Statistical Analysis of Non-stationary Waves off the Savannah Coast, Georgia, USA

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1 1 Statistical Analysis of Non-stationary Waves off the Savannah Coast, Xiufeng Yang School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA, Received: October 16, 2014; Accepted: January 12, 2015 Abstract The statistical characteristics of short-term wave records collected from a wave buoy deployed off the coast of Savannah, are analyzed. The statistical distribution of observed surface elevation is derived and compared with theoretical estimations. The minimum length of time series for stationary conditions is specified using a synthetic approach. A new technique that considers the variability of different dimensionless parameters has been introduced to improve the theoretical distributions of wave crests, troughs and heights. The observed distributions of wave crests and troughs are computed and compared with the Rayleigh distribution as well as the Tayfun distribution. The characteristic wave heights of measured data are compared with those estimated from the Boccotti distribution over different time intervals. The empirical estimation of expected maximum wave heights as a function of the number of waves is examined using a statistical approach, and compared with results from both Boccotti and Tayfun models using the newly introduced technique. Conclusions concerning the efficacy and applicability of theoretical distributions to real ocean wave conditions are summarized. Key words: US east coast; statistical analysis; non-stationary waves; ocean surface waves; 1. INTRODUCTION In a perfect Gaussian random sea, the surface elevation is composed of many independent wave components. Given a power spectrum, each wave component has a random phase and deterministic amplitude that can be calculated from the power spectrum. Under perfect circumstances, the probability distribution of the wave surface elevation will be exactly Gaussian in the following form: (1) Where η is the surface elevation, η is the mean surface elevation and σ 2 η is the variance. However, non-gaussian behavior of sea surface elevation, although very weak, has been observed in various field measurements and also described in various literatures (e.g., [Goda, 2000]). Real observed waves have higher and sharper crests and shallower and flatter troughs than what the Gaussian law predicts. Non-linear wave-wave interactions have already been shown theoretically by Volume 6 Number

2 2 Statistical Analysis of Non-stationary Waves off the Savannah Coast, [Longuet-Higgins, 1963] to be proportional to the slope of the wave. A number of studies have been carried out to describe the non-gaussian statistical distribution of the wave surface elevation and led to their better understanding. Some of these studies are performed by [Longuet-Higgins, 1963] and [Huang and Long, 1980]. The theory of second-order interactions of waves in a random directional sea has been available for some time and been discussed by other researchers such as [Forristall, 2000]. In a Gaussian sea, the crest height distribution follows the Rayleigh distribution, and according to [Boccotti, 2000] the Probability Density Function (PDF) is written as (2) where σ H is the standard deviation and is the crest height. Due to symmetry of the Gaussian sea state, both crest and trough distributions should follow the Rayleigh law for narrow-band spectra. However, in reality, ocean waves are nonlinear and the PDF tends to deviate from the Gaussian form. Tweaked shape of ocean waves due to nonlinear effect has a profound impact in ocean engineering, and many previous studies have examined such impacts. For example, its influence on the design of offshore structures has been discussed by [Jahns and Wheeler, 1973]. [Longuet-Higgins, 1963] found that the first approximation of the theoretical crest or trough distribution should correspond to the ordinary Gaussian distribution, but higher approximations are described by the Gaussian law multiplied by certain polynomials, which correspond in fact to successive terms in a Gram-Charlier series. [Tayfun, 1994; 2002] corrected Rayleigh distribution by adding nonlinearity into his expressions of wave crest/trough distribution, which leads to higher and more pointed crests, and shallower and more rounded troughs. In this paper, we study the statistical distribution of wave surface elevation using wave data collected by a directional wave buoy deployed off the coast of Savannah Georgia in USA, and then compare the results with different theoretical distributions. We also study the distribution of wave crests and wave troughs from data and compare them with Rayleigh as well as Tayfun distribution. Further, a new technique that takes into consideration the variability of dimensionless parameters has been presented to improve the accuracy of various theoretical distributions. The expected maximum wave heights as a function of the number of wave heights are computed from observed wave data and compared with theoretical results based on Gumbel distribution using the new technique. We also investigate the relationship between the length of the time series and the stationary conditions of the sea state using synthetic approaches. The work in this paper examines the statistical characteristics of the ocean waves near the coastal area of, and results from this study help evaluate different wave statistics theories. 2. SHORT TERM STATISTICS A Gaussian sea state is given by the linear superposition of a large number of elementary waves with distinct frequencies, given amplitudes, and random phases which are uniform in [0, 2π] and stochastically independent. The mathematical form of Gaussian wave elevation is represented as (3) Where amplitude, S(w) is the power spectrum as a function of the frequency w, and ε j is random phase in [0, 2π]. Ocean waves of Gaussian seas have many statistical properties, which will be detailed in following sections. International Journal of Ocean and Climate Systems

