Characterizing Freak Waves with Wavelet Transform Analysis

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1 Characterizing Freak Waves with Wavelet Transform Analysis Paul C. Liu 1 and Nobuhito Mori 2 1 NOAA Great Lakes Environmental Research Laboratory, Ann Arbor, MI , USA 2 Central Research Institution of Electric Power Industry, Abiko, Chiba Japan Abstract. This paper presents an analysis of a set of available freak wave measurements gathered from several periods of continuous wave recordings made in the Sea of Japan during by the Ship Research Institute of Japan. The analysis provides an ideal opportunity to catch a glimpse of the incidence of freak waves. The results show that a well-defined freak wave can be readily identified from the wavelet spectrum where strong energy density in the spectrum is instantly surged and seemingly carried over to the high frequency components at the instant the freak wave occurs. Thus for a given freak wave, there appears a clear corresponding signature shown in the time-frequency wavelet spectrum. Since freak waves are primarily transient events occurring unexpectedly, wavelet transform analysis on continuous, long duration wave measurements clearly represents the most ideal approach to discern the localized characteristics of freak waves for further exploration Introduction Perhaps one of the weakest and most difficult aspects for the explorations of rogue or freak waves in the oceans is the ostensive scarcity of actual field measurements of rogue wave events. Because of the uncertain and unpredictable nature of the occurrence of rogue waves, the conventional, discrete kind of wave measurements at fixed time intervals have not been conducive in capturing actual freak wave episodes. A viable approach for making comprehensive rogue waves measurement does not seem to be presently available. The current literatures on rogue waves are predominantly comprised only with conjectured mechanisms such as those variously reported in Dean(1990), Yasuda et al.(1992), Trulsen and Dysthe(1997), Lavrenov(1998), White and Fornberg(1998), among others. This paper presents an empirical analysis of available wave measurements collected during in the Sea of Japan where freak waves are known to have observed. Since freak waves are primarily transient events, conventional frequency spectrum analysis is clearly incapable in effectively processing rogue waves in the frequency domain. We applied wavelet transform analysis here to analyze the time series and examine the localized freak wave characteristics in the generalized time-frequency domain.

2 2 The Wave Measurements As freak waves are basically rare and unexpected occurrence, for the conventional wave measurements which typically making minutes recordings hourly, it would be entirely possible that an incidence of a freak wave be overlooked if it occurs at in between recording times. It is only through extended and continuous recordings would plausible expectations to capture an event of freak waves be realized. There are conspicuously fewer continuous wave measurements available, hence very few freak wave studies are based on actual measurement. Wave measurements used in this study were made from the Sea of Japan, at a location 3 km off the Yura fishery harbor in 43 m water depth. The instruments used in the measurement were ultrasonic type wave gages. Five sets of sea surface fluctuations data, recorded at 1 Hz sampling frequency, are used. Each of the data sets has over 20 to 40 hours of continuous recordings. Wind measurements were also available, which shown the occurrence of freak waves are generally during steady wind conditions (Mori, et al., 2000, Yasuda, et al., 1997). The measurement was originally made by the Ship Research Institute of the Ministry of Transport of Japan. 3 The Wavelet Spectrum Wavelet transform analysis, developed during the last two decades, is an ideal tool for the study of the measured time series data of nonstationary, transient phenomenon such as freak waves. As the last decade marked an explosive publication of wavelet related books and articles, details on wavelet transform can now be found in many widely available introductory articles and texts (e.g. Liu, 1994 or 2000). Therefore we shall just present a very brief summariaztion of the formulation of continuous wavelet transform here. For a given function or data signal X(t), which is assumed to be square integrable, its Fourier transform, ˆX(ω), is given by ˆX(ω) = X(t)e iωt dt, (1) which transforms the function in the time domain to the frequency domain. In order to examine the characteristics of the function in the frequency domain as well as the time domains, a direct approach of extending Eq.(1) to the timefrequency domain can be obtained by including a time windowing function g(t) such that: ˆX(ω, τ) = X(t)g(t τ)e iωt dt. (2) Formulating analougously and replacing the window function with a new family of functions, and discretizing the time, t, and frequency, ω, with position, a, and scale, b, respectively, then it readily leads to the wavelet transform: X(a, b) = X(t) a 1/2 ψ ( t b a ) dt. (6)

