short term statisitics of wave observed by buoy

7/28/2019 Short Term Statisitics of Wave Observed by Buoy

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SHO RT-TERM STATISTICS OF 10,000,000 WAVE SOBSERVED BY BUOYSMerce Casas Prat1, Leo H. Holthuijsen2 and P. H. A. J. M. van Gelder3

Time records of the surface elevation measured by four Waverider buoys in theMediterranean Sea off the coast of Spain have been analysed to inspect the statistics ofcrest heights and wave heights. By concatenating the normalised records we obtained along, quasi-stationary record of 10,000,000 waves, permitting a verification of theRayleigh distribution and its theoretical variations at rather low levels of probability(wave heights up to 10 times the standard deviation of the surface elevation). The crestheights were almost perfectly Rayleigh distributed over the entire range of observation.The distribution of the wave heights is close to a Rayleigh distribution with scale factor0.88 (rather than 1 as in the conventional Rayleigh distribution), but it is betterapproximated with a Weibull distribution with a shape factor 2.162 (rather than 2, as fora Rayleigh distribution). Supplementary observations with laser altimeters in the NorthSea (10,000 waves) showed nearly identical results in the range of overlap (thenormalised crest heights were slightly higher, showing a nonlinear behaviour).

I n t roduct ionThe short-term statistics of wind-generated waves in deep water are usuallybased on the assumption that the sea surface elevation is a stationary, Gaussianprocess, resulting in the conventional Rayleigh distribution for wave heights andcrest heights (Longuet-Higgins, 1952). However, field observations indicate thatthis Rayleigh distribution over-predicts the significant wave height by 7 - 8 %(e.g. Forristall, 1978; Holthuijsen, 2007) whereas the maximum crest height in agiven duration is reasonably well predicted by the same theory with less than 2%error (Cartwright, 1958). With our rather long wave records of up to 10 millionwaves (obtained by concatenating normalised observed records of 20 minuteseach), we can revisit the Rayleigh distribution at very low probabilities. M ost ofour data were measured with W averider buoys in the M editerranean Sea. To seethe effect of using a fixed instrument, we supplemented our data with wavesobserved with laser altimeters from an offshore platform in the North Sea.O b s e r v a t i o n sMost of our time series were measured by 4 Waverider buoys off theCatalan coast of Spain during a 15-year period (see Fig. 1 and Table 1). We arewell aware of the fact that in the time records of the buoys the peaks of steep

1 Escola Tecnica S uperior d' Enginyers de Camins , Canals i Ports, Techn ical University ofCatalonia (UPC ), Jordi Girona 1-3, 08034 Barcelona, Catalonia, Spain2 Faculty of Civil Engineering and Geo sciences, Delft University of Techno logy, Stevinweg1, 2628 CN , Delft, the Netherlands3 Faculty of Civil Engineering and G eoscience s, Delft University of Technology, Stevinweg1, 2628CN, Delft, the Netherlands

560


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COASTAL ENGINEERING 2008 561waves tend to be flatter than they actually are and that buoys may be draggedthrough or swerve around the three-dimensional peaks of waves. However, mostwave measurements at sea are made with buoys and understanding the statisticsthus obtained is important.

Figure 1. The locat ion of the four WAVERIDER buo ys of f the Catalan co ast .

~T7io>

55 *

Figure 2. The locat ion of the EDDA plat form in the North Sea ( f rom Al lender et a l .1989).


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562 COASTAL ENGINEERING 2008

Ta b le 1 . Spe c i f i c s o f the buoy obs e rv a t ion s o f f the Ca ta la n c oa s t a nd o f thea l t ime te r obs e rv a t io ns in the Nor th Se a .

coordinatesdepth (m)diameter(m)sampleinterval (s)resolution(m)

instrumentrecordlength (min)period(years)

