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Field Measurements of Rogue Water Waves [Journal of Physical Oceanography]
[September 16, 2014]

Field Measurements of Rogue Water Waves [Journal of Physical Oceanography]


(Journal of Physical Oceanography Via Acquire Media NewsEdge) ABSTRACT This paper concerns the collation, quality control, and analysis of single-point field measurements from fixed sensors mounted on offshore platforms. In total, the quality-controlled database contains 122 million individualwaves,ofwhich3649areroguewaves.Geographically, themajorityofthefieldmeasurementswere recorded in the North Sea, with supplementary data from the Gulf of Mexico, the South China Sea, and the North West shelf of Australia. The significant wave height ranged from 0.12 to 15.4m, the peak periodranged from 1 to 24.7s, the maximum crest height was 18.5m, and the maximum recorded wave height was 25.5m. This paper will describe the offshore installations, instrumentation, and the strict quality control procedure employed to ensurea reliable dataset. An examination of sea state parameters, environmental conditions, and local characteristics is performed to gain an insight into the behavior of rogue waves. Evidence is provided to demonstrate that rogue waves are not governed by sea state parameters. Rather, the results are consistent with rogue waves being merely extraordinary and rare events of the normal population caused by dispersive focusing.



(ProQuest: ... denotes formulae omitted.) 1. Introduction A rogue water wave is an unexpected large individual wave given the underlying sea state in which it arises. Rogue waves are also referred to as freak or abnormal waves, and they are important for several reasons. One of the key design criteria of an offshore structure is the elevation of the deck. Placing the deck too low will in- undate the superstructure and lead to enormous wave- in-deck loads, while a deck that is too high may lead to a prohibitively expensive structure. New offshore struc- tures are designed to avoid wave-in-deck loads by main- taining an effective air gap. In determining the deck elevation, it is necessary to have a good understanding of the short-term variability of the crest height, which will be influenced by the presence of rogue waves. The unexpectedness of a rogue wave can often be more dan- gerous than the amplitude itself. For example, mariners may interpret an interval of calm as a decreasing sea state (Gemmrich and Garrett 2008), only to be suddenly hit by a large wave event. In addition, maintenance on an offshore platform may be sanctioned in a given sea state without the consideration of an unexpected rogue wave event. Therefore, rogue waves have important health and safety implications.

There have been numerous observations, photographs, and mariners' tales of rogue waves from ocean-going vessels (Nikolkina and Didenkulova 2011). The stories often recount deep holes opening up in the ocean and walls of water (Gibbs and Taylor 2005). These are very useful for postulating theories of the physical mechanism behind rogue waves. However, without actual instrument measurements of the water elevation, any attempts to quantify these effects are difficult to believe. Indeed, for some time the occurrence of rogue waves was thought to be a nautical myth. However, the measurement of the Draupner New Year wave in 1995 provided clear evidence that rogue waves actually exist (Haver and Anderson 2000). It is important to note that the Draupner New Year wave was associated with wave loading on the underside of the deck, which provided corroborating evidence of the height of the crest and ruled out the possibility of instrument measuring error.


The Draupner New Year wave has been analyzed in great detail (Walker et al. 2004; Cherneva and Guedes Soares 2008; Adcock et al. 2011; Clauss and Klein 2011). While it was a significant event, one must bear in mind that it is only a single rogue wave and concrete conclu- sions of a stochastic system such as the ocean cannot be drawn from a sample of one. However, obtaining a large sample of rogue waves is inherently difficult because they are such rare events and field measurements are generally only recorded at single points in the vast ocean. Indeed, Haver and Anderson (2000) have commented that the scarcity of rogue wave measurements makes it difficult to determine whether they conform to standard statistical models.

To address this, it is necessary to accumulate a very large database of field measurements and hope that it will capture some rogue wave events. However, this necessitates storing the raw time history of the water surface elevation. In the past, only the sea state pa- rameters were saved and the raw measurement of the water surface elevation was discarded due to computa- tional storage constraints. Indeed, rogue waves were also often removed from the time trace, as they were consid- ered to be anomalies due to instrument errors. In recent times, with an abundance of cheap computational stor- age, the time history of the raw wave measurements is also saved. This step change has permitted the detailed analysis of rogue waves, without which the present study could not have been undertaken.

As field measurements are associated with instrument errors, a vast amount of quality control (QC) is required. To produce a reliable database, a strict QC procedure is necessary; however, this has the undesired effect of re- ducing the amount of data available for analysis. Con- sequently, the only way to form both a large and reliable database is to collect vast amounts of raw wave data. The importance of a reliable database that can be trusted cannot be stressed enough; no matter how sophisticated the analysis, an unreliable database will lead to wrong conclusions.

The present study describes the formation of such a large and reliable database and its subsequent exam- ination. This paper continues in section 2 with a brief summary of the background on rogue waves and field measurements. For a more detailed review, the reader is directed to Dysthe et al. (2008). Section 3 describes the offshore installations, instrumentation, and quality control procedure for the field measurements. An ex- amination of the influence of sea state parameters, environmental conditions, and local parameters on the formation of rogue waves is then presented in section 4. Conclusions are discussed in section 5.

2. Background There are several definitions of a rogue wave within the literature; however, all are based on the wave height or crest height being larger than some multiple of the significant wave height. The present study defines a rogue wave within a 20-min sample according to Haver (2000): ... (1) where hc is the crest height (defined as the maximum elevation between zero crossings), H is the wave height, and Hs is the significant wave height, which is the value based on the zeroth spectral moment m0. Note that ei- ther the crest or wave height criteria by themselves are sufficient for the definition of a rogue wave; there is no requirement that both of these criteria are satisfied such as given by Tomita and Kawamura (2000).