3 Xiufeng Yang Distribution of surface elevation The wave elevation of a Gaussian sea conforms to equation (1). However the effect of non-linearity is excluded in equation (1). [Tayfun, 1994] used the second-order random wave theory in finite water depth and the characteristic function expansion method to derive a new expansion for the statistical distribution of the wave surface elevation. According to [Tayfun, 1994], the probability density of the dimensionless wave surface elevation in deep water is written as: (4) where dimensionless wave surface elevation is, and the skewness is. The above formula is exactly the same as the Gram-Charlier series obtained by [Longuet-Higgins, 1963] truncated at the third-order cumulants and depends only on the parameter λ Distribution of crests and troughs Consider a time interval τ, the expected number of h-upcrossings can be expressed as a function of joint probability density [Adler, 1981]: (5) (6) in which is the covariance matrix: (7) (8) Since the process is Gaussian, we have (9a) (9b) (9c) where m 0 is the variance, and m 2 is the 2 nd order spectral moment. Thus equation (6) can be simplified to (10) Plug equation (9) into (10) and we have Volume 6 Number

4 4 Statistical Analysis of Non-stationary Waves off the Savannah Coast, (11) There is a one-to-one correspondence between the number of zero-upcrossings and the number of waves. Thus the expected number of waves is equal to EX(0). For very large h, one-to-one correspondence exists between the number of crests and the number of h-upcrossings. Therefore the exceedance probability of wave crests is represented as (12) Above equation conforms to Rayleigh law. The probability density of wave crests can be obtained by differentiating equation (12), which yields equation (2). Due to symmetry, the wave troughs conform to the same distribution. However wave crests and troughs are observed to be asymmetric due to nonlinear interactions. The nonlinear surface elevation from the mean-zero level is described by [Tayfun, 1994] as (13) where A represents the random amplitude or envelope function, and φ (t) is the random phase. The joint density of scaled envelope is presented by [Tayfun, 1994] as: (14) (15) Where, and λ 3 is the skewness coefficient. The conditional density of ζ, given that η > 0 or η < 0, will be defined as p + (η) and p (η) respectively. The theoretical expressions are given by [Tayfun, 2002] as (16) where p ζ is Rayleigh distribution. If as in the linear theory. The Exceedance distribution corresponding to p + or p follows by integration equation (16) as (17) In which E(ζ) follows Rayleigh law. International Journal of Ocean and Climate Systems

5 Xiufeng Yang 5 3. DATA ANALYSIS A general definition of stationary sea state is that the averages of the wave elevations to the 1 st and the 2 nd order over different realizations at any time are approximately the same. For a real sea state, the stationary characteristic only holds for a short period of time. Over time, the statistical properties of waves will change. Even if wave data are normalized to have zero mean and unit variance, the statistical characteristics of waves in different period of time are still different due to non-stationarity. To account for such variability, a new technique is introduced in the following in computing wave statistics. Let us consider a non-stationary time series of length t with standard deviation σ variable in time. Since we normalize the time series within each chunk t first by executing ξ = (η η ) / σ, we have already accounted for the variability of σ. Therefore we only need to consider the variability of ψ * within each chunk. ψ * is the first minimum of the normalized non-dimensional covariance function. Let and be the mean and variance of the series of ψ *. Then, the probability for the wave height is given by (18) where p ψ* (x) is the PDF of ψ * estimated from time series and (19) which is the theoretical exceedance probability of wave height distribution. Assume that p ψ* (x) is concentrated around its mean and is the variance of ψ *. Then we can expand in Taylor series the term F (ξ, x) as (20) Then equation 18 simplifies to (21) 3.1 Estimation of the wave height distribution There are different probability density functions for wave heights. One good example is Boccotti distribution [Boccotti, 1989; 2000] which assumes the form of (22) We consider the variability of ψ * so that (23) Volume 6 Number

