On Correlation between Wind and Wave Storms

: The paper is focused on the formulation of an adequate criterion for associating wave storm events to the generating wind storm ones, and on the study of correlation between their characteristic parameters. In this context, the sea storm definition commonly used for storm identification from significant wave height data is adapted for wind storm, by processing wind speed data. A sensitivity analysis is proposed as function of the storm thresholds aiming at identifying optimal combination of wind speed and significant wave height thresholds allowing the association of relatively large number of events ensuring high correlation between wind and wave storm parameters. The analysis is carried out using as input data wind speeds and significant wave heights from four meteorological (buoys and anemometers) stations of the National Data Buoy Center moored off the East Coast of the United States. Results reveal that an optimal threshold combination is achieved assuming both wind speed and significant wave height threshold as 1.5 time their respective averages.


Introduction
The study of offshore climate plays a key role for any activity developed in offshore sea environment, including navigation, oil and gas, and offshore wind and wave farms. In this context the research community focuses on a variety of topics that ranges from the forecasting of environmental data [1][2][3][4][5] to the investigation of climate changes [6][7][8][9], energy resource assessment [10][11][12][13][14], simulation methods [15,16], and design and operation of offshore systems [17][18][19]. With regard to offshore structures and energy systems, the knowledge of climatic events occurring and their impact on any device is paramount both at design and at operational stages. Usually, one refers to environmental states within a time duration compatible with the stationarity assumption, described by two sea state parameters-which could be the significant wave height and the spectral peak period-and a wind state with given average wind speed and turbulence intensity. Among all the possible states, the most representative ones are selected from in-situ scatter diagram at site. However, it could be interesting to investigate the system response under nonstationary conditions, identified as sequences of environmental states exceeding critical thresholds and defined as storms. In order to do so, an in-depth study of such events and their characteristics is needed. The separation of wind events from long wind speed data sets is a crucial step for conducting both statistical and wind-excited structural response analyses. The approaches employed for this kind of extrapolation differ from each other if the phenomenon is investigated from a meteorological or engineering perspective. In the first case, wind events identification is performed by analyzing related meteorological parameters [20][21][22][23][24][25][26][27]. In the second approach, wind peaks are extrapolated from long wind speed time series with a systematic selection method without any consideration of climatic aspects, and are used for long-term statistical estimations and design value calculation; among these selection criteria are: Gumbel's method [28] in which the largest values per fixed time period (e.g., one year) is selected; its extension with the selection of the r-largest value over a given time duration [29]; the method of independent storms [30,31] that uses a period of wind speeds below a selected (low) threshold to separate storms; and the Peak-over-threshold (POT) method that extracts maxima from sample data series to produce a series of extreme values above a chosen, adequately high, threshold [32,33].
In this paper, a criterion for associating a wave storm to its generating wind storm is formulated, starting with results achieved through wind and wave data analysis. The last is based on the storm concept intended as sequence of stationary wind or wave states exceeding pre-established critical thresholds. Specifically, it is aimed at identifying the wind and wave thresholds such that it is possible to associate a quite large number of events ensuring a high correlation between the storms' parameters. Trends between the wind and wave storm parameters are investigated as well. The paper is organized as follows: section 2 describes the data used and the site, section 3 introduces the storms identification and association criterion, section 4 shows data analysis on the basis of the proposed criterion, and section 5 illustrates some concluding remarks. Results of this analysis are useful for survivability, reliability, and operability assessments of combined wind and wave systems, which take advantage of the complementarity of these resources. In this regard, the understanding of how nonstationary wind and wave events follow each other, their shift in terms of beginning time, time lag between peaks, and correlation over common time window where both persist over certain ranges of values, represent key elements in the development of combined technologies. Further, they can provide preliminary information on how to couple wind and wave states over longer time interval than those commonly considered in response assessments of offshore wind systems, for which wind and waves represent the main load factors.

