Wave Energy 73 The function "bðaÞ can be estimated by regression of b on a Usually, the regression is of the form "b ẳ x expyaị, where x, y > so that "bðaÞ is always positive, which may not be the case if linear regression is used The distribution of the maximum wave in N years is then given by the distribution of the maximum wave in a random storm, raised to the expected number of storms in N years: PrfHmax h j Ng ẳ ẵPrfHmax ≤ h j r:s:gŠ vN ½164Š where v is number of storms exceeding threshold per year Finally, the return period, N(h), for a given wave height is given by NðhÞ ¼ ν ⋅ PrfHmax > h j r:s:g ½165Š So to summarize, the procedure to follow to arrive at an estimate of the distribution of the maximum wave or crest height in N years can be broken down into the following steps: Identify storm maxima Calculate Borgman integral for each storm and find ETS Estimate "bðaÞ and T" z ðHs Þ from the data, using regression Estimate pA(a) using POT Calculate distribution of the largest wave or crest height in a random storm Calculate distribution of the largest wave or crest height in N years References [1] Rice SO (1944) The mathematical analysis of random noise Bell Systems Technical Journal 23: 282–332 [2] Rice SO (1945) The mathematical analysis of random noise Bell Systems Technical Journal 24: 46–156 [3] Young IR and van Vledder GP (1993) A review of the central role of nonlinear interactions in wind-wave evolution Philosophical Transactions of the Royal Society of London A 342: 505–524 [4] Holthuijsen LH (2007) Waves in Oceanic and Coastal Waters Cambridge, UK: Cambridge University Press [5] Bretschneider CL (1959) Wave variability and wave spectra for wind generated gravity waves Technical Memorandum No 118 Washington, DC: Beach Erosion Board, US Army Corps of Engineers [6] Pierson WJ and Moskowitz L (1964) A proposed spectral form for fully developed wind seas based on the similarity theory of S A Kitaigorodskii Journal of Geophysical Research 69(24): 5181–5190 [7] Hasselmann K, Barnett TP, and Bouws E (1973) Measurements of wind wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP) Deutsche Pharmazeutische Gesellschaft A8: 1–95 [8] Ochi MK and Hubble EN (1976) Six-parameter wave spectra In: Proceedings of the 15th Coastal Engineering Conference, pp 301–328 Honolulu, HI: ASCE [9] Bouws E, Gunther H, Rosenthal W, and Vincent CL (1985) Similarity of the wind wave spectrum in finite depth Part 1: Spectral form Journal of Geophysical Research 90(C1): 975–986 [10] Young IR and Babanin AV (2006) The form of the asymptotic depth-limited wind wave frequency spectrum Journal of Geophysical Research 111: C06031 [11] Guedes Soares C (1984) Representation of double-peaked sea wave spectra Ocean Engineering 11: 185–207 [12] Torsethaugen K (1993) A two peak wave spectrum model In: Proceedings of the 12th International Conference on Offshore Mechanics and Arctic Engineering, vol 2, pp 175–180 New York: ASME [13] Torsethaugen K and Haver S (2004) Simplified double peak spectral model for ocean waves Proceedings of the 14th International Offshore and Polar Engineering Conference Toulon, France [14] Boukhanovsky AV and Guedes Soares C (2009) Modelling of multipeaked directional wave spectra Applied Ocean Research 31: 132–141 [15] Cartwright DE (1963) The use of directional spectra in studying output of a wave recorder on a moving ship In: Ocean Wave Spectra: Proceedings of a Conference (National Academy of Sciences), pp 203–219 23–28 May 2004 Englewood Cliffs, NJ: Prentice Hall [16] Ewans KC (1998) Observations of the directional spectrum of fetch-limited waves Journal of Physical Oceanography 28: 495–512 [17] Hwang PA, Wang DW, Walsh EJ, et al (2000) Airborne measurements of the wavenumber spectra of ocean surface waves Part II: Directional distribution Journal of Physical Oceanography 30(11): 2768–2787 [18] Young IR, Verhagen LA, and Banner ML (1995) A note on the bimodal directional spreading of fetch-limited waves Journal of Geophysical Research 100(C1): 773–778 [19] Donelan MA, Hamilton J, and Hui WH (1985) Directional spectra of wind-generated waves Philosophical Transactions of the Royal Society of London A 315: 509–562 [20] Hasselmann DE, Dunckel M, and Ewing JA (1980) Directional wave