Proceedings of the ASME nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France

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1 Proceedings of the ASME nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France OMAE WEST AFRICA SWELL SPECTRAL SHAPES Michel Olagnon Ifremer Plouzané, France Kevin Ewans Sarawak Shell Berhad Kuala Lumpur Malaysia George Forristall Forristall Ocean Engineering, Inc. Camden, Maine, ME Marc Prevosto Ifremer Plouzané, France ABSTRACT Wave spectra measured at sites off West Africa are dominated by the constant presence of one or several swell wave systems. The West Africa Swell Project WASP JIP) was carried out to propose and assess parametric models for the shapes of the swell components. Bias, variability, and dispersion of estimates are affected by the length/stationarity compromise of the record lengths and the window-tapering used to reduce their variability. In particular, shapes with sharp angles are strongly smoothed, for instance a triangular peak would appear round and reduced by 15 to 25% with rectangular or Tuckey windowing. Models that consider each wave system individually, and an arbitrary number of those, were preferred to global ones. Partitioning of directional spectra is thus a prerequisite, and needs to be tuned taking account some prior knowledge of the swell characteristics. Triangular, log-normal, Gaussian and Glenn- Jonswap shapes were considered. Sampling variability makes it difficult to distinguish between those shapes as far as swells are concerned. The models also indicate that the width of the spectrum in frequency should be inversely proportional to the peak frequency. Directional spreading width shows a similar trend. Address all correspondence to this author. Fits to the measurements established proportionality factors for each location. INTRODUCTION The spectrum of a sea-state is a stochastic description that can theoretically only be exactly computed if the sea-state extends to infinity in time and in space. In practice, sea-states are only stationary for durations of the order of magnitude of a few hours and measurements are often even shorter. The properties of the spectrum must thus be estimated from a short segment of time-history that appears as an instance within the set of all the possible sea-states that may derive from that theoretical spectrum. Variability within that set leading to random uncertainty) is large, and processing techniques are used to improve the quality of the estimate of the spectrum. Those techniques, smoothing and averaging, lead to biases and other epistemic uncertainties, and though engineers are usually well aware of the existence of the problem, they may sometimes satisfy themselves with the fact that they use a reputable processing method and do not consider the actual range of uncertainties and its consequences on the significance of results. Swell spectra exhibit uncommon narrowness, both in fre- 1 Copyright 2013 by ASME

2 quency and direction, whereas most spectral models consider wind seas of wider bandwidth. A first section of the present paper is thus devoted to biases and uncertainties in spectral estimation that appear specifically when dealing with narrow spectra. In many applications where a sea state is to be represented by its power spectral density, parametric models are prefered to non-parametric ones, given the ease of use of the former for statistics, design specifications, or input into standard software. The spectrum S f,θ) is usually represented as the product of a frequency spectrum E f ) with a directional spreading function D f,θ). A general class of single peaked parametric models is the Γ-spectra, which, in dimensionless form are: Fx) =Np,q)x p e p q x q 1) with x = f f p and Np,q) a normalization constant that may be expressed using the Gamma function. The most common frequency spectrum models used for engineering are the Pierson- Moskowitz one [1] obtained for q = p 1 = 4, that is often found convenient under the form recommended by the ISSC [2]: E f )= H2 s T 4 z f 5 e f 4 Tz 4 2) and the one that was obtained by modifying the Pierson- Moskowitz spectrum to represent fetch-limited wind seas of the North Sea in the JONSWAP project [3]: E f )=α g2 2π) 5 f 5 e 5 f 4 fp ) 4 γexp 1 f fp 2σ 2 ) 2 with σ set to 0.07 for the ascending part of the spectrum f < f p ) and 0.09 for the descending part f f p ). Another specific form of the Γ-spectrum is that where q is set to 4, it is called the Wallops spectrum [4]: Fx) = βg2 f 5 p fp f ) p e p 4 where p = 4λ + 1. A popular form for the directional spreading function is the cos-2s distribution: fp f ) θ Dθ)=Kcos 2s θ0 2 ) 4 3) 4) where K is a normalization factor such that Dθ) has unit integral over [0,2π]. Those models and the other available ones were designed with wind waves in mind, and a number of limitations appear when trying to extend them to swell dominated sea states as are commonly found in West Africa regions. First, observed spectra are made of several swells and an occasional wind sea, so each of those components needs to be modeled separately. Then, the frequency peakedness of swell spectra is very high, their tails exhibit a cut-off since swells are devoid of short waves, and the parameters of conventional models do not allow separate fitting of peakedness and of high-frequency tail drop-off. Lastly and similarly, directional focusing is very high, and swell energy is unlikely to approach the location of interest from outside a narrow directional sector. Parametric models with two peaks aim at being used to represent likely conditions for applications. They usually combine a swell wave system with a wind sea one. The most well-known are the ones proposed by Ochi and Hubble [5], Guedes Soares 1984, 1992) [6, 7], and Torsethaugen 1993, 1996) [8, 9]. The Ochi-Hubble spectrum is the combination of two Wallops, yielding a six-parameter function. The Guedes Soares model is the combination of two Jonswap ones, with restrictions that γ is fixed to 2, thus leaving only 4 parameters to estimate. The choice of a single and same value of γ for the two peaks is however questionable, since swell spectra are supposed to be much more narrow-banded than wind sea ones. The Torsethaugen model is more flexible, it combines a Jonswap-enhanced Γ-spectrum with a standard Pierson-Moskowitz one, and the γ of the Jonswap-like spectrum is fitted differently depending on whether the spectrum is recognized to be swell or sea dominated. Yet, the parameters of the Γ-spectra reflect compromises between peakedness and tail drop-off, and those parameters vary very rapidly with peakedness, making their fit difficult for swell spectra. In the following, we investigate other models, and the conditions of their definition and assess them with the help of the data available in the WASP [10] project. A preliminary stage to parameterization of the spectra consists in their partitioning into distinct wave systems. The objective of the swell parameterisation work is to obtain the most appropriate spectral description of the swell components in the West African sea-states. This involved partitioning each wave spectrum into wind-sea and swell components, and then fitting each individual swell partition with the frequency spectrum shapes normal, lognormal, triangle, and JONSWAP-Glenn. In addition, for those data for which a directional analysis was possible, a Wrapped-normal directional distribution was fitted to the directional distribution. The partitioning is described, followed by the parameterisation of the spectra. Assessment of the significance of the differences between spectral shapes, either within the range of theoretical models in- 2 Copyright 2013 by ASME

3 vestigated, or with field observations, is also carried out. BIASES AND UNCERTAINTIES IN SPECTRAL ESTIMA- TION Frequency In the estimation of the spectral density from measurements, a bias is introduced on the spectral shape due to two reasons. First the effect of windowing and secondly the non stationarity of the sea-states coming to the time evolution of the frequency of the swells. psd Effects of processing on spectral shape Tp = 16 s triangular spectrum, m=6 sweeping fp df=.0012hz rectangular windowing Tukey windowing Averaging and window tapering To increase the number of degrees of freedom in the spectral estimator or in other words to decrease the variance of the estimator, the time series is split into pieces, with or without overlapping, and the spectral estimator is obtained by calculating the average of the individual spectral estimates. For each estimate, this is theoretically equivalent to multiplying the theoretical infinitely long stationary time series with a window rectangular or others). So the operation of windowing modifies the spectral density of the time process in a way which can be calculated as the convolution between the true spectrum and the Fourier transform of the window see for example Brillinger [11]. This effect is particularly crucial when the bandwidth of the spectral peaks