6 6 Statistical Analysis of Non-stationary Waves off the Savannah Coast, Plug equation 22 into equation 23 and after simplification, we get (24) [Tayfun and Fedele, 2007] also present a formula for the probability of non-linear wave height, which reads the following: (25) Where γ 40 = 12( 1 2γ 0 )λ 3 2, and γ 0 is a non-negative dimensionless parameter defined by [Tayfun, 1986]. If we assume γ 0 is zero (which is reasonable because γ 0 is usually very small) and consider the variability of λ 3, the probability reads (26) 3.2 Estimation of wave crest distribution Calculations can be done with the Tayfun distribution [Tayfun, 1990] of wave crest which also accounts for the variability of the skewness λ 3 or steepness. In this case we assume that (27) where p µ (x) is the PDF of µ estimated from time series and (28) Where. We then have (29) Where (30) (31) International Journal of Ocean and Climate Systems

7 Xiufeng Yang 7 (32) Calculation yields (33) 3.3 Estimation of expected maximum wave heights In ocean and offshore engineering, engineers sometimes are more interested in the maximum wave heights that can happen in the ocean because design standards are always relevant to the worst possible events. Various theoretical forms of distributions of wave heights have been proposed and reviewed. For example, [Tayfun and Fedele, 2007] reviewed different analytical wave height models, and derived a formula for calculating expected maximum wave heights based on Gumbel distribution. According to [Tayfun and Fedele, 2007], we consider the same time series of N independent wave heights (h 1 ) j (j=1,2,,n) described by the exceedance distribution functions (EDF) in the following equation with f=1: (34) We let ξ max = max {(h 1 ) j, j = 1,2,..., N}. For large N, the EDF of ξ max is given by Gumbel distribution (35) For Boccotti distribution, (36) After plugging Boccotti distribution into equation (35), we get We assume there is value of h N for which (37) (38) so that we have (39) Volume 6 Number

8 8 Statistical Analysis of Non-stationary Waves off the Savannah Coast, Let z = z N + u, where u is very small so that z is in the neighborhood of z N, then (40) The mean of ξ max based on Boccotti distribution then can be written as (41) where γ e is the Euler constant which is equal to More precisely, above equation should be written in the form of conditional probability as (42) Similarly the expected maximum wave height can be written as =F N, (43) We can also apply Tayfun distribution ([Tayfun, 1990]) instead of Boccotti distribution ([Boccotti, 1989]) into equation (35). Tayfun distribution for wave height reads (44) In which, and is the Hilbert transform of ψ *.To further simplify the equation by approximation as [Tayfun and Fedele, 2007] did for Tayfun 2 model (refer to Table 1 in [Tayfun and International Journal of Ocean and Climate Systems

9 Xiufeng Yang 9 Fedele, 2007]), equation (44) can be written as (45) Therefore (46) Following the same derivation from equation (38) to equation (43), we get (47) Where are the mean and variance of respectively. 4. COMPARISONS WITH MEASUREMENTS 4.1 The measurements Since July 2004, a Triaxys wave buoy has been deployed 10km offshore near Savannah,. The mean water depth of the site is 13m and the mean tidal range is 2.1m. The site features diurnal tides and the mean depth-averaged flow velocity for the period under consideration was about 20cm/s. Instruments were programmed to collect nearly twenty minutes of wave burst data every hour. The original record was arranged in hourly files, and the whole data set includes 743 files, each file containing 1382 data points sampled at the time interval of 0.78 seconds. In a quasi-gaussian sea, the significant wave height can be approximated as, with σ 2 being the variance. Fig. 1 shows the approximated significant wave heights throughout the month of Figure 1: Significant wave heights as a function of time with A, B and C indicating the positions of three storms during the observation period. Volume 6 Number