spectra observed during JONSWAP 1973 Journal of Physical Oceanography 10: 1264–1280 [21] Mitsuyasu H, Tasai F, Suhara T, et al (1975) Observations of the directional spectrum of ocean waves using a cloverleaf buoy Journal of Physical Oceanography 5(4): 750–760 [22] Ewans KC (2001) Directional spreading in ocean swell In: Proceedings of the Ocean Wave Measurement and Analysis, pp 517–529 San Francisco, CA: ASCE [23] DTI (2007) Preliminary wave energy performance protocol DTI Report 07/807 London: DTI [24] Smith GH, Venugopal V, and Falsham J (2006) Wave spectral bandwidth as a measure of available wave power Proceedings of the 25th International Conference on OMAE 2006 Hamburg, Germany, 4–10 June [25] Kerbiriou MA, Prevosto M, Maisondieu C, et al (2007) Influence of sea-states description on wave energy production assessment Proceedings of the 7th European Wave and Tidal Energy Conference Porto, Portugal, 11–13 September 2007 [26] Tucker MJ and Pitt EG (2001) Waves in Ocean Engineering London: Elsevier Science Ltd [27] Forristall GZ, Barstow SF, Krogstad HE, et al (2004) Wave crest sensor intercomparison study: An overview of WACSIS Journal of Offshore Mechanics and Arctic Engineering 126(1): 26–34 [28] Pedersen T and Siegel E (2008) Wave measurements from subsurface buoys In: Proceedings of the IEEE/OES/CMTC 9th Working Conference on Current Measurement Technology, pp 224–233 Charleston, NC: IEEE [29] Kahma K, Hauser D, Krogstad HE, et al (2005) Measuring and analysing the directional spectra of ocean waves EU COST Action 714, EUR 21367, ISBN 92-898-0003-8 74 Resource Assessment for Wave Energy [30] Collard F, Ardhuin F, and Chapron B (2005) Extraction of coastal ocean wave fields from SAR images IEEE Journal of Oceanic Engineering 30(3): 526–533 [31] Schulz-Stellenfleth J, Konig T, and Lehner S (2007) An empirical approach for the retrieval of integral ocean wave parameters from synthetic aperture radar data Journal of Geophysical Research 112: C03019 doi: 10.1029/2006JC003970 [32] Schulz-Stellenfleth J, Lehner S, and Hoja D (2005) A parametric scheme for the retrieval of two-dimensional ocean wave spectra from synthetic aperture radar look cross spectra Journal of Geophysical Research 110: C05004 doi: 10.1029/2004JC002822 [33] Violante-Carvalho N, Robinson IS, and Schulz-Stellenfleth J (2005) Assessment of ERS synthetic aperture radar wave spectra retrieved from the Max-Planck-Institut (MPI) scheme through intercomparisons of year of directional buoy measurements Journal of Geophysical Research 110: C07019 doi: 10.1029/2004JC002382 [34] Abdalla S, Bidlot JR, and Janssen P (2006) Global validation and assimilation of Envisat ASAR wave mode spectra Proceedings of SEASAR 2006: Advances in SAR Oceanography from Envisat and ERS Missions Frascati, Italy: European Space Agency 24–26 January 2006 [35] Gulev SK and Grigorieva V (2004) Last century changes in ocean wind wave height from global visual wave data Geophysical Research Letters 31: L24302 doi: 10.1029/ 2004GL021040 [36] Gulev SK and Grigorieva V (2006) Variability of the winter wind waves and swell in the North Atlantic and North Pacific as revealed by the voluntary observing ship data Journal of Climate 19: 5667–5685 [37] Gulev SK, Grigorieva V, Sterl A, and Woolf D (2003) Assessment of the reliability of wave observations from voluntary observing ships: Insights from the validation of a global wind wave climatology based on voluntary observing ship data Journal of Geophysical Research 108: 3236 doi: 10.1029/2002JC001437 [38] James ID (1986) A note on the theoretical comparison of wave staff and wave rider buoys in steep gravity waves Ocean Engineering 13: 209–214 [39] Longuet-Higgins MS (1986) Eulerian and Lagrangian aspects of surface waves Journal of Fluid Mechanics 173: 683–707 [40] Joosten HP (2006) Wave buoys and their elastic mooring International Ocean Systems 10(3): 18–21 [41] Rademakers PJ (1993) Waverider–wavestaff comparison Ocean Engineering 20(2): 187–193 [42] Forristall GZ (2000) Wave crest distributions: Observations and second-order theory Journal of Physical Oceanography 30(8): 1931–1943 [43] Prevosto M, Krogstad HE, and Robin A (2000) Probability distributions for maximum wave and crest heights Coastal Engineering 