Investigating the use of the Hilbert-Huang transform for the analysis of Freek Waves

Date 2012

Author Rene H.M. Huijsmans,Jan Maris and Marios Ctiristou.

Address Delft University of Technology

Ship Hydromechanics and Structures Laboratory Mekelw/eg 2, 2628 CD Deift

Delft University of Technology

TUDelft

Investigating tlie use of the Hilbert-Huang

transform for the analysis of Freak Waves

by

R.H.M. Huijsmans, J . Maris and M. Christou

Report No. 1 8 5 0 - P 2012

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Proceedings of the ASiVIE 2012 31st International Conference on Ocean, Offshore and Arctic Engineering OMAE2012 July 1-6, 2012, Rio de Janeiro, Brazil

OMAE2012-8DD69

INVESTIGATING THE USE OF THE HILBERT-HUANG TRANSFORM FOR THE

ANALYSIS OF FREAK WAVES

Jan Maris

Shell, Rijswijk, The Netherlands Email: jan.maris@shell.com

Marios Christou Shell, Rijswijk, The Netherlands Email: marios.christou@sheli.com

René Huijsmans TU Delft, The Netherlands Email: r.h.m.huijsmans@tudelft.nl

ABSTRACT

This paper presents the Hilbert-Huang Transform (HHT) with Empirical Mode Decomposition (EMD), which is an oppor-tune method to analyse non-stationary traces of nonlinear waves, with freak waves as a case i n point. Comparison to conventional methods such as the Fourier, Wavelet and Stockwell Transform is undertaken. Furthermore, an investigation of different E M D schemes (Ensemble E M D or Smoothed E M D ) is undertaken to improve the interpretability of the results. Ultimately the Draup-ner New Year Wave is examined by using the HHT. A f t e r all, on successfully applying the HHT, a new perspective on analysing waves is suggested.

INTRODUCTION

Rogue waves (or freak waves) have been frequently ob-served in the past. They are often labelled as freak waves, as they are likely to occur beyond the theoretical expectation models of maximum wave and crest height. The definition suggested by Haver [1] is commonly used:

Hrogue y 2

ani id/or ^c,rogiie ^ J 25 ₍₁₎

Hrogue and r\c,wgue respectively indicate freak wave height and freak crest height and Hg is the significant wave height of a 20 minute time trace.

Offshore structures are at high risk, when they are not designed or operated with these extreme waves in mind. Therefore it is

of high interest to investigate the occurrence of rogue waves, in order to enhance design models. This problem has been widely studied by statistic analysis of spectral parameters and by doing numerical or tank simulations.

Wolfram et al. [2] concluded that rogue waves are generally 50% steeper than the significant wave steepness and neighbouring waves with heights equal to the significant wave height -only have steepness values of half the significant wave steep-ness. Christou and Ewans [ 3 , 4 ] found that similarly steep waves were observed without being a rogue wave, hence there is no strict relation between steepness and the rogue wave definition. Veltcheva and Guedes Soares [5] concluded that when rogue waves appear individually, they appear to be more asymmetri-cal than when a group of rogue waves occur.

Numerical simulations were done by Trulsen [6], Osborne [7] and Slunyaev et al. [8]. They have used NLS equations to gener-ate rogue wave events based upon the existence of the Benjamin-Feir instability. However, this theory presumes unidirectional and narrow-banded wave groups i n deep water conditions. Tank simulations were carried out by Clauss and Klein [9] and Buch-ner et al. [10]. They simulated a wave group from which a rogue wave emerged three times at different locations in the tank [9] and revealed that extreme waves can develop f r o m a moderate wave into an extreme event i n only half a wave length [10]. These simulations have all been based on theoretical bases in-volving mechanisms that do induce rogue waves, but these mod-els and their presumptions are not applicable to any location where rogue waves have been measured. Therefore, this study aims to get better understanding of the underlying mechanisms of rogue waves by investigation of the real-measured time traces of rogue wave events using signal processing methods. There

t[s]

Figure 1. Fragment of ttie time trace of tine Draupner New Year Wave.

are two complications i n this objective. First, measurements of rogue waves are scarce such that only a few samples can be anal-ysed. The Draupner New Year Wave (Figure 1) is well known in the literature, and is therefore also analysed i n this study. Sec-ond, conventional signal processing methods such as the Fourier Transform (FT) do not transform non-stationarity and nonlinear-ity i n a meaningful way. Christou et al. [11] used FT and the Wavelet Transform to demonstrate the presence of linear focus-ing and high frequency components befocus-ing amplified durfocus-ing the extreme event. Adcock et al. [12] argued based upon Fourier spectral analysis that nonlinear interaction of crossing waves may cause a rogue event.