10 10 Statistical Analysis of Non-stationary Waves off the Savannah Coast, observation. From Fig. 1, it is easy to observe three storms happened in the deployment site (marked as A, B and C). One happened on Aug 7, The next one, and also the largest one, happened on Aug 25, After that, on Aug 29, another relatively smaller storm happened. Fig. 2 shows one-hour-time wave power spectra during the three storms identified above. It is obvious that those three wave spectra have very similar profiles with two peak frequencies. The higher peak frequency is roughly 0.18 Hz, and the lower peak frequency is about 0.04 Hz. It is also observed that in the range of high frequency, the spectral confirm very well to S=f -4. Figure 2: One-hour wave power spectra for each of the three storms identified in Fig Minimal length of time series for stationary conditions The real sea state evolves over time, and there is no such sea state that can be strictly considered as stationary. However, for practical purposes, if the evolving time is short enough that the change of sea state is very small and negligible, we can still consider the sea state as stationary within that time period. Therefore a long non-stationary sea state can be divided into short stationary time chunks. Within each chunk, the sea state is considered as stationary and statistical theories for stationary sea state can apply. To investigate the approximate length of time for stationary conditions, the following experiment is carried out. The whole time series is equally divided into shorter time series. The length of each chunk t ranges from 5 minutes to 20 hours. For each case, the time series is normalized by within each time chunk. The ratio of between normalized wave heights and Rayleigh height is calculated and plotted against the exceedance probability, as is shown in Fig. 3. From Fig. 3, we can get an overview of the stationary status of the sea state within each chunk of time series. It is clear that the plots corresponding to equal to from 5 minutes to up to 30 minutes are very close to each other without much deviation. The plots corresponding to time intervals longer than 60 minutes, however, have shown considerable deviation from previous curves. When becomes as large as 10 hours, the deviation become phenomenal. Thus we can briefly estimate the minimal length of time series for stationary conditions to be around 30 minutes. However, we should keep mind that the deviation of curves away from stationary conditions change gradually and there is no such strict number to define stationary conditions. The cut-off minimum length of 30 minutes is not exact. International Journal of Ocean and Climate Systems

11 Xiufeng Yang 11 Figure 3: Ratio of H/HR as a function of exceedance probability, where HR is the Rayleigh wave height. Therefore, for safety purpose, we d better choose a time interval smaller than 30 minutes in order to better preserve the piece-wise stationary characteristics within each time chunk. As we already know, the data files are arranged with each file containing 1382 data points with time interval of 0.78 seconds. Therefore each data file contains a wave data set of about 18 minutes, which can be conveniently used as the length of time chunks. In the following sections, this length will be employed to normalize the time series. 4.3 Distribution of dimensionless parameters Application of Tayfun distribution in calculating wave surface elevation and wave crests requires computing the parameter of wave skewness, as is shown in equation (4) and equation (17). There are practically two means to compute λ 3 directly from wave data. According to definition, λ 3 can be computed as (48) in which ζ is normalized wave elevation. Another means of computing λ 3 is based on the following relation (49) in which.. Through either way, the problem is that the sea state is not stationary. So it is unreasonable to calculate the skewness by averaging the whole time series. One reasonable solution is to calculate skewness according to equation (24) or (25) for each chunk of time series and the length of each chunk is Volume 6 Number

12 12 Statistical Analysis of Non-stationary Waves off the Savannah Coast, arranged to be 18 minutes, as is described in previous section such that the sea state within each chunk of time period is stationary. Then we average the results from all chunks to get an averaged skewness, which will be more representative of the sea state characteristics. Fig. 5 plotted the PDF of λ 3 using two different approaches identified above. It is clearly seen that two different methods yield very different results. The averaged λ 3 from two methods are and 0.24 respectively. Comparison between two PDFs in Fig. 4 shows that the estimates of λ 3 directly from time series are statistically unstable, since the time series are shorter due to the constraint of stationarity. To improve the estimates we need longer chunk of time series but then stationarity is violated. Therefore, λ 3 estimated based on the upcrossing period Tm is more accurate and will be used in all following calculations. Figure 4: Probability density functions of λ 3 calculated using two different methods. Fig. 5 plots the normalized covariance functions at the three storms peaks indentified in Fig. 1 using one-hour wave data. Fig. 6 shows the PDF of ψ * which is the first minimum of normalized φ(t). From Fig. 6 we can see the bimodality due to swell and wind interactions. International Journal of Ocean and Climate Systems

13 Xiufeng Yang 13 Figure 5: Normalized covariance functions at the three storm peaks identified in Fig. 1. Figure 6: PDF of that shows the bimodality due to swell and wave interactions. Volume 6 Number