40: 329–360 [44] Barrick DE, Lipa BJ, and Steele KE (1989) Comments on theory and application of calibration techniques for an NDBC directional wave measurements buoy: Nonlinear effects IEEE Journal of Oceanic Engineering 14(3): 268–272 [45] Steele KE, Lau J, and Hsu YH (1985) Theory and application of calibration techniques for an NDBC directional wave measurements buoy Ocean Engineering 10(4): 382–396 [46] Allender J, Audunson T, Barstow SF, et al (1989) The WADIC project: A comprehensive field evaluation of directional wave instrumentation Ocean Engineering 16(5–6): 505–536 [47] Joosten HP (2006) Directional wave buoys and their elastic mooring International Ocean Systems 10(4): 18–21 [48] Steele KE (1996) Ocean current kinematic effects on pitch–roll buoy observations of mean wave direction and nondirectional spectra Journal of Atmospheric and Oceanic Technology 14: 278–293 [49] Krogstad HE, Barstow SF, Aasen SE, and Rodriguez I (1999) Some recent developments in wave buoy measurement technology Coastal Engineering 37: 309–329 [50] de Vries J, Waldron J, and Cunningham V (2003) Field tests of the new Datawell DWR-G GPS wave buoy Sea Technology 44(12): 50–55 [51] Rodríguez G, Guedes Soares C, and Machado U (1999) Uncertainty of the sea state parameters resulting from the methods of spectral estimation Ocean Engineering 26: 991–1002 [52] Massel SR (2001) Wavelet analysis for processing of ocean surface wave records Ocean Engineering 28: 957–987 [53] Benoit M, Frigaard P, and Schaffer HA (1997) Analysing multidirectional wave spectra: A tentative classification of available methods Proceedings of the IAHR Seminar: Multidirectional Waves and Their Interaction with Structures, 27th IAHR Congress San Francisco, CA, USA, August [54] Longuet-Higgins MS, Cartwright DE, and Smith ND (1963) Observations of the directional spectrum of sea waves using the motions of a floating buoy In: Ocean Wave Spectra: Proceedings of a Conference (National Academy of Sciences), pp 111–136 Englewood Cliffs, NJ: Prentice Hall [55] Kuik AJ, Van Vledder GP, and Holthuijsen LH (1988) A method for the routine analysis of pitch–roll–heave buoy wave data Journal of Physical Oceanography 18: 1020–1034 [56] Pawka SS (1983) Island shadows in wave directional spectra Journal of Geophysical Research 88: 2579–2591 [57] Oltman-Shay J and Guza RT (1984) A data-adaptive ocean wave directional-spectrum estimator for pitch–roll type measurements Journal of Physical Oceanography 14: 1800–1810 [58] Krogstad HE, Gordon RL, and Miller MC (1988) High-resolution directional wave spectra from horizontally mounted acoustic Doppler current meters Journal of Atmospheric and Oceanic Technology 5: 340–352 [59] Lygre A and Krogstad HE (1986) Maximum entropy estimation of the directional distribution in ocean wave spectra Journal of Physical Oceanography 16: 2052–2060 [60] Kobune K and Hashimoto N (1986) Estimation of directional spectra from the maximum entropy principle In: Proceedings of the 5th International Offshore Mechanics and Arctic Engineering Symposium, vol 1, pp 80–85 Tokyo, Japan: ASME [61] Brissette FP and Tsanis IK (1994) Estimation of wave directional spectra from pitch–roll buoy data Journal of Waterway, Port, Coastal, and Ocean Engineering 120(1): 93–115 [62] Kim T, Lin L, and Wang H (1994) Application of the maximum entropy method to the real sea data In: Proceedings of the 24th International Conference on Coastal Engineering, pp 340–355 Kobe, Japan: American Society of Civil Engineers [63] Nwogu OU, Mansard EPD, Miles MD, and Isaacson M (1987) Estimation of directional wave spectra by the maximum entropy method Proceedings of the 17th IAHR Seminar Lausanne, Switzerland 1–4 September 1987 [64] National Data Buoy Center (NDBC) (2003) Handbook of automated data quality control checks and procedures of the National Data Buoy Center Technical Document 03-02 Hancock County, MS: Stennis Space Center [65] National Oceanic and Atmospheric Administration (NOAA) (2005) Second workshop report on the quality assurance of real-time ocean data CCPO Technical Report Series No 05-01, July 2005, Norfolk, VA, 48pp [66] Krogstad HE, Wolf J, Thompson SP, and Wyatt LR (1999) Methods for intercomparison of wave data Coastal Engineering 37: 