A substantial effort has been made to try to explain the occurrence and nature of extraordinarily large wave events. Two almost contradictory mechanisms have been offered for crests substantially higher than pre- dicted by design codes. The first is that the dynamics of unidirectional or very long-crested waves are qualita- tively different from directionally spread waves. The fully nonlinear calculations of Gibson and Swan (2007) show that in unidirectional seas the largest waves can be much higher than would be predicted by second-order theory. Several studies based on experimental mea- surements support this conclusion (Onorato et al. 2006; Fedele 2008; Cherneva et al. 2009, 2013; Latheef and Swan 2013). On the other hand, Donelan and Magnusson (2005) have suggested that crossing seas might lead to ex- ceptionally high crests. Second-order simulations (Toffolli et al. 2006; Christou et al. 2009) do show slightly higher crests in crossing seas. As such, field measurements in a variety of environments must be studied to provide evi- dence of physical mechanisms creating rogue wave events.

One problem with field measurements is that most routine and operational wave measuring programs em- ploy surface-following buoys. Although these provide reliable estimates of the usual sea state parameters, such as significant wave height and wave periods, they tend to underestimate the crest height of individual waves (Krogstad and Barstow 2000; Casas-Prat and Holthuijsen 2010). Furthermore, in the extreme cases, it is thought that the buoy may be dragged through the crest or skirt around a short-crested wave (Seymour and Castel 1998). There have been some large databases of field measure- ments recorded by surface-following buoys. Pinho et al. (2004) measured 1.2 3 106 waves in the Campos basin offshore of Brazil, and Casas-Prat and Holthuijsen (2010) analyzed 10 3 106 individual waves measured off the Catalan coast in Spain.

With 122 3 106 individual waves, the present study has an order of magnitude more data than Casas-Prat and Holthuijsen (2010). Furthermore, the present database was collected from fixed instruments on offshore plat- forms, which are capable of accurately measuring crest heights. Dysthe et al. (2008) commented that measure- ments from fixed instruments give the most reliable in- formation on rogue waves. Indeed, this is the reason why only data from fixed instruments have been collected in the present study. As far as the authors are aware, the largest database of raw wave data from fixed instru- ments was collected by Olagnon and van Iseghem (2000), who measured 1.6 3 106 waves using a Plessey wave radar on the Frigg platform in the North Sea. Therefore, the present database is two orders of mag- nitude larger. As mentioned in the introduction, a large and reliable database is essential to be confident of the conclusions on rogue wave behavior.

3. Description of field measurements a. Offshore installations Raw measurements of the water surface elevation were gratefully provided by the following sponsors of the Cooperative Research on Extreme Seas and Their Impact (CresT) Joint Industry Project: Anardarko, BP, ConocoPhillips, Shell, Statoil, Total, and Woodside. In addition, there were sources of data from the Wave Crest Sensor Intercomparison Study Joint Industry Project (WACSIS JIP) and the public domain. The majority of the data were recorded from fixed jacket structures. However, in a minority of cases the mea- surements were taken from tension leg platforms (TLPs) or spars, for which the raw wave measurement was ad- justed to account for the recorded motion of the floating platforms. In total, there were 22 installations in the North Sea, 5 in the Gulf of Mexico, 5 in the South China Sea, and 1 on the North West shelf of Australia. The source, geographic location, platform name, water depth, and sensor type for the whole dataset are presented in Table 1.

As far as the period of the dataset is concerned, the earliest sample is from August 1969 and the latest is from April 2008; however, this is by no means continu- ous. Most of the raw wave samples from the North Sea and the South China Sea (which represent the majority of the dataset) are continuous over a span of 3 yr. The datasets from the Gulf of Mexico and the North West shelf of Australia are sporadic, and the data provided were only recorded during large storms and hurricanes (or tropical cyclones). The installations were located in mean water depths ranging from 7.7 to 1311 m.

b. Instrumentation The majority of the data were recorded on Shell in- stallations using the Saab radar, which accounted for approximately 94% of the measured wave samples in the whole database. For that reason, this section will principally describe the Saab WaveRadar REX and only briefly mention the other sensors employed.

The Saab WaveRadar REX (hereafter referred to as Saab radar) is manufactured by Saab Rosemount, and it was first introduced in 1994 as a derivative of their TankRadar system. It uses a low power (,0.5 mW) frequency, modulated (9.7 to 10.3 GHz, with a linear sweep), continuous microwave signal. The transmitted signal changes frequency during the time taken for the reflected signal to return. The frequency of the received signal reflected from the surface is compared to the transmitted signal frequency, and the difference in fre- quency is proportional to the distance to the surface of the water. Digital filtering and FFT calculations are used to counter disturbances and maintain measurement ac- curacy. During the recording cycle of 10 Hz, a number of measurements are produced and an average distance recorded. The manufacturer specifications quote an ac- curacy that depends on the range, which is between 3 and 65 m from the TRL/2 adaptor (the measurement datum). For a range less than 50 m, the accuracy is 66 mm, whereas for a range greater than 50 m, the ac- curacy is 612 mm. The beamwidth has a 108 included angle. For the installations in the present study that employed the Saab radar, the distance from the in- strument to the mean water level ranged between 14.2 (South China Sea) and 34.3 m (North Sea); conse- quently, the accuracy is 66 mm. In addition, because of the antenna beam pattern and subsequent signal pro- cessing, the target footprint on the ocean surface is generally constrained to 628. The manufacturer claims that every Saab radar is calibrated by Rosemount in a special calibration facility. Because of the design and construction of the electronic and microwave unit, the Saab radar calibration is extremely stable and periodic recalibration is not required. The Marex radar and Endress1Hauser FMR130 radar were also employed as instrumentation on some of the offshore installations. These are both based on similar principals as the Saab radar.