14 14 Statistical Analysis of Non-stationary Waves off the Savannah Coast, 4.4 Distribution of wave crests and troughs Fig. 7 shows the probability densities of wave surface elevation from data, Gaussian distribution and Tayfun distribution, respectively. In the middle range of the surface elevation, namely between -2 and 3, both theoretical distributions are doing very well in terms of accurately simulating the observed data. However, at two ends of the distribution, results are diverse. It is seen that the Gaussian distribution is symmetric with respect to zero elevation, which reflects the linear wave conditions. The Tayfun distribution, on the other hand, shows evident asymmetry with respect to zero wave elevation. The negative end of the Tayfun plot curves inwards, while the positive end curves outwards. This is due to the involvement of non-linearity in Tayfun distribution that corresponds to higher crests and shallower troughs. In the positive end, the Tayfun distribution fits better with the data than Rayleigh distribution. However, what surprises us is that the data fit better with Rayleigh distribution in the negative end. The data plot doesn t show evident non-linear asymmetry between positive and negative wave elevation. Figure 7: PDF of surface elevation from: a) data (black dots); b) Gaussian distribution (solid line); c) Tayfun distribution (dashed line). Fig. 8 shows the probability densities from data, Rayleigh and Tayfun distributions for both crests and troughs. The PDF of crest from Tayfun is higher than the one from Rayleigh, which is in correspondence with the general scenario of sharper crest due to non-linearity. In the extreme tail of the data plot, there are a few data points that go beyond the scope of Rayleigh law, and Tayfun distribution seems to be more capable of fitting data in the extreme tail. It seems that for small crest heights, Rayleigh law fits very well with the data. But when the crest height becomes very large, the data start to deviate from Rayleigh distribution and get closer to Tayfun distribution. As to trough distribution, the Tayfun plot curves inward and corresponds to the non-linear scenario of shallower troughs. When the trough height is small, the Tayfun distribution fits greatly with the data. However when the trough height get very large, neither Rayleigh nor Tayfun distribution can accurately predict the data. International Journal of Ocean and Climate Systems

15 Xiufeng Yang 15 Figure 8: Exceedance probability of wave crests and troughs from data and from theories. Fig. 9 shows the exceedance probability of wave height from both Boccotti distribution and from data. Different plots are created with different lengths of intervals of time series. The three time intervals used are 3 minutes, 20 minutes and 60 minutes. It is noticed that plots of data-3min and Boccotti-3min match very well with each other. Plots of data-20min and Boccotti-20min also match very well with each other. However data-60min and Boccotti-60min don t match so well and a considerable deviation is observed compared to previous two cases. Remembering that the rough length limit for stationary Figure 9: Ratio of H/HR as a function of probability from both data and Boccotti distribution. Volume 6 Number

16 16 Statistical Analysis of Non-stationary Waves off the Savannah Coast, conditions is determined to be around 30 minutes based on results from Fig. 3, time intervals shorter than 30 minutes will maintain stationary conditions very nicely while time intervals longer than 30 will lose stationarity. Comparison from Fig. 9 verifies our conclusion about the minimum length of time series for stationary conditions. 4.5 Estimation of expected maximum wave heights Gumbel theory can be employed to estimate expected maximum wave heights. There are different models for this purpose (e.g. [Naess, 1985], [Boccotti, 1989], [Tayfun, 1990]). In our study we investigate two representative models: Rayleigh model and Boccotti model, and compare the modeling results with observational data. Fig. 10 shows expected maximum wave heights as a function of the number of wave heights N. The blue dotted line is from the observed data. The blue dashed lines outline the upper and lower stability limits of the data. The black solid line marked with R is the theoretical maximum wave heights obtained from Rayleigh model. The black solid line marked with B is obtained from Boccotti model and the dashed line marked with T is from simplified Tayfun model. The result from Rayleigh model has an evident deviation from observed data. At N=100, the absolute difference between Rayleigh and data is 1.005, which is about 18% of the data. As N increases, the absolute difference between Rayleigh and data increases. At N= , the absolute difference rises to and the percentage rises to 23%. It is also seen that the Rayleigh plot is way out of the stability limits. Figure 10: Expected maximum wave heights as a function of the number of wave heights from both data and theories: R-Rayleigh; B-Boccotti; T-Tayfun. Plots from Boccotti and Tayfun models are quite close to each other, and they all fit the data very nicely and stay within the stability zone. As N increases from a very small value, the accuracy of these two models in predicting data also increases. When N reaches around the gap between theory and data reaches minimum. However as N keeps rising, the gap starts to increase. Above comparison shows both Boccotti and Tayfun models are much more accurate and stable than Rayleigh model in estimating the expected maximum wave heights. As the number of wave heights N increases from very a small number, the Boccotti and Tayfun models tend to be more accurate as N International Journal of Ocean and Climate Systems

