About the Frequency of Occurrence of Rogue Waves

Transcription

1 About the Frequency of Occurrence of Rogue Waves Michel Olagnon IFREMER Centre de Brest B.P. 70, F Plouzané, France Abstract. Some articles dealing with rogue waves state in their introduction that rogue waves are more frequent than common statistics would predict (see for instance [12]). However, supporting evidence that taking into account second-order non-linearities would fail to be sufficient to explain the deviation from Rayleigh distribution is very hard to find. To be fair, the opposite thesis, that extreme waves would not be freak, is also difficult to validate. Nearly 20 years ago, we ([10]) did not observe in a large 2 million individual waves dataset more high waves than expected from the Rayleigh distribution. We extended the dataset to about 3.3 million waves in [8] without any change on those conclusions. In the present study, we analyse the Alwyn dataset, where some authors have detected rogue waves [6, 14] but for which we are not aware of any global statistical analysis. We detail the techniques that we apply to validate the data on the grounds of physical limits on the water velocities, accelerations and other quantities. We conclude that here again extreme waves are not more frequent than common statistics would predict, and that the problem to be addressed is not the number of their occurrences, but their precise times and locations that are still unpredictable as far as we can tell from the available references. Introduction A number of articles dealing with rogue waves state in their introduction that rogue waves are more frequent than statistics would predict (see for instance [12]). To find supporting evidence for this statement, the reader is commonly refered to the introductions of other similar papers or to papers such as that by Dean ([1]) or the Gorm field analysis by Sand et al ([11]) in the same workshop. However, when returning to those insightful papers, one finds out that for instance Bob Dean was very cautious about the statistical significance of the observations, and only concluded that rogue waves might be defined by being more numerous than expected from the Rayleigh distribution and that non-linearities might then be the explanation. Samewise, Sand et al analysed waves of 10 to 18 m in only 40 m water depth, and found that those largest waves departed

2 from linear statistics. Yet, there is no evidence in those papers that taking into account even only second-order non-linearities would fail to be sufficient to explain deviations from Rayleigh distribution, nor indeed to be fair that it would succeed. We do not infer that there would be lack of evidence that extreme and dangereous waves exist: their gigantic size has been testified by many shipmasters reports and their dangerousness has been proven by damages on ships and offshore structures, some examples of which are described in Ersdal & Kvitrud (1999) [2] or Kjeldsen (1997) [5]. One should note however that in the very short term, other waves are bound to appear disproportionately small in the vicinity of an extreme one. Whether in a long-term statistical sense such extreme waves are normal waves or abnormal ones is however still an unsolved problem. Moreover, Robin ([10]) analysed about 2 million individual waves at a location where water depth non-linearities were unlikely to have any significant influence and did not observe more high waves than expected from the Rayleigh distribution. We extended the dataset to about 3.3 million waves in [8] without any change on those conclusions. The present paper investigates another dataset, in order to help decide whether extreme crest-trough waves are actually more frequent than expected from linear theory. Crest-trough heights distributions are almost unaffected by conventional non-linearities. Of course, if some non-conventional phenomenon occurs, it might also leave crest-trough heights unaffected. Yet, if nothing abnormal is detected on long-term crest-trough heights distributions, we can at least conclude that crest-trough height is not an appropriate parameter to characterize rogue waves. That dataset analysed here is the Alwyn one, where some authors have detected rogue waves [6,?] but for which we are not aware of any full statistical analysis. Especially, Wolfram et al [14] do not detail the validation and selection processes for the 2762 records used in their bivariate distribution fitting, nor did they conclude about the frequency of extremes. 1 Validation of the dataset 1.1 General description The initial dataset consists of minute records of water surface elevation by a wave altimeter (EMI monitor) located on the Alwyn North platform in the North Sea, below the flare boom on the south of the platform. Records were periodically made at 5 Hz. Continuous recording was triggered when significant wave height was about or exceeded 3.5 meters. The dataset covers most of the period, with a few interruptions. As it is unfortunately usual with marine data, many records show signal histories that can not sensibly correspond to actual water surface elevation ones. Validation is thus necessary, and needs to be at least semi-automated given the high number of records. Care should however be taken in the validation process that criteria should not result in discarding abnormal but actual values.