is narrow and when the peaks are very close to each other. The choice of the optimal window is then a compromise between the bias on the peak bandwidth and the leaking effect of the neighbouring peaks. The properties of a great number of windows have been studied by Harris [12]. The bias introduced on the triangular spectral shape see further Eqn. 10) with different peakedness coefficients has been calculated for a rectangular and a Tukey 50-percent cosine taper) window. It is shown in the top panel of Fig. 1 for the triangle peakedness parameter m=6. The effect is important on the amplitude of the peaks and on the flanks, transforming the triangular shape to a smoothed one, and even more for higher peakedness. Non stationarity The duration of the time series used to estimate the spectral density must correspond to a stationary period of time of the process. Of course when measuring waves, this hypothesis is never verified since the mean frequency and H s are always changing. In the case of a wind sea, the sweeping of the mean frequency is not really a problem as it is relatively small compared to the frequency bandwidth of the spectrum. In the case of swell, it could be different and so the effect of a sweeping frequency must be analysed. A maximum of the interval of sweeping frequency during 3 hours, df=.0012hz, has been evaluated from the hindcast data base. It shows red line in Fig. 1) that in the cases of triangular shapes, the bias that is introduced is very small and that it could be neglected compared to the effect of the window tapering. psd frequency Hz) Effects of processing on spectral shape Tp = 16 s triangular spectrum, m=6 sweeping fp df=.0012hz rectangular windowing Tukey windowing frequency Hz) FIGURE 1. BIASES AT THE PEAK AND FLANK OF TRIANGU- LAR SPECTRUM - Tp=16s, m=6 Direction For point measurements, the Fourier expansion of the spreading function is only available up to second order. Only 1 2π 1 + n=2 n=1 r n cosnθ θ n ) is available to fit a Dθ) and the first σ 1 ) and second-order σ 2 ) directional spreads are given by σ 1 = 21 r 1 ) and σ 2 = 1 r2 )/2. Krogstad and Barstow [13] point out that for small values of σ 1, σ 2 = σ 1 and the various common forms of the spreading function: wrapped-normal, cos-2s, sech-2, are undistinguishable. For engineering purposes, it is thus of little importance which is used. In those applications, directions are commonly not distinguished within sectors of 10 or 15 deg., so when swell is more concentrated its exact spread is of reduced importance. ) 5) 3 Copyright 2013 by ASME

4 TABLE 1. TYPICAL SWELL DIRECTIONAL SPREADING σ d r 1 r access to are of too short duration, or too short stationary duration, to provide significant parameters from the fit on a single measured estimate that is subject to the biases and uncertainties described in the previous section. Conventionally, one chooses a parametric model, fits it to the available spectra, and studies the residual. The main problem is: How does one build the parametric model? In order to get rid of this problem, the assumption is made of a unique underlying spectral model, valid for all sea states, but undefined at this stage, of the form: For narrow spreads, a problem may stem out at the estimation of σ 1 from r 1. S f )= m ) 0 f F f c f c 7) r 1 = 1 2 σ 2 1 6) where f c is a frequency characteristic of the sea state, for instance f 02 or modal frequency f p, and F is normalized in such a way that: so for small values of σ 1, r 1 needs to be estimated with an extreme accuracy to provide a meaningful estimate of σ 1. Many measurement systems use different sensors for the heave measurement and for the horizontal or slope quantities, and small calibration biases then lead to huge errors in σ 1 that results from a correlation of heave with the other motions. Table 1 illustrates the relation between σ, r 1 and r 2. σ 2 is approximately equal to σ 1 for narrow spreads and should be somewhat more robust. One can also use some other method, for instance rotary component spectral analysis [14], to assess the swell directional spread in case it is not sufficient to state that it is small. SPECTRAL MODELS SUITED TO SWELL Swell peaks are much narrower than the wind sea peaks that are observed in the North Sea because of frequency dispersion on long propagation paths. Modelling such narrow peaks by a Jonswap implies that γ take much higher values than those in the commonly used range of 1 to 7. Very high values of γ for swell are somewhat in contradiction with the construction of the Jonswap shape, where that parameter is used to reflect the nonsaturation of a fetch-limited wind sea, and they lead to a risk of unnoticed numerical accuracies in the practical computations. Following the considerations in Prevosto et al. [15] and in order to keep the complexity of the fitting and reconstruction processes within reasonable limits, simpler formulations than the Jonswap one were thus also sought for, via the method described in Olagnon [16] and briefly recalled hereafter. 