17 Xiufeng Yang 17 becomes larger, while the Rayleigh model goes the other way. However when N reaches certain number, the gap between model results and data reaches minimum. Further increase of the value of N won t yield more accurate results. 5. SUMMARY AND CONCLUSIONS The results from this article reveal some basic statistical characteristics of the non-stationary surface waves off the coast area of Savannah, Georgia. Spectral analysis of waves during storm seasons shows that the stormy wave spectra usually consist of two leading frequencies, which are in correspondence with the swell and the wind-driven waves respectively. It is also seen that the high frequency range of the wave power spectrum tends to confirm to the negative fourth power of the frequencies. A nonstationary wave series can be equally divided into a series of shorter quasi-stationary time series. Analysis suggests that the length of time series within which waves can be considered as stationary should be less than 30 minutes. After normalization, wave series from different times at the same location are still distinct due to variability in various dimensionless parameters. Such variability is taken into consideration in computing wave statistics in a newly introduced technique. Different dimensionless parameters can be used to account for such variability, and commonly used ones include wave skewness λ 3 and the first minimum of normalized covariance function ψ *. Results show that wave skewness calculated from wave period is statistically more stable than calculated directly from time series. Distribution of ψ * reflects the bimodality due to swell and wind interactions. The results from wave elevation distribution suggest that non-linear Tayfun distribution yields satisfactory predictions for positive, especially extremely large wave elevations. But Tayfun distribution doesn t show nice fit with the distribution of negative wave elevations, which surprisingly fits very well with linear Rayleigh distribution. In terms of predicting the wave crest distribution, the Tayfun distribution does a better job than Rayleigh distribution in that it takes into account nonlinear effect. However the prediction of wave troughs from Tayfun distribution is still less satisfactory than Rayleigh distribution. Studies both on wave elevation distribution and crest/trough distribution suggest that non-linear Tayfun model describes statistics of positive wave displacement better while linear Rayleigh model describes statistics of negative wave displacement better. Such findings suggest positive wave displacement preserves more non-linear features than negative wave displacement. However, such mismatch could be due to the fact that buoy instrument may have filtered out some portion of non-linear signals from the data. Further studies would be needed to ascertain if such mismatch can be reduced by improving instrument precision. In terms of estimating the exceedance probability of wave heights, the Boccotti distribution can do a reasonably nice job as long as the length of time series is within the stationary limit, which is found to be about 30 minutes. Both Boccotti model and Tayfun model derived from Gumbel formula using the new technique describe the trends of maximum wave heights as a function of the number of wave heights quite accurately compared to Rayleigh model, which is shown to be inaccurate and unstable. However, model results are not necessarily more accurate with larger number of N. When the number of wave heights reaches certain limit, the discrepancy between model results and data reaches minimum. Further increase of the number of wave heights won t yield more accurate results. ACKNOWLEDGMENTS Many thanks to Dr. Francesco Fedele for his valuable comments and suggestions. I also want to thank Dr. Paul Work for kindly sharing the wave data. REFERENCES Adler, R. J. (1981), The Geometry of Random Fields, John Wiley, New York. Boccotti, P. (1989), On mechanics of irregular gravity waves, Accademia Nazionale dei Lincei. Boccotti, P. (2000), Wave mechanics for ocean engineering, Elsevier Science. Volume 6 Number

18 18 Statistical Analysis of Non-stationary Waves off the Savannah Coast, Forristall, G. Z. (2000), Wave crest distributions: Observations and second-order theory, Journal of Physical Oceanography, Goda, Y. (2000), Random sea and design of maritime structures, 2nd edn. Advanced series on ocean engineering, World Scientific, Singapore. Huang, N. E., and S. R. Long (1980), An experimental study of the surface elevation probability distribution and statistics of wind-generated waves, Journal of Fluid Mechanics, Jahns, H., and J. Wheeler (1973), Long-term wave probabilities based on hindcasting of severe storms, Journal of Petroleum Technology, Longuet-Higgins (1963), The effect of nonlinearities on statistical distributions in the theory of sea waves, Journal of Fluid Mechanics, Naess, A. (1985), On the distribution of crest to trough wave heights Ocean Engineering, Tayfun, M. A. (1986), On narrow-band representation of ocean waves: 1. Theory, Journal of Geophysical Research: Oceans, 91(C6), Tayfun, M. A. (1990), Distribution of large wave heights, Journal of waterway, port, coastal, and ocean engineering. Tayfun, M. A. (1994), Distributions of envelope and phase in weakly nonlinear random waves, Journal of Engineering Mechanics. Tayfun, M. A. (2002), Distribution of nonlinear wave crests, Ocean Engineering, Tayfun, M. A., and F. Fedele (2007), Wave-height distributions and nonlinear effects, Ocean Engineering, International Journal of Ocean and Climate Systems

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