3 where a > 0, < b < +, and the asterisk superscript indicates the complex conjugate. Suffice to assert that the wavelet spectrum, based on the continuous wavelet transform, represents a natural extension of the familiar, conventional Fourier spectrum analysis. While Fourier transform is based on the concept of frequency, the wavelet transform is based on the concept of scale and time. As scale and frequency are inversely related, thus instead of results presented in a conventional plot of energy versus frequency for Fourier energy spectrum, the wavelet spectrum is known to be three-dimensional in nature and plotted in the timefrequency domain with the equivalent energy density appears in terms of contour levels. This provides an ideal opportunity to examine the process of energy variations where the freak waves occur locally and abruptly in time. Note that we present mainly the results of continuous wavelet transform using Morlet wavelet here, similar result of characteristic features shown here can in general also be obtained from using different mother wavelet, or with the application of discrete wavelet transforms. Since there is presently no available studies examining the localized characteristics during the occurrence of freak waves, and neither is there any theoretical implications as to how a freak wave might have behaved in the time-frequency domain, our results are therefore necessarily exploratory and tentative. On the other hand, the qualitative nature of the results also presents challenging implications that are inviting for rational interpretation. From the five sets of continuously recorded surface wave data which comprise over 200 hours of measurement, only a few episodes of freak waves have been identified both in the field and by the time series data. These cases generally resembles the ideal freak wave time series case given by the North Sea measurement (Sand et al., 1990). Figure 1 presents such a characteristic freak wave time series and its corresponding wavelet spectrum. The freak wave episode shown in Figure 1 is represented by a plotting of 10 minutes time series segment that contains the occurrence of the freak wave along with a panel of corresponding contour plotting for the wavelet spectrum. It appears that for the well-defined freak wave as shown in the time series plot, it can also be readily identified from the wavelet spectrum where strong energy density in the spectrum appears instantly surged at the onset of the freak wave and the energy density seemingly carried over to the high frequency components at the freak wave instant. Therefore, for a given freak wave, there emerges a clear corresponding signature shown in the time-frequency domain of the wavelet spectrum. However, for another similar characteristic freak wave time series shown in Figure 2, there is no corresponding instantaneous energy surge feature appear in the wavelet spectrum as those in Figure 1. So it is somewhat uncertain in this case whether a freak wave identified only in the time series can be really considered as a freak wave or not. On the other hand, it is of interest to note that the time series in Figure 1 at the onset of the freak wave its profile appeared rather asymmetric with respect to the mean level, whereas the freak wave profile in

4 8 Yura Y Surface Waves Surface Elevation (m) Time (s) 0.6 Wavelet Spectrum of Yura Y Surface Waves Frequency (Hz) Fig. 1. Freak wave time series and its wavelet spectrum for data set Yura Yura Y Surface Waves Surface Elevation (m) Time (s) 0.6 Wavelet Spectrum of Yura Y Surface Waves Frequency (Hz) Fig. 2. Freak wave time series and its wavelet spectrum for data set Yura Y Figure 2 was generally symmetric 1. So the difference in wavelet spectrum might 1 We are indebted to Prof. Douglas Faulkner for pointing out this feature to us.