0 Fu)du = 1 8) One may note at this stage that this assumes that bandwidth is proportional to frequency, which was verified as a general feature of the WASP database. The validation of the assumption that no other parameter is necessary to describe the spectra is only performed in retrospect at the end of the study. In order to simplify the analysis and to give more chances of validity to the assumption, only singlepeaked spectra should be used. In cases such as West Africa, where most spectra exhibit several peaks, only the largest peak in the range of swell periods will be considered, and the characteristic frequency is set to the corresponding modal frequency f p. For all measured spectra, the frequency scale is normalized with that peak frequency f p and the spectra are then normalized by their value at f p. From the above uniqueness assumption and spectral estimation theory, it stems out that for each normalised frequency f = f f p, the observed spectral density is a random variable with a χ 2 distribution with ν degrees of freedom of average value F f ). Since we use the same estimation parameters for each spectral estimate, and as the χ 2 distribution is fully defined by a single parameter, each characteristic of the distribution of the spectral densities mean, mode, percentiles) should be in constant ratios with respect to each other and to the actual spectral density: C F f ), as long as the above assumption that all have the same normalized shape is verifed. Methodology It is not easy to validate or define a spectral shape, even restricted to a given location or area. Measurements that one has Triangle Shape Figure 2 shows the resulting shapes for the swell T p > 8s) peaks at the Ekoundou location see Forristall 2013) [10] for the 4 Copyright 2013 by ASME

5 energy norm. dens average 10% 30% median 70% 90% reference norm. freq FIGURE 3. SCHEMATIC EXPECTED SWELL SPECTRAL SHAPE and where the parameter m is related to the Goda peakedness factor by m = 3Q p+2 4, with the usual definition of Q p : Q p = 2 m fs 2 f )df 9) norm. freq. FIGURE 2. SPECTRAL SHAPE INFERENCE description of datasets) blue: median, 25%, 75% percentiles, dotted red: 10% and 90% percentiles, black: average, green: proposed shape. Given that the influence of other neighbouring peaks cannot be eliminated by the method, it was decided to use a triangular shape displayed in green) to model the peak rather than some function that would follow more closely the black empirical one outside of the interval [0.9 f p, 1.1 f p ]. A schematic construction of the spectral shape is provided in the WASP report [17]. Using a simple model of a fixed remote storm, growing then decaying, and emitting waves towards the location of measurement, one may expect a roughly triangular spectral shape for swell at a given time of measurement as shown on Fig. 3. Also, propagation theory implies that the spectral shape must exhibit low and high cut-off frequencies, and even though smooth models may have steep decay, they do not fully correspond to that physical constraint. Similar triangles were found at the other WASP locations, but of varying widths, so in practice the swell spectra can be chosen within the family of triangles defined, in non-dimensional coordinates, by the 3 x,y)-points m 1 m m, 0), 1, 0), and m 1,0) S f )= 2mm 1) 2m 1 S f )= 2mm 1) 2m 1 Hs 2 16 f p m f Hs 2 16 f p ) f p m 1) m m 1) f f p ) m 1 m S f ) =0 elsewhere f p < f < f p f p < f < m 10) m 1 f p The same triangle was tried for hindcast spectra and gives a good agreement, which seems to confirm that the shape is determined by a propagation transfer function. One may note that the empirical shapes exhibit a thick tail in the high frequency range, yet the effect of the spectral window acounts for a large part of it. JONSWAP-Glenn shape A variation of the JONSWAP function called the JONSWAP-Glenn spectrum is given in terms of three parameters significant wave height, H s, peak frequency, f p, and the peak enhancement factor, γ,by ) G f )=C H2 S f 5 e 5 f 4 fp 16 f p f p ] ) 4 [ γ exp f fp)2 2σ 2 f 2 p where σ = 0.07 for ff p and σ = 0.09 for f > f p and 11) C = γ ) 1 12) γ) 5 Copyright 2013 by ASME