235–257 [67] Donelan MA and Pierson WJ (1983) The sampling variability of estimates of spectra of wind-generated waves Journal of Geophysical Research 88: 4381–4392 [68] Carter DJT and Tucker MJ (1986) Uncertainties in environmental design criteria Underwater Technology 12: 2–8, 33 [69] Young IR (1986) Probability distribution of spectral integrals Journal of Waterway, Port, Coastal, and Ocean Engineering 112(4): 338–341 [70] Forristall GZ, Heideman JC, Leggett IM, et al (1996) Effect of sampling variability on hindcast and measured wave heights Journal of Waterway, Port, Coastal, and Ocean Engineering 122(5): 216–225 Discussion: Goda Y 124(4): 214 [71] Fu LL and Cazenave A (2001) Satellite Altimetry and Earth Sciences: A Handbook of Techniques and Applications International Geophysics Series, vol 69 San Diego, CA: Academic Press [72] Chelton D, Walsh EJ, and MacArthur JL (1989) Pulse compression and sea level tracking in satellite altimetry Journal of Atmospheric and Oceanic Technology 6: 407–438 [73] Quartly GD (1997) Achieving accurate altimetry across storms: Improved wind and wave estimates from C band Journal of Atmospheric and Oceanic Technology 14(3): 705–715 [74] Quilfen Y, Tournadre J, and Chapron B (2006) Altimeter dual-frequency observations of surface winds, waves, and rain rate in tropical cyclone Isabel Journal of Geophysical Research 111: C01004 doi: 10.1029/2005JC003068 [75] Quilfen Y, Chapron B, and Serre M (2004) Calibration/validation of an altimeter wave period model and application to TOPEX/Poseidon and Jason-1 altimeters Marine Geodesy 27: 535–549 [76] Mackay EBL, Retzler CH, Challenor PG, and Gommenginger CP (2008) A parametric model for ocean wave period from Ku-band altimeter data Journal of Geophysical Research 113: C03029 doi: 10.1029/2007JC004438 Resource Assessment for Wave Energy 75 [77] Mackay EBL, Retzler CH, Challenor PG, and Bahaj AS (2008) Wave energy resource assessment using satellite altimeter data In: Proceedings of the ASME 27th International Conference on Offshore Mechanics and Arctic Engineering (OMAE2008-57976) Lisbon, Portugal: ASME [78] Carter DJT, Challenor PG, and Srokosz MA (1992) An assessment of Geosat wave height and wind speed measurements Journal of Geophysical Research 97(C7): 11383–11392 [79] Challenor PG, Foale S, and Webb DJ (1990) Seasonal changes in the global wave climate measured by the Geosat altimeter International Journal of Remote Sensing 11(12): 2205–2213 [80] Young IR and Holland GJ (1996) Atlas of the Oceans: Wind and Wave Climate, 241pp Oxford: Pergamon Press [81] Tournadre J (1993) Time and space scales of significant wave height Journal of Geophysical Research 98(C3): 4727–4738 [82] Caires S and Sterl A (2003) Validation of ocean wind and wave data using triple collocation Journal of Geophysical Research 108(C3): 3098 [83] Challenor PG and Cotton PD (2002) The joint calibration of altimeter and in situ wave heights World Meteorological Organization Document Number WMO/TD-No 1081, JCOMM Technical Report No 13 Geneva, Switzerland: WMO [84] Gower JFR (1996) Intercalibration of wave and wind data from Topex/Poseidon and moored buoys off the west coast of Canada Journal of Geophysical Research 101: 3817–3829 [85] Janssen PAEM, Abdalla S, Hersbach H, and Bidlot JR (2007) Error estimation of buoy, satellite and model wave height data Journal of Atmospheric and Oceanic Technology 24: 1665–1677 [86] Queffeulou P (2004) Long-term validation of wave height measurements from altimeters Marine Geodesy 27: 495–510 [87] Cheng CL and Van Ness JW (1999) Statistical Regression with Measurement Error London: Arnold [88] Tolman HL (1998) Effect of observation errors in linear regression and bin-average analyses Quarterly Journal of the Royal Meteorological Society 124: 897–917 [89] Krogstad HE and Barstow SF (1999) Satellite wave measurements for coastal engineering applications Coastal Engineering 37: 283–307 [90] Barstow S and Krogstad HE (1995) Wave climate assessment by satellite remote sensing Proceedings of the 5th International Offshore and Polar Engineering Conference (ISOPE ’95) Hague, The Netherlands [91] Queffeulou P and Bentamy A (2007) Analysis of wave height variability using altimeter measurements: Application to the