With 6.2% of the total waves recorded in the data- base, the laser sensors were the second most abundant type of instrument used for the present field measure- ments. The EMI laser is a pulsed range finder operating in the near-infrared region. Narrow pulses of light are produced by a laser diode, and the radiation from the target is used to stop a time interval measurement. The time of travel is converted to an analog voltage pro- portional to the distance to the reflector. The response of the optical unit is 10 to 15 Hz, but the output is usually filtered by a 2-Hz Butterworth filter to eliminate high- frequency noise. Angevaare (1986) specifies that the in- strument can measure to within an accuracy of 0.1 m for wave heights between 0 and 10 m and to within 1% for wave heights between 10 and 50 m. The transmitter has a maximum beamwidth of 18, and therefore the EMI laser has a much narrower footprint than the radars. However, Crabb et al. (1983) report measurements made from a platform in Christchurch Bay with an EMI laser, during which the sensor was affected by fog, occasionally dem- onstrating a low-amplitude noise-dominated signal due to fog (backscatter), but independent of wave height. The other laser employed was the Optech laser, which is a pulsed infrared laser and can measure wave height in the range of 0.3 to 50m with an absoluteaccuracy of 2cm and operational accuracy of 5 cm.

A typical instrument for wave measurements from platforms in the Gulf of Mexico is the Baylor wave staff, which consists of a pair of stainless steel wire ropes separated by insulators about 20 cm long. The transducer measures the natural frequency of the inductive loop made by the two wires and the sea surface, from which the length of the loop is found. The instrument is robust and relatively immune to fouling.

Finally, the Vlissingenbaak step gauge is a wave and tide measuring system produced by Etrometa Holland and is based on the measurement of conductivity to sea- water of equally spaced electrodes. It consists of a 15-m- long metal pole with electrodes isolated from the pole and connected via a cable connector combination to the electronic unit where the highest wet electrode is de- tected and where a check is performed on defective electrodes. Measurements are performed at 10 Hz.

An important consideration for the instrumentation is their location on the platform, such that they measure only the undisturbed wave field (Forristall 2005). As such, the instruments were placed as far away as feasibly pos- sible from the platform legs and upwave of the dominant mean wave directions.

c. Quality control of raw wave data Field measurements are associated with instrument errors such as lock-ins, dropouts, and spikes. To produce a reliable database that can be trusted, a strict QC pro- cedure is necessary to identify and remove these errors. As mentioned above, the quality of the database is es- sential, and no matter how sophisticated the analysis performed, poor data will lead to incorrect conclusions. The first step of any automatic QC procedure for raw wave data consists of defining a series of error flags. The number of occurrences of these error flags within a wave sample then determines whether the sample passes or fails the QC check. The field measurements were first divided into 20-min wave samples in order to ensure stationarity. Then for each sample, the error flags de- scribed below were employed: Flag 1: Number of occurrences of 10 consecutive points of equal value.

Flag 2: Number of zero upcrossing wave periods longer than 25 s.

Flag 3: Number of occurrences that the limit rate of change is exceeded. The limit rate of change of the water surface elevation Sy is given by Sy 5 (2ps/Tz ) 2lnNz , where s is the standard de- viation, Nz is the number of zero upcrossing periods, Tz 5 N/(srNz) is the mean zero upcrossing period, with N being the number of points in the time history of the surface elevation and sr being the sample rate.

Flag 4: Number of consecutive errors of flag 3.

Flag 5: Number of crest elevations greater than 5 times the standard deviation of the water surface elevation.

Flag 6: Number of consecutive errors of flag 5.

Flag 7: The amount of energy in the spectrum below 0.04 Hz is greater than 5%.

Flag 8: The amount of energy in the spectrum above 0.60 Hz is greater than 5%.

During operations at offshore installations, the prin- ciple quantities of interest are the sea state parameters and the maximum wave and crest heights. Consequently, all the error flags above can occur a predefined number of times before a raw wave sample is rejected. However, as the present study is performing a detailed analysis of the raw wave data, it was decided to use a very strict auto- matic QC procedure. This involved rejecting raw data that contained any error flags, and consequently the wave samples that passed did not contain any of the errors described above. This ensured a high-quality dataset that is very reliable, which can be interrogated to test any particular hypothesis, and the results can be interpreted with a high level of confidence that the underlying data are error free. In doing so, it was found that flag 3 was too strict, and therefore the double temporal derivative of the water surface elevation was also checked against Sy, and only when both were larger than the limit did the sample fail.

With the error flags defined and the very strict method employed, the key steps of the QC procedure are as summarized below: (i) Divide the raw wave data into 20-min samples, demean the time trace, and calculate the parame- ters (H, hc, Hs, Tp, ...) using the zero upcrossings method.

