In ocean engineering designs, random extreme events are the most difficultly determined environmental load factors. Appearances of freak waves in the normal sea states are just such events. To reduce their potential destruction of marine structures, a forecast model is studied to predict whether freak waves occur in the wave propagation. In this model, the modified fourth-order nonlinear Schrödinger equation is employed as the deep-water wave model, which controls the evolution of complex wave envelopes. The measured wave train is set as its initial condition which needs to be changed into its corresponding complex wave envelope by use of the Hilbert transform method to be input into the wave model. During the evolution of complex wave envelope, its corresponding wave heights are estimated and compared with the definition of freak waves. If a freak wave is captured, its occurrence position and time are given. Three cases of observed, simulated and laboratory wave trains as initial conditions are performed to predict the generation of freak waves. Results show that measured wave trains can be simply and accurately input into this forecast model through the Hilbert transform method and this model can predict the generation of freak waves within some space and time in its traveling. In addition, larger groups than usual group height or length increase the probability of freak wave generation. This forecast model may provide marine activities with a safety warning in some open seas.

## Introduction

In the ocean, a freak wave is usually a high and steep giant wave, which can “suddenly” appear and “instantly” disappear in the background wave field. People's original knowledge about freak waves comes from the prevailing myths and legends of sailors. In recent years, freak waves have been believed to be the cause of an increasing variety of shipwrecks. In particular, for those marine casualties which have taken place in good weather on the open sea, many scientists have attributed them to freak wave events. To reduce their potential destruction of marine activities, many oceanographers have investigated the probability and mechanism of freak wave occurrences and discussed some relevant environmental factors of freak wave generations.

Yasuda and Mori (1997) analyzed the measured wave data from the surrounding Japan Sea, Chien et al. (2002) estimated the observed wave data from stations around Taiwan, researchers of the Max Wave project (Jenkins et al., 2002) processed the data from satellite radar monitoring of the global seas and Feng et al. (2014) analyzed wave heights from the Norwegian Sea. They considered that the wave height distribution of freak waves tended to be similar to Rayleigh or Weibull distributions, but were slightly different from either of them. Compared with ordinary waves, freak waves represent a high concentration of wave energy, so some researchers study the mechanism of freak wave formation from the viewpoint of energy focusing (Dysthe, 2000). Pelinovsky et al. (2000) suggested that temporal-spatial focusing, which comes from spatial distribution of component waves using dispersion and frequency modulation, could be used to explain some freak wave phenomena. Dysthe (2000) and White and Fornberg (1998) pointed out that spatial focusing due to refraction by bottom topography or current gradients was a well-known reason for freak wave generation. Stokes waves are unstable to sideband disturbances, whose frequencies deviate slightly from the fundamental frequency of carrier waves. This behavior of Stokes waves was called Benjamin–Feir instability (Benjamin and Feir, 1967), which can cause nonlinear focusing of waves to produce a very large wave. Based on this mechanism, Osborne et al. (2000), Osborne, 2001), Onorato et al. (2001, 2006), Shukla et al. (2006), Zhang et al. (2009a,b), Li et al. (2015) and Xia et al. (2015) studied performances and formations of freak waves and considered this instability as a possible mechanism of freak wave generation. In addition, ocean storms transferring energy to surface waves, wave-current interaction or soliton collision might arouse freak waves (Chien et al., 2002; Kharif et al., 2009).

Outcomes of the above investigations on mechanism, probability, or environmental factors of freak wave generation are all overviews. For a series of wave, whether its heights fulfill probability distribution of freak waves, its behaviors meet a certain mechanism of freak wave generation, or its external environments satisfy conditions of freak wave formation, any one of them is not enough to judge the generation of freak waves. Thus a question is put forward: How can we predict the formation of freak waves by measured wave data in the realistic sea states?

Considering the transient existence of freak waves, it once seemed inconceivable that men could predict their occurrences. However, Trulsen (2000) pointed out that freak waves likely do not suddenly appear from nowhere. In his study, he developed an iterative technique—namely a bandpass-filtered method—to extract the spectrum of linear free waves and exactly reconstruct the most energetic part of the complex spectrum. He employed the “New Year” freak wave as an initial condition and simulated its forward and backward evolution in space. Using the same method, Slunyaev et al. (2005) initialized four freak wave trains from the North Sea and gave their upstream and downstream propagation within a range of 500 m. Lifetimes and distances of these freak waves were determined by the simulation results. In these prediction evolutions, measured wave trains are not easy to be input the wave model because they need to be transformed into their corresponding complex envelopes by an iterative bandpass-filtered method.

Based on the above investigation of measured freak wave evolutions, we study an approach of forecasting freak wave generation which can be applied to measured wave trains. We employ the Hilbert transform method (Fan and Cao, 2012) to initialize three cases of wave trains and simulate their performances using the modified four-order nonlinear Schrödinger equation (Lo and Mei, 1987). In the evolution of wave trains, variation of wave heights are compared with the definition of freak waves (Klinting and Sand, 1987) recognized by many researchers. Once thresholds of this definition are met, appearance location and time of freak waves are given immediately.