Wind and Wave Data and Study Area
Data used as input of the analysis proposed in this work are those provided by the National Data Buoy Center (NDBC) (https://www.ndbc.noaa.gov/ (Accessed on 8 December 2021)). The NDBC has deployed buoys around the USA coastline and Great Lakes since 1970s. From that date, the NDBC has deployed at least eight directional buoys and over a hundred wave measurement stations for operational and experimental use, covering areas from the western Atlantic to the Pacific Ocean around Hawaii, and from the Bering Sea to the South Pacific. Stations have been subject to modifications throughout the years, due to instrumental improvements and changing data storage systems as well. The NDBC time series contain wave data in terms of significant wave height, dominant (peak spectral) and average (mean) periods, mean wave direction at the peak frequency, spectral wave density data, the four Fourier directional parameters for spectral wave direction data, and meteorological data as wind direction, wind speed, wind gust, air pressure at sea level, air temperature, and sea surface temperature. In this work only significant wave height and wind speed data are considered. Regarding the wind speed data, it is worth to mention that anemometer heights on NDBC buoys vary according to buoy type. Anemometers on 3 m discus and 6 m Navy Oceanographic and Meteorological Automatic Device (NOMAD), buoys are located approximately 5 m above the waterline. Anemometers on 10 and 12 m buoys are located 10 m above the waterline. Anemometer heights at Coastal-Marine Automated Network (C-MAN) stations vary widely, depending on site structure and elevation above sea level. The NDBC adjusts wind speeds to conform to the universally accepted reference standard of 10 m. NDBC also adjusts wind speeds to 20 m, a height closer to that of typical ship anemometers. These standardized wind speeds are stored in a different file than the one which can be downloaded in the historical data page, where only unadjusted data are included. The latter are those used in this paper for the coupled analysis of wind and wave storm events. Four buoy stations were selected off the East Coast of the United States: 44004, 44008, 44014, and 41025. The time duration covered by each time series data and station characteristics are summarized in Table 1. This study area was selected because of the great interest faced by companies in proposing new offshore wind farm projects and it is already involved in pilot projects. Among them, the Coastal Virginia Offshore Wind (CVOW) located about 43 km (27 mi) off the coast of Virginia Beach, Virginia, US, whose initial phase, was a 12 MW pilot project constructed in 2020, consisting of two 6-megawatt offshore wind turbines [34,35]. Another example is Sunrise Wind [36,37], a 880 MW offshore wind farm to be developed in New York state, about 48km east of Montauk Point on Long Island.
The results of the analysis carried out in this work could be useful in the context of reliability and operability assessment of existing offshore wind farms, for designing and planning new wind farms, and for the integration of systems with wave energy converters and the design of combined technologies.

Methodology: Wind and Wave Storm Identification and Association Criteria
In this section the method used for identifying storm events and the association criteria of a wave event to its preceding wind one will be described. Note that in the context of this work the term "storm" refers to a nonstationary either wind or wave event, which is characterized by growth, peak, and decay stages (see Figure 1). In this regard, for what concerns the wind storm extrapolation, it is performed regardless to Beaufort's scale [38][39][40]. Specifically, the sea storm definition adopted in long-term analysis via Storm Models [41][42][43] is adjusted for extrapolation of wind storms from wind speed data time series. Following the above-mentioned approach, a sea storm is a sequence of sea state during which the significant wave height Hs exceeds a given threshold named critical and does not fall below it for a time interval greater than twelve hours. This definition takes into account that a calm period may occur for a maximum duration of twelve hours and the storms characterized by this condition may have more than one peak. In this paper, the above definition is slightly reviewed removing the calm condition and considering only the part of the storm characterized by the sequence of sea states above the critical threshold hcrit. Thus, when Hs falls below the critical threshold it is considered the end of the storm. The adaptation of this definition to wind storm events simply consists in repeating the sea storm definition by replacing the significant wave height Hs and the critical threshold hcrit with the average wind speed u and critical wind speed ucrit, respectively. The critical thresholds of both average wind speed and significant wave height are fixed as equal or multiple times that of their averages (calculated from the whole time series) at site. The criterion adopted in this paper for storm identification and association is described in the flow chart represented in Figure 2. It involves as first steps, the calculation of average values of wind speed and significant wave height at site and the related critical thresholds ucrit, hcrit. Subsequently, the identification of sequences of wind storm events according to the definition above is performed. Then, for each wind event the related generated sea storm, if any, is extrapolated from Hs data. Specifically, considering the time interval (tin wind, tfin wind) (being tin wind, tfin wind the starting time and final time instants of the wind storm, respectively) covering the whole wind storm evolution (growth, peak and decay) if an excessed of at least one value of Hs over its critical threshold is detected it is assumed the wind storm generates a wave storm. Otherwise, no wave storm is generated and the considered wind storm is excluded from the correlation analysis. Furthermore, only events characterized by a duration of more that twelve hours are considered as relevant ones. Another important aspect is related to the time spacing between two successive storm events (either wind or wave). In this regard, it is assumed that only events with a minimum time separation of 48 hours are relevant, in agreement with previous studies [31,44].