(ii) Determine whether or not the sample contains a rogue wave according to the definition of Haver (a2r0o0g0u).ewaveaccordingtothedefinitionofHaver (iii) There are now two paths that can be followed: 1) Normal wave sample-If there are no rogue waves in the sample, apply the very strict auto- matic QC method described above. If the sample passes, then further analysis can be performed; a failed sample is discarded.

2) Rogue wave sample-If there is a rogue wave in the sample, first apply the very strict automatic QC method, but without imposing flag 5, as this would automatically fail due to the large crest elevations. If the sample passes, then it moves on to a visual QC check; a failed sample at this stage is discarded. The visual QC check con- sists of looking at the water surface elevation of the rogue wave and deciding if it looks plausible. If the sample passes, further analy- sis can be performed; a failed sample is dis- carded.

Clearly, the visual QC check performed on the rogue waves is a highly subjective criterion, but there was no automatic procedure that could be applied with confi- dence. Furthermore, one could argue that the shape of rogue waves is not known for certain. However, during the process of looking through vast quantities of raw wave data from numerous installations around the world, the authors accumulated good experience of detecting the typical errors of the various wave sensors. This was especially useful in performing the visual QC checks of the rogue waves. The quality control pro- cedure has been both extensive and exhaustive.

Overall, 82% of the normal wave samples passed the QC checks, leading to 528 475 20-min sea states. Whereas only 16% of the rogue wave samples passed, leading to 3649 20-min sea states. This created a reliable data- base of 122 million individual waves. Within the quality- controlled database, the significant wave height covered the range 0.12 m # Hs # 15.4 m, the peak period was 1 s # Tp # 24.7 s, the maximum crest height was 18.5 m, and the maximum recordedwave height was 25.5m. The majority of the raw wave data were measured in the North Sea (81% of normal waves and 92% of rogue waves).

The ratio of rogue waves to total waves is largest in the Gulf of Mexico followed by the North West shelf of Australia. The raw wave data from these two regions were only provided during times when hurricanes (or tropical cyclones) passed close to the offshore instal- lations. In contrast, the North Sea measurements were supplied in the form of continuous datasets. Conse- quently, there were approximately 100 million waves recorded in the North Sea, whereas the Gulf of Mexico consisted of only 85 400 waves. The difference of three orders of magnitude in sample size may well explain the larger ratio in the Gulf of Mexico compared to the North Sea; clearly the latter dataset is considered to provide a more reliable estimate of the ratio of rogue to normal wave samples. Another possible explanation though is that steeper seas are more likely during hurricanes (or tropical cyclones), which may be more conducive to creating rogue wave events. The works of Latheef and Swan (2013) and Cherneva et al. (2013) have demon- strated that sea state steepness is important, and as such, its effect on rogue wave samples will be examined fur- ther in the following section.

As the conditions in the North Sea vary depending on location, it can be further subdivided into its three main regions-northern, central, and southern. In doing this, the northern North Sea only contains 10.9% of the total waves and 19.9% of the rogue wave samples in the whole database. Consequently, the majority of the data were measured in the central and southern North Sea. There- fore, the results that will be discussed in the next section are representative of waves propagating in intermediate water depths within extratropical locations.

4. Discussion of results a. Sea state parameters This section will examine whether there is a link be- tween the characteristic sea state parameters and the likelihood of rogue wave events. An interesting param- eter to examine first is sea state steepness, which could be an explanation for rogue wave events occurring, espe- cially extreme crests. This is because the design distri- butions for crest heights are based on second-order theory, whereas it is known that nonlinear wave-wave interactions beyond second order can provide significant increases in the crest elevation, especially in long-crested seas (Gibson and Swan 2007).

The sea state steepness is typically represented by plotting the significant wave height Hs against the peak period Tp, which is shown in Fig. 1a. This figure repre- sents all sea states that passed the QC procedure with a single dot, and the rogue wave samples are distin- guished from the normal wave samples. While Fig. 1a shows that the cloud formed by rogue wave sea states are clustered more toward the steep wave part of the plot, they nevertheless fall completely within the large num- ber of normal wave samples. It is also clear from the figure that there are many normal sea states that are steep; thus, wave steepness cannot be the sole cause of these rogue wave events.

Figures 2a and 2b illustrate the maximum crest heights for each 20-min sample as a function of the mean steep- ness and skewness, respectively. Each of these is a sea state parameter that may indicate a link to the formation of rogue waves, and they will be considered in turn. First, consider the mean steepness S1 as defined by ... (2) where g is the gravitational acceleration, and T1 is the mean wave period calculated from the ratio of the first two moments of the wave spectrum m0/m1. This is pre- sented in Fig. 2a and is similar to Fig. 1a, using the mean wave period T1 instead of the peak wave period Tp to take advantage of the increased stability of T1. It is clear that Fig. 2a is not materially different from Fig. 1a and does not change our conclusion that sea state steepness plays no governing role in the formation of rogue waves.

Second, Fig. 2b illustrates the influence of the skew- ness of the sea state. Once again, the skewness of the rogue wave samples falls within the bounds of those for normal sea states. Therefore, there is no indication that the skewness of the sea state alone might indicate the likelihood of the occurrence of a rogue wave.