R o s e s03 11.99 E42 10.79 N

4 60 .7

1/2.56

0.01sca la rb u o y

2 02 0 0 1 - 2 0 0 6

T o r d e r a02 48.93 E41 38.81 N

7 40 .7

1/2.56

0.01sca la rb u o y

2 02 0 0 2 - 2 0 0 6

L lo b re g a t02 08.48 E41 16.69 N

4 50.7

1/2.56

0.01sca la r buoy

2 02 0 0 1 - 2 0 0 4

T o r t o s a00 58.89 E40 43.29 N

6 00 .9

1/1.28

0.01d i re c t i o n a l

b u o y2 0

1 9 9 1 - 1 9 9 72 0 0 1 - 2 0 0 6

E d d a03 28E56 28N

7 01

0 .001laser a l t ime te rs

1 7 min 4 s5 -6 /1 1 /1 9 8 5

2 1 - 2 3 / 1 2 / 1 9 8 5

We subjected the buoy data to some rigorous tests. Of the original records, weaccepted only those for which (in order of testing):1. The record has the nominal length (see Table 1).2. All absolute values of the vertical accelerations are


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COASTAL ENGINEERING 2008 563

8. The mean wave length (determined with the linear wave theory from theabove mean frequency) is smaller than V2 the local water depth.We removed all records that did not pass these tests - we did not try to repair anyrecords. In addition, we removed 16 unusual looking records. With these 9criteria, we accepted 42,377 buoy records with a total of approximately 10million waves. The highest individual wave height in this data set is 8.53 m andthe highest significant wave height is 5.38 m.

To find some indication of the differences with a fixed instrument, wesupplemented our data with 92 time series obtained by altimeter observationsduring the WADIC project (Allender et. al, 1989; see acknowledgements) at thePhillips Edda platform (see Fig. 2). We did not censor these records - we onlyconsidered the requirement of shallow water (see above criteria), which caused23 records to be rejected (including the ones with thehighest significant waveheight in November 1985). The remaining 69 records contained about 10,000waves. The highest individual wave height in these records is 13.55 m and thehighest significant wave height is 8.82 m.TheoryIn the linear approach of Longuet-Higgins (1952), the crest height isRayleigh distributed and, because the wave height is assumed to be twice thecrest height, the wave height too is Rayleigh distributed:

P(n m) = Veres, e X p ( - X C S , ) = Pnml ,Rayleish ( 1 )and

p(H) = y*Hex?(-KH 2) = Pfi,Ra,e,gh C)in which the normalized crest height is fjcresl = rjmsl I yjm 0 and the normalizedwave height is H = HI yfm^ and m0 is the variance of the surface elevation. Ifthe wave height is not assumed to be twice the crest height (instead thecorrelation between crest height and trough depth may be taken into account),the Rayleigh distribution for the wave height should be scaled:

p(H) = rexp4a v 8 a 2 y (3 )with the scaling factor a depending on the spectrum, e.g.,a

2=\-{y%n

2-y 2)v

2in which the spectral width v = ^m0m2/mf-\(Longuet-Higgins, 1980) and m 0,m x and m2 are the zero-th, first- and second-

order moment of the variance density spectrum. Or, a2 = /2(\ - p) inwhich pis the correlation between the crest height and the trough depth of a wave (Naess,1985). For large wave heights Vinje (1989) multiplies this distribution (with thesame value for a ) with J%(l-p~ ]) and Tayfun (1990) multiplies this again


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564 C O A S T A L E N G I N E E R I N G 2 0 0 8

with |l + (/7 2 - l ) / ( 4 p # 2 ) l . For weakly nonlinear waves, adaptations of theRayleigh distribution for the crest height have been suggested, e.g. by Mori andJanssen (2006)Pin**,, ) = [l + X ( 4 | - 3) ( l - , + X , ) ] P lc ,Ray,Clg>, (4)which depen ds on the kurtosis XA of the surface elevation or, Tayfun (1994)

P(flcrest) : l + c lA ifj cresl(-fjles ,-2)/j2 l{\-c^)p^msiMyleigh (5 )which depends on the sk ew ne ss/^ of the surface elevation ( q and c2 areconstants).The cumulative distribution function of the maximum crest height ormax imum wave height in a record of JV w aves is readily ob tained from theabove, assuming that crest heights or wave heights are independent. Theexpressions are identical for the wave height and the crest height. Representingeither of these as x,

^{x jnax


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P(ricresl) P(H)1-10

a = 0.878

o buoy observations(this study)scaled Rayleigh distributionswith scaling actor a

I Jm,10H

12HI,'crest Icrest ' \ "*0 -*-* i J ' \"*0

Figure 3. The dis t r ibut ions of the normal ized crest height and wave height asobserved by the buoys in the present s tudy, the best - f i t Rayle igh dis t r ibut ions,theoret ical and empir ical d is t r ibut ions. Convent ional Rayle igh dis t r ibut ions a = 1adde d for reference.