Wave storm identification
Wave storm associated to the identified wind storm are selected from Hs data. In particular: If in the time interval (tin wind, tfin wind) at least one Hs value exceeds hcrit, it is assumed that the considered wind storm generates a wave storm. Otherwise, no associated wave storm.

Nwind total number of wind storm
Nwind with wave number of wind storm generating a wave stotm

For each of the Nwind with wave
The whole time history of the associated wave storm is identified as: Starting from the related time instant t*, data are processed backward to identify the time instant tin wave preceding t* at which Hs<=hcrit;  Starting from the related time instant t*, data are processed forward to identify the time instant tfin wave following t* at which Hs<=hcrit.

Data Analysis
This section proposes data analysis of wind and wave storms based on the criterion presented in the previous section. Specifically, wind and wave storms are extrapolated from time series of average wind speed and significant wave height, respectively, according to the proposed criterion. Then, for each sample wind and wave storm, correlation between relevant storm parameters is investigated. This analysis is essentially divided in two main parts: the first part deals with the proposition of a sensitivity analysis to the couple of thresholds (hcrit, ucrit) aiming to identify their most adequate combination; the second shows the correlated storm parameters trends when the optimal couple (hcrit, ucrit) is considered in storms identification and association.

Variability of Storm Number and Correlation Parameters Assuming Different Thresholds
In this section, the sensitivity of wind and wave events to their critical thresholds (hcrit, ucrit) is investigated for our storm sample. In this regard, both critical wind speed and significant wave height thresholds are correlated to the average value of related variables (see Table 2). In particular, the following ratios between critical thresholds and average values of both wind speed and significant wave height are considered: 1, 1.5, 2, 2.5. The effects of all the possible combinations of these thresholds are examined. For each couple of thresholds (hcrit, ucrit) the analysis proceeds as follows: • Wind storms are identified as a "sequence of wind states with average wind speed exceeding the critical threshold ucrit". Those having a too short duration (less than 12 hours) and of those not satisfying the condition of established minimum time separation (48 hours intended as time distance between peaks) are removed. Note that, when two wind storm peaks have a time separation less than 48 hours, the one characterized by the highest value of maximum wind speed is kept in the sample and the other one is eliminated.
• The storm sample prepared following the procedure described above is divided in two subsamples: one including the wind storms during whose time window none of the significant wave height values exceed the critical threshold hcrit, and the other constituted by all wind storms during which at least one significant wave height value is over hcrit. The last subsample is assumed to be made up of those wind storms which generate a wave storm.
• For each wind storm in the second subsample the corresponding complete storm event is extrapolated from the significant wave height time series. Specifically, starting from the time instant at which the maximum Hs value is detected in the wind storm time window, Hs data are processed backward and forward the preceding and the following time instants at which Hs is below hcrit. Note that, the last two time instants define the time window over which the wave storm event evolves and the related duration.
• Finally, some parameters are calculated to characterize the events. They are: maximum wind speed umax in the wind storm, wind storm duration Dwind, maximum significant wave height Hsmax in the wave storm, wave storm duration Dwave, time distance Δt umax−Hs max between umax, and Hs max. Note that, Δt umax−Hs max is calculated as t (Hsmax)−t(umax), thus if Hs max occurs after umax it is greater than 0, negative if umax occurs after Hs max, and equal to 0 if they occur simultaneously. At this stage of the analysis, for each couple (hcrit, ucrit) the following quantities are investigated: 1.
6. Number of events characterized by time distance Δt umax−Hs max greater than one day in absolute value (Figures 3-6e).    and Dwave, (d) number of events in which the peak of wave storm occurs before that of wind storm, (e) number of events with a time distance Δt Hs max -umax between wind storm and wave storm peaks that is more than one day.
A comparison among these Figures shows that, for the investigations at points 1, 4, and 5 general considerations can be developed, while for points 2 and 3 the trend is different from one location to another. In general, from Figures 3-6a it is seen that the events number decreases as the threshold increases, as expected. For a fixed ratio u u crit / < 2, the number of wind events generating a wave storm increases as the as the ratio  The assumption of the last option above leads to a good correlation of maximum wind speed umax and significant wave height Hsmax, which in such situation is always between 0.6 and 0.7 (see Figures 3-6b). In this regard, it is important to say that the correlation of Hsmax and umax does not exhibit a regular trend and varies from one location to another. However, for wind ratios u u crit / < 2 it keeps itself always above 0.5 (see Figures 3-6b). The correlation between wind and wave storm duration is completely irregular and does not show any particular trend: sometimes it is very high and sometimes it can be very low. Then, focusing on the time distance Δt umax −Hs max between umax and Hsmax, from Figures 3-6d it could be noticed that the peak of wind speed could proceed or follow that of significant wave height. Another insight into Figures 3-6d is that the number of events characterized by Δt umax−Hs max (Hs peak occurring before u peak) decreases with increasing u u crit / and s crit H h / , and for u u crit / 2  does not depend upon the significant wave height ratio s crit H h / . Lastly, observing Figures 3-6e it is seen that Δt umax−Hs max can be greater than one day and the number of events characterized by such distance between peaks decreases when both u u crit / and s crit H h / increase up to 2, while for higher values it tends to zero. This result is quite intuitive because with such high ratios the sample includes only extreme events in which it is expected wind speed and significant wave height peaks to be very close to each other (see Figure 1). In light of the results analyzed in this section, a suggestable combination of wind and wave ratios u u crit / and s crit H h / could be achieved by assuming both of them to be equal to 1.5. In this regard, it is worth mentioning that the value of 1.5 is the one generally used for sea storm identification. Furthermore, any combination of u u crit / and s crit H h / in the range (1-1.5) is adequate for reliable analysis.