Figure 3a presents the maximum crest heights for each 20-min sample as a function of the kurtosis of the sea state. In this case, there appears to be a pattern with all the rogue wave samples having a value greater than three. However, this is misleading, as a rogue wave by definition has a water surface elevation significantly larger than the rest of the waves in the sample, and the record will have a high kurtosis value as a consequence. As such, Fig. 3a is merely reiterating the definition of a rogue wave. Evidence for this is provided by Stansell (2004), who analyzed field measurements and also found that sea states containing rogue waves had large kurtosis values. On removing the rogue wave from the sea state and recalculating the kurtosis of the remaining water surface elevation, Stansell (2004) found that the kurtosis returned to normal levels, as expected. This same pro- cedure was performed on the present database and re- sults in Fig. 3b. As the cloud for the kurtosis of the rogue wave samples completely lies within that of the normal wave samples, Fig. 3b confirms the findings of Stansell (2004). Therefore, a high kurtosis simply indicates the presence of a single rogue wave in a sea state, rather than being a measure of the likelihood of observing a rogue event.

In addition, the shapes of the variance density spectra of rogue wave samples were visually examined for the presence of uni- or bimodal conditions. However, there were no observable trends with both unimodal and bi- modal frequency spectra occurring. This suggests that rogue waves can form in pure wind seas, pure swells, or a combination of wind seas and swells. Therefore, cross- ing seas that have been hypothesized as the cause of rogue waves (Adcock et al. 2011) cannot be the sole ex- planation of these rare events. Once again, it does not appear that the underlying sea state conditions influence the formation of rogue waves.

b. Environmental conditions This section examines whether the rogue waves within the database were created in similar environmental conditions, which would indicate a possible physical mechanism for their formation. This involves examining the wind, wind-sea waves, swell waves, currents, and re- lative conditions between these environmental condi- tions during the rogue wave events. It is important to note that the environmental conditions that generate rogue waves cannot be analyzed in isolation from those that cause normal wave samples. This is because if a certain environmental condition that results in a rogue wave can also lead to a normal wave sample, then it cannot be a factor conducive to forming rogue waves. This would only confirm that there must be an additional mechanism in place for forming the rogue wave. Therefore, the en- vironmental conditions present during normal and rogue wave events were both examined.

As mentioned above, the majority of the rogue waves occurred in the North Sea, and therefore the focus was on this region. For the present study, the waves and winds were obtained from hindcast data by Ocean- weather, Inc. (Cardone and Cox 2011). The hindcast sim- ulation was specifically run for the present study over the period of 1 January 2000 to 31 December 2008. The frequency-direction spectra from the hindcast database were then partitioned into wind-sea and swell com- ponents using a program called XWaves (Hanson and Phillips 2001). The hindcast grid point that was geo- graphically closest to the offshore installation was cho- sen. Furthermore, as the hindcast spectra were output in 1-hourly intervals and the rogue wave samples were 20 min in duration, the hindcast spectrum predicted at the closest time to the rogue wave event was employed. In terms of the current, the tidal component was calcu- lated using the POLPRED software (Proctor et al. 2004). The residual current was not calculated, since the cur- rents in the North Sea are dominated by the tides, and the residual should have a negligible influence on the results.

The absolute environmental conditions at the Gold- eneye platform (see Table 1) are presented in Fig. 4 and the relative conditions are presented in Fig. 5. These figures illustrate the empirical probability den- sity function (pdf) of the environmental conditions that created normal waves and of those that formed rogue waves. The hindcast dataset produced 77 083 hourly sea states that represented the normal wave samples. In contrast, at the Goldeneye platform there were 743 roguewaveevents,whichwerethemostrogueevents at an installation in the database and as such are used as an example. Therefore, the normal wave samples outweigh the rogue waves by two orders of magnitude. The vast difference in sample size should be considered when interpreting Figs. 4 and 5. It can be observed that the pdfs for the normal waves are smoothly varying, whereas those for the rogue waves are quite erratic, which is simply due to the difference in sample size. As mentioned in the introduction, this goes back to the issue of having a large enough database to examine the occurrence of rogue waves and ensuring reliable results.

Figures 4a, 4d, 4g, and 4j present the pdfs for the sig- nificant wave height Hs, peak period Tp, mean direction of propagation um, and directional spreading ss of the wind-sea partition, respectively. Similarly, Figs. 4b, 4e, 4h, and 4k illustrate the pdfs for the same parameters for the swell partition. Figures 4c and 4f show the pdfs of the wind speed and direction, respectively. Likewise, Figs. 4i and 4l present the pdfs of the current speed and di- rection, respectively.

Figures 5a-c illustrate the pdfs for the relative con- ditions between wind-sea and swell components for the parameters Hs, Tp, and um, respectively. Finally, Figs. 5d and 5e show the relative direction of propagation be- tween wind sea and current and between swell and current, respectively. It is important to note that out of the 743 rogue wave events at Goldeneye, there were only 189 that contained both wind-sea and swell com- ponents; as mentioned in section 4a, uni- and bimodal spectra were both present for rogue wave samples. Therefore, the pdfs of the rogue wave samples for the relative conditions in Fig. 5 are even more erratic than for the absolute environmental conditions. As before, this should be taken into consideration when comparing the pdfs for normal and rogue wave samples.