5 also be a result of the difference in freak wave profiles. It is possible that the single large wave height shown in Figure 2 represents merely a maximum wave from the extreme statistics. Or alternatively it is indeed a freak wave but generated from a different process from the case shown in Figure 1. Just as various different conjectures all can be shown to produce freak waves, freak waves certainly can be generated from different physical processes. Time series alone clearly may or may not be relied on for distinguishing freak waves. While wavelet transform applied to the time series can provide further discernible features, it is still beyond the scope of the wavelet transform to readily comprehend the differences in possibly different processes. Undoubtedly more detailed measurements than just surface time series would be needed in order for proper and practical study of freak waves. At any rate, since the mechanism of freak wave formation is understandably diverse, it should not be surprising that different freak waves exhibit different qualitative features. Both cases in Figures 1 and 2 can be freak waves, or only the case in Figure 1 represent a freak wave. As idealized and more comprehenive measurements encompass all possible relevant parameters are unavailable at the present, wavelet transform analysis is nevertheless the ideally suited approach to study the available freak wave time series that may be used to clarify the occurrence of the freak waves as well as their general characteristics and statistical properties. 4 Concluding Remarks Rogue or freak waves have always been a fascinating subject for contemplation and speculation, but the hazardous of severe damages it may inflect upon ships and mariners can not be over stated. While the existence of freak waves has been accepted by scientists and engineers, one can not overlook the fact that the existence of freak waves is basically only based on eyewitness accounts. It is of utmost importance that concerted field efforts should be implemented to provide actual detailed measurements and analysis and thus to ascertain its veritable existence. The working group on breaking and freak waves at the NATO Advanced Research Workshop (Torum and Gudmestad, 1989) had recommended over 10 years ago that future research needs should be on full scale freak wave measurement, correlation of meteorological information and freak wave occurrence, and extended analysis of existing data. These recommendations are certainly wellfounded, unfortunately none of the advices has been carried out since then. The continuous measurements of longer duration in time, made by the Ship Research Institute of Japan, which had led to possibly the only available field recordings of freak waves, are of extreme importance and usefulness. Analysis using wavelet transform would probably be the one of the versatile tools available that is truly ideal for the study of freak waves data. Hopefully the exploratory results presented in this paper may serve to demonstrate the useful effects of wavelet transform and freak waves and thereby enticing more particularized freak wave measurements and wavelet transform applications.

6 5 Acknowledgement The authors wish to express their gratitude to the Ship Research Institute of the Ministry of Transport of Japan for providing and allowing the use of their data. References Dean, R.G., 1990: Freak waves: A possible explanation. In Water Wave Kinematics Torum and Gudmestad, eds., , Kluwer. Lavrenov, I. V., 1998: The wave energy concentration at the Agulhas current off South Africa, Natural Hazards, 17, Liu, P. C., 1994: Wavelet spectrum analysis and ocean wind waves. In Wavelets in Geophysics E. Foufoula-Georgiou and P. Kumar (Eds), , Academic Press. Liu, P. C., 2000: Wavelet transform and new perspective on coastal and ocean engineering data analysis. In Advances in Coastal and Ocean Engineering, Vol.6, P. Liu (Ed), , World Scientific. Mori, N., T. Yasuda and S. Nakayama, 2000: Statistical Properties of Freak Waves Observed in the Sea of Japan, ISOPE2000, 3, Sand, S. E., N.E.O. Hansen, P. Klinting, O.T. Gudmestad, and M.J. Sterndorff, 1990: Freak wave kinematics. In Water Wave Kinematics Torum and Gudmestad, eds., , Kluwer. Torum, A. and O.T. Gudmestad, Eds., 1990: Water Wave Kinematics, 771pp. Kluwer. Trulsen, K. and K. Dysthe, 1997: Freak waves A three-dimensional wave simulation, Twenty-first Symposium on Naval Hydrodynamics, , National Academy Press. White, B. S. and B. Fornberg, 1998: On the chance of freak waves at sea, J. Fluid Mech., 355, Yasuda, T., N. Mori, and K Ito, 1992: Freak waves in a unidirectional wave train and their kinematics, In Proceedings, 23rd International Conference on Coastal Engineering, , ASCE. Yasuda, T., N. Mori, and S. Nakayama, 1997: Characteristics of giant freak waves observed in the Sea of Japan. In Ocean Wave Measurement and Analysis, B. Edge and M. Hemsley (Eds.), , ASCE.