However, the rapid evolution and the steep character of rogue waves implies high non-stationarity' and strong nonliiiearity re-spectively. Therefore, advanced wave analysis is required. A novel method called the Hilbert-Huang Transfonn ( H H T ) [13] has been employed as this method is capable of interpreting both non-stationaiy and nonlinear signals. The H H T has been applied on rogue wave analysis before by Huang, Tiulsen, Veltcheva and Ortega et al. [14, 15, 16, 17, 5]. It was found that the H H T with the Empirical Mode Decomposition (EMD) is a unique method which is able to locally transform a single wave. Most of these studies mainly focused on analysis of the decomposed modes rather than the complete spectral behaviour as it was found d i f f i -cult to interpret. This study has further investigated the possibil-ities of spectral interpretation.

Other methods to investigate non-stationarity and nonlinear ef-fects of the Draupner wave were used by Cherneva and Guedes Soares [18], who applied higher order time-frequency spectra.

THE HILBERT-HUANG TRANSFORM AND DEVELOP-MENTS

The Hilbert-Huang Transform (HHT) with Empirical Mode Decomposition ( E M D ) was first published by [13], who proposed the E M D as a method which decomposes signal into i n -trinsic rather than analytical modes, that latter being produced by conventional Fourier-based methods. After the decomposi-tion through E M D the HHT-EMD employs the Hilbert Trans-f o r m (HT) to transTrans-form each intrinsic component locally into the frequency domain, whereas Fourier-based methods such as the

Wavelet Transform globally define the frequency content. As a result the Hilbeit Spectrum returns highly instantaneous com-ponents representing contour levels of ampHtude in the time-frequency domain. The H H T - E M D is well suited to provide new insights in the analysis of nonlinear signals such as ocean waves [14].

The E M D is the most critical part of the process. First, the E M D w i l l briefly be explained. Second, several developments which have been published w i l l be examined.

Empirical Mode Decomposition (EMD)

The E M D decomposes the signal into Intrinsic Mode Func-tions (IMF) [13, 19] by a linear decomposition process which is characterised by a sifting procedure. The sifting procedure gen-erates intrinsic components using cubic splines of the extrema (maxima and minima) of the signal. By doing so it removes rid-ing waves and the resultrid-ing IMFs with zero mean are smoothed such that they ami to be so-called monocomponent [20] signals. The E M D first extracts the high-frequency content from the sig-nal into the first IMFs, fisig-nally after the lowest frequency con-tent has been assigned to the final I M F a monotone residue re-mains. Hence the signal is decomposed i n a linear summation of all IMFs and the remaining (monotone) residue (Figure 2).

The laborious sifting process is sensitive to various settings and boundary conditions, such that two executions of E M D with dif-ferent settings may result i n extremely difdif-ferent decompositions. It is difficult to interpret or trust these different results. Moreover, two erroneous effects occur due to the empirical sifting proce-dure:

(i) E n d effects - The construction of cubic splines as a start-ing point for siftstart-ing suffers problems at the ends of the time trace. A t the ends, the lack of construction points results i n deflections of the spline. As a result, a number of additional IMFs are needed to 'store' the eiToneous content that has been generated. Furthermore, the H T induces some minor end effects when the kernel of the H T integral reaches the edge of the signal.

(ii) Neighbouring frequencies - When two linear (freely propa-gating) wave components are close i n frequency, the E M D is unable to decompose these two components accurately. The amplitude and frequency obtained f r o m the IMFs have ac-curate mean values, but fluctuate rather than being constant. First, the end effects can be reduced by constructing additional boundary points at the ends [21]. Through simply duplicating the outer extremes of the I M F or extrapolating them using the signal slopes at the edges, one can construct a more meaningful spline at the end. However, i n some cases the slope technique induces severe deflections of the spline, such that the end effects become even larger.

Second, i n [13] it was mentioned that the E M D is not able to

IMFs _{HHT (EMD) HHT (EMD)}

Figure 2. Draupner wave (Figure 1): first 6 IIVIFs from EMD.

accurately decompose a bichromatic function consisting of two sinusoid close i n frequency. Spline fitting limitations were held responsible for this flaw. In [21] further investigation was un-dertaken and a relation was established to govern ratios of the amplitude (^1,02) and frequency ( / i i / a ) of a bichromatic wave, such that the accuracy of the result is sufficient.