As expected, because the wave height is not twice the crest height, theobserved distribution of the wave heights deviates considerably from theconventional Rayleigh distribution ( a = 1). A Rayleigh distribution with a scalefactor a = 0.878 does fit the data reasonably w ell but the observations show agentle S-curve around this (best-fit) Rayleigh distribution with slightly highervalues for the low wave heights. A fit over a lower range of normalised waveheights (e.g., from 0 to 8), would therefore give a larger scaling factor than overthe complete range of the observations (from 0 to 10). This may explain (part of)


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the discrepancy between the above scaling factor a = 0.878 and the slightlylarger scaling factors of Forristall (1978) and Holthuijsen (2007). The data arebetter approximated with the Weibull distribution suggested by Forristall (1978)with one coefficient slightly adapted (Forristall's coefficient value a = 2.126 isreplaced here by a = 2. 16 2; for a R ayleigh distribution a = 2). In the lineartheories the distribution of the wave heights deviates from the conventionalRayleigh distribution only in terms of a re-scaling (except the correction byTayfun, 1990 which modifies the shap e of the distribution). With the averageobserved values of the spectral width v and correlation p (determined perwave record and averaged over all records), we find values of a = 0.94 anda = 0.89 for the theories of Long uet-Higgins (1980) and N aess (1985)respectively. Obviously, the latter result agrees better with our observations( a = 0.878) than the former.

The observed distribution of the crest heights agrees almost perfectly withthe conventional Rayleigh distribution (Fig. 3): the data cluster along a straightline with only slight deviations over the entire range of observation and thescaling factor of the best-fit Rayleigh distribution is an almost perfect witha = 1.0001 (fitted to the data shown in Fig. 3). In the nonlinear theories, thedistribution of the crest heights deviates from the conventional Rayleighdistribution but the average skewness and kurtosis of our data are so close tothose of a Gaussian population that the differences predicted by these theoriesare barely visible in Fig. 3.The above analysis was repeated for the observations with the laseraltimeters of the WADIC experiment and the results are shown in Fig. 4. Thebest-fit Rayleigh distributions through the altimeter data have scalesa = 1.048 and a = 0.869 for the crest heights and w ave heigh ts respec tively.Obviously, the crest heights are slightly higher in the altimeter observations thanin the buoy observations (confirming the dynamic effect of the buoys on theobservations) but perhaps more remarkable: the wave heights in the altimeterobservations are almost identical to the wave heights in the buoy observations,thus confirming the Weibull-type deviation from the Rayleigh distribution in thebuoy data.For the buoy data, we also considered the maximum crest height and waveheight in a sequence of N consecutive waves from the normalised records (themean of n sequences, each obtained by concatenating observed normalisedrecords of nom inal duration). The results of this and the theoretical estimates aregiven in Fig. 5.As expected, because the wave height is not twice the crest height, theobserved average of the maximum wave heights (upper set of data in Fig. 5)deviates considerably from the conventional Rayleigh distribution ( a = 1). Butthe line based on a scaled Rayleigh distribution with a = 0.863 fits theobservations almost perfectly. This scaling is close to the scaling based on theobserved wave height distribution in Fig. 3 in which the best-fit is achieved with


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COASTAL ENGINEERING 2008 567a = 0.8 78 . The observed average of the maximum crest heights (lower set ofdata in Fig. 5) agrees very well with the conventional Rayleigh approach( a = 1): the data cluster along the line based on this approach with only slightdeviations over the entire range of observation and the scaling factor a = 0.991is close to unity (fitted to the data shown in Fig. 5). Again, the deviationspredicted by the nonlinear theories are so small that they are barely visible inFig. 5.