Examining Figs. 4 and 5, it can be observed that for all parameters the pdfs of the rogue wave samples overlap those for normal waves, especially considering the two orders of magnitude difference in the sample size of the two datasets; there are some differences, but they are generally small. Figure 4c has relatively larger differences and one may be tempted to speculate that higher wind speeds, which affect the wave age and thus steepness of the sea state, could lead to rogue waves. Similarly, one may be tempted to speculate from Fig. 5a that rogue waves formed in bimodal sea states have wind seas with relatively higher Hs when compared to the swell partition. However, both of these ob- servations may also simply be a manifestation of the difference in sample size, but they warrant further in- vestigation in a future study. Aside from this, these figures present evidence to suggest that the environ- mental conditions that generate normal waves, which are the prevalent conditions in a region, also form rogue waves. Once again, it does not appear that physical processes that operate on a sea state level in- fluence the formation of rogue waves. The next section will examine the influence of local parameters on the appearance of rogue waves.

c. Local parameters It has been demonstrated that there is no convincing evidence to suggest a link between the sea state condi- tions and the likelihood of a rogue wave event. Conse- quently, the local parameters in the vicinity of the rogue wave will now be examined to ascertain whether these have a strong influence. Consider first the individual wave steepness, which is illustrated in Fig. 1b. This figure plots the individual wave height against the wave period for all waves that passed the QC checks and distin- guishes between the normal and rogue waves. The com- monly applied, limiting, one-seventh curve is also shown simply as an indication of the individual wave steepness, though strictly this limit is only applicable for deep-water regular waves. A more rigorous approach for all water depths takes the form of a Miche-Stokes-type limit, as described in Tayfun (2008) and Cherneva et al. (2009, 2013), who have demonstrated that field and experi- mental measurements of large waves do not exceed this limit. In contrast to the sea state steepness, Fig. 1b dem- onstrates that the rogue waves are typically steep with some very close to the one-seventh steepness. However, while the rogue waves are steep, there is still a large quantity of normal waves that are at least as steep that do not become rogues. Therefore, local steepness alone, and consequently increased nonlinearity, does not appear to explain the formation of rogue waves.

Another local aspect of interest is the average shape of rogue waves and how it differs from the largest normal waves. It has long been known that the average shape of the largest waves in the ocean is the scaled autocorrela- tion function (Lindgren 1970; Boccotti 1983, 2000), which takes the form of a focused crest event and is also referred to as a new wave event (Tromans et al. 1991). This has already been validated against field measurements for normal wave samples (Jonathan et al. 1994; Jonathan and Taylor 1997).

Figure 6 presents the average shape of the rogue wave crests compared to the largest 1% of normal waves within the present database (corresponding to 5286 samples); the surface elevation being normalized by Hs and time by Tp. This figure has several interesting fea- tures. First, it illustrates that the average rogue wave shape also takes a new wave form in the same manner as the largest normal waves. This provides yet another as- pect in which normal and rogue wave events do not differ. Second, the average rogue wave shape has higher crests and deeper troughs than the largest normal waves. This implies that the difference cannot be attributed to effects described by Walker et al. (2004), who developed a nonlinear new wave surface profile, as this creates larger crests but shallower troughs, which is not con- sistent with Fig. 6. This is not meant to suggest that nonlinear effects are not present, only that the differ- ence between the average rogue waves and largest normal waves cannot be attributed to nonlinear effects. Third,thenormalizedwave periodsof the averagerogue event in Fig. 6 are almost identical to those for the highest normal waves. This indicates that the average rogue wave event is locally steeper than the largest normal waves, which agrees with Fig. 1b and the pre- vious discussion. However, as noted above, the differ- ence in shape of the average rogue wave event is not consistent with nonlinear transformations. This could be explained by adifferencein the spectral bandwidth,with the rogue wave samples being more narrowbanded than the highest 1% of normal waves. This can be determined by assessing the spectral width parameter in terms of « or n as defined by Cartwright and Longuet-Higgins (1956) and given by ... (3) ... (4) where m0, m1, m2, and m4 are the zeroth-, first-, second-, and fourth-order spectral moments, respectively. Figure 7 presents the empirical probability density functions of «andnforthehighest1%ofnormalwavesandtherogue wavesamples.Itcanbeseenthatthetwodistributionsfor «are verysimilar.As thisparameterisrelatedtothe local maximum and minima of the water surface elevation, it appears that there is little difference between the largest normal waves and rogue waves in this respect. This is not surprising given that extreme waves tend to be more narrowbanded and thus more regular (Tayfun 2008). In contrast, the rogue waves demonstrate lower values of the n parameter, indicating that they are more narrow- banded. This provides evidence to suggest that the spec- tral bandwidth does play a role in distinguishing rogues waves from the largest normal waves.

An additional explanation could be that during the formation of a rogue wave, a larger proportion of fre- quencycomponentsareinphase.Thiscouldalsoinclude the high frequencies, which will have a negligible effect on periods within the wave group, but they can lead to considerable increases in amplitude if they are in phase with the other frequency components. To examine this in more detail, the local spectrum and phase information in the vicinity of the rogue wave events was analyzed using a time-frequency signal processing technique. In the present study, the wavelet transform was selected along with the complex Morlet wavelet (Liu 2000; Krogstad et al. 2006; Christou et al. 2008). Before ap- plying the wavelet transform, it is first informative to present water surface elevations for three rogue waves events that are representativeof the broad rangeof rogue wave samples from within the database. On its left axes, Fig. 8 presents the water surface elevation [h(t) 5 a(t)cosu(t)], its Hilbert transform [h^(t) 5 a(t)sinu(t)], and the upper and lower wave envelopes all normalized withrespect tothe root-mean-squareof thewater surface hrms. In addition, on its right axes, Fig. 8 illustrates the time-varying phase u(t). In this figure, the maximumcrest occurs at tmax, and the abscissa is normalized with respect to the peak period of each of the sea states in which the rogue waves formed. All three rogue waves demonstrate a maximum crest coinciding with the maximum wave envelope and corresponding to a value of h^ 5 0andu 5 0. This is in keeping with the findings of Tayfun (2008) based on second-order theoretical models.