3 1.2 Validation of the data In a first stage, moment statistics (average, standard deviation, skewness, kurtosis) were performed over the signals in each record, and wave identification and count followed by wave statistics (zero-crossing period, H 1, H 1, H max, and the 3 10 associated periods). A first selection was made by discarding any record for which either: Computation of the moments failed Skewness is larger than 5 Kurtosis is larger than 10 H σ (4 times standard deviation) is larger than 20 meters Identification of the zero-crossing waves failed H 1 is less than 1 meter 3 T Z is larger than 25 seconds T Z is smaller than 2 seconds T H 13 is larger than 25 seconds H σ is more than twice H 1 3 H 1 3 is more than twice H σ Upon examination of a few of the remaining records, we felt that to carry out moment statistics on the first and second time derivatives of the signal (velocity and acceleration of the water surface elevation) and apply similar criteria would be more efficient than making the above ones more stringent. The following criteria were added: Computation of the moments of the velocity failed Standard deviation of the derivative is larger than 10 m/s Kurtosis of the derivative is larger than 10 Random visual examination showed problems Computation of the moments of the acceleration failed Kurtosis of the acceleration is larger than 10

4 time elevation Fig. 1. A signal that visual examination discards elevation time /03/ fsi Fig. 2. A signal that visual examination keeps despite doubts Eventually, the records with the highest ratios Hmax H S in the remaining set were visually examined, and those for which the signal did not look sensible were discarded, until the 100 records with highest ratios were validated. It appeared that many of the visually faulty cases showed a highly varying mean level over the record duration. A moving average over a 2-minute window was thus computed, and those records for which standard deviation of the moving average was more than 0.3 meter were discarded. A typical record that this criterion dis-

5 cards is shown on figure 1. Visual examination of a dozen of the records discarded according to that criterion randomly selected in the subset where Hmax H S > 2 confirmed that the threshold at 0.3 meter only removed invalid time-histories from that subset. After applying all criteria, only records remained, i.e. 66% of the initial database. The distribution of the records with respect to significant wave height originating from an independent source (ERA-40 and NOAA hindcasts) is shown on figure 3. Malfunctions of the system are more frequent for low H s than for higher ones, and for routine records than for threshold triggered ones. Therefore, validation might introduce a bias, with more weight given to high H s situations. However, statistics of extreme waves for low H s have little practical use, especially as they can be biased by, for instance, ship wakes and other phenomena of an artificial origin. We can thus consider in the following that the sampling bias towards high H s is no problem for practical applications. sea states discarded validated database Hs Distribution of records Fig. 3. Distribution of records with respect to hindcast H s

6 1.3 Distribution of the ratio Hmax The empirical distribution of the ratio r = Hmax is compared to several Rayleigh approximations, i.e. based on the assumption that the maximum height in a record is the same as the the maximum of N W independent random values distributed with a Rayleigh distribution, where N W is the number of waves in the record: P(r) = (1 e 2r2) N W This formula assumes narrow bandwidth, the extremes are slightly reduced when a larger bandwidth is taken into account. The first approximation assumes that all sea states have the same number of waves, taken as the average (168= /47426) over the database. P(r) in the above formula is then the distribution model for all observed ratios. A further approximation is to assume that P(r) can be replaced by its asymptotic Gumbel limit: P(r) G l (r) = e e (r 0.5ln(N W ))(2 2ln(N W )) Eventually, a reference Rayleigh distribution was computed through 2000 simulations of the set of sea states with their actual numbers of waves (independent Rayleigh random values). This simulation also provides a confidence interval for the wave distribution to follow the Rayleigh model, thanks to the quantiles of the simulated cumulative distribution.