6 FIGURE 4. TRIANGLE AND JONSWAP SPECTRAL SHAPES FIGURE 5. NORMAL TOP) AND LOG-NORMAL BOTTOM) SPECTRAL SHAPES The function is plotted in Fig. 4 for H s =2m,γ = 3.3, and T p = 6s, 10s, 16s, together with a number of JONSWAP spectra specified at different distances along a fetch for a wind speed of 10 m/s. Figure 4 shows the differences between the standard JONSWAP and the JONSWAP-Glenn spectra. In particular, the JONSWAP spectrum has a one-to-one relationship between the peak frequency and the variance or H s, which is not the case for the JONSWAP-Glenn spectra. Gaussian shape The Gaussian spectral form is also expressed as a function of three parameters, significant wave height, H s, peak frequency, f p, and a standard deviation, σ, as follows Lognormal shape A variation of the Gaussian spectral form is the lognormal. It allows for some asymmetry in the shape. It is given by G f )= f pm 0 e ln f ) μ) 2 2s 2 d 14) fs d 2π where the lognormal parameters μ and s d can be expressed in terms of f p and to the standard deviation σ of the normal distribution as follows ) s d = σ 2 ln fp ) G f )= m 0 σ f fp) 2 2π e 2σ 2 13) Examples of this spectral form are given in Fig. 5. μ = ln f p )+s 2 d 16) s d is also related to the Goda peakedness parameter Q p defined at Eqn. 9 by Q p = 1 s d π. Examples of this spectral form are given in Fig Copyright 2013 by ASME

7 normalized density 1.00 triang. median triang. ref gauss median gauss ref log-norm median log-norm ref jonswap median jonswap ref FIGURE 7. PARTITIONED DIRECTIONAL SPECTRUM Comparison of various shapes FIGURE 6. norm. freq. COMPARISON OF SPECTRAL SHAPES Spectral Fitting Figure 6 shows application of the model choices described above to simulated data from the four shapes corresponding to the same frequency bandwidth. It shows for each the reference theoretical spectral shape from which the data were simulated and the median of the computed spectra. The shapes are difficult to discriminate, and the windowing bias mentioned earlier furthermore reduces the differences. One should especially be cautious that smooth shapes may fit mostly the smoothing effect of the window, and that the high-frequency tail of the JONSWAP- Glenn shape may fit mostly energy that does not actually belong to the considered swell but is merged with it by the partitioning software. For engineering purposes, it is thus of little importance which one is prefered. This is convenient because some engineering analysis tools do not have much flexibility in choosing spectral shape. Directional distribution A common description of the directional distribution for a unimodal swell component is given by the so-called Wrapped- Normal distribution, defined as follows. [ k=+ 1 Dθ)= σ wn 2π exp 1 ) ] θ θ0 2kπ 2 2 σ wn k= 17) where θ 0 is the mean wave direction, σ wn is the standard deviation or angular width and gives a measure of the spreading. The summation over k ensures the distribution is continuous through 360. In practice it is not needed for the narrow angular widths associated to swell. Also, as mentioned earlier, the various common shapes for the directional distribution are undistinguishable for those narrow widths. Still, it is physically sensible to want a distribution that is restricted to a finite angular sector: though the initial sector under which the generating storm can be seen may be modified in the course of propagation, it should remain limited. A distribution such as a raised cosine should thus also be considered: for which )) Dθ) = 2π 1 p 1 + cos θ θ0 ) p Dθ) =0 θ θ 0 < p π elsewhere 18) ) π σ 2 2 θ = p ) 2 p 2 19) PARTITIONING AND PARAMETERIZATION OF SPEC- TRA The objective of the swell parameterisation work is to obtain the most appropriate spectral description of the swell components in the West African sea-states. This involved partitioning each wave spectrum into wind-sea and swell components, and then fitting each individual swell partition with the frequency spectrum shapes Normal, lognormal, triangle, and JONSWAP- Glenn. In addition, for those data for which a directional analysis was possible, a Wrapped-normal directional distribution was fitted to the directional distribution. The partitioning is described in the next section, followed by the parameterisation of the spectra. 7 Copyright 2013 by ASME