Figure 9 illustrates the results of applying the wavelet transform to the same three example rogue wave events as presented in Fig. 8. In each subfigure, the local spec- trum is shown along with the phase of all frequency components at a given instant in time; the ordinates of the former are displayed on the left y axis and of the latter on the right y axis. Figures 9a-c illustrate the local spectrum and phase one peak period before the rogue wave event; Figs. 9d-f are at the time of the rogue wave event; and Figs. 9g-i are for one peak period after the rogue wave event.

It can be observed from Figs. 9d-f that at the time of the rogue wave event almost all frequency components with nonzero spectral values are approximately in phase withone another; this isalso true for the high frequencies and confirms the discussion above concerning the average shape of rogue waves. Figure 9d is an example of a fo- cused crest event, with the phases around 08. Figure 9e has phases at approximately 308 and Fig. 9f has phases around 608. These three figures illustrate that at the time of the rogue wave event there is focusing of the energetic frequency components. They may not always correspond to a phase of zero, but the energetic frequency compo- nents are in phase with each other. These three figures present the range of phases about which focusing oc- curred within the database. Considering Figs. 9a-c as well as Figs. 9g-i for the phases one peak wave period either side of the rogue wave event, there is a positive gradient before and a negative slope of the phase after all of these focused events. This is a very familiar pattern that occurs due to linear dispersion.

To quantify the degree to which frequency components are in phase with each other at the time of a maximum event, the standard deviation of the phases at that instant may be calculated. This corresponds to the phases within Figs. 9d-f for the three example rogue waves considered above. This calculation was performed for all rogue events and the highest 1% of normal waves and will be referred to as sphase hereinafter. A low value of sphase indicates that the frequency components are in phase with each other and represents a focused wave, whereas a high number suggests a large variation of the phases and a nonfocused event. The value of s phase is a function of the number of frequency components included in its calcula- tion. If only the peak frequency is taken this value would equal zero, whereas if all frequency components resulting from the wavelet transform were included, then sphase would be very large, as this would include frequencies with very low variance density, whose phases are erratic.

Therefore, sphase will be calculated for an increasing number of frequency components. The number of fre- quency components nphase is determined by considering only those with a variance density within a given per- centage of the spectral peak value; this percentage will be denoted as the variable a hereafter. When a has a low value, this represents only the frequency components very close to the spectral peak. In contrast, a high value of a includes most of the variance-density spectrum. For example, a 5 10% represents the frequency compo- nents with variance density greater than or equal to 0.9Sp, where Sp is the peak spectral value. Similarly, a 5 95% considers all frequencies with variance density greater than or equal to 0.05Sp.

Figure 10 presents empirical probability distributions of sphase for the highest 1% of normal waves and the rogue events as a function of a. It can be observed that for low values of a in Figs. 10a-c both probability dis- tributions are very similar. In contrast, as a increases to larger values, as in Figs. 10g-i, the probability distribu- tions differ with a higher number of rogue wave events having lower values of sphase. This is summarized in Fig. 11a, which presents the mean of sphase as a function of a for the highest normal waves and rogue events. This figure clearly demonstrates that as a increases, the dif- ference between the normal waves and rogue events increases, with the rogue waves having lower mean values of sphase. This suggests that rogue waves are generated when a larger number of frequency compo- nents are in phase with each other.

This difference in standard deviation could be due to the number of frequency components nphase in the cal- culation of sphase. This is illustrated in Fig. 11b, and it can be observed that for a less than 60%, this value is identical for both the highest normal waves and the rogue waves. Consequently, this cannot be responsible for the difference in mean sphase that is observed in Fig. 11a, but it indicates that the components are more in phase at the time of a maximum event than for the highest 1% of normal waves. Note that the difference in the mean value of sphase between the normal and rogue events is not that large. However, it is sufficiently large to cause the rogue wave events to exceed the threshold defined by the rogue wave criteria. This further indicates that rogue waves are simply rare occurrences of the normal wave population.

Therefore, evidence from the present study has demonstrated that rogue wave events are not governed on a sea state level, but are caused by local effects and in particular by dispersive focusing. They are differenti- ated from the highest normal waves due to increased focusing and a greater number of frequency components being in phase with each other, which is indeed a rare occurrence.

d. Expected number of rogue waves A crest elevation that looks unlikely when viewed in the context of a 20-min or 1-h interval in which it was found may well be expected in the context of the whole dataset (many intervals of storm history over many lo- cations). The objective of this section is to put the results into a long-term, multilocation context.

Let P be the probability that a random wave sample of N waves contains at least one rogue event: ... (5) where F is the chosen probability distribution of the normalized crest elevation h/Hs. Now, assuming each record is independent of all other records, the probability b of seeing k records in a total of n is given by the bi- nomial distribution, assuming P is constant for all records: ... (6) The most probable number of records can then be taken as the number of rogue wave events we might expect. To perform this calculation, it is necessary to assume a probability distribution for the normalized crest eleva- tion F. For a linear calculation, the Rayleigh distribution can be employed. In this case the probability F of ob- serving h/Hs . 1.25 is constant, as it only depends on the significant wave height. An improved calculation can be made by using the second-order crest distribution of Forristall (2000). In this case, the probability F is no longer constant, as it will vary because the Forristall dis- tribution depends on the mean steepness and the Ursell number and therefore differs for each sea state. There- fore, the required approach is to calculate the permuta- tions of all values of P for each sea state. For a database with more than half a million sea states, this calculation is prohibitive. Consequently, an efficient approximation is to determine the Forristall distribution for all sea states and then calculate the median distribution. This median Forristall distribution represents F, and the binomial distribution can then be applied in the same manner. For both the linear and second-order approaches, the value of N is taken as the mean number of waves in a sea state based on the average value across the entire database.