Trend and Correlation between Wind and Wave Storm Parameters Associated with Selected Thresholds
Based on the analysis proposed in previous section, the trends of relevant storm parameters are investigated, taking as wind and wave ratios u u crit / and s crit H h / with a value of 1.5. Starting from our wind and wave storm sample, for each event, the following parameters are determined: 1. Maximum wind speed umax; 2. Wind storm duration Dwind; 3. Maximum significant wave height Hsmax; 4. Wave storm duration Dwave; 5. Time distance Δtu max−Hs max between wind storm and wave storm peaks (Hsmax, umax); 6. Standard deviation of wind speed σwind; 7. Standard deviation of significant wave height σwave; 8. Correlation coefficient ρ(u(t), Hs(t)) between time histories of wind speed and significant wave height over a time window including the whole evolution of both wind and wave storms. In this regard it is important to specify that this time window could include a time interval in which either the wind speeds or the significant wave heights are below their critical thresholds.
The estimations listed above are used to draw graphics to elucidate some interesting trends. Specifically, Figures 7-10a-d, show wind storm duration Dwind versus maximum wind speed umax, wave storm duration Dwave versus maximum significant wave height Hsmax, wind speed standard deviation σwind versus maximum wind speed umax, and significant wave height standard deviation σwave versus maximum significant wave height Hsmax. Data summarized in these figures essentially confirm some well known trends. For instance, both wind and wave storm durations increase with the storm intensity and standard deviations of both wind speed and significant wave height increase as well.  Figures 7-10g σwave versus σwind, in Figures 7-10h Δt umax -Hs max versus umax, and in Figures 7-10i ρ(u(t), Hs(t)) versus umax. Looking at this last group of figures, some general considerations can be developed: • The wave storm peak Hsmax increases with the wind storm peak umax; • The wave storm duration Dwave increases with the wind storm duration Dwind; • Considering a given storm event the significant wave height standard deviation σwave is always less than the wind speed σwind standard deviation. This result is interesting in the context of offshore renewables energies because it indicates which energy source is more regular and justifies the interest in combined systems to the aim of smoothing the power output.

•
The time distance Δt umax−Hs max may be greater than one day, in most of the storms shorter than 10 hours in absolute value; • The correlation ρ(u(t), Hs(t)) increases with the wind storm intensity umax.

Conclusions
In this paper, a criterion is proposed to identify and associate wind and wave storm events, based on a sensitivity analysis as a function of wind and wave critical threshold. An optimal combination of such thresholds is suggested as a result of the proposed analysis. It consists of assuming both wind and wave critical thresholds as 1.5 times the average of wind speed and significant wave height, respectively. Thanks to this assumption, it is possible to associate a large number of correlated wind and wave storm events. Furthermore, the relationship between wind and wave storm parameters are investigated with these thresholds, elucidating that, as the wind speed increases the significant wave height increases as well. However, the peak of the significant wave height does not always occur after that of the wind speed. In general, the structure of a wind storm is very similar to that of a wave storm. In addition, the wind speed is characterized by a higher variability with respect to the significant wave height. All the results mentioned above could be useful both for sample extrapolation in the context of a statistical analysis, offshore energy resource assessment, and wind and wave condition combination for response analysis of offshore structures.