## Abstract

The Hilbert–Huang transform method is used in this work to analyze wave records from the North Sea, which include abnormal waves. The analysis of the characteristics of abnormal waves is based on the local decomposition of wave data by using the intrinsic mode functions. The variations of amplitude of intrinsic mode functions around the time of the occurrence of abnormal waves are examined in order to investigate the contribution of different intrinsic mode functions on the abnormal waves’ profile, in an attempt to find relationships between them. The changes of local frequency of each intrinsic mode function in the vicinity of abnormal waves are examined. The essential nonlinearity of abnormal waves is considered as intrawave frequency modulation, a variation of the instantaneous frequency within one oscillation cycle. The Hilbert spectrum is used as a detector of abnormal waves.

## 1. Introduction

Abnormal, freak, or rogue waves are local, primary transient phenomena with uncertain and unpredictable occurrence. A while ago, Draper (1964) referred to the existence of freak waves, but until recently they were thought to rarely occur and very little was known about their characteristics. During the last several years, various research groups have studied freak, rogue, or abnormal waves. While in earlier publications these waves were designated as freak, later the designation of rogue waves was adopted. However, as these waves are being defined as the ones that are outside of the normal population of expected waves, they are also called abnormal waves. In general all these terms have been used by different authors to designate essentially the same type of waves.

Dean (1990) was probably the first who defined abnormal waves as the ones with a height that would be outside the normally expected ones within the established linear wave theory. Based on 20-min duration and the Rayleigh model for the probability distributions of wave heights, he concluded that in cases where the wave height was greater than twice the significant wave height, the wave would be considered abnormal. The value of this ratio, which was later called the abnormality or amplification index (AI), is thus related to the duration of a sea state and a probability level, although there is no strong reason why 2.0 was chosen instead, for example, of 1.9, 2.1, or a similar number. In fact, different authors tend to adopt different reference numbers, and a review of the various criteria being used can be found in Clauss (2002) and in Guedes Soares et al. (2003).

The study of this type of wave has been given a new impetus by the identification of such waves in full-scale wave records as reported by Haver and Karunakaran (1998), Mori et al. (2002), and Guedes Soares et al. (2003, 2004a), among others.

The effort to understand the nature of these waves has shown that various mechanisms can generate this type of very large waves as reviewed by Kharif and Pelinovsky (2003). In general, it is accepted that they are a result of nonlinear interactions that transform local wave groups in such a way that at a certain occasion one single large wave is formed and disappears after a while. Several authors, such as Trulsen and Dysthe (1997), Osborne (2000), and Slunyaev et al. (2005), have studied the propagation of wave groups using different mathematical theories that are able to explain how they develop into a very large wave.

An investigation of the local properties of sea waves is necessary in order to try to identify any mechanism that may develop prior to the creation of an abnormal wave and in this way contribute to the understanding of what happens in reality. The widely used Fourier spectral analysis is inappropriate to analyze the local conditions associated with the occurrence of abnormal waves, as it provides no useful information about time variation of wave characteristics. Also, in Fourier-based analysis, as well as in almost all of the conventional methods for wave data analysis, the wave process is assumed to be stationary and linear, while an abnormal wave is typically a nonstationary and nonlinear event. Furthermore, the parameters of water waves, determined by Fourier spectra, are integral characteristics of sea states and their local time variations are impossible to be tracked.

The time distribution of energy is needed, complementary to the frequency distribution for the detection of the occurrence of abnormal waves both in the frequency and time domains. There are several Fourier-based methods for frequency–time distribution of the energy like time window–width Fourier transform (Guedes Soares and Cherneva 2005), evolutionary spectrum, principal component analysis, or wavelet analysis (Liu and Mori 2000).

These methods suffer from the demerits of Fourier spectral analysis for the purpose of determining the characteristics of abnormal waves, such as the global definition of harmonic components and their linear superposition for decomposition of the data, thus assuming the stationarity and linearity of data. There is also a resolution problem inherent in all Fourier-based analysis due to the Heisenberg uncertainty principle. Wavelet analyses provide some improvement of the resolution by utilization of adjustable windows, but the problems caused by the nonadaptive nature of these windows and the leakage of energy have still not been completely solved.

Wavelet transform analysis has been used by Liu and Mori (2000) to analyze abnormal wave records, and although the abnormal wave is identified from a wavelet spectrum, there are some limitations of wavelet analysis, as discussed by Schlurmann (2002). The preliminary choice of a mother wavelet leads to some subjective, a priori assumption on the characteristics of the investigated phenomenon. Wu and Yao (2004) analyzed the laboratory-generated abnormal waves, interacting with currents, and stressed that the detection of higher, steeper freak waves using the Fourier spectrum is a challenging task.

Cherneva and Guedes Soares (2007) have adopted nonlinear spectral analysis and have shown that with the use of bispectrum it is possible to identify second-order nonlinear wave interactions in sea states with high skewness that can occur during severe storms. However, since the power spectrum suppresses all phase relations, it cannot be used to analyze the phase coupling. The nonlinear interactions between harmonic components induce certain phase relations that can be studied by the local frequency even in deep-water waves (Cherneva and Guedes Soares 2001), which was shown not to follow a uniform distribution in the sea states with abnormal waves.

An adequately chosen method for the analysis of abnormal wave records is necessary in order to obtain the correct knowledge of this transient event. A nonlinear and nonstationary method for data analysis is probably better suited for the study of the nonlinear and nonstationary nature of abnormal waves. This was the motivation to apply the Hilbert–Huang transform (HHT) method, a comparatively new method for nonlinear and nonstationary time series analysis to study abnormal wave records.

The HHT method, introduced by Huang et al. (1998), differs from all previous methods for nonstationary and nonlinear data analysis by its unique approach for data processing. The key part of this method is the empirical mode decomposition (EMD), which identifies the specific local time scales and extracts them into intrinsic mode functions (IMFs). The instantaneous frequency, determined by the Hilbert transform of IMF, provides a much sharper identification of the embedded events. The frequency–time distribution of the wave energy, designated as a Hilbert spectrum, provides an overall view of the variation of energy with time and frequency.

In comparison with wavelet and Fourier analysis, the HHT method offers much better temporal and frequency resolution. The Hilbert spectrum pointed to the local, not global, peculiarities of energy and is therefore a perfect tool for identification of abnormal wave occurrence in time and frequency. It stresses that the recent developments of the HHT method, namely, the confidence limit of EMD, the normalized HHT, and the statistical significance of IMF, presented in Wu and Huang (2004), Huang et al. (2003), and Huang (2005), which are adopted here, improve considerably the precision and reliability of the HHT method.

In this study, the HHT method for nonlinear and nonstationary time series analysis is used to study wave records measured in the North Sea, which included abnormal waves. The essential nonlinearity of abnormal waves is examined by the HHT method. The Fourier-based methods presented nonlinearity by introducing many harmonics to describe the distorted waveform. These harmonics are mathematically necessary but have no physical meaning. The nonlinearity of sea waves is considered by the HHT method as intrawave frequency modulation, which is a variation of the instantaneous frequency within one oscillation cycle. Wu and Yao (2004) found a strong correlation between the magnitude of intrawave instantaneous frequency modulation and the wave nonlinearity in their study of freak waves by the HHT method.

The characteristics of the largest waves in the storm sea states had been examined by Guedes Soares et al. (2004b) and new parameters, characterizing the asymmetry and steepness of abnormal waves, were proposed and examined by Guedes Soares et al. (2003). In this paper six of those time series are examined, four with single abnormal waves and the remainder with an abnormal wave imbedded in a group. As the various numerical studies referred to earlier have shown, single abnormal waves are the result of the evolution of a group, and it is considered that records that do not show a single wave but present a group are simply in a different state of the evolution process. The choice of the two types of wave situations aimed at verifying whether different characteristics would be apparent between the two types of situations.

For each of the six wave records used in this study, a confidence limit for EMD is initially determined according to Huang et al. (2003). The characteristics of abnormal waves are examined on the basis of local decomposition of wave data into IMF. The significance of the information extracted in different IMFs is verified by comparing the characteristics of each IMF with those of white noise. The variations of the amplitude of the IMFs around the time of occurrence of abnormal waves are examined in order to investigate the contribution of the different IMFs to the abnormal wave profile, and an attempt is made to find a relationship between them. The changes of local frequency of each IMF in the vicinity of an abnormal wave are examined. The instantaneous frequency is determined with precision by using the normalized HHT procedure. The peculiarities of the decomposition of an abnormal wave appearing in a group of several high waves and as an isolated single wave are examined here. The Hilbert spectrum is proposed to identify abnormal waves in wave records.

## 2. Hilbert–Huang transform method

The Hilbert–Huang transform method provides a novel approach for nonlinear and nonstationary data processing. The HHT consists of the EMD and the Hilbert spectral analysis. The EMD relies on data sifting according to the time scale. A time interval between successive extremes in a time series is defined as a time scale. This is the essence of the EMD. The main idea of EMD is first to identify the time scale that will reveal the physical characteristics of the process recorded as a time series and to extract them into IMFs. The EMD is a data-sifting process to eliminate locally riding waves as well as to eliminate locally the asymmetry of the time series profile. A procedure for EMD and several applications of the HHT method are presented in Huang et al. (1998, 1999).

The time series *X*(*t*) is first decomposed by EMD into a finite number *n* IMF *C _{j}*,

*j*= 1,

*n*, which extract the energy associated with various intrinsic time scales and residue

*r*. The superposition of the IMF and the residue reconstruct the data record:

_{n}An IMF is defined as a function, having the same number of extremes and zero crossings, and at any point the mean value of the upper envelope, defined by the local maxima and the down envelope defined by the local minima, is zero. These upper and lower envelopes are determined by using cubic splines. The important condition for an IMF is that only one maximum or minimum exists between successive zeros. The empirical mode decomposition is an iterative process where envelopes and their mean values are used to decompose the original data into frequency components, in sequence from the highest to the lowest frequency. The sifting process of the EMD is repeated until some conditions, known as a stopping criterion, are satisfied.

The stopping criterion is an important step in the HHT method in order to obtain physical meaningful IMF. Generally, there are many ways to decompose one given time series into different components. Even using EMD, the different sets of IMF can be obtained by changing the stopping criteria. The first stoppage criterion, proposed by Huang et al. (1998), is similar to the Cauchy convergence test, but later (Huang et al. 1999) it was replaced by the so-called simplified stoppage criterion, which on its side was revised by Huang et al. (2003) and substituted by a confidence limit for the EMD. The effect of the stopping criterion on the characteristics of the extracted IMFs was examined by Pascoal et al. (2005). The last criterion, that is, the confidence limit for the EMD, is applied here.

The set of IMFs is unique and specific for the particular time series, since it is based on and derived from the local characteristics of these data. The components of the EMD are usually physically meaningful, as the characteristic scales are defined by the physical data. The IMF could be considered as a more general case of the simple harmonic functions, but it can be claimed that, due to their specific derivation, IMFs also have a physical meaning in addition to the mathematical one.

In the second step of data analysis, the Hilbert transform is applied to these IMF:

where *P* indicates the Cauchy principal value. The amplitude *a _{j}*, the phase

*ϕ*, and the instantaneous frequency

_{j}*ω*are calculated by

_{j}which is considered as a generalized form of the Fourier expansion. Here both amplitude *a _{j}*(

*t*) and instantaneous frequency

*ω*(

_{j}*t*) are functions of time

*t*in contrast with the constant amplitude and frequency in the Fourier expansion. With the IMF expansion, the amplitude and frequency modulations are clearly separated.

The intrinsic mode functions, by definition, always have positive frequencies, because the oscillations in IMF are symmetric with respect to the local mean and consequently they admit well-behaved Hilbert transforms. The IMFs provide sharp identifications of imbedded structures in the data time series. The instantaneous frequency, determined by (5), is precise in both time and frequency domain; it breaks through the limitation of the uncertainty principle, inherited in the Fourier transform pairs or Fourier-type transform pairs, such as the wavelet transform. The previous attempts to determine the instantaneous frequency by performing the Hilbert transform directly from the original data time series (Melville 1983; Huang et al. 1992; Cohen 1995; Cherneva and Veltcheva 1995) led to negative local frequencies. The real advantage of the Hilbert transform for determination of the instantaneous frequency became obvious after Huang et al. (1998) introduced the EMD.

The frequency–time distribution of the amplitudes or squared amplitudes was designated by Huang et al. (1998, 1999) as a Hilbert amplitude spectrum or Hilbert energy spectrum, respectively. In this study, the Hilbert energy spectrum, determined as frequency–time distribution of the squared amplitude, is used. For simplicity, the Hilbert energy spectrum is denoted as Hilbert spectrum.

The HHT is a direct method for the analysis of nonlinear and nonstationary data, and its basis is a posteriori, adaptive, and may or may not be linear; the frequency is derived by differentiation rather than by convolution; therefore, it is not limited by the uncertainty principle.

The basic development of the HHT method has been followed by some improvements in the area of the normalized Hilbert transform and the statistical significance of the IMFs.

The Hilbert transform exists for any function, but the obtained phase function will not always yield a physical meaningful instantaneous frequency as discussed by Veltcheva and Guedes Soares (2004). In addition to the requirement of being an IMF, which is only a necessary condition, additional limitations have been summarized in two theorems, the Bedrosian theorem and the Nuttal theorem. To satisfy those requirements, Huang (2005) proposed the normalization of the IMF by dividing it by the envelope of the IMF. As an important measure of the correctness of the extracted IMF, a time variable error index EI is introduced as

where *A _{j}*(

*t*) =

*C*(

_{j}*t*)/

*R*

_{Cj}(

*t*) is the normalized IMF

*C*and

_{j}*R*

_{Cj}(

*t*) is an envelope of the

*C*IMF.

_{j}The statistical significance of the IMF is another important question. It is necessary in data processing to ensure confidence in the separation of the noise from the useful information. This question was addressed by both Flandrin et al. (2004) and Wu and Huang (2004) through the study of noise signals. Wu and Huang (2004) studied statistical properties of the scattering of the Gaussian white noise data and deduced a 99% bound for the white noise. They concluded that when a dataset is analyzed with EMD, if the mean period and rms values of IMF exist within the noise band, the extracted components most likely represent noise. On the other hand, the IMF, whose energy and period exceed the noise bound, is considered to contain statistically significant information.

## 3. Data

Six records of sea surface elevation containing abnormal waves are used in this study. This dataset was collected at the North Alwyn fixed steel jacket platform in the northern North Sea at a depth of about 130 m and has already been studied by Guedes Soares et al. (2004b). During the storm of 16–22 November 1997, 421 recordings using a laser altimeter of the northeast corner monitor were stored. Each record is 20 min at a sampling rate of 0.2 s.

The largest wave in the record appears as a very high wave in the vicinity of comparatively lower waves in four wave records. For convenience this type of abnormal wave is called a single abnormal wave. Four out of the six records contain this type of abnormal wave. Figure 1a presents a wave record (NA9711200151) with abnormal wave of about 375 s, which is well distinguished with large wave height and contrasts with the waves immediately before and after it.

The extreme wave in the other two wave records is accompanied by high waves that precede and succeed the extreme wave. The appearance of this type of abnormal wave is called an abnormal wave group and an example record (NA9711200731) is show in Fig. 1b. The highest wave in the record is surrounded by neighboring high waves. The profile of the abnormal wave is more symmetric relative to the zero level, if the abnormal wave appears as the largest wave in a group. The single large wave is characterized by a sharp crest with a height higher than the preceding or succeeding trough height.

## 4. Results of HHT data analysis

The EMD method was applied to six wave records and an IMF set was determined for each of them. First, the confidence limit for the EMD is determined. The sifting process of EMD was stopped when the number of extremes equaled the number of zero crossings for *S* successive sifting steps. Here EMD was performed for 10 selected numbers of stopping number *S*. The squared deviations (sds) of the individual cases from an ensemble mean for the 10 Hilbert spectra of records (NA9711200151 and NA9711200731) are presented in Fig. 2.

The orthogonality of the obtained basis is quantified as the orthogonal index (OI)

and is presented as a function of the stopping number *S* also in Fig. 2. The OI provides a criterion to reject those IMF sets that are nonorthogonal.

The range of stopping number *S*, where the squared deviation has a minimal difference from the computed ensemble mean, is determined for each record. For the record NA9711200151, a minimal sd range of parameter *S* is 7–10, while for record NA9711200731 it is 8–10. The OI also has low values for these values of *S* and thus the obtained sets of IMF could also be considered orthogonal for these values of *S*. All wave records are first analyzed in this way to determine the confidence limit for the empirical mode decomposition.

The detailed results of the decomposition of record NA9711200151 into 10 IMF and the residue are presented in Fig. 3, and the sea surface elevation data are shown in Fig. 3a. The sifting is performed for *S* = 9 successive times.

The information extracted in each IMF is checked in order to estimate its statistical significance. Using the relation between energy and the period of components of decomposition of white noise obtained by Wu and Huang (2004), each IMF is checked according to 99% white noise boundary. The results of the significance test for the records NA9711200151 and NA9711200731 are presented in Fig. 4. The energy of the first component *C*_{1} is below the 99% boundary so it can be considered that *C*_{1} contains little useful information and it can be assumed that *C*_{1} comes from noise, which is valid for most records. On the basis of this assumption, the energy of *C*_{1} can be assigned on the 99% line and it can be used to rescale the rest of the IMF. The results of these tests are shown by dots in Fig. 4.

The statistical significance of the decomposition of the records [NA 9711200151 (single) and NA9711200731 (group)] shows that according to a priori test four IMFs contained statistically significant information: *C*_{2}, *C*_{3}, *C*_{4}, and *C*_{5}. Under the rescaled posteriori criterion, however, *C*_{1} and *C*_{6} also contain some useful information since all of the first six IMFs are bordering the statistical significance at the 99% confidence limit.

The energy and time characteristics of individual IMF are estimated and compared with wave characteristics in order to estimate quantitatively the contribution of different IMFs to the wave data. The zeroth moment *m*^{Cj}_{0} is proposed as a measure of integrally determined energy of *C _{j}* IMF, while the peak frequency

*f*of spectrum

_{p}*C*is used as the representative frequency.

_{j}The characteristics of records NA9711200151 and NA9711200731 and their IMFs are presented in Table 1. The energy of the IMF with index higher than five is significantly lower than the energy of the first five IMFs. From an energetic point of view, the first five or six IMFs are the main contributors to the energy contents of these wave data. The third IMF *C*_{3} has the highest energy in the decomposition of record NA9711200151 with a single abnormal wave. The peak frequency of the *C*_{3} IMF is 0.088 Hz.

The record NA9711200731 with an abnormal wave group is characterized with a dominant *C*_{2} IMF, followed by the third *C*_{3} IMF. The peak frequency of *C*_{2} and *C*_{3} components are close to each other, respectively, 0.098 and 0.088 Hz. The presence of two dominant IMF components with close frequencies of oscillations in the decomposition of wave records was found to be related to wave groupiness by Veltcheva (2002). The energy hierarchy of the IMFs in the decomposition of a wave record clearly reflected the specific peculiarities of the particular wave data as also obtained by Veltcheva and Guedes Soares (2004).

Next, the peculiarities of the decomposition are examined locally around the time of the abnormal wave occurrence. The zoom of the IMF variation around the time of abnormal wave occurrence is shown in Fig. 5. The third IMF, *C*_{3}, shown by the dashed thick line in Fig. 5a, has the largest amplitude among the IMF; *C*_{3} has also the highest energy in the decomposition of the whole record. There is a kind of focusing of the first four IMF at the crest of single abnormal waves at *t* = 375 s (Fig. 5a). The crest of the *C*_{3} IMF coincides with the crest of the *C*_{2}, *C*_{4}, and *C*_{1} components. In the vicinity of wave troughs, preceding and succeeding the crest of the single abnormal wave, the second IMF *C*_{2} is out of phase with *C*_{3} and *C*_{4}. The decomposition of record NA9711200731 around the time of the abnormal wave group occurrence is presented in Fig. 5b. The second *C*_{2} IMF (dotted line) has the largest energy in the decomposition, followed by the *C*_{3} IMF (dashed thick line). The decomposition of the abnormal wave group in Fig. 5b does not present such a type of focusing of different IMFs at the time of occurrence of the wave crest of the abnormal wave. The second IMF *C*_{2} is in phase with the abnormal wave, but a shifting of *C*_{3} and *C*_{4} can be observed. These arrangements of the phases of *C*_{2}, *C*_{3}, and *C*_{4} contributed to the reconstruction of the specific peculiarities of the two types of abnormal waves. Figure 6 presents the reconstruction of the profile of the abnormal wave by adding up the first six IMFs, in ascending order, starting with the IMF with the highest energy. The second and third components are the main contributors for the reconstruction of the abnormal waves as the sum of *C*_{2} and *C*_{3} is close to the wave profile. However, for the complete reconstruction of the abnormal wave profile all components are necessary. The arrangements of being in and out of phase of *C*_{2}, *C*_{3}, and *C*_{4} in Fig. 5a resulted in correct reconstruction of the larger sharp crest and flat troughs of the single abnormal wave in Fig. 6a. For the case of the abnormal wave group (Fig. 6b), the phase-shifted *C*_{2}, *C*_{3}, and *C*_{4} reconstruct the observed more symmetric wave profile of abnormal wave at *t* = 373 s.

The variations of local frequency within the vicinity of the abnormal wave are examined. The instantaneous frequency is determined by the normalized HHT and the correctness of its estimation is checked by means of the EI. The time variation of the instantaneous frequency of the second *C*_{2} and third *C*_{3} components, which have the largest energy in the decomposition of records NA9711200151 and NA9711200731, are shown in Figs. 7b,e and 8b,e, respectively. Additionally, the sea surface elevation and corresponding IMF are shown for reference in Figs. 7a,d and 8a,d, while Figs. 7c,f and 8c,f present the time variation of the EI of each IMF. Generally, the estimated local frequency of different IMF in the area of the abnormal wave is correct as the EI has small values.

The asymmetrical waveform of the single abnormal wave shown in Fig. 7a is accompanied by increasing the instantaneous frequency of the second IMF (*C*_{2}) up to 0.45 Hz in Fig. 7b. The local frequency of *C*_{3} in Fig. 7e also varies within the time of occurrence of the abnormal wave but not so much. The variation of local frequency of the second IMF (*C*_{2}) (Fig. 8b) within the abnormal wave group is small, while the frequency of *C*_{3} (Fig. 8e) does not change at all during the abnormal wave occurrence. Quantitatively, the intrawave frequency modulation can be estimated by the difference between the instantaneous frequencies of IMF, which contribute mainly to the profile of the abnormal wave, as suggested by Wu and Yao (2004). The difference Δ*ω*_{23}(*t*) = *ω*_{2}(*t*) − *ω*_{3}(*t*) between the instantaneous frequency *ω*_{2}(*t*) and *ω*_{3}(*t*) of the second and third IMF has been verified. From Figs. 7b,e the difference Δ*ω*_{23} for a single abnormal wave is 0.32 Hz, while for an abnormal wave group (Figs. 8b,e) Δ*ω*_{23} it is 0.11 Hz. The larger intrawave frequency modulation is associated with a higher asymmetry of the profile of the single abnormal wave in contrast to the more symmetrical abnormal wave in the group. Depending on the mechanism of generating abnormal waves, Wu and Yao (2004) reported a different trend in the correlation between Δ*ω*_{23} and wave steepness.

The peculiarities in the amplitude and frequency variations of the IMF are reflected in the frequency–time distribution of the energy. The Hilbert spectrum *H*(*ω*, *t*) of the record NA9711200151 is shown in Fig. 9a, while the *H*(*ω*, *t*) of record NA9711200731 is presented in Fig. 10a. The 3D plot of the zoomed Hilbert spectrum in the vicinity of the abnormal wave is shown in Figs. 9b and 10b. The color bars next to the diagrams indicate the energy scale.

The abnormal waves are clearly shown in time and frequency as the maximum energy of the record. An interesting peculiarity is traced by detailed examination of the zoomed Hilbert spectrum. The instantaneous frequency increases considerably during a single abnormal wave around 375 s in Fig. 9a as the variation of the local frequency is the highest for the whole record. This is an indicator of the intrawave frequency modulation.

The Hilbert spectrum of the abnormal wave group in Fig. 10a also has a very well distinguished peak, but without high frequency variations. Liu and Mori (2000), analyzing the wavelet spectrum of abnormal wave data, also observed that the symmetrical waveform of an abnormal wave does not produce energy in the high frequency range in contrast with the wavelet spectrum of steep abnormal waves. The intrawave frequency modulation, a variation of the local frequency within one wave, is connected with the large asymmetry of the profile of a single abnormal wave and produced the sharp increase of local frequency.

The Hilbert spectrum *H*(*ω*, *t*) provides an overall view of the variation of energy with time and frequency and points to the local time–frequency distribution of the energy. The EMD is a nonlinear decomposition and the Hilbert spectrum contains all the nonlinear waveform distortions, such as intrawave frequency modulation. This nonlinear representation does not need the harmonics to fit the waveform.

## 5. Conclusions

The Hilbert–Huang transform method for nonlinear and nonstationary time series analysis is used in this study to investigate abnormal waves. The characteristics of abnormal waves are examined on the basis of local decomposition of wave data into IMF. The confidence limit for EMD is determined first and the significance of information, extracted in a different IMF, is verified in detail.

Two types of abnormal waves, isolated single wave and abnormal wave group, produce a different type of decomposition. The third IMF (*C*_{3}) has the highest energy in the decomposition of the record with single abnormal waves, while the record with abnormal wave groups is characterized with two dominant IMFs (*C*_{2} and *C*_{3}) with close peak frequencies, a fact related to the wave groupiness.

The peculiarities of the decomposition are examined locally around the time of the abnormal wave occurrence. The profile of the abnormal wave is reconstructed by summing up the first six IMFs in ascending order starting with the IMF with highest energy in the decomposition. The dominant components are the main contributors for the reconstruction of abnormal waves as their summed total is close to the wave profile.

The changes of local frequency of each IMF in the vicinity of abnormal waves are examined. The larger intrawave frequency modulation is associated with a higher asymmetry of the profile of single abnormal waves in contrast to the more symmetrical abnormal waves in a group.

The Hilbert spectrum easily detected the appearance of abnormal waves both in the time and frequency domains. The instantaneous frequency increases considerably during a single abnormal wave occurrence as the variation of the local frequency is the highest for the whole record, while the Hilbert spectrum of the abnormal wave group has very distinguished peaks due to the occurrence of the abnormal wave but without intrawave frequency modulation. The asymmetry of abnormal waves is correlated with the magnitude of intrawave frequency modulation.

## Acknowledgments

The data used in this study have been obtained during the project “Rogue Waves—Forecast and Impact on Marine Structures (MAXWAVE),” which was partially funded by the European Commission under Contract EVK3-CT2000-00026.

This study has been partially financed by the Portuguese Foundation for Science and Technology (FCT) under the Pluriannual funding to the Unit of Marine Technology and Engineering.

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## Footnotes

* Current affiliation: ECOH Corporation, Yokohama, Japan

*Corresponding author address:* C. Guedes Soares, Unit of Marine Technology and Engineering, Technical University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal. Email: guedess@mar.ist.utl.pt