8 Partitioning The partitioning of the spectra into wind-sea and swell partitions was performed using the program APL Waves, developed by the Applied Physics Department of Johns Hopkins University Hanson and Phillips 2001) [18]). The input to the program is a data file of wave frequency-direction spectra. The program then partitions the 3D spectrum into separate peaks as in the example of Fig. 7. It may be noted that other partitioning methods exist, see for instance [19], yet since swell peaks are well-identified their results would be very similar. If wind data are not available to determine the wind sea component, the wind speed and direction is estimated from the spectrum by means of an iterative process based on the method of Wang and Hwang 2001) [20]. Frequency-direction spectra are required for input to the partitioning analysis. These are directly available in the case of the hindcast dataset, but they must be derived in the case of the measured data. The standard approach for doing this is to follow the technique proposed by Longuet- Higgins et al. 1963) [21], in which the directional distribution is expanded as Fourier series. Unfortunately, most directional wave measurement systems, including heave-pitch-roll buoys, provide a limited number of Fourier Coefficients, and the determination of the directional distribution is not accurate. As a result there has been a number of possible ways proposed to compute the directional distribution to try and optimize the estimate, and the choice of a particular method is somewhat arbitrary. Perhaps the most popular methods are the Maximum Likelihood Method MLM) and the Maximum Entropy Method MEM). These two were used to derive the frequency-direction spectra for some test data for comparison, but the MEM was found to provide swell sources estimates with better directional resolution than the corresponding MLM estimates. In addition, MLM-based directional spread is always biased high. Accordingly, the MEM estimates were used for the partitioning analyses. Swell groups are constructed from swell partitions that are likely to originate from the same region and source winds. Swell events are identified from these groups and the generation time and location of the event is calculated. Parameterization Four spectral functions, as described previously, were evaluated. This involved fitting spectral functions to the partitioned spectra, and then comparing the fits with the original spectrum. Following partitioning of the spectra, the frequencydirection spectrum is partitioned into separate regions each partition corresponding to a swell or a wind-sea component, with of course a maximum of one wind-sea partition per spectrum. The frequency spectrum of each partition is calculated, by integrating over direction, and fit with each of the four spectral functions Gaussian, lognormal, triangle, and JONSWAP-Glenn, as described earlier. The fit was performed by the method of least FIGURE 8. RMS SPECTRUM AND SCATTER INDEX FOR THE BONGA QSCAT POINT squares. The fit spectral functions are summed over the partitions, resulting in four model estimates of the frequency spectrum of the total sea-state. In all cases the JONSWAP-Glenn spectrum is used to describe the wind-sea; the four models then consist of a JONSWAP-Glenn wind-sea added to a Gaussian, lognormal, triangle, or JONSWAP-Glenn fit to all the swell components. The model spectra are then compared against the original spectra to deduce the best function for describing the swell. The analysis was performed for the WANE OPR and QSCAT) hindcast data sets and the measured spectral data. The frequency spectrum comparisons involve a comparison of the spectral amplitudes at each frequency. For example, Fig. 8 gives scatter index plots of the model spectral amplitudes against the measured spectral amplitudes for the QSCAT Bonga) hindcast grid point. They confirm that spectral shapes are difficult to discriminate in practice, especially given the aforementioned biases and leakages that might ironically be the most influential factors in the choice of a model. Nevertheless, the lognormal model was selected for the study, since it appeared in general superior, especially on the more reliable hindcast data. PRACTICAL MODELING OF SWELL SPECTRA Frequency bandwidth The objective is to provide a parametric description of the swell spectra off West Africa, based on the assessment of the goodness-of-fits of the model spectral comparisons with the spectra, both in terms of the direct spectral comparisons and also in terms of the response amplitude comparisons. The spectral forms have the general form G f )=G f ;H s,t p, p) where p is a parameter relating to the spectral width. Thus, the spectral de- 8 Copyright 2013 by ASME