1

6Ö

« l Y 3 «1 11

02] 20 fl2 6 (2)

This effect is clearly visible when a bichromatic wave, consist-ing of two sinusoids with equal amplitude, are examined for different frequency ratios (Figure 3). Both sinusoid components have constant frequencies over time. However, i n the case that the component frequency ratio is small, modulations are present in the Hilbert Spectium. In fact a nonlinear wave, which is conventionally described by a carrier wave with bound harmonics, is a series of completely focused sinusoids and hence

^ 1 0 50 100 150 t[s] (a),/2//i = 1.5,ni/fl2 = l 0 50 100 150 t[s] (b) / 2 / / l = 3 , f l , / « 2 = l Figure 3. The effect of neighbouring frequencies for a bichromatic wave.

the H H T interprets it as a single component with modulating frequency.

EMD variants

Apart from the minor adjustments to the E M D settings and boundary conditions, alternative decomposition methods have been suggested. The Smoothed Empirical Mode Decompo-sition (SEMD) and Ensemble Empkical Mode DecompoDecompo-sition (EEMD) have been compared i n this study. The SEMD [22] ap-plies additional smoothing during the sifting process in order to create smoother and therefore more narrow-banded I M F signals. However, [22] predominantly analysed the character of the IMFs rather than the Hilbert spectral information. The present study has shown that the H T after SEMD produces high amounts of en-ergy i n the low frequency domain, as results show i n Figure 10. Moreover, the selection of smoothing parameters is very critical to the result.

Alternatively, the E E M D [23] is another variant on E M D , which involves less thorough sifting procedures in order to reduce the negative side effects of it. The E E M D adds randotn white noise to the input signal, which is decomposed by an E M D with a fixed number of sifting iterations (10 - 20) to obtam each IMF. This E M D operation with added noise is repeated a number of times (100 — 200), each time with a randomly generated white noise, such that eventually an ensemble of innque E M D solutions is obtained. A l l these E M D solutions a set of n IMFs and a residue -is averaged, which -is the final E E M D solution (Figure 4). Unlike i n the case of E M D , the IMFs are assigned after a fixed number of sifting iterations, and not once the I M F fulfils the sift-ing criteria. However, ensembUng the large number of E M D so-lutions of uniquely, sUghtly deformed input signals is considered to give a stable solution which has a strongly intrinsic character and approximates the conditions of the I M F definition. I n order to ensure the confidence of the solution, a statistical significance study can be perfonned on the E E M D solution [24, 16].

IMFs 2"'^-order Stokes wave CM o CO o in Ü CD O

Figure 4. Draupner wave (Figure 1); first 6 IMFs from EEMD performed with NE = 200, s = 15 and N,,id = 2. IMFs c\ and C2 have readily filtered the added white noise.

EXPERIMENTAL ANALYSIS WITH HHT

In order to compare it to conventional signal processing methods, the H H T has been applied on theoretical signals, i n which the content is known exactly. The results of two of these wave signals w i l l be presented below:

10 0 -10 50 100 Cross Chirp 150 20 -ê- 10 0 !=- -10 20 0 -20 50 100 Wave 1 150 50 100 t [ s ] 150

Figure 5. Wave 1: a non-stationary Cross Chirp embedded in a nonlin-ear 2"''-order Stokes wave.

Bi-modal spectrum 0 0.05 0.1 0.15 0.2 0.25 0.3 f[Hz] Wave 2 -600 -400 -200 0 200 400 t[s]

Figure 6. Wave 2: a simulation of a stationary bi-modal spectrum con-sidering a linear wave model, including an extreme event at ? = Os.

Wave 1 : Cross Chirp embedded in 2"''-order Stokes wave

A wave consisting of a summation of a non-stationary (chirp) and a nonlinear (Stokes) component (Figure 5);

Wave 2: Bi-modal Spectrum wave

A simulation of a random-phase linear wave was generated f r o m a combined swell (Gaussian) and wind-wave

(JON-SWAP, peak enhancement factor y = 3.5) spectrum (Figure 6). The extreme event at f = Os is caused by random linear focusing of the components.