Mediterranean Sea Journal of Atmospheric and Oceanic Technology 24: 2078–2092 11–16 June 1995 [92] Woolf DK, Cotton PD, and Challenor PG (2003) Measurement of the offshore wave climate around the British Isles by satellite altimeter Philosophical Transactions of the Royal Society of London A 361: 27–31 [93] Young IR (1994) Global ocean wave statistics obtained from satellite observations Applied Ocean Research 16: 235–248 [94] DTI (2004) Atlas of UK marine renewable energy resources: Technical report DTI Report No R.1106 ABPmer, Met Office, Garrad Hassan and Proudman Oceanographic Laboratory London: DTI [95] ESB International (2005) Accessible wave energy resource atlas: Ireland Report No 4D404A-R2 Marine Institute/Sustainable Energy Ireland Dublin, Ireland: Marine Institute [96] Dingemans MW (1997) Water Wave Propagation over Uneven Bottoms Part 1: Linear Wave Propagation; Part 2: Non-Linear Wave Propagation Advanced Series on Ocean Engineering, vol 13 Singapore: World Scientific [97] Komen GJ, Cavaleri L, Donelan M, et al (1994) Dynamics and Modelling of Ocean Waves Cambridge, UK: Cambridge University Press [98] WAMDI Group (1988) The WAM model – A third generation ocean wave prediction model Journal of Physical Oceanography 18(12): 1775–1810 [99] Tolman HL, Balasubramaniyan B, Burroughs LD, et al (2002) Development and implementation of wind-generated ocean surface wave models at NCEP Weather and Forecasting 17: 311–333 [100] Hasselmann S, Lionello P, and Hasselmann K (1997) An optimal interpolation scheme for the assimilation of spectral data Journal of Geophysical Research 102(C7): 15823–15836 [101] Lionello P, Gunther H, and Janssen PAEM (1992) Assimilation of altimeter data in a global third-generation wave model Journal of Geophysical Research 97: 14453–14474 [102] Young IR and Eldeberky Y (1998) Observations of triad coupling of finite depth wind waves Coastal Engineering 33(2–3): 137–154 [103] Booij N, Ris RC, and Holthuijsen LH (1999) A third-generation wave model for coastal regions Part 1: Model description and validation Journal of Geophysical Research 104(C4): 7649 (98JC02622) [104] Benoit M, Marcos F, and Becq F (1996) Development of a third generation shallow-water wave model with unstructured spatial meshing In: Proceedings of the 25th International Conference on Coastal Engineering (ICCE 1996), pp 465–478 Orlando, FL: American Society of Civil Engineers [105] Danish Hydraulic Institute (1996) MIKE 21, Elliptic-Mild-Slope Wave Module, Release 2.6, User Guide and Reference Manual [106] Cavaleri L, et al (2007) Wave modelling – The state of the art Progress in Oceanography 75: 603–674 [107] Janssen PAEM (1998) On error growth in wave models ECMWF Technical Memo 249, p 12 Reading, UK: ECMWF [108] Rogers WE, Wittmann PA, Wang DWC, et al (2005) Evaluations of global wave prediction at the Fleet Numerical Meteorology and Oceanography Center Weather and Forecasting 20(5): 745–760 [109] Feng H, Vandemark D, Quilfen Y, et al (2006) Assessment of wind-forcing impact on a global wind-wave model using the TOPEX altimeter Ocean Engineering 33: 1431–1461 [110] Tolman HL (2002) Alleviating the Garden Sprinkler Effect in wind wave models Ocean Modelling 4: 269–289 [111] Ponce de Leon S and Guedes Soares C (2005) On the sheltering effect of islands in ocean wave models Journal of Geophysical Research 110: C09020 doi: 10.1029/ 2004JC002682 [112] Tolman HL (2003) Treatment of unresolved islands and ice in wind wave models Ocean Modelling 5: 219–231 [113] Tolman HL (2002) Validation of WAVEWATCH III version 1.15 for a global domain Technical Report 213 Camp Springs, MD: NOAA National Centers for Environmental Prediction [114] Cavaleri L (2006) Wave modelling – Where to go in the future Bulletin of the American Meteorological Society 87(2): 207–214 [115] Liu PC, Schwab DJ, and Jensen RE (2002) Has wind–wave modelling reached its limit? Ocean Engineering 29: 81–98 [116] Janssen PAEM (2008) Progress in ocean wave forecasting Journal of Computational Physics 227: 3572–3594 [117] Bidlot JR, Holmes DJ, Wittmann PA, et al (2002) Intercomparison of the performance of operational ocean wave forecasting systems with buoy data Weather and Forecasting 17: 287–310 [118] Cox AT and Swail VR (2001) A 