9 TABLE 2. FREQUENCY BANDWIDTH sd = a Name Latitude Longitude a b FIGURE 9. FIT OF SPECTRAL WIDTH sd = a scription is a function of three parameters, two H s and T p relating to the sea-state, and the third possibly related to location. It is the specification of the third parameter that is of interest here. Since the choice was made of the lognormal spectral description, a parameterisation for p for this spectrum has been established. A good fit of the lognormal standard deviation parameter to the data is given by the function sd = a The curves resulting from the best fit of this function to all the QSCAT hindcast and measured data are given in Fig. 9 for comparison. The figure shows that with the exception of the Chevron, 8m water depth curve, which is likely showing shallow water effects, the lines corresponding to the measured data are consistent with those from the QSCAT data sets. In addition, the plot shows a general trend for the standard deviation of the lognormal to increase with increasing latitude, although this effect does not become significant until latitudes south of 10 degrees South - the Central Angola grid point. The parameters of the QSCAT best-fit curves are given in Tab. 2. Kudu Namibia Central Angola Exxon Nemba Gabon Marathon Côte d Ivoire Ekoundou Bonga Directional spreading The ten QSCAT data sets show that the function σ θ = a provides a good description of the variation of the standard deviation as a function of peak period. The best-fit lines for all ten QSCAT data sets are given in Fig. 10. As was observed with the frequency spectrum, increasing directional spread with increasing southern latitude is apparent from this figure. The values of the parameters, a and b are given in Tab. 3. CONCLUSIONS WASP analyzed a large quantity of existing wave data with the goal of producing an improved description of the characteristics of swell off West Africa. The data base included both measured and hindcast wave data from several locations from Côte FIGURE 10. FIT OF DIRECTIONAL SPREAD σ θ = a d Ivoire to Namibia. The main focus of the project was to test various descriptions of the shape of the spectrum of swell, but many other features of the swell environment were also studied. Several spectral forms triangle, normal, lognormal, Glenn- Jonswap) were fitted to the hindcast and measured spectra. Considerations of sampling variability show that it is difficult to distinguish between these forms for real data. The choice of the best spectral shape for swell depends on the application, and has little importance in many of them. The JONSWAP allows a larger high frequency tail, which is sometimes seen in swell 9 Copyright 2013 by ASME

10 TABLE 3. DIRECTIONAL SPREADING σ θ = a Name Latitude Longitude a b Kudu Namibia Central Angola Exxon Nemba Gabon Marathon Côte d Ivoire Ekoundou Bonga spectra, but in these cases the swell component may be contaminated with some high frequency energy from another source and processing effects. Simple theoretical arguments show that the standard deviation of the spectral density function should vary inversely with the peak period. In addition, the standard deviation should decrease as the distance from the source of the swell increases. Both of these hypotheses are borne out by the detailed calculations and observations. Table 2 gives the parameters which allow the calculation of the standard deviation of the spectral density function for each of the WANE grid points. We recommend that these functions be used for the calculation of the shape of the swell spectrum for any site off West Africa at a similar latitude to those grid points. The directional spreading of the swell components can be described by its directional standard deviation σ θ, and for the low values of σ θ that are observed with swell, about 10, the various possible shapes cos-2s, wrapped-normal, raised cosine, etc.) of the directional spreading are undistinguishable. Calculations show that the standard deviation of the spreading is inversely proportional to the peak period, and that the directional spreading becomes narrower as the distance from the source of the swell increases. Parameters of the spreading function for each of the WANE grid points are given in Tab.3. The partitioning of the spectra often led to more than one swell partition in addition to a wind sea