Based on this approach, for the current database the expected number of rogue crests from the Rayleigh distribution is 455 and 1980 for the median Forristall distribution. In contrast, the actual number of observed rogue crests within the database is 745 for h/Hs . 1.25; note that this does not include the events with only H/Hs . 2. Consequently, the measured value lies in between the linear and second-order estimates. Given that the Rayleigh distribution underestimates crest elevation, it is logical that this will also predict a reduced number of rogue waves. The second-order Forristall crest distribu- tion should provide a better estimate; however, it vastly over predicts the number of observed rogue waves. This may be in part due to the approximation of the median Forristall distribution employed to determine the number of rogue events. It may also be due to effects beyond second order such as further nonlinear amplification and wave breaking as demonstrated by Latheef and Swan (2013); the latter effect of breaking leads to crest eleva- tions that lie between the Rayleigh and Forristall distri- butions. In particular, the result supports the conclusion that the waves are behaving with statistical characteristics somewhat stronger than Gaussian but less than second order. This supports the conclusion of Fedele (2012) and Fedele et al. (2013) that realistic oceanic rogue waves behave statistically as quasi-Gaussian in contrast to the strongly non-Gaussian waves generated in laboratory investigations (Fedele 2008).

5. Conclusions This paper has described the collation, quality control, and analysis of single-point field measurements from fixed sensors mounted on offshore platforms. In total, the quality-controlled database contains 122 million in- dividual waves, of which 3649 are rogue waves according to the criteria of Haver (2000). A database of this size is necessary in order to ensure a large enough sample of rogue waves, as they are very rare. The majority of the data were measured in the central and southern North Sea, and therefore the results are representative of waves propagating in intermediate water depths within extratropical locations.

To determine physical mechanisms for rogue waves, the sea state parameters were examined. It was found that the occurrence of rogue waves was not a function of the steepness or skewness of the sea state; the kurtosis of the sea state of rogue wave samples had values greater than three. However, the latter is not an indicator for rogue waves, as it is the rogue wave that causes the high kurtosis, which was also demonstrated in the present study following the work of Stansell (2004). The variance- density spectrum for all the rogue waves was also ex- amined, but there was no evidence of a common trend with both uni- and bimodal frequency spectra present. Therefore, neither could be judged to be a mechanism behind rogue waves. On examining the environmental conditions for each sea state, there was no indication to suggest that there was any particular combination of wind sea, swell, wind, or current that is particularly conducive to the formation of rogue waves.

Having determined that the sea state parameters do not play a role in the formation of rogue waves, the local parameters were then examined. It was found that the steepness of the individual rogue waves was greater than the bulk of normal waves. However, there were also normal waves that were as steep as or steeper than rogue waves. Consequently, a rogue wave is generally steeper than normal waves but not all steep waves are rogues. The average shape of rogue waves was also shown to be a new wave profile (Tromans et al. 1991) similar to the largest normal waves and corresponding to a focused crest event. Furthermore, the average rogue wave shape had higher crests as well as deeper troughs than the highest 1% of normal waves. It was also demonstrated that rogue waves were slightly more narrowbanded than the highest 1% of normal waves based on the n spectral bandwidth parameter.

Finally, a wavelet analysis was performed on all rogue wave samples to ascertain the temporal evolution of the variance-density spectrum and phases. The rogue wave samples exhibit dispersive focusing, resulting in the majority of frequency components coming into phase with each other at the time of the rogue wave events.

In conclusion, the present study has presented evidence to suggest that rogue waves are merely extraordinary and rare occurrences of the normal population that are caused by dispersive focusing.

Acknowledgments. We acknowledge the participants of the CresT JIP for their continuous support and interesting discussions during the project, without which the present study would not have been possible: ABS Consulting, Aker Solutions, Anadarko, SBM Offshore, BHP Billiton, Bluewater, BP, Bureau Veritas, ConocoPhillips, Det Norske Veritas, Floatec, Forristall Ocean Engineering, HSE, Imperial College London, JOGMEC, Lloyd's Register, MARIN, MMS, Ocean Wave Engineering, Oceanweather, Petrobras, Shell, Sofec, Statoil, Total, Woodside, and WorleyParsonsSea. We are also grateful to the insightful and constructive feedback from the reviewers, who have helped to im- prove the overall manuscript.

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MARIOS CHRISTOU* Shell Global Solutions International B.V., Rijswijk, Netherlands KEVIN EWANS Sarawak Shell Berhad, Kuala Lumpur, Malaysia (Manuscript received 14 September 2013, in final form 17 April 2014) * Current affiliation: Department of Civil and Environmental Engineering, Imperial College London, London, United Kingdom.

Corresponding author address: Marios Christou, Department of Civil and Environmental Engineering, Skempton Building, Impe- rial College Road, South Kensington Campus, Imperial College London, London, SW7 2AZ, United Kingdom.

E-mail: [email protected] (c) 2014 American Meteorological Society

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