P{V crest)P(H)

,^ R a y l e i g ha = \

o buoy observations(this study)A altimeter observations(this study) scaled Rayleigh distributions

with scaling factor a

H = HlJm nFigure 4. The distributions of the normalised crest height and wave height asobserved by the buoys and the altimeter in the present study (over the range of thealtimeter observations) and the Rayleigh distributions that best fit the altimeterobservations (compare with Fig. 3).D iscuss ionThe above results for the buoy observations show that the observeddistribution of the crest heights agrees exceptionally well with the expected(conventional, i.e., a = 1) Rayleigh distribution. T his supports the linear


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approach of Longuet-Higgins (1952) and it is also predicted by the nonlineartheories in the sense that these theories reduce to the linear theory for Gaussiansea states (which our composite buoy data set is, with kurtosis = 3.01 andskewness = 0.022). The observed wave height distribution is obviously not theconventional Rayleigh distribution. Here, the correlation between crest heightand trough depth in a wave is important and, given the average observedcorrelation in the observed time series p = -0.57, the theory of Naess (1985)with a = 0.89 gives a better agreement with the observed value a = 0.878 thanthe theory of Longuet-Higgins (1980) with a = 0.94 (using the averageobserved spectral w idth v = 0.41 ). How ever, the slightly adapted Weibulldistribution suggested by Forristall (1978) fits the observed distribution evenbetter.

1 0 -

E\H.

K Rayleigh a = 1

0

a = 0.991Mori and YasiidaX, = 3 . 0 1

[ 95% confidence intervalbuoy observations(this study)for scded Rayleigh distributionswith scahngfactor a

1 0 10 J 10 5 10 s N 10 'Figure 5. The expected values of the maximum wave height and crest height(normalised) observed by the buoys as a function of the number of waves in asequence, and the predictions.For the crest heights observed by the laser altimeters the agreement with theconventional Rayleigh distribution is not as good as for the crest heightsobserved by the buoys but still reasonable if some scaling is allowed (Fig. 4): theshape of the observed distribution is close to that of a Rayleigh distribution witha scaling factor a = 1.048. This scaling may be indicative of some degree ofnonlinearity in the waves, probably obscured in the buoy observations by thebuoy dynamics (see section Observations). The distribution of the wave heightsobserved by the laser altimeter is remarkably similar to the distribution observedby the buoys (Fig. 4). In fact, they are almost identical except for the veryhighest values, suggesting that the statistics of wave heights observed by buoys,


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in contrast to those of the crest heights, are not seriously affected by the buoymotion.It would be convenient if the theories could be used to predict the abovestatistics for arbitrary conditions, and preferably from the spectrum, which canbe predicted from wind fields. A necessary condition then is that these statisticsdepend on the correlation, the spectral width, or the skewness or the kurtosis ofthe surface elevation. To inspect this, we ranked all records in ascending order ofkurtosis and divided them into 5 groups (each containing 20% of the totalnumber of records). Then we computed the corresponding conditional expectedmaximum in a sequence of N waves. The result for this kurtosis selection isshown in Fig. 6 in which the indicated values of the kurtosis are the averages foreach group.

E\H mE\f]cre. Rayleigh a = 1.

Rayleigh a = l-.

. - -c

a = 0.863

tcrest.max. O O-

bvoy observations(this study)

= 3.25= 3.08= 2.99= 2.92= 2 80

10= 10' 10 s 10 6 N 10 'Figure 6. The dependency of the average maximum crest height and wave height onthe number of waves in a sequence and kurtosis (in groups in ascending order,lines in same order from top to bottom as in legend).

It is obvious that the observed maxima are well organised in the sense that,at each value of N , the observed values rise monotonically with rising kurtosis.We found similar well behaved dependencies for the spectral width, correlationand skewness.Mori and Janssen (2006) estimate the kurtosis of the sea surface elevationfrom the spectrum with the Benjamin-Feir index (Janssen, 2003, 2005)

fh t int4 =3 + (xlj3\BFI2, where the Benjamin-Feir index BF1 = Qpi, Ol


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which the spectral width parameter is Qp =2m 02 f fE 2(f)df (Goda, 1970)and in which the wave steepness s tmOl with the mean wave number

\ o i estimated from the mean frequency with the dispersion relationship fordeep water km0l = (2nfm0i) Ig. This suggests that the kurtosis can bedetermined with the BFI. Unfortunately, we found no correlation between thekurtosis and the BFI thus computed (see Fig. 7). (It must be noted that thetheory of Mori and Yasuda has been derived for unidirectional, i.e., long-crestedwaves. When applied to real, short-crested waves, the effects of their theory aregreatly reduced. In fact, for young sea states with a typical directional spreadingof 25 - 30, the spectral estimate of the kurtosis would be reduced by as much asa factor 5.)

1.4Figure 7. The kurtosis computed f rom the observed surface e levat ion (one datapoint cor res pon ds to one buoy record) as a fun ct io n of the Benjamin-Feir index( BFI) compu ted from the spec t rum.