partition. One way of dealing with this complexity is to consider the response of simple systems to the spectra, see Ewans et al. 2013) [22]. The joint statistics of systems should also be investigated. Especially, there is evidence that systems should not be independent, since the largest observed global H s values [23]) or responses [22]) are not obtained from the combination of several systems but from single energetic swells. To this aim, the swell systems should be analysed over their history of successive observations. ACKNOWLEDGMENT We thank the participants of the WASP Joint Industry Project for their financial support and technical input. The participating companies were ChevonTexaco, ExxonMobil, Ifremer, Marathon Oil Co., Shell International Exploration and Production, Statoil ASA, and TotalFinaElf. REFERENCES [1] Pierson, W., and Moskowitz, L., A proposed spectral form for fully developed wind seas based on the similarity of S.A. Kitaigorodskii. J. Geophys. Res., 6924), pp [2] ISSC, Report of the environmental conditions committee. In Proceedings of the 2nd International Ship and Offshore Structures Congress, H. Jaeger and J. D. Does, eds., Vol. 1, Delft, The Netherlands. [3] Hasselmann, K., and et al., Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project. Deutsche Hydrographische Zeitschrift, 812). [4] Huang, N., Long, S., Tung, C.-C., Yuen, Y., and Bliven, L., A unified two-parameter wave spectral model for a general sea state. J. Fluid Mechanics, 112, pp [5] Ochi, M., and Hubble, E., Six-parameter wave spectra. In Proceedings of 19th International Conference on Coastal Engineering, pp [6] Guedes Soares, C., Representation of double peak spectra. Ocean Engineering, 11, pp [7] Guedes Soares, C., Spectral modeling of sea states with multiple wave systems. Journal of Offshore Mechanics and Arctics Engineering, 114. [8] Torsethaugen, K., A two peak wave spectrum model. In Proceedings of 12th International Conference on Offshore Mechanics and Arctic Engineering, Vol. 2, pp [9] Torsethaugen, K., Model for a doubly-peaked wave spectrum. Tech. rep., SINTEF. Ref. STF22 A [10] Forristall, G., Ewans, K., Olagnon, M., and Prevosto, M., The West Africa Swell Project WASP). In Proceedings of the 32nd Int. Ocean, Offshore and Arctic Eng. Conf., ASME. Paper number [11] Brillinger, D., Time Series: Data Analysis & Theory. Soc. for Industrial & Applied Math, p [12] Harris, F., On the use of windows for harmonic 10 Copyright 2013 by ASME

11 analysis with the discrete fourier transform. Proc. IEEE, 661), pp [13] Krogstad, H., and Barstow, S., Directional distributions in ocean wave spectra. In Proceedings of the 9th Int. Offshore and Polar Eng. Conf., Vol. III, ISOPE, pp [14] Gonella, J., A rotary component method for analysing meteorological and oceanographic vector time series. Deep Sea Research, 1912), pp [15] Prevosto, M., Ewans, K., Forristall, G., and Olagnon, M., Swell genesis, modelling and measurements in West Africa. In Proceedings of the 32nd Int. Ocean, Offshore and Arctic Eng. Conf., ASME. Paper number [16] Olagnon, M., Representativity of some standard spectral models for waves. In Proceedings of the 11th Int. Offshore and Polar Eng. Conf., Vol. III, ISOPE, pp [17] Olagnon, M., Prevosto, M., Van Iseghem, S., Ewans, K., and Forristall, G., WASP west africa swell project final report and appendices. Tech. rep., Ifremer - Shell IEP. See URL [18] Hanson, J., and Phillips, O., Automated analysis of ocean surface directional wave spectra. J. Atmos. Oceanic Technol., 18, pp [19] Ewans, K., Bitner-Gregersen, E., and Guedes Soares, C., Estimation of wind-sea and swell components in a bi-modal sea state. Journal of Offshore Mechanics and Arctics Engineering, 128, pp [20] Hwang, and Wang, An operational method for separating wind sea and swell from ocean wave spectra. J. Atmos. Oceanic Technol., 18, December, pp [21] Longuet-Higgins, M., Cartwright, D., and Smith, N., Observations of the directional spectrum of sea waves using the motions of a floating buoy. In Ocean Wave Spectra, Prentice-Hall, pp [22] Ewans, K., Forristall, G., Olagnon, M., and Prevosto, M., Response sensitivity to swell spectra off West Africa. In Proceedings of the 32nd Int. Ocean, Offshore and Arctic Eng. Conf., ASME. Paper number [23] Guédé, Z., Olagnon, M., Pineau, H., François, M., and Quiniou, V., Fatigue analysis of an FPSO under operational sea states with multimodal spectra. In Proceedings of the 28th Int. Ocean, Offshore and Arctic Eng. Conf., ASME. Paper number Copyright 2013 by ASME