40-year global reanalysis of ocean waves 1958–1997 (GROW) Journal of Geophysical Research 106(C2): 2313–2329 [119] Mackay EBL, Bahaj AS, and Challenor PG (2010) Uncertainty in wave energy resource assessment Part 1: Historic data Renewable Energy doi: 10.1016/j.renene.2009.10.026 [120] Falcão AF (2002) Wave-power absorption by a periodic linear array of oscillating water columns Ocean Engineering 29: 1163–1186 [121] Millar DL, Smith HCM, and Reeve DE (2007) Modelling analysis of the sensitivity of shoreline change to a wave farm Ocean Engineering 34: 884–901 [122] Mackay EBL, Bahaj AS, and Challenor PG (2010) Uncertainty in wave energy resource assessment Part 2: Variability and predictability Renewable Energy doi: 10.1016/ j.renene.2009.10.027 [123] Mackay EBL (2009) Wave Energy Resource Assessment PhD Thesis, University of Southampton [124] Monbet V, Ailliot P, and Prevosto M (2007) Survey of stochastic models for wind and sea state time series Probabilistic Engineering Mechanics 22(2): 113–126 [125] Athanassoulis GA and Stefanakos CN (1995) A nonstationary stochastic model for long-term time series of significant wave height Journal of Geophysical Research 100(C8): 16149–16162 [126] Cunha C and Guedes Soares C (1999) On the choice of data transformation for modelling time series of significant wave height Ocean Engineering 26: 489–506 [127] Guedes Soares C and Cunha C (2001) Bivariate autoregressive models for time series of significant wave height and mean period Coastal Engineering 40: 297–311 [128] Hardle W, Horowitz J, and Kreiss JP (2003) Bootstrap methods for time series International Statistical Review 71(2): 435–459 76 [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] Resource Assessment for Wave Energy Deo MC, Jha A, Chapekar AS, and Ravikant K (2001) Neural network for wave forecasting Ocean Engineering 28: 889–898 Makarynskyy O (2004) Improving wave predictions with artificial neural networks Ocean Engineering 31: 709–724 Woolf DK, Challenor PG, and Cotton PD (2002) The variability and predictability of the North Atlantic wave climate Journal of Geophysical Research 107(C10): 3145 Longuet-Higgins MS (1952) On the statistical distribution of the heights of sea waves Journal of Marine Research 9: 245–266 Tayfun MA (1980) Narrow-band nonlinear sea waves Journal of Geophysical Research 85(C3): 1548–1552 Longuet-Higgins MS (1980) On the distribution of the heights of sea waves: Some effects of nonlinearity and finite bandwidth Journal of Geophysical Research 85: 1519–1523 Onorato M, Cavaleri L, Fouques S, et al (2009) Statistical properties of mechanically generated surface gravity waves: A laboratory experiment in a three-dimensional wave basin Journal of Fluid Mechanics 627: 235–257 Toffoli A, Benoit M, Onorato M, and Bitner-Gregersen EM (2009) The effect of third-order nonlinearity on statistical properties of random directional waves in finite depth Nonlinear Processes in Geophysics 16: 131–139 Forristall GZ (2005) Understanding rogue waves: Are new physics really necessary? In: Proceedings of the 14th ‘Aha Huliko’a Hawaiian Winter Workshop, pp 29–35 Honolulu, HI: University of Hawaii Tayfun MA and Fedele F (2007) Wave-height distributions and nonlinear effects Ocean Engineering 34: 1631–1649 Boccotti P (1981) On the highest waves in a stationary Gaussian process Atti Accademia Ligure di Scienze e Lettere 38: 271–302 Fedele F and Arena F (2005) Weakly nonlinear statistics of high random waves Physics of Fluids 17(2): 026601, 1–10 Tayfun MA (2006) Statistics of nonlinear wave crests and groups Ocean Engineering 33(11–12): 1589–1622 Coles S (2001) An Introduction to the Statistical Modelling of Extreme Values London: Springer-Verlag Al-Humoud J, Tayfun MA, and Askar H (2000) Distribution of nonlinear wave crests Ocean Engineering 29: 1929–1943 Ferreira JA and Guedes Soares C (1999) Modelling the long-term distribution of significant wave height with the Beta and Gamma models Ocean Engineering 26: 713–725 Mathiesen M, Goda Y, Hawkes P, et al (1994) Recommended practice for extreme wave analysis Journal of Hydraulic Research 32(6): 803–814 Guedes Soares C and Scotto MG (2004) Application of the r largest-order statistics for long-term predictions of significant wave height