## Numerical forecast model of freak wave generations

### Numerical scheme of wave evolution in deep water

Equations (1)–(4) contain a higher order nonlinear equation and can be solved efficiently by a split-step, pseudo-spectral Fourier method (Lo and Mei, 1985). This method is based on Fourier transform and employs the first-order complex wave envelope as the initial condition. Equations (2)–(4) can be solved analytically in Fourier space and then Equation (1) is solved by the finite difference method to receive the complex envelope at a forward space interval. Utilizing this method, you can get the evolution of complex wave envelopes in the whole space.

*c.c*. denotes a complex conjugate function.

### Initialization of the measured wave train

In signal processing, it is often necessary to extract envelope information. The measured wave train is also a type of signal. If it is input into Equations (1)–(4), its complex envelope information is also required. A common method of extracting the envelope information is application of the Hilbert transform, which is an all-pass filtering method that allows envelope information of signals to be picked up quickly and easily.

where represents the Fourier transform.

### Discrimination of freak waves

## Numerical experiments

### Experimental procedure

The measured wave trains are applied to the forecast scheme, and their water depth and bandwidth is required to meet the limit of the wave evolution model. Then they are initialized by the Hilbert transform and input into the governing equation. The energy conservation law is verified in the process of wave evolution.

### Experiment of the observed initial wave train

#### Initialization of the “New Year” wave train from the North Sea

At present, the most famous and comprehensive record of freak waves is the “New Year” wave which attacked the “Draupner” oil platform on 1 January 1995 on the North Sea (Haver, 2004). Its wave height was 25.6 m, its crest height 18.5 m and its period about 12 s, while the significant wave height was only 11.9 m.

Part of this wave train record is selected as an initial condition, which is about 480 s long. Its carrier frequency is given as . The spectrum of this wave train is analyzed, and its bandwidth is about 0.25, which is constrained to meet the bandwidth limitation of the numerical wave model. To reduce the numerical noise of wave evolution generated at the boundary, this wave train is processed periodically. Its complex envelope is reconstructed by the Hilbert transform method. Let the wave train be in deep water, and its dimensionless complex wave envelope evolved by the governing equation. The corresponding surface elevation is reproduced via Equation (6).

As shown in Figure 1, the surface represented by circles is the measured one, , and the solid line is the reproduced one, . To verify the constructed complex wave envelope, , the figure also displays the envelope curve of the reproduced surface elevation (dashed line). The rebuilt wave train almost completely coincides with the input wave train, and the envelope curve also fits the two wave trains perfectly. Figure 2 shows the real and imaginary part curves of this complex envelope.

Figure 3 shows energy spectrum curves of the input and reconstructed surface elevations, which are perfectly consistent.

#### Evolution of the “New Year” wave train and forecast of freak wave generation

The initial complex wave envelope constructed above is input into the numerical simulator and evolved along the wave propagation direction. Corresponding wave trains in space are shown in Figure 4. With the increase of evolution distance, the height of the “New Year” wave gradually decreases, and the shorter group with higher wave height develops to be the longer one with the relative lower height. Comparing evolved wave heights with the freak wave definition, no freak wave is found in the propagation of the initial wave train but only at its initial location.

### Experiment of the simulated initial wave train

#### Initialization of the simulated wave train

Suppose that the water wave is a stationary random process, its surface elevation can be obtained by the linear superposition of component waves with cosine forms. Let the sea states be described by the observed JONSWAP spectrum where peak frequency is 0.1 Hz and Phillips parameter and enhancement coefficient are 0.0162 and 4, respectively. Fifty frequencies are selected with the equal frequency interval in the frequency domain of this spectrum and 50 initial phases are set randomly in the domain.

The simulated surface series is shown in Figure 5. Its significant wave height is where is the 0-order moment of spectrum and its carrier wave frequency is set at . By means of the analysis of the wave power spectrum, the bandwidth of this wave train is 0.087, indicating a narrow spectrum. The Hilbert transform method is applied to this wave train to catch its complex envelope .

Figure 5 shows the input and reconstructed wave train and their corresponding envelope. The rebuilt wave train almost completely coincides with the input one, and the envelope curve also fits the two wave trains perfectly. Figure 6 shows the real and imaginary part curves and envelope shape of the complex wave envelope. Correspondence among them is very good. Figure 7 shows the energy spectrum curves of the input and reconstructed surface elevations, and they are perfectly consistent.

#### Evolution of the simulated wave train and forecast of freak wave generation

The evolution of the established dimensionless complex envelope is carried out by the deep-water wave model, and some corresponding wave trains are shown in Figure 8. Within the time domain from 0 s to 140 s, there is a longer wave group with higher wave height in this wave train. The long wave group contains three short wave groups on the left, middle, and right, with the height of the middle group being the largest, followed by the right. With the development of the wave train in space, three short wave groups gradually form, in which the left becomes lower and longer, and the middle and the right become higher and shorter (see free surfaces at ). When the wave train propagates to the middle group becomes shorter and lower and the right continues to be shorter and higher. Here the shorter group on the right contains a large wave, which does not satisfy the given freak wave definition at this point.