These wave signals have been used to: examine the results of H H T in comparison to conventional methods including the Fourier Transform (FF), Wavelet Transform (WT) and Stockwell

Transform (ST); compare the outcomes resulting f r o m the dif-ferent decomposition methods (EMD, SEMD and E E M D ) . The spectra with time-resolution are characterised by spectral contour levels indicating the relative amplitude, as indicated i n Figure 7. A marginal spectrum is obtained from the contour spectra by averaging the contour spectrum over time, resulting in a conven-tional amplitude spectrum.

low high ampiitude scale

Figure 7. Ttie spectral colours of the relative ampiitude contour levels.

HHT versus Fourier-based methods

For Wave 1 (Figure 5) two components are expected in the spectrum: the chirp component evolving i n amplitude and frequency; and the Stokes component constantly modulating in frequency and amplitude. The FT (Figure 8) does not give a sensible representation of Wave 1 at all, as it returns constant frequency components which are present throughout time. The W T and ST return more meaningful spectra, however the compromise between time and frequency resolution results in a blurred spectrum. Neither the W T nor ST distinguishes the Stokes wave as being a nonlinear wave as only a single constant frequency contour is visible. A higher harmonic representing the bound packet can not be distinguished i n the contours of the spectrum. The H H T gives the most meaningful representation of Wave 1. A distinct contour line - hogging i n frequency, sagging in amplitude - is reairned for the chirp component. The Stokes wave is well distracted and represented by a modulating spectral component, which corresponds to the HHT-representation of an individual Stokes wave as was shown by [13]. However when the chirp frequency comes close to the stokes frequency, the H H T suffers from the neighbouring frequency effect. The neighbouring frequency effect and end effects are clearly disturbing the spectrum i n the edges, but seem to have no impact on the mid-point of the time trace. Due to these flaws energy is re-distributed in the spectrum, such that amplitude is no longer preserved.

Wave 2, a stationary model of linear wave components, is correctly transformed by FT as it shares the linear and stationary definitions (Figure 9). The H H T clearly distinguishes the swell component and the wind component. However, the swell is represented as a modulating wave group, which is explained by the neighbouring effect of the many adjacent frequency

FT WT

ST HHT (EMD)

Figure 8. Examination of Wave 1 by HHT compared to conventional transforms FT, WT and ST.

components i n the spectnim. The wind spectrum is represented by widely scattered components i n the { r , / } - s p e c t r u m and the H H T gives a high concentration of energy i n the vicmity of the extreme event due to the locally performed HT. Similar misinterpretations of non-stationarity have been found with the W T [11] and ST. Nevertheless in the time-averaged marginal spectnim the shapes of the two spectral components are rather well distinguished, although severe energy flaws are present between the two spectral components.

ConventionaUy FT is used f o r wave analysis, however the FT gives no resolution i n time. As a result it is unable to detect non-stationarities i n a signal. Furthermore, nonlinear waves are transformed to high-frequency harmonics which relate to the freely propagating carrier component by nonUnear theory. As the Fourier decomposition is a linear summation and considers all harmonics to be freely propagating, no distinction can be made between bound harmonics (nonlinear) and freely propagating waves (linear).

W T and ST [25, 26] are Fourier-based methods which do use a time integration window such that non-stationarity effects are detected by the regional transformation. Nevertheless, as the (arbitrary) window-width decreases, the time-resolution enhances but the frequency-resolution becomes poorer. This compromise i n resolution is reduced i n the ST, where a

pro-gressive frequency-resolution fits the window-width to the transformation frequency. As a result, the high frequency band is transformed with higher time-resolution than the low-frequency band. Furthermore, by normalising the integration window, the ST preserves amplitude instead of energy [27]. However, the deep impMcation of the linear Fourier decomposition means that neither W T nor ST interpret nonlinearity i n a meaningful way. The H H T with E M D [13] is a method which considers the signal intrinsically rather than analytically like Fourier-methods do. The signal analysis therefore has an a posteriori base. Hence a nonlinear wave w i l l be transformed by H T into a monocomponent wave that modulates i n frequency rather than being decomposed into a carrier wave with higher harmonics. As the H T locally transforms the signal, the computed frequency and amplitude are highly instantaneous. The energy conservative H H T also preserves amplitude i n the core of the spectnim (no end effects) and provided that the IMFs are monocomponent, which means that (2) should be adhered to when applying the H H T on a bichromatic wave.

A l l transformation properties are summarised i n Table 1.

F T W T S T H H T

Method priori priori priori posteriori

Analytical

+

+

+

Scale global regional regional local Resolution frequency

_+/-

_+/-

+

+

-Table 1. Hilbert-Huang Transform compared to conventional transforms. Scale indicates tlie time-scale on which the transformation is performed. The conservation of amplitude indicates if nominal amplitude values in the spectrum are meaningful, whereas conservation of energy means that the sum of components at each point in time give a correct representation of the energy of the signal.

-100 0 100 t[s] HHT (EMD) 0 0.1 0.2 f[Hz] HHT (EMD) -100 0 100 t[s] HHT (SEMD) 0 0.1 0.2 f[Hz] HHT (SEMD) 0 -100 HHT (EEMD) HHT (EEMD) 100

EMD, SEMD and EEMD

The literature on E M D and its variants generally discusses the shape and character o f the resulting IMFs [22, 23]. Addi-tionally, spectral results by Fourier and Hilbert Transforms on the IMFs have been investigated i n this study.

Figure 9. Examination of Wave 2 by FT compared to HHT with EMD, SEMD and EEMD. / } - s p e c t r a (left) and time-averaged marginal spec-tra (right).

Results o f applying H H T with E M D on Wave 1 and Wave 2 have been analysed in the previous section. Accordingly, these wave signals have been examined with H H T when employing the SEMD and E E M D instead o f the EMD. Spectral results after performing H T to the decompositions are shown in Figure 9. With the SEMD the smoothing operations squeeze the energy down to lower frequency to such an extent, that the relative amplitude level i n the spectrum becomes meaningless. On the other hand, with E E M D the Hilbert Spectrum has become more distinctive than with E M D (Figure 9). The swell component again modulates, but clearly is more stable than it is in the Hilbert Spectnim with standard E M D . Also the spectral peak at f = Os and the spectral peaks o f the swell and wind-sea partitions in the marginal spectrum give a better approximation of the theoretical peak frequencies. Moreover, the shape of the marginal spectrum of the HHT-EEMD is more similar to the actual spectral shape (Figure 6) with only small flaws in the lower band and different peak amplitudes for the wind-sea and swell partitions. However, the H H T with E E M D still miscalculates the relative amplitude scale o f the two spectral components. As w i l l be shown later, performing FT after (E)EMD results i n a correct amplitude scale for both peaks. Hence, the mcorrect transformation by H H T is due to the inabiUty o f H T to transform focused groups: as a result the frequency is slightly downshifted and the nominal amplitude is underestimated inherent to an increased number o f focused components.

It has been clearly observed how the H H T relocates energy i n the marginal spectrum as there is sometimes a discrepancy with the actual (theoretical) content. This effect is readily a result o f the H T returning a single frequency for signals containing more components.

The E M D is a Unear decomposition and so are the SEMD and EEMD. A l l resulting IMFs and the residue together f o r m a linear summation equal to the input wave signal. Nevetheless, significant spectral differences have also been observed when the E M D alternatives were employed. Certainly for SEMD it ap-pears that by the extensive smoothing substantial low frequency content is invoked, hence probably should not be labelled as intrinsic. In order to investigate this, Fourier analysis on the IMFs is done (Figure 10). As the FT is an analytical and Unear transformation it gives identical results for a transformation o f the signal directly and the complex integration o f results f r o m transformations o f all IMFs individually. However, the phase information can be removed from the FT results by taking the absolute amplitude spectra of IMFs. Accordingly, when summing all I M F amplitude spectra, the results wiU exceed the amplitude spectrum obtained f r o m FT to the input signal directly. This is explained by the fact that an I M F may contain Fourier components which cancel out by their corresponding components in other IMFs due to phasing. However, the E M D aims to decompose a wave only into linear intrinsic components

and therefore any component that cancels out due to phasing has no physical meaning.

A f t e r analysing this effect for the different variants of E M D on Wave 2 (Figure 10), it appears that the SEMD contains enormous amounts o f non-intrinsic energy after decomposition. The E M D does rather well, but the E E M D returns the most intrinsic result with negligible amounts o f non-intrinsic energy around the peaks and only little flaws i n the zero-energy bands. Moreover, it is important to notice that the H H T is presumed to produce higher amounts of energy i n the low frequency band, as nonlinear bound waves are represented by modulations around the carrier frequency rather than by high frequency harmonics [28]. However, this theory should be employed carefully after the observation o f the E M D technique generating non-intrinsic waves with low frequency - even i n the case o f a linear wave model. These non-intrinsic waves w i l l also appear in the marginal spectrum, hence the high amplitude scale in the low-frequency domain of the spectrum should not regardless be labelled as nonlinear content.

Cumulative Fourier amplitude of all IMFs

0 0.05 0.1 0.15 0.2 0.25 0.3 f[Hz]

Figure 10. Fourier analyis of the different Empirical Mode Decomposi-tions on the Bi-modal Spectrum wave prove that non-intrinsic components are generated.

EEMD parameters

Although the E E M D process [22] involves a briefer sifting operation, some arbitrary parameters are introduced which again highly affect the result o f the E E M D :

NE: The number of E M D simulations that is used for the

ensemble average (typically 100 — 200).

s: The number of sift iterations that is performed to establish

an I M F (typically 1 0 - 2 0 ) .

Ns,d'- The standard deviation o f the added white noise,

relative to the standard deviation o f the signal (typically 0 . 5 - 2 . 0 ) .

The ensemble number NE is presumed to give better results for higher numbers, as a large number of simulations w i l l give a more stable average. However, cases have been found in which a relatively small ensemble - an 'unstable' E E M D solution - gives a better representation of the signal content than a stable solution obtained with a large ensemble.

After analysing the spectral results for different types o f waves the following correlations for A',^,^ and i have been noticed:

• The optimal Ns,d is positively correlated to the optimal s per IMF. Hence when choosing a high Ns,d, a large value of j is recommended.

• Optimal values for {s,Nstd} are relatively low for a regular signal and high for an irregular signal.

• Optimal values for {s,Nstd} are relatively low i n order to accurately transform the lower band of a spectrum and high to resolve the higher band.

It must be noted that these correlations are based upon only a few experiments, hence further investigations to confirm these correlations need to be carried out.

FREQUENCY SHIFTS DURING THE FREAK WAVE EVENT

In this section the Draupner wave [29] w i l l be analysed by examining the spectral evolution in the vicinity of the freak wave event. H H T involves the unique empirical decomposition result-ing i n a limited number of intrinsic components (IMFs). It has been observed in the above, that different outcomes result when employing different variants of the E M D . Therefore, the individ-ual behaviour o f these IMFs is also investigated. As the SEMD turned out to be a doubtful method, only the E M D and E E M D are compared.

Frequency shifts of individual IMFs

In Figure 11 a fragment of the Hilbert spectra in the vicinity of the extreme event is shown for both the E M D and E E M D . The E M D (Figure 2) with its extensive sifting procedure results in three substantial components {c\ — C3). However, after H T in the Hilbert spectra (Figure 11) only two distinct spectral

HHT (EMD)

0.2 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ H

-100 - 5 0 0 50 100 t[s]

Figure 11. Draupner wave: fragment of the Hilbert Spectrum after HHT with EMD and EEMD.

contours are observed, which means that I M F components by H T have been merged again. Before the extreme event both spectral components drop down in frequency and come together at about t = —25s. From this point the two components intrinsically grow by increasing their instantaneous amplitude and frequency.

The E E M D (Figure 4) decomposes the signal into only two

substantial components ( 0 3 , 0 4 ) . The Hilbert Spectrum after

E E M D gives a more distinct representation than the E M D . Three spectral contours (03 — C5) are clearly observed, o f which the frequencies of C4 and C5 interfere before the occurence o f the extreme event. The modulating component (C4) clearly has the highest amplitude, which further increases during the extreme event. However, during the extreme event also C3 attains a higher amplitude, whereas C5 slightly shifts down i n frequency.

On the one hand, when observing the I M F modes (Figure 2 and 4), it appears that linear focusing o f crests - certainly for the E E M D - plays an important role in the occurrence of the extreme event. On the other hand, the severe changes of amplitude and frequency indicate nonlinear mechanisms have a dominant contribution. The cumulative figures o f instantaneous amplitude and phase obtained f r o m the E M D (Figure 12) and E E M D (Figure 13) have been analysed. The cumulative phase indicates the randomness o f Unear focusing, whereas the cumulative amplitude represents intrinsic wave growth, which is presumably a nonlinear process. In the E M D results, during

the extreme event extreme focusing clearly occurs. However, the enormous peak of instantaneous amplitude at r = Os suggests that the intrinsic wave growth is more a unique event than the linear focusing. Similar observations and conclusions, but even stronger, were made based upon the E E M D results. The phasing during the extreme event i n this case is not a unique event at all as strong focusing occurs throughout time. These observations suggest that the Draupner wave is caused by (nonlinear) wave growth rather than linear focusing of waves. However, it should be mentioned that this is merely an interpretation of HHT, as it is most likely that some I M F components still consist of linearly focused components, which could not have been decomposed by the E M D / E E M D . The IMF, consisting of such linearly focused components, w i l l be transformed by H T into modulating frequency and amplitude as was seen for the bichromatic wave and Wave 2.

Cumulative instantaneous ampiitude

Cumulative instantaneous amplitude E, 20

--200 0 200 400 600 800 Cumulative instantaneous phase

-200 0 200 400 time [s]

600 800

Figure 12. Draupner wave: cumulative figures of the instantaneous am-plitude and phases of IMFs c i — cg obtained from EMD.

Spectral Evolution

In the above, severe evolution o f I M F components was noticed during the extreme event. Now the evolution of the complete spectrum w i l l be investigated. For both E M D (Figure 14) and EEMD (Figure 15) the spectral evolution f r o m one minute before up to one minute after the extreme event is calculated i n steps of 20s.

The marginal spectra are normalised to the maximum spectral level that is attained during the two minutes. Hence it is ob-served, that with E M D a substantial amount of energy emerges f r o m 40s before the extreme event, which has disappeared again

-200 0 200 400 600 800 Cumulative instantaneous phase

200 400 time [s]

600 800

Figure 13. Draupner wave: cumulative figures of the instantaneous am-plitude and phases of IMFs C3 - obtained from EEMD.

20s after the event. The sharp spectral peak readily shifts down to 0.03 - 0.04Hz. It is remarkable that the FT hardly presents any energy i n this band. The FT is presumed not to remove or relocate content, hence i t is concluded that the H H T and its flaws have severely down-shifted this energy peak, which therefore is an indication of either a strongly focused or highly nonUnear wave group.

Spectral evolution after E E M D looks rather different. I n i -tially at / = - 6 0 s the marginal spectrum looks similar to the reference Fourier Spectrum with two peaks representing swell (0.06Hz) and wind waves (0.07 - 0.08Hz). Accordingly, as the extreme event occurs, the spectrum has diverged into a small but distinct peak around 0.02 - 0.03Hz and a negatively skewed spectrum with its peak remaining between 0.07 - 0.08Hz. Remarkably the E E M D also shows significant energy being present at frequencies higher than 0.1 Hz. These humps between 0 . 1 2 - 0 . 1 3 H Z would be regarded as 2"''-order nonlinear sim-frequency harmonics, but the way H H T interprets nonlinearity

suggests these components are independentiy propagating waves at high frequency.

NEW PERSPECTIVE ON WAVES

The H H T transforms the wave signal i n a total different way than the Fourier-based methods. The E M D (or E E M D ) intro-duces the empirical character of the H H T by empiricaUy decom-posing the signal into a limited number of intrinsic modes rather than analytical functions. The effect of these intrinsic modes ap-pear to be significant i n the above examples, however the mean-ingfulness is still disputable. This section w i l l discuss how to

1 0.5 O t = -60 s t = -60 s 0.05 0.1 0.15 0.2 t = -40 s 0.05 0.1 0.15 t = 40 s 1 0.5 O 1 0.5 O 0.05 0.1 0.15 0.2 t = 6 0 s 0.05 0.1 0.15 f[Hz] 0.2

Figure 14. Draupner wave: Spectral evolution by HHT-EMD. marginal spectrum calculated witti a 60s-window, FT amplitude spectrum of the complete record (black-dotted) as a reference.

O 0.05 0.1 0.15 t = -40 S 1 0.5 O O 0.05 0.1 0.15 0.2 t = -20 s 0.05 0.1 0.15 t = 4 0 s 1 0.5 O 1 0.5 O

K,./v.

Figure 15. Draupner wave: Spectral evolution by HHT-EEMD. marginal spectrum calculated with a 60s-window, FT amplitude spectrum of the complete record (black-dotted) as a reference.

employ the H H T results i n getting further understanding of non-stationary and nonlinear waves such as freak waves.

Intrinsic waves

The essence of the intrinsic character of IMFs Ues in the sifting procedure that is used f o r the decomposition. This sifting procedure has appeared to be incapable of decomposing focused

components when their frequencies are close together. Neverthe-less, this 'inabiUty' also results i n the unique characterisation of a nonlinear wave, in fact being a focused series o f bound har-monics, which is represented as a single component modulating in frequency.

Hence the idea o f (nonlinear) waves being built of sinusoids with bound harmonics should be rejected when applying the H H T

with E M D . Waves are presuinably decomposed into indepen-dently propagating, intrinsic components by the linear decompo-sition of E M D . Accordingly these intrinsic components are pre-sumed to be monocomponent, however in practice w i l l often still contain several wave components.

A l l i n all, the E M D returns intrinsic wave components in which extreme events are caused by either nonUnear bound waves or strong linear focusing. Therefore the conventional distinction of Unear waves versus nonlinear bound waves can not be applied on the E M D result itself Nevertheless, examination of the intrin-sic waves on their instantaneous behaviour could provide more insights i n the mechanisms causing extreme waves f r o m a new perspective.

Instantaneousness

The instantaneous behaviour is determined by transforming the IMFs obtained from E M D by HT. The H T has perfect proper-ties for monocomponent signals, as discussed i n the comparison of H H T with conventional methods. However, interpretation of the analytical transformation of H T becomes problematic when wave signals include more than one periodic component. It has been observed that, certainly for realistic sea states including wave groups of many components, the amplitude response is blurred i n the spectnam and for narrow-banded wave groups modulations result.

The interpretation of these modulations deserves further re-search. Intra-wave modulations - i.e. a modulation of frequency within the intrinsic wave period of a component - have been associated with an assymetric or nonlinear wave profile. On the other hand, interwave modulations - i.e. long-period frequency modulations - are presumed to represent either spectral evolution (non-stationarity) or narrow-banded wave groups.

experiments to confirm the correlations which were found i n this study.

The E M D highly determines the result of the HHT. The de-composition with the sifting procedure becomes problematic when multi-component waves are strongly focused as is the case for narrow-banded wave spectra. The Hilbert Transform (HT), calculating a single instantaneous frequency and ampUtude for the subjected signal, w i l l return unstable (modulating) frequencies for such a multi-component signal. Moreover, the marginal amplitude of components is no longer preserved when H T is applied on a multi-component signal.

As modulating frequencies were presumably associated with nonlinearities the interpretation of HHT-results becomes ambiguous when narrow-banded wave groups are analysed. However, it is suggested that a new perspective on wave mech-anisms by studying the instantaneous behaviour of intrinsic waves be employed, rather than using analytical distinctions of nonlinear versus linear waves. For instance, the character of modulations (intra-wave versus interwave) could give more insight to the degree of nonlinearity and non-stationarity. Further study into how theoretical bi-modal wave spectra with wave groups of various bandwidths behave i n the HHT-results should give more insights i n how to interpret HHT-results of real measurements.

The analysis of the Draupner wave has shown that H H T characterises the rogue wave event predominantly by intrinsic wave growth. With E E M D the spectrum widens during the extreme event with the substantial amount of energy on higher frequencies. On the contrary, E M D returns a distinct energy peak shifting downwards i n frequency at the time of the extreme event. Furthermore, E M D retums a uniquely focused event, whereas with E E M D the linear focusing during the extreme event does not stand out compared to the rest of the wave record.

SUMMARY

The Hilbert Huang Transform (HHT) with Empirical Mode Decomposition ( E M D ) is a method that is capable of isolating nonlinear waves as a single wave rather than a carrier wave with bound harmonics. The H T calculates instantaneous frequency and amplitude such that it is highly able to detect non-stationary behaviour of nonlinear waves, provided they are monocom-ponent waves. To this end, the E M D aims to decompose a multicomponent wave into monocomponent Intrinsic Mode Functions (IMF). It was shown that the E M D is highly sensitive to parameter selection and also introduces non-intrinsic content due to end effects. The Ensemble E M D (EEMD) turned out to invoke less non-intrinsic content, but the decomposition of the E E M D is less extensive than that of E M D as a smaller number of substantial IMFs is decomposed by E E M D . Optimal parameter setting for E E M D has been suggested, however it needs further

ACKNOWLEDGMENT

This sfitdy has been performed i n the context of an MSc graduation project, which was supported by: Janou Hennig ( M A R I N , The Netherlands), Marcel Zijlema, Peter Naaijen and Pepijn de Jong (Delft University of Technology). Furthermore we would like to thank Sverre Haver (Statoil, Norway) for shar-ing the time trace of the Draupner New Year Wave, Karsten Trulsen (SINTEF, Norway) for sharing his E M D computation files and Norden Huang (National Central University, Taiwan) for sharing his E E M D computation files.

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