7 cum. prob R^168 Gumbel simulated Rayleigh (average) simulated Rayleigh (min) simulated Rayleigh (max) measured ratio Hmax/H1/3 Fig. 4. Distribution of Hmax Figure 4 shows that the R 168 model does not differ significantly from the average. On the opposite, the Gumbel approximation is much higher, and it may be assumed that the number of waves in a sea state is not sufficient to allow the use of that asymptotic approximation. The measured ratios are significantly lower than the Rayleigh approximation on most of the database. It can be seen on figure 5, a zoom on the highest part of the previous figure obtained by using Gumbel coordinates, that only the highest ratio in the database matches Rayleigh theory, and on figure 6, another zoom, that those highest measured values are within reasonable confidence limits about the Rayleigh. Thus Rayleigh theory is certainly not underestimating the numbers of extremely high crest-trough waves as long as only a few millions of waves are concerned.

8 gumbel scale R^168 Gumbel simulated Rayleigh (average) simulated Rayleigh (min) simulated Rayleigh (max) measured ratio Hmax/H1/3 Fig. 5. Distribution of Hmax - Gumbel scale cum. prob simulated Rayleigh (average) simulated Rayleigh (min) simulated Rayleigh (10%) simulated Rayleigh (20%) simulated Rayleigh (40%) simulated Rayleigh (60%) simulated Rayleigh (80%) simulated Rayleigh (90%) simulated Rayleigh (max) measured ratio Hmax/H1/3 Comparison to Rayleigh Fig. 6. Distribution of Hmax - zoom Conclusion When considering the carefully validated Alwyn database, the overall number of extreme crest-trough wave heights are not in excess of the Rayleigh linear theory predictions.

9 Crest-trough height alone is thus not an appropriate parameter for characterizing rogue or freak waves. Rogue or freak waves should on the opposite be characterized by their potential damage, and research should consider the velocities in the wave, the steepness, the pressures on a hull, and similar parameters. In addition, it should be sought reasons why extreme heights are not expected at the time and place where they occur whereas they are not abnormal from the long term statistics point of view. References 1. Dean, R. Freak Waves: a Possible Explanation Proc. Water Waves Kinematics, NATO ASI Series E, Vol. 178, pp Ersdal, G. & Kvitrud, A. Green sea on Norwegian production ships Proc. Airgap Workshop, HSE/E&P Forum, London C. Guedes Soares, Spectral Modeling of Sea States With Multiple Wave Systems, Journal of Offshore Mechanics and Artics Engineering, vol International Association for Hydraulic Research List of Sea State Parameters Supplement to bulletin No 52 (1986). 5. Kjeldsen, S.P. Examples of heavy weather damages caused by giant waves it Bulletin of the Society of Naval Architects of Japan, 820, , pp Linfoot, B., Stansell, P. & Wolfram, J. On the characteristics of storm waves Proc. Int. Offshore and Polar Engineering Conf., ISOPE, Olagnon, M. & Krogstad, H.E. Observed short- and long-term distributions of wave steepness, Proc. Int. Offshore and Polar Engineering Conf., Vol. 3, Montréal (1998), pp Olagnon, M. & van Iseghem, S. (2000), Some observed characteristics of sea states with extreme waves, Proc. 10th International Offshore and Polar Engineering Conf. ISOPE, 3, pp Podgórski, K., Rychlik, I., Rydén, J & Sjö, E. How big are the Big Waves in a Gaussian Sea?, Int. J. of Offshore and Polar Engineering, Vol.10, No.3, (2000), pp Robin, A. & Olagnon, M. Occurrence of Extreme Waves with respect to Significant Wave Height Proc. Offshore Mechanics and Artic Engineering, OMAE Vol.2a, 1991, pp Sand, S.E., Ottesen Hansen N.E., Klinting, P., Gudmestad, O.T. & Sterndorff, M.J. Freak Wave Kinematics Proc. Water Waves Kinematics, NATO ASI Series E, Vol. 178, pp Solli, D.R., Ropers, C., Koonath, P., & Jalali, B. Optical rogue waves Nature, Vol.450, Dec. 2007, pp van Iseghem, S., Deleuil, G. & Guerin, P. Improved characterizattions for design waves Proc. Int. Offshore and Polar Engineering Conf., ISOPE, Wolfram, J., Linfoot, B., Stansell, P. (2000) Long-and short-term extreme wave statistics in the North Sea Proc. Rogue Waves 2000.