C o n c l u s i o n sOur analysis of 10,000,000 waves observed with buoys in the M editerraneanSea shows that the crest heights are almost perfectly Rayleigh distributed withoutscaling ( a = 1.0001). Our analysis of 10,000 waves observed with laseraltimeters in the North Sea shows an almost equally good correspondence for thecrest heights but with a scaling factor of a = 1.048, indicative of some degree ofnonlinearity (not seen by the buoys, probably because of the buoy dynamics).The distribution of the observed wave heights in both data sets is close to a


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COASTAL ENGINEERING 2008 571Rayleigh distribution with a scale factor a = 0.873 (average of the buoy dataand the altimeter data). However, it is closer to a Weibull distribution with ashape parameter that is slightly larger than for a Rayleigh distribution (2.162instead of 2), confirming a similar conclusion of Forristall (1978) for waves infive hurricanes in the Gulf of Mexico.

The maximum crest height observed by a buoy in a sequence of wavesagrees correspondingly with the Rayleigh distribution of the linear approach ofLonguet-Higgins (1952) and the maximum wave height agrees similarly with thescaled Rayleigh distribution.Grouping the buoy data shows that the statistics correlate well with

skewness, kurtosis, spectral width and the correlation between crest height andtrough depth, suggesting that these statistics can be predicted with spectralparameters (presently under investigation).A cknow ledgemen tsWe are grateful to the Xarxa dTnstruments Oceanografics i Meteorologiesde la Generalitat de Catalunya (XIOM) and the participants of the WADICexperiment for their permission to use their data. We also w ant to thank Joan PauSierra Pedrico of UPC , Jesus Gom ez Aguar of UPC and Steven Barstow ofFUGRO-OCEANOR for their kind and very helpful assistance in accessing thedata.ReferencesAllender, J., Audunson, T., Barstow, S. F., Bjerken, S., Krogstad, H. E.,Steinbakke, P , Vartdal, L., Borgman, L. E. and Graham, C. 1989. TheWADIC project: a comprehensive field evaluation of directional waveinstrumentation. Ocean Engineering, Vol. 16, 5/6, 505-536 .Cartwright, D. E. 1958. On estimating the mean energy of sea waves from thehighest waves in a record. Proceedings of the Royal Society of London, Vol.247, A, 22-48.Forristall, G. Z. 1978. On the statistical distribution of wave heights in a storm.Journal of Geophysical Research, Vol. 83 , C5 , 2353-235 8.Goda, Y. 1970. Numerical experiments on wave statistics with spectralsimulation. Port and Harbour Research Institute, Vol. 9, 3Goda, Y. 1988. On the methodology of selecting design wave height.

Proceedings of the 21s' International Conference on Coastal (Malaga),New Y ork, ASCE, pp. 899-913.Holthuijsen, L. H. 2007. Waves in oceanic and coastal waters. CambridgeUniversity Press.Janssen, P. A. 2003. Nonlinear four-wave interactions and freak waves. Journalof Physical Oceanography,. Vol. 33 . 4. 863-884.Longuet-Higgins, M. S. 1952. On the statistical distribution of the heights of seawaves. Journal of Maritime Research. Vol. XI, 3, 245-267.


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Longuet-Higgins, M. S. 1980. On the distribution of the heights of sea waves:some effects of nonlinearity and finite band width. Journal of GeophysicalResearch. Vol. 85, C3, 1519-1523.Mori, N. and Janssen, P. A. 2006. On kurtosis and occurrence probability offreak waves. Journal of Physical Oceanography, Vol. 36, 7, 1471-1483.Naess, A. 1985. On the distribution of crest to trough wave heights. OceanEngineering, Vol. 105, 3, 221-2 34.Tayfun, M. A. 1990. Distribution of large wave heights. Journal of Waterways,Ports and Coastal Engineering, ASCE, Vol. 116, 6, 686-707.Tayfun, M. A. 1994. Distributions of envelope and phase in weakly nonlinearrandom -waves. Journal of Engineering Mechanics, Vol. 120, 5, 1009-1025.Vinje, T. 1989. The statistical distribution of wave heights in a random seaway.Applied Ocean Research, Vol. 11,3, 143-152.

short term statisitics of wave observed by buoy

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