Coastal Engineering 51(5–6): 387–394 Sobey R and Orloff L (1995) Triple annual maximum series in wave climate analyses Coastal Engineering 26: 135–151 Van Vledder G, Goda Y, Hawkes P, et al (1993) Case studies of extreme wave analysis: A comparative analysis In: Magoon OT and Hemsley JM (eds.) Ocean Wave Measurement and Analysis, pp 978–992 New York: American Society of Civil Engineers Guedes Soares C and Scotto MG (2001) Modelling uncertainty of long-term predictions of significant wave height Ocean Engineering 28: 329–342 Goda Y, Hawkes P, Mansard E, et al (1993) Intercomparison of extremal wave analysis methods using numerically simulated data In: Magoon OT and Hemsley JM (eds.) Ocean Wave Measurement and Analysis, pp 963–977 New York: American Society of Civil Engineers Elsinghorst C, Groeneboom P, Jonathan P, et al (1998) Extreme value analysis of North Sea storm severity Journal of Offshore Mechanics and Arctic Engineering 120(3): 177–183 Ferreira JA and Guedes Soares C (1998) An application of the peaks over threshold method to predict extremes of significant wave height Journal of Offshore Mechanics and Arctic Engineering 120(3): 165–176 Naess A (1998) Statistical extrapolation of extreme value data based on the peaks over threshold method Journal of Offshore Mechanics and Arctic Engineering 120: 91–96 Taylor PH and Goh JYK (2000) Extrapolation of long-term storm statistics, the POT method and data transformation Proceedings of the 19th International Conference on Offshore Mechanics and Arctic Engineering (OMAE00-6127) ASME New Orleans, 14–17 February 2000 Guedes Soares C, Ferreira RG, and Scotto MG (2003) On the prediction of extreme values of significant wave heights Proceedings of the 22nd International Conference on Offshore Mechanics and Arctic Engineering (OMAE2003-37478) Cancun, Mexico, 8–13 June 2003 Caires S and Sterl A (2005) 100-Year return value estimates for ocean wind speed and significant wave height from the ERA-40 data Journal of Climate 18(7): 1032–1048 Morton ID, Bowers J, and Mould G (1997) Estimating return period wave heights and wind speeds using a seasonal point process model Coastal Engineering 31: 305–326 Anderson CW, Carter DJT, and Cotton PD (2001) Wave climate variability and impact on offshore design extremes Report for Shell International and the Organization of Oil & Gas Producers, 90pp http://info.ogp.org.uk/metocean/JIPweek/att/WCEReport_2sided.pdf (accessed 23 December 2011) Leadbetter MR, Lindgren G, and Rootzen H (1983) Extremes and Related Properties of Random Sequences and Processes New York: Springer-Verlag Tawn JA (1988) An extreme-value theory model for dependent observations Journal of Hydrology 101: 227–250 Tromans PS and Vanderschuren L (1995) Response based design conditions in the North Sea: Application of a new method Proceedings of the 27th Offshore Technology Conference (OTC 7683) Houston, TX, 1–4 May 1995 Stefanakos CN and Athanassoulis GA (2006) Extreme value predictions based on nonstationary time series of wave data Environmetrics 17(1): 25–46 de Zea Bermudez P and Kotz S (2009) Parameter estimation of the generalized Pareto distribution – Parts I and II Journal of Statistical Planning and Inference doi: 10.1016/j.jspi.2008.11.019 and doi:10.1016/j.jspi.2008.11.020 Hosking JRM and Wallis JR (1987) Parameter and quantile estimation for the generalized Pareto distribution Technometrics 29: 339–349 Dupuis DJ and Tsao M (1998) A hybrid estimator for generalized Pareto and extreme-value distributions Communications in Statistics – Theory and Methods 27: 925–941 Mackay EBL, Challenor PG, and Bahaj AS (2010) A comparison of estimators for the generalised Pareto distribution Ocean Engineering 38(11–12): 1338–1346 Zhang J (2007) Likelihood moment estimation for the generalised Pareto distribution Australian & New Zealand Journal of Statistics 49(1): 69–77 Efron B and Tibshirani RJ (1993) An Introduction to the Bootstrap New York: Chapman & Hall Gumbel EJ (1958) Statistics of Extremes, 375pp New York: Columbia University Press Cunnane C (1978) Unbiased plotting positions – A review Journal of Hydrology 37(3–4): 205–222 Jonathan P and Ewans KC (2007) The effect of directionality on extreme wave design criteria Ocean Engineering 34: 1977–1994 Jonathan P and Ewans KC (2008) Modelling the seasonality of extreme waves in the Gulf of Mexico Proceedings of the 27th International Conference on Offshore Mechanics and Arctic Engineering (OMAE2008-57131) Estoril, Portugal, 15–20 June Jonathan P, Ewans K, and Forristall G (2008) Statistical estimation of extreme ocean environments: The requirement for modelling directionality and other covariate effects Ocean Engineering 35: 1211–1225 Mackay EBL, Challenor PG, and Bahaj AS (2010) On the use of discrete seasonal and directional models for the estimation of extreme wave conditions Ocean Engineering 37: 425–442 Cardone VJ, Jensen RE, Resio DT, et al (1996) Evaluation of contemporary ocean wave models in rare extreme events: The “Halloween Storm” of October 1991 and the “Storm of the Century” of March 1993 Journal of Atmospheric and Oceanic Technology 13: 198–230 Jensen RE, Cardone VJ, and Cox AT (2006) Performance of third generation wave models in extreme hurricanes Proceedings of the 9th International Workshop on Wave Hindcasting and Forecasting Victoria, BC, Canada, 24–29 September Brooker DC, Cole GK, and McConochie JD (2004) The influence of hindcast modelling uncertainty on the prediction of high return period wave conditions Proceedings of the 23th International Conference on Offshore Mechanics and Arctic Engineering Vancouver, BC, Canada, 20–25 June Mackay EBL (2011) Correction for bias in return values of wave heights caused by hindcast uncertainty Proceedings of the ASME 30th International Conference on Offshore Mechanics and Arctic Engineering (OMAE2011-49409) Rotterdam, The Netherlands, 20–23 June 2011 Carter DJT and Challenor PG (1990) Metocean Parameters – Wave Parameters Part 2: Estimation of Extreme Waves UK Department of Energy, OTH89-300 London: HMSO Forristall GZ (2008) How should we combine long and short term wave height distributions? Proceedings of the 27th International Conference on Offshore Mechanics and Arctic Engineering (OMAE2008-58012) Estoril, Portugal, 15–20 June Battjes JA (1970) Long-term wave height distribution at seven stations around the British Isles National Institute of Oceanography, Internal Report No A44, 31pp Resource Assessment for Wave Energy [182] [183] [184] [185] [186] [187] [188] [189] [190] [191] 77 Tucker MJ (1989) Improved ‘Battjes’ method for predicting the probability of extreme waves Applied Ocean Research 11(4): 212–218 Krogstad HE (1985) Height and period distributions of extreme waves Applied Ocean Research 7: 158–165 Arena F and Pavone D (2006) The return period of non-linear high wave crests Journal of Geophysical Research 111: C08004 Arena F and Pavone D (2009) A generalized approach for long-term modelling of extreme crest-to-trough wave heights Ocean Modelling 26: 217–225 Boccotti P (2000) Wave Mechanics for Ocean Engineering Oxford: Elsevier Science Krogstad HE and Barstow SF (2004) Analysis and applications of second-order models for maximum crest height Journal of Offshore Mechanics and Arctic Engineering 126: 66–71 Borgman LE (1973) Probabilities for highest wave in hurricane Journal of the Waterways, Harbors and Coastal Engineering Division 99(WW2): 185–207 Fedele F and Arena F (2010) Long-term statistics and extreme waves of sea storms Journal of Physical Oceanography 40(5): 1106–1117 Saulnier J-B, Clément AH, Falcão AF de O, et al (2011) Wave groupiness and spectral bandwidth as relevant parameters for the performance assessment of wave energy converters Ocean Engineering 38: 130–147 Cotton PD and Carter DJT (1994) Cross calibration of TOPEX, ERS-1, and Geosat wave heights Journal of Geophysical Research 99(C12): 25025–25033 ... 1.206 1.1 98 1. 183 1.165 1.151 1.140 1.130 1.122 1.115 1.405 1.3 38 1.295 1. 285 1.264 1.240 1.221 1.206 1.193 1. 181 1.172 18 Resource Assessment for Wave Energy 8. 03. 2.2.5 Standard shapes for the...12 Resource Assessment for Wave Energy 8. 03. 1 Introduction The assessment of the wave energy resource is an important step in the planning of a wave energy project The process... 1 986 –3 0 December 1 989 January 200 0–2 6 November 20 08 25 September 1992 8 October 2005 15 January 2002–present July 20 08 present August 199 1–2 June 1996 29 April 1995–present 24 September 2002–present