At each position, every wave in the wave train is compared with the freak wave definition, and the large wave in the shorter group grows into a freak wave at which is entirely consistent with the definition. This freak wave occurs between 100 s and 120 s after the initial wave train reaches this point, and then keeps on propagating until The huge wave travels a distance of about 3 m in 0.36 s. After that, the freak wave develops to become a shorter group including a large wave, see the surface elevations at 2570.48 m, 3084.57 m. At a freak wave is captured again, and holds a distance of about 255 m with a period of 32.66 s to reach the position Then the higher and steeper short wave group reduces its height, is gradually combined with the successive wave group, and becomes a group with a larger height and length similar to that of the initial wave train.

### Experiment of the laboratory initial wave train

#### Initialization of the laboratory wave train

The laboratory wave data came from wave focusing experiments in the State Key Laboratory of Coastal and Offshore Engineering of Dalian University of Technology. In these experiments, wave tank length and width are 50 m and 3 m, respectively, wave depth is 0.5 m and sample frequency is 0.02 Hz. Since the experimental dimensionless water depth is about 2, these experiments were performed in infinite deep water. To achieve good focusing waves, slope topography was set at the focusing point. These experiments in fact don't satisfy our numerical model and it is very difficult for us to obtain a random wave train to be able to generate freak waves, so we choose a wave train before the focusing point as the initial wave train to study the capture of freak waves.

The selected wave train is shown in Figure 9. Its significant wave height is 0.07376 m and spectral peak frequency is 0.998 Hz. Suppose that it is a deep-water wave train, we cut the part wave train within the focusing time region of 70–110 s as the initial inputting wave train. Its bandwidth is about 0.22, which meets the bandwidth limitation of the numerical wave model. The complex envelope of this wave train is also obtained by the above Hilbert transform method and input the numerical model to reproduce its input wave train. Figure 10 displays the laboratory wave train, reproduced wave train and corresponding wave envelope in which reproduced wave train are fit well with laboratory wave train.

Figure 11 shows the real and imaginary part curves and envelope shape of the complex wave envelope. Correspondence among them is good. Figure 12 shows the energy spectrum curves of the input and reconstructed surface elevations, and they are perfectly consistent.

#### Evolution of the simulated wave train and forecast of freak wave generation

The dimensionless complex envelope is input the deep-water wave model and the evolution of wave trains are shown in Figure 13. No freak wave occurs within the region before 2.37 m. At , a freak wave appears and propagate to which holds about 1 s. Then the freak wave evolves a high wave group. This wave group forms a freak wave in the region of 3.71–4.05 m which holds about 1.37 s. After that, only an instantaneous freak wave occurs at and no freak wave again appears until , the simulated ending point. In the process of freak wave generation, there is always a large wave group before and after their occurrences and their occurrences seem to be symmetrical about freak waves, such as wave groups at and , and , and so on.

## Discussions and conclusions

In this article, the Hilbert transform method is applied to a wave train to reconstruct its surface elevation and complex wave envelope. Constructed complex envelope is input into evolution equation, and every wave height in the corresponding surface elevations is compared with the freak wave definition. When a freak wave occurs, its generation location and time are given. Thus the forecast of freak wave generation for this wave train is realized.

Through three forecast simulations of measured, simulated and laboratory initial wave trains, the following conclusions can be drawn:

1) The Hilbert transform method is a simple and accurate method to extract the complex envelope information of wave train;

2) The combination of the Hilbert transform method and the modified four-order nonlinear Schrödinger equation can predict freak wave generations in the propagation of a wave train;

3) Occurrences of higher or longer wave groups may increase the probability of freak wave generation;

4) Though lifetimes of freak waves seem instantaneous, some of them still exist in a certain space and over a measurable timescale;

5) After the freak wave first appears in its wave train, it may occur once again with the evolution of the wave train.

While people often regard freak waves as rare events, it is not the scarce generation of freak waves, but the necessary wave conditions for their formation that are rare. If these conditions are met, there may be multiple occurrences of freak waves, such as freak waves presented in Case 2, freak waves on the southeast coast of South Africa, possible freak wave events successively attacking the Indonesian coast on 17 May 2007 (http://www.ce.cn/xwzx/gjss/gdxw/200705/20/t20070520_11420006_2.shtml), and so on.

In addition, this prediction simulation was carried out on the open sea in deep water without other external environment factors such as wind, flow, etc. considered, and the wave evolution model employed is limited to a narrow spectrum assumption. Therefore, the application of this forecast model presents certain restrictions. To obtain more realistic forecasts, this model still needs to be further improved.

## Acknowledgements

Statoil is acknowledged for permission to use the observed freak wave record from the North Sea.

## Funding

Partial support from the National Nature Science Foundation of China (Grant No.10902039, 41106031) is acknowledged, and support of the open topic (1215) from State Key Laboratory of Ocean Engineering in Shanghai Jiao Tong University is also acknowledged.

